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THE THEORY OF HARMONY.

HANDBOOKS FOR MUSICIANS.

Edited by ERNEST NEWMAN.

THE

THEORY OF HARMONY

AN INQUIRY INTO THE NATURAL PRINCIPLES
OF HARMONY, WITH AN EXAMINATION OF THE
CHIEF SYSTEMS OF HARMONY FROM RAMEAU TO

THE PRESENT DAY.

MATTHEW SHIRLAW.

Mus.D., F.R.C.O.

(Lecturer in Music, University of Edinburgh;
Lecturer on the Theory of Music, Heriot-Watt College, Edinburgh, etc.).

London: NOVELLO & COMPANY, Limited.

New York: THE H. VV. GRAY CO.. Sole Agents for the U.S.A.

MADE IN ENGLAND,

Music
Libraty

Ml
S55t

TO MY FRIEND,

The Rev. R. Sangster Anderson, M.A.,

MINISTER OF THE BARCLAY CHURCH,
EDINBURGH.

CONTENTS.

PART I.

Preface.

Chapter I.

PAGE.

The Consonan'ces of Polyphony : Figured Bass

Schools and the Classification of Chords ... i — 28

Chapter II.

Gioseffo Zarlino (1517-90) and the Generation
of Harmony : Nature and Influence of his Work
as a Theorist: Rene Descartes (i 596-1650)
{Compendium Miisicae, 161 8) ... ... 29 — 62

Chapter III.

Jean Philippe Rameau (168 3-1 764): Traite de
VHarmonie : Harmonic Generation and the
Inversion of Chords : The Minor Harmony :
Origin of the Theory of Chord Generation by
means of Added Thirds : Chords of the Seventh :
Chords by " Supposition " : Resolution of Dis-
sonances: Resume of Rameau's Theories of
Chord Generation ... ... ... ... 63 — 97

Chapter IV.

Rameau's Traite de FHarmonie (continued) : The
Fundamental Bass : The Cadence • Nature and
Functions of Chords : Determination of "Key " :
Necessity for Dissonance in Music : Melody
has its Origin in Harmony : The Nature and
Constitution of the Scale ... ... ... 98 — 133

PART II.

(Chapter V.

Rameau's Nouveau Systeme de Musique
Theorique : Relationship of Harmony and
the Fundamental Bass to Mathematical Pro-
gressions : Theory of the Subdominant : The
Major and Minor Modes : The Chromatic Scale :
Nature and Origin of Tone-Systems ... ... 134 — 154

CONTENTS.
Chapter VI.

PAGE.

Rameau's Generation Harmonique and Demon-
stration du Principe de VHarmonie : Harmony |
a Physico-Mathematical Science : Objections to
Rameau's Theories by Berlioz, Fetis, and others i 5 5— 1 8 1

Chapter VII.

Rameau's Generation Harmonique and Demon-
stration {continued): The Diatonic System
(Major Mode) : Harmonic Dissonance : ' Double
Employment" of Dissonance: Chord of the
"Added Sixth": Examination of Rameau's
Views concerning the Origin and Nature of the
Key-System : Difficulties in Connection with
the Subdominant : Helmholtz's Theory of the
Origin of Scales: Difficulties Connected with
the Tritone: Chord of the "Added Sixth":
" False Intervals " of the Scale : Temperament,
its Theory and Practice 182 218

Chapter VIII.

Rameau's Generation Harmonique and Demon-
stration (continued): The Minor Harmony:
The Minor Mode : Further Development of
Rameau's Views respecting the Minor Harmony :
Relationship of the Major and Minor Modes :
Anticipation of Helmholtz's Theory of the
Minor Harmony : the Chromatic (ienus : Origin
of the Chromatic Scale : the Enharmonic Genus :
the Use made of Quarter-tones in Modern
Music: Other Aspects of Rameau's Theory ... 219—254

Chapter IX.

Other Theoretical Works of Rameau : Code de
Musique Pratique : Nouvelles Reflexions sur
la Demonstration du Principe de VHarmonie :
Nouvelles Reflexions sur le Principe Sonore :
The Minor Harmony: The Subdominant:
Generation of the Major Mode : Origin of
Dissonant Chords: Contemporary Criticism
of Rameau's Doctrines : Rameau and the
"Encyclopedists" ••• 255—285

CONTENTS.

PART III.
Chapter X.

PAGE.

Development of the Theory of Harmony from the/
time of Rameau up to the present day : Tartini'S
Trattato di Musica (1754) .• The Principle of
Harmony is independent of human will : Resul-
tant Tones : Physical Root of the Harmonic
System : The Arithmetical as well as the
Harmonic Division of the Octave, and of the
Fifth, is a necessity of the Harmonic System :
The Minor Harmony arises from the same
Principle as the Major : Origin of the Key-
System : Origin of Harmonic Dissonance.

G. A. '&o\\ov.{Vorgemach der Musikalischen
Kompositioii, 1745-47): Generation of Chords:
Theoretical Importance of Chord of the
Dominant Se:verrth.

F. \V. Mar>wg (Systematische Einleitung
in die mnsikalische Setzkinist, 1757,
Handbiich bei dem General basse und der
Composition, 1755-58, etc.): Development of
the Added-Third Theory of Chord Generation :
Chords of the "Ninth," "Eleventh," and
"Thirteenth": Sorge and Marpurg on the
Origin of the Chord of the Diminished Seventh 286 — 316

Chapter XL

J. P. KiRNBERGRk (1721-83): Reaction against
Marpurg's doctrines : Significance of the Leading-
note : All Harmony comprised in the Triad and
Chord of the Seventh : " Essential " and " Non-
essential " Discords : Origin of the Chord of the
Augmented Sixth.

Other Theorists of the end of the i8th
and beginning of the 19th centuries: P. J.
Roussier ; Abbe Vogler : G. Weber : L. A.
Sabbatini : C. S. Catel, etc.

F. J. Fetis (i 784-1 871) and the Doctrine of
Tonality : The " Laws]' of Tonality ; Change
from the Old to the New World of Harmonic
Music brought about by Monteverde's introduc-
tion of the Chord of the Dominant Seventh :
The Chord of the Dominant Seventh a Natural
Discord: Chord Relationship and Succession:
" Altered " and " Chromatically Altered " Chords 3 1 7 — 3 5 1

CONTENTS.

Chapter XII. page.

MoRiTz Hauptmann ( 1 792-1 868) and the Nature of
Harmony : Rejection of the Partial-tone Theory :
The real Basis of Harmony : Octave, Fifth and
Major Third the only "directly intelligible"
Intervals : The Key-system ; Secondary Triads
of the Key-system : Origin of Discords :
Diminished Triads and the Chord of the
'Added Sixth ": Resolution of Dissonant Chords :
The Chromatic Scale : The Minor Harmony :
Chord Relationship and Succession.

H. L. ¥. Helmholtz (The Sensations of
Tone, 1863): The Major Harmony: Minor
Harmony and the Chord of the '.' Ad4ed Sixth " :
Origin of Dissonant Chords : Tonality : Theory
of Consonance and Dissonance.

Ottingen and the Dual Nature of Harmony.

Dr. H. RiEMAtii^ {Die Natur der Harmonik,
1882; Harmony Simplified, 1893, etc.): The
"Under-tone Series": The "Tonal Functions"
of Chords: "Overklangs" and " Underklangs " :
The Key-system : Characteristic Discords :
Parallel-klangs : Melodically altered Chords :
Chord Succession ... ... ... ... 352 — 410

Chapter XIII,
English Theorists : Day's Treatise on Harmony
(1845) : Diatonic and Chromatic Harmony : The
Key-system : Natural Discords : Day's
Fundamental Bass : System of " Roots " : Minor
Harmony and Minor Mode : The Subdominant :
The Augmented Triad.

Macfarren's Rudiments of Harmony
(i860) and Six Lectures on Harmony (1867).

Ouselev's Treatise on Harmony (1868).

Stainer's Theory of Harmony (1871) : The
Tempered Scale : Added Third Generation of
Chords : The Third "The Basis of all Harmony."

E. Prout {Harmony : its Theory and
Practice, 20th ed., 1903) : Resuscitation of
Day's Theory : Subsequent Abandonment of the
Harmonic Series as the Basis of Harmony :
Origin of Discords : the Subdominant :
Secondary Discords : Chord Succession :
" Tonality " and the " Melodic Tendencies "
of the Sounds of the Scale ... ... ... 411 — 452

Chapter XIV.
RESUME and Conclusion ••• ... ... ••• 453 — 484

PREFACE

The present work might be described as, to some
extent, a contribution to the history of the theory of
harmony. Notwithstanding the extensive and highly
speciahzed hterature which we possess, deahng with the
history and the art of music, a hterature which of late
years has been enormously enriched, comparatively little
attention has been given to the history and development of
the theory of harmony, which, ever since the time of Rameau,
has been considered b}" musicians themselves to be intimateh-
connected with the art of music. Coussemaker's Histoire
de rharmonie du moyen-dge has to do with the history of
the art of music during the Middle Ages, rather than with
its theory. The Esqnisse de I'histoire de I'harmonie of Fetis
is a real histor\- of harmonic theory, and of harmonic
systems. But it is, from various points of view, in-
adequate. It would be quite impossible, for example, to
gain from the brochure of Fetis any real acquaintance with
Rameau's theoretical achiev^ements, or the nature of his
researches in the domain of harmony ; while Zarlino is
dismissed with the remark that "he is unable to present
to us any synoptic science of chords " ! With regard to
Dr. Riemann's important work, Geschichte der Musiktheorie
ini IX. -XIX Jahrhnnderi, it is evident that its author
does not consider it to be a history of the theory of harmony,
since he makes no attempt even to summarize the systems
of such distinguished theorists as Tartini, Hauptmann, and
Fetis, as well as of other theorists.

The real object of the present work, however, is not to give
a mere colourless exposition of the most important and

viii THE THEORY OF HARMONY

representative systems of hamiony, but rather to ascertain,
as far as possible, what constitutes the true basis of a theory
of harmony, and especially whether, or to what extent,
harmony can properly be said to have a physical basis.
Ever since the time of Pythagoras, music and harmony have
been related to mathematical science. But in all times
there have also been found theorists who were sharply opposed
to the \-iew that the underlying principles of harmony are
natural or mathematical principles. The art of harmony,
they have contended, rests not on physical, but on meta-
physical principles : music, they point out, is the expression
of man himself, that is, it is man-made, and has nothing
to do with an3d:hing external to man, nor, especially, with any
natural phenomena, acoustical or otherwse.

It was Jean Philippe Rameau (born 1683 at Dijon, died
1764 in Paris), the famous musical theorist, and one of the
most distinguished composers of his time, who first proposed
a theory of harmony based on acoustical phenomena.
Rameau made it his principal task to demonstrate, not
only that all music, whether melodic or harmonic, is governed
by certain laws, but that these laws are derived from
" natural principles," which, he endeavoured to prove,
reside in musical sound itself, and are neither more nor less
than the natural relations which may be observed to exist
in a sonorous body capable of producing an appreciable
musical sound. Rameau was followed by the scarcel}^ less
distinguished ItaUan theorist and composer Giuseppe Tartini
(1692-1770), who, working independenth', nevertheless arrived
in his Trattato di Musica at results which, in the main,
were strikingly similar to those obtained by the illustrious
Frencliman. Since the latter part of the eighteenth century,
and up to the present day, a vast number of works on harmony
have made their appearance, in which the theory of harmony
is related to acoustical phenomena. In these we find attempts
to develop still further the theories of Rameau, or to evolve
fresh theories. F. W. Marpurg (1718-1795), for example,
the author of what he termed the Rameau-Marpurg
System, exerted himself to remedy, as he imagined, the
principal defects of the Rameau system, to bring it
" up-to-date," and to provide the musical world with a good
working and practical theory of harmony. The distinguishing
features of such works on harmonv by the successors of

PREFACE ix

Rameau, are, undoubtedly, the extraordinary exploitation
of the harmonic series for the purposes of scale and chord
generation, and the no less extraordinary development of
the theory of chord formation b}' means of added Thirds.

On the other hand, there appeared works whose most
conspicuous feature was the delinite abandonment of the
harmonic series as a principle of harmony. These, and
especially the writings on harmony of J. P. Kirnberger
(172 1 -1 783), may be regarded as being, in a sense, a protest
against such theoretical absurdities as those presented in the
Rameati-Marpurg System. But even in works on harmony
by some of the most eminent theorists and musicians
of their time, we find the opinion, expressed with the utmost
emphasis, that a rational theory of harmony based on
acoustical phenomena is impossible. Witness, for example,
the Tmite de I'harmonie of Fr. J. Fetis (1784-1871), and
the Harmonik und Metrik of Moritz Hauptmann (1792-
1868). In this country, the well-known system of harmony
of Dr. Alfred Day has long held a foremost place, notwith-
standing that it was \-igorously opposed by such a musician
as Sir John Stainer, who himself proposed a new " theory
of harmony based on the tempered scale." Of late years,
however, the Day system has fallen into discre'dit ; at least
the number of those who still place their faith in it is daily
diminishing. The late Professor Prout, who at first closely
adhered to Day's S5^stem of harmony, finally discarded
acoustical phenomena as the basis of the theory of harmony,
without, however, being able to find for it any other adequate
basis, or to e^•olve any independent theory. The examination
of the numerous works in existence which treat of the theory
of harmony reveals the fact, not only that these tend to
•contradict each other, but that they exhibit, more frequently
than not, decided inner contradiction, as well as contradiction
with the facts of musical experience. It is not surprising that
at the present day the greatest uncertainty and misgiving
exist, not only with respect to the theory of harmony itself,
but even as to what constitutes the proper basis of such
a theory.

Nevertheless, it need hardly be said, the results of the
strivings of generations of musical theorists by no means
represent so much time and labour wasted. Their researches
have already borne fruit, and are destined to bear, we believe,

X THE THEORY OE HARMONY

much greater fruit. At any rate, these researches no student
of the subject can possibly afford to neglect.

In our examination, in the present volume, of the various
important works which treat of the subject of harmony,
much prominence has been given to the theoretical works of
Rameau, who is generally spoken of as having " laid the
foundations of the science of harmony." Rameau was a
real theoretical genius. He was not only one of the greatest
theorists of his time, but one of the greatest of all the theorists
who have at any time endeavoured to elucidate the mysteries
of harmonv, and to discover its laws. In his own day, he
was acclaimed as the " Newton of harmony." Before many
years had passed, however, his theories began to be considered
as inadequate, and insufficient for the explanation of the
many new harmonic combinations which had been sanctioned
by the practice of composers of genius. His system of the
fundamental bass, regarded by his contemporaries as his
greatest theoretical achievement, was judged to be out-of-date.
Ed. J. Fetis, in his Esqiiisse de I'hisioire de I'harmonie
(1840), as well as in his Traiie de I'harmonie, made a severe
attack on Rameau's system. He asserted that Rameau, in
his theory of chord generation, had totally disregarded the
principle of Tonality, that the chords thus generated appeared
as isolated entities, destitute of connection. In order to
remedv these defects, he had invented his fundamental bass.
This bass, however, was itself arbitrar^^ and irrational ; its
rules, further, were insufficient for a multitude of cases, and
its defects had become more and more apparent since a great
quantity of strange harmonies, unknown in Rameau's time,
had been introduced into music. Finally, his theory' of
" double employment " {double emploi), and his pretended
fundamental chord of the " Added Sixth " were sufficient to
destroy his theory from top to bottom.

After this onslaught of Fetis, than whom few wielded
greater influence as a musical critic, historian, and theorist,
nothing appeared to be wanting in order to consign Rameau's
theory finally to oblivion. Nevertheless, we find Helmholtz
in his work. The Sensations of Tone (1865), making use of
Rameau's principles in connection with his own theories,
not without acknowledgment of the great value and im-
portance of Rameau's theoretical researches and discoveries,
lie thinks, with Rameau, that harmony has a p]i3-sical basis ;

PREFACE xi

he commends " his fine artistic feeling," which so " fully
corresponded with the facts in nature " ; he energetically
supports his theory of " double employment," and with
respect to the chord of the "Added Sixth," thinks that
Rarrieau has a much clearer insight into the nature of this
chord than the great majority of modern theorists. The
ghost of the " fundamental bass " peers out from many a
page of Helmholtz's work. Rameau's influence has been
widespread and powerful, and even those who have rejected
his doctrines have not hesitated to borrow his principles.

But most remarkable of all, and a striking testimonv to
Rameau's importance as a theorist, is the fact that certain
of Rameau's doctrines, which have been long neglected,
or misunderstood and even ridiculed, are, in our own day,
springing into new hfe. Thus Dr. Riemann, in his work.
Harmony Simplified, or Theory of the Tonal Functions of
Chords (1893), has not only utiHzed certain of the most
essential of Rameau's doctrines, and, in particular, the
fundamental bass, as the foundation of his system, but has
made a notable attempt to develop them. Dr. Riemann 's
work might well be described as the apotheosis of Rameau's
fundamental bass. In the face of these facts, it would be
rash to assume, like Fetis, that Rameau's works are antiquated,
or that they possess little significance for present-day theorv.

But notwithstanding Rameau's great importance as a
theorist, no adequate exposition and examination of his
theoretical researches, embodied in his numerous works
on harmony, have ever been given, whether in this or in any
other country. The only work of Rameau which has been
translated into English is the third book of his Traite de
rharmonie, the least important, from the point of view of
harmonic theory, of all the four books comprised in the
Traite and from which alone it would be quite impossible
to acquire any adequate knowledge of the nature of the
theoretical principles contained in this, the first, of Rameau's
works on harmony. But even the most complete acquaint-
ance with all the books of the Traite would not entitle us
to assume that we were familiar with Rameau's theor\- of
harmony. Rameau has embodied the results of his reflections
on the subject, not in one onh', but in several important
theoretical works, a fact not always remembered by his
commentators. His ideas on the subject of harmonx' were

xii THE THEORY OF HARMONY

in a state of constant flux, and of continuous development.
For this reason it would be a somewhat difficult task to
give a synopsis of Rameau's theories on half a sheet of note-
paper. The explanation of the minor harmony, for example,
given in the Traite is essentially different from that given
in the Generation Harmonique ; while, in his De-monstration
du Principe de riiarmonie, Rameau's ideas on the subject
have undergone still further development, and he in effect
there anticipates that explanation of the minor harmony
which is generally attributed to Helmholtz.

For a similar reason it has been thought proper to deal
with Rameau's works separately. Such a method has,
besides, other advantages. It is instructive to trace the
gradual development of Rameau's ideas ; while the nature
of the problems which arise, and the difficulties which attend
their solution, are more adequately realised, and more
clearly understood. On the other hand, the attempt to give
an exposition of Rameau's theorj^ as a whole could only
lead to inadequate and even false conceptions with respect
to his work as a theorist. An exposition of this kind would
be noteworthy, not so much for what it contained, as for
what was necessarily omitted. Such a work is d'Alembert's
Elements de Miisique snivant les Principes de M. Rameau
(1752), which is frequently described as a concise and lucid
exposition of Rameau's theory of harmony. It is certainly
the only exposition we possess worthy of the name. But
in this work, d'Alembert has found it necessary to proceed
by a process, not only of selection, but of elimination ; he
selects what he considers to be most important and essential,
and eliminates the rest. The result is, that no adequate
knowledge of Rameau's theoretical researches, nor just
appreciation of his achievements as a theorist, can be gained
from the perusal of d'Alembert's work.

In the present \x)lume, we huve gi\'en not only a complete
exposition of the theoretical researches of Rameau, but have
also subjected his theories to a careful examination. In the
course of this examination fresh light has not only been
thrown on certain important aspects of Rameau's theory,
l)iit results have been arrived at which, the writer believes, are
of importance, not only with respect to Rameau's work, but
for the theory of harmony in general. It might well be
imagined, for example, that little or nothing remains to be

PREFACE xiii

said in connection with such a well-worn theme as Rameau's
theory of the inversion of chords, familiar, we may suppose,
to every musician. But Rameau's title to be considered as
the author of this theory has, especially of late years, been
seriously called in question. It has been contended that
this theory in reality originated with the figured bass prac-
ticians of the seventeenth century. An attempt has been
made in the present work to settle this question, with which
are connected considerations of much more than merely
historical importance. It is extremely doubtful whether,
at the present day, the real significance of Rameau's theory
of harmonic inversion is properly understood. Theorists
appear, for the most part, to have overlooked the fact that
Rameau's theory of harmonic inversion is inseparably bound
up with his theories of harmonic generation and of the funda-
mental bass, and have consequently failed to appreciate the
significance which such a fact possesses for the theory of
harmony, and how it affects the question as to whether the
theory of harmony has a physical basis. One curious result
of this has been that theorists who, like Fetis, are totallv
opposed to the conception that harmony has such a basis,
and who altogether reject Rameau's theories of harmonic
generation, and of the fundamental bass, have nevertheless
considered themselves at liberty to benefit from, and to utilise,
his theory of harmonic inversion. In justification of such
theorists, however, it may be remarked that Rameau himself
did not perceive to anything like its full extent the great
theoretical significance of his theory of harmonic inversion.
Here Rameau " builded better than he knew."

The theory of Rameau has its roots in the theoretical
principles elucidated by Zarlino and Rene Descartes. To both
these great men Rameau was indebted to an extent hitherto
almost unsuspected. But the origin of Rameau's theory
may be traced much farther back than Zarlino. Several
centuries before Christ, the Greeks made the discovery
(attributed to Pythagoras) that the Consonances or harmonies
of the Octave, Fifth, and Fourth, which formed the basis of
their musical system, could all be expressed by the ratios
I : 2, 2 : 3, and 3 : 4, or, more accurately, corresponded exactly
with the determinations given by these ratios. This repre-
sents— although to many it may appear a startling statement
— the first solid achievement in musical theory which led

xiv THE THEORY OF HARMONY

directly to the fundamental principle of harmony of Zarlino,
Descartes, and Rameau. In these facts elucidated by the
Greeks, Gioseffo Zarlino (1517-1590) discovered a definite
principle of harmonic generation, and, adding to the con-
sonances of the Greeks the major and minor Thirds, which by
his time had been recognised as consonant, he demonstrated
(although the minor Sixth proved a difficulty) that all the
consonances, which formed the sole constitutive elements of
pol}' phony, were comprised in, and generated from, the
scnario, or arithmetical series of numbers 1:2:3:4:5:6.

The senario of ZarHno formed the starting point for Rameau
in his theoretical researches : it was his principle of harmonic
generation ; of the fundamental bass ; the foundation for
his theory of harmonic inversion — his principle of principles.
After the publication of his Traite dc I'harmonie he discovered,
to his inexpressible astonishment, that this principle was not
merelv a mathematical but a natural principle. Harmony
actually existed in nature ; it had its source in musical sound
itself. These facts are suggestive. As is famihar to every
reader of Helmholtz's Sensations of Tone, the circumstances
relating to the mathematical determination of the con-
sonances are exhaustively investigated by this distinguished
scientist. Helmholtz, like Pyi:hagoras, is of opinion that the
Octave is determined b}' the ratio 1:2, and the Fifth by
the ratio 2:3, and is by no means prepared to allow that
the Greek theory of determination of the consonances
of the sixth century B.C. has no significance for the theory of
harmony of the nineteenth century a.d.

The researches of Zarlino are, in themselves, of extreme
importance, not only for the theory of music in general, but
for the theory of harmony in particular. Zarlino 's position
as a theorist, and especially the bearing which his researches
have on the theory of harmony, have ne^•er been properly
determined. Dr. Riemann, in his Geschichte der Musiktheorie,
has credited ZarHno with certain extraordinary discoveries,
and arrives at certain conclusions regarding ZarHno 's work
as a theorist which are by no means borne out by the facts.
At the same time, he has overlooked some of the most sig-
nificant of ZarHno 's theoretical achievements. In the present
work, an attempt has been made, not only to give an adequate
exposition of Zarlino's theoretical principles, but to indicate
clearly what he actually accomplished.

PREFACE XV

If Rameau owes mucli to Zarlino and Descartes, his influence
on his successors, on the other hand, lias been all-powerful.
In tracing the influence of Rameau on his successors, it has
been necessary to trace the development which the theory of
harmony has undergone in every important work on the
subject which has appeared since his time. The examination
of these works by no means induces the opinion that they
supersede the theories of Rameau, but tends rather to em-
phasise the value, even for present-day theory, of the work
performed by the illustrious Frenchman.

The whole subject is one, not merely of musical, but of
scientific and philosophical importance. It has a direct
bearing on Esthetics and Psychology. When Rameau set out
to penetrate the obscurities which surrounded the domain of
harmony, he set out, it may be thought, on a somewhat
Quixotic adventure, for his object was to demonstrate that
music and harmony were based on natural principles, and on
natural laws as invariable and steadfast as those which govern
the planets in their courses. He may, at first, have expected
too much from his science, and from the rules of composition
which it enabled him to deduce. But he had too great an
insight not to perceive that genius may transcend " the rules."
He was not one of those who see in every new and startHng
development of human activity the threatened destruction of
all the law and the prophets. Genius came, not to destroy
the artistic law, but to fulfil it. The principle of harmony was
independent of the human will. Music was not a mere play of
sensations, having no better origin than human caprice, than
the propensity of the human animal to sport. Truth and
beauty were no vain chimeras. Even in his artistic en-
deavours, man, although he might imagine himself to be free,
was nevertheless not left wholly to his own imaginings. He
had, fortunately, a guide. The result of Rameau's researches
was his conviction that he had discovered " the invisible
guide of the musician," and that, left to his own devices, man
might indeed attempt to build up artistic works, but in vain,
because he had no foundation on which to build.

In his endeavours to demonstrate the truth of his prin-
ciples, Rameau encountered serious difficulties. These diffi-
culties none of his successors have been able to remove. It may
be partly owing to this fact that theorists, at the present day,
are forsaking acoustical phenomena, and turning towards

xvi THE THEORY OF HARMONY

psychology for an explanation of the problems connected with
harmony. But it should be noted not only that psychology
has its own problems, but that psychologists are seeking in
music and harmony (consonance) and its effects on the mind,
for a solution of some of these problems. It may prove
eventually that, instead of musical theorists finding their
difficulties removed by means of the science of psychology,
psychology itself will be advanced by means of discoveries
made in the domain of the theory of harmony.

A word remains to be added in connection with the
preparation of this work, which has entailed the careful
examination and study of a very large number of volumes
and treatises on the subject of harmony and its theory. It
is a word expressing grateful acknowledgment of the courtesy
of the library officials of the British Museum, and of the
Music Class-room, University of Edinburgh.

THE THEORY OF HARMONY

CHAPTER I.

THE CONSONANCES OF POLYPHONY. FIGURED BASS SCHOOLS
AND THE CLASSIFICATION OF CHORDS.

The earliest examples of polyphonic music, which date
from about the end of the ninth century, are based solely
on the consonances already known to and recognized as
such by the Greeks of the time of Pythagoras, namely, the
Fourth, Fifth, and Octave. Soon other intervals made
their appearance — dissonances, as well as imperfect con-
sonances. The consonant nature of the latter was not
at first perceived, or at least admitted, by writers on music,
who were doubtless considerably influenced by Greek theory,
but were described as Dissonances, and later as Imperfect
Dissonances, that is, occupying an intermediate position
between Consonance and Dissonance ; ultimately they were
recognized as Consonances.

It is instructive to note the different stages in the gradual
evolution towards the complete theoretical recognition of
the consonant nature of the Thirds and Sixths. Thus
Franco of Cologne ^ groups the intervals into two main
classes. Consonant and Dissonant. ^ The consonant intervals
are of three kinds : —

Perfect — Unison and Octave.

Intermediate — Fourth and Fifth.
Imperfect — Major Third and Minor Third.
The dissonant intervals are of two kinds : —

Perfect — Semitone ; Tritone ; Major Seventh ; Minor
Sixth.

Imperfect — Major Second ; Major Sixth ; Minor Seventh.

' First half of thirteenth century : Fetis, in his Biographic Universelle
des Musiciens, gives the date of Franco's activity as more than a century
earher.

2 Ars Cantiis Mensurahilis, Cap. XI. (Coussemaker, Scriptores I.),

2 THE THEORY OF HARMONY

On the other hand, the classification of the intervals given
by the writer of the Compendium Discantus,^ a contemporary
treatise, is essentially different from that given above. He
says : " There are six pure dissonances, namely, the minor
Second, major Second, Tritone, minor Sixth, minor and
major Sevenths. Of the consonances, three — the Unison,
Octave, and Fifth — are in themselves perfect {per se
perfectae) ; three are consonant by virtue of their relation-
ship to perfect consonances, namely, major Third proceeding
to perfect Fifth ; minor Third to Unison ; and major Sixth
to perfect Octave." The perfect Fourth, although in itself
consonant, has the effect of a dissonance ; a statement which
is noteworthy in so early a treatise. The minor Sixth was
still for some time regarded as dissonant. The Ars Contra-
pundi secundum Johannem de Aluris,^ written in the first
half of the fourteenth century, treats the major Sixth as a
consonance, but the minor Sixth as a dissonance. In a
treatise^ which is appended to the above, however, the
minor Sixth is placed on the same footing as the major ;
thus both the Sixths, as well as the Thirds, are ultimately
recognized as imperfect consonances.

It is evident, then, that the practice of harmony of the
early contrapuntists was largely a question of intervals ;
and this is true also of this entire period of polyphonic music.
Thus the chord c-e-g was considered to arise from the union
of the major Third c-e with the perfect Fifth c-g. The chord
g-c'-e , which we know as the second inversion of the chord
c-e-g, could not, however, be employed except as a suspension,
as it contained the dissonant interval g-c ', a Fourth. Com-
paratively early, parallel successions of perfect consonances,
such as characterized the first attempts at polyphonic music,
are prohibited; also the rules for the treatment of the
various intervals are clearly defined. As a general rule it
was laid down that an imperfect consonance should be
followed by a perfect one ; while a dissonance should be
followed by a consonance.^ It is evident then in the second

• Ccussemaker, Scriptores I. - Coussemaker, Scriptores III. •' Ibid.

^ Thus Guilelmus Monachus (c. 1450) directs that the dissonance
of the Second be followed by the consonance of the Third ; the Tritone
by the Fifth ; the Seventh by the Sixth, and — a remarkable circum-
stance— the Fourth by the Third I The Fourth, a perfect interval, is
dissonant, and requires to be resolved ! (See also p. 23.)

THE CONSONANCES OF POLYPHONY 3

place that the harmonic art of this period had, as its basis,
Consonance. The consonances are the pillars of the harmonic
structure ; the dissonances, on the other hand, are notes
of ornament, resulting from the figuration of the melody,
or they are notes of suspension, as of the Third by the Fourth,
of the Sixth by the Seventh, or passing-notes, etc. By the
middle of the sixteenth century we find Zarhno treating
of the inversion of intervals. Zarlino also attaches significance
to the bass (not of course the Fundamental Bass as under-
stood by Rameau, but the lowest note in every interval or
chord, whether inverted or not) as the real support and
foundation of the harmony. In the concluding Cadence,
Zarlino directs that the bass proceed to the Final of the mode,
whether the Tenor do so or not ; here the Bass may descend
a Fifth, or ascend a Fourth, to the Final, while the highest
part, or at least one of the upper parts, proceeds from the
semitone below the Final to the Final itself. ^ This corre-
sponds in every way with our Perfect Cadence, which as
we shall see is a fact of great importance for Rameau, and
for the theory of harmony. The only harmonies generally
practised during this epoch of polyphonic music which
culminated in the works of Palestrina and Lassus, at the
close of the sixteenth century, are those of the Third and
Fifth ; of the Third and Sixth ; of the Third or Sixth and
Octave ; of the Fifth and Octave, or of the Third, Fifth,
and Octave. At the same time the second inversion of the
consonant major or minor harmony, that is, the Perfect
Fourth combined with the major or minor Sixth, might be
employed much in the same way as at the present day, as
a suspension of the consonant triad on the same bass note.
Occasionally also the combination of consonant intervals
above a bass note with a suspension in one of the voices led
to some extremely curious harmonic results, as in the following

1 The semitone below the Final is required by Joh. de Muris as
early as the fourteenth century, even in cases where it is foreign to the
constitution and character of the mode, as in the Dorian and Mixolydian
modes : —

Dorian.
(*)::§: -<tsp (h-&- -Q. Mixolydian.

JZH

^^ Q

4 THE THEORY OF HARMONY

passage from the Gloria of Palestrina's Missa Papce
Marcelli : —

At the third minim in the first bar, we find that the notes
actually present are a, c, e, g. It is difficult to explain i- as a
non-harmonic note which merely retards the /# immediately
following ; for if /# be the real harmony note, then there
results the harmony a, c, ej^. In fact, the last three chords
in this passage correspond to what we at the present day
understand as the chord of the Seventh on the Supertonic,
followed by a Dominant Tonic Cadence, in G major.

But, as is known, the music of this period is of a
nature essentially different from that of a later time, and
of our own day. What is the nature of this difference ?
We are frequently told that the older art, based as it was
on the Ecclesiastical Modes, had its roots in Melody ; that
is, its harmony was the result of the concurrence of the
various melodic voice parts. Our modem music, on the
other hand, has as its foundation Harmony; melody,
instead of being the determining factor, as was the case in
the older art, is itself harmonically determined. As to this,
one may say that the view that, in the music of the polyphonic
period, harmony was determined as the result of the, presum-
ably, fortuitous concurrence of the different melodies, is a
very superficial one. The harmonies or consonances which
at first formed the basis of polyphony, namely the Octave,
Fifth, and Fourth, were known and their mathematical
ratios (1:2,2:3 and 3 : 4) even discovered by Pythagoras
fourteen or fifteen centuries before polyphony was thought
of. The Church Modes themselves depended for their
definition on these same consonances. The Octave deter-
mined the compass of the mode ; while the Fourth and
Fifth were necessary for the division of the modes into
Authentic and Plagal. The harmony of polyphony was not
arbitrarily determined ; on the contrary the melodies
were shaped so as to produce a pleasing harmony. If in

THE CONSONANCES OF POLYPHONY 5

monophonic music the individual melody was apparently
able to pursue its own free unfettered course, this was no
longer possible in a union or community of melodies. Nothing
but chaos could be the result. Such a union was possible
only when each melody, in seeming surrender of its liberty,
and out of consideration for its neighbours and for the
general well-being, so to speak, of the community of sounds,
submitted itself to a certain guiding and immanent principle,
and thus took its indispensable part in bringing about those
immeasurably richer and grander artistic creations which
form the imperishable glory of musical art. This guiding
principle was Harmony.

For those who hold that the harmony of early polyphony
had its origin in melody, it is a distinctly disconcerting circum-
stance that the composers of that time altered the Ecclesiastical
Modes in order to obtain a proper harmony. ^ And yet these
Modes had been consecrated by the traditions of centuries,
and especially by their use in the sacred services of the
Church. In short, the constitutive elements of the harmony
of polyphony which Zarlino, the theorist par excellence
of the polyphony of his time, has expressly stated to
consist of nothing but the Perfect and Imperfect Con-
sonances, are the constitutive elements of the harmonic
art of our day. Our Perfect Consonances are, in every
respect, the Perfect Consonances known to the Greeks
of the time of Pythagoras. The art of music exhibits
itself as an organism ; and the history of music and
of harmony is the history of a gradual, continuous, and
consistent development. It is somewhat unphilosophical,
therefore, to explain the harmony of the early polyphonic
period as having its source in melody, but to maintain that
in our modern music exactly the opposite is the case ; that
melody has its source in harmony, while harmony itself now
becomes apparently inexplicable.

Still it remains true that the music of the early
polyphonic period is in its nature different from that
of more modern times. To the modern ear, the progression
from harmony to harmony is determined by certain relation-

' Hence, in order to avoid the tritone, and to obtain a true Cadence,
the use of the so-called Musica fida, that is, alterations, by means of
sharps or flats, expressed or understood, of the notes of the
Ecclesiastical Modes.

6 THE THEORY OF HARMONY

ships existing between the harmonies themselves. It would
be untrue to assert that in the older art harmonic relationship
was non-existent — very much the reverse ; on the other hand,
in its movement from consonance to consonance, and from
harmony to harmony, we do not find that definiteness of
harmonic significance, those principles of chord succession
which especially gather up as in the music of a later time
the whole harmonic material into a certain unity — the
Key-system. This alone accounts in great part for the
peculiar and characteristic effect of the older music.

The change from the old art to the new is frequentlv
assumed to have been accomplished at the beginning of the
seventeenth century. This however is an assumption not
altogether justified by the facts. The change which occurred
was the result, not of sudden revolution, but of gradual
development. Many influences had already been at work
tending towards the overthrow of the old modal system.
On the other hand, composers did not rid themselves so
easily of the influence of estabUshed traditions, and our
modern tone-systeni did not become finally fixed until much
later than the first decade of the seventeenth century. But
the gradual development and transformation of the Church
Modes to the Major and Minor Modes of our own day, the
beginnings of which can be traced back to a period even
before the time of Palestrina, received a powerful impetus
from the rise of accompanied monody towards the end of
the sixteenth and beginning of the seventeenth centuries
as well as from the invention, about the same time, of the
Basso Continuo or thorough bass. This bass appears to
have been devised for the sake of convenience in the accom-
paniment of poljAphonic music in order to obviate the
difficulty, on the part of the cembaUst or organist, of reading
a great many parts at one time. Unlike the vocal bass
part, which was frequently interrupted, this instrumental
bass was continuous, and represented always the lowest
moving voice part ; hence the term Basso Continuo. This
bass was made use of for the accompaniment of Recitative,
which was the most characteristic feature of the new style
which now arose.

The invention of Recitative, as is known, coincides with
the rise of the Opera, and represents an attempt to
resuscitate the musical declamation of the poetic text of

FIGURED BASS SCHOOLS 7

ancient Greek tragedy. For such a dramatic recitation,
in which the natural accent and appropriate expression
of the words were all important, the highly elaborate
polyphonic music of the Church composers was rightly
judged to be unsuited. The means towards this end was
therefore sought for and found in a solo melody which should
imitate the accents of speech — -the Recitative. So great
importance being attached to the words, it can be easily
imagined therefore that the musical element in the first
attempts at opera played a very subordinate part. Hence
the accompaniment to the Recitative was of the simplest
possible kind, consisting of a few chords serving as a harmonic
support to the voice, which were indicated simply by a bass
part — the Basso Continuo above mentioned. To this bass
figures were added, and placed above the different notes of
which the bass was composed ; these figures — from 2 up to
9, and even to 12 and 13 — indicating the intervals, reckoned
from the bass upwards, of the harmony to be employed.
This Figured Bass it is evident was not a theoretical but
a practical device, a kind of musical shorthand, and of great
convenience to the accompanist. Hence every contrivance
which could facilitate sight-reading and simplify matters for
the figured bass player was adopted. Before long therefore
the figures 10, 11, 12, 13, representing compound intervals,
were discarded in favour of the more easily apprehended
simple form of these intervals, represented by the figures
3, 4, 5, and 6. This substitution of the simple for the com-
pound form of the interval — except in the case of the Ninth,
and the recognition of their identity, as regards their harmonic
significance, was a distinct gain not only from a practical
but from a theoretical point of view. That the Ninth was
an exception, and could not be represented by the simple
form of the interval, was owing to the nature of its employ-
ment as the retarding note in the suspension 9-8, already
made long famiUar by the practice of composers.

Most noteworthy was the peculiar position assigned to the
Triad, especially the consonant triad, which alone of all the
harmonies employed required no figuring. The reason for this
cannot have been wholly in order to facilitate practice. From
the outset the consonant triad, both in its major and minor
form, appears to have been regarded as of peculiar importance,
and as possessing qualities shared by no other harmony.

THE THEORY OF HARMONY

The term irias harmonica is, according to Dr. Riemann,^
used by Joh. Lippius ^ as early as 1609. Before the
middle of the seventeenth century one finds the major
and minor common chord referred to as I'accordo perfetto
among musicians in Italy. Later the same term, I'accord
■parfait, apparently borrowed from the Italians, appeared
in France ; although as early as 1636 Mersenne, in his
Harmonic Univcrselle, speaks of the harmonic parfaitc which,
he informs us, is an expression in general use. The EngUsh
name common chord is found in Gottfried Keller's Rules for
Playing a Thorough-Bass (1707), although it is likely to have
been in use before this date. In Germany the consonant
triad was designated in various ways. Joh. D. Heinichen ^
makes use of the terms Hauptaccord, Ordinaraccord (common
chord), and Trias Harmonica, and remarks: — "The chief
and most excellent combination of consonances from which
a musical harmony can arise is that known to all musicians as
the trias harmonica, which consists of a bass note. Third
and Fifth." The sounds composing this chord could be
arranged in any order above the bass without altering the
essential nature of the harmony. The three different orders
of distribution are thus given by Heinichen : {a) . But many
other arrangements were possible, as at {b) : —

(«)

-fS-

22:

-&«-

zz:

:gr

-Gt-

221

etc.

rj fn

~rjr

33:

22:

221

22;

331

22:

In the same way any number of voices or instruments
might take part, without radically changing the nature of
the chord : (c) .*

' Geschichte der Musiktheorie im IX.-XIX. Jahrhundert.

* In his three Musical Disputations.

■' Neu erfundene und griindliche Anweisung, etc., 1711.

^ Heinrich Albert, in the Preface to the second part of his Arien
(1643) says : — " It is known that all musical harmony, even although
a hundred voices take part in it, consists of three sounds only."
Mattheson makes the same remark in his Neueroffnete Orchester (1713).

FIGURED BASS SCHOOLS 9

All this represented a marked advance towards a truer
appreciation of the nature of harmony. Further, it was
observed, as it could scarcely fail to be, that a close relation-
ship existed between a chord and its inversions, seeing that
all were composed of practically the same sounds. Add to
this that Zarlino had already treated of the inversion of inter-
vals. It was known for example that the Sixth represented
the inversion of the Third, the Fourth of the Fifth, the
Fifth of the Fourth, and so on. Are we therefore entitled
to assume that composers and writers on music of this period
were acquainted with the nature of Harmonic Inversion ?
Dr. Riemann ^ cites a passage from the Hodegus Curiosus
(1687) of Andreas Werckmeister which, he is of opinion,
not only treats specifically of fundamental chords and their
inversions, but already embodies the complete theory of
the inversion of chords. The passage is as follows : —
" Harmony consists of the union not of like, but of unUke
or diverse elements. As all consonances are of good effect,
and please us because of their clearness, we try to arrange
them in every possible order. Therefore we may take the
Third, the natural position of which is above the ground-
tone (this ground-tone, occupying the lowest position, being
reckoned as the root) and use it instead of the ground-tone,
which then appears as a Sixth above it ; for if the Fifth
or Third is not present in any combination {Satz), then
the regular series of ordinal numbers has been departed from,
and we have, as it were, a borrowed fundamental note "
(" erhorgtes Fundament clavis "). Dr. Riemann would almost
appear to be justified in pointing to this passage as a proof
that the theory of the inversion of chords w^as in reahty no
discovery of Rameau, but gradually revealed itself to the
consciousness of composers and of cembaUsts in their
practice of figured bass accompaniment. But if this is so,
why then was this theory not made use of ? Writers on
figured bass were becoming more and more embarrassed by
the new and strange chords which were every day being
added by composers to the large number already existing,
and were diligently searching for the key towards that inner
relationship which, they felt, ought to exist between the
numerous and otherwise isolated harmonic combinations.

Geschichte der Musiktheorie. (Footnote, p. 431.)

lo THE THEORY OF HARMONY

Here, in the theory of Harmonic Inversion, was the only
possible key towards a rational system of chord classification.
Why was it not immediately taken advantage of, and why
was it necessary to wait for the appearance of the Traite de
r Harmonie of Rameau ?

In the meantime it may be observed that Werckmeister
presents us with nothing that was really new. His remarks
on the consonances, on the nature of harmony and of its origin,
are only an echo of what had already been said by ZarUno ^
in treating of intervals and no more than Zarlino does he
speak of "fundamental chords" and their inversions, but
only of intervals. It is just the importance which not only
Werckmeister but his contemporaries attached to the interval —
for each interval had its own peculiar harmonic significance —
which gave rise to so much confusion and uncertainty as to
which chords should be regarded as original, and which as
inverted. For Werckmeister the fundamental note of the
chord e-g-c' is not c but e. Like Zarhno, Werckmeister explains
the consonances as arising successively from the numerical
series 1:2:3:4:5: 6 — (8). But while the Fourth (3 : 4)
arises directly from this series, the Sixths (3 : 5 and 5 : 8)
arise accidentally.^ It is necessary to include the number 8,
even if it introduces a gap in the series of numbers, for other-
wise the Minor Sixth cannot be found.^ This is not Rameau's
view. Rameau expressly declares that neither the Fourth
nor minor Sixth should be regarded as an " original " but
as a " derived " interval. What Werckmeister is chiefly
concerned to point out is, not that the major harmony produces

1 Compare, e.g., the passage from Zarlino's Istituzioni harmoniche,
Pt. III., Cap. 29, which begins : — " Consciosiache molto ben
sapeuano, che I'Harmonia non piui nascere se non da cose tra loro diuerse,
discordanti at contrarie et non da quelle ch'in ogni cosa conuengono."
Also Pt. I., Cap. 15. (Delle proprietii del numero senario & delle
sue parti cS: come tra loro si ritroua la forma d'ogni consonanze
musicale).

2 " Wenn wir die Musicalischen Proportional Zahlen
betrachten, so . . . sehen wir erstlich daraus den rechten Sitz aller
Consonantien, da wir denn befinden dass die Octava erst, darnach die
Quinta, dann die Quarta und Tertia Major und minor folge, die Sexten
aber stecken zufiilligerweise in diesen Zahlen 1:2:3:4:5: 6 — 8."
(Musicalisches Memorial, 1697, Cli. i.)

^ •' Wenn diese Zahl 8 nicht dabey wiire, so konten wir keine Sextam
minorem in dieser Ordnung haben, als 5:8." (Musicalische Paradoxal-
Discourse^ 1707- Ch. 19.)

FIGURED BASS SCHOOLS

all the consonances, but that the consonances, arranged in
a certain order, give rise to the major harmony, which is
exactly the view taken by Zarhno. Further, too much impor-
tance need not be attached to Werckmeister's use of the
term " root." He describes the series 1:2:3:4:5:6 as
a series of " roots " (" Radices ") ; and further tells us that
by means of the addition of the first four terms of this series
there results the number 10, which is the " root " of the minor
harmony 10 : 12 : 15.

Keller would appear to be familiar with the theory of
the inversion of chords, when he says ^ :— " To make some
chords easie to your memory you may observe as follows :

{a) A common chord to any note makes a ;• to the
Third above it or Sixth below it, as —

-IdA

A common chord makes a
Fourth below it," as —

to the Fifth above it or

Here Keller might describe the notes e and g, in the lowest
part, as " borrowed fundamental notes." But that he is
merely elucidating the method of figuring, and not explaining
the process and nature of inversion, is clear from what

immediately follows : " (6) A common chord makes a f;
to the Sixth above it, or the Third below it," as —

-7--

In this case it is quite impossible to consider the note a as
bringing about an inversion of the original chord, c-e-g.

Heinichen employs the same term as Werckmeister {funda-
mental clavis) to designate the lowest note in all chords,

' Rules for Playing a Thorough Bass.

THE THEORY OF HARMONY

whether inverted chords or not. He recognizes quite clearly
that chords may consist of different intervals, and yet be
composed of practically the same sounds. Nevertheless,
how completely he fails to grasp the difference between
fundamental and inverted chords is evident from the following
passage : " The chord {Haupf accord) d-e-g^-h is capable of
the foUowing three changes of its harmony : (i) e-d-gjlj^-b ;
(2) gj^-e-d-h ; and (3) h-d-e-g^." ^ Here the chord described
as original, as a Hauptaccord, from which the others would
appear to be derived, is itself a derived chord, namely, the
last inversion of the chord of the Dominant Seventh, e-g^-b-d.
Even more striking is the example which he thus explains :

9 s

"If now we invert the chord ";. , so that the Sixth appears
in the bass, we obtain a syncopation of the Fourth, thus " ^ : —

{«)

-<5>-

(b)

22:

-ry-

-s>-

-Gt-

ilOt

-<5>-

Accord.

Verkehrung.

9 8 4 3

In this case Heinichen riot only " inverts " an inverted
chord, but describes the fundamental position of the chord
at (b) as an inversion, which is of course exactly the reverse
of the real state of matters.

Like almost every author who has before or since written
on the subject of the theory or practice of harmony, Heinichen
in the first part of his book Der General-bass in der Composition
(1728) devotes a chapter to the consideration of the different
intervals (Ch. i). In addition to the table of diatonic
intervals, consonant and dissonant, which played such an
important part in the works of the older theorists, we find
several new ones, both diatonic and chromatic. Chief among
these are the diminished Fifth (as b-f) and the augmented
Fourth (as f-b) : the first being found in the first inversion

' Der General-bass in der Composition, Part II., Ch. i.
* Ibid.. Part I., Ch. 3.

FIGURED BASS SCHOOLS

of the chord of the Dominant Seventh ; the second in the
third inversion of the same chord. Thus the dreaded Mi
contra Fa, the great stumbhng-block of an older generation
of composers, had become by its incorporation in the chord
of the Dominant Seventh the chief ornament of the new
music. There are also the augmented intervals of the Second,
Fourth, Fifth, and Sixth ; and the diminished intervals of
the Fourth, Fifth, and Seventh. The only compoimd
intervals mentioned are the major and minor Ninths.

The second chapter treats of the consonant triads of the
Major Mode {triades harmonicae). The third chapter deals
with the inversions of these triads,^ and with all other harmonic
combinations used in figured bass practice. Beginning with
the chords of the Sixth, Heinichen proceeds to treat of the
various dissonant chords, among which he includes the
chord of 4, the second inversion of the consonant triad.
Two systems of chord classification are adopted. In the
first, a distinction is made between chords which are consonant
and those which are dissonant. In the second, the dissonant
chords are classified according to the species of the interval
which forms a dissonance with the bass, and according to the
order in which the intervals are arranged in Ch. i. " The
[interval of the] second is the first dissonance " : therefore the
first dissonant chords to be considered are those which
contain the interval of a second between the bass and an
upper part. They arise for the most part from a suspended or
" syncopated "bass, which is duly prepared and resolved thus: —

{a) (b) (c) (i)

n I I.I II.

-G»-

hSK

-Gh-

:g:

-rJ Q-

-O"

-rD~

-<s>

-<5>- -o-

in~

r If

^Hi^

1C2I

231

6
4
2

5
4
2

5
2

In examples {a) and (b) the second chord is the third inversion
of the chord of the Seventh on the Supertonic and Dominant

' Including the first inversion of the diminished triad on the leading
note.

14 THE THEORY OF HARMONY

respectively. At (c) the second chord is the chord of the Domi-
nant Seventh, the third of the chord being retarded in the
bass. At (d) we have merely, in the second bar, the first in-
version of the Tonic triad, the Third of which is retarded in
the bass. But notwithstanding the widely divergent harmonic
conditions which obtain in these examples, Heinichen, so far
as the theoretical aspect of the question is concerned, treats
them all alike : they are all dissonant chords of the Second,
arising from a " syncopated " bass. Such a system of chord
classification is, of course, quite inadequate and misleading.
Nor is Heinichen able even to draw an effective distinction
between the two main classes of consonant and dissonant
chords. For among the former he includes several which
are dissonant, such as the first inversion of the Diminished
triad, and the second inversion of the chord of the Dominant
Seventh. In the latter chord, which is introduced among
the consonant chords of the Sixth, we have the dissonance
of a Second occurring between the notes / and g : —

But as the intervals which compose the chord are reckoned from
the bass note upwards, that of the Second cannot be included,
for according to this theory of chord formation the only
intervals present are those of the Third, Fourth, and Sixth.^

1 Johann Mattheson, in his Kleine Generalbass Sckule, 1735, thus
defines the term chord : — " A chord is the union of several sounds,
from two up to eight or more, which are either pleasing, or harsh
and discordant, according to their relation with the ground tone
[bass note]." As therefore, in the figured bass schools of this time, all
chords are considered to arise from a combination of various intervals,
and are, theoretically at least, regarded as consonant or dissonant
according to the nature of the intervals which occur above the bass,
chords such as the following are a source of considerable perplexity : —

-&-

■3<S>-

Scz

6 6 6„ 6

4 3| 5

3 * 3

for the intervals of which they are composed, reckoned from the
bass upwards, are all consonant. As for the chord at *, while Mattheson
regards it as consonant, Heinichen recognizes its dissonant nature.

FIGURED BASS SCHOOLS 15

Heinichen explains the chord thus : "To the minor Third
and major Sixth [as in the chord of the Sixth d-f-b'\ may be
added the perfect Fourth. This Fourth appears in the previous
chord, and may be allowed to remain in the I; chord " : thus : —

This Fourth, the real fundamental note of the chord, is
described as merely an accessory note {Hulffs-stimme). It
should be noted that although Heinichen regards the
Diminished triad as a dissonant chord,- he considers its
first inversion to be consonant.

Of the numerous dissonant chords treated of, we find
various chords of the Seventh and their inversions, principally
those on the second, fifth, sixth, and seventh degrees of
the major scale ; and on the second, fourth, fifth, and sixth
degrees of the minor. The chord of the Diminished Seventh,
with its inversions, occurs in the Minor Mode ; also the
Augmented triad on the third degree of this mode, in its
first inversion. Of chromatic chords there are the three
forms of the chord of the Augmented Sixth. Examples
are also given of the pedal point, and of the suspensions
of the major and minor Ninth, which may be accompanied
by simultaneous suspensions in one or more of the other

parts : as 2», %%, or tI. Heinichen does not treat
specifically of the triads proper to each degree of the

and — a noteworthy circumstance — even determines that c is the
dissonant note, in which case he can have been guided solely by his
ear. For he is quite unable to explain why this note c, which makes
with the bass a perfect Fifth, one of the smoothest of consonances,
should be regarded as dissonant. He is of opinion however that
the perfect Fifth, although in itself consonant, may still be employed
" after the manner of a dissonance " (Ch. 3, § 34).

• Heinichen understands quite well the exceptional nature of the
Diminished triad on the leading note, and refuses to give it a place
among the other triads of the major scale. His employment of this
triad is noteworthy. It never appears except with the addition of
the minor Sixth, as {b-d-f-g) ; therefore as Third, Fifth, and Seventh
of the chord of the Dominant Seventh. Even when the chord is taken
in its first inversion Heinichen prefers, as is evident from what has
been said above, to add to it the perfect Fourth, again obtaining
Dominant Seventh harmony.

i6 THE THEORY OF HARMONY

minor scale, but except in the case of the Augmented
triad on the third degree and the Diminished triad, which
may form part of the chord of the Diminished as well
as of the Dominant Seventh, seems to imply that they
are to be used in a way similar to those of the Major
Mode. In the second part of his work he devotes a
lengthy chapter to the treatment of dissonances (beginning
with the Second, and proceeding up to the Ninth) and
their resolutions, peculiar to the free or dramatic style
of composition. Here, as well as in the other sections of
the work, which are taken up chiefly with the consideration
of the various circumstances relating to the melodic figuration
of the parts above the figured bass, as well as of the bass
itself, Heinichen shows much sagacity and musical insight.
Such then was the harmonic material in most common use
among composers when Rameau pubHshed his Traite de
I'Harmonie. " These are," says Heinichen, " the most
usual signatures of general-bass." But dissonant chords, he
proceeds, " are so to speak innumerable, and may by good
practicians be daily varied and invented."

Joh. Mattheson (1681-1764), in his Kleine General-bass
Schick, strives even more assiduously than Heinichen to
arrive at a rational system of chord classification. He adopts
not one but several methods. First of all he distinguishes,
like Heinichen, between consonant and dissonant chords.
Then he classifies all the chords, both consonant and dis-
sonant, according to the interval which distinguishes each,
beginning with chords of the Second, of the Third, of the
Fourth, etc., and concluding with those of the Ninth.
Subsequently he gives still another arrangement, and divides
the chords, which number seventy in all, into three classes
or orders. The first class comprises " the most common
and most harmonious chords," tw^enty-four in number ; the
second class, those w^hich are less common ; and the third,
those which are least frequently used. The last two classes
consist of dissonant chords only. Mattheson is even less
successful than Heinichen in correctly distinguishing between
consonant and dissonant chords. Among the former he
includes the first inversion of the chord of the Dominant
Seventh (j,^) and of the Diminished Seventh (i^^) ; and the
first inversion of the chord of the Seventh on the Supertonic of
the major scale {':). With respect to the Diminished triad

FIGURED BASS SCHOOLS 17

•on the seventh degree, he at first rejects it, altering it
chromatically so that it appears as a minor triad {b-d-fj^) ;
afterwards however placing it on an equal footing with the
■other triads of the major scale, i.e.. as a consonant triad.
Mattheson's description of some of the chords just mentioned
i js peculiar. Thus the chord e-g-h\^-cj^, which we understand
as the first inversion of a chord of the Diminished Seventh,
he terms an augmented chord ; the augmented interval being
ik-C#! So also with the "consonant" chord e-gjf^-c i^%),
'"which is described as a chord of the Diminished Sixth —
the diminished interval being e-c — whereas the real
•diminished interval is that of the Fourth, \'iz., ^#-c.
It can only have been on theoretical grounds that
-Mattheson described such chords as consonant, for he
was too good a musician not to perceive their dissonant
effect.

The common chord, major or minor, is termed a perfect
harmonic triad. It is the presence of the Tonic and Dominant,
the principal notes of the scale, in the common chord on
the Ionic which gives this chord its perfection, the Third
being added as a matter of course in order to complete the
harmony. The other triads of the scale, which are hkewise
composed of the intervals of the Third and Fifth, are then
perfect by their analogy with the Tonic triad. Hence
Mattheson's vacillation in respect of the Diminished triad on
the seventh degree.

After an examination of the twenty-four chords contained
in his first table, he proceeds : "So far we have been deahng
with the most common consonances, [!] now we have to treat
of dissonances, or the less usual chords : and first of aU,
the Second." These chords of the Second are produced by a
suspended bass-note, and are of four different kinds :
(i) Chords of the diminished Second ; (2) chords of the
minor Second ; (3) chords of the major Second ; (4) chords
of the augmented Second ; and are to be distinguished thus
according to the variety of the interval of the Second occurring
between the bass note and an upper part. The diminished
Second, according to Mattheson, is the Semitone, which may
be either diatonic or chromatic ; the minor Second corresponds
to the ratio 9 : 10 ; the major Second to 8 ; 9. A distinction

is made between the 4 chord (last inversion of chord of the

THE THEORY OF HARMONY

Dominant Seventh) in which the bass note is prepared, and
that in which it is not prepared, as : —

('')

:2=5

-?-^

(^)

^^r

@— b^-

-«>

-«s>-

:l322:

6
2

6
4
2

In the first case, " the Fourth is merely an accessory- note
beside the Second " : in the second case, " the Fourth becomes the
chief note : the Second is the accessory note " (A^ebenklang) I
The Fourth, which may be diminished, " major " (perfect)
or augmented, " has 14 different resolutions. There is not
space to deal with them all here . . . the theory of these
resolutions is treated of in Heinichen's Der General-bass in der
Composition." So also " the Ninth may be resolved in eight
different ways." Mattheson examines minutety the various
circumstances relating to the appropriate treatment of
each of his seventy chords ; whence arise an extraordinary
number and diversity of rules, exceptions from rule, and
the like ; and as if the rules were not already numerous
enough, barren distinctions are drawn between chords
identical in their nature and manner of employment. On
the other hand he is quite aware of the great change
which the art of music had undergone, in that it was
no longer based on the twelve Church modes, but
made use of two only, namely, our major and minor
modes. Heinichen insists on the same fact, and even
urges a reform of the illogical method, then in use,
of indicating the key-signature. Thus in the major mode
the sharp necessary for the leading note was not included
in the kev-signature ; nor in the minor mode the flat
which indicated the minor Sixth ; so that, for example,
E major had for key-signature three sharps instead of
four, and C minor had only two flats instead of three : the
degrees of the scales in question being chromatically altered

FIGURED BASS SCHOOLS 19

by means of the necessary accidentals.^ Heinichen says :
" It cannot be denied that, for example, in the E major
mode the major Seventh t/# is as natural and essential
as is the bt{ in the C major mode ; nevertheless in practice
this sharp is seldom included in the key signature, but for
the most part is indicated by an accidental placed before
the note, which itself appears to have an accidental character.
The minor Sixth [in the minor mode] is indicated in a similar

I The effect of this practice, which prevailed well into the eighteenth
century, i.e. , up to the time of Bach and Handel, was to give to the
major scale in sharp keys the same form, the same order of tones and
semitones, as the seventh Church mode, the Mixolydian ; and to the
minor mode, in fiat keys, the same form as the first Church mode — the
Dorian ; thus : —

E Major. (#) Mixolydian Mode.

C Minor. t>. • iv/r j

,_s Dorian Mode.

The exclusion from the key-signature of the flat necessary for the
sixth degree of a minor scale was in conformity with the traditional
practice in respect of the Dorian mode, where a Minor Sixth — B[> — was
required in order to avoid the tritone, / — b ; but which was not
written. If, therefore, we add to the Dorian mode the flat necessary
in order to indicate the minor Sixth, and to both Dorian and Mixolydian
modes the sharp necessary for the seventh degree, in order to obtain
the semitone below the Final required for the Final Cadence, we obtain
our modern major and minor modes : —

(Major Mode.)
Mixolydian. (|;)

-o ^

o ^^

(Minor Mode.)
^°"^°- (P) (#)

-~^^' — .. ^ <^ '^ '' "-4

All this throws an interesting light on the manner of development
of our major and minor modes from the old modes of the Church.
It should be noted that, even in our own day, the sharp necessary for
the seventh degree of the minor scale is never included in the key
signature, but invariably prefixed as an accidental.

20 THE THEORY OF HARMONY

way " ^ (by a flat prefixed to the note in question). But
although Heinichen pleads for a rational method of indicating
the key-signature, he himself, none the less, names the various
degree^ of the /# major scale as follows : /#, g^, b\^, 6tl, c|:,
djlf, f, /# ; and those of b\} minor thus : b\^, c, c#, djf^, f,
/#' &#' ^b- Characteristic of the time, also, is Mattheson's
description of the minor triad on ^#, for example, as
^#"/#'^b ' while the chord of the Dominant Seventh on
e\f, is e\^-g-b\}-c^ ; but the chord of the Augmented Sixth

on g, — g-b-F-

Of other works of the time treating of figured bass, there
may be mentioned the Principles of Accompaniment at the
Clavecin, 1727, by J. F. Dandrieu, a Parisian organist and
composer. Mattheson especially commends its system of
chord classification, a system which he himself adopted.
A work which appears to have been held greatly in esteem,
and which according to Spitta was familiar to J. S. Bach,
was the Miisikalische Handleitiing of Friedrich Erhardt Niedt,
the first part of which was pubhshed in 1700. This treats
of chords and their signatures : of cadences, and simple
formulas of modulation. A second part (1706) describes the
different methods of varying the bass part ; instead of
moving stiffly from one to another harmony note, this may
by means of scale and arpeggio figures, of passing and
auxiliary notes, be made more melodically interesting. In
the same way, the upper parts are susceptible of a great
many forms of variation. The third part (1717) treats
of Counterpoint, Canon, and various forms oif vocal com-
position. A second edition of the second part of the work
was given by Mattheson in 1721, in an enlarged form. The
great merit of Niedt, according to Fetis, is that " he for
the first time presents the chords of the Dominant Seventh
and Dominant Ninth in their true character, i.e., as capable
of being taken without preparation." - Fetis however
cannot understand why Niedt, having made such a notable
advance in the science of harmony, should frequently resolve
the Seventh in the chord of the Dominant Seventh upu'ards,
instead of allowing it to descend one degree, which is its
natural resolution. He says : — " The ninth chapter, which
treats of these chords, presents us with several examples

1 Dey Generalhass in der Composition, p. 150.
- Esquisse de I'histoire de Vharmonie.

FIGURED BASS SCHOOLS

2 1-

of a false ascending resolution of the Seventh. This fault
is frequently repeated. It is remarkable that ^lattlieson,
to whom we owe a second edition of the second part of
Niedt's book, should have said nothing of this irregularity."
This " irregularity " is indeed for Fetis not only a remarkable
but an awkward fact. But it is not astonishing that
Mattheson should have said nothing concerning Niedt's
false resolution of the Seventh, seeing that he frequently
does the same thing himself ; as for example in the following
succession of chords from that section of the Kleine General -
bass Schule which treats of chords of the Seventh : —

z#s

--^

_c_

-^-

:S=

Although Niedt's work is regarded by Fetis as having gi\'en
" a wholesome impetus to the theory of harmony," it never-
theless brings forward no new theoretical principles ; the aim
of its author is rather to present, in as clear and concise a form
as possible, the rules relating to the science of figured bass.^

The same traditional \-iews respecting the nature of
harmony, and the attempt to apply to chords and their
treatment the old contrapuntal rules originally designed
to apph' only to intervals, characterize also the other works
of this time which treat of Figured Bass and Accompaniment ;
such as those of G. M. Bononcini (// pratico mitsico, 1673) ;
J. A. Delaire {Traite de V accompaniment, 1690) ; J. Boyvin
{Traite abrege de I' accompaniment, 1700) ; and Fr. Gasparini
[L'armonico pratico al cimbalo, 1683). But an important
consequence, especially in its influence on the theory of
Rameau, of the practice of accompaniment not only from
a figured but also from an unfigured bass was the gradual
development of what became known as the " rule of the
octave" {Regitla deU'ottava ; R>-gle de V octave), a simple,
concise harmonic formula which prescribed for each note

1 This is to some extent indicated by the title : Miisikalische
Handleitung, oder Grnndlicher Unterrichl, vermittelst wetchem ein Lieh-
haber edlen Miisik in kurzer Zeit sich so iveit peyfectioniren lian, etc.

22 THE THEORY OF HARMONY

of the scale or key system its appropriate harmony. The
chords employed for this purpose consisted of, or were
derived from, the three principal harmonies of the key
namely, Tonic, Dominant, and Subdominant.^ This formula
was made the basis of a system of instruction in composition
by Fr. Campion, in his Traite de Composition selon la ri'gle de
r octave (1716). But such works as those of Gasparini,
Delaire, Heinichen, and Mattheson, all of which treat of
the " rule of the octave," were evidently regarded as schools
of composition, as well as of accompaniment from a figured
or unfigured bass.

From the foregoing a sufficiently adequate idea may be
obtained of the state of matters which prevailed when
Rameau set out to discover the fundamental principles
of harmony. It was clearly perceived that the Church
Modes had given place to our Major and Minor Modes. It
was recognized that between the simple and compound
forms of the interval no theoretical distinction, as regards
their harmonic significance, need be maintained. The
consonant triad is given a place by itself, and assigned a
special name {common chord, trias harmonica, I'accordo
perfetto) ; that on the Tonic being regarded as of peculiar
importance. The relationship existing between a chord and
its inversions, in so far as all are perceived to be composed
of the same sounds, is understood ; while the lowest note
of every chord, whether inverted or not, is described as the
fundamental note of the chord. But most noteworthy of
all and dominating, if not the practice of harmony, at least
every conception as to the nature and properties of chords
is the theory, not only implied but expressed, that chords
are the result of the (arbitrary ?) union of intervals, and are
consonant or dissonant, pleasing or the reverse, according
to the nature of the intervals which occur between the bass
and the upper parts. This was the outlook on harmony
which undoubtedly prevailed up to the time of Rameau. Is
it unworthy of being described as a theory, or regarded as
a principle of chord generation ? On the contrary, it is
not only a very old theory but a very respectable one, and
plays a most important part in the theory of the generation of
chords as this is understood by many even at the present

1 See also pp. 118-120.

FIGURED BASS SCHOOLS

day.^ It dates from the first beginnings of polyplionic music.
But however adequate it may have been then, or in the time
of Dufay, of Depres, or even of Palestrina and Lassus, when
the harmonies in use were few, simple, and for the most part
consonant, it was quite inadequate for the new harmonic
conditions which had arisen during the course of the seventeenth
and the beginning of the eighteenth centuries. The conception
of harmony as arising from the arbitrary addition of sounds
above any bass note, besides having become insufficient for
the explanation of the facts, had led to the most contradictory
results, even to the extent of admitting, as consonant, chords
that were dissonant, and turning into dissonances, intervals
that were consonant. Thus in the following chords : —

c, the Fifth above the bass note, was considered to be the
dissonant note in the first chord,- and e, the Sixth, the
most characteristic dissonance in the Second. But both
Fifth and Sixth are consonant intervals. It was therefore
concluded that these intervals, although in themselves
consonant, might nevertheless be employed " after the
manner of dissonances." But other intervals were a source
of equal embarrassment. Especially was this the case with
the Fourth. The Fourth, tlieoretically recognized, and
rightly so, as a perfect consonance — or at least the most
perfect consonance after the Fifth — was perceived in practice
to produce a dissonant effect,'^ as it had been by generations

^ Thus the triad is said to result from the union of two Thirds, or of a
Third and a Fifth ; a Third added above this triad giv-es the chord of
the Seventh ; a Third added above this chord of the Seventh gives
the chord of the Ninth, and so on.

- So \vith Heinichen and Keller ; Mattheson, however, considers
this chord to be consonant.

^ Thus Gasparini says : " E veramente la Quarta posta fra le
Consonanze, si da gli Antichi, come da' Moderni vien considerata
per Consonanza perfetta, nia fii disapprovata di itsarla per foudamento.
Onde per tal ragione, e per il nostro proposito la chiameremo
Dissonanza." {L'armonico pratico al cimbalo, Ch. 7.)

24 THE THEORY OF HARMONY

of composers before this time, and has been up to the present
day. It was therefore placed among the dissonant intervals.
Being dissonant, it required to be resolved, and to be followed
by the Third. Hence we have this remarkable result of the
interval theor}' of chord formation ; the Fourth, one of
the smoothest of consonances, produces even when used
alone, and not in combination with other intervals, a dissonant
effect ; further, its most natural resolution is on the Third,
an interval which is not nearly so consonant as itself !

It was this view of the nature and constitution of chords
which, in part at least, prevented writers on music and
theorists of this period from anticipating Rameau's theorv
of harmonic inversion. Acquainted as they were with the
intimate relationship existing between Octave sounds, as
well as with the fact that in the case of a chord and its
inversions each chord consisted 'of practically the same
sounds, it must appear strange, until all the circumstances
are taken into account, that they did not make a practical
use of the knowledge they possessed for the purpose of
simplif\'ing their signature tables, of reducing the number
of their rules, and making their application less difficult
and obscure. Rameau refers to this matter as follows :
" The knowledge of inverted chords," he remarks, " has
been gradually acquired ; but as this knowledge has been
gained b\' experience alone, the principle has been lost sight
of ; whence has arisen an infinit\' of exceptions, equivocations,
subterfuges. Inverted chords have been regarded as original,
while terms, intervals, chords, their progressions and
properties, have all been jumbled together." ^

So then, although it was quite clearly perceived that in
a chord and its inversions each chord was composed of the
same sounds, this did not shed much light on the question.
For was not each chord composed of different inter\-als ;
and could it be maintained that there was anything in common
between a Third and a Fourth, or between a Fifth and a
Sixth ? Each chord, then, must be considered to have its
own fundamental note, for was it not from this fundamental
note that the intervals placed above it were determined ?

This question of a fundamental note is intimately connected
with the whole subject of harmonic inversion. In the

I Traitf de Vharmovie, Bk. II., Ch. iS, Art. I.

FIGURED BASS SCHOOLS 25.

figured bass schools of the latter part of the sev^enteenth
century the conception of a fundamental note appears to
have been so natural, and its necessity so obvious, that
although the term {Fondamenio ; Fnndament-clavis) was
new it was either not defined at all, or explained merely
as the lowest note of an interval or chord. But if the term
was new, the principle it stood for was not. It is quite
wrong to suppose, as is frequently done, that the custom
of regarding chords as arising from the addition of intervals
above a bass note was the necessary outcome of figured bass
practice ; that is, of the use of figures, as .!;, ;.;, etc., to
designate the intervals of which the chord was to be composed.
Such a theory of chord formation had long been in use. From
the time of Zarhno. and indeed before his time, composers
had reckoned their inter\'als from the bass upwards ; thus, in
the case of the interval c-g, g was regarded as F^ifth of c,
and not c as Fifth oig. This is a fact not without significance
for the theory of harmony, for intervals might quite as easily
be reckoned downwards.

It is not in the figured bass system that we must seek for
the origin of such a custom. On the contrary, it was this
theory of the bass as fundamental note which lay at the
root of the whole figured bass system. The bass was the
bearer of the harmony ; the sound from which all the other
sounds composing the chord were determined. But, as has
already been pointed out, this conception of the fundamental
note as the lowest or bass note of every chord was quite
inadequate for the purpose of determining the consonant
or dissonant character of a chord, seeing that not a few
dissonant chords were composed of intervals all of which
were consonant with the bass. So also it was inadequate
for the purposes of a theory of harmonic inversion : more
accurately, it made such a theory absolutely impossible.
It completely barred the way.

Before Rameau published his Traite de Vharmonie, it was
considered — and this cannot be too frequently insisted on —
that in the case of three chords, such as c-e-g. e-g-c' , g-c' -e' ,
each chord had a different fundamental note : in the first
chord it was c ; in the second e ; and in the third g. But
before the theory of harmonic inversion could be established,
it was necessary to prove that all three chords had in reality
but one and the same fundamental note. It was necessary

26 IHE THEORY OF HARiMONY

to give to the term fundamenial a new meaning and
definition ; above all, to draw a sharp distinction between
bass note and fundamental note. How was this to be done ?
This question may well be asked, notwithstanding the
intimate knowledge we of the present day possess of the
nature of harmonic inversion. Was nothing more required
than to point out the identity, in respect of harmonic
significance, of Octave sounds ? By such means, it is true,
it is easy to demonstrate that the Fourth is the inversion
of the Fifth. But it is quite as easy to demonstrate, by the
same means, that the Fifth is the inversion of the Fourth.
Each interval, then, is "original" and "fundamental"
■ — seeing that each may be regarded in turn as the foundation
of the interval which arises from it by inversion — and each
has its own fundamental note. This represents exactly
the state of matters which prevailed among the theorists
and practicians of the figured bass schools. The mere
recognition of the identity of harmonic meaning of octave
sounds left matters where they were, in so far as the principle
of harmonic inversion was concerned. Rameau's task was
to demonstrate not only that both the intervals in question had
the same fundamental note, but also to show that while the
Fourth was derived from the Fifth, it was altogether opposed
to a rational conception of the nature of harmony to describe
the Fifth as derived from, or as the inversion of, the Fourth.
What is true of intervals in this connection is true of
chords. Rameau hits the nail on the head when he states
that the fundamental error among theorists of the figured
bass schools was that they described " derived chords as
original," notwdthstanding that they must have perceived
the similarity, in respect of harmonic significance, between
a chord and its inversions. It was necessary to prove that
in the case of a chord and its inversions there existed an
original and fundamental chord from which the inverted
chords were derived, and that all had but one and the
same fundamental note. Rameau had therefore to disco^'e^
what was the real Fundamenial Bass, or Basis, of these
chords. But nothing of this could be done until there had
been brought to light the principle of harmonic generation, or
generation of chords. In no other way could the " funda-
mental note " be established as the basis, source, and
foundation of the harmony.

FIGURED BASS SCHOOLS 27

It is evident tlien that the principles of Harmonic Inversion,
of the l^'undamental Bass, and of Harmonic Generation, are
all closely linked together, and in fact cannot be dissociated
from each other ; and it is no mere accident that Rameau
treats first of the principle of Harmonic Generation, and only
subsequently of the Fundamental Bass and of the inversion
of chords. Rameau's task was not quite so easy as might
be imagined. Theorists have too lightly assumed that all
that was necessary in order to establish the principle of
harmonic inversion, and of the inversion of chords, was to
demonstrate the identity of meaning of Octave sounds.
But if this had been all, then the credit for the discovery
of the principle of harmonic inversion belongs not to
Rameau, but to Zarlino. Such indeed is the view taken
by Dr. Riemann, who thinks that Zarlino knows all about
harmonic inversion.

These remarks relating to the inversion of chords represent
facts which, especially with regard to their theoretical sig-
nificance, have never been properly elucidated. Nothing
is more common than to find musical theorists who,
although they accept Rameau's theory of inverted chords,
nevertheless reject his principles of Harmonic Generation
and of the Fundamental Bass ; a fact which, if it does
not argue on the part of such theorists an insufficient
acquaintance with Rameau's theory of harmony, proves at
least that they have inadequately grasped the nature of
the intimate connection existing between the principle of
harmonic inversion and those principles from which it
naturally proceeds.

It was, then, the inabihty to draw an effective distinction
between fundamental and bass note which was mainly
responsible for the confusion, obscurity, and uncertainty
which prevailed concerning the rules and their appHcation ;
a confusion quite well recognised by Mattheson, who says :
" These things "—that is, chords, and the rules for their
treatment — " bring to the mind more darkness than light,
presenting themselves to us as they do in a compUcated
and disorderly fashion, and leading the thoughts often quite
away from what is really essential, which is directly opposed
to a good system of instruction." It was in order to discover
a means for the more systematic treatment of chords that
Mattheson made his various attempts at chord classification,

THE THEORY OF HARMONY

all of which of course were quite futile. One of his methods
of classification is as follows : the table of chords is peculiarly
interesting, showing as it does the chords in use at the
beginning of the eighteenth century, and also to some extent
the nature of the difficulties which confronted Rameau : —

2nds.

3ids

4ths.

4
5

oths.

Sifinaturen

4
2

4
3

6 2

b5 i b5

Full-

SthuiHcii

6
4

5
3

3 I

' i '

5ths.

o6 m

3 3

8 I 8

6 6 6

i 6

6ths.

I [

^ js i 6 6 „^ „, _ i 65 ; [>6 [76
4 ^ f, 5 bo ^-^ ^^^ 43 $ i

3
6

3 :
8 8
6

8 i 8 8 « I 8 ^ 8 > « « ^

6th s.

7ths

6
54

5 b"'

7
2

4
5

5
3

7Ss

7
4

7
4
2

4
2

7
5

b7
b'5

7
5
2

b">

3 3

7ths.

8ths.

9ths.

1 1
4 ' 5
•2 4

76 ; b7 i b7
^6 b5 65

87 '8b7

I Q Q 9 98 ' , q„ q„ 98

9 98 ^ ^„ 7 76 'yl*' ^^ 76

^ ' 4 ^6 ^^ ^•■' ; 56

3 3

3 3 5 3 38 38 I 3

CHAPTER II.

GIOSEFFO ZARLINO (1517-1590), AND THE GENERATION OF
HARMONY : NATURE AND INFLUENCE OF HIS WORK AS

A THEORIST.

In his search for the " natural principles " of Harmony
Rameau was wise enough not to trust solely to his own
reflections, but availed himself of every additional ray of
light which might help to dissipate some of the obscurity
which beset his path. He appears to have studied diligently
everything within his reach which had already been written
on the subject of harmony. Of the authors quoted by
Rameau in his Traite, the following are the most important : —
Gioseffo Zarlino [Istituzioni harmoniche, Venice, 1558, and
Dimostrazioni harmoniche , 1571) ; Rene Descartes {Com-
pendium musicae, 1618 ; Rameau made use of a French
translation of this treatise, entitled Abrege de la mitsiqne) ;
Charles Masson {Nonveau traite des regies pour la composition,
1694) ; Marie Mersenne {Harmonic Universelle, 1636-1637,
two volumes of over 1,500 pages ; the first part only of this
work — Livre I . De la musique the'oriqiie, is mentioned by
Rameau. It contains numerous musical illustrations, and
was published under the pseudon\'m of " Desermes," which
likewise is the name given by Rameau when he quotes
this author). Further, the important work b}' Sebastian
de Brossard, which must have pro\-ed of great service to
Rameau {Dictionnaire de musique, first edition in 1703,
frequentl}^ referred to as the first musical lexicon), ^ and a
text-book by Sr. Frere [Les transpositions de musique de
toutes les manieres). Of these authors something must be
said before we proceed to the examination of Rameau's
Traite' de I'harmonie. Of especial importance are the

^ This honour, however, would really appear to belong to the work
Terminorum musicae dijfinitoriuni of Johannes Tinctoris (d. 1511).

30 THE THEORY OF HARMONY

theoretical researches of Zarhno and Descartes. It has been
found necessary to devote a considerable amount of space
to the consideration of the theoretical works of Zarlino.
For this no apology need be made. The acquaintance with
the theoretical achievements of Zarlino is indispensable for
a right understanding of the development of the theory of
harmony, even of the nature of harmony, and its employrnent
in harmonic music. It has been necessary to define clearly
Zarlino's theoretical position, and to shovv exactly what he
accomplished. We find it frequently stated, and generally
credited among musicians, that the theory of harmony begins
with Rameau. This is, to say the least, an exaggerated
statement. It would be more in accordance with the facts
to describe Rameau's works as a link, one without doubt
of extreme importance, in the chain of the development of
harmonic science. It is difticult to imagine that the works
of Rameau constitute an exception to those general laws
of development which may be observed to mark the progress
of every other art and science. Such a notion in fact is
altogether erroneous. In the theoretical researches of
Zarlino and Descartes we find beyond all question the roots
of the theory of harmony of Rameau ; how much Rameau
was indebted to both these theorists will soon be evident.

Extremely lucid are the definitions of the two Modes,
Major and Minor, given by Brossard and Masson. The
former says : — " In every mode there ought to be distinguished
three essential notes, namely, the Final, the Dominant, and
the Mediant. . . . The INIediant divides the interval between
the Dominant and the Final into two Thirds ; whence arises
what is known as the Triad or Trio harmonique. ... As
every one of the diatonic or chromatic semitones within the
compass of the Octave admits of a major Third being placed
above it, there are therefore twelve Major Modes, and as
each of these may bear a minor Third, there are also twelve
Minor Modes." Masson expresses himself in much the same
terms. It is clearly understood that the mode is major or
minor according as the common chord on the Final or Tonic
is major or minor, and that the essential notes of each mode
are the notes of the Tonic chord. \\'hence it follows that
the mode, the scale, has as its basis not Melody but
Harmony. Herein also is the root of the doctrine so vigorously
expounded bN- Rameau that Melody arises from Harmony.

ZARLINO AND THE GENERATION ()¥ HAKiMONV 31

Even more important however, in respect of its undoubted
influence on Rameau, is the definition given by Brossard of
the Iriade harmonique, or common chord. This chord, he
says, " is composed of three essential sounds, heard simultane-
ously ; none of these sounds being an Octave apart, but two
of them lying a Fifth and a Third above the sound which serves
as their fundamental. In a word, it is a chord composed of
a Third and Fifth, as Sol-si-re', or La-iit-mi. . . . The term
harmonique is without doubt given to it because of the
marvellous nature of the perfect Fifth, which naturally divides
itself into two Thirds, both of which are excellent and verv
harmonious : . . that sound which divides the Fifth so admir-
ably and agreeably into two Thirds is called the Harmonic
Mean, or Medius Harmonicus. The division of the Fifth into
two Thirds can be made in two ways : (i) harmonically,
when the major Third is at the bottom, and the minor Third
at the top (as c-e-g), then the triad is perfect and natural ;
(2) arithmetically , when the minor Third is at the bottom,
and the major Third at the top (as a-c-e), then the triad
is imperfect and minor." The striking resemblance between
the language of Brossard and that used by Rameau will soon
be evident. Too much importance however need not be
attached to Brossard 's use of the word " fundamental."
With him, as with Heinichen and Mattheson, " fundamental "
and "bass" are equivalent terms. On the other hand his
conception of the nature of the " harmonic triad " is note-
worthy. He regards it as arising, in the first place, from the
interval of the Fifth, which naturally divides itself into two
Thirds — not the result therefore of the arbitrary addition of
Third and Fifth above a bass note — the Medius Harmonicus
being then the determining factor in respect of the major or
minor character of the triad. ^

Rameau's references to Mersenne are chiefly in connection
with various acoustical phenomena. Mersenne points to
the natural tones produced by certain wind instruments,
from the first six of which there arise in succession the intervals
of the Octave, perfect Fifth, perfect Fourth, major Third,
and minor Third. This natural order of consonances corre-
sponds to the arithmetical series of numbers i, 2, 3, 4, 5, 6,

^ Brossard understands the harmonic division of the Fifth as

follows : — The Fifth, of which the proportion is 2 : 3 = 4 : 6, has, as
harmonic mean, 5 : whereby the Fifth is divided into a major Third
4 : 5 -|- a minor Third 5:6.

32 THE THEORY OF HARMONY

which represent proportionally the vibrations of the sounding
body : —

I 2 3 4 5 ^ ^^^ ..

-<5>-

'Tzr

"Cjt

Mersenne is well aware that the natural harmonic sounds
of the trumpet or horn do not stop at the number 6,
and can disco\'er no satisfactory reason as to why the con-
sonances should be limited by this number. He cannot
understand why, at least, the number y should be regarded
as introducing a dissonance, and is of opinion that the
interval b : 7, which is slightly smaller than the minor Third
5 : 6, should be regarded, as consonant. Rameau follows
Mersenne in his use of the arithmetical series of numbers,
but applies this to the division of the monochord.^ In this
of course he acts wrongly, for it is not the arithmetical
but the harmonic series, i, h, h i> h> h which, apphed to
the division of the monochord, produces the consonances
in the order given above. Mersenne is much occupied with
the phenomenon of sympathetic vibration, and is disposed
to make the degree of perfection of consonances dependent
on the extent of the co-vibration of strings. But between
the sounds which constitute the perfect Fourth, as well as
the minor Third or the compound forms of these intervals,
no power of sympathetic vibration exists. Yet both these
intervals are consonant.

But it is especially Zarlino to whom Rameau constantly
refers throughout his theoretical works. Zarlino, he says,
is known as the " prince of musicians " (musical theorists),
yet is it not Zarlino we have to blame for all the confusion
which prevails in musical theory at the present day ?
Zarlino, with his Church Modes, his endless rules for the
progressions of the parts, for the syncopation of notes and
the resolution of dissonances, his wrong use of proportions.

* In his Traite. In his Generation Hartnonique, however, and subse-
quent works he makes use of the harmonic series in treating of the
major harmony.

/ARLINO AND THE GENERATION OF HARMONY 33

his failure to show clearly that melody results from harmony,
and not harmony from melody ! But Rameau never seemed
to have clearly grasped the fact that he lived in a different
epoch from that of Zarlino, and that the harmonic art of
his time was somewhat different from that which existed
in the time of the Church composers. Unquestionably he
owes ZarHno a great deal more than he appears willing to
confess. Zarlino's achievements as a theorist are highly
important, and his strong influence on Rameau may easily
be traced.

In the I stiticzioni H armonic he a.nd Dimostrazioni Harmoniche
of Zarlino, numbers, proportions, etc., play a great part,
Zarlino discusses the relationship which exists between the
science of music and arithmetical, harmonic, and geometric
proportions, with allusions to Pythagoras, Euchd, Plato,
and Aristotle. He shows reasons why music ought to be
considered as subordinate to arithmetic. From arithmetic
music borrows numbers, and from geometry mensurable
quantity.^ He apphes to the monochord a great variety
of different measurements, and compares at considerable
length the various intervals thus obtained. In his Soppli-
inenti Musicali (1588) he brings forward a scheme of equal
temperament, in which by means of a diagram of the
strings of the lute he demonstrates how the Octave may be
divided into twelve equal semitones. He however concludes
that music is neither purely mathematical nor purely natural
in its essence ; it is partly both, and may consequently be
said to be a medium between the one and the other.^

Zarlino considers harmony to be the result of the union
not of Hke, but of unhke or diverse elements. Thus from
the union of two intervals of the same species, whether
perfect or imperfect, there result inharmonious, that is,
dissonant combinations : —

ifsjg

^ " La Scienza della Musica piglia in prestanza dall' Arithmetica i
Numeri & dalla Geometria la quantita misurabile." (1st. Harm., Pt. I.,
Cap. 20).

2 Compare with Zarlino's definition of music that of Beethoven :
" Music is the link which connects the spiritual with the sensuous
life." Here it is not the theorist who speaks but the tone-poet.

34 THE THEORY OF HARMONY

The Octave however is an exception, because its sounds so
completely assimilate. In this necessity for diversity in
harmony he also finds the reason for the bad effect of
consecutive Octaves and Fifths — like must not be followed by
hke. Zarlino quite consistently extends this prohibition
to the Imperfect as well as to the Perfect intervals.
Therefore two major Thirds should not be taken in immediate
succession, nor even two minor Thirds : —

^ <^^^^ H^

-<^-

Still, two minor Thirds may on occasion be permitted, as
they are " so far removed from the perfection of the perfect
consonances." In the progressions by Fifths and major Thirds,
also, it will be observed that each voice proceeds by the equal
step of a whole-tone ; but it is only when one of the parts
proceeds by a whole-tone, and the other by a semitone, that a
good effect is produced. This half-tone step constitutes " the
principal ornament of harmony," and where it is absent
every modulation in harmony (that is, progression from one
to another interval within a mode), sounds harsh and as it
were dissonant.^ Like the other theorists of and before

^ " . . . del semituono maggiore, nel quale consiste tutto'l buono
nella Musica, & senza lui ogni Modulatione & ogni Harmonia e dura,
aspra, & quasi inconsonante." {1st. Harm., Pt. III., Cap. 27.)

In an excellent work by William Holder, D.D., Fellow of the Royal
Society', written " for the Sake and Service of all lovers of Musick,
and particularly the Gentlemen of Their Majesty's Chapel Royal,"
and entitled A Treatise of the Natural Grounds and Principles of
Harmony (1694), we find views similar to those of Zarlino with
respect to the immediate succession of imperfect intervals of the
same species. The author remarks : " It is a Rule in composing
Consort Musick, that it is not lawful to make a Movement of two
Unisons, or two Eighths, or two Fifths together : nor of two Fourths
unless made good by the addition of Thirds in another part ; but we
ma}' move as many Thirds or Sixths together as we please. \\Tiich
last is false, if we keep to the same sort of Thirds and Sixths."
(Ch. 4.) He admits, however, like Zarlino, that the effect of two
minor Thirds in succession is not unpleasing, but explains this in a
somewhat different way. He says : " In a Third minor, which hath
two Degrees or Intervals, consisting of a Tone and Semitone, the
Semitone may be placed either in the lower Space, and then generally
is imited to his Third major (which makes the Complement of it to
a Fifth) downward, and makes a sharp {i.e., major] Key : or else it
may be placed in the upper Space, and then generally takes his Third

ZARLINO AND THE GENERATION OF HARMONY 3 5

his time, Zarlino devotes much attention to the various
movements of the parts, to the laws of part-writing. In the
following examples of hidden consecutive Octaves and Fifths,
(a) (6) (c) {d)

■-€?-

--0.

231

32:

r-l&-

"ry

-(^-

he considers the descending progressions at (a) and
(b) to sound better than those ascending, as at (c) and
(d). It is characteristic of Zarlino that he endeavours
to find for this a scientific explanation ; he thinks that the
second interval at {a) and (b) is more easily apprehended
by reason of the comparatively slower vibrations of the
sounds which form it.-

Zarlino's importance as a theorist has been duly emphasized
by Dr. Riemann in his Geschichte der Musiktheorie.^ Dr.
Riemann points to the noteworthy fact that ZarHno has
demonstrated the possibility of a two-fold generation of
harmony ; that the major harmony may be shown to result
from the harmonic division of a string (by means of the
numbers or proportions i, \, \, I, ^, J), the minor harmony,
on the other hand, from its arithmetical division (by means
of the proportions i, 2, 3, 4, 5, 6). Dr. Riemann however
has permitted his enthusiasm for Zarhno to carry him too
far, and has made several statements which are not warranted
by the facts. He attributes to the ItaHan maestro a number
of theoretical discoveries with which he cannot properly
be credited. In so doing he gives an erroneous idea as
to what ZarUno actually accomphshed, and causes to be
overlooked the real significance of some of his theoretical
achievements. Zarlino, he tells us, is acquainted not only
with the inversion of intervals, but also with the inversion

major above, to make up the Fifth upward, and constitute a flat
[i.e., minor] Key. ... I say, if the Semitone in the Third minor
be below, then the Third major Ues below it, and the Air is sharp.
If the Semitone be above, then the Third major Hes above, and the
Air is flat. And thus the two minor Thirds join'd in consequence of
Movement, are differenc'd in their Relations, consequent to the place
of the Semitone : which Variety takes off all Nauseousness from the
Movement, and renders it sweet and pleasant." (Ch. 4.)

2 1st. Harm., Pt. III., Cap. 36.

3 Zarlino unci die Aufdeckuiig der dtialen Natttr der Harmonie,
pp. 369-406.

36 THE THEORY OF HARMONY

of chords ; he understands in its full theoretical significance
the nature of harmonic inversion ; and he has laid down the
principle that besides the major and -minor harmonies no
other fundamental harmonies exist. ^

Dr. Riemann even gives to Zarhno a place among the
theorists of our own day (Hauptmann, Ottingen, Riemann).
Zarhno has demonstrated, he says, that the intervals of the
Third and the Fifth are the sole constitutive elements of
composition ^ ; further, that he distinguishes only one species
of Third, namely the major Third, as the constitutive
element of the minor as well as of the major harmony, and
that he has expressly informed us that in the major harmony
the major Third (5 : 4) occupies the lower position, but in
the minor harmony, on the contrary, the higher position ;
that is only the major, not the minor Third is, in Haupt-
mann's language, a " directly intelHgible " interval.'' Such

1 " Die 1571 erschienenen Dimostrazioni harmoniche beseitigen
aber auch den letzten Zweifel daran, dass Zarlino eine vollkommene
klare Vorstellung von der Identitat der nur durch Oktavversetzungen
(Umkehrungen) von einander verschiedenen Accordbildungen hatte,
und beweisen zugleich, dass er dieselben bereits in den Istitntioni
ebenso meinte, wie er sie hier widerspruchslos darlegt. ... S. 87
(Ragion. II., Defin. XVII.) ist bereits die sehr wichtige Behauptung
aufgestellt, dass die kleine Sexte innerhalb des Senario zwar nicht
wirklich, aber doch ' in potenza ' enthalten sei und darum konsoniere !
die kleine Sexte hat bekanntlich die Proportion 8:5, und die 8 liegt
ausserhalb des Senario .- da aber die 8 nur eine ' replica ' der 4 und
2 ist, so ist doch die kleine Sexte ' potentiell ' im Senario inbegriffen.

" Daniit ist thatscichlich die Identitat der Bedeiitiing alter im Verhaltnis
der Umkehning stehenden Harmoniebildungen aufgestellt, und Zarlino 's
Satz, dass alle Verschiedenheit der Harmonie in der Einstimmimg der
Terz herithe, gewinnt den fundamentalen Sinn, dass es ausser dem
Dur- und Mollaccord keine Grundharmonien giebt." (Gesch. der
Musiktheorie, pp. 372-373.)

2 " Dass Zarlino mit den Replicate wirklich alle Oktavversetzungen,
auch die der Terz und Qiiinte unter den Grundton oder doch die des
Grundtones ilher die Terz und Quinte meint, ist zweifellos : sonst ware
ja auch nicht verstandlich, wie so er das gesamte Wesen der Harmonie
auf diese beiden Intervalle (Terz und Quinte) konnte zuriickfiiliren
wollen " (p. 371).

^ " Dass aber Zarlino auch bereits ebenso wie nach ihm Francisco
Salinas, Kameau, Tartini, u.a., und in unserem Jahrhundert Moritz
Hauptmann nicht zweierlei Terzen, sondern nur eine und dieselbe
Grdsse der Terz (5 : 4) als konstitutives Element sowohl der Dur- als der
Mollharmonie annimmt, habe ich bereits anderweit mehrfach betont :
Zarlino sagt ausdriicklich, dass im Duraccord (der Divisione harmonica)
die Terz (5 : 4) unten, im Mollaccord (der Divisione arithmetica)
dagegen oben liegt " (p. 373).

ZARLINO AND THE GENERATION OF HARMONY 37

then are Dr. Riemann's statements. They have a direct
bearing on the subject of this inquiry ; and it remains to
be seen whether or not, or to what extent, Dr. Riemann is
justified in making them.

It has been pointed out that the harmonies in use in the
time of Zarhno were few and simple. But it is only to one
trained to regard music from our present harmonic standpoint
that such harmonic resources appear to be meagre and in-
sufficient. The older art, although it was not on harmony
alone that it depended for its aesthetic effect, was neverthe-
less capable of a very high degree of harmonic expressiveness.
Composers of that time did not consider that there was any
lack of harmonic material ; for them a rich means of harmonic
variety existed in the various consonances, and in the various
wavs of combining them. Not only so, but by different
arrangements of these consonances it was possible to obtain
a great many different tone-combinations which varied in
harmonic effect and expressiveness : a delicate and subtle
art which has since been to a great extent lost. For example,
the following harmonies represent to us but a single chord,
the chord or harmonv of c. But such was not the view of

-y 1

^ — ^ —

f-i

¥f — ~. —

&»

G»

—^^

■~

\>i

-&-

-<s»-

-s»-

-<s>-

-&-

-<s»-

etc

1 1 f^

{(*)•

^■^±>— <r>

^-1

the matter taken b}^ the composers of the period in question.
For them, these harmonies represented individual tone-
combinations, differing in effect, and produced by a varied
disposition and combination of the various consonances ;
of a Third, a Fifth, an Octave, a Tenth, Twelfth, Double
Octave, and so on. If then at the time of Zarlino the
harmonic material did not comprise a great variety of chords,
it consisted on the other hand of a great variety of intervals,
simple and compound, and dissonant as well as consonant,
for dissonant intervals were made use of as notes of suspension
or syncopation, or as passing or auxiliar}' notes. This
large assemblage of intervals constituted for Zarlino a
theoretical problem not unUke that which confronted Rameau

312084

38 THE THEORY OF HARMONY

at the beginning of the eighteenth century ; only where
the former had to do with intervals, the latter had to do
with chords. It is worthy of note that Zarlino proceeded
much in the same way as Rameau ; that is, he set himself
to classify the various tone combinations in use, and to
discover their principle of generation.

In the first place, Zarlino makes a sharp distinction
between consonant and dissonant intervals. Not the dis-
sonances, but the consonances, are the constitutive harmonic
elements of polyphonic composition.^ A dissonance has no
real or separate existence apart from the consonance to
which it is related. It not merely retards this consonance,
but may even be said to define it more clearly. ^ Thus
Zarlino disposes of the dissonant intervals. This was not
a new theory ; it had long been held as an article of faith
by theorists and composers.

In dealing with the consonances, Zarlino points to the
fact that these correspond to certain simple numerical
ratios or proportions. He refers to Pythagoras, who had
demonstrated that all the perfect consonances maj' be
expressed by means of the first four numbers. Thus the
ratio which determines the Octave is. i : 2, the Fifth 2 : 3,
and the Fourth 3 : 4. For Zarlino therefore the principle
which determines the consonances is a mathematical principle
■ — the arithmetical series of numbers. The principle or
source of numbers is Unity ; unity is not itself a number,
but it is in unity that all things have their origin. /<The
varying degrees of perfection of the consonances are deter-
mined by the varying degrees of simplicity of the ratios
which express them ; the most consonant intervals are those
whose ratios are most simple, that is, are nearest to unity.
Thus the most perfect consonance is the Octave ; the Fifth
is less perfect than the Octave, and the Fourth less perfect
than the Fifth. The Octave unites itself so closely with the

1 " Le Compositioni si debbono comporre primieramente di
Consonanze & dopoi per accidente di Dissonanze." {1st. Harm.,
Pt. III., Cap. 27.)

2 " La Dissonanza fa parer la Consonanza, la quale immediamente
la segue piii dilettevole." {Ibid.)

■' " Ma la Vnita, benche non sia Numero, tuttavia e principio del
Numero : & da essa ogni cosa, 6 semplice, 6 composta, 6 corporale,
<) spirituale che sia, vien detta Vna." {1st. Harm., Pt. I., Cap. 12.)

ZARLINO AND THE GENERATION OF HARMONY 39

principal sound, represented by unity, that when both are
sounded at the same time they give the impression of a single
sound ; the reason for this being the nearness of relationship
of the Octave sound represented by 2 to the principal
sound represented by unity.^ The Octave, then, may be
considered as the replica of the principal sound. All intervals
larger than an Octave are therefore merely replicas of those
contained within the Octa\-e. Of the intervals which remain
there are, in addition to the perfect consonances already
mentioned, only the major and minor Thirds and Sixths —
the imperfect consonances. The ratio of the major Third
is 4:5, while that of the minor is 5:6. All the
perfect consonances therefore, as well as the major
and minor Thirds, may be expressed by the numbers
I, 2, 3, 4, 5, 6.

These are for ZarHno all the simple or, as he styles
them, "elemental " {elementali) consonances, which he defines
as those consonances whose terms do not differ by anything
greater than unity, that may therefore be expressed by any
two consecutive terms in the senario, or series of six numbers.
The major and minor Sixths are not considered by Zarlino
to be simple or elemental intervals ; neither are they replicati,
because they do not exceed the compass of an Octave.
Zarlino gives them the name of " composite " intervals
{composte), because they are formed from the union of two
simple intervals. The ratio of the major Sixth, 3:5, is
capable of a middle term, which is 4 ; the major Sixth,
then, is seen to arise from the union of the perfect Fourth
3:4, with the major Third 4:5.^ The minor Sixth (8 : 5)
is also a composite interval, and arises from the union of
the perfect Fourth and minor Third, corresponding to the

1 " Et e in tal maniera semplice la Diapason, che se ben e contenuta
da sue Suoni divcrsa per il sito : diro cosi. paiono nondimeno al senso
im solo : percioche sono molto simili : & cio aviene per la vicinita del
Binario all' Vnita " (1st. Harm., Pt. III., Cap. 3.)

- " L'hexachordo maggiore e Consonanza composta, percioche i
minimi termini della sua proportione, che sono 5 & 3, sono capaci d'un
mezano termine che e il 4." {Isi. Harm., Pt. I., Cap. 16.)

" Vedesi oltra di questo l'hexachordo maggiore, contenuto in tale
ordine tra questi termini 5 & 3, il quale dico esser Consonanza
composta della Diatessaron & del Ditono : percioche e contenuto tra
termini, che sono mediati dal 4." [1st. Harm., Pt. I., Cap. 15.)

40 THE THEORY OF HARMONY

ratios 8:6:5.^ These two composite intervals, although they
are not actually {in alto) found among the simple consonances
comprised within the senario, nevertheless exist there poten-
tially {in potenza), seeing that they result from the union of
simple consonances which actually exist in the senario.^ The
minor Sixth (8 : 5) it is true causes Zarlino some little embarrass-
ment, for 8 lies outside the senario ; still, he thinks,
this 8 may be regarded as the cube of the first number 2, a
number which " actually " exists in the senario ; in any case
we know that this minor Sixth results from the union of
Fourth and minor Third, both of which are simple intervals.^

Thus Zarlino concludes his classification of the consonances.
He distinguishes three kinds of consonant intervals : (i) those
larger than an Octave {Replicati) ; (2) simple or " elemental "
consonances, and (3) " composite " consonances.^

Most remarkable is Zarlino 's explanation of the origin of
the Sixth. He does not explain the Sixth as arising from
the inversion of the Third, but accounts for it in quite a
different way. It is not only the minor Sixth (5 : 8) which
he considers to exist only " potentially " within the senario,
but the major Sixth (3 : 5) as well ; both have their origin
in the union of two of the simple consonances, the Fourth,
and the major or minor Third. Not only in the Istitnzioni,
but also in the Dimostrazioni, he insists that both the Sixths
are to be explained in this way.^ And yet in the latter

1 " Alquale aggiungeremo il minor Hexachordo, che nasce dalla
congiuntione della Diatessaron col Semiditono. . . . Imperoche
ritrouandosi tal proportione tra 8 & 5, tai termini sono capaci d'un
mezano termine harmonico ch'e il 6 ; il quale la divide in questa
maniera 8:6:5, in due proportioni minori : cioe, in una Sesquiterza
& in una Sesquiquinta." (1st. Harm., Pt. I., Cap. 16.)

2 " Pero dico . . . die nel Senario, cioe, tra le sue Parti, si ritroua
in atto ogni semplice musical consonanza, & anco le Composte in
Potenza. {1st. Harm., Pt. I., Cap. 16.)

3 " Et benche la sua forma non si troui in atto tra le parti del
Senario, si troua nondimeno in potenza : conciosiache veramente
la piglia dalle parti contenute tra esso ; cioe, dalla Diatessaron & dal
Semiditono : perche di queste due consonanze si compone : la onde
tra'l primo numero Cubo, il quale e 8, viene ad hauerla in atto." (Ibid.)

■♦ Zarlino, however, regards the Sixths also as " simple " intervals,
in the sense that they do not exceed an Octave.

"^ " L'hexachordo maggiore, anco il minore, nascono della con-
giuntione della Diatessaron col Ditono, o Semiditono : come diligente-
mente habbiamo dimostrato nel secondo Ragionamcnto delle Dimos-
trazioni harmoniche." (1st. Harm., Pt. I., Cap. 13.)

ZARLINO AND THE GENERATION OF HARMON\' 41

work Zarlino proves that he is quite famiUar with the inversion
of intervals. He shows that the Fourth is the inversion of
the Fifth, the Sixth of the Third, and the Seventh of the
Second. He even demonstrates that the mverted interval
partakes somewhat of the nature of the inter\'al of which
it is the inversion. Thus perfect intervals when inverted
give rise to other perfect intervals, imperfect give rise to
imperfect, and dissonant to dissonant intervals. For this
reason he considers the Fourth to be consonant, for it is the
inversion of ■ the Fifth. It must therefore appear strange
that Zarlino should have accounted for the Sixths in the
way he does ; for there seems to have been no reason why
he should not have explained the Sixths as arising b\^ inversion
from the Thirds. By relating as he does the major Sixth
to the major Third, and the minor Sixth to the minor Third,
he takes the most effective means of totally obscuring the
relationship of inversion which actualh^ exists between the
Thirds and the Sixths. For the major Sixth is not related
bv inversion to the major Third, but to the minor Third ;
and the minor Sixth is not related to the minor, but to the
major Third. It may be thought that ZarHno might have
explained at least the major Sixth as a " simple " and not
a " composite " interval, and as arising directly from the
senario, seeing that its ratio is 3:5, both of which numbers
exist " actually " within the senario. But he could not do
this without contradicting his principle of generation of
the consonances. This principle is the arithmetical progres-
sion I, 2, 3, 4, 5, 6, where the consonances find their exact
determination in the successive tenns of the progression.
It is not from this progression therefore that the major
Sixth can be generated. The major Sixth could arise directly
only from a new mathematical and arithmetical progression,
namely, i, 3, 5, 7, etc. But Zarlino, as might be expected,
is by no means prepared to abandon his first progression
in order to substitute for it the second. Hence his explanation
of the major Sixth as a " composite " interval consisting of
the proportions 3:4:5, w^hich proportions then are repre-
sented by successive terms of the senario. The minor Sixth
he attempts to account for in a similar way. Its middle
term, he tells us, is 6, and the interval is properly represented
bv the proportions 5:6:8. an explanation with which he
himself does not appear to be quite satisfied.

42 THE THEORY OF HARMONY

Why then does not Zarlino, instead of referring the minor
Sixth to the minor Third, with which it has nothing to do,
explain the minor Sixth as the inversion of the major Third ;
why does he not consider the minor Sixth, to use Rameau's
language, as a " derived " interval, of which the major
Third represents the " original " and " fundamental " form ?
The answer to this question throws a remarkable light not
only on Zarlino's real position ^^dth regard to inverted intervals,
but on the subject of harmonic inversion in general. Here
we find Zarlino in possession of a quite consistent theory of
interval inversion by means of the Octave ; even maintain-
ing, in despite of all objections to the contrary, that the
interval of the Fourth is consonant, because it is the inversion
of the Fifth. But Zarlino's theory of the inversion of intervals
by means of the Octave, while it enables him to show that
the Fifth when inverted becomes a Fourth, and that a
Fourth is the inversion of a Fifth, cannot prevent it from
being maintained that the Fifth is an inverted Fourth, or
that the major Third is an inverted minor Sixth. That is
Zarlino, notwithstanding his theory of inversion, is unable
to draw any effective distinction between " original " and
inverted intervals, for the simple reason that the inverted
intervals may themselves be regarded as " original." By
no means can Zarlino prove that the minor Sixth is not an
"original" interval, but is merely "derived" from an
interval which is " original," namely, the major Third.
Instead therefore of explaining the minor Sixth as the
"inversion" of the major Third, and as derived from it,
Zarlino prefers to consider this interval as " original," and
to give it quite a different explanation, even if this involves
him in the greatest embarrassment and difficulty.
'yf ZsLvlino however shows no desire, and does not even
'attempt, to make any such distinction between the various
intervals. He considers all the consonances to be " original "
and " fundamental." Each consonance has its own pecuHar
character, and Zarlino regards this as a happ}- circumstance ;
for, as he repeatedly insists throughout his works, it is by
the use of the consonances, each of which produces its own
characteristic effect, that the composer is able to obtain a
great variety of the harmon}'. In short, although Zarlino
explains the consonances as arising successively from term
to term of the senario, he nevertheless looks on each of the

ZARLINO AND THE GENERAllON OF HARMONY 43

intervals thus generated as having its own harmonic founda-
tion, its own " fundamental note." As he himself tells us,
the bass is " the foundation of the harmony." For Zarlino,
therefore, while c is the fundamental note of the harmony
c-e, the fundamental note of its inversion e-c' is not c but
e ; and this is why he describes the major Third, as c-e, as
a very good consonance, but its inversion e-c' as a very poor
one. How great is the difference here between the point
of view of Zarlino and that of Rameau, for whom both
consonances represent but different aspects of the same
harmony, that is, have the same harmonic meaning. The
reason is, of course, because the latter theorist perceives
that c is the fundamental note of both harmonies, and in
both cases relates e to c. Zarlino on the other hand feels
that the lowest note e is the foundation of the harmony
e-c' — although it is not the real " fundamental note " in
Rameau's sense of the term — and relates c' to e, whereby
the harmony e-c' obtains, as it needs must, a quite different
harmonic meaning and character from that of the harmony
c-e, and this quite apart from any question of key, or of
the position which the interval e-c' may have in the scale.
This aspect of the matter was one quite overlooked by Rameau.
So also with the interval of the Fourth. No sooner has
Zarlino affirmed this Fourth to be consonant, seeing that it
is the inversion of the Fifth, than he treats it as a dissonance :
it may be used between two upper parts (a), but is
dissonant if heard between the bass and an upper part (b) : —

(«) -0- (b) :§:

~<:r

In the same chapter we read that the Sixth, especially the
minor Sixth, almost approaches a dissonance in effect.^
And yet there is little question but that Zarlino, and other
composers of and before his time, were quite well aware
of the resemblance in harmonic effect existing between the
harmony c-e-g and the harmony e-g-c' ; in this connection

. ^ " Imperoche si come la Sesta per sua natura non e molto consonante,
& e men buona della Terza, massimente della maggiore : come si vede
che non la lasciate ne i Contrapunti dimorare in un luogo per molto
tempo, perch offende il senso." (Dimos. Harm. Ragion Seconda
Def. X.)

44 THE THEORY OF HARMONY

the part played by the " Faux-bourdon " in the evolution
of polyphonic music is of especial significance. Yet Zarlino
treats the Fourth and Sixth as " original," i.e., non-inverted
intervals. The reason is obvious. Both intervals possess their
own peculiar effect, and both are generated from the senario.
It is as impossible for Zarhno to explain the Fourth as having
its " origin " in the Fifth, or the minor Sixth in the major
Third, that is, as arising from the principle of inversion, as
it is for him to consider the ratios 3 : 4 and 5 : 8 to have
their " origin " in the ratios 2 : 3 and 4:5. And if this is
true of intervals, it is even more true of chords. But Zarhno,
as will soon be evident, has no suspicion that such things as
inverted chords exist.

In Chapter 10, Part III., of the Istituzioni we find a
noteworthy passage in which Zarlino shows us that he
considers that arrangement of the consonances which
corresponds to the harmonic progression of numbers to
be the only natural one ; the other (arithmetic) is, so to
speak, contrar}^ to the natural order. In this chapter he
asks why some melodies or compositions [Cantilene) sound
bright and cheerful, while others are somewhat sad or
plaintive in effect. He also distinguishes between the ]\Iodes
in a similar manner. Some of the ]\Iodes are bright in
character {allegro), the others are somewhat mournful [mesto) ,-
that is, he demonstrates the major or minor character of the
Modes according as the major or minor Third is heard above
the Final of the Mode. " The reason is," he says, " that in
the first the major consonances appear above the Final, as in
the ist, 2nd, 7th, 8th, gth and loth Modes or Tones ; thus
these Modes are bright in character ; for in them we see
the consonances arranged according to the nature of the
sonorous number, that is to say, the Fifth is divided harmoni-
calty into a major Third and a minor (4:5:6), which is
extremely pleasing to the ear. I say that here the consonances
are arranged according to the nature of the sonorous number,
for then the consonances appear in their natural places. . . .
In the other I\Iodes, which are the 3rd, 4th, 5th, 6th, nth
and 12th, the Fifth is placed in the opposite direction, that
is divided arithmetically (6:5:4), so that many times we
hear the consonances arranged contrary to the nature of
the number in question. In the first (the Modes first
mentioned), the major Third is frequently placed below the

ZARLINO AND THE GENERATION OF HARMONY 4 5

minor; whereas in the second the contrary is the case
[that is, the minor Third is placed below the major], and a
certain mournful or languid effect is produced, so that, the
whole melody has a certain softness of character {molle)."

With regard to these Church ]\Iodes, it must be understood
that Zarlino's classihcation of them is as follows : —

II.

III. _0- IV.

-^- VI.

!1^[:

1i=:t

H— K

lonius. Hypoionus. Dorius. Hypodorius. Phrygius. Hypo-

phrygius.

VII. .Q. VIII.

H^^

•^Ot4

IX.

-tta

XI.

XII.

-K

Lydius. Hypo- Mixo- Hypomixo- Aeolius. Hypo-

lydius. lydius. lydius. aeolius.

in which the odd numbers represent the Authentic, and the
even numbers their corresponding Plagal ]\Iodes ; the ist,
2nd, 7th, 8th, gth and loth Modes have a major Third and
perfect Fifth above the Final, whereas the 3rd, 4th, 5th, 6th,
nth and 12th Modes, have a minor Third and perfect Fifth.
It is not difficult to understand how -it is that Zarlino
comes to make this new and important distinction between
the Modes. He is struck with the mysterious, somewhat
mournful effect of the minor Third which appears above the
Final, which so strongly contrasts with the bright major
effect of the major Third when heard above the Final. He
endeavours to find a reason for this, but he cannot find it
in any principle of harmonic generation which he has so far
brought to light. He has shown that all the simple conson-
ances, including the minor Third (5:6), result from the
scnario, as well as from the harmonic division of the Octave
and of the Fifth. But in neither of these principles of
generation of consonances is he able to discover any explana-
tion of the peculiarly minor effect which may be produced
by the minor Third. For one thing, he clearly observes
that it is only in certain circumstances and under certain
conditions that this minor Third produces a minor effect.
For example, the minor Third c-g is present in the
harmony ^ . ^ - ^ > nevertheless the effect of this harmony is not

46 . THE THEORY OF HARMONY

minor, but major ; the same minor Third e-g is present in
the harmony ^.'^if ; in this case the harmony is minor.

Both harmonies are composed of the same intervals ; each
consists of a major Third (4 : 5) and a minor Third (5 : 6)
which together make up the perfect Fifth (2 : 3). Zarhno
however perceives that while in the first chord the minor
Third occupies the higher position, in the second chord it
occupies the lower position. The difference in the effect of
the two harmonies is therefore, he considers, owing to the
difference in the disposition of the Thirds of which they are
composed ; the minor Third does not in itself invariably
produce a minor effect, for the minor Third is present in
the major harmony ; this can only happen when it occupies
the lower position in the harmony. So then, Zarlino remarks,
while the Fifth never changes but has always the same

proportions (as '^'_% or f:f), the Thirds do change, not
with regard to their proportions, but with regard to their
position within this Fifth. If the major Third occupies the
lower position, the harmony is Major {allegro) ; if on the
other hand it occupies the higher position, the harmony
is Minor {mesta). The difference in the harmony is there-
fore owing to the difference in the disposition of the two
Thirds.^

1 " Ma perche gli estremi della Quinta sono invariabili et sempre si
pongono contenuti sott' una istessa proportione (lasciando certi casi
ne i quali si pone imperfetta) pero gli estremi delle Terze si pongono
differenti tra essa Quinta, non dico pero differenti di proportione ma
dico differenti di luogo : percioche (come ho detto altroue) quando si
pone la Terza maggiore nella parte graue I'Harmonia si fa allegra : &
quando si pone nell' acuto si fa mesta. Di modo che dalla positione
diuersa delle Terze, che si pongono nel Contrapunto tra gli extremi
della Quinta, . . . nasce la varieta dell' Harmonia." {1st. Harm.,
Pt. III.^ Cap. 31.)

Dr. Riemann has unfortunately failed to quote Zarlino correctly ;
he makes him say : " pero gli estremi della Terza si pongono differenti
tra essa Quinta," etc. (Geschichte der Musiktheorie, p. 373). In this
case, Zarlino appears to refer to one Third only, and some colour
is certainly given to Dr. Riemann's assertion that he distinguishes
only one kind of Third (4:5). Zarhno, however, uses not the singular
but the plural number {delle Terze), and speaks not of one but of
both the Thirds. The whole passage presents not the slightest
difficulty. What Zarlino actually tells us is, that while the Fifth
never alters (except in the case of the diminished Fifth) but is always
represented by the same proportions, the Thirds which are placed
within this Fifth do undergo alteration, not with regard to their

ZARLINO AND THE GENERA ITON OF HARMONY 47

It is from this passage that Dr. Riemann has argued that
ZarUno actually distinguishes only one kind of Third, namely
the major Third, and has given him on this account a place
among the representatives of our newest school of modern
theorists (Hauptmann, Ottingen, Riemann, etc.). " I have
on frequent occasions," Riemann remarks, " called attention
to the fact that Zarlino, in the same way as Francisco Salinas,
Rameau, Tartini, etc., and in our own day Moritz Haupt-
mann, distinguishes not two kinds of Third, but only one
and the same proportion of Third (5 : 4) as the constitutive
element of the minor as well as of the major harmony." 1
Dr. Riemann however has no better grounds for this
assertion than a line or two from Zarlino, which he misquotes,
and in which Zarlino is made to speak of one Third only,
when in reality he refers to both the Thirds. It must
certainly appear astonishing that Zarlino should make such
an assertion as that there is but one species of Third which
divides the Fifth either harmonically or arithmetically,
for this reason, among many others, that this startling
statement occupies only a line or two of the chapter in which
Dr. Riemann supposes it to occur, and not only is not
repeated in any other portion of his works, but m,eets on
the contrary, with the most positive contradiction. 2 Nowhere
does Zarlino state, or even suggest, that there is but one
species of Third ; throughout his works he repeatedly and

proportions, but with regard to their position, and the whole matter
becomes perfectly clear by a glance at the diagram which appears
immediately above the passage in question (see p. 49). In the one
harmony the major Third appears in the lower part, and in the other
in the higher part ; while the minor Third is in the higher part in
the one harmony, and in the lower part in the other. But while the
Thirds thus alter their positions, they do not alter their proportions.

^ Geschichte der JMusiktheorie, p. 373.

- Rameau is perfectly familiar with this chapter, and in treating
of the major and minor harmonies he uses language very similar to
that of Zarlino. He frequently states that the only difference between
the major and the minor harmonies is in the different disposition of
the Thirds. " The only difference is in the disposition of the Thirds
which together make up the Fifth; the Third which is major in one
case being minor in the other." {Traiti', Bk. I., Ch. 8, Art. 2.)
" As for the harmonic and arithmetical proportions, .the first divides
the Fifth so that the major Third is at the bottom and the minor
Third at the top ; whereas, according to the second proportion, the
minor Third is at the bottom and the major Third at the top." {Traiti',
Bk. I., Ch. 3, Art. 5.)

48 THE THEORY OF HARMONY

expressly asserts that there are two kinds of Third, a major
and a minor. Even in the same chapter as that from which
Dr. Riemann quotes, we read : " We may secure greater
variety in the harmony (although this is more necessary in
composition for two voices than in that for several voices) by
placing the different Thirds in the following manner. Having
first taken the major Third, which arises from the harmonic
division, we rriay take after it the minor Third, which arises
from the arithmetical division." ^ Here Zarlino not only
considers the minor Third to be a distinct species of interval,
but explains it as being different in its origin from the major,
and as resulting from the arithmetical division of the Fifth.

Having satisfied himself that the strongly contrasted effect
of the minor as compared with the major harmony is owing
to the different disposition of the Thirds which together
make up the Fifth, Zarlino now finds his way clear. Already
Glarean {Dbdecachordon, 1547) ^^^ pointed out that the
time-honoured division of the Modes into Authentic and
Plagal was one which was theoretically justifiable. As is
known, every Authentic and Plagal Mode was considered
to consist of a pentachord and a tetrachord ; but while in
the former the tetrachord occupied the higher position, in
the latter it occupied the lower position ; that is, the positions
of pentachord and tetrachord were reversed :

tetrachord.

Dorian (Authentic).

-'S>-

.c^

fY*X^

1 — H-

r^ s ^ -^

=-'

'^l^-

-J ^-> '■'

__<

pentachord. tetrachord.

Hypodorian (Plagal).

In other words, the Authentic Mode consisted of a Fifth and
a Fourth ; the Plagal, on the other hand, of a Fourth and a
Fifth. But such an arrangement, Glarean pointed out, exactly
corresponded to the harmonic and the arithmetical division of
the Octave ; for the harmonic division of the Octave d-d' ,

' Se adunque noi porremo variar I'Harmonia & osseruare piu che
si puo la Regola posta di sopra nel Cap. 29 (ancora che nelle com-
positioni di piii voci non sia tanto necessaria, quanto e in quelle di
due) e di bisogna, che noi poniamo le Terze differenti in questa maniera :
c'hauendo prima posto la Terza maggiore, che faccia la mediatione
Harmonica, Doniamo dopoi la minore, che farn la divisione Arithmetica."
(1st Harm.. Pt. III., Cap. 31.).

ZARLINO AND THE GENERATION OF HARMONY 49

gives the following result : 5th 4th ; while the Octave a-a' ,

d - a - d'

arithmetically divided, is 4th 5th

a - d - a

Zarlino, for the theore-

tical explanation of the new distinction he is drawing between
the Modes, now carries this process a step further, and applies
it to the Fifth. Both major and minor harmonies consist
of a major and a minor Third, but with positions reversed.
This exactly corresponds to the harmonic and arithmetical

division of the Fifth, for the Fifth

harmonically

Maj. 3rd Min. 3rd
divided, is c — e — g
Fifth

while the Fifth a-e, arithme-

Min. 3rd Maj. 3rd
tically divided, is a — c — e

Fifth

This distinction constitutes for ZarHno a new means of
obtaining variety of the harmony. He has frequently
pointed out that harmony is the result of the union, not of
like, but of unlike or diverse elements. The composer
should bear this in mind, for it is in the variety or
diversity of the harmony that its perfection consists. But
the variety of the harmony, or harmonic material, at the
disposal of the composer consists not only of the various
consonances which arise from the senario ; another means of
variety consists in the arithmetical as well as the harmonic
division of the Fifth. Zarlino explains this in a passage
to which great prominence is given by Dr. Riemann, and
which it is necessary to quote. In the chapter from which
the passage is taken, Zarlino gives the following diagram : —

Harmonica.

Ditono .

Arithmetica.

Semiditono.

:c2:

_cz:

isz:

Semiditono.

-f^-

-&-

Ditono.

so THE THEORY OF HARMONY

and remarks : " The variety of the harmony does not
consist solely in the variety of the consonances which two
voices form with each other, but also in that variety of the
harmony which is determined by the position which the
Third or the Tenth occupies above the lowest note of the
chord. Either this Third is minor, and the harmony to
which it gives rise is determined by or corresponds to the
arithmetical proportion ; or it is major, and the harmony
corresponds to the harmonic proportion. It is on this
variety that all the diversity and perfection of the harmony
depend. Perfect harmony demands that the Third and
Fifth, or their compounds (the Tenth and Twelfth) be actually
{in aito) present ; for besides these two consonances the ear
desires no further sounds which could render the harmony
more perfect." ^

Of this passage Dr. Riemann has given a free, a
somewhat too free, translation. He imagines that Zarlino
here states that " the essential content of polyphonic
music is to be found, not in the numerous consonances, but
rather in the distinction between the two possible forms
of harmony " - (that is, the major and minor harmonies) ;

^ " Conciosia che la varieta dell' Harmonia in simili accompagna-
menti non consiste solamente nella varieta della Consonanze che si
troua tra due parti, ma nella varieta fl»co dell' Harmonia, la quale consiste
nella positione della chorda che fa la terza, ouer la Decima sopra la
parte graue della cantilena. Onde, ouer che sono minori & I'Harmonia
che nasce e ordinata o s'assimiglia alia proportionalita 6 mediatione
Arithmetica, ouer sono maggiori & tale Harmonia e ordinata ouer
s'assimiglia alia mediocrita Harmonica : & da questa varieta dipende
tutta la diversita & la perfettione dell' Harmonia. Conciosiache e
necessario (come diro altroue) che nella Compositione perfetta [or
Harmonia perfetta, cf. note p. 54] si ritrouino sempre in atto la
Quinta & la Terza ouer le sue Replicate, essendo che oltra queste
due Consonanze I'Udito non puo desiderar suono che caschi nel mezo
ouer fuori de i loro estremi che sia in tutto differente & variato da
quelH." {1st Harm., Pt. III., Cap. 31.)

2 " Nicht in der Mannigfaltigkeit der Konsonanzen, welche je
zwei Stimmen bilden, .sondern vielmehr in der Unterscheidung der
beiden m5glichen Formen der Hai-monie der eigentliche Inhalt des
mehrstimmigen Tonsatzes zu suchen ist " {Gesch. der Musiktheorie,
p. 369). By this passage Dr. Riemann evidently means to sav that
Zarlino recognizes the major and minor harmonies to constitute the '
sole harmonic material of polyphonic music ; otherwise his language is
meaningless ; for " the essential content of polyphonic music" cannot
be held to consist in a mere " distinction " between two different kinds
of harmony. A little later, however, he makes his meaning clear, when

ZARLINO AND THE GENERATION OF HARMONY 51

and that " the Third and Fifth, and their inversions, constitute
the sole (harmonic) elements of composition." ^ Dr. Riemann,
however, reads into Zarlino's language what it certainly
does not contain, and gives a wrong impression both as to
what ZarUno has actually said, and as to what he actually
means. ZarUno does not state, nor even impl}-, that the
major and minor harmonies constitute the sole harmonic
material of polyphonic composition. He is speaking of the
variety of the harmony, and of the means by which this
variety may be obtained. This variety does not consist
solely {solamente) in the various consonances, but also {anco)
in the quaUty of the Third which appears above the lowest
note of a chord. In order to give to his assertion some degree
of probability. Dr. Riemann is obliged to assume that
Zarlino is acquainted with the inversion of chords, and that he
distinguishes between chords which are fundamental and
chords which are inverted. (See p. 36.) But these are
mere assumptions ; they have no basis in fact, nor is Dr.
Riemann able to bring forward any real evidence in support
of them. He thinks that by Replicati ZarUno understands
inversions. (See p. 36.) But ZarUno distinctly defines
Replicati as " intervals which are larger than an Octave,"
that is, the compound forms of simple intervals, and nowhere
throughout his works does he attach any other meaning
to the term.- Nor does Zarlino anywhere suggest that he
considers the Third and Fifth to be " the only elements
of composition." He says expressly the opposite.^ " The
elements of composition " [contrapunto), he states, " are of
two kinds. Simple and Compound {Replicati). The simple

he states that Zarlino's words can only be interpreted in the sense that
" except the major and minor chords, no other ground-harmonies exist "
(" Zarlino's Satz, dass alle ^^erschiedenheit der Harmonie in der
Einstimmung der Terz beruhe, gewinnt den fundamentalen Sinn, dass
es ausser dem Dur- und Mollaccord keine Grundharmonien giebt.")
{Ibid. pp. 372-373-)

1 " Die Terz und Quinte oder ihre Oktavversetzungen sind die
alleinigen Elemente der Komposition." {Ibid., p. 370.)

- " La onde dico, che gli Elementi del Contrapunto sono di due
sorti : Semplici & Replicati. I Semplici sono tutti quelli Intervalli
che sono minori della Diapason : com' e I'Vnisono, la Seconda, etc. . .
et li Rephcati sono tutti quelli che sono maggiori di lei : come sona
la Nona, la Decima, etc. (1st. Harm., Pt. III., Cap. 3.)

3 Ibid.

52 . THE THEORY OF HARMONY

intervals are all those which are less than an Octave ; as
the Unison, the Second, the Third, the Fourth, the Fifth,
the Sixth, the Seventh, and the Octave [!] ; the compound
intervals are all those which are larger than an Octave ;
as the Ninth, the Tenth, the Eleventh, the Twelfth, and so
on." But in fact Dr. Riemann, in his eagerness to include
Zarlino as one of the foremost representatives of the " newer
school " of harmonic science, not only quite mistakes the
real drift of his remarks, but fails to grasp the real nature
of the important theoretical pronouncement which he makes.
What Zarlino is chiefly concerned to demonstrate is that
there is a certain position of the harmony which excels all
others — the Compositione- or H armonia-perfetta. In this
the ear desires no further sound which could render the
harmony more perfect.

Rameau, to whom this passage was well known, employs
Zarlino's language, and borrows his terms. In the " perfect
harmony " {accord par fait) he states, we find only the Third
and Fifth, or their compounds. It is so called because it is
" the most perfect that the ear can imagine."

Further, Dr. Riemann has no ground whatever for his
extraordinary assertion that Zarlino recognizes the highest
note, that is the Fifth, of the minor harmony, as well as
the lowest note of the major harmony, to be the fundamental
note. It is true, and it is important to note, that Zarlino
defines the bass as " the Basis or foundation of the harmony,
because it forms the support of all the other parts." ^ But
Zarlino has nothing to do with " ground-harmonies " or
" fundamental notes " in our or in Rameau's sense of the
term. For like the figured bass practicians a century later
he regards the bass as the foundation of every combination
heard above it, whether this represents an inverted chord
or not. And if Zarlino was unable to distinguish correctly
the fundamental note of an inverted major harmony, it is
unlikely, to say the least, that he should prove himself to
be a more advanced theorist than Rameau himself, and
even of Helmholtz, in respect of the minor harmony.

1 " Et si come la Terra e posta per fondamento de gli altri Element! :
cosi '1 Basso ha tal proprieta, che sostiene, stabilisce, fortifica, & da
accrescimento all' altre parti : conciosiache e posto per Basa & fonda-
mento dell Harmonia : onde e detto Basso, quasi Basa, & sostenimento
dell' altre parti." {1st. Harm., Pt. III., Cap. 58.)

ZARLINO AND THE GENERATION OF HARMONY 53

If we summarize the foregoing, we find that ZarUno is
acquainted with the principle of Octave inversion, but does
not explain any of the intervals as arising from this principle ;
while of inverted chords he knows nothing. He does not
consider any consonance or harmony to arise from the principle
of harmonic inversion. He is not to be regarded as the real
progenitor of the Hauptmann-Ottingen, etc., school of
modern theorists who recognize only one species of Third
as " directly inteUigible." He does not consider the Fifth
of the minor Triad to be its fundamental note. He does not
state that the Third and Fifth are the only elements of
composition. Finally, he knows nothing of " ground-
harmonies," nor does he state that the only fundamental
harmonies which exist are the major and minor chords. In
short, it is impossible to consider ZarUno as a more advanced
theorist than Rameau himself, or as one of the most illus-
trious exponents of the " newer school " of harmonic science^

Still, the real theoretical achievements of Zarlino are
of much importance ; and it remains to be stated, as
briefly as possible, what it was that ZarUno actuaUy accom-
pUshed. In the first place, ZarUno classifies and systematizes
the harmonic material in use in his time. This consisted
of a large number of intervals, dissonant as well as consonant. ^. ; ^'
The dissonant intervals, Zarlino demonstrates, have no real y
separate existence apart from the consonances ; nevertheless
the dissonant intervals have a well-defined function, for they k,-^
not only retard but enhance the harmonious effect of the '\j
various consonances. Of the consonant intervals, some are
compound {Replicati) and are to be regarded merely as
repetitions of the simple intervals. The identity of harmonic
significance existing between a compound and a simple
interval is owing to the nature of the Octave, which resembles,
and may be said to represent, the principal sound. All the
consonances arise either directly or indirectly from the
senario, the most perfect being those which are nearest to
Unity. The consonances therefore do not arise arbitrarily,
but depend for their origin on a certain fixed and definite
principle, which at the same time determines their varying
degrees of perfection. This principle is a mathematical
one, and is contained in the senario. For ZarUno therefore
the senario is the " natural principle " of harmony, and of
harmonic generation.

54 THE THEORY OF HARMONY

Zarlino also shows that while some of the Modes are major
in character, the others have a minor effect. This is owing
to the quality of the Third which appears above the Final.
Either the Third is major, and it arises from the harmonic
division of the Fifth ; or it is minor, and arises from the
arithmetical division. The minor harmony is less harmonious
and perfect than the major ; the reason being that in the
minor harmony we find the consonances arranged " contrary
to the nature of the sonorous number."

Although Zarlino does not treat of " chords " in our sense
of the term, but of consonances, and of the various waj^s of
combining them,^ he nevertheless recognizes that there are
certain combinations of consonances which sound fuller
and more harmonious than any other.- The_most_perfect
•combination is that which consists 'of a Third and
Fifth, or their replicas (the Tenth and Twelfth). This
combination is regarded by Zarlino as being worthy"
of a distinctive name. He calls it the harmonia perfetta.
It is noteworthy that he assigns as the reason for it§
" perfection " not the blending of its sounds together in
such a way as to convey to the mind the impression of a
harmonic unity, but the " diversity " of its sounds, which
produce on the mind a sense of the greatest possible harmonic
" variety." ^ He advises the composer to make use of this
" perfect " harmony wherever possible. It is true, he admits,

^ Thus Rameau says : — " The error of ZarUno in the application of
liis rules is, that he considers not more than two parts at a time."
{Traite de I'harmonie, Bk. IL, Ch. 14.)

^ See the concluding part of the quotation from the 1st. Harm., p. 50
(footnote) .

* " Oltra di questo e da auertire, che quella Compositione si puo
■chiamar Perfetta, nella quale in ogni mutatione di chorda, tanto
uerso '1 graue, quanto uerso I'acuto, sempre si odono tutte
quelle Consonanze, che fanno varieta di suono ne i loro estremi.
Et quella e veramente Harmonia perfetta ch' in essa si ode tal consonanze ;
ma i Suoni 6 Consonanze che possono far diversita al sentimento sono
■due, la Ouinta & la Terza, ouer le Replicate dell' una & dell' altra :
percioche i loro estremi non hanno tra loro alcuna simiglianza, come
hanno quelli dell' Ottava : essendo che gli estremi della Ouinta non
movono 1' Udito nella maniera, che fanno quelli della Terza, ne per il
contrario. . . . dobbiamo per ogni modo (accioche habbiamo perfetta
cotale harmonia) cercare c6 ogni nostro potere, di fare udir nelle nostre
Compositioni queste due consonanze piu che sia possibile, ouer le loro
Replicate." {1st. Harm., Pt. III., Cap. 59.)

ZARLINO AND THE GENERATION OF HARMONY 55

that many compusers put the Sixth in place of the Fifth.
This is quite permissible, and is even to be recommended ;
but it is quite evident that he considers the Sixth to be much
less harmonious and " perfect " than the Fifth. ^ In thus
considering the Sixth to be a somewhat imperfect substitute
for the Fifth, Zarlino demonstrates how far he is removed
from any conception of inverted chords, or from any suspicion
that the chord of the Sixth, e.g., c-e-a, represents, not an
" altered " major harmony c-c-g, but is itself the first inversion
of the " perfect " minor harmony, viz., a-c-e.

Further, Zarlino defines the bass to be the real support
and foundation of the harmony. It is the " basis " of the
harmony, because it resembles the earth, which forms the
support of the other elements. This, it is true, is not a
sufficiently exact theoretical explanation. Nevertheless, this
recognition of the nature and function of the bass represents
a fact of the greatest importance for the science of harmony.
Zarlino's definition may quite well have been, and indeed
was in reality, the expression of what had been gradually
revealing itself to the consciousness of composers. But, as
already stated, the " foundation of the harmony " of Zarlino
has not the same meaning as the " fundamental note " of
Rameau. For Zarlino " fundamental note " and " bass
note " are equivalent terms.

The historical position of Zarlino is quite well understood.
He stands just at the close of the great potyphonic period
of music ; his works constitute a vast exposition of the
principles and practice of the masters of composition of that
period. But what exactly is his theoretical position ?
We have seen how he recognizes that there is a certain
harmonic combination which excels all others. It takes
a place by itself. It is the " perfection " of harmony ; the

^ " E ben vero, che molte volte i Prattici pongono la Sesta in luogo
della Quinta, & e ben fatto. Ma si de auertire, che quando si porra
in una delle parti la detta Sesta sopra'l Basso, di non porre alcun' altra
parte che sia distante per una Quinta sopra di esso : percioche queste
due parti uerrebono ad esser distanti tra loro per un Tuono, ouer per
un Semituono, di maniera che si udirebbe la dissonanza. (See also foot-
note on page 50). . . . Osseruara adunque il Compositore questo,
c'ho detto nelle sue compositioni : cioe, di far piii ch'ello potra, che
si ritroui la Terza, & la Quinta, & qiialche state la Sesta in luogo di
questa, 6 le Replicate : accioche la sua Cantilena venghi ad esser
sonora & plena." {1st. Harm., Pt. III., Cap. 59.)

56 THE THEORY OF HARMONY

Harmonia Perfetta. We have seen also that the chief work
of Zarlino has been to classify and to reduce to its ultimate
source the entire harmonic material of polyphony, consisting
of the various intervals, consonant and dissonant. What is
the net result of his labours, and what is the net result, so
far as harmony is concerned, of the strivings of generations
of composers, of the artistic labours of centuries ? It is this
Harmonia Perfetta, for as ZarHno himself points out, if all
the sounds represented by the terms of the senario be heard
together, there results from such an arrangement of the
consonances, not a clashing of sounds, but a harmony of
the most pleasing character.^

Zarlino 's position as a theorist, indeed, is in entire accord
with the nature of polyphony itself. The essence of polyphony
is its diversity, and the problem of polyphony is to bring
together those diversified elements in such a way that there
shall result a certain harmoniousness of character and of
effect. Zarlino's task as a theorist was to reduce the great
" variety " and " diversity " of already existing harmonic
elements to a definite and rational principle. That the result
of his labours should be the Harmonia Perfetta, represented
by the terms of the senario, is evidently for him a wonderful
circumstance. He cannot explain it, for it is a result he
certainly did not contemplate when he set about his task
of reducing the harmonic elements of polyphony to a rational
order. But it is for Zarlino a circumstance of deep signifi-
cance. It is a circumstance no less remarkable for the
history of music than for the science of harmony. For the
Harmonia Perfetta, the consummation from the harmonic
point of view of this great polyphonic period of music, the
end also of Zarlino's work as the theorist, the greatest and
most representative, of the polyphony of his time, is the
starting-point of the new harmonic period of music which
was shortly to be ushered in ; it is the starting-point also
of Rameau's theory of harmony, his principle of
principles.

^ " Et sono queste parti in tal modo ordinate, che quando si pigliassero
sei chorde in qual si voglia Istrumento, tirate sotto la ragione de i
mostrati Numeri, & si percuotessero insieme, ne i Suoni, che
nascerebbono dalle predette chorde, non solo non si udirebbe alcuna
discrepanza, ma da essi, ne uscirebbe una tale Harmonia, che I'Vdito
ne pigliarebbe sommo piacere." {1st. Harm., Pt. I., Cap. 15.)

DESCARTES {COMPENDIUM MUSIC AE). 57

These are important facts, the signiticance of which, by
musical historians and musical theorists generally, has been
passed over unobserved. We have here a picture which
touches the imagination ; here if an\^vhere we find a
veritable romance of musical history and science. Zarlino,
the learned and pious maestro, stands like an aged Simeon
between two great epochs ; he holds in his arms the fruit of
the striving of centuries, the principle from which shall
proceed a new artistic creation. He himself belongs to the
old order of things, but he looks forward into the new.

Rene Descartes (i 596-1650) {Compendium Musicae).

The Compendmm Musicae (1618) of Rene Descartes
appears to have been as famiUar to Rameau as the more
voluminous writings of the learned \'enetian master. This
little treatise, written when its author was only twenty-two
years of age, is in many respects a remarkable work.^
Descartes, as might be expected, proves himself to be
possessed of an acute faculty of precise scientific observation.
He refers in se\'eral places to the natural phenomena of
harmonics and of sympathetic vibration. Thus of the
overtone of the Octa\'e he says : " We never hear a [musical]
sound but its upper Octave appears also to strike the ear
in a certain measure." Not only so ; this Octave sound
reinforces the fundamental sound ^ (combination tones!).
The Octave is the first and most perfect of the consonances ;
not only is it the first consonance to arise from the senary
division of a string, that is, of a string divided successively
by the first six numbers, but in such instruments as the flute
it is the first harmonic sound to be obtained ; the Fifth
(Twelfth) arises only after the Octave. There is no conson-
ance which is in reality larger than an Octave ; for intervals
which exceed the Octave are " composite " intervals, and
consist of an Octave and a simple interval. Further, all
the consonances are contained within the Octave ; for

1 It was not published until after the death of its author in 1650.

2 Unde praeterea sequi existimo nullum sonum audiri, quin hujus
octava acutior auribus quodammodo videatur resonare, unde factum
est etiam in testudine, ut crassioribus nervis, qui graviores edunt sonos,
alii minores adjungerentur una octava acutiores. qui semper una
tanguntur cS: efficiunt, lit gvaviores distinctius awliantur." (De Octava.)

58 THE THEORY OF HARMONY

from the harmonic division of the Octave there arise the
Fifth and the Fourth, and from the harmonic division of
the Fifth, the major and the minor Thirds.^ Descartes also
points to the fact that if a string be set in vibration, other
strings more acute, representing the Octave, Fifth (Twelfth)
and major Third (Seventeenth) of the first sound, will be
made to vibrate, and to sound along with it.

It is by means of the arithmetical division of a string,
Descartes states, that we obtain all the consonances. But
in dealing with these consonances he proceeds in a very
different way from that of Zarlino. For he considers some
of the consonances to arise directly, the others only by
accident {per accidens). This it is true he cannot prove,
any more than could Zarlino, from the division of a string
by the first six numbers. He therefore adopts another
method. As, he remarks, we never hear a sound but we
hear at the same time its upper Octave, as c-c' , therefore we
never hear a Fifth but the Fourth also is heard to be

present thus 5th 4th So that, to use Rameau's language,
c—g — c'

the Fifth is to be regarded as the " original " interval ;
the Fourth, on the other hand, as " derived " from it. This
Fourth is in reality merely the " shadow " of the Fifth ; it
displeases, for it is the " shadow " and not the substance ! ^
The Fourth indeed is the most imperfect {infelicissima) of
all the consonances. Like the Fourth, which arises from the
harmonic division of the Octave, the minor Third also is a
consonance per accidens. " The minor Third arises from the
major Third, as the Fourth from the Fifth." Descartes
means that just as the Octave may be harmonically divided,
so also may the Fifth ; from the harmonic division of the
Fifth there arise two intervals, the major and minor Thirds,
of which the first is direct, and the second " accidental."
Similarly the major Third may be harmonically divided ;
of the two intervals which arise from its division, the first,
the major tone (8 : g) is direct ; the second, the minor tone

1 De Octava.

- " ideoque maxime quarta illi displiceret, quasi tantum

umbra pro corpore, vel imago pro ipsa re foret objecta." {De Quarta.)

DESCARTES {COMPENDIUM MUSICAE) 59

^9 : lo) accidental (!). These distinctions, Descartes proceeds,
are not merely imaginary ; they are confirmed by the
phenomenon of sympathetic vibration ; for " in the lute and
other such instruments, if one of the strings be set in vibration,
it will also cause to vibrate and to sound along with it other
strings which represent the Fifth and major Third above it,"
but none other. Whence it is manifest that the Octave,
Fifth, and major Third are the only direct or " original "
consonances ; the Fourth and the minor Third which proceed
from these are consonances only per accidens. So that there
are but three " sonorous numbers " namely, 2, 3, and 5,
the first of which represents the Octave ; the second, the
Fifth ; and the third, the major Third ; ^ the numbers 4
and 6 are merely compound forms of 2 and 3.

These observations of Descartes must have proved in the
highest degree illuminating for Rameau. Descartes makes
a noteworthy advance in the direction of Rameau's theory
of inversion. Not all the intervals have an independent
origin; some are "derived" intervals. His explanation of
the Fourth as the " shadow " of the Fifth, an expression
which is quoted by Rameau, is also Rameau's explanation,
but with a difference. For Descartes, strange to say, imagines
that this explanation accounts for the dissonant effect of
the Fourth ; on the contrary, the Fourth, as the " shadow "
or inversion of the F^ifth, ought to appear as one of the best
of the consonances, and not the worst. The Fourth, when

heard along with the Fifth, thus 5th 4th sounds almost

as consonant as the Fifth. Descartes then, although like
Rameau he considers the Fourth to be " derived " from the
Fifth, nevertheless fails to perceive that the Fourth when it
represents the Fifth is a good consonance, and has a harmonic
meaning similar to that of the Fifth.

In his treatment of the Fourth the resemblance between
Descartes' theory of " original " and " derived " intervals
and Rameau's theory of inversion begins and ends. As for

1 " In ilia enim advertendum est tres esse duntaxat numeros
sonoros 2, 3, & 5, numerus enim 4, & numerus 6 ex illis com-
ponuntur." [De Octava.)

6o THE THEORY OF HARMONY

the two Sixths, although Descartes makes passing mention
of the fact that the minor Sixth is the Octave complement of

the major Third as 3rd 6th he does not explain it, as

c — e —c'

he might be expected to do after his treatment of the Fourth,
as derived by inversion from the major Third ; possibly
for the reason that he would be unable to explain, in the
same way, the major Sixth as derived from the minor Third,
seeing that the minor Third is itself a " derived " interval.
He does not regard either of the Sixths as " derived," but
explains them, like ZarUno, as composite intervals. The major
Sixth, he tells us, arises from the union of the major Third and
the Fourth ; the minor Sixth from the union of minor Third
and Fourth. Descartes relates the major Sixth to the major
Third, from which it proceeds ; these intervals, he says, are
similar in nature and effect ; and in the same way he relates
the minor Sixth to the minor Third. ^ In proceeding thus
he succeeds, like Zarlino, in totally obscuring the real relation-
ship of inversion which exists between the Sixths and the
Thirds. Nevertheless, it was in the observations and
suggestions thrown out by the philosopher Descartes that
Rameau discovered some of the ideas from which were
evolved the main principles which lie at the root of
his theory of harmony. Descartes' treatment of the
Fourth, and his statement that the only " sonorous
numbers " are 2, 3, and 5, were for Rameau of the utmost
significance.

Finally, it was Descartes' version of Zarlino 's theory
of the senario that furnished to Rameau his chief " funda-
mental principle " of harmony. The words of Descartes are
thus quoted by Rameau at the beginning of his Traite de
I' harmonic : " Sound is to sound as string to string ; but
each string contains in itself all others which are less than
it, and not those which are greater ; consequently every
sound contains in itself those sounds which are higher, but
not those which are lower. Whence it is evident that the

1 " Sexta minor eodem modo fit a tertia minore ut major a ditono, &
ita tertiae minoris naturam & affectiones mutuatur, neque ratio est
quare id non esset." (De Ditono, Tertia minore, <~ Sextis.)

DESCARTES {COMPENDIUM MUSICAE) 6i

higher term should be sought for by the division of the lower,
and this division should be an arithmetical one, that is, one
consisting of equal parts. If then in the following figure
A-B represent the lower term in which I wish to find the

A D C E B

t ! t I I

higher, in order to form the first of the consonances, then
I divide it in two (this being the first number), as has been
done at the point C ; then A — C, A — B, are removed from one
another by the first of the consonances, which is called
Octave, or Diapason. Likewise if I wish to have the other
consonances, which follow immediately on the first, I divide
A — B into three equal parts, from which will result not one
acute term only, but two, namely A — D and A — E, giving
two consonances of the same kind, a Fifth and a Twelfth.
I can still further divide the line A — B into four, five, or six
parts, but not more, because the capacity of the ear does not
extend beyond this point " ^ (that is, the comparison of

^ De Consonantiis. The actual words of Descartes are : — " quia
scilicet aurium imbecillitas sine labore majores sonorum differentias
non posset distinguere." Descartes' meaning according to Dr. Riemann
{Gesch. der Musiktheorie, p. 456) is as follows : — From the harmonic
division of the Octave there result the intervals of the Fifth and Fourth ;
from the harmonic division of the Fifth, there result the major and
minor Thirds (4:5:6), and from the harmonic division of the major
Third, the major and minor tones (8:9: 10). Beyond this we
cannot go, because already' the diatonic semitone 15 : 16 arises as the
difference of the Fourth and major Third, and the chromatic semitone
24 : 25 as the difference of the diatonic semitone and minor tone. As
the chromatic semitone is the smallest interval known to melody, it
is evident that the complex of consonances must be limited by the
number 6, and that 7 and all higher intervals are theoretically inadmis-
sible. This reasoning, however, is faulty ; for if the two smallest
intervals 15 : 16 and 24 : 25 are to be determined by the comparison
of intervals derived from the harmonic division of the chief consonances,
then the Pythagorean division of the monochord by the first four
numbersonly will furnish these intervals. The Fifth 2: 3, harmonically
divided, produces the major Third 4 : 5, and the minor Third 5 : 6. If
we compare these two Thirds, their difference will be the chromatic
semitone 24 : 25, while the diatonic semitone 15 : 16, will represent the
difference of Fourth 3 : 4, and major Third 4 : 5.

But Descartes, in the passage in question, is not thinking of the
harmonic division of the consonances at all. What he really means is
that from the comparison of the consonances which arise from the
senario there result the smallest intervals which the ear is capable of

62 THE THEORY OF HARMONY

the different consonances arising from such a division of the
monochord gives the smallest intervals of tone which the
ear is capable of readily appreciating). With this statement
of Descartes as his starting-point, then, Rameau proceeds
to build up his theory of harmony. It is important to note
that his point of departure was a mathematical, not strictly
speaking an acoustical one. Of the series of overtones or
harmonics, resulting from the natural divisions of a string
or other sonorous body, he did not at this time appear to
have been aware, at least they are not mentioned in the
Traite, although they figure prominently in all his subsequent
works. He indeed refers to the co-vibration of strings,
but only in deahng with the Octave, or for the purpose of
comparing the consonances of the Octave and Fifth.

appreciating without difficulty, and which are actually made use of for
the degrees of the scale. Thus, in the chapter of his work entitled,
" De gradibus sive fonts musicis," he says, "Est aulem probandum
gradus sic spectatos ex imsqualitate consonantiarum generari." So that
the major tone, 8 : 9, is the difference of Fourth and Fifth ; the minor
tone 9 : 10 is the difference of Fourth and minor Third ; the diatonic
semitone 15 : 16, the difference of Fourth and major Third ; and the
chromatic semitone 24 : 25, the difference of major Third and minor
Third ; this chromatic semitone being the smallest interval obtainable
by such a comparison of the consonances, and the smallest melodic
interval in use. Any smaller interval could be appreciated by the ear
only with great difficulty. Understood in this sense, the argument of
Descartes is much more convincing. Hut it does not adequately
explain why the consonances should be limited by the number 6.

CHAPTER III.

JEAN PHILIPPE RAMEAU (1683-1764). TRAITE DE LHARMOXIE,

The firstfruit of Rameau's reflections on the fundamental
principles of harmony appeared in 1722, in which year he
published his Traite de I'/iannonie reduite a ses principes
naturels, in some respects his most important work. It is
divided into four Books. The first book treats of chords,
ratios, and proportions, and the relationships which exist
between them ; the second, of the Fundamental Bass and
of the nature and properties of chords ; the third, of the
Principles of Composition ; and the fourth, of Principles
of Accompaniment. An examination of Rameau's work
inevitably leads to the conclusion that it is the result, not
of one or two onl\-, but of many years of reflection and re-
search.^ x\lthough it does not represent his fully-matured
theory — for some of his ideas are still in an embryonic
state — it nevertheless contains the most essential of his
principles, such as the Generation and Inversion of Chords,
the Fundamental Bass, chords by " Supposition," and the
relationship of Melody to Harmony. The reader who sets
out to master the contents of the somewhat bulky Traite de
I'harmonie has not an easy task before him. Rameau has
poured out his ideas in a pell-mell confusion, \nth little order
or arrangement. If as a composer his instrumental style
is distinguished by the greatest clearness and precision, his
literar}- style on the other hand is difticult, obscure, and
diffuse. This, however, is evidently owing not so much
to lack of literary skill as to the difficulties of the subject ;
for elsewhere Rameau could express himself in the most
definite and lucid manner.

1 The words of the Motet which Rameau has appended to the third
book might be considered, in this connection, to be amusingly suggestive.
They begin thus : — " Laboravi damans, rancae factae sunt fauces meae."

64 THE THEORY OF HARMONY

The Preface to the Traite begins thus : " Whatever progress
the art of music may have made amongst us, it would appear
that the more the ear becomes sensible to its marvellous
effects the less is the desire manifested to understand its
true principles, so that one may say that reason has lost
its rights, while experience alone has acquired any authority.
The writings which remain to us of the ancients ^ sufficiently
prove that reason alone has procured for them the means
of discovering the greater part of the properties of music ;
nevertheless, although experience makes us still approve of
the majority of the rules which they have given us, we
to-day neglect all the advantages that we might derive from
reason in favour of empirical methods which relate solely
to practice." Rameau has attempted, with more or less
success, to cast his theory into a scientific form. He has
approached his task in the spirit of the scientist, of the savant.
His theoretical principles are to be natural principles ; they
must have their source in Nature and have, therefore, all
the certainty of natural laws. He has endeavoured, as he
himself tells us, to free himself from all preconceived notions
respecting the nature of harmony, all fettering constraint
imposed b}' rules derived merely from tradition, from " custom
and authority." Reason, truth, fidehty to Nature, these
were the guides that he felt himself impelled to follow. In all
this Rameau was undoubtedly strongly influenced by the
intellectual forces of his age. It was indeed a time of
brilHant intellectual achievement and progress, especially
on the side of philosophy and mathematics. Before the
end of the eighteenth century the discoveries and researches
of Sir Isaac Newton in physics, of Harvey in physiology,
of Locke in philosoph}', had become known aU over Europe.
In Holland there were such names as Christian Huyghens, ^
mathematician and astronomer, who defined the wave theory
of light ; and the brilHant philosophical genius Spinoza, who
in his Ethica had already proclaimed to the world those
philosophical propositions and demonstrations " for which,"
as Hume remarked, " he had become so universally infamous."

1 That is, before the time of Zarlino, as Rameau himself explains.

2 Huyghens must also be included in the ranks of musical theorists.
He wrote Xovus Cyclus Harmonicus, a work treating of musical
temperament : also Cosmotheros, in which he treats of prohibited
consecutives.

RAMEAU'S TRAITE DE L'HARMONIE 65

In Germany, Leibnitz, the apostle of the " pre-estabUshed
harmony," had discovered the differential calculus. In
France the influence of Descartes, whose Discours de la
Methode had appeared in 1637, had become especially wide-
spread, and had penetrated far beyond the learned and
scientific circles of Europe. Boileau, whose influence in
literature and belles leitres was as powerful as that of Descartes
in philosophy, had in his L'art poitique laid down the principle
that rein nest beau que le vrai ; le vrai seul est aimable. The
poet should take reason, not imagination, as his guide ;
and hi? aim should be — " fidelity to Nature."

In the anxiety of Rameau to rid himself of all prejudices,
all preconceptions respecting the nature of harmony
derived merely from tradition, the influence of the Cartesian
"method" may easily be traced. "Has anj^one so far
sought in Nature," he asks, " some invariable and steadfast
principle from which one may proceed with certainty, and
which would serve as the basis of melody and harmony ?
Not at all ! It has been a case rather of fumbling about,
of compiUng facts, of multiplying signs. After much time
and trouble all that there was to show was a collection
of phenomena without connection, and without succes-
sion ; . . . besides, the use of these phenomena is so arbitrary
that he who is most familiar with them derives little
instruction therefrom. Such was the state of matters when,
astonished at the difficulty I experienced in acquiring what-
ever [theoretical] knowledge I had, I attempted to discover
the means whereby such knowledge might be made more
easy of attainment to others, and the art of composition
rendered more certain and less laborious. It seemed to
me that I could hardly fail, if I were successful in the one
direction, to be successful also in the other, and that progress
in the science of sounds would be assuredly less laborious
when its principles were more certain. Enlightened by the
Methode of Descartes, which I had fortunately read, and with
which I was much impressed, I began by subjecting myself
to a process of self-examination. I attempted to put myself
in the place of a child who tries to sing for the first time ;
essayed various fragments of melody, and examined what
avere the effects produced on my mind and by my voice," ^

^^Demonstration du principe de I'harmonie (1750), pp. G-S

66 THE THEORY OF HARMONY

At the beginning of his Traite de Vharmonie, Rameau
quotes in full the passage from the Compendhim Musicae of
Descartes, which has already been given on p. 60, and draws
therefrom the following conclusions : —

" That all the consonances are determined by the first
six numbers ; for the sounds produced by the whole string
and its different divisions correspond to the notes C, c, g, c', e' ,g'
(if C be taken to represent the sound produced by the entire
string) in which, if the Octave c" be added, all the consonances
will be found ; for this reason all the force of harmony has
been attributed to numbers".

" That the origin and degrees of perfection of these
consonances are determined by the order in which the numbers
arise. Thus the Octave is the most perfect consonance ;
after it comes the Fifth, which is not so perfect as the Octave,
then the Fourth, and so on.

" That the sounds which arise from these divisions of the
string give, when heard together, the most perfect harmony
that one can imagine.

" That all these sounds are generated from the whole-
string, or from its parts ; but just as numbers must be related
to Unity, which is the source of numbers, so must the different
divisions of the string be related to the entire string in which
they are contained ; and the sounds arising from these
divisions must be considered as being generated (engendrez)
from the first or fundamental sound, which is therefore the
source and foundation of all the other sounds. The harmony
therefore resulting from the consonant intervals produced
by the entire string and its divisions is not perfect unless
this fundamental sound is heard below the other sounds :
for this sound must appear as the principle or source of
these consonances, and of the harmony which they form ;
it is their base and foundation." ^

Following the examples set by Zarhno and Descartes,
Rameau now examines at much length the nature and
quahties of the consonant intervals which have thus arisen.
Of the consonances generated from the principal sound, the
Octave, the first and the most perfect consonance, is only
a replica or repetition of this sound. Every replica is thus
merged in its principle, and represents it. Male and female

1 Traite, Book I.. Ch. 3.

RAMEAU'S TRAITE DE UHARMONIE 67

voices or men and boys, singing this Octave, appear to
sing the same somid. In flutes and other such instruments
this Octave depends on the pressure of wind (a shghtly
increased pressure of breath on the part of the player producing
the harmonic sound of the Octave). Further, the perfection
of the Octave is evident from the fact that it remains the
Octave (or represents the same sound) whether one divides
or doubles the term which represents it. Thus the same
sound may be represented by 2, i, or 4. Therefore the
Octave ought not to be regarded as really differing from the
fundamental sound from which it is derived ; although
naturally this fundamental sound has the greater importance
attached to it.

From this identity of the Octave with the fundamental
sound there arises the principle of inversion. Thus the
Fourth is only a consequence of the Fifth, and is immediately
derived from it. In the same way the minor Sixth is the
inversion of the major Third, and the major Sixth is the
inversion of the minor Third. This requires some further
explanation. If we compare with the fundamental sound
the other sounds which arise successively from the senario
(excluding the Octave sounds, which are merely repetitions
of sounds already existing) it will be found that the only
intervals or consonances which thus occur are those of the
Twelfth and Seventeenth. But as all that exceeds the
Octave is merely the replica of what is contained within the
Octave, consequently it is possible to reduce every interval
to its smallest terms ; therefore the Twelfth (reduced by one
Octave) and the Seventeenth (reduced by two Octaves) are
but the Fifth and Third. The Fifth and major Third, then,
are the only consonances which arise directly from the
fundamental sound. The Fourth and minor Sixth are derived
from the Fifth and major Third by inversion. What then of
the minor Third and major Sixth ? These may be explained as
follows : — The major Third divides the Fifth into two Thirds,
a major and a minor. But as the major Third, in thus
dividing the Fifth, necessarily generates at the same time
the minor Third, this minor Third, and not only the major
Third, must be considered to be generated directly ! ^ The
major Sixth, then, is derived by inversion from the minor

^ Traite, Book I., Ch. 3., Art. 5.

68 THE THEORY OF HARMONY

Third. The three primary consonances are therefore the
Fifth and the two Thirds ; and the three secondary
consonances derived from these are the Fourth and the
two Sixths.

Not only intervals but chords may be inverted. Thus
in the major harmony (as c-e-g), which is represented by
the numbers 4:5:6, if we place 4 an Octave higher we
obtain the first inversion of the harmony, that is, a chord of
the Sixth {e-g-c'), represented by the numbers 5:6:8. If in
the same way we place 5 an Octave higher, we obtain the
second inversion of the harmony, a chord of the Fourth and
Sixth {g-c'-e'), represented by the numbers 6:8: 10. We
cannot however here carry the process of inversion further,
for if we place 6 an Octave higher, we get a chord represented
by the numbers 8 : 10 : 12. But this proportion is the same
as 4:5:6, and indeed represents the original harmony
itself. The first chord is called Perfect ; the two chords
derived from it are called Imperfect ; for in the case of
these derived chords the fundamental sound, c, is not in
the bass ; it is transposed, and represented by another
sound, namely, its Octave.^

This principle of inversion is the ke}^ to the
diversity which characterizes harmony. Such inversion
will modify the interval or chord, luithoiit destroying its
foundation.

Already Rameau has treated of three of the most important
principles of his theory of harmony, namely, the principles
of Harmonic Generation, of the Fundamental note or Bass,
and of the Inversion of Chords. He naturally deals first
of all with the principle of Harmonic Generation. If there
be no such principle, if music and harmony have no better
origin or foundation than mere human caprice, there can
be no intelligible system of harmony. We have seen how-
ever that the consonances which are actually used in music,
which form the material of harmony and constitute the
ultimate basis of all rational musical systems, do not depend
for their origin on caprice. On the contrary, they are deter-
mined by certain numerical proportions which are as definite,
precise, and invariable as any natural or scientific laws
whatever. It has been objected that if the impression on

1 Traite, Book I., Ch. 8.

RAMEAU'S TRAITS DE L HARMON IE 69

the senses made b}' a certain harmony or interval depends
on a certain definite and determined numerical relationship
existing between the sounds which compose this harmony, one
is utterly unconscious of it, until the attention has been
directed to the fact by observation of the physical properties
of the sonorous bodies themselves. This no doubt is quite
true. In the same way, the eye may receive different
impressions of colour without the mind being aware that
these arise from luminiferous vibrations of varying rapidity.
But ignorance of a fact does not necessarily imply its non-
existence. One may pursue a fairly equable existence as
a constitu,ent portion of the universe without having heard,
much less understood, anything of the correlation of forces.
Even a child may sing various intervals in perfect tune
without being aware that these intervals correspond to
certain numerical ratios ; just as there are many persons
who could draw quite correctly all sorts and sizes of triangles,
who would nevertheless be extremely nonplussed if they
were asked to describe three angles which together should
be equal to two right angles. But it should be noted that
the question here is not primarily as to whether the effect
produced on the ear and mind by harmony, or by the various
consonances, is owing to the proportions which determine
these consonances. The question is, does harmony arise
arbitrarily, or from a fixed and definite principle ? Zarlino,
Descartes, Rameau, have all contended that harmony
does arise from such a principle, which is certainl}^
sufficiently definite, namely, the senario or series of
numbers i, 2, 3, 4, 5, 6. This principle of the determina-
tion of the consonances which are accepted as such by
the ear is constant and invariable. The consonances are
judged by the ear to be in perfect tune only when they
correspond accurately ^^|J:h the acoustical determinations
given by this principle. When this is not the case the
consonance is said to be " out of tune," and when this
" out-of-tuneness " is sufficiently pronounced, the effect
produced on the ear is that of actual physical pain. This
physical sensation of pain, in which the ear is torn, as
it were, between the contending sounds, has its counterpart
in Nature in the remarkable acoustical phenomenon of
beats. Beats are, we might say, Nature's protest against
the " false " consonance.

70 THE THEORY OF HARMONY

Of course we may, if we please, cherish the beUef that
all this is mere coincidence ; that it is a mere chance that
the consonances happen to correspond with the numerical
series i, 2, 3, 4, 5, 6, and that they are actually present in
musical sound itself, even that of the human voice ; but
such a belief demands a much greater amount of credulity
than is possessed by the average musician.

It has however been objected that in the prevailing
system of equal temperament musicians constantly make
use of intervals which are actually out of tune. But this
is merely to confirm the laws relating to the acoustical
determination of these intervals. Otherwise, in what sense
can the tempered intervals be said to be " out of tune " ?
It is significant that a " tempered " interval is almost
universally understood to mean, not an interval whose
natural " out-of-tuneness " is removed by a process of
" tempering," but one which, naturally in perfect tune,
is placed very slightly out of tune, that is which differs, even
if only to a small extent, from its acoustically determined
proportions. However musicians may agree as to the
necessity for equal temperament, few of them would contend,
notwithstanding that they have been bred and brought up
on the tempered scale, that a tempered Third or Fifth sounds
better than the natural and untempered interval. A
tempered major Third or perfect Fifth, in short, stands for
and represents to the ear and mind the " natural " major
Third or Fifth. A tempered major harmony, at the same
time, is a poor substitute for the natural one, the almost
ethereal effect of which, especially when produced by a
capable body of singers, once experienced is not readily
forgotten. Such a harmony, as Rameau has said, is as
" perfect as can be imagined."

It is this " perfect harmony " {accord parfait), the Harmonia
perfetta which represents the consummation of Zarlino's
researches in the domain of harmony, that forms the
starting-point of Rameau. While Zarlino argues from the
consonances to the Harmonia perfetta, Rameau argues from
the Harmonia perfetta to the consonances. Zarlino cannot
arrive at unity except through diversity ; Rameau cannot
understand diversity except through unity. It is instructive
to compare the first diagram given by Zarlino in his first
theoretical work, the Jst. Harmoniclie (Pt. I., Cap. 7.)

RAMEAU'S TRAITE DE L'HARMONIE

with that given by Rameau at the beginning of his
Traite :■—

(Zarlino.)

18 12 9 6

diapente ^^diatessaron^\ diapente

(Rameau.)

rt 8111

1 1 1 1

Sol ft 1 I

, 1 , Fourth

\Ti 5 1 !

Minor
, 1 third

T't I 1

^Nlajor
, , third

sol 3 1

1 Fourth

TTt ^

1 Fifth

Ut I

Octave

Of this diagram Rameau remarks : — " It should be noted that
the numbers indicate always the division of Unity " ; and
if, like Zarlino, he afterwards treats of the various consonances,
it is to show that these consonances are all derived from the
" perfect " harmony, and that they have no other harmonic
foundation. Rameau's addition of the number 8 to the

72 THE rHEORY OF HARMONY

senario is in order to demonstrate that all the consonant
intervals, including the minor Sixth (5:8), have their origin
in the " perfect " harmony. But, as we have seen, Rameau
does not consider the minor Sixth to be an "original"
interval, but as derived by inversion from the major Third.
This however is merely the first instance of the contra-
dictions which abound in the Traiie, just as his adding the
number 8 so as to leave a gap in the arithmetical series
between the numbers 6 and 8 is our first proof of the inferiority
of Rameau as a logician to both Zarhno and Descartes.
To Zarlino the distinguishing characteristic of harmony
was its " diversity." Rameau on the other hand
recognizes almost from the outset its essential unity, and
this becomes more and more clear to him as he proceeds.^
It is noteworthy that Descartes, in his generation of the
consonances, points expressly to the relationship which the
Octave and Fifth — the major Third he includes later — bear

' Rameau seems to have considered that this view of harmony
was shared bv Zarlino. " ZarUno," he says, " has remarked that
music is subject to arithmetic, and that Unity, which is the principle
of numbers, represents to us the sonorous body, from which one-
derives the proof of the relationship of sounds ; also, that the Unison
is the principle of the consonances." Rameau then cites several
chapters from Zarlino's Istitutioni. On examining these chapters,
we find that Zarlino compares the Unison to unity. Unity, he proceeds,
is not a number, but it is the beginning or source (principio) of numbers ;
so, likewise, the Unison is not a consonance, but it is the beginning,
source, or starting-point of the consonances. The number 2, which
expresses the Octave, consists of two unities, or unity doubled .- the
number 3, which express the Twelfth, is unity trebled, and so on.
Rameau, therefore, is mistaken if he imagines that Zarlino considered
the unison to represent " the fundamental sound, in which all the
other sounds are contained " ; just as he would have been mistaken
had he considered Zarlino to have been familiar with the principle of
the Fundamental Bass, or of " klang-representation." Zarlino leaves
us in no doubt as to his actual meaning. In one of the chapters cited
by Rameau he remarks : " The unison which is represented by unity
is to Music what the point is to Geometry " ; and he goes on to explain
that just as geometricians have defined the line as consisting of a
series of points, so Music may be said to consist of a succession of
unisons. {1st. Harm., Pt. III., Cap. 11.) Nevertheless, Zarlino's
statement that unity represented the beginning or source of numbers
and the Unison the source of the consonances, indicates how nearly
he approached to the principle which forms the basis of Rameau's
theory ; at any rate, for Rameau it was a statement of intense
significance ; for him it possessed a meaning which it did not have
for Zarlino.

RAMEAU'S TRAITE DE L'HARMONIE

I y

to the principal sound, and regards this relationship as proved
by the power of co-vibration existing between these two
sounds and the principal sound in which they are contained.^
But Descartes, as we have seen, is by no means of opinion
that all the consonances generated from the senario have
the same Fundamental Bass. The major Sixth he con-
siders to be derived from the major Third : and the minor
Sixth from the minor Third, by means of the addition of
the Fourth, which he considers to represent a sort of imperfect
Octave {octava deficiens & imperfecta) so that the Sixths
appear as compound or " composite " forms of the Thirds.

For Rameau however, impressed as he is with the essential
unity of the major harmony, the statements of Zarlino and
Descartes have a new meaning. Zarlino had said that "the
Unison is the source of the consonances, as unity is the source
of numbers " ; Rameau seizes upon this idea, and carries
it firmly to its ultimate and strictly logical issue. Descartes
had said, "As string is to string, so sound is to sound " ; and
Rameau finds in Descartes' application of the senario to
the divisions of the string exactly what he stands in need
of for the demonstration of the unity of the major harmon}-,
and the relationships of its sounds. The sounds which arise
in succession from the senario do not to Rameau, as they do
to Zarlino, represent so many " unities." As the relationship
of the half to the whole string in which it is contained, so
is the relationship of the sound produced by this half to
the sound of the whole string ; and so for the other divisions.
All the sounds of the major harmony are contained in, or
proceed from, a single sound. This sound is the fundamental
sound to which all the other sounds are related ; it is the
fundamental note, or Fundamental Bass, of the harmony.
It is evident that Rameau was not at this time acquainted
with the natural series of harmonics resulting from the
resonance of a sonorous body ; other\nse he would hardly
have failed to point to it as a wonderful confirmation of
his theory.

^ Secundum ex duobus terminis, qui in consonantias requiruntur,
ilium, qui gravior est, longe esse potentiorem, atque alium quodammodo
in se continere : ut patet in nervis testudinis, ex quibus dum aliquis
pulsatur, qui illo 8™ vel quinta acutiores sunt, sponte tremunt &
resonant, graviores autem non ita, saltern apparenter : cujus ratio
sic demonstratur. Sonus se habet ad soniim tit nervus ad nervnm, etc.
De Coiisoiiaiitiis. (See p. 60.)

74 THE THEORY OF HARMONY

Rameau's theories of harmonic generation and of the
fundamental note are thus seen to be closely interwoven ;
it is, in fact, impossible to separate them.

As for his theory of inversion, this is rightly regarded as
one of his greatest achievements. Without such a theory, no
intelligible system of haripony can be imagined. In what
respect does Rameau's theory of inversion differ from that
of Zarlino, if Zarlino's demonstration of the fact that each
interval has its Octave complement may be described as a
theory of inversion ? Is it not accomplished by the same
means, namely, by means of the^ Octave ; by the similarity,
the almost identity of effect existing between Octave sounds ?
We find at the outset this very great difference, that Rameau
considers some of the consonances to be original, or funda-
mental, and the others to be derived from them. For ZarUno,
on the other hand, all the consonances are equally fundamental;
they are to be considered as arising successively from the
senario, or as composed of its parts. Rameau's theory
makes it impossible to consider a fundamental interval as
other than it is ; it can never represent an inverted interval ;
thus the Fifth can never be considered as an inverted Fourth.
But Zarlino, if he regards the Fourth as the Octave comple-
ment of the Fifth, regards also the Fifth as the Octave
complement of the Fourth. In the same way, he is unable
to show why the major Third should not be considered as an
inverted minor Sixth. Strictly speaking, ZarUno has no
intervals which he can describe as inverted, for the reason
that they are for him all equally " original " and " funda-
mental." While then Rameau considers, for example, the
minor Sixth to have the same harmonic foundation as the
major Third, to represent the inversion of this Third and,
what is most important, to have the same harmonic meaning
as this Third, Zarlino considers the minor Sixth to be an
independent interval, explains its origin in a way altogether
different from that of the major Third, and considers it to
possess a quite different harmonic effect ; compared with
the major Third it is much less consonant, almost resembhng
a dissonance.

With Zarhno, the inverted interval changes its meaning ;
the reason for this being that it changes its fundamental
note. ZarUno instincti\'ely regards tlie lowest note as the
basis and foundation of the harmony, even if he is unable

RAMEAUS TRAITE DE L' HARMON IE 7 5

to explain why this should be so. While, therefore, in
the case of the major Third, as c-e, he relates tJ to c ; in
its inversion e-c' , he relates c' to e, and considers e to
be the base or foundation of the harmony. In this he
does quite rightly ; the minor Sixth c-c' retains its " original "
effect only so long as the ear regards c as the fundamental
note ; if, on the other hand, it represents the major Third,
and has the same harmonic meaning as this Third, the ear
relates ^ to c as the fundamental and determining note of
the harmony. But long before Rameau's time the minor
Sixth, in addition to retaining its original meaning, had
acquired a new one. Musicians perceived that the minor
Sixth might produce on the ear much the same effect as
the major Third. But although they percei\-ed this, they
were unable to account for it ; and by their failure to
recognize the cause of this change of effect they were led
into all sorts of theoretical difficulties and contradictions.
Rameau found the true explanation. In the minor Sixth
£-c' which is derived from the major Third c-e, not e but c
must be regarded as the fundamental note, for the ear relates
e to c, and not the reverse. The minor Sixth therefore
must in this case be regarded, not as an original interval, but
as derived from the major Third ; and in the same way,
the Fourth is derived from the Fifth. On the other hand,
it is impossible to consider the major Third to be " derived "
from the minor Sixth. The minor Sixth, in itself, is almost
a dissonance, as ZarHno has sho\\-n ; its inversion, the major
Third, cannot represent this "original" character of the
minor Sixth ; nor can it have the same harmonic meaning.
Similarly, the Fifth cannot be considered to be deri\-ed from
the Fourth.

It is no accident that Rameau treats of Harmonic Generation,
of the Fundamental note, and of the Inversion of Chords, at
one and the same time. They are all connected in the closest
possible way. In short, unless connected %nth some principle
of harmonic generation, and of a harmonic fundamental
or determining note, the inversion of intervals or of chords
has no meaning for the science of harmon}-. This is a fact
which has not always been duly appreciated by musicians
and musical theorists. Nothing is more common than to
find musicians who entirely reject acoustical phenomena as
the basis of harmony. All, however, accept and utiUze

THE THEORY OF HARMONY

Rameau's principle of harmonic inversion. The question
therefore arises, have musicians sufficiently considered what
this principle of harmonic inversion impHes, and especially
how it affects the whole question as to whether harmony
has a physical basis ? Inversion by means of the Octave
does not in itself imply identity of harmonic significance ;
for unless the}^ proceed from a common source and can
be referred to a common fundamental note, " derived "
or inverted intervals will retain, as they did for Zarlino,
and must retain, their "original" character; all will
be equally " fundamental," not only for the science of
harmony, but also for the ear. As a famihar instance of
the two-fold aspect which an interval or chord may assume,
take the second inversion of the major or minor harmon\-,
which long w^as a puzzle to theorists ^ : —

('')

(b)

:z3;

-Gf

-O-

-O

1251

j^_

-<s>-

22_

At («) the ^ chord is almost dissonant in effect ; hence
Heinichen gives it a place among the dissonant chords,
its sounds e and c merely retard the Dominant
harmony ; the ear relates these sounds to or the
Dominant, and not to c the Tonic ; that is, the
ear understands g as fundamental note. At {b), on the
other hand, the chord has a different effect ; in this case
it has the same meaning as the Tonic chord, which it
represents ; here the ear understands c to be the fundamental.

From the foregoing it is comparatively easy to understand
how the predecessors of Rameau so completely failed ta
make use of any knowledge they may have had of inverted
intervals or chords for the simphfication of their chord

1 Rameau himself could not explain the dissonant effect of this chord.
He considers it in one aspect only, namely, as representing the harmony
of c. See remarks on this subject, p. 4S1.

RAMEAU'S TRAITH DE L'HARMONIE 77

tables. Within a few years of the appearance of the Traiie
de I'harmonie, however, Ramcau's theory of the inversion
of chords became universally accepted, and has been for
long regarded as a commonplace of the theory of harmony.
Rameau's predecessors attached the greatest importance-
to the interval, and were unable to distinguish between the
bass note and the fundamental note of an interval or chord ;
the term fundamental being applied to the note which, for
the time being, occupied the lowest position in the chord, no'
matter what this might be. Rameau, on the other hand, lays
the greatest possible stress, not on the interval which an upper
part forms with the bass, but on the fundamental note, and
distinguishes carefully between fundamental and bass note.
In thus relating the inversion of chords to his principles of
Harmonic Generation and of the Fundamental Bass, Rameau
firmly establishes the theory of Harmonic Inversion.

So far Rameau has done admirably. His treatment of
the minor Third and the major Sixth, however, fails to
convince. After saying that the only intervals directly
generated from the fundamental sound are the Fifth and
major Third, he tells us that the minor Third also must be
considered to be generated directly, for it arises from the
harmonic division of the Fifth. He then considers that the
major Sixth is " derived " from the minor Third. But then
might not the Fourth, in the same way, be considered to be
generated directly by means of the harmonic division of
the Octave ? The major Sixth cannot be considered to be
a " derived " interval until the minor Third has been proved
to be fundamental. Unhke the Octave, Fifth, and major
Third, the minor Third and major Sixth are not generated
directly from the fundamental sound. They are not, in
Hauptmann's language, " directly intelligible " intervals.
Both intervals may retain their major as well as produce a
minor effect, but they depend for their definition as con
stitutents of the major harmony on a third sound, the
fundamental note, which is nothing less than the terzo suono
(combination tone) of Tartini. This third sound may not
only be understood, but is actually present whenever either
interval is sounded. In treating of the two intervals in
question, Rameau does not push his researches far enough ;
but had he been acquainted with this phenomenon of the
combination tones, he could hardly have failed to adduce it

78 THE THEORY OF HARMONY

as a striking confirmation of his theory. Rameau's anxiety
to make the minor Third appear as a fundamental interval
can be well understood. He requires it for the explanation
of the minor harmony-
Further, Rameau is unable to explain why in the generation
of the consonances by the first six numbers the next number,
that is 7, should introduce, as he tells us, a dissonance. The
reason alleged by Rameau, which he borrows from Descartes,
namely, that the comparison of the consonances produced
by the senary division gives the smallest intervals which
the ear is capable of appreciating, is not adequate. Consonance
is not determined by the extent to which the ear can appreciate
minute differences of tone. Nor can the smaller interv^als
of tone actually used in music be said to be limited by those
intervals which are recognized to be consonant. The Greeks
recognized only the Perfect consonances, which could all be
expressed by the first four numbers. Nevertheless they not
only distinguished but made use of quarter tones, and were
acquainted with such a small interval as the Pythagorean
limma (243 : 256).^ In modern music, also, smaller intervals
than the chromatic semitone (24 : 25) are distinguished. Thus
a species of quarter tone (125 : 128) arises from the enharmonic
change.- It is quite evident however that in deciding that
the number 7 introduces a dissonance, Rameau is influenced
less by theoretical considerations than by the judgment of
his ear. Yet even by allowing the ear to become the sole
arbiter, the matter could not thus be placed beyond the range
of controversy or of individual opinion.^ On the other

1 Obtained by comparing the Fourth wdth two major tones,
i.e., [t]-Xi=243:256.

2 As, for example, bj^ enharmonically changing the augmented
Fifth as c-g^ (f XM=M) into the minor Sixth c-a^ (5 : 8). The
difference between these intervals is -V^ X -^, = 1 ff . The same quarter
tone arises from the comparison of diminished Fourth {g^-c) with
major Third {a\f-c).

* Thus Mersenne, in his Livre I. de la musique thJorique, with
which Rameau was acquainted, is of opinion that there is no reason why
the consonances should be Umited by the number 6, and that the
proportions 6 : 7, and 5 : 7 represent consonant intervals ! Even
Helmholtz [Sensations of Tone) cannot account for the exclusion
of the number 7 on physical grounds. " As a matter of fact," he
says, " the chords of the natural or sub-minor Seventh 4 : 7, or of the
sub-minor Tenth 3:7, in many qualities of tone sound at least as

RAMEAU'S TRAITE DE L'HARMONIE 79

hand, Ramcau might have maintained that it was time enough
to treat of this " natural Seventh," and to give it a place in
the musical system, when musicians actually begin to make
use of it. This is by no means the most serious of the
difficulties with which Rameau is soon to find himself
confronted.

The Minor Harmony.

Rameau's first great difficulty is to account for the Minor
Harmony. He sees clearly that although the senario pro-
vides him with a major harmony it does not provide him
with a minor one. He imagines that this difficulty can be
o\-ercome by pro\dng that it is only in appearance and not
in reahty that the minor Third is generated indirectly from
the fundamental sound. " From the union of the [major]
Third and Fifth," he says, " there immediately arises another
interval, the minor Third. The Fifth, then, being composed
of two Thirds, it suffices to determine the nature of the lower
Third, in order to determine at the same time the nature of
the upper Third (that is, if the lower Third is major, the
upper Third must be minor, and vice versa, as c-e-g, c-e'^-g),
for the interval of the Fifth remains the same, no matter
whether the major Third is at the bottom or at the top."
Rameau now considers that he is at liberty to place this
minor Third either at the top, as in the major harmonj^
or at the bottom, as in the minor harmony. After all, he
remarks, the only difference between these two harmonies
lies " in the different disposition of the Thirds, which together
make up the Fifth," ^ for whether the minor Third occupies
the lower or the higher position, " this makes no difference
in the character of the Fifth, which has always a Third on
one side or the other." ^ Such in brief is the explanation
of the minor harmony given in the Traite.

well as the minor Sixth, 5 : 8, and the sub-minor Tenth really sounds
better than the minor Tenth 5 : 12." It is only " a circumstance of great
importance for musical practice which gives the minor Sixth an advan-
tage over the intervals formed Avith the number 7," namely, the
fact that the minor Sixth is the inversion of the major Third.
(Sensations of Tone. Part II., Ch. 12.)

1 Traitr, Book I., Ch. 8., Art. 2. 2 Traite', Book I., Ch. 3., .\rt. 5.

8o THE THEORY OF HARMONY

Rameau however casts a longing glance at the inverted
series of numbers, by which Zarlino explained the minor
harmony. The minor Third, he says, must be considered
to be generated directly from the fundamental sound; if
not, " this Third could never alter its position, but must

always take a middle place in chords (as c-c-g-c'), and never

appear in the extremities (as c-e\^-g-c') ; which would be
contrary to the nature of the Arithmetical and Harmonic
proportions, the first (4:5: 6) dividing the Fifth into a major
and a minor Third ; the second (6:5:4) dividing the Fifth
into a minor and a major Third." ^ But it is only for a
moment that Rameau hesitates. In this inverted series he
sees his fundamental principles, which he has discovered
by the division, not the multiphcation of a string, en-
dangered. He thereupon devotes a long chapter to the
discussion of this inverted proportion, in which he accuses
Zarlino of having, by his use of it, obscured the principles
of harmony and reversed the natural order of numbers,
of intervals, and of the harmony resulting therefrom. He
says :— " Zarhno has remarked that music is subordinate to
arithmetic. But he forgets all this in his rules and demon-
strations. By adopting the Harmonic klescending] pro-
portion, he reverses not only the natural progression of
numbers, but all the beautiful order of harmony which
presents itself at once in the division of the string . . . for
the numbers mark, in this case [that is, the descending
progression] the multiplication of the string, which is the
sonorous body representing Unity, and not its division.
In the descending progression 6, 5, 4, 3, 2, i, the number 6
cannot represent Unity, nor serve as the source or foundation
of the harmony. The foundation of harmony is therefore
destroyed ; it remains without a basis, or bass."

In short, Rameau, in the Traiti, will have nothing to do
wth the descending progression. His fundamental sound,

1 Traits, Book I., Ch. 3, Art. 5. Rameau here uses the terms
Arithmetical and Harmonic in a sense contrary to that which is
generally accepted, and to that which he himself adopts in his later
works. As used by Rameau, the terms apply not to string-divisions,
but to sound vibrations : only in this sense can the Arithmetical series
I, 2, 3, 4, 5, 6, represent the major harmony, and the Harmonic series,
1, h'\' i- I- «' ■the minor harmony.

RAMEAU'S TRAITE DE UHARMONIE 8i

which forms the foundation of the harmony ; the Perfect
Cadence, on whose importance he rightly lays so much stress ;
his Fundamental Bass ; — all these form the very foundation
of his system, and these he regards as the natural fruits of
the ascending progression. But in rejecting the descending
progression of numbers, he finds himself totally unable to
give any rational account of the origin of the minor harmony.

Origin of the Theory of Chord Generation

BY MEANS OF AdDED ThIRDS.

But Rameau not only considers himself at liberty to place
the minor Third, as well as the major Third, wherever it
suits him ; he imagines that he can add one Third to another.
Thus he remarks : — " The Fifth and Thirds not only divide
the principal chords, the}' also compose them, whether by
their squares or by their addition." ^ Rameau now thinks
that he has discovered a new and satisfactory way of
accounting for the minor triad : — " Thus the addition of
a major and a minor Third gives us the ratio 20 : 30 [that is
2:3 = perfect Fifth]. The difference between them is
24 : 25, and, according as we take the proportions 20 : 25 : 30
[ = 4:5:6] or 20 : 24 : 30 [ = 5:6 + 4:5], we obtain
the major or the minor harmony. Likewise the squares of
the major Tliird," he goes on to say, " and of the minor
Third, give us respectively the augmented triad [as c-e-gjlff],
and the diminished triad " [as b-d-f]^.

Here we find the germ from which has been developed
the theory of the generation of chords by adding Thirds
together. But, it will be noticed, Rameau not only makes
use of the addition of Thirds, but also of the squaring of
Thirds and of other intervals, in order to explain the con-
struction of chords. However extraordinary this latter
process may appear to us, it may nevertheless be asked :
If it is lawful to add intervals in order to generate chords,
why is it absurd to square intervals in order to obtain a
similar result ? The one process is theoretical^ as good,
or as bad, as the other. And why not also, while one is
about it, make use of multiphcation and subtraction ? In

1 Traite, Book L, Ch. 7. 2 /^/^

82 THE THEORY OF HARMONY

this respect at least Rameau is strictly logical. If he makes
use of mathematics, he does so in a whole-hearted fashion
and does not exclude either subtraction or multiplication.
Here, for example, is one of his ways of generating the chord
of the Submediant Seventh (major mode).^

]\Iinor Third =5:6

Fifth =2:3

By multiplication =10 : 18

By subtraction =12 : 15

Result : chord of Submediant Seventh = 10 : 12 : 15 : 18.

It is not only the Thirds which are manipulated in this
way. For example, we are told that the square of the Fourth
produces a Seventh (f5=T%=minor Seventh), and the square
of the Fifth, a Ninth (^'=i=major Ninth). It is only
the addition of Thirds however which plays any essential
part in the further development of Rameau's theory. On
this point Rameau is quite definite. He says : — " In fact,
to form the ' perfect ' chord, it is only necessar}^ to add one
Third to another, and for dissonant chords it is necessary
to add three or four Thirds to one another, the difference
between these chords arising only from the different situation
of the Thirds." ^

Thus Rameau brings in a second and entirely new principle
of harmonic generation. It is evident that he cannot
successfully run both theories together side by side. Yet he
attempts to do so, with the inevitable result that he faUs
into the grossest absurdities. His new principle of chord
generation is introduced with the express object of proving
that certain chords, including the minor triad, are in reality
fundamental chords. Nevertheless, while he accepts the
minor triad as a fundamental chord, he rejects the augmented
and diminished triads. These, he tells us, are not fundamental
chords.^ But how does Rameau discover this fact ? Accord-
ing to his new theory of harmonic generation, these chords
must be regarded as fundamental : for there is nothing in

1 Traits. Book I., Ch. 7. « Ibid. » Ibid.

RAMEAU'S TRAITE DE L'HARMONIE 83

this theory to indicate that a chord formed by the addition
of two minor Thirds, as in the diminished triad, or two
major Thirds, as in the augmented triad, is not equally
fundamental with a chord formed by the addition of a major
and a minor Third (as in the minor triad) : especially as
Rameau considers himself at liberty to add a major or a
minor Third wherever he thinks the circumstances require it.
Thus Rameau, having thrown his first principle of harmonic
generation overboard, now finds himself compelled in turn to
reject the consequences of his new principle. He is, in fact,
in an extremely awkward predicament. Having brought
forward a reason why chords generated by means of the
addition of Thirds should be regarded as fundamental, he
has now to discover a reason why such chords should be
regarded as non-fundamental. He thinks this may be done
by reverting to his original principle of chord generation (!)
where not the Third, but the Fifth, is the first sound generated
after the octave. The (perfect) Fifth, therefore, is proved to
be more important than the Third ; where then this Fifth
does not dominate in a chord, such a chord is not fundamental :
" its foundation is inverted, supposed or borrowed." ^

Rameau has now to determine what is the principal or
primary constituent of harmony. Is it the Fifth : or is it
the Third ? According to his first principle of chord genera-
tion, it is the Fifth ; according to his second principle, it
is the Third. He requires the first principle in order to
prove that certain chords generated by the second
principle (the augmented and diminished triads, as well as
certain chords of the Seventh) are non-fundamental ; he
requires the second principle because his first principle is not
sufficient to determine the fundamental or non-fundamental
nature of a chord. But the absurdity does not stop here :
for Rameau sets out to prove that not only the Fifth, but
also the Third, must be regarded as the primary constituent
of harmony. He begins thus : — " The Fifth is the primary
constituent in all chords."- In the same chapter he says :
" In order to make matters more easily understood, we may
for the present [!] consider the Thirds to be the sole
constituents of all chords ; in fact, in order to form the

^ Traits, Book I., Ch. 7.

^ " La Quinte est le premier objet de tous les accords." TraiU,
Book I., Ch. 3, Art. 5 ; Ch. 7, etc.

84 THE THEORY OF HARMON V

' perfect ' [common] chord, it is only necessary to add one
Third to another." ^ And again, "If we have considered
the Fifth to be the primary constituent in all chords, we
ought none the less to attrilDute this quahty to the Thirds,
of which it is composed." -

In short, the more Rameau endeavours to explain how it
is that the Fifth, and at the same time the Third, should
be regarded as the primary constituent of harmony, the
more contradictory becomes his language, as might be
expected ; he cannot have it both ways. It is unfortunate
for Rameau's principle of chord generation by the addition
of Thirds, that if there is one thing more necessary than
another for his theory of harmony, and especially his theory
of the Fundamental Bass, it is that the Fifth should be
considered to be more perfect than the Third, and to be
the primary constituent of harmon}-. For this reason
whenever Rameau is not immediately concerned with the
generation of chords from added Thirds, he lays all possible
stress on the greater perfection of the Fifth as compared
with the Third. This is so, not only in the Traite, but also
in his later works. In the Demonstration du Principe de
I'Harmonie (1750), he remarks : " The difference between
these two proportions [namely, those of the major and mihor
harmonies] consists of a transposition in the order of their
Thirds, which produce in each case a Fifth ; whence it is
evident that the Fifth alone constitutes harmony, and that
the Thirds vary it." ^

Rameau's juggle with the Thirds, so that a major or a
minor Third may now appear at the bottom, and again
emerge at the top, does not constitute a " natural principle "
of harmonic generation. It is a purely arbitrary process,
and one which he is totally unable to justify.

^ " Pour se rendre les choses plus familieres, Ton peut regarder a
present les Tierces comme I'unique objet de tous les accords ; en effet,
pour former I'accord parfait, il faut ajouter una tierce a I'autre."
Traite, Book I., Ch. 7.

^ " Si nous avons regarde la Quinte comme le premier objet de
tous les accords, nous ne devons pas moins attribuer cette qualite
aux Tierces, dont elle est composee." Traite, Book II., Ch. 5.

3 " La difference de ces deux proportions consiste dans une transposi-
tion d'ordre entre les deux Tierces, dont la succession forme de chaque
c6te la Quinte : d'oii il est evident que la seul Quinte constitue I'harmonie,
& que les Tierces la varient." Demonst. du Principe de I'Harmonie, p. 23.

RAMEAU'S TRAIT E DE L' HARMON IE 85

Chords of the Seventh.

Thus far Rameau has treated of the major and minor
harmonies. The Diminished and Augmented Triads, about
which more is said later, he does not consider to be funda-
mental chords. His next task is to explain the origin or
generation of the dissonant chords of the Seventh. In the
Traite he has a plethora of means whereby chords of
the Seventh may be generated, namely, by the multiplica-
tion, subtraction, addition and squaring of intervals. That
which he most fa\'ours, and which he mainly adheres to in
his later works, is the addition of Thirds. " If there are other
chords besides the preceding," he says, "it is necessary that
they should be formed from a major or a minor Triad, and
one of its parts ; that is, one of its Thirds. For example,
the addition of a Third to a Fifth gives us the interval of the
Seventh, and their subtraction will give us the complete
chord. Thus the addition of Fifth and minor Third = 10 : 18,
their difference is 12 : 15, and the complete chord has the
proportions 10: 12: 15: 18 (a). Proceeding in a similar way
for the Fifth and major Third we obtain the proportions
8 : 10 : 12 : 15 (6)" :

Chords of the seventh.
[b) (a) (a) (h)

[c] (a) ,</)

mode. ^-— |-— 1= § ^ §— ^ ^

Minor
mode.

I . Most important of the chords of the Seventh is that on the
Dominant. This chord is formed by adding a minor Third
above the major triad. In general, it is better that the
minor rather than the major Third should occupy the acute
position in chords. " This arises from the natural order
which has been at first prescribed for these Thirds, where we
find the major Third at the bottom, whereas the acute position
is occupied by the minor Third. "^ The chord of the Dominant

1 Traite, Book I., Ch. 7.

86 THE THEORY OF HARMONY..

Seventh is the most perfect of all the dissonant chords,
although the diminished Fifth occupies a prominent place ;
this circumstance however seems to render still greater
the perfection of the consonant chords which ought to follow
it," namely, the Tonic triad, or its inversions.^ The ratios of
this chord are 20 : 25 : 30 : 36 ! (c).

2. A chord of the Seventh formed by adding a major Third
above a major triad (8 : 10 : 12 : 15) [h), and another by
adding a minor Third above a minor triad (10 : 12 : 15 : 18)
{a). The first of these chords is accidental in its origin,
and the Ninth is always understood ; that is, it forms
part of the chords " by Supposition," c — e-g-b-d, or
f^a-c-e-g, where e and a are the real fundamental notes,
and c and / the fundamental notes " by Supposition."
(See p. 87.)

3. A chord of the Seventh formed by adding a minor Third
below a minor triad, as b-d-f-a. " This chord differs from the
chord of the Dominant Seventh in that the major Third, which
in the latter chord was at the bottom, is here at the top [!] " -
The addition of a minor Third below the minor triad is
evidently in order to avoid making the diminished triad
b-d-f the foundation of the chord. The ratios of this chord

are 25 : 30 : 36 : 45 (^)-

4. A chord of the Diminished Seventh, formed by adding a
minor Third above the diminished Fifth divided harmonically
(as g^-b-d-f). Such at least is Rameau's first explanation of
the origin of this chord. The other chords of the Seventh are
fundamental, for all are derived from the major or minor
triad by means of a Third added above or below. But
such is not the case with the chord of the Diminished Seventh,
for this consists of three minor Thirds added together, so
that the " perfect " chord is neither at the bottom (as in
the chord of the Dominant Seventh) nor at the top (as in the
chord of the Seventh on the Submediant of a major key) . This
leads Rameau to his second explanation of this chord. It
must be understood as an altered Dominant Seventh Chord :
"it is derived from the chord of the Dominant Seventh, by
raising the fundamental note of this chord a semitone. In
this shape the chord is said to be 'borrowed' [emprunte),
because it borrows its perfection from a sound which does

1 Traiti, Book I., Ch. 8, Art. 3. '^ Ibid., Art. 6.

RAMEAU'S TRAIT E DE U HARMON IE 87

not appear in it." ^ Thus the chord f-g^-b-d is borrowed
from the ]3ominant Seventh chord e-g^-h-d, by substituting
/ for t'. Rameau insists that this is the original form of the
chord : "It might appear that the chord should have the
form, gH^-b-d-f ; but as it is not ^#, but e which represents
the fundamental note of the chord, the form f-g^-h-d is the
correct one " {e).- (See Example, p. 85.)

Rameau, then, places a chord of the Seventh on each degree
of the major scale, as well as on the minor scale, except the
first and third degrees. The chords of the Seventh on these
degrees, each of which contains the augmented triad, as

{a-c-e-g^ : c-e-g^-b) he does not consider to be fundamental
chords. " In the chord of the augmented Fifth we can only
understand a chord by Supposition, the lower sound being
regarded as supernumerary." All the chords of the Seventh
treated of are comprised within the compass of the octave,
and all give rise to three other derived chords, or inversions
{':, t, and i).

Chords by " Supposition."

Rameau has laid down the principle that no chord can exceed
the compass of an Octave. The Octave, as Zarlino has said,
" is the mother of all the interv^als," and all intervals larger
than an Octave are merely repetitions of those contained
within the Octave. Therefore a Ninth and an Eleventh are
but the compound forms of a Second and a Fourth. Rameau
however is aware that there are chords which do exceed the
compass of an Octave ; it is necessary to account for such
chords. " We have seen," he says, " that the foundation
of harmony exists in the lowest sound of the ' perfect ' chord ;
even if we have added a Third above this chord, in order to
form the dissonant chord of the Seventh, still this does not
contradict our principle ; for this chord does not exceed
the extent of an Octave, and it is divisible into Thirds. But
if another Third be added, so as to form a chord of the Ninth,
or still another Third, so as to form a chord of the Eleventh,
everything becomes confused, and the basis of the harmony is

1 Traite. Book I., Ch. 8, Art. 7. ^ Ibid.

88 THE THEORY OF HARMONY

made obscure (a). The compass of the Octave is exceeded,
and as the Ninth and Eleventh are merely the compound forms
of the Second and Fourth, the chord is no longer divisible
into Thirds (b) :

Chord of Eleventh.
(«) {b)

:q'

^<s»-

-Gh- _ -oQ

"If then a Fifth sound cannot be added above a chord of
the Seventh, it must be added below. This added sound will
suppose the fundamental sound, but the real fundamental
sound will be immediately above it." ^ Thus in the chord
of the Eleventh at (a) g is the real fundamental sound, while c
is the " supposed " fundamental. In all " chords by Supposi-
tion," 2 Rameau considers that the essential harmony is the
chord of the Seventh. The sound which is added a Third or
Fifth below is non-essential : it is merely " supernumerary."
In the above chord, therefore, the essential harmony is the
chord of the Seventh g-h\}-d-f ; while c is the "super-
numerary " sound. In the chord of the Ninth Hkewise,
as G — b-d-f-a, the essential harmony is the chord of the
Seventh b-d-f-a. In this chord Rameau evidently considers
that b is the real fundamental sound, while G represents the
" supposed " fundamental. In the chords by Supposition
the chord of the Seventh lying immediately above the
added sound is capable of the various inversions ; the
added sound itself, the fundamental by Supposition,
cannot however participate in these inversions, but must
always occupy the lowest position as a supernumerary
sound, which does not alter the natural progressions of the
notes of the chord of the Seventh lying immediately
above it : —

1 Traite, Book II.. Ch. lo.

2 The term Supposition, Rameau informs us, " has been used up
to the present time to designate ornamental or grace notes, which
form no essential part of the harmony or chord in which they occur ;
the term, however, ought more correctly to be applied to those sounds
which alter the perfection of chords, in making them exceed the extent
of an Octave." Table of Terms.

RAMEAU'S TRAITE DH L'HARMOME 89

-j^'

:S-

-^^

ggg^

1C2:

^a_

There are but two chords by Supposition, that of the Ninth,
obtained by adding a Third below the chord of the Seventh ; and
that of the Eleventh, obtained by adding a Fifth below. The
chords of the Ninth in most common use are the following : — -

C — e-g-b-d
G — h-d-f-a
F — a-c-e-g
^—e-g^-b-d

Chord of Ninth on Tonic of Major Key
,, Dominant ,,
,, ,, ,, Subdominant

,, Mediant of Minor Key

(Chord of the superfluous Fifth)
,, ,, Mediant of Minor Key : — C — f-g^-b-d
(Chord of the superfluous Second)

Of chords of the Eleventh there are : —

Tonic of ^lajor Key

-C-g-b-d-f

Dominant

— G — d-f-a-c

Supertonic ,, .,

— D — a-c-e-g

Submediant ,,

: — A — e-g-b-d

Mediant

E b-d-f-a

Tonic of Minor ,,

— A — e-g^-b-d

(Chord of the Superfluous Seventh)
„ ,, ,, ,, Tonic of Minor Key : — A — f-gj/^-b-d

(Chord of the superfluous Second)

The above, however, is not a complete list. " In practice," says
Rameau, " other chords of the Ninth and Eleventh are used."
Many of these chords of the Ninth and Eleventh sound harsh
when all the notes are present,^ consequently the Third or
Fifth, or both Third and Fifth, of the chord of the Seventh

* Traill, Book III., Ch. 29, et seq.

90 THE THEORY OF HARMONY

(l^dng immediatel}^ above the fundamental note by sup-
position) must sometimes be omitted. For the same
reason the Seventh should be omitted from the chord of
the Ninth on the Tonic of the major key.

Resolution of Dissonances.

The Seventh is the source of all the dissonances, and the
chord of the Seventh is the source of all dissonant chords. Of
all the chords of the Seventh, that on the Dominant is the
most important. It is by means of the resolution of the chord
of the Dominant Seventh on the Tonic chord — its most natural
resolution — that we discover the proper way in which to
treat all dissonant chords. In this chord there are two dis-
sonances which demand resolution : one, between the fun-
damental note and Seventh ; the other, between the Third
and Seventh. The first is a minor dissonance, and should
fall one degree ; the second is a major dissonance, and should
ascend one degree. The former is the source of all the minor
dissonances ;

the latter of all the major dissonances. In reality,
however, the Seventh is the origin of all dissonances,
whether major or minor : for the (minor) Seventh which is
added above the "perfect "chord (siSg-b-d-f) forms a dissonance
not only with the bass (g-f) but also with the third of the
chord (b-f). The minor dissonance is so called because it is
formed by the addition of a minor Third above the " perfect "
chord ; and the major dissonance because this is a major
Third above the fundamental note.^ The major dissonance
is always the leading note of a key, the Third of the chord of
the Dominant Seventh, and occurs only in this chord ; in all
other dissonant chords the dissonance is a minor one, namely,
that of the Seventh, or its inversion, the Second.

1 Traiie, Book II., Ch. i8, Art. i.

RAMEAU'S TRAITE DE L'HARMONIE 91

Up till now the greatest uncertainty and confusion have
prevailed in respect of the proper treatment of dissonances.
" Theorists tell us that the Seventh may be resolved on the
3rd, the 5th, the 6th, the 8th; that the diminished Fifth may be
resolved on the 3rd, the 4th, the tritone, the 9th ; thus science
is made obscure ; particular cases are cited, but no simple and
intelligible rule has been formulated for the treatment of
dissonances. If the Seventh may be resolved on different
intervals, this arises only because of the different progressions
of the bass " ^ (that is, because of inverted chords). In the
resolution of the chord of the Dominant Seventh on the Tonic
chord, however, " we find a sure and certain rule for the
resolution of dissonances. . . . This rule permits of no
exceptions, and proves that the fundamental harmony
subsists only in the ' perfect ' chord, and that of the
Seventh." -

Further, the old rules concerning the s^nicopation of notes
and the preparation of dissonances are merely a source of
embarrassment : " Here are two simple rules which suffer
no exception : I. To prepare b\' means of a consonance every
dissonance which admits of preparation ; and, II. to take
[by step] after a consonance, whether by ascending [a],
or by descending (b) that dissonance which cannot be
prepared " ^ : —

{«) I xi W , X,

-f^-

""23

Thus the confusion which has prevailed in respect of the
treatment of dissonances disappears ; a confusion largelv due
to the fact that " theorists have been accustomed to reckon
dissonances from the bass upwards," whereby the dissonant
nature of such chords as c-e-g-a or a-c-d-f, which consist
entirely of consonant intervals above the bass, could not be
determined.

By means of the natural resolution of the chord of the
Dominant Seventh on the Tonic "chord, we find the proper
way in which to treat not only all other chords of the Seventh,

1 Traite, Book II., Ch. 8. 2 IhiJ., Ch. 18, Art i.

3 Ibid.. Ch. 16, Art. 4.

92 THE THEORY OF HARMONY

but all chords by Supposition as well (chords of the Ninth and
Eleventh) . For in every chord by Supposition the essential har-
mony is that of the chord of the Seventh which is placed above
the lowest sound of the chord (the " supposed " fundamental).
This chord of the Seventh will be treated in the usual way,
according to the rules prescribed for chords of the Seventh,
and will be capable of all the different inversions ; the sound
added below, however, cannot be inverted, but will always
occupy the lowest position ; thus —

r^=^^^r=s;

*- .5:

-J-

:?5:

:^±

A striking example of the correctness of the theory of
inverted chords,^ is the fact that a dissonant chord and its
inversions are all resolved in the same way ; for example, the
chord of the Dominant Seventh and its inversions are all re-
solved most naturally by the chord of the Tonic. It is clearly
evident then, that the chord of the Seventh is the source
of all the dissonant chords. Other writers have distinguished
a great many varieties of dissonant chords, as chords of the
Second, of the Tritone, and so forth ; these however are not
independent, but " derived " chords, which arise through
in version. 2 Note then, concludes Rameau, that there exist
in harmonic music but two chords which are " original " and
func^amental, namely, the " perfect " chord, and the chord
of the Seventh.

1 Rameau should add :• — and of the manner in which inverted
chords may retain the harmonic significance of the original and funda-
mental chord from which they are derived.

2 Traiti, Book TI., Chs. 8. and 17.

RAMEAU'S TRAITE DE L'HARMOXIE 93

Resume OF Rameau's Theories of Chord Generation.

Rameau's task in dealing with chords has been similar to
that of the scientist who, finding himself confronted with a
multitude of diversified and apparently unrelated phenomena,
has to discover whether there may not be some hidden connec-
tion between them, whether indeed it may not be possible that
even such a bewildering variety of species has had a common
source in some simple and primiti\-e o't?H»s. Of the innumerable
chords which ma}- be used in harmon}-, Rameau, b}' means of
his theory of inversion, finds that there are but two, fundamen-
tal and original, from which all others are derived, namely
the " perfect " chord (the major and minor harmonies) and
the chord of the Seventh ; and in place of the infinity of
rules, exceptions from rule, etc., relating to the employment of
these chords he brings fonvard one or two simple and com-
prehensive rules which are based on the natural resolution of
the chord of the Dominant Seventh. Whether Rameau's
explanation of the chord of the Seventh and of " chords by
Supposition " be accepted or not, there is no doubt that
his theory of inverted chords had already produced splendid
results, representing an achievement for which he was fully
deser\'ing of the eulogies which were bestowed on him when
once his theoretical principles were sufficiently understood.
His fine ear and musical penetration, also, are apparent in
his treatment of the augmented and diminished triads, which
he refuses to consider as fundamental chords ; and in his
explanation of the chord of the diminished Seventh as being
derived from the chord of the Dominant Seventh.

But no sooner has Rameau set out to estabHsh
his " natural principles " of harmony, than he finds himself
plunged into difiiculties and contradictions. He starts
\\'ith a " natural principle " of harmonic generation which in
itself is perfectly intelligible and consistent. It is from this
principle, he tells us, that all chords derive their origin. But
this principle has furnished him \nth one chord and one only,
namely, the major harmony ; neither the minor harmony, nor
a single chord of all the dissonant chords which he considers
to be fundamental, is to be found in the division of a string
bv the first six numbers. Rameau therefore finds himself
compelled at the very outset to abandon the principle of

94 THE THEORY OF HARMONY

harmonic generation which he has at first proposed. The
generation of chords by means of the addition of Thirds,
which he brings forward in its place, is not a "natural principle "
of harmony at all, but merely a device to get rid of a difficulty.
He assumes that the " perfect " chord may be considered
to be generated by the process of adding one Third to another,
thus placing himself in contradiction with his original principle,
whereby the first sound generated after the Octave is the
Fifth (twelfth) and only afterwards the Third (seventeenth).
But, says Rameau, does not the " perfect " (major) chord
result from the harmonic division of the Fifth, whereby
there arise two Thirds, one major, and the other minor ?
Rameau however cannot have it both ways. Even if he
correctly explains the major harmony as arising from the
harmonic division of the Fifth, he is still unable to show us
how it can be regarded as resulting from the addition of
Thirds, and still less to explain whence he derives the liberty
of adding sounds to this harmony so as to form other chords.

In abandoning his original principle of harmonic generation,
Rameau necessarily gives up at the same time his theories of
the fundamental Bass, and of the inversion of chords. He
leaves his chords without a harmonic foundation, without
a Fundamental Bass. This is true even of the chord of the
Dominant Seventh, which he recognizes to be the most
important of all the dissonant chords. Rameau, of course,
might have anticipated here the methods of some of his
successors, and derived the chord by means of the number
seven, applied to the division of the monochord. That he
does not avail himself of this method as a possible means of
escape from his difficulties is a proof of his perspicacity. If
he admits it, he will destroy his whole system of harmony.
But in rejecting it, he rejects at the same time the only possible
means whereby the chord of the Dominant Seventh can be
made to appear as fundamental, that is, in his own words,
" generated from the first sound, which sound is consequently
the principle and foundation " of all the other sounds of the
harmony heard above it.^

Having assumed that chords are formed by the addition
of Thirds, he makes the further assumption that, in order to
form the chord of the Seventh, he is at liberty not only to add

1 Book I., Ch. 3, Art. i.

RAMEAU'S TRAITE DE U HARMON IE 95

either a major or a minor third to the " perfect " chord, but
to place the added third either above or below. The results
of this last process are somewhat peculiar ; for example, in
the chord of the Seventh on the leading note of a major key, as
b-d-f-a, the note b cannot be regarded as the fundamental note
of the chord, because not a perfect but a diminished Fifth
is heard above it. This b must then be considered to be
added beloiv the minor harmony d-f-a ; whereby d, the Third
of the chord b-d-f-a, becomes its fundamental note.

Thus a new interval is formed, the Seventh, and in attempting
to make this Seventh appear as fundamental, Rameau gives
utterance to the most contradictory statements. He explains
the Seventh as resulting from the square of the perfect Fourth.
Not content with this — although the squaring of intervals
is quite as justifiable as their addition — he argues that just as
the harmonic division of the Fifth gives us two Thirds, each
of which is fundamental, so the harmonic division of the major
Third produces two Seconds, a major and a minor, from the
inversion of either of which will arise the interval of the
Seventh. " If fundamental chords, and if the fundamental
progression of the bass consist solely of the intervals of the
Third, Fifth and Seventh, then these intervals must also
be regarded as fundamental. The best authors have proposed
to us the Third and Fifth as the fundamental intervals
{pour principe), but have always forgotten the Seventh,
which is the first of its species. For does not this Seventh
arise by inversion from the harmonic division of the major
Third ? Therefore it must be regarded as fundamental or
excluded from the dissonances." ^

Rameau has from the first maintained that there is only one
dissonance, that of the Seventh ; here he repeats this state-
ment, and then almost in the same breath informs us that
the Seventh is really an inverted Second, a contradiction
which Mattheson was not slow to seize upon.^ Again we

1 Traite, Book II., Ch. 17, Art. 3.

- In his Kleine General-bass Schule Mattheson remarks : — " I must
mention that in many places M. Rameau makes his beloved Seventh
the origin of all the dissonances. But he also remarks that it arises
from the addition of two Fourths, which however he soon contradicts,
and asserts that the tone forms the Second, and from the inversion
of this Second arises the Seventh. How then can it be the origin of
all the dissonances, when it is itself derived from the Fourth, and is
an inverted Second ! "

THE THEORY OF HARMONY

read : — " If we are sometimes obliged to distinguish the dis-
sonance by different names [Seventh, Second], this is only
in order to facilitate practice ; for at bottom there is only
one dissonance, from which all the others are derived."^
Finally, Rameau informs us that " the interval of the Seventh
owes its origin more to good taste than to Nature, since it is
not found in the most natural operations, as a part of the
harmonic body, hke the intervals which compose the ' perfect '
chord." '^ This last statement is a confession of failure, and
proves that Rameau was unable to satisfy even himself that
his explanation of the origin of the chord of the Seventh was
a reasonable one.

As for his theory of the generation of chords of the Ninth
and the Eleventh (chords by Supposition) by adding a Third or
Fifth below a chord of the Seventh so that each chord has two
fundamental notes, a real and a " supposed " fundamental —
little need be said. Rameau's procedure in respect of these
chords is extremely ingenious, but of course purely arbitrary.
It is singular that, although he treats the chords of the Ninth
and Eleventh as fundamental chords, he is nevertheless quite
well aware that they arise from the principle of the suspension
or retardation of notes. Of the following example he
remarks : — " It is certain that the chords by Supposition serve
only to suspend the sounds that ought naturally to be
heard: thus the sounds A suspend the sounds B"^: —

-Gh-

-(S>-

-Gt-

Basse-continue .

-&-

-G>-

Basse-f ondamentale .

1 Supplement.

" " Cet intervale devant son origine an bon gout plus qu'a la nature,
puisqu'il ne se trouve point dans les operations les plus naturelles,
faisant partie du corps Harmoniquc, de meme que les autres intervales
qui composent I'accord parfeit." (Book II., Cli. 17, Art. i.)

3 Book III., Ch. 31.

RAMEAU'S TRAITE DE IJ HARMON IB 97

It is singular also that after explaining the chord of the
Diminished Seventh on the leading note of a minor key
(as ,i,'#-/>-^-/) as an altered Dominant Seventh chord, Rameau
should not have explained exactly in the same way the chord
of the Seventh on the leading note of a major key (as b-d-f-a).
Both chords are formed in an exactly similar way, although
they do not consist of the same intervals. Of the chord of
the Augmented Sixth, the different forms of which are already
known to and mentioned by Heinichen, Rameau does not
speak. His silence in respect of this chord can be understood.

Do Rameau's efforts, then, to explain the generation of
chords represent merely so much time and labour wasted?
Not altogether. They may instead lead to a positive result
of the greatest importance for the science of harmony.
Rameau, one of the greatest of theorists in the domain of
harmony, is unable with all his ingenuity to discover anj^
natural principle of harmony which will furnish him with
more than one chord — the " perfect " chord. He is
quite unable to justify in any way the theory of the
generation of chords by means of the addition of Thirds.
Finally, he is quite unable to explain chords of the
Seventh, Ninth, and Eleventh as fundamental, that is,
as consisting of sounds all of which are directly related
to and arise from the lowest and fundamental sound of
the chord. Rameau indeed demonstrates that the Second,
which arises from the harmonic division of the major Third,
has more right to be considered as fundamental than the
Seventh. These are not merely negative results. It is a
matter of the greatest consequence for the science of harmony
if it can be proved — and Rameau's failures go far to prove
it — that with the exception of the major harmony and that
of the minor, to be further discussed, no others exist as a
constituent and essential part of our modern tonal and
harmonic system. It is not alone Rameau who has failed
to discover them ; the most strenuous endeavours of those
who, even up to the present day, have sought to explain other
chords as fundamental, in the sense given above, have met
with no better success.

98 THE THEORY OF HARMONY

CHAPTER IV.

Rameau's Traite de I'Harmonie [contd.).

THE FUNDAMENTAL BASS.

Strictly speaking, the Fundamental Bass of Rameau is of
more than one kind. There is the fundamental bass which
is the direct result of his theory of the inversion of chords,
and which has been in practical use in nearly every text-book
of harmony since his time. This bass, which always represents
the fundamental note of the harmony, is to be distinguished
from the actual bass, the basso continiw, in which the bass
note may have the Third, Fifth, etc., of the chord. But
admirable and useful as this bass may be, the species of
Fundamental Bass which Rameau evolves in Book H. of
the Traite de Vharmonie, is even more important. By it
Rameau endeavours to explain, on logical and scientific
grounds, the laws which govern harmonic succession.
A real science of harmony, Rameau perceives, must not
be satisfied with the explanation of chords as isolated
entities ; it must also take into account harmonic suc-
cession : it must try to discover the underlying principles
which govern the progressions from one harmony to another,
and which render these intelhgible.

It is the bass on which everything, as regards harmonic
succession, depends ; it is the bearer of the harmony, and
its foundation ; its progression therefore will determine
the harmony which is to follow. Rameau lays stress on
this point. He remarks : " Zarlino has compared the
bass to the earth, which serves as a foundation for all

RAMEAU'S TRAITE DB U HARMON IE 99

the other elements. It is called the bass of the harmony,
because it is the basis and foundation of it. If the
foundation were to fail, that would be as if the earth were
to fail : all the beautiful order of Nature would fall into
ruin ; every piece of music would be filled with dissonance
and confusion. When then one wishes to compose a bass,
it is necessary to proceed by movements somewhat slow and
separate. The higher parts may move more quickly and in
diatonic [conjunct] progression." ^

This principle, Rameau says, cannot be too strongly
insisted on, and it receives the greatest possible confirmation
from the arithmetical division of a string, on which his theory
is based. " The string with its divisions furnishes us with a
perfect harmony, the bass of this harmony resulting from
the entire string, which is the source and foundation of all
the other sounds. If now we wish to determine the pro-
gression of the bass, it is evident that we ought to make it
proceed by those consonant intervals given us by the first
divisions of the string. Each sound therefore [that is, of this
fundamental bass] will accord with that which has preceded
it, and will bear a harmony like that which we have received
from these first divisions. ... It is the Fifth [the first
sound generated after the Octave] which best suits the
progression of the bass ; in fact, one never hears a Final
[Perfect] Cadence where this progression does not appear ;
the bass descending a Fifth, or, what is the same thing,
ascending a Fourth. But as the Fifth is composed of two
Thirds, the bass may proceed by this interval also [that is,
by a Major or a minor Third] as well as by the interval of
the Sixth, which is the inversion of the Third. All the
progressions of the Fundamental Bass should therefore
be comprised in these consonances. Sometimes, however,
dissonance obliges us to make the bass ascend a tone or a
semitone. But this can only occur by a licence, as in the
Deceptive Cadence. It should be observed that this tone
■or semitone is the inversion of the Seventh." -

It is not essential, of course, that every bass note should
at the same time be the fundamental note — the Fundamental
Bass — of the harmony. Inversions may be made use of,
where the bass note is the Third, Fifth, or Seventh of the

TraiU, Book II., Ch. i. 2 y/,/j

lOO

THE THEORY OF HARMONY

chord, for by this means a great div^ersity of movement
and harmony is obtained. Nevertheless, in such cases, the
correct progression of the harmony can always be verified
by comparing the Fundamental Bass with the actual bass
{basso continno) thus : —

...:(

Fundamental {^.-
Bass. ~5^-

6
4

4
2

-<s>-

-&-

"3:21

In thus determining the progression of the Fundamental
Bass, we at the same time determine the progression of the
upper parts, which for the most part, as Zarlino has said,
should be diatonic : that is, these upper parts should proceed
to the nearest harmony notes of the following chord. " Hence
there will arise an agreeable succession of chords, without
our being obliged to have recourse to any other rule. Nature
herself being here our guide as to what is most appropriate
and beautiful." ^ The resolution of the chord of the Dominant
Seventh on the Tonic chord shows clearly how the progression
of the bass, which here descends a Fifth, determines the
progression of the other notes of the harmonv : —

1 Traiie, Book II., Ch. 2.

RAMEAU'S TRAIT E DE L HARMON IE

lOI

Even in cases wlicre tlie Fundamental Bass is not present,
it will nevertheless be understood ; as in the following passage
from Zarlino : —

Zarlino's
example.

Fundamental
Bass added.

Perfect
Cadence.

izz;

:±

—&- _

Perfect
Cadence.

Here it is evident tliat the most natural resolution of the

h - c ■ f - C

tritone r , , and its in\ersion, the dimmished Fifth ■{

J ~ L 0 ~ 0

is in complete accordance with the most perfect progression
of the Fundamental Bass, which is to descend a Fifth.
Although the descending progression of the bass is better
than the ascending, nevertheless the same intervals may be
taken also in ascending progression. But it is only by means
of a descending Fundamental Bass that dissonance can be
prepared and resolved. When the bass falls a Fifth or,
which is the same thing, rises a Fourth, the Third of the
chord prepares and resolves the dissonance {a) ; and if the
bass falls a Third or a Seventh (or rises a Second),

i It should be noted that Rameau regards this chord d-f-a as in
effect dissonant, that is, as a chord of the Sevpnth d-f-a-c.

I02

THE THEORY OF HARMONY

the Fifth and the Octave resolve this dissonance (b)
and (c) : —

(6)

5th.

Ic)

8va.

S^^

zsz:

-<s»->-

^-3 o

s/c /

22:

(d)

(/)

^^=^

^B=r

^g=

-<s>-

But according to the most natural progression of the harmony,
the Third should be regarded as the only consonance which
can serve as the resolution of dissonance (as at (a)). On the
other hand, should the bass rise a Third, Fifth, or Seventh
(or fall a Sixth, Fourth, or Second), the dissonance can neither
be prepared nor resolved {(d), {e), (/) ).^

The fundamental progression of the bass, then, ought to
be comprised in the intervals of the Third, Fifth, and Seventh ;
of which that by the Fifth is the best, then the Third, and
lastly the Seventh. These same intervals which suit best the
progression of the Fundamental Bass, ought also to accompany
it ; that is, each note of this Bass should bear the " perfect "
chord, or the chord of the Seventh."- Such, in brief, is the
Fundamental Bass, which Rameau explains at great length,
and with much diffuseness and repetition, in the second as
well as in the third and fourth books of the Traite de Vharmonie
and which \\e ha\-e now to examine more closelv.

1 Train, Book II., Chs. i ^, 17. etc.

I hid.. Ch. 2.

RAMEAUS TRAITE DE LHARMOSIE 103

In developing this part of his system, Rameau steps out
with greater confidence, and does not betray the hesitation
and uncertainty so conspicuously e\-ident in the first book
of the Traite, in which he has explained the generation of
chords. He eWdently regards his Fundamental Bass as a
great achievement, as in some respects it imdoubtedly is.
He feels that he has grappled, not without success, with the
two great central problems of harmonic science, namely,
the generation of chords and the laws which govern their
succession : that he has evolved a real science of harmony,
and proved that all the bewildering variety of harmonic
phenomena arises from a fundamental principle, the most
simple and natural that one can imagine. With all the
exultation of one who. after long combating and stri\"ing.
has at length reached his goal, he exclaims : " How
mar\ellously simple it all is ! . . . The principle of harmony
exists solely in the ' perfect ' chord, from which is formed
the chord of the Sexenth : more precisely, in the fundamental
sound of these two chords, wliich is, so to speak, the Harmonic
Centre {Centre Harmoniqite) to which all the other sounds
are related.^ ... So that all this infinite diversity of
harmony and melody, all these artistic ideas expressed with
so much nobihty and truth, proceed from two or three
inter\als disposed in Thirds, the principle of which is contained
in a single sound, thus : —

Fundamental Third Fifth Seventh "
I 3 5 7

One cannot grudge Rameau these few words of self-
congratulation. Although he has failed to explain the
generation of the \arious chords, he has ne\"ertheles5 in other
directions succeeded to a surprising extent. In the "' natural
principle " of harmon\- presented to him by Descartes
there did not appear at first sight to be much which could
suggest to him his theories of the fimdamental note and the
inversion of chords ; certainly Descartes was far from
deriving such consequences from his own principle. But
who would have imagined that Rameau woijd seek in this

^ Here Rameau use? the term " Harmonic Centre," not in the sease of
a fundamental sound or Tonic which is the central sound of the key-
s\-stem, but in the sense that each sound of the fundamental bass is
itself a "■ Harmonic Centre." — Traite. Book II.. Ch. 17.

I04 THE THEORY OF HARMONY

same principle of the mathematical division of a string for
the origin and explanation of the laws which govern harmonic
succession ? That from such a division of a string tliere
should arise the most " perfect " of all harmonies is in itself
an astonishing fact ; but that from such a harmony there
should in turn arise the principles which determine the pro-
gression from one harmony to another, appears at first to be
well-nigh incredible. Rameau has from the outset maintained
that the whole principles of harmony have their origin in the
division of a string by the first six numbers. If he succeeds
in proving his theory of the Fundamental Bass, this will
undoubtedly represent the highest achievement of his genius.
Rameau tells us that the most " perfect " progression
of the Fundamental Bass is to descend a Fifth, as in the
Perfect Cadence : —

ZCSZ2Z

z?:2z

Most musicians will agree that the most directly and easily-
intelligible of all harmonic successions is that from the
Dominant to the Tonic hanuony ; it is in this sense, evidentlv,
that Rameau makes use of the term " perfect progression."
How then does he account for this ? Because, he says,
the Fifth is the first interval (that is, after the Octave)
generated from the division of the string. The " perfection,"
then, of the fundamental progression in question would appear
to be owing to the " perfection " of the consonance of the
Fifth ; this would explain also the comparative inferiority
of the fundamental progression by the Third, which is
generated after the Fifth. But this is not a sufficient
explanation. " Perfection " of consonance and " perfection "
of the fundamental progression of the bass do not necessarily
mean the same thing, and we are not entitled to infer that
the " perfection " of the one arises from the " perfection "
of the other. Let it be granted, however, that Rameau
fullv understands what he nevertheless fails to communicate
to us, namely, that the " perfection " of the descending
progression by the Infth, as in the Perfect Cadence, is due

RAMEAU'S TRAITE DE L'HARMONIE

I () •

to the directness and closeness of relationship existing between
the two sounds which constitute this Fifth. The lower
sound of this Fifth is understood as the fundamental sound ;
the hit,'her sound is a dependent sound, which has its meaning
as Fiitli determined by the fundamental sound in which
it has its origin. Such a closeness of relationship existing
between the two sounds of Fifth and Fundamental, it follows
that the same closeness of relationship will exist between
their harmonies : —

-rz'

3151

zc:iL

221

Fifth. Fundamental.
Something of this appears to have been in Rameau's mind,
for he makes the noteworth}- statement that " in the
Perfect Cadence the Fifth returns, as it were, to its source. "^
But even if we accept the above as an adequate and
complete explanation of a Dominant-Tonic harmonic succes-
sion, as in the Perfect or Authentic Cadence, or of a Tonic-
Dominant succession, as in the Imperfect Cadence or Half-
Close, what of the other cases in wliich the Fundamental Bass
descends a Fifth ? —

-C3>-

'5'

rgzzS:

-<s«-

iSr:

-/s-

-<s-

-&-

-o-

"o~

-«5»-

-?y-

-(S»-

* *

* ♦

1 Tvaite, Book II., Ch. i8.

io6 THE THEORY OF HARMONY

Rameau seems to imagine that these admit of an
explanation similar to that of the Perfect Cadence.^
Would then Rameau say of examples {a) and {d) that here
the Fifth returns to its source ? If so, and if these harmonic
successions are to be explained in the same way as the Perfect
Cadence, how then do they differ so greatly from it in effect ?
Rameau's explanation does not suffice. Further, the
ascending Fifth progression of the bass at (b) and {e) cannot
be considered to be inferior to the descending progression
at (a) and (d). Nor, finally, can the progression of the
Fundamental Bass by Thirds (c), be considered to be inferior
to the descending Fifth progression at (a) . Moritz Hauptmann
considers it to be greath' superior. For Hauptmann, " the
succession of two triads is intelligible only in so far as both
can be referred to a common element which changes meaning
during the passage." - This " common element " consists
in the community of sounds existing between the two
triads. The harmonic succession at (d) is, according to
Hauptmann, intelligible by virtue of the common notJ c,
but is rendered more directly intelligible b}- means of the
mediating triad a-c-e. as at (/), between which and the
triads c-e-g, andf-a-c. there are found two notes in common.
Hauptmann therefore plainly considers the Fundamental
Bass in Thirds to be more " perfect " than that in Fifths.

The position here taken up b}^ Hauptmann (chord-relation-
ship by community of sounds) does not differ essentially
from that of Helmholtz, in his Sensations of Tone (chord-
relationship by community and relationship of upper partial
tones) ; but although the latter follows the former in his
conclusions as respects chord-relationship, he is much less
consistent. Helmholtz, in fact, thinks with Hauptmann
that those chords are most closely related which have most
notes in common ; and also with Rameau that the closest
relationship existing between any two sounds is that between
a note and its Fifth ! He says expressly : — " When two
chords have two notes in common they are more closelj'
related than when they have only one note in common.
Thus c-e-g and a-c-e are more closely related than c-e-g and

g-b-d." ^ In the same chapter however he makes the

^ Traitc, Book II., Ch. i8. ^ Harmony cuid Metre.

^ Sensations of Tone, Part III., Ch. 13.

RAMEAUS TRAITH DE LHARMOSIE 107

following pronouncement with regard to the Fundamental
Bass rising, as well as falhng, a Fifth : " The closest and
simplest relation of the tones is reached in the major mode,
when all the tones of a melody are treated as constituents
of the compound tone of the Tonic, or of the Fifth above or
the Fifth below it. By this means all the relations of tones
are reduced to the simplest and closest relation existing in
any musical system — that of the Fifth." Helmholtz
apiparently prefers to have it both ways. " The chord of
the Tonic C," he proceeds, " is somewhat differently related
to the chord of G, the Fifth above it, and to the chord of F,
the Fifth below it. When we pass from C-E-G to G-B-^,

we use a compound tone G, which is already contained in
the first chord, and is consequently properly introduced,
while at the same time such a step leads us to those degrees
of the scale which are most distant from the Tonic, and have
only an indirect relationship with it. Hence this passage
forms a distinct progress in the harmon}-, which is at once
well assured and properly based. It is quite different with
the passage from C-E-G", to F-A-c. The compound tone F

is not prepared in the first chord, and it has therefore to be
discovered and struck. The justification of this passage,
then, is not complete on the ground of close relationship
between the chords, until it is felt that the chord of F contains
no tones which are not closely related to the Tonic C. Hence,
in this passage from the chord of C to that of F, we miss that
distinct and well-assured progression which marked the passage
from the chord of C to that of G. But as a compensation,
the progression from the chord of c to that of F has a softer
and calmer kind of beauty, due perhaps to its keeping within
tones directly related to the Tonic C." ^

Let it be observed that Helmholtz is here explaining the
chord successions by virtue of the upper partial tone relation-
ship existing between the harmonies ; that the first succession
is that of the Fundamental Bass rising a Fifth, and the second
that of the Fundamental Bass falhng a Fifth. Helmholtz
considers that the first chord succession, in which the bass
rises a Fifth, is more " distinct and well assured " than the
second, in which the bass falls a Fifth. But in this latter
case we have exactly the same succession of harmonies as

^ Sensations of Tone, Part III., Ch. 15.

io8 THE THEORY OF HARMONY

in the Perfect Cadence. If Helmholtz refuses to consider
that the progression from the Dominant harmony g-h-d to the
Tonic harmony c-e-g is to be explained in the same way
as the progression c-e-g to f-a-c, then lie is unable to find
any explanation whatever of the progression from the
Dominant to the Tonic harmony. As matters stand
Helmholtz says in effect that of the two Cadences, the Perfect,
in which the Fundamental Bass falls a Fifth, and Imperfect
(Half-Close) in which it rises a Fifth, the second is in reality
the more perfect, for we find in it that " distinct and well-
assured progression " which we miss in the former. This
is a conclusion exactly the opposite of that arrived at by
Rameau ; it is also one which no musician will entertain
for a moment.

Further, in the passage which follows the above, Helmholtz
thus treats of the Plagal Cadence, of the Subdominant-Tonic
harmonic succession, in which the Fundamental Bass rises
a Fifth. '• The Plagal Cadence," he says, " corresponds to a
much quieter return of the music to the Tonic chord, and
the progression is much less distinct than before." ^ Here
Helmholtz completely reverses his former statement regarding
the harmonic progression in which the Fundamental Bass
rises a Fifth. There the progression to the Fifth above was
a " distinct and well assured " progression ; here the pro-
gression produces quite a different effect. In the first
instance, it was the bass descending a Fifth which gave to
the harmonic succession " a softer and calmer kind of
beauty "; it is now the opposite progression of the bass
rising a Fifth which produces this effect. To be sure the
aesthetical impressions made upon us by these harmonic
successions are as Helmholtz describes them ; but Helmholtz
makes it quite evident that it is not by means of his theory
of chord relationship that such harmonic successions are to
be explained.

On the other hand, it is equally evident that Rameau has
not sufficient grounds for asserting that the most " perfect "
progression of the Fundamental Bass is to descend a Fifth.
It is contrary to the facts ; while it is true of the Perfect
Cadence it is not true in the case of many other harmonic
successions. In certain cases the descending Fifth progression

' Sensations of Tone, Part III., Ch. 1.5.

RAMEAU'S TRAITE DH L'HARMOSIE 109

is even inferior to the ascending Fiftli progression, as well
as to the progression by the Third. It is to be noticed that
in allowing the Fundamental Bass to ascend as well as
descend, by the intervals of the Third, Fifth, and Seventh,
or their inversions, Rameau accords to it the hbert\' to
fall by any inter\-al, large or small, to be found in the diatonic
scale. It may proceed by means of a semitone, tone,
minor or major Third, perfect Fourth or Fifth, major
or minor Sixth, major or minor Seventh, perfect Octave
— an extremely satisfactory arrangement, no doubt, for
by this means the Fundamental Bass is made to fit in
\\-ith every conceivable harmonic progression in the
diatonic scale. Where has Rameau discovered these
intervals, and whence does he think he has derived the
liberty of making use of them for the progressions of his
Fundamental Bass ? From the division of a string by the
first six numbers ? So he apparently imagines.

Rameau, in fact, proceeds here almost exactly in the same
way as he has already done in the case of the generation of
chords. He brought forward the senario as the true principle
of Harmonic Generation, and no sooner had he done so than
he abandoned it in favour of a process of chord formation
by means of added Thirds. He now brings forward this
same senario as the true principle of harmonic succession,
and immediately abandons it in favour of a theor}- whereb}-
the Fundamental Bass is allowed to progress by intervals,
or tone relations, which are not found in the senario at all.
In the Train de I'liarmonie Rameau does not understand
his own theory of the Fundamental Bass. He quite loses
sight of the fact that the actual sounds which arise from
the arithmetical division of a string are these, and these
only : —

1 2 3 4 5 6

W-

(in this case C, the lowest sound, represents the sound produced
b\- the entire length of string). Excluding the octave

I lO

I'HE THEORY OF HARMONY

sounds, the Fundamental Bass may proceed from c to g,
or from c to e, or back again from e or o^ to c. But there
are no other sounds to or from which it can proceed. It cannot,
for example, proceed from c to /, for there is no / for it to
proceed to. Rameau also forgets what he has expressly
stated to be one of the principal conditions of his Fundamental
Bass, namely, that each sound of this Bass should bear a
chord " similar to that which we have received from the
divisions of the string." Although Rameau imagines differ-
ently, the only harmony we have thus received is the major
harmony. If then Rameau allows his Fundamental Bass
to proceed an3'where except to the Fifth above (Dominant),
or the Fifth below (Subdominant) he will immediately find
himself outside of the key system : — ■

v?v — * ^ * — ^ * — ii. * — i* —

Hence when several years later Rameau pubhshed his
Generation Harmonique, we find that he has very considerabl}^
modified his views with respect to his theory of the Funda-
mental Bass. It may now proceed in two ways only : by the
Fifth, or by the Third. From the Fifth progression oi the
Fundamental Bass, he tells us, there arises the Diatonic
system ; and from the Third progression, the Chromatic
system.

In summing up we find that Rameau, by his theory
of the Fundamental Bass, furnishes us with the means
of explaining two diatonic chord-successions, and two only,
viz., those of the Perfect Cadence and of the Tonic-Dominant
Cadence. This is by no means such a meagre result as might
at first sight be imagined. No better explanation of these
chord successions has ever been discovered. Rameau does
not enter sufficiently into the question of rhythm and accent,
but apart from this he finds for these successions a rational
and scientific, even an jesthetical explanation.

RAMEAU'S TRAITE DE L HARMON IB

f I I

Thk Cadence.

The progression of all dissonant chords, whether these are
inverted chords, chords by "Supposition," or "borrowed"
chords, is comprised in three Cadences, which are the Perfect,
the Interrupted, and the Irregular Cadences : —

Perfect. Interrupted. Irregular.

I— &—

1 — /r> — "P

^r — ^ —

r^ t

— 1^ —

frr\ ■ *-*

"W r^

r'j

jCl.

-&-

-Q_

■g"

-G>-

'.- \ •

,-o

''^

1 — ,^ 1

-U

In the Cadence the penultimate chord should be dissonant ;
this dissonant chord will render the consonant effect of the
final chord still greater, and accentuate the impression of
repose.

Thus in the Perfect Cadence, in which the Fundamental
Bass proceeds from Dominant to Tonic, " it appears natural
that the penultimate chord should be distinguished by some-
thing which renders it less perfect ; for if two perfect chords
follow one another in a Perfect Cadence, one is unable to
judge which of these chords is the true chord of repose." ^

The Interrupted, or Deceptive Cadence. — " If we alter the
progression of one of the sounds of the first chord which
forms part of the Perfect Cadence, this change of progression
will interrupt the conclusion ; hence the term Interrupted
Cadence (Cadence Romptie). In this Cadence the Funda-
mental Bass, instead of descending a perfect Fifth from
Dominant to Tonic, will ascend diatonically one degree,
namely, to the sixth degree of the scale. With the exception
of the progression of the bass, this cadence differs in nothing
from the Perfect Cadence ; the other notes of the chord of
the Dominant Seventh have the same progression as formerly.
It should be remarked that it is better to double the Third,
rather than the fundamental note, in the chord on the sixth
degree of the scale, because this third ' supposes,' or takes
the place of the true fundamental sound. "^ . . . The
progression of the bass in the Interrupted Cadence is due

* Traits, Book II., Ch. 2.

2 IhicL, Ch. 6

112 THE THEORV OF HARMONY

to a licence. A dissonance can be resolved only by the
Fundamental Bass descending a Fifth ; if then the bass
descends a Seventh or, which is the same thing, rises a
Second, it is only by means of a Hcence that this can be
effected. For this interval of the Seventh owes its origin
more to good taste than to nature, since it is not found
among the sounds arising from the division of a string ; it is
this interval of the Seventh which gives rise to such a licence,"
and, consequently, to the Interrupted Cadence. ^

The Irregular Cadence. — In the Perfect Cadence the
progression is from Dominant to Tonic ; in the Irregular
Cadence it. is from Tonic to Dominant. As in the other
two cadences, the first chord will be dissonant ; but the
dissonance in the Irregular Cadence consists, not in the
Seventh added to the " perfect " chord, but in the Sixth
added (chord of the "Added Sixth"). This Sixth, it is true,
is consonant with the bass, but it forms a dissonance with the
Fifth of the chord. Unlike the Seventh, it resolves upwards ;
it has therefore an irregidar resolution. But, in common with
the Seventh, it resolves on the Third of the following chord.
The Irregular Cadence frequently occurs also in the progression
from Subdominant to Tonic. ^ The Subdominant should, in
fact, naturally bear the chord of the Fifth and Sixth (Ji).^

But, Rameau proceeds, " the Sixth added to the first
chord in this cadence is a supernumerary sound, sanctioned
only by good taste " [!] It does not, therefore, determine the
progression of the bass. The chord which it forms is not
a fundamental chord ; that is, it cannot be regarded as
being derived from a chord of the Seventh (first inversion of
the Supertonic Seventh), because its resolution is different.'*
Nevertheless, in the Supplement to the Traite, Rameau is of
opinion that the chord of the Added Sixth must actually be
regarded as a fundamental and original chord, although
he cannot well reconcile this with his former statement
that all fundamental chords consist of a series of Thirds.
In short, Rameau contends that if the chord of J.; on
the Subdominant resolves on the Dominant harmony, it

1 Traite, Book II., Ch. 6.

- In all his subsequent works, Rameau treats the chord of the
" Added Sixth " exclusivel}- as a Subdominant discord.
3 Traits, Book 11., Ch. 7. * Ibid.. Ch. 17, Art. 3.

RAMEAU'S TRAITE DE L' HARMON IE 113

is the chord of the " grande Sixte," the first inversion of
the chord of the Seventh on the Supcrtonic ; if, on the
other hand, it resolves on the Tonic harmony, as in the
" irregular " cadence, it is the chord of the " Added Sixth,"
an original chord, and not derived from any other chord.
In the first case the dissonant note is the Fifth (the
Seventh of the fundamental chord) ; in the second case the
dissonant note is the Sixth, and the fundamental note of
the chord is in the bass. Thus in the chord of the " grande
Sixte " f-a-c-d (the first inversion of the chord of the Seventh
d-f-a-c), the dissonant note is c, and the fundamental note
d ; but in the chord of the " Added Sixth " f-a-c d, the
dissonant note is d, and the fundamental note /. Rameau's
theory of the chord of the " Added Sixth " has fared rather
badly at the hands of some of his successors, who have
described as the chord of the " Added Sixth " what Rameau
expressly stated was not such. Even in our own day there
are theorists who have explained the chord f-a-c-d, as a
" Dominant Discord," as the inversion of the chord d-f-a-c,
but who nevertheless have given to it the name of the chord
of the " Added Sixth." Rameau, on the contrary, insists that
the chord of the " Added Sixth" consists of a major Sixth added
above the Subdominant harmony ; that it is a Subdominant,
not a Dominant discord. Unless this view as to the origin
of the chord of the "Added Sixth" be accepted, it is obviously
incorrect to describe it as such : the name becomes altogether
meaningless.

The theoretical importance of this chord, and the re-
markable theoretical acumen evinced by Rameau in dealing
with it, will soon be commented upon : in the meantime,
only a passing notice need be taken of the gross contradictions
in which Rameau finds himself involved in his attempts to
explain the chord of the "Added Sixth" as original and
fundamental. He has maintained that there is but one
dissonance, that of the Seventh, and that the Second is
derived from the Seventh by inversion : he now concludes
that the Seventh and the Second are really the same :
" they are all one ; the dissonance arises from the division
of the major Third." ^

He is now satisfied that he has accounted for the two

1 Supplement to Traite.

114

THE THEORY OF HARMONY

most characteristic discords of the key-system. One is a
Dominant cUscord (a), and is obtained by adding a Seventh
above the Dominant harmony ; the other is a Subdominant
discord {b), and is obtained by adding a Sixth above the
Subdominant harmony -. —

zas-

Rameau's principal object, however, in adding a Sixth
above the Subdominant chord is to give to this chord
a determined progression, to make its resolution on
the Tonic chord an absolute necessity. But he has
himself informed us that this chord may present a
two-fold aspect ; it may be considered not only as a chord
of the " added Sixth," but as the first inversion of the chord of
the Seventh on the Supertonic. In reaUty, the effect of this
added Sixth is to accentuate the tendency of the Subdominant
harmony, not towards the harmony of the Tonic, but towards
that of the Dominant. Of the two resolutions of the chord
f-a-c-d at (a) and (&), of which the first is Rameau's example
of the " Irregular Cadence," it cannot be said that the second
is any less " natural " than the first ; on the contrary, the
second may be regarded as the more " natural " resolution.
Thus Rameau, instead of giving to this chord a determined
progression, only succeeds in defeating his own object : —

(a)

(^)

-e>

iSi

HS<-

-G>-

-e*-

1221

It is by means of the three Cadences that the treatment of
all dissonant chords is determined. Rameau indeed is of
opinion that all harmony is nothing else than a succession
of cadences.i xhe cadences, then, prove afresh " that there

1 Traite, Book III., Ch. 27.

RAMEAU'S TRAITE DE L' HARMON IE us

are but two chords whicli are essential and fundamental,
namely, the Perfect chord and the chord of the Seventh ;
and that all the rules of harmony are based on the progressions
natural to these two chords." ^ It is " from the Perfect
Cadence that the principal and fundamental rules of
harmony are derived."

Further, it is by means of the Cadence that the key is
determined. The Irregular as well as the Perfect Cadence
may serve to determine the key. In this respect however
the Irregular Cadence is less definite than the Perfect.
Indeed it is only when the major and minor dissonances
are heard together and resolved as in the Perfect Cadence
that the key can be said to be properly fixed.- In the Perfect
Cadence we find all the notes of the scale except the " sixth
note " (submediant) ; this sixth note however appears in
the Irregular Cadence.^

Again, it is by means of the Perfect Cadence that we are
able to modulate into other keys. " We cannot proceed
naturally from one key to another except by a consonant
interval, so that, after beginning a piece in a certain key,
we may modulate into another that is a 3rd, 4th, 5th or 6th
above or below " the original key-note or Tonic. ^ In other
words, Rameau . considers that those keys are related whose
tonics are consonant with each other. He is of opinion
that, for example, the keys of E major with four sharps,
and A[7 major with four flats, are more closely connected
with the key of C major than is D major, which has only
two sharps, or Bj? major which has only two flats in the
key-signature. In this question of key-relationship Rameau
proves himself to be far in advance of his time. Few, if
any, of his contemporaries were of opinion that the keys
of E and A[7 major were closely related to C major. Even
a whole century later, such views were by no means
prevalent among orthodox theorists. Hence, when Beethoven
introduced in some of his works in Sonata form the
second subject in the key of the mediant major,^ tlieorists
were considerablv embarrassed in order to account for

1 Traitc, Book II , Ch. 5. 2 ji^^^^ Book III., Ch. 14.

3 Ibid., Book II., Ch. 21. * Ibid., Book III., Ch. 23.

■"' See, for example, the first movement of his Pianoforte Sonata
in C major Op. 53 (the Waldsteiii).

ii6 THE IHEORV OF HARMONY

such an innovation. It is only since Beethoven's time
that the old rules applying to the relationship of keys
have been found to be inadequate. The teaching of our
present-day theorists, namely, that those keys are related
whose tonics are consonant with each other, has in fact
been necessitated by the practice of the great composers.
But it is a noteworthy fact that these views were first
enunciated by Rameau, who had no such advantages of
experience, but who based his conclusions mainly on
theoretical grounds, at the beginning of the eighteenth
century.

Nature and Functions of Chords : Determination of
" Key " : Necessity for Dissonance in Music.

In the course of the second, third, and fourth books of the
Traits, and especially in treating of the use of the Fundamental
Bass in composition, Rameau throws out a number of observa-
tions respecting the nature and functions of chords, which
raise questions of the utmost importance for the theory of
harmony. Rameau is of opinion that composition by means
of the Fundamental Bass is an easy matter. " We might
speak," he says, " of the experience of several persons, who,
by means of the fundamental bass, and after reading through
our rules once or twice, have composed a harmony as perfect
as one could wish." He lays down a principle which has
since been almost universally followed in text-books of
harmony, namely that the learner should from the outset
write his exercises in four-part harmony, for it is only in this
way that harmony can be properly taught. " Zarlino has
said on the subject of four-part harmony that it can scarcely
be taught on paper, and that he leaves it to the discretion
of composers, who should be guided by the rules given for
composition in two or three parts. On the contrary, harmony
can be properly taught only in four parts, in which all
particulars are comprised in two chords ; it is then easy
to reduce these four parts to three or two parts." ^

1 Traite, Book II., Ch. 19. Here again it is ev'ident that Rameau was
little aware of the change which the art of music had undergone since
the time of Zarhno. It is surprising that Rameau, good contrapuntist

RAMEAU'S TRAITE DE L'HARMONIE 117

As to the kind of harmony which ought to be assigned to
each note of the Fundamental Bass, that is, the species of
chord proper to each degree of the scale, Ramcau directs
that the " perfect " chord should be placed only on the
Tonic. He states further that every note which bears the
" perfect " chord must be regarded as a Tonic. ^ The reason
for this is that " the consonant progression of a fundamental
l^ass, above which only ' perfect ' [therefore consonant]
chords are heard, presents to us as many different keys as
there are sounds in this bass. ... It is certain that every
sound above which the ' perfect ' chord is heard conveys to
the mind the impression of its key."- Rameau however
finds himself obUged to modify, and indeed to contradict
this statement, and to admit that other degrees of the scale
besides the Tonic may bear the " perfect " chord. He first
concedes that this chord may appear on the Dominant, ^ and
afterwards allows the same liberty to the Subdominant. •
"The 'perfect' chord may be given only to the key-note,
its Fifth, and its Fourth."^ A Uttle later he makes a statement
in which he appears to include other notes of the scale besides
the three already mentioned. ' ' The ' perfect ' chord," he says,
" may be taken in a diatonic progression of the bass " (as
for example in the Deceptive or Interrupted Cadence). It
is in fact impossible to discover precisely on which degrees
of the scale Rameau considered that the " perfect " chord
should be placed. If he says expressly in one place that the
harmony should be that of the " perfect " chord, he says
no less positively in another place that the chord of the
Seventh should be taken. But it is not difficult to understand
the reasons for his perplexity.

though he was, did not perceive that ZarHno's rules appHed to counter-
point, not harmonv, in the modern sense of the term, and that he did
not distinguish between the two forms of composition. As Rameau
maintained that melody has its origin in harmony, we must infer that
he also considered harmony to be the basis of counterpoint, and that
an acquaintance with the rules of harmony, as well as a certain
degree of facility in harmonic composition, should precede the study
of counterpoint.

^ Traitc, Book III., Chs. 23 and 26 ; Book IV., Ch. 7, etc.

- Ibid., Book II., Ch. 22.

3 Ibid.. Book II., Ch. 21.

* Ibid.. Book III., Ch. iS.

Ii8

THE THEORY OF HARMONY

His version of the " rule of the octave " is as follows : —
The Tonic takes the " perfect " chord.
The second degree^ — the second inversion of the Dominant

Seventh chord.

The Mediant

— the first inversion of the Tonic
chord : " the Mediant always
represents the principal note — the
Tonic."

The Fourth degree^ — the chord of the " grande sixte "

(f ) when it rises a degree ; and
the last inversion of the Dominant
Seventh chord when it falls a degree.

The Dominant — the " perfect " chord : or the chord

of the Dominant Seventh.

The Sixth degree ^ — the chord of the sixth, in rising a

degree ; and the chord of f
(" petite sixte ") in falling a degree.

The Leading note (" note sensitive ") — the first inversion

of the Dominant Seventh Chord in
rising a degree : and the chord of
the sixth in falling a degree.
That is : —

-e^

,€?-

-(^5>-

-f^^h-

-e^

-iS»-

~Z2~

-G>,

'«2

-s»-

— I — f-_jf-

-<S-

251

~r5~

-Gt-

221

4
3

6
5

~o —

_Q_

231

-C5>-

:g-

-o-

-«s»-

-(£^-

-O-

'j,Gh-

-iS>-

~Q'

-O-

^ These are the terms employed by Rameau in the TraitL

RAMEAU'S TRAIT E DH L'HARMOXIH

I 19

Rameau however apparently considered the harmony he
assigns to the descending sixth degree to be stiff and unnatural
in its effect, for later (Book III., Ch. 11) he changes Flq to F:{f
so that the harmon\- of this section of the scale appears in the
key of the Dominant : —

«agj

-Q_

--%^--

This alteration he also made use of in his later works.
It will be observed that he permits the " perfect "
chord to appear not only oa the Tonic but on the
Dominant as well. He does not make mention of the
circumstance which, according to some theorists, gave to the
" rule of the octave," as practised not only in France, but
also in Italy, Germ my, and England at the latter part of
the seventeenth and beginning of the eighteenth centuries ^
its real import for the science of harmony ; the circumstance,
namely, that all the harmonies comprised in it consist either
of those of the Tonic, Dominant, and Subdominant, or may
be considered to be derived from these three. The awkward
necessity of being obliged to borrow, from the Dominant
key, a harmony appropriate for the descending sixth degree

^ The versions of the " rule of the octave " given by Gasparini
{L'armonico pratico al cimbalo) — which was indeed that constantly
in use in the Italian school of violinists of which Corelli was chief, — and
by Mattheson {Kleine Generalbass Schule) are substantially the same
as that of Rameau.

Gasparini.

:^~-r,

~g?~

22:

-<s>-

,0-

-&-

$6 6 6 I #6 $ 6 6-^6

P %

221

-<s»-

22:

-Gt-

-Gh-

-^?-

# 6
*2

6 #6

I20

THE THEORY OF HARMONY

may have proved a difficulty. ^ But it was in his Xouveau
Systeme de Musiqiie Thcorique that Rameau first grasped
the theoretical significance of the Subdominant, and gave
this name to the fourth degree of the scale.

To the important chord of the Dominant Seventh Rameau
gives the name of " Dominant-Tonic ," seeing that this chord
is most naturally followed by that of the Tonic. To the
other chords of the Seventh he gives the name of " Dominants,"
as they require for their resolution that the Fundamental
Bass should descend a Fifth, as in the Dominant-Tonic chord.
As now in composition we ought to prefer those progressions
of the Fundamental Bass which are most perfect, that is.

^ ^ -^ -^^ -^ <^

Matheson.

iri\ •T

r^ ^^

t-^

l<^-^*-lr> <-3

•^ <> V^

V^^ll jt o

IT II

6
4
3

6 6 6 6 16
5 5 4

r3 tf<^ 1'^' -^^<S'-7,o

4
2

6 6

(^ra>*

LW ., >3

~m —

C-^

^^ ''^ r-j

^w--'> <--'

: ^ 1

? 6

Iff 54 6 it6

46 6 6

*4 5

9,
Keller (Rules for playing a thorough bass) gives the ascending form
of the major scale thus : —

-• *-

6 6 6 6

In the descending form of the scale, he prefers like Mattheson,
Gasparini, ana Rameau, to make a modulation to the Dominant key : —

:l=t

6 ^6

^ In his later works, Rameau's difficulties in connection with the
logical and systematic harmonization of this section of the scale
increased rather than diminished, and gave rise to his well-known
doctrine of the " double employment of dissonance."

RAMEAUS TRAITE DE LHARMOXIE

I 2 I

those which proceed by the interval of a Fifth descending
or Fourth ascending, we obtain by this means a series of
harmonies which are closely linked together. " Commencing
with the Tonic chord," he remarks, " we can quit this only
by passing to another chord connected with it, and so on,
b}' means of a linked succession of harmonies, we are finally
obhged to return to the Tonic chord, and to conclude " ;^
as for example : —

IC2I

JZC

=s=

_C2I

22;

-<&>-

-<S>-

'/zr

7 7 7

So that " harmonic succession is nothing but a connected
series {enchainement) of Tonics and Dominants." - It is
indeed little else than a succession of cadences, in which,
the Fundamental Bass descending a Fifth or ascending a
Fourth, we find an imitation of the Perfect Cadence. The
effect of the cadence may ne^•ertheless be avoided by adding
a Seventh to the second chord ; dissonance then destroys
the effect of repose, and impels the harmony onwards, for
every dissonant chord urgently demands resolution. " Every
Dominant chord," he says, " should resolve in the same way
as the Dominant-Tonic chord, the fundamental bass rising
a Fourth or falling a Fifth. This progression represents
a species of Perfect Cadence. The Perfect Cadence, although
imitated in this wav, should however be avoided by adding
a Third [Seventh] above the second chord forming this
cadential progression, thus making the second chord a
Dominant in its turn, e.g- : — ■

-CS-

instead of

-fe»-

:g=i

zaz

1 Traite, Book II., Ch. 22.

2 //,/(/., Book III.,Ch. 27,

122 THE THEORY OF HARMONY

This is necessary, as the ' perfect ' chord should be heard
only on the Tonic." ^

The manner in which this works out in practice may be
seen from the following example, taken from the same section
of the Traite [De la maniere d'eviter les Cadences, en les
imitant), and which is surely as heavy and inflexible a
piece of music as one could well devise ; —

8 O^ I g r:; f—G> V=^-~

_i2

^2;

^S^

-s>-

-&*-

izo:

-<s>-

Continuo.

-o-

-<&-

7 7 7

Fundamental Bass.

i?7

C3-

-^>-

"?3"

6(!)

7 i?'

■Gh-

-^GL

?Q-

"g" Q^~

^— ?^-

"C?"

-O-

-S»-

-<s»- -<&>- -,

:^2z

-<s>-

'3ZS1

M5>-

-?€^

231

When we compare this with his music for the clavecin,
or his operatic music, both of which are distinguished by the
greatest clearness, directness, and refinement of the harmony,
Rameau appears to us like a doctor who has not the courage,
or is too wise, to follow his own prescriptions.

1 Traite, Book II., Ch. 9.

RAMEAU'S TRAITH DH L'HARMONIE 133

It is not difficult to understand why Ramcau regards
the " perfect " chord, wherever found, as a Tonic chord,
and why, although he finds himself obliged in his " rule of
the octave " to place this chord not only on the Tonic but
on the Dominant and Subdominant, and although in his own
music for the stage he makes use of the " perfect " chord
not only on these three degrees of the scale, but on other
degrees as well, he nevertheless persists in asserting repeatedly
throughout the Traitc that the " perfect " chord should
be placed only on the Tonic ; and this notwithstanding
the manifest contradiction involved, and the obvious im-
possibility of reconciling his principles with his practice.
In the first place, Rameau stood much nearer than musicians
of our day to the polyphonic music of the Church composers,
in which successions of consonant harmonies were frequently
used in such a way that the key, in our sense of the term,
was quite undetermined.

Take for example the following passage, selected at
random from a work by Palestrina : —

ill

V2-

-C^- -O*.

:!2i

^AJ.

-<^-

-&^-

-G>-

-S-

-m» <S»-

-(S-

-(S>--»--(^-

— I— j-*-^-*-

r:> ■

0 '

-P

1
4

1
1

-•— 1

1 1

^^V-

~^=^

etc.

Rameau's meaning then, when he states that such a series
of harmonies represent so many Tonics, is clear.

In the second place, if it be true that every succession of
consonant harmonies represents as many Tonics as there are
sounds in the Fundamental Bass, then such a fact supplies
Rameau with a reason for the use which is made in music of

124 THE THEORY OF HARMONY

dissonance, and of dissonant chords of the Seventh. Both in
the Traite and in his later works, Rameau makes his position
here perfectly clear. Without the use of dissonance, he tells
us, and of dissonant chords, the key cannot properly be
determined. Thus a Dominant-Tonic succession, in which
both harmonies are consonant, does not constitute a real
Perfect Cadence, that is, one which properly determines the
key ; for " if two ' perfect ' chords follow one another in a
Perfect Cadence, one is unable to judge which of these chords
is the true chord of repose." ^ It is dissonance then that
determines the key, and it is dissonance that obliges the
chord of the Dominant Seventh to resolve on the Tonic chord.
" As soon as the leading-note appears in a dissonant chord
it is certain that it determines a conclusion of melody, and
therefore it must be followed by the " perfect " chord upon
the key-note ; whereas if the leading-note does not appear
in a dissonant chord, the conclusion is not determined."^
It is only, in fact, when the leading-note appears as a dissonant
note, and as a constituent of the chord of the Dominant
Seventh, that it has the power to determine the key ; its
leading quality is due to this circumstance. " The major
dissonance can never appear without the minor. "^ If the minor
dissonance (the Seventh of the chord of the Dominant
Seventh) is not actually present in the first chord of the
Perfect Cadence, it must be understood.* Dissonance, then,
and the necessity for its resolution, determines the Perfect
Cadence, and consequently the key. In taking up this
position, Rameau appears to have completely forgotten his
former definition of the Perfect Cadence.^

Here we are at the source of the doctrine of Tonality so
vigorously propounded by Fetis, to whom the works of
Rameau were known, and who considered that the tonahty
of our modern music has been determined by the necessity
for resolving the two dissonant notes (the Third and Seventh)
of the chord of the Dominant Seventh ; and of the theory,
also so widely disseminated by Fetis, that the revolution which
marked the change from the old to our present harmonic
art was brought about by Monteverde, who is supposed to
have first made use of the chord of the Dominant Seventh.

1 See p. III. 2 Traiie, Book III., Ch. 14. » Ibid., Book IV., Ch 15.
* Book III., Ch. 13. 5 See p. 105.

RAMEAU'S TRAIT E DE E HARMON! E

I 2

It is in the chord of the Dominant Seventh that we find
both dissonances, the major dissonance, which rises a semitone,
and the minor, which falls a semitone or tone. Rameau
repeatedly dwells on this fact, as though it contained for the
theory of harmony some hidden significance the full import
of which he is unable completely to fathom. He gives
various examples of this harmonic progression in which he
points to the natural, almost irresistible tendencies of the
dissonant notes of the Dominant Se\'enth chord, one of which
is impelled upwards, the other do\\-nwards, a degree : —

-&> —

=g:

--&-

~rjr

Not onl3' does Rameau la\' great stress on this fact, he tries
to discover a reason for it. He thinks that the old rules of
the contrapuntists regarding the progression of intervals
furnish him with a solution of the problem. He quotes the
rule given by Zarlino to the effect that ever}- major inter\-al
should be followed by one which is greater, and every minor
interval bv one which is less,^ thus : —

therefore in the chord of the Dominant Seventh, as g-b-d-f. b
laturally rises, while / falls a semitone. What is true of
major and minor inter\'als is true also of augmented and
diminished intervals ; therefore the Tritone f-b, which
includes both the dissonant notes of the Dominant
Seventh chord, should be followed by the minor sixth e-c;

1 Traite, Book II., Ch. 5.

I 26

THE THEORY OF HARMONY

while its inversion, the diminished Fifth, should be
followed by the major Third c-e : —

ZZ21

— a(S>-

^g:

This explanation, if it be an explanation, of course does
not suffice. For the Tritone may resolve in other ways, as,
for example : —

-s>>-

'<r>

:z2z

Here it is followed by the perfect Fifth, which is a much more
consonant interval than the minor Sixth, and which ought to
be considered to provide a much more perfect resolution.
So also Rameau's explanation supplies no reason why the
chord of the Dominant Seventh should not resolve quite
naturally on other chords besides that of the Tonic : —

~rD-

~ry-

=«&>-

-^■

-e>-

zac

-o-

~f~S'

--^-

_C2_

Such being the case, it is evident that it is not the chance
addition of a dissonant note above the Dominant harmony
which is the cause of the tendency it undoubtedly has towards
the harmony of the Tonic. It should also be remembered that
Rameau is quite unable to give any adequate explanation
as to the principle which should govern the formation of
dissonant chords. For the Dominant discord he adds a
Seventh ; for the Subdominant discord a Sixth, above the
"perfect" chord; but he is unable to tell us why the dissonance
of the Dominant, as well as of the Subdominant discord, should
not be that of the Added Sixth. As matters stand, the fourth
degree of the scale appears to have strayed by chance into \
a harmony with which it has nothing to do, and of which it
forms no part.

RAMEAU'S TRAITE DE L'HARMONIE

I 27

In arguing as he does, Rameau does not appear to
observe that he is bent on destroj'ing his system of
the Fundamental Bass. It is no longer the Fifth pro-
gression of the Fundamental Bass that gives to the
Dominant-Tonic harmonic succession its " perfection " ;
the origin e\'en of such a succession is to be found
in Dissonance and the necessit}^ for its resolution ; and
if Rameau is still prepared to maintain that the cadence
in question is more " perfect " than any other harmonic
progression, then this must be owing to some undefined
quahty in the dissonance which necessitates the chord of
the Dominant Seventh being succeeded by that of the
Tonic. Xor does he notice that he puts himself in
dangerous contradiction with his cherished principle that
melody has its origin in harmonj', that it is harmony
which determines the notes of the scale, and not the
notes of the scale nor the melodic tendencies, real or
imagined, of such notes, which determine harmony "and
harmonic succession.

Such then, according to Rameau, is the explanation of the
necessity for Dissonance in music. Consonance is the
attribute of the Tonic alone ; only the Tonic ma}' bear
a consonant harmon\'. In every other case the chord
should be dissonant ; and where, in such cases, the
dissonance is not actually present in the chord, it must
nevertheless be understood. In the following : —

-€J-

-&--

-Q-

-&-

~ry

jc2-

Rameau would regard the Dominant triads as in reality
representing chords of the Dominant Seventh ; while the
Subdominant chord he would consider to represent the
first inversion of the chord of the Seventh on the Super-
tonic (chord of " grande sixtc "). He not only repeatedly
states that every chord ex'cept that of the Tonic must be

128

THE THEORY OF HARMONY

regarded as dissonant, whether the dissonance is actually
present or not, but he also remarks : "In composition of
two or three parts, it frequently happens that only the
consonant notes of a dissonant chord are actually heard ; . . .
we have already said that a consonant chord can appear
only after a discord in which the leading note is present "
(that is, in the Perfect Cadence), " for otherwise we pass
from one discord to another, as appears from our rules
of the Seventh ; and this is a Kttle difficult to discover
in compositions of two or three parts." ^

In fact, every chord except that of the Tonic is or
represents a chord of the Seventh. To test the truth of
this statement it is only necessary to add the dissonant
Seventh to all the chords of the above example, except
that of the Tonic. The result can hardly be said to
improve matters : —

? — r3

r-^^

^—

-^W-

-H-

''^

-B— ^

^ -g- "

Jfi^'

r^ ^-^

(W.

Vi^

In the same waj^ if the following successions of harmonies : —

:x2:

-^-

-€>-

221

rjC2i

"p~

iS:

-e>-

-Gt-

really represent chords of the Seventh, we immediately find
ourselves in a sea of difficulties. According to Rameau, we
ought to understand the passage as at {a) ; or as at (&)

1 Traite, Book III., Ch. i8.

RAiMEAU'S TRAITE DE LHARMONIE

1 29

where the chords are Hnked together as closely as
possible :■ —

(«)

(«>)

C-^

fj 1

»5> ^-.

' '

Jl{

<5 r^

fff\ — _

fn CP

v^ '-'

•5 /^

r:i

•^ H

-f^-

-^^

1-^

//V^•

^~^

-^ r^

— &

— __

— s,

But here, as formerly, the Seventh cannot be added without
flagrantly transgressing Rameau's own rules for the prepara-
tion and resolution of discords. The simple successions of
harmonies, which in themselves are easily understood
by the ear, are made unrecognizable and unintelHgible.
But Rameau himself, as we have 'seen, found it impossible to
apply this part of his theory to practice, and was obliged
to admit that the Dominant, Subdominant, and indeed other
degrees of the scale as well as the Tonic may bear a consonant
harmony.

Yet Rameau, in dealing with this subject, proves himself
to be possessed of an extraordinarily fine ear, and of
a keen musical perception. He clearly perceives that the
" perfect " chord in itself has no tendency one way or
another ; it is consonant, in a state of rest, and such is the
impression which it conveys to the mind. He also perceives
that it is only when this chord appears on the Tonic that the
impression of complete repose is produced. It is onh' then,
and especially when it is the second chord of the Perfect
Cadence, that the ear is fully satisfied, and desires nothing
more to follow. On the other hand, when the " perfect "
chord is not apprehended as a Tonic-chord, as a harmonic
centre, it does not produce, like the Tonic harmony, the
feeling of complete repose ; the mind is not satisfied, but
desires and expects some other harmony to follow ; thus we
are obliged to proceed from chord to chord until we
again reach the Tonic chord, and attain a satisfactory
conclusion.

It is not surprising that Rameau should be unable to give
a clear presentation of this fact, nor that he should give the

130 THE THEORY OF HARMONY

wrong explanation of it. ^^'hat is surprising is that Rameau,
the first to grapple, in any real sense, with the mysteries
of our hannonic system, should have been the first to bring
to light facts of such importance for the science of harmony.
Of the part played by Rhythm in music, Rameau does not
treat to any considerable extent, although his remarks on the
subject show that he realizes its importance. " So great is the
influence of Rhythm in music," he says, " that it alone is
capable of exciting in us the various passions w^hich we
generally consider to demand, for their expression, the use of
harmony." ^ He points to the fact that the Cadence depends
for its effect on Rhythm as well as harmony, but he does not
attempt to explain w^hy this should be so. Noteworthy is
the remark that the principle of Rhythm, or Metre {Mesure)
in which the numbers two, three, and four are made use of,
is the same as that of harmony.^ This is the root-idea from
wliich ]\Ioritz Hauptmann has evolved his metrical system
in his important work Harmonik und Metrik (1853).

Melody has its Origin in Harmony : The Nature and
Constitution of the Scale.

" It would at first seem," says Rameau, " that harmony
has its origin in melody ; that it was the result of the union
of melody notes produced by the different voices. But it
is necessary to determine the course which each of these
voices must follow in order that the\^ ma}^ accord together ;
and this can onty be effected by means of the rules of harmony.
It is therefore Harmony, not ]\Ielody, which guides us."^
Again, it is the fundamental principle of hamiony, the arith-
metical division of a string, which suppUes the essential
notes of the Mode or scale, that is the Tonic, Mediant,
and Dominant. It is true that the intervals thus obtained
do not suffice ; for the scale and for melody smaller intervals
are necessary. But these smaller intervals, which are the
tone and semitone, although they cannot be directly derived

1 TraiU, Book II., Ch. 23. ^ /;,,j _ q^i 23.

' Ibid., Chs. 19 and 21.

RAMEAU'S TRAITE DE UHARMONIE 131

from such a principle, are nevertheless derived indirectly,
by means of the progression of the Fundamental Bass. Thus
the two Cadences, the Perfect and the Irregular Cadences,
furnish us with all the notes necessary for the formation of
the scale. Therefore, concludes Rameau, it is evident that
Melody has its origin in Harmony.

He is nevertheless quite well aware of the fact that purely
melodic music was for many centuries the only kind of music
in existence, that it was extensively practised long before
composers began to make use of hannonic music, and that
the melodic music in use among the Greeks, the Romans, and
in the early Church was based on a well-defined system of
Modes or scales, which like our modem modes were diatonic,
consisting of five tones and two semitones. This fact causes
him considerable uneasiness. " The ancients," ^ he says,
" have defined perfectly well the properties of the ]\Iodes,
to which they subjected both harmony and melody, of the
nature of which however they were ignorant. The effect
of all melody, they considered, depended on these Modes.
Why then should they have altered them, Especially in the
Cadences, and tried to iinitate the notes of our perfect
system. 2 According to the ancients, there was no sub-
semitone in the modes on Re, Mi, Sol and La [that is, the
Dorian, Phrygian, Mixotydian and ^Eolian modes], and
j-et they considered that it was necessary for a proper Cadence
to alter these modes chromatically so as to obtain this sub-
semitone [leading note !]. If they had paid some respect
to harmony, they would not have fallen into errors so gross.
Thus we see how vainly people \\dthout taste, fuU of the rules
of the ancients, of which the true meaning is unknown to
them, attempt to furnish a good and agreeable harmony to
different kinds of Plain Chant. . . . The blindness of
these old musicians is apparent also from the manner
in which they divided their modes into Authentic or
principal, and Plagal or collateral. The harmonic and
arithmetical proportions ought to be apphed only to harmony,
not to melody. ... If Zarlino had been of the same
mind as Plato, who as he himself tells us considered

1 Rameau is here referring to composers and theorists of and before
the time of Zarhno.

- That is, our major and minor modes.

132 THE THEORY OF HARMONY

that melody has its origin in harmony, he would have
sought for the foundations of modulation [scale, mode] in
this harmony." ^

It is evident that the music of the " ancients " is for
Rameau a source of the greatest perplexity. He cannot
understand it : it refuses to accommodate itself to the rules
of his Fundamental Bass. But although Rameau finds
himself at fault with regard to the music of the polyphonic
period prior to the 17th century, it by no means follows
that the harmony of polyphony had its source in melody.
Rameau furnishes a strong argument against such a view
when he points to the fact that the melody of the polyphony
of this time was not arbitrary, for it was necessary to shape
the various melodies in such a way that they should har-
monize together. It is frequently stated that the music of
early polyphony ought to be considered from a horizontal
point of view, that is, as so many voice-parts or melodies
woven together so as to produce harmony. We of the
present day, on the contrary, regard music from a vertical
point of view. There is much truth in this, if it be taken
to mean that polyphony is the art of combining melodies,
of fitting them together in such a way as to produce harmony,
and not that the harmony is the result of the fortuitous
concurrence of the various melodic parts. Rameau does not
in his Traite enter into further particulars as to the nature
and origin of the scale, whether major or minor. He does
not treat of the Minor Scale as an independent scale, but
relates it to the major ; the rules for the Major Mode, he tells
us, are appUcable also to the minor. So also one is at liberty,
if the expression demands it, to substitute for a Major Mode
its Tonic minor, " as frequently happens in Chaconnes and
Passacaglias."

So then, we are asked to observe, we find that all harmony,
whether major, minor," consonant or dissonant, all scales,
modulation and melody, the Fundamental Bass and the
rules for harmonic succession, the Cadences, the Major
and Minor Modes — " all that is necessary for a good and
agreeable harmony " are " derived from our fundamental
principle which is based on the first divisions of a
string " ; a somewhat large claim \\-hich, as we have
. • ■ ^

2 Traite, Book II., Ch. 21,

RAMEAU'S TRAITE DE UHARMONIE

' jj

seen, Rameau is far from being able to establish ; although,
in his laborious researches, he has succeeded in bringing
to the light results of real and lasting value for the science
of harmony.

In 1726, four years after the appearance of the Traite de
I'hannonie, Rameau pubUshed the Notiveau Systeme de
Mnsique Theoriqtte, written " in order to serve as an intro-
duction " to the former work. We therefore find again,
although in a much more concise form, the main outlines of
the theoretical principles already set forth in the Traite.
But Rameau's ideas on the subject of harmony have in
the interval undergone some development.

134 THE THEORY OF HARMONY

PART II.

CHAPTER V.

ra:\ieau's xouveau systeme de musique
th^orique (1726).

From the Preface to liis Nouveau Systeme we learn that
during the time which has elapsed since the publication of
the Traiie, Rameau has made himself acquainted with the
results of the acoustical researches of ]\Iersenne and Sauveur/
especially as regards the natural resonance of sonorous bodies
(harmonics, partial-tones) . He has disco\'ered that the sounds
arising from the first divisions of a string, which have furnished
him with his fundamental principle of harmony, are actually
present in the string during its vibration, and may actually
be heard ; that the string or other sonorous body not only
vibrates in its totahty, that is, throughout its whole length,
but in librating jiaturally divides itself into sections,
(segments), which vibrate independently ; these sections
corresponding exactly wdth those resulting from the harmonic
division (i, |, ^, J, ^, |, etc.) of the sonorous body.

This is for Rameau a wonderful fact, as in truth it is.
Having followed him thus far in his operations, we can well
appreciate with how much force such an acoustical phenomenon
must ha\'e impressed itself upon him. He had toiled with
" the help of mathematics," " by reason alone," to discover
the natural principles of harmony, and to prove that

1 Joseph Sauveur (1653-1716), a distinguished French acoustician
and mathematician, was one of the first (although deaf from his birth)
to investigate the phenomena of partial tones. His works include :
Principes d' acoustique et de miisiqiie (1700) ; Application des sons
harmoniqiies a la composition des jenx d'orgiie (1702) ; Rapports des
sons des cordes d' instruments de musique atix fleches des cordes, et
nouvelles determinations des sons fixes {171 3) ; also other works treating
of systems of temperament.

RAMEAU'S NOUVEAU SYSTEME

harmony has a natural basis, while all the time a greater
artificer had been at work before him ! What a
revelation ! Harmony is the direct gift of Nature. From
this sounding body, which breaks itself up into sections,
there proceeds " the most perfect harmony " of which the
mind can conceive. This now becomes the fact of primar\'
importance for Rameau. From this time onwards the nature
of the sonorous body is the theme which is hardly ever
absent from his lips. It becomes his battle-cry in the contro-
versies with Rousseau, d'Alembert, and others of the French
" Encyclopaedists," in which he was shortly to find himself
engaged. He champions it as the key to the theory of
harmony. That a musical sound is not simple, but composite,
and that in a well developed musical sound we hear not onlv
the primary fundamental tone, but other secondary tones,
which unite with it, and together form the " perfect harmony "
— Zarlino's Harmonia Perfetta — the harmon}'' of Nature, that
it is in this natural phenomenon that the whole art of
music and of harmony ha\'e their origin, and from which they
take their development — on these things Rameau lays the
greatest possible stress, and dwells on them with almost
tiresome iteration.

Rameau is overjoyed at what he considers to be such a
striking confirmation, given by Nature itself, of the correct-
ness of his theoretical principles, and of the Fundamental
Bass. " How can one fail to be convinced," he says, " of
the truth of the Fundaritental Bass, seeing that the three
fundamental sounds from which it is developed naturally
arise not only from the divisions of a string, but are found in
a [musical] sound of the human voice itself. . . . There is
indeed within us a germ or principle of harmony. Harmony
may be said to be natural to us ; and in the fact that these
three sounds are heard in a sound produced by the voice the
author has found the strongest possible confirmation that
here was the true principle of the Fundamental Bass, whose
discovery he owes to experience alone." ^

A considerable part of the Nouvean Systeme is occupied by
Rameau in citing every acoustical phenomenon which he
imagines can in any way serve to demonstrate the correctness
of his theories. He lays greatest stress however on the

^ Preface.

136 THE THEORY OF HARMONY

fact that the three sounds which together form the perfect
chord (that is, a fundamental sound together with its upper
partial tones of the Twelfth and Seventeenth), must be
regarded as a product of Nature. These sounds may be
distinguished not only in the tones produced bv the strings
of the \'ioloncello, but may also be heard " in cymbals, and
in the lowest sound of the trumpet, in bells, etc." He points
also to sympathetic vibration as a proof of the close relation-
ship which exists between the three sounds in question.
" Those," he says, " who are unwilling to trust their ears
may however accept the evidence of their eyes. For if
one takes three strings of the violoncello, and tunes one of
them a twelfth and the other a major seventeenth above
the lowest string, the latter in being made to vibrate
powerfully v\-ill make the other two strings vibrate also, in
such a manner that these vibrations may be perceived by
the eye as well as by the ear." ^

Strings which are tuned to the Octave or Unison of the
fundamental sound are most readih- affected in this way ;
but although they are made to \'ibrate more powerfully
than the sounds of the Twelfth and major Seventeenth, they
are not so distincth' heard. " The greater the uniformity
existing between sounds, the less easily can they be dis-
tinguished from one another." ^ The following statement,
which Rameau considers to have a bearing on temperament,
is not quite accurate : " It is yet to be remarked that a
Fifth, if slightly diminished, will «till co-\dbrate ; whereas
the Unison, Octave and [major] Third will not co-vibrate
if altered to the shghtest extent ; whence we must conclude
that this slightly diminished Fifth cannot be displeasing
to the ear, as is proved by experience, in the tuning of organs
and clavecins." ^ Rameau ought rather to have concluded
that although the correspondence which exists between the
judgment of the ear and acoustical fact may be, and is, of
importance for physical science and for the theory of harmony,
the ear does not suspend its judgment in respect of the con-
sonance or dissonance of intervals until it has been ascertained
whether this correspondence exists, but promptly decides
for itself. All the intervals he mentions and not the Fifth
only may be shghtly altered, and still co-vibrate. Rameau,

1 Nouveau Systeme, Ch. i. - Ibid. ^ Ibid.

RAMEAU'S NOUVEAU SYSTEME 137

although one of the most honest of theorists, occasionally
permits his scientific observation to be affected by the
necessities of his theory. Here the necessity is one connected
with temperament. In the Nouveau Systeme Rameau
favours the unequal or mean-tone temperament, in which
the Fifths are shghtly altered, the minor Thirds likewise but
to a larger extent, while the major Thirds are made as just
as possible. This system he abandoned later in favour
of equal temperament.

Relationship of Harmony and the Fundamental Bass
TO Mathematical Progres'Sions : Theory of the

SUBDOMINANT.

Rameau does not in the Nouveau Systeme attempt to
express, by means of ratios, the relative degrees of consonance
and dissonance pertaining to the various chords used in
harmony ; he contents himself with quoting Descartes to
the effect that " objects, in order to please, ought to be
disposed in such a manner that they do not appear confused
to the senses, so as to cause these to labour in order to dis-
tinguish them." 1 For this reason the Octave (i : 2) is a
more perfect consonance than the Third (4:5). Nevertheless,
Rameau cannot give up the idea, which forms his starting-
point in the Traite, that a close connection exists between
mathematics and music. Accordingly, we find this idea
taking shape in a new form.

He says: "From the three sounds [Fundamental,

Twelfth „„ J Seventeenth"! , . , ,^ , ^,

and which result from the resonance

o > 5 J

of a sonorous body, we obtain the following proportions or
progressions : — (ist) the Arithmetical progression 1:3:5,
which is determined by the difference between i and 3,
and from which the perfect [major] harmony is derived ;
(2nd) the Geometrical progressions determined by the
relationship between i and 3, and between i and 5 ; that is,
a triple progression i, 3, 9, 27, etc., or progression by Fifths ;
and a quintuple progression i, 5, 25, 125, etc., or progression
by Thirds." ^ (for the Twelfth i : 3 is merely a compound

^ Nouveau Systeme^ Ch. 2. ^ Ibid-, Preliminaries.

138 THE THEORY OF HARMONY

form of the Fifth 2 : 3, and the Seventeenth 1:5a compound
form of the Third 4:5). " The arithmetical progression
[1:3:5] gives us the most perfect of all the chords ; the
Geometrical progression [1:3:9] gives, in its turn, the
most perfect harmonic succession." ^ " From chords and
their progressions will arise modes, modulation, and melody.'"^
Rameau's theory now takes a notable development. In
the Traite he is quite aware of the unique position
occupied by the Tonic chord. It alone is truly consonant ; it
alone can produce the effect of finality, of absolute repose. He
is also aware of the importance which attaches to the harmonies
on the Fifth and on the Fourth degrees of the scale. The
progression of the first of these two chords towards the Tonic
harmony produces the Perfect (Authentic) Cadence, while
the progression of the second towards the same harmony gives
us the " Irregular " (Plagal) Cadence. But while the Perfect
Cadence may be explained as a harmonic progression in
wliich " the Fifth returns to its source," " the Irregular "
Cadence does not admit of a similar explanation. Neverthe-
less, Rameau is convinced, not only the chord on the fifth
degree, but also that on the fourth degree of the scale is a
Dominant : both chords " announce " the Tonic harmony.
Here now the thought strikes him : — if both chords are
Dominants, is not the relationship to the Tonic harmony in
each case that of the Fifth ? It is he finds indeed so. The
harmony on the fifth degree of the scale is that of the Fifth
above the Tonic ; while the harmony on the fourth degree
of the scale is that of the Fifth below the Tonic. Every
Tonic therefore has two Dominants, an upper Dominant
and a lower Dominant, or Subdominant. This Rameau
considers to be fuUv demonstrated by the triple progression

, or progression in Fifths, in which the central

G D

term ^ may be taken to represent the Tonic, the

r
Dominant, and the Subdominant. Here the real

relationship of both Dominants towards the Tonic is clearly
perceived, while at the same time the Tonic is determined
as the central sound, and the Tonic harmony as the Harmonic

1 Noiivcau Sysfanc. Ch. 2. * /ft/^.^ Ch. 4

RAMEAU'S NOUVEAU SYSTEMS 139

Centre to which both Dominants are related, and towards
which both tend to proceed.

This geometrical progression, 1:3:9, Rameau tells us,
not only determines the fundamental bass in Fifths, it
determines also harmonic succession. " In order," he says,
" that the fundamental sound, the Tonic, may be at
hberty to proceed either upwards or downwards, we shall

assign to it the number 3, and describe it as G : thus —

C (^ r\ n

This fundamental sound will commence and
1:3:9 3

finish the harmonic progression, and will proceed in

D C

differently either to or It is necessary to remark,

C D

however, that and cannot immediately succeed one

another without detriment to the triple progression. . . .
As the Fifth 9 is heard or at least understood when 3
is sounded, one naturally prefers that 3 should proceed to
9, its Fifth above, rather than to i, its Fifth below. There-
fore the most perfect progression of the fundamental sound
is to proceed to its Fifth above. On the other hand, the
most perfect progression of this Fifth is to return to the fun-
damental sound, for returning then as to its source one
desires nothing further after such a progression, which arises
from the inversion of the first."

Thus for the first time in the history of musical theory we
find the fourth degree of the diatonic scale described as the
Subdominant, and defined as the Fifth below the Tonic.
Rameau evidently regards his discovery of the Subdominant
as of the greatest importance ; it is for him the cope-
stone of his theory. As a matter of fact, Rameau's explana-
tion of the relationship existing between the Tonic and its
two Dominants not only forms one of the most important
features of his own theory of harmony, but has had the
greatest possible influence on the theory of harmony as
a whole. The term Subdominant was soon in general
use. At the present day it is constantly used by musicians
even who are unacquainted with Rameau's explanation
of it : and, what is more remarkable still, by musicians
and theorists who, although acquainted with Rameau's
theory of the Subdominant, nevertheless reject it, and give to
the Subdominant a Dominant " root."

I40

THE THEORY OF HARMONY

It is then, according to Rameau, not only between the
Tonic and the fifth degree of the scale, and between their
harmonies, that there exists a Fifth relationship, while the
fourth degree of the scale with its harmony appears to
occupy an isolated position : —

i^^^m

The fourth as well as the fifth degree has this Fifth relation-
ship, the one appearing above and the other below the Tonic.
Thus not only is the relationship which both bear to the Tonic
correctly determined, but the Tonic itself, standing midway
between its two Dominants, appears in its true character as
principal or determining note or chord. Could anything be
more symmetrical, more natural ? —

Is it not, asks Rameau, in entire accordance with
" experience," with the judgment of the ear? Whether
this be so or not we must ask : — What exactly does
Rameau mean when he speaks of a Subdominant
which is a Fifth below the Tonic ? His reference to the
Fifth above the Tonic can be understood ; this Fifth or
Twelfth, he has pointed out, forms part of the resonance
of the fundamental sound or Tonic. But where is the
Subdominant, or any sound which represents it, to be found ?
It forms no part of the resonance of the fundamental sound.
In short, the Tonic has no Fifth below. Of the two sounds
F-C, F cannot be regarded as Fifth of C ; on the contrary C
must be considered as Fifth of F. In the Subdominant
harmony, F-A-C, Rameau himself considers F to be the
fundamental note of which C is the Fifth.

It is chiefly in order to obviate this difficulty that Rameau
has recourse to the geometrical progression 1:3:9; here,
he tells us, 3 may represent the principal sound or Tonic,

RAMEAU'S NOUVEAU SYSTEM H 141

whereby the Fundamental Bass has the hbertv to proceed
from this central sound either upwards to its Dominant (9),
or downwards to its Subdominant (i). But Rameau
merely assumes what he wishes to prove. He assumes
that the Tonic has a lower Dominant, and is unable to
state where this lower Dominant is to be found. He
assumes that his mathematical progression must be limited
to three terms only, whereas it might consist of an indefinite
number of terms. He assumes that the middle, rather
than one of the extreme terms, should represent the Tonic.
He first states that the Fundamental Bass mav proceed
" indifferently " from the central sound to either of its
Dominants, but afterwards corrects this statement, and
points out that the natural tendency of the Fundamental
Bass is to the upper rather than to the lower Dominant.
But it is certain that there is nothing in the mathematical
progression 1:3:9, which indicates any such tendency or
preference on the part of the Fundamental Bass. Of the
quintuple progression, or progression in Thirds, Rameau
does not state whether he considers that this, hke the other
progression, should consist of three terms only, of which
the middle term should represent the Tonic. Obvioush^
there are considerable difficulties in the way of such an
arrangement : — •

Rameau, then, does not appear to reap much benefit
from his new use of mathematics ; his triple and quintuple
proportions do not enable him to throw much additional
light on the mysteries of harmonic science. Rameau owes
his discovery of the Fundamental Bass to his musical
intuition, to his keen observation of harmonic facts, and not
to mathematics. It is true no doubt that if such a bass,
once it has been fixed and conclusively proved, can be
shown to correspond with a certain geometrical progression,
such a circumstance may be of the highest scientific import-
ance. It is a most important scientific fact that the major
harmony corresponds with the first six numbers. But
although Rameau has little difficulty in showing that his

142 THE THEORY OF HARMONY

triple progression corresponds with a Fundamental Bass
in Fifths, and that it may even be considered to determine
such a bass, he is quite unable to show that there is anything
in this progression which corresponds with or even suggests
his theory of a Subdominant. The progression would appear
to indicate, on the contrary, that there is no Subdominant.
We have seen that Rameau selects the middle term, 3, of the

progression j • -. • q (more correctly ^ . 1 . ij to represent

the principal sound or Tonic. But the only term which
Rameau can make use of with any show of reason
for this purpose is the first, that is unity. In formulating his
mathematical progressions, in the Preliminaires de MatJie-
matique, which is prefixed to the Nouveaii Systeme, he
bases all his operations on the principle that i, or Unity,
represents the fundamental and principal sound to which
the other sounds are related. In the geometrical progression

"~^"7 1 , should therefore represent the Tonic ; 1 is

its Fifth, while 1 is the Fifth of i- It is evident, also, that
¥ 3

1 cannot represent the Tonic, because it is itself a related

' ■ C

and a determined sound : it is the Fifth of . Rameau's

J- «

difficulty however can be easily understood. For if in
the triple progression i, or unity, represents the principal
sound, the Tonic, it will be impossible to discover the
Subdominant.

Still, the Subdominant is a harmonic fact, a fact
of experience. Of Rameau's successors not one has suc-
ceeded in finding for it an adequate explanation. Some do
not reaUse the nature of the difficulties connected mth it.
Others appear to be unconscious of the part played by the
Subdominant in the estabhshment of our key system, and to
be unaware that any explanation is required. Until the
nature of the Subdominant is known, it might be rash to
assume that Rameau's proportions and progressions have no
significance for the theory of harmony.

RAMEAU'S NOUVEAU SYSTEM E

143

The Major and Minor Modes : the Chromatic Scale.

From the triple progression, or Fundamental Bass in
Fifths,—

-(&-

'O-

-Bi

JQ.

we obtain all the notes of the major scale in their correct

,, G — A — B — C— D— E — Fit- G

proportions, thus i-^^ : 27 : 30 : 32 : 36 : 40 : 45 : 48

The Minor Mode, Rameau says, arises from a Fundamental
Bass in Thirds ; that is —

-jOl.

where the first chord represents the major Tonic
chord, and the second the Tonic chord of its

E G

relative Minor Mode. "The numbers . r indicate to us

the relationship existing between the major and minor modes,
and explain the liberty we possess of passing from the one
mode to the other, by means of a Fundamental Bass in
Thirds." ^ Rameau however does not explain whence he
derives the liberty of making the Fundamental Bass descend
a minor Third. His quintuple progression i : 5 : 25 is
composed exclusively of major Thirds.

In order, then, to obtain the notes of the minor scale,
Rameau takes the chord E — G — B as a Tonic chord : E now
represents the middle term of a triple progression, and maj^
proceed as in the Major Mode to B its Fifth above, or to

^ Nouveau Systeme, Ch. 6.

144 THE THEORY OF HARMONY

A, its Fifth below. But now a difficulty arises, for if these
sounds are assigned the same proportions as in the Major
Mode, they cannot represent the triple progression. The

te™s 27 : 40 : 60 «^' 27 : 80 : 240 ^° "°^ correspond

with those of the triple progression 1:3:9; the ratio

27 : 40 does not represent the correct proportions of the

perfect Fifth (2:3), but one diminished to the extent of

a comma (80 : 81). If, on the other hand, the proportions

assigned to the sounds of the Fundamental Bass are

^ E B

o ^, so as to make them conform to those

27 : 01 : 243

of the triple progression, we find that E, the Tonic, which in
the relative major scale has the term 80, is now assigned the
term 81 ! Although Rameau is aware of these defects, he
imagines that they only furnish proof of the necessity for
temperament. " If," he says, " we do not discover the triple
progression in the terms given to the sounds A-E-B [i.e-,
27 : 80 : 240] this is because we have given to E the proportion
80, the repHca of 5, instead of 81 ; otherwise we could have

\ ^ E B

found it in ' -81 • -74^* ^^^ have done this purposely,

however, as it shows the necessity for temperament."^ It
is thus that Rameau, whose entire system is based on the
acoustical correctness of intervals, on just intonation, takes
refuge in temperament, that blessed haven of many a distressed
theorist. Although he is unable to discover a Fundamental
Bass for the Minor Mode, he nevertheless assigns to the
degrees of this mode in its Melodic form the following

E— F# — G— A — B— C#— D#— E
proportions-g^ : 9S : 96 : 108 : 120 : 135 : 150 : 160

In the descending form of this scale the sixth and seventh
degrees, both of which are lowered a (chromatic) semitone,

should have the proportions Jo . J]. He says little

as to why the ascending form of this scale should differ from
the descending form. "The sounds C#, D|:," he remarks,
" can only be used in the ascending scale ; in descending, the

1 " Nous n'avons affecte ce defaut que pour en preparer les voyes ;
d'autant qu'il est absolunient necessaire dans le temperament."
{Nouveau Systeme, Ch. 6).

RAMEAU'S NOUVEAU SYSTEME 145

scale contains the same notes and the same diatonic proportions
as the relative major scale ; whereby we see the close relation-
ship which exists betweeru-the two modes." ^ Here we find
a new explanation of the relationship existing between the
major and minor Modes. Rameau has previously explained
this relationship as arising from a descending Third-progression
of the Fundamental Bass.

Rameau's procedure in respect of the chromatic scale
is no less extraordinary. He derives this from the minor
scale. This is possible, " because we may place indifferently
either the major or minor Third on the fundamental sounds." ^
The proportions he assigns to the various degrees of this
sc'ale begin with 480 : 512, and terminate with 900 : 960.
He is of opinion that the proportions which distinguish the
three scales indicate the relative degrees of perfection of
these scales. " The major sj'stem commences at 24 ; the
minor, in which the major is again found, at 80 ; and the
chromatic, in which both the major and minor systems
find themselves repeated, at 480. . . . The fact that the
major system is composed of more simple proportions than the
other two svstems ... is a proof that this system is the
most perfect." ^

He now imagines that he is in possession of a coherent
system of modes, in which everything revolves round a
firmly established centre. " In the minor and the chromatic
svstems we find that the principal sounds of each form
a minor chord, namely, E-G-B, in which the Tonic of
the major scale holds the central place : this place it also

(^ Q Y)

holds in the triple progression, so that this

G I • 3 ■ 9

sound determines on every side the modulation " —

Rameau means the system of modes, as well as harmonic

succession within a mode — "whether in the same mode, or

n
in the passage from one mode to another."* Thus the

C ^
Third of the minor Tonic chord, and the Fifth of . becomes

the central note (Tonic) of the ]\Iinor, as well as of the Major
Mode !

^ Notivean Svsteme, Ch. 6.

2 Ibid, Ch. 6". » Ibid. « Ibid.

146 THE THEORY OE HARMONY

Rameau however is convinced that both the major
and minor diatonic systems have their origin in a
Fundamental Bass which consists of the three fundamental
sounds, Tonic, Dominant, and Subdominant. The Tonic,
together with its third and fifth upper partial tones, furnishes
us with the three essential sounds of the mode : —

Subdom. Tonic. Dom.

J=5) i~^)

^^=£211):

_-s5:i):

that is, the first, third, and fifth degrees of the scale ; the

Dominant, in the same way, gives us the second and seventh

degrees, while from the Subdominant we obtain the fourth and

sixth degrees of the scale. We are therefore, considers

Rameau, in possession of the diatonic scale of G major,

consisting of eight sounds arranged in the following order

g-a-h-c'-d'-e'-f^'-g'. But why does Rameau arrange the sounds

in this order ? ^ is fifth of c. It is c, therefore, which

ought to serve as the foundation and starting-point of

the whole diatonic succession- of sounds, which ought to

appear thus : c-d-e-fji^-g-a-b-c' . Unfortunately, as Rameau is

aware, such a scale has no place in our harmonic system.

Even if we accept Rameau 's assurance that there is

really a Subdominant ; even if we accept his explanation as

to the Major Mode, it is important to note that Rameau's

substitution, in the Nouveati Systeme, of the physical

principle of harmonic resonance for the mathematical

principle of the senary division of a string does not lessen

his difficulties with regard to the minor harmony and

the Minor Mode. On the. contrary these become well-nigh

insuperable. Rameau considers the major harmony as

a natural harmony which results from the resonance of a

sonorous body. Is not the minor harmony, however, also a

natural harmony ? But where in Nature is such a harmony

to be found ? The harmony which is formed between the

prime tone of a musical sound and its first series of partial

RAMEAU'S NOUVEAU SYSTEMS 147

tones is invariably major, andean never be minor. Rameau
is dealing with an immutable natural law. In the minor
harmony c-e\^-g, therefore, regarded from the point of view
of physical resonance, e\f appears as a sound which is foreign
to the natural harmony of c ; not only so, it contradicts
this natural harmony, for e\^ contradicts the true harmonic
sound, which is c\:\. Nevertheless, Rameau considers that
he is at liberty to give to the sounds of his Fundamental
Bass, now a major and now a minor harmony wherever
lie thinks fit. In the Major ]\Iode the fundamental sounds
have major harmonies. It might be considered then that
in the Minor ]\Iode all three fundamental sounds should bear
minor harmonies. This of course does not suit Rameau. He
places here a major harmony, and there a minor one, while
the same fundamental sound may at one time bear a minor,
and at another time a major harmony, according to the
exigencies of the mode : —

y — =W=(W)=

^^m

_o_

Dominant and Subdominant Discords : Development of
THE "■ Added Third " Theory of Chord Formation.

In the XoHveaii Systeme we find that Rameau's \-iews on
the subject of the Dominant and Subdominant discords,
and their use in defining the key, have also undergone some
development. " Since we cannot hear a tone," he remarks,
" without the ear being at the same time affected by its
Fifth and its Third (these three sounds forming the essential
notes of the mode), we cannot therefore hear it without being
at the same time impressed with its key. ... It follows that
each of the three fundamental sounds which constitute a
mode can in turn impress us wth the idea of its key, for
each bears a harmony equally perfect. In moving from one

148

THE THEORY OF HARMONY

to another of these fundamental sounds, there is formed a
species of repose. . . . hence there arises an uncertainty as
to the key, an uncertainty which can only be removed b}'
the skill of the composer." ^ The means which should be
taken to preserve the impression of the key are Dissonance
and Accent {Mestire). The harmony which is heard on the
strong beat of a bar affects us more strongly than that which
occurs on a weak beat. Hence the impression of the key is
strengthened by placing the chord of repose on the strong
beat."^ Accent by itself, however, is not enough. The above
principle works both ways, and may be employed to confirm
not only the Tonic or central key, but also the keys of its
two Dominants, thus : — •

KeyC.

KeyG.

KeyF

/T rh

fm ^^

\S\J

-Gt-

f^ri\'' f^

r:^

r:.

\i*) .

^-^

r-j

From this difficulty we are helped by Dissonance, for if we
add a Seventh above the Dominant harmony, and a Sixth
above the Subdominant harmony, these chords can no
longer be regarded as chords of repose, that is, as Tonic
chords. As now only the Tonic appears as a consonant
chord, and as further this Tonic harmony should be
heard on the accented beat, both accent and dissonance
will co-operate to confirm the impression of the key. This
is the true explanation of dissonance, which is a necessity
in music, and not merely the result of choice or caprice.
" The introduction of dissonance into music cannot be
justified, unless it is meant to serve some useful purpose ;
so far however the only explanation advanced has been
that by its means a greater variety of harmony has been
obtained." ^

In the chord of the Dominant Seventh, as d-j%-a-c, Rameau
proceeds, we find both Dominants, D and C, united in a single

Nouveau Systeme, Ch. 7. " Ibid. * Ibid., Ch. 11.

RAMEAU'S NOUVEAU SYSTEME 149

chord. These two Dominants " appear indeed to become
united so as to make even more marked that tendency
towards a Tonic Close which each of them singly possesses."^
Such being the case with regard to the Dominant harmony
d-f^-a, it would appear that the Subdominant harmony c-c-g
ought to be characterized in a similar way. But we cannot
add the upper Dominant d to the Subdominant harmony ;
we can only add the Fifth of this Dominant, so that the
Subdominant discord appears as c-e-g-a, that is, as the chord
of the Added Sixth. Hence the superiority which marks
the Perfect Cadence, as compared with the Irregular Cadence.
" The fruits which we can derive from the union of the Sub-
dominant with the harmony of the Dominant would vanish
as soon as we tried to combine this Dominant with the harmony
of the Subdominant, since the one is more perfect than the
other, and the cadence announced by the one is more perfect
than that of the other. It is necessary that the subordination
of the one to the other should be preserved. Instead, then,
of both Dominants being united in the Subdominant discord,
we find onlv the Fifth of the upper Dominant,, which takes
its place and, so to speak, represents it." -

This is an important development of Rameau's theory in
respect of these chords. It is of course an after-thought :
for both chords have their functions determined, and are
discussed at considerable length, in the Traite. As Rameau
in his later works lays great stress on this part of his theory,
fuller examination of it may for the present be reserved.
In the meantime it need only be pointed out that the explana-
tion of the origin of these chords given in the Xouveaii
Systeme is practically the same as that given in the Traite.
The chord of the Dominant Seventh is still considered to
arise from the addition to the "perfect" chord of one of its
parts ; that is, a minor Third is added above the " perfect "
chord. As regards the resolution of the Seventh in this chord,
Rameau is still of opinion that its natural tendency to fall
a semitone is owing to what he calls " the natural progression
of the minor Third. "^ He even thinks that in the :\Iinor
]\Iode the Third of the Tonic chord, which forms a minor
Third with the Tonic, tends to proceed a semitone
downwards, so as to form a chord of the " Added Sixth "

1 Nouveau Systhne, Ch. 12. '^ Ibid., Ch. 13. ^ See p. 125.

I50

THE THEORY OF HARMONY

with the Subdominant harmony, rather than rise a tone
thus:-

221

-«^-

-Q-

6
5

and remarks : ^ " After having pointed out that the Seventh
is derived from the minor Third, it is no longer the Seventh
that we ought to consider in order to understand its pro-
gression, but the consonance [the minor Third], which is
the cause of it." - This needs no comment.

The manner in which Rameau appHes his principles to some
of the simplest of harmonic progressions not infrequently
produces extraordinary results. In the Traite he distorts
beyond recognition simple contrapuntal passages by Zarlino.
In the Nottveaii Systeme he gives a revised version of a
series of progressions by Corelli : —
CoRELn's Bass.

"SZC

-o-

etc.

5656 5 6 56 56

which amount to nothing more intricate than a
succession of triads and chords of the Sixth above
a bass ascending by diatonic degrees, at the same
time censuring Corelli for not having indicated more clearly,
by means of the figured bass, the real nature of the harmony.
"'What does one think," he says, " of this method of figuring
se\'eral notes which ascend diatonically, where nearly every
5 and every 6 should indicate quite a different chord, as
follows": — Rameau's version.

•f^—

t^i

1 rj

_C,>_

6
5

4
3

6
5

6
4
etc.

Fundamental f(a)' 1 1 1 . —

— ^

P i -

H«c.o ^^

-&-

— \ —

— •—

—f-

— 1

— ^ —

__L —

— • —

-A-—

— ^ —

' !

1
7

^ Notti'caii Sysfciiie, Ch. 15.

Ibid.

RAMEAU'S NOUVEAU SYSTEMS 151

There are few, we imagine, who would not prefer CoreUi's
version to that of Ranieau.

So also with respect to his theory of chord generation by
means of added Thirds, we read : — " The greater the number
of Thirds in a chord, the greater is the variety of possible
combinations of these Thirds ; as may be observed in the
chord of the Seventh. Experience permits us to place whatever
Thirds we wish above one another, provided that two major
Thirds do not occur in succession : then the chord of the
Seventh admits of the follo^\^ng five different combinations : —

All these are chords of the Seventh, although the order of
Thirds differs in each. " ^ Comment here is needless !

Nature and Origix of Toxe-systems.

In the ninth chapter of the Noiiveau Systeme {De la
Melodie natiirelle) Rameau attempts to demonstrate that
melody is natural to us. This, he thinks, can be proved by
a simple experiment. " Take any one," he says, " destitute
of musical knowledge or experience, it is almost certain that
the first sound he sings \\\\\ be regarded by him as a Tonic,
or at least as the Fifth above this Tonic. If the first sound
be taken as Tonic, he will then ascend to the Fifth or Third
of the Tonic harmony ; if however the first sound be regarded
as Fifth, the notes of the Tonic harmony wall be taken in
descending : all this will be done instinctively." The reason
of which is, that " we cannot hear a sound without being
at the same time affected b\' its harmony." It is soon apparent
however that the singer in question is not so destitute of
experience as Rameau imagines him to be. It appears that
he knows something of the Fundamental Bass. " If this
person sings indifferently the Fifth or the Fourth after the
first sound [Tonic], remark that he follows here the progression
of the fundamental sounds " ; also that he is acquainted with
our modem major and minor scales, preferring the semitone

Notiveait Systeme, p.

152

THE THEORY OF HARMONY

lying immediately below the Tonic (leading-note) to the
whole-tone ; " in singing a scale, this person will always sing
the whole-tone above the Tonic, and most frequently the
half-tone below . . . for the tone above the Tonic and the
semitone below are nothing but the Fifth and major Third
respectively of the Fifth of the Tonic : whereas the whole-tone
below is the minor Third of its Fifth, which minor Third
is less natural than the major." Also " he wih prefer the
major mode to the minor, because the major mode is the most
natural."

Rameau, however, does not tell us which of the old Modes
on which the Church melodies were based he considers to
be the most natural. He should also have mentioned the
fact that our major and minor modes, of which alone he
takes account, did not become natural to us until the end
of the seventeenth and beginning of the eighteenth centuries.
In the chapter which immediately follows [Que nous trouvons
naturellement la basse-fondamentale de tons les repos inserez
dans un chant) Rameau endeavours to show that harmony
also is natural to us ; and even remarks that we naturally
discover the Fundamental Bass appropriate to any given
melody ; which, if it were really so in Rameau's time, is
certainly not true in the case of many students of harmony
of our own day. All the same, it is not difficult to understand
what it is that Rameau wishes us to infer — that melody, scales,
harmony, etc., cannot have arisen arbitrarily, but must
have had their source in some definite, even natural principle.
This is his real meaning, and it is necessary to a very large
extent to agree with him, even if he is unable fully to explain
how such developments have been brought about.

Rameau proceeds to raise questions of the greatest im-
portance for the science of harmony. He remarks, for
example, that "it is difficult to sing three whole-tones in
succession, because such a succession of sounds does not
conform to the [natural] progression of the Fundamental
bass " : —

^^=^=

m—^—

1 8 \

1 b^' II

RAMEAU'S NOUVEAU SYSTEM E

3 J

Here the Fundamental Bass cannot proceed further
Jthan the note a ; if however a semitone be taken above
this note, instead of a whole tone(*), the Fundamental
Bass is at liberty to descend a Fifth ; the succession of
sounds, f-g-a-b\}, now becomes easy to sing. He also dwells
on the importance of the Cadences, which arise from the
progression of the sounds of this Fundamental Bass. " Even
the most experienced musicians," he says, " must agree that
the only Cadences which they can make use of — except the
Interrupted Cadence, which however is derived from the
Perfect Cadence — must always have as their real bass the
fundamental sounds, thus : —

22:

2a:

-«>>-

-o-

"CT"

-€>»-

r^i S>-

m^i

-nr

JZU

"cy

_Q_

-«'9>-

fej^ I rj [I n y ^i

I ^- II -

-<s*-

-<s>-

9 II 3

" If, then, the most consummate of musicians can do
nothing other than what is dictated to us by the principle
proposed . . . and if this principle is based solely on a
•single fact of experience, which proves to us that we cannot
intone a sound without being at the same time affected by
its harmony . . . this is unquestionably to presume that
the different inflexions of our voice [in singing] arise from
this principle, and that they are natural to us, since this
principle is natural. We may go further, and say that not
•only are they natural to us, but that they alone are natural
to us.i For if, in order to find greater variety, we add
something to the progression by the Fifth, we can only add
the progression by the Third. From this progression by the
Third there will arise the Chromatic semitone, thus : —

'if^

-I^

-231

-<'S<-

^ Rameau has evidently forgotten, for the time being, the
■existence of the minor harmony and the Minor Mode.

154 'I'HE THEORY OF HARMONY

"Is it necessary to go further and try to discover smaller
degrees than the chromatic and diatonic semitones ? This
would be against the natural order of things. For we must
not forget that a single string furnishes us with three different
sounds ; that Harmony and its progression can result only
from these three sounds, and that Melody is only a consequence
of the progression of Harmony.

"If we abandon this principle, nothing will be easier than
to imagine any kind of interval : than to appropriate these
intervals to harmony, to its progression, even to the voice.
If we deny this principle everything will be good ; the
Enharmonic Diesis which divides the semitone, the Comma
which divides this Diesis, the semi-comma which divides
this comma — in fact, everything that presents itself will be
equally good. ... It may be said that the degrees most
natural to the voice are those which it can intone with the
least difficulty ; that we get accustomed to these degrees ;
but that if we were accustomed to others, these would appear
equally natural. . . . But it is not to frequent use that
we owe the inflexions which we remark as natural to the voice ;
custom, it is true, may render them more familiar to us, but
if they were not natural, in vain would we force ourselves
to sing them. Not even the most experienced musician,
however flexible his voice, can accurately determine a quarter
of a tone . . . because it is not natural to the voice, and
the reason why it is not natural is, that one cannot understand
the progression of the two fundamental sounds whose harmony
furnishes to us this quarter of a tone. . . . The Greeks, it is
true, had an Enharmonic system, in which this quarter-tone
is found, but this system was with them only a theoretical
system. By it they demonstrated the composition of certain
intervals. We could, in much the same way, form an entire
system from the commas which compose the whole-tone." ^

^ Nouveau Systeme, Ch. 9.

'55

CHAPTER VI.

RAMEAU'S GENERATION HARMONIQUE AND DtMONSTRAl ION
DU PRINCIPE DE L'HARMONIE.

In the two important works which we have now to examine,
the Generation Harmoniqiie, 1737, and Demonstration dii
Principe dc I'Harmonie, 1750, we may consider Rameau's
views on the subject of harmonic science to be fuHy matured,
and his theory of harmony completely developed.^ The
second work, which is smaller than the first, was presented
by Rameau (December, 1749), in the form of a memoir
or communication addressed to Messieurs de I'Academie
Royale des Sciences, was " approved " by that learned body,
and a report dealing with the main principles of Rameau's
theory, drawn up and signed by three distinguished
members of the Academy, MM. de Mairan, Nicole, and
d'Alembert — a report which was subsequently appended by
Rameau to his Demonstration — placed among the " registers "
of the Academy.

In the Generation Harmonique, which is also inscribed to the
members of the Academie des Sciences, we find that Rameau
now considers harmony to be a physico-mathematical science,
of sufficient scientific importance to merit the attention of
the most eminent savants of his day. " Music," he remarks,
" is for most people an art intended only for amusement ;
as respects artistic creation and the appreciation of artistic
works, this is supposed to be only a question of taste ; for
you however Music is a Science, established on fixed principles,
and which, while it pleases the ear, appeals also to the reason.
Lcng before Music had attained its present degree of perfection,
several savants had deemed it to be worthy of their attention
and investigation, and almost since its origin it has had the

• See, however, his Nonvelles Reflexions sur le Principe sonore
(p. 264).

156 THE THEORY OF HARMONY

honour of being regarded as a physico-mathematical science.
One may say that it has this singular advantage, of presenting
simultaneously to the mind and to the senses every conceivable
proportion {rapport) by means of a vibrating sonorous body ;
while in other departments of mathematics the mind is not
usually helped by the senses in appreciating such proportions."

Since the publication of the Nouveau Systeme, Rameau
has continued to investigate the nature of various acoustical
phenomena, especially those relating to the resonance of a
sounding body, which he considers to have a direct bearing
on the science of harmony. In these investigations he appears
to have had the assistance of two eminent French physicists,
MM. de Mairan and de Gemaches. "It is ten or t^^elve
years ago," he says, " since M. de Mairan, whose name alone
commands respect, in the course of a conversation with
regard to my system of harmony, communicated to me this
idea concerning the particles of air. . . . But as at this time
I had not given to the subject any great consideration,
and besides did not perceive how I could derive any advantage
from it, I had almost forgotten about it when M de Gamaches
recalled M. de IMairan's con^'ersation to my memory, and had
the kindness, for which I cannot sufficiently thank him, to
point out to me the bearing it had on the principles on which
my system is based." ^

In the first chapter of the Generation Harmonique,
Rameau brings forward a number of theses {Propositions),
and observations relating to the physical properties of a
sonorous body (Experiences), some of which, as Dr.
Riemann {Geschichte der Musiktheorie) remarks, are calculated
to make physicists and physiologists even of the present day
raise their eyebrows in astonishment. Some of them fore-
shadow and may even be said to anticipate some of the
discoveries in physical and physiological science which have
been generally understood to belong to a later time. The fibres
connected with the basilar membrane in the cochlea of the
ear [Fibres of Corti) are referred to thus : —

XI P. Proposition : " What has been said of sonorous
bodies ought equally to be understood of the Fibres which
line the base of the shell (conqite) of the ear ; these fibres
are so many sonorous bodies, to which the air communicates

^ Gen. Harm., Ch. i, Prop. III.

RAMEAU'S GENERATION HARMON IQUE 157

its vibrations, and by means of wliicli the sensation of sound
and of harmony is transmitted to the brain."

Some of the propositions touch closely on the question
of the nature of consonance and dissonance. The power which
two or more sounds may possess of blending together, or fusing
into what appears to be a single sound, is shown to depend on
the degree of commensurabilty existing between them.

V^. Proposition .• "A sonorous body set in motion
communicates its vibrations not only to the particles of air
capable of the same vibrations, but to all the other particles
commensurable to the first."

VIP. Proposition: "Those sounds are most commen-
surable which communicate their vibrations most easily
and most powerfully ; whence it follows that the effect
cf the greatest common measure between sonorous bodies
which communicate their vibrations by the medium of the
air ought to outweigh that of every other aliquot part, since
this greatest common measure is the most commensurable."

X. Proposition: " The more nearly an aliquot part
approaches to the ratio of equality, the more its resonance
becomes united with that of the entire [sonorous] body ; this is
a fact of experience to be observed in the Unison, Octave, etc."

Of beats and their connection with the problem of conson-
ance and dissonance, Rameau treats thus : —

V'^. Experience: ". . The air marks a harsh disagreement,
dissonance [between two sounds] by the frequency and
rapidity of the beats (battements) which arise ; their con-
sonance is marked by the cessation of these beats."
Further: "The rapidity of the beats increases as the
two sounds in question approach towards a state of
consonance." This reads like a passage from Helmholtz's
Sensations of Tone. Let it be observed, however, that
Rameau does not discover in beats, or their absence, the
catise or explanation of the phenomena of consonance and
dissonance. He merely points to the connection existing
between the two. On the contrary, Rameau explains
consonance as resulting from the degree of commensurability
existing between sounds. In this, some able physicists and
acousticians of the present day ^ would consider that Rameau

1 See Karl Stumpf's Tonpsychologie (1890) and Uber Konsonam und

Dissonanz (1898).

158 THE THEORY OF HARMONY

shows himself to be more penetrating and more scientifically
accurate than Helmholtz.^

Concerning the sympathetic vibration of strings, we read :

II'. Experience : Take a Viola or 'Cello, and tune accurately
two of its strings at the distance of a Twelfth from each
other ; sound the lower string, and you will not only see
the higher string vibrate, but you will hear it sound. . . . Set
now the higher string in vibration, and you will not only
observe the lower string to vibrate throughout its whole
length, but also to divide itself into three equal parts, forming
three segments {ventres) with two nodes or fixed points."

IIP. Experience .- " Sound one of the lower strings of the
Viola or 'Cello, and you will hear, if you hsten attentively,
not only the fundamental sound, but also its Octave, double
Octave, Twelfth and Seventeenth above [here Rameau is
treating of upper partial tones] which are related to the
fundamental sound in the follo^^dng proportions, i, h, \, \, ^,|.
The seventh harmonic, which has the proportion 4, may also be
distinguished — to say nothing of other sounds — but it is so
faint that it is seldom noticed." In Experience IV., reference
is made to the " mixture stop" on the organ, as showing how
sounds related to one another in the proportions i, -|, i, may
combine so as to produce the impression of a single sound.

By such Experiences Rameau demonstrates the
essentially compound nature of musical sound. "It is
harmonious, and its harmony produces the proportion
I. h i • which finds itself reproduced in the proportion
I. 3. 5. by virtue of the power of sympathetic vibration,"
possessed by acute sounds on lower sounds which are com-
mensurable to them. The proportion 1, I, ^ represents the
major harmony, consisting of fundamental note, TweWth,
and (major) Seventeenth above ; the proportion i, 3, 5
represents the minor harmony, consisting of fundamental
note, Twelfth, and (major) Seventeenth below : —
Major harmony. Minor harmony.

9- m-n

^^¥F

321

W-

-12-

z'ym-TT

1 See. however, pp. 383-385.

KAMEAUS GENERATIOX HARMOiMOUE 159

Wlien the fundamental note is set in vibration, the Twelfth
and Seventeeth above may be heard to sound along with
it (major harmony) ; at the same time the Twelfth and
Seventeenth below, according to Rameau, are made to co-
vibrate, although they are not heard to sound (minor
harmony). But here Rameau finds himself confronted by
a difficulty. For other harmonics than the Twelfth and
Seventeenth above a fundamental note may be heard to
sound along with it ; while also other sonorous bodies
than those which correspond to the Twelfth and Seventeenth
below are capable of being acted upon by the resonance
of the fundamental sound. For example, in addition to
the third and fifth upper partial tones (the Twelfth and
Seventeenth above) or, as Rameau calls them, " harmonics,"
the seventh upper partial tone may frequently be quite
clearty distinguished as a constituent of the compound
tone of the fundamental sound. As Rameau himself
points out, other sounds still higher in the harmonic
series may also be distinguished. \\'hy then should not
at least this seventh harmonic, as well as the third and
fifth, be included as an essential and constituent part
of the harmony which Rameau considers to be generated
from the resonance of the fundamental sound ? Rameau is
aware of this difficulty, and attempts to remove it. He
thinks that a musical sound, to be appreciated as such bv the
ear, should not contain more than the three sounds above
mentioned (i, \, ^), for where higher harmonics than these
strike distinctly on the ear, the ear becomes confused, and is
unable to appreciate any sound clearly. This mav be proved,
Rameau proceeds, b\^ a very simple experiment.

Vr. Experience : " Suspend a pair of tongs by means of a
thin piece of twine and apply to each ear an end of the twine.
Now strike the tongs smartly, and nothing \\-ill be heard
but a confused jumble of sounds, which will make it difficult
for the ear to appreciate any one sound clearl\-. Soon
however, the more acute of these sounds will begin to
die -dway . . the lowest sound, that of the entire sonorous
body, will then begin to make itself heard, sounding in the
ear like one of the low tones of an organ. In addition,
there ma}' be heard along with this fundamental sound its
harmonics of the Twelfth and Seventeenth major." It is
also owing to the confusion of the ear produced by too

i6o THE THEORY OF HARMONY

great a number of the higher harmonics that one is
unable to appreciate clearly the lower sounds produced
by the i6 and 32 feet pipes of the organ. These low-
tones however, although inappreciable by themselves,
may be rendered appreciable by combining with them
tones an Octave higher. ^ (F*. Experience.) But -it is not
only the presence of too great a number of these
higher harmonics, but also their entire absence which,
according to Rameau, renders musical sound as such
inappreciable to the ear. Very acute sounds are also in-
appreciable by themselves. This is because, owing to the
extreme smallness of the sonorous body producing such an
acute sound, it is impossible for it to draw any harmonics
from its aliquot parts. By combining however such a
sound with its Octave below, the sound is rendered appreci-
able : this lower sound supplying the Octaves of the harmonics
required.- " Hence," concludes Rameau, " musical sound
is inappreciable without the help of the resonance of a certain
fixed number of its aliquot parts." Th's number " is limited
to the three different sounds which have the proportion
I, ^, ^ . . . since without the resonance of ^ and ^, or at
least one of the two, the sound is no longer appreciable by
itself ; and the same is the case if the sounds produced by
the smaller aliquot parts strike on the ear too distinctly ;
everything then becomes confused." ^

In these Propositions and Experiences Rameau is of
opinion that he has at last found the means whereb}' he can
give to his theoretical principles a firm and sure foundation.
Thus in the Preface he begins : I have at last succeeded,
if I do not deceive myself, in obtaining the proof of this
principle of harmony, which had been, suggested to me only
by means of experience ; this Fundamental Bass, the sole
compass of the ear, the invisible guide of the musician, which
he has unconscioush' followed in his artistic productions,
but which he has no sooner become acquainted with than

1 Cf. Helmholtz : Sensations of Tone, Part II., Ch. 9. — " When
we continually descend in the scale, the strength of our sensation
decreases so rapidly that the sound of the prime tone, though
its vis-viva is independently greater than that of its upper partials,
as is shown in higher positions of a musical tone of the same
composition, is overcome and concealed by its own upper partials."

^ Gen. Harm., "Conclusions." Ch. i. ^ Ibid.

RAMEAU'S GENHRAriON HARMON lOUE i6i

he has claimed it as his own. . . . Such a Fundamental
Bass is indeed one of those natural sentiments which only
reveal themselves to us clearly the moment we begin to think
about them." So enamoured is Rameau of his new ideas,
and of the physical properties of the sonorous body, that he
does not hesitate to afhrm, notwithstanding his express
declaration to the contrary in the Traite de I'hannonie.
that in his previous efforts he has been guided by " experience "
alone, and that only now for the first time is he enabled
to give to his principles a scientific foundation.

What then does Rameau alhrm to be the net result of
his acoustical researches? In the Traite he has sought for
his theoretical principles a mathematical basis ; in the
Generation Harmoniquc his endea\^our is to demonstrate that
these principles are not only intimately connected with,
but have their origin in the physical properties of the sonorous
body itself. " We must regard harmony," he says, " as a
natural effect resulting from the resonance of a sonorous
body ; it is from this that it derives its origin ; musical
sound as such is not in its nature simple, but hannonious,
and its harmony produces the proportions i, ^, ^, and
1:3:5. . . . ; the proportion i, |, ^, is just what has
always been known by the name of the Trias Harmonica,
the Harmonic Proportion.'^ Here we find something new,
of which we have already observed signs in the Nouveau
Svsteme. Rameau has, however, changed his standpoint.
In his previous works he regards the major harmony — his
principle of principles, upon which he has sought to build
up a complete system — as resulting from the division of a
sonorous body by the first six numbers, representing a certain
fixed mathematical proportion ; now this major harmony is
shown to be a property inherent in the sonorous body itself ;
it is no longer considered to be the result of a mathematical
proportion, but itself produces this proportion. This re-
statement of the connection existing between harmony and
mathematics or proportions, shows that Rameau has not
completely succeeded in satisfying himself that his use of
mathematics and, especially in the Nouveau Svsteme, of
proportions, is free from serious objection, as assuredly it
is not. In the Generation Harmoniquc he expressly states

1 Gen. Harm., "Conclusions." Ch. i.

1 62 THE THEORY OF HARMONY

that he has endeavoured, in the course of that work, to avoid
a too arbitrary use of such proportions.^

The position which Rameau takes up in his Experiences
is quite clear and definite. He makes everything depend on
the " appreciabilty," as he calls it, or " non-appreciabihty "
of musical or ^«as^-musical sounds. He endeavours to
demonstrate that a musical sound, to be appreciable as such,
as well as in respect of its pitch relationship, must consist of
neither more nor less than the three sounds of the fundamental
tone, with its third or fifth upper partial tones (the second and
fourth upper partials being considered as replicas of the
fundamental tone). All other sounds are non-appreciable as
musical sounds. Such is Rameau's argument, which has
at least the merit of being extremely ingenious. But
Rameau, his ingenuity notwithstanding, cannot dispose
of the matter in this way. He makes an observation relating
to the physical nature of a vibrating sonorous body, which has
the fatal defect of being demonstrably inaccurate, indeed
false. A very large number of musical sounds which are
appreciable, both as such, and in respect of their pitch
relationship, contain more upper partials than those of the
Third and Fifth ; such are the musical tones of the voice,
those of the organ and pianoforte, the majority of orchestral
instruments, etc. , indeed, as Helmholtz has informed us,^ nearly
all sounds which are useful for musical purposes.

This question of the appreciability of musical sounds appears
to have been brought forward by Rameau with the object
also of getting rid of a difficulty of which he is quite
aware, namely, that the natural series of upper partial tones
arising from the resonance of a ^•ibrating sonorous body is

^ " Nous ne sommes point conduits en consequence de cette pro-
portion [i.e. I, J, -^j or I : 3 : 5] nous avons feint de I'ignorer, & nous
avons attendu que la nature meme des corps sonores nous la
rendit, pour etre convaincus par nous-meme qu'elle est effectivement
I'unique arbitre de I'liarmonie." [Gen. Harm.. "Conclusions," Ch. i.]
" La proportion harmonique peut bien etre regardee comme un principe
en musique, mais non pas comme le premier de tous : elle n'y existe
qu'k la faveur des differens sons qu'on distingue dans la resonance
d'un corps sonore. . . . Se servir a propos des proportions, meme des
progressions, les appliquer a leur objet, rien n'est mieux : mais
vouloir en tirer leur principe meme, et ses dependances, c'est
s'exposer infailliblement a I'erreur." (Gen. Harm., Preface.)

' Sensations of Tone. Part II., Ch. 10.

RAMEAU'S GENERATION HARMONIQUE 163

not limited to the first five or six of such tones. (See his
remarks on the natural sounds of the trumpet, Gen. Harm.,
Ch. 6, Art. 4.) It is necessary for his purpose to admit
I' \> i' for these produce the major harmony, but it is
necessary to exclude the ^. Rameau's reasons for ex-
cluding this ^ are not convincing. In many musical
sounds this ^ is distinctly audible, nevertheless the
compound tone of which it forms a part is be3^ond
question quite appreciable, both with regard to pitch and
character. Another reason brought forward by Rameau for
the exclusion of the ^ is that it is not in tune ! Speaking
of instruments, such as the trumpet, capable of producing
the natural series of harmonics, he says : " The sounds of
the ^, Y^y and ^^, being harmonic nejther of i nor of 3, are
always false in these instruments." And yet Rameau's
lifelong task, his task even at the moment when he is describing
the natural Seventh as false, and out of tune, is that of
endeavouring to prove that we receive harmony directly
from Nature !

It would indeed appear as if the net result of Rameau's
digression into the realm of physical science was to make
matters rather worse than they were before. For the
arithmetical division of the monochord, which forms his
starting point in the Traite, he now substitutes the natural
division of the sonorous body. He may argue, with reason,
that in the senary division of the monochord, as he finds it
explained by Zarlino, and in the lucid theorem of Descartes,
we discover the principle and origin of harmony, in the
sense that it supplies us with all the consonances. But unfor-
tunately for Rameau the natural division of the sonorous
body does not stop where he wants it to stop. Rameau would
place his finger on the number 6, and would say to Nature :
" Thus far, but no further ! " He will have nothing to do
with any harmonics beyond this number. Further, he has
a grievance against Nature — the very first of such harmonics
is out of tune.

In the Traite, Rameau is of opinion that the minor harmony
arises from the same principle as the major, and altogether and
emphatically rejects Zarlino 's explanation of the minor
harmony as arising from the Arithmetical proportion. In the
Generation Harmonique he abandons his former views respecting
the nature and origin of this harmony. He now sees clearly

1 64 THE THEORY OF HARMONY

that while the first harmonic sounds resulting from the
resonance of the sonorous body may be considered to constitute
the major harmony, such an order or arrangement of sounds
can never constitute a minor one. That is, the harmony
" we receive directly from Nature " is always major, and can
never be minor. He therefore now relates the minor harmony
to another principle, namely, the sympathetic vibration of
strings, 1 and to the arithmetical proportion ; that is, he accepts
ZarUno's explanation of it. At the same time he claims to
have discovered for the minor harmony its real physical
basis. "Those who hke ourselves," he remarks, "have made
use of the Arithmetical Proportion have done so only for the
sake of convenience, and without a harmonic foundation." ^
This new and extremely important feature of Rameau's
theory will be fully discussed later.

Objections to Rameau's Theories.

In thus claiming for harmony a physical basis, and in making
the science of harmony to depend on the physical properties
of musical sound itself, Rameau has found many adherents
and imitators. At the same time, his methods in this
respect have met with severe criticism,^ and it may be as
well, before proceeding further, to examine to what extent
such criticism actually" affects Rameau's position as a
theorist.

Berhoz, in an analysis which he has made of Rameau's
theory of harmony,^ raises objections of a kind which has
found voice in many quarters, and at ever-recurring intervals
even up to the present day. — He says : — " The whole system
of Rameau is based on a natural fact which he had very
badly observed, as one will see, namely the harmonic reson-
ance of the sonorous body." ... " He speaks incessantly
of the resonance of such a body, which is for him a stretched

^ See p. 219 ^ See p. 80.

* Ed. Fetis. — Esquisse de I'histoire de I'harnwnie ; Traiie complet
de la thiorie et de la pratique de I'harmonie. Kirnberger. — Kunst des
reinen Satzes {die wahren Gnmdsdtze zimi Gebrauch der Harmonie).
Hauptmann. — Die Nalur der Harmonik und der Metrik, etc.

* De Rameau et quelques uns de ses ouvrages. {Gazette Musicale,
Paris, 1842.)

RAMEAU'S GENERATION HARMON IQUE 165

string or an organ pipe. But lie seems to believe that the
resonance of all other sonorous bodies gives the same results,
which is false. On the contrary, there are those which
give only frightful discordances, which nevertheless can be
called their harmonics, and which owe their existence to the
same law as sounds called harmonic and musical. Why then
are all these sonorous bodies disinherited in favour of the
stretched string and the organ pipe ? They also are in
Nature." ... " It is astonishing to hear at the present
day such expressions as ' this is beautiful, because it is in
Nature ! ' Nothing could be more absurd ! There is really
nothing which is not in Nature. Vocal music is in Nature,
because it is Nature that produces the voice. In this
case, let us include the cries of animals : these are as
natural as the accents of the human voice. Instrumental
music, then, is not natural, because Nature does not make
instruments !

" Since Rameau admits dissonances, although they are
not natural . . . what then does it matter that the harmonic
resonance of a string gives the perfect chord ? . . . But here
Rameau appears to be ignorant of a fact which is of favourable
import for his theory. For in the string and organ pipe may
be observed the natural production of dissonance ! He believes
'that they produce only the Fifth and Third, when manifestly
they produce also the minor Seventh and major Ninth, and
several other harsh dissonances arranged diatonically. . . .
Again, he goes to extraordinary lengths in order to naturalize
the Minor Mode. O unfortunate Rameau ! not to have noticed
that the majority of large bells make us hear quite distinctly
the minor Third above its fundamental tone ! How this
fact would have consolidated his theory ! . . . Here is a
musician . . . who pretends to derive harmon}^ from a
natural phenomenon, and yet who does not know the real
power which this phenomenon has in affecting, in a favourable
way, his theory ; and who, if he knew it in its entirety,
would be forced to admit combinations as harmonious which
are really insupportable, or to avow that musical harmony
is the result of a choice of sounds, according to the different
impressions that they make on our ear in such and such
combinations, with particular conditions as to their successive
connection, and to recognize finally that the science of chords
has no other raison d'iirc than that of our organization.

1 66 THE THEORY OF HARMONY

and no other basis than that which he denies to it, namely.
Experience." ^

More restrained in language, as well as more accurate in
statement, are the observations of Helmholtz. He says : —
" In the middle of last century, when much suffering
Arose from an artificial social condition, it might have been
enough to show that a thing was natural, in order at the
same time to prove that it must be also beautiful and desirable.
Of course no one who considers (for example) the great
perfection and suitability of all organic arrangements in the
human body would, even at the present day, deny that
when the existence of such natural relations has been proved
as Rameau discovered between the tones of the major triad,
they ought to be most carefully considered, at least as starting
points for further research. And Rameau had indeed quite
correctly conjectured, as we can now perceive, that this
fact was the proper basis of a theory of harmony. But that is
by no means everything. For in nature we find not only
beauty but ugliness ; not only help, but hurt. Hence
the mere proof that anything is natural, does not suffice to
justify it aesthetically. Moreover, if Rameau had listened
to the effects of striking rods, bells, and membranes, or
blowing over hollow chambers, he might have heard many
a perfectly dissonant chord, quite unlike those obtained
from strings and musical instruments. i\nd yet such chords -
cannot but be considered equally natural." ^

^ It would seem that the article by Berlioz from which the above
is taken, and which appeared in the Parisian Revue et Gazette
Musicale, of which Fetis was editor, was to a large extent inspired
by Fetis himself, who, in previous numbers of the periodical, had
devoted some space to an examination of Rameau's theory of
harmony. The expressions used by Berlioz are in many respects
similar to those used by Fetis. Further, Berlioz makes use of the term
first coined by Fetis to describe the inner relationship existing between
sounds and chords, namely, the term Tonality; and he in effect accepts
the views of Fetis as to what constitutes the real basis of the theory of
harmony. Thus he remarks : " The great law of Tonality, which
appears to dominate all our harmonic edifice, has attracted Rameau's
attention very little ; he ignores it even in cases where it manifests
itself most clearly." But it is not at all clear that Berlioz had
himself grasped the essential points of Rameau's theory.

- More correctly, discords, or dissonances.

' Sensations of Tone. Part II., Ch. 12.

RAMEAU'S GENERATION HARMON IQUE 167

Rameau was, however, quite well aware that there were
sonorous bodies which produced what Berlioz calls " frightful
discordances." He had himself, as we have seen, examined
such bodies and had rejected them, not because their reson-
ance was not " natural," but because they produced not
musical sound but a " jumble of sounds."

Rameau might quite well have inquired what reasonable
prospect existed of discovering the source of harmony in
sonorous bodies which were capable of producing only
" frightful discordances," otherwise noise, and might quite
well have considered that it was time enough to base a
theory of harmony on the phenomena presented by the
resonance of such bodies, w^hen it had become the prevailing
custom among composers to write artistic works for an
orchestra composed of " striking rods " and " hollow
chambers." But we have not yet arrived at this stage
of symphonic development.

Nevertheless, it is clear that Rameau is unable to make
harmonic resonance the basis of a theor}^ of harmony on
the ground that such resonance is " natural." When he
speaks, as he frequently does, of harmony being " a natural
effect," he does not appear to make use of the term in any
specific or restricted sense. But as Helmholtz points out,
the mere fact of a thing being natural does not suffice to
justify it aesthetically.

Rameau's standpoint with regard to the problem of
Consonance is better and more correctly appreciated by
E. F. F. Chladni (1756-1827), well known as one of the most
eminent acousticians of his time, and who in various works
has given the result of his researches and experiments in
connection with many different kinds of sounding bodies.
Chladni does not agree with Rameau's explanation of con-
sonance or of harmony. Rameau, he considers, was led
astray through ignorance of the laws of vibration of sonorous
bodies, and by his belief that the resonance of all such bodies
gave similar results, that is, that the partial tones in every
case arose in the same order as those of a string. There is
no doubt that such was Rameau's opinion.

Chladni says : " Many theorists, for example Rameau and
his adherents, ha^•e thought that the principle of consonance
and dissonance, and indeed of harmony in general, was to
be found in the presence — or absence — of higher partial

1 68 THE THEORY OF HARMONY

tones which arose, corresponding to the natural series of
numbers, from a prime or ground tone, this ground tone being
regarded a^ unity. They have even gone so far as to imagine
that the presence of such higher tones constituted the chief
difference between musical sound and noise. The origin
of such an error lies in the fact that in a string there is nothing
to prevent its aliquot parts as well as the string as a whole
from vibrating, and because, mainly through ignorance of
the laws of vibration of other sonorous bodies, they have
supposed that the order of partial tones produced by such
bodies must be the same as that of a string. On the contrary,
many other sonorous bodies, as rods, discs, bells, etc., produce
quite a different order of partial tones. (In bells, circular
vessels, etc., of uniform thickness, the proportions of the
partial tones which result from their resonance are as the
squares of the numbers 2, 3, 4, 5, etc,, or, if we regard the
lowest tone as unity, as i, 2j, 4, 6|, etc.).

"It is evident, therefore that consonance and dissonance
cannot be explained in this way. The attempt to do so
leads to many absurd consequences ; for example, in a
harmonic bell [Harmonika-glocke] the Ninth 4:9 is the
first consonance ! " ^

It will be perceived that Chladni does not, like Helmholtz,
make the whole question turn on Rameau's use of the word
" natural." Altogether apart from the meaning which may
be attached to this term, Chladni considers that the facts
are against Rameau.^ It is true that Helmholtz points to
the inharmonic partial tones, the dissonances, which result
from the natural resonance of such bodies as striking rods.
But it can hardly be supposed that Helmholtz means, like
Chladni, to advance this- as a proof of the impossibility, the
absurdity, of claiming harmonic resonance as the real principle
of harmony. On the contrary, it is important to note,
Helmholtz supports Rameau's view, and considers that he
was right in his conjecture that in harmonic resonance we
discover the proper basis of the theory of harmony. In*
referring as he does to the " dissonant chords " produced by

^ Kurze Uebersicht der Schall- und Klanglehre, nebst einem Anhange
die Entwickelung und Anordnung der Tonverhaltnisse betreffend (1827).

^ It should be noted, however, that Chladni explains consonance as
due to the simplicity, or comparative simplicity, of the ratio which
determines it (i : 2 ; 2 : 3, 3 : 4, etc.).

RAMEAU'S GENERATION HARMON IQUE 169

such bodies as striking rods, Helmholtz therefore can only
mean to indicate that the mere fact of their being " natural "
does not suffice to justify them aesthetically. Rameau to
be sure never evinced any desire to justify them t-esthetically ;
nor has any theorist of repute since his time shown any great
eagerness to accept them or to give them a place in the
theory of harmony. The difftculty, indeed, with many of the
theorists who have made acoustical phenomena the basis
of harmonic science and of " natural discords " has been,
and still is, not so much to discover a reason for accepting
the "natural discords" of which Helmholtz speaks, as to
discover a reason for rejecting them ; for if, as Helmholtz
points out, we find in Nature not only help but hurt, it
must be remembered that in music we find not only concord
but discord. In this respect at least music holds the mirror
up to Nature.

As is known, Helmholtz considers the effect of dissonance
to be due to the phenomena of beats, that is, rapid pulsations
arising from the alternate reinforcement and enfeeblement
of sound. These beats interrupt the steady uniform flow
of the sound, and produce an intermittent effect on the ear,
corresponding to the effect produced by a flickering light
on the eye. The effect of such sensations is unpleasant, and
this unpleasantness of effect is owing to the intermittent
excitement such sensations produce in the nerves of hearing
and of sight. The physical or physiological explanation
of dissonance, therefore, is to be found in the jolting or jarring
of the auditory nerve by means of beats. On the other hand,
consonance is distinguished by the absence of beats. The
tones which form a consonance co-exist undisturbed in
the ear, and there is nothing to interrupt the smooth,
continuous flow of sound. This is the physical explanation
of consonance. 1

In order that two or more sounds, when heard simul-
taneously, should affect the ear with the sensation of
consonance, it is necessary that they form with one another
perfectly definite intervals. But the proportions which
exactly determine such intervals are found in the natural
relations which may be observed to exist in the resonance
of sounding bodies such as a stretched string or organ

1 See also pp. 383-385-

I70 THE THEORY OF HARMONY

pipe. Such natural relations, then, may be said to constitute
the proper basis of a theory of harmony, not because they
are natural, but because they determine the consonances^
Is Helmholtz able, then, by means of his theory of Conson-
ance as resulting from the absence of beats, to draw an
effective distinction between intervals which are consonant
and those which are dissonant, a distinction which is necessar}'
in music, and one which is made by every musician ? The
relative degrees of " harmoniousness " or consonance pertain-
ing to the various consonant intervals Helmholtz illustrates
by means of the following table : —

Octave

Twelfth ..

Fifth

Fourth

•• 3

Major Sixth

Major Third

.. 4

Minor Third

•• 5

Of these consonant intervals, the Octave and Twelfth are
the smoothest in effect. On the other hand, the major and
minor Thirds exhibit a decided roughness of character,
this roughness being due to the presence of beats arising
between the upper partial tones of the two sounds forming the
interval. \Vhat, then, of the minor Sixth? This interval
Helmholtz finds to be so rough, indeed dissonant in character,
that he is unable to explain it as a consonance, which no
doubt accounts for its exclusion from his table of consonant
intervals. Indeed the minor Sixth, it appears, is frequently
less consonant than the " natural " Seventh, 4:7. " The
sub-minor Se\'enth 4 : 7," Helmholtz remarks, " is very often
more harmonious than the minor Sixth 5:8; in fact, it is
always so when the third partial tone of the note is strong^
compared \\dth the second, because then the Fifth has a
more powerfully disturbing effect on the intervals distant
from it by a Semitone, than the Octave or the sub-minor
Seventh, which is rather more than a whole-tone removed from
it." - Later in his work, however, Helmholtz is of opinion
that this " natural " Seventh is sufficiently dissonant to

^ Sensations of Tone, Pt. II., Ch. 11. 2 jj,,\/ Ch. 10.

RAMEAU'S GENERATION HARMON IQUE 171

form the Seventh of the chorcl of the Dominant Seventh.^
Finally, he tells us that the reason why this sub-minor
Seventh is not used as a consonance in music is because
" when combined with the other consonances in chords it
produces intervals which are all worse than itself." -

Helmholtz however is quite well aware that the minor
Sixth is actually employed in music as a consonance. This,
he thinks, can onl}- be explained by the fact that the minor
Sixth is the inversion of the major Third. But in this case
what becomes of the " jarring of the auditory nerve " produced
by the beats which so distinctly characterize the minor Sixth ?
Are these beats no longer present ? On the other hand we
find that the Fourth is a better consonance than the major
Third, and that it is unnecessary to explain its consonant
character as arising from the principle of inversion. But
unfortunately for Helmholtz's theory, this Fourth, ever since
within one or two centuries of the first rude beginnings of
harmonic music, has been consistently treated by musicians
as a dissonance, except when it represented the inversion
of the Fifth, and this apart from any question of modulation,
key, or tonal order. It is unnecessary to dwell here on the
unsatisfactory results obtained by Helmholtz in treating
of the consonance of the minor harmony. Only one other
point need be noticed. From the table given above it appears
that the Fourth and major Sixth are superior, as consonances,
to the major and minor Thirds. It follows therefore that the
* position of the major harmony is in effect not only more
consonant than the minor harmony in fundamental position,
but more consonant than the fundamental position of
the major harmony itself, and this apart from any question
as to the " tonal function " of the chord. It is scarcely
credible that Helmholtz should attempt to vindicate this
extraordinary result of his theory. Nevertheless we read :
" For just intervals the Thirds and Sixths decidedly disturb
the general harmoniousness more than the Fourths, and
hence the major chords of the Fourth and Sixth are more
hannonious than those in the fundamental position." ^

Helmholtz concludes his researches into the nature of con-
sonance and dissonance with the remark that it is impossible

^ Sensations of Tone, Part III., Ch. 17. - Ibid.. Part II., Ch. 10.

' Ibid., Ch. 12.

172 THE THEORY OF HARMONY

to draw any sharp line between the two, and that the distinc-
tion between consonant and dissonant intervals " does not
depend on the nature of the intervals themselves, but on the
construction of the whole tonal system " ! ^ — It cannot be
denied that a close connection exists between beats and the
phenomenon of consonance : even the most experienced
tuner has daily reason to be profoundly grateful to Nature
for the assistance she renders him in the practical work
of " tuning." Nevertheless, it can scarcety be maintained
that Helmholtz provides us with an adequate solution of
the problem of Consonance.

The conclusions arrived at by Helmholtz in his investigations
concerning the nature of scales, tone-systems, consonance
and dissonance, and of harmony in general, have been regarded
in many quarters as authoritative and final. They have,
however, by no means met \\dth universal acceptance. On
the contrary many of them, and especially within recent
years, have met with a vigorous opposition from musical
theorists and psychologists. It is in great part owing to the
unsatisfactory nature and inadequacy of the theoretical
results arrived at by Helmholtz in dealing with some of the
most fundamental problems of harmonic science that many
have concluded that the construction of any rational theory
of harmony on the basis of acoustical phenomena is an
impossibility.

Thus Dr. Riemann is of opinion that this reaction against
acoustical theory represents a decided gain for the theory of
harmony : - while speaking of the difficulties presented by the
minor harmony, he remarks: — "The principle of ' klang-
representation ' {KJang-V ertretung) has really to do not with
physical science, nor with physiology, but with Psychology.
If it is a fact of experience that we are able to understand
a tone as the representative of a minor, as well as of a major
chord. . . . then this is a scientific fact, which forms as
good a foundation to build upon as acoustical phenomena.
Once this fact has been thoroughly established and under-
stood, we need not concern ourselves further with the
physical basis of the minor harmony." ^

* Sensations of Tone, Part, II. Ch. 12.

* Geschichte der Miisiktheorie, p. 502.
^ Die Natur der Harmonik, p. 29.

RAMEAU'S GENERATION HARMON IQUE 175

But it is Fetis who is the most vigorous and uncompromising
exponent of the doctrine that all harmony, scales, tone-
systems, etc., have not a physical but a psychological basis.
In the preface to his Traite de I'harmonie (1844), Fetis tells
us that he had made the subject of harmony a life-long study,
and that before venturing to publish the treatise in question,
he had, in the course of twenty years, read and studied no
fewer than 800 works deahng with the subject, as well as
analysed musical compositions of every epoch. Fetis is
well aware of the reach of his subject, and during these twenty
years the Pythagorean notion of a universal harmony, of the
"harmony of the spheres," seems to have taken up not a
little of his attention. " These ideas," Fetis remarks, " con-
cerning music, this primordial art which alone of all the arts
has been accounted worthy of a divine origin, we find again
with certain modifications in different parts of the Orient ;
one principal idea however runs through the centuries,
namely, that of a harmony which rules and directs the move-
ments of the heavenly bodies, and of which the music of men
is but an imperfect imitation. The Hebrews borrowed the
notion from the Chaldeans and the Sadducees who, attentive-
observers of the course of the stars, attributed to them an
influence direct, supreme, and eternal on the whole of
the universe. This led the Hebrews to *the conception of
particular intelligences, the Angels, who presided over
the harmony of the stars, and whose songs, in which they
praise and glorify the Eternal Being, are formed by the
motions of the celestial spheres. ... It is this same
idea of a power, inferior to that of the Creator of the
universe, but which gives life and movement to his work,
which Pythagoras borrowed from the peoples of the Orient,
the idea of a universal harmony. To Pythagoras it is the
soul of the world, and he attributes to it harmonic proportions
with which Plato makes us acquainted in a somewhat obscure
passage of his Timceus, and which are those of the musical
scale of the Greeks. . . . The notion of a universal harmony
did not stop here ; propagated from century to century,
accepted and modified by the school of Alexandria,
reproduced in the writings of Plutarch, of Cicero, Ptolemy,
and many others, it again emerges after the Renaissance in
the works of Plato's commentators, and ends by leading
astray the powerful intellect of Kepler, just at the time

174 THE THEORY OF HARMONY

this learned man had discovered the fundamental laws of
astronomy." ^

Helmholtz also refers to this subject, and adds that " even
in the most recent times natural philosophers may still be found
who prefer such dreaming to scientific work."- For Helmholtz
there is no mystery whatever. "The enigma," he remarks,
" which 2,500 years ago Pythagoras proposed to science —
' Why is consonance determined by the ratios of small whole
numbers ? ' — has been solved by the discovery that the ear
resolves all complex sounds into pendular oscillations, accord-
ing to the laws of sympathetic vibration. Mathematically
expressed, this is done by Fourier's law, which shows how
any periodically variable magnitude, whatever be its
nature, can be expressed by a sum of the simplest
periodic magnitudes. The length of the periods of the
simply periodic terms of this sum must be exactly such,
that either i, 2, 3, or 4, etc., of their periods are
equal to the period of the given magnitude. This, reduced
to tones, means that the vibrational numbers of the partial
tones must be exactly once, twice, three times, four times, and
so on, respectively as great as that of the prime tone. Ulti-
mately, then, the reason of the rational numerical relations
of Pythagoras is to be found in the theorem of Fourier, and in
one sense this theorem may be considered as the prime source
of the theory of harmony." ^ It should be noted that when
Helmholtz speaks of vibrational numbers corresponding to
the terms i, 2, 3, 4, etc., he means, of course, i, 2, 3, 4, 5, 6, 7,
8, 9, 10, II, 12, 13, 14, 15, and so on ad infinitum. In this
series we find the terms 9 and 15, whose vibrational numbers
are exactly nine and fifteen times respectively as great as that
of the prime tone. The first represents a major Ninth or
major Second ; the second represents a major Seventh. Are
these intervals consonant, seeing that they are comprehended
in Fourier's law ; and is Helmholtz here presenting us with
a new theory of Consonance ? On the contrary, as Helmholtz
knows well, they are dissonant ; and indeed, if the terms of
the harmonic series be extended much further, we meet
with as large and varied an assemblage of dissonances
as the ear could well conceive of. It is difficult to believe that

^ Traits de V harmonic. Preface.

"^ Sensations of Tone, Part II., Ch. 12. 3 Ibid.

RAMEAU'S GENERATION HARMON IQUB 175

Helniholtz really succeeded in satisfying himself that we ha\-e
here the solution of the problem of Consonance, or of the
enigma proposed by Pythagoras.

It is evddent that Fetis is not much enamoured of the
idea of a " harmony of the spheres," of a " harmony of
nature." It is, no doubt, a grand and subhme conception.
Perhaps the morning stars sing together, and perhaps there
is an ear to hear their music. But this, he thinks, has
nothing to do with the theory of harmony. It may be good
poetry, but it is bad science. What, he asks, in such a
case, becomes of human liberty and free-will? "Let it be
supposed that nature has determined a fixed and invariable
order of sounds, and that man is conscious of this immutable
order, it would be necessary to admit that any variet}^ in
the character of music and of harmony is impossible, and
that the impressions produced by the combinations of these
sounds ought to be identical in the case of all individuals
gifted with the organ of hearing." ^

Wliat then is the actual basis of music, of harmony ?
It is, Fetis answers, the scale : and, in the tonal relationships
of the notes of the scale, in a word, in Tonality, we find the
source and explanation of harmony, of harmonic relationship,
and harmonic succession. Scales are, however, by no means
a product of nature. Nature does not make scales ; she only
supphes the raw material, as it were, from which scales may
be formed. " If it be asked," says Fetis, " what is the
principle of these scales, and what determines the order
of their sounds, if it be not acoustical phenomena, and the
laws of calculation, I reply that this principle is purely
metaphysical, and that such an order, and the melodic and
harmonic phenomena to which it gives rise, are conceived by
us as the necessary consequence of our conformation and
our education. It is something which exists for us by itself,
and independently of every cause outside ourselves. . . .

"Nevertheless, we seek in acoustical phenomena for the
explanation of a tonal order, of a tonaUty, which lies ready
to our hand ! It is necessary to point out that these acoustical
phenomena, badly understood as they often are, have not
the significance that one so carelessly attributes to them.
For example, the major harmon}^ which has been observed

1 Trait t' de Pharmonie, Preface.

I 76 THE THEORY OF HARMONY

to result from the resonance of certain sonorous bodies, is
accompanied by other more feeble sounds. It is the same
in the case of other sonorous bodies which produce other
harmonies. Besides, it has been proved that one and the
same body is capable of different modes of vibration, giving
rise to diverse phenomena. It has been demonstrated, for
example, that the interval of the tritone discovered in the
resonance of a square metal plate is the result of the vibration
of the plate in the direction of its diagonal ; whereas other
forms of vibration of the plate give rise to other phenomena.
Let it be supposed that in course of time we discover acoustical
phenomena which furnish us with all the harmonies possible
in our system ; must we conclude that these unknown
phenomena are the origin of the harmonies discovered a priori
by the great composers ? Truly this would be a rude blow
dealt at our philosophic liberty ; a somewhat perverse
application of the doctrine advanced by certain sophists of
the influence of occult causes on the determinations of the
human will." ^

The case against Rameau certainly appears to be a strong
one. How, asks Berlioz in effect, are we really able to
distinguish between consonance and dissonance, and what
means do we have of discriminating between sonorous bodies
which produce concord, and those which produce only
discord ? Is not the ear the sole judge ? If not, what other
means do we have ? There is no other. Such being the
case, why then not admit that " harmony is the result of a
choice of sounds, according to the different impressions that
they make on our ear " ? To this Rameau might have
replied that even if it be granted that the ear is the sole means
we possess of distinguishing between consonance and dis-
sonance, it does not. necessarily follow that it is free to choose
the intervals it may apprehend as consonant, and which
constitute harmony, nor does it follow that, in the words
of Fetis, " harmony is something which exists for us by
itself, and independently of every cause outside ourselves."

The question has another aspect. In all ages, as Fetis
himself points out, men have thought that they discerned
in music a faint echo, as it were, of some far-off celestial
harmony, and have regarded it not merely as a mode of

i Esquisse de Vhistoire de I'harmonie.

RAMEAU'S GENERATION HARMON IQUE 177

expression of the human soul, but, hke Beethoven, as con-
necting in some mystical way the individual soul with the
great universal Over-soul. They have gone even further,
and have considered music to be essentially related to that
mysterious power which guides the stars in their courses.
But all this according to Fetis is a delusion : vain out-
pourings of childish or over-heated imaginations ; sentiments
which, at the best, are mere mysticism, incapable of clear
definition, or of being expressed in scientific language. The
music of man has nothing to do with any pretended " harmony
of nature," or " music of the spheres." All music, harmonic
or melodic, has its origin in scales ; but scales are man-made ;
nature makes no scales. Why waste time in the attempt
to identify the fundamental principles of harmony with the
constitution and course of nature ? It is a mere idle dream,
unworthy of the scientist and the philosopher. Man cannot
reach the stars ! He may long, like Goethe, after the infinite
soul of Nature, but he cannot grasp it !

It is evident that, on the hypothesis of Fetis, we are con-
fronted not only with a serious theoretical, but by a no
less serious gesthetical difficulty.

Rameau strives to identify music, to some extent at least,
with reality, with objective truth. Fetis, on the other hand,
can offer no reason whatever why music should not be con-
sidered merely as a play of sensations, the mere chance occasion
of a passing pleasure. There are many, it is true, whose
philosophy does not forbid such a view. Such a philosophy
at least, we are told, does not go beyond the facts so far as
these are known to us ; in any case, there is much in it
calculated to soothe and restrain the too ardent spirit. Why
need we on this account, it is asked, compare the charming
art of music to " a tale told by an idiot, mere sound and fury,
signifying nothing"? Is the delight we find in music nothing
in itself? Is it nothing that music, with its soothing influence,
should help us to bear with greater equanimity the ills of
life ? May we not even reverence those gifted men who, out
of their genius, have created for us such beautiful phantasies ?
But if this is the conclusion at which we must arrive, it is. a
sorry conclusion. It means, at least ultimately, the certain
degradation of music. Music becomes a mere titillation of
the aesthetic palate, a pleasure which of course, at least at
first, should be regarded as being a Httle higher in the scale

178 THE THEORY OF HARMONY

than that to be derived from eating and drinking. But
soon even this place of honour is lost, and music, which
early Christianity considered to be the handmaid of rehgion
becomes, as among the degenerate Greeks and Romans,
a means for enhancing the pleasures of the table. Such a
philosophy no doubt is not much disturbed by " idle
dreams." This, however, is only what might be expected.
No dreams, idle or otherwise, disturb the placidity of
artistic death.

One ma3^ here refer in passing to the opinion expressed by
certain philosophers of the present day, that if music has
as Rameau claims for it its source in Nature, then all music
becomes but the chance occasion of a passing pleasure, and
the creative artist merely a kind of aesthetic cook. The
considerations we have just advanced would go to prove
that exactly the opposite was the case. But we have here,
probably, nothing more than a misunderstanding, due to
some confusion of ideas.

It is unnecessary to enter at present into an examination of
the theory of harmony which Fetis has propounded in his
Traiie. Only one other point need be discussed here. Let
it be assumed that we are in complete agreement with Fetis
and BerHoz in their contention that harmony has nothing
to do \\dth acoustical phenomena, but that its principle is
purely psychological, and that, in the full enjoyment and
exercise of our " philosophic liberty," we select those sound-
combinations which impress us as being harmonious or
consonant. Let us take those intervals which from the
earliest times, and among all peoples who have possessed any
developed tone-system, have been regarded as consonances,
namely, the Octave, Fifth, and Fourth. We find, however,
as did Pythagoras in the sixth century B.C., that these
consonances which have been undoubtedly selected by man
in the free exercise of his genius, or at least without
any conscious dependence on mathematical law, are all
expressed by means of the proportions 1:2:3:4. Not
only so, they arise according to a quite definite mathe-
matical principle, which determines their respective degrees
of perfection (Octave =1:2, Fifth = 2:3, Fourth = 3:4).
Such a fact might well cause Fetis to rub his eyes, and
ask himself whether some " occult influence " had not
indeed been at work here.

RAMEAU'S GEXERATIOiX HARMON IQUE 179

Further this mathematical principle accords not only
with the order of these consonances, but with the historical
development of harmony, in which first the Octave was used,
as in the magadizing of the Greeks, and many centuries
later, the Fifth and Fourth, as in the ecclesiastical organum.
These consonances constituted the basis of the whole system
of ecclesiastical modes. Authentic and Plagal. By the
time of Zarhno, and indeed much earlier, the " natural "
Thirds determined by the ratios 4:5:6. had been apprehended
as consonant. Accordingly we find Zarlino drawing a new-
distinction between the modes, and classifjang them as
major or minor according to the nature of the Third which
appeared above the Final. As we shall see more clearly
later on, the introduction of the " natural " Thirds led in
great measure to the ultimate overthrow of the old modes,
to the emergence of our two modes of major and minor, and,
consequently, startling as the statement may appear, to
the gradual decay of an old artistic world, and the rise of a
new period of harmonic music. Rameau, then, would seem
to have some ground for his belief that harmony is " not
arbitrary, hut arises from a definite principle."

As to the manner in which this principle has influenced the
course of harmonic development, there is nothing " occult "
whatever. Descartes had already remarked that we hardly
ever hear a musical sound without at the same time hearing
its Octave. This Octave, indeed, forms part of the resonance
of the fundamental sound. But what is true of the Octave is
true also of the Fifth and Fourth, as well as of the natural
Thirds. As Helmholtz himself informs us, all sounds suitable
for musical purposes are richly endowed with upper partial
tones. 1 In every musical sound, then, produced by the
human voice, these consonances were to be heard, sounding
now faintly, now powerfully, but ever present to the sensitive
and attentive ear. First the Octave, Fifth and Fourth
(1:2:3:4) were apprehended, and later the natural Thirds
(4:5:6). Rameau therefore might well claim, not only
that " harmon}' arises from a definite principle," but
that " this principle resides in musical sound itself." But
these are, in fact, the main points for which Rameau has
all along been contending.

^ Sensations of Tone, Part II., Ch. 10.

i»o

THE THKORV OF HARMONY

What, then, of the seventh upper partial tone, the "natural
Seventh," which may also be distinguished, though with greater
difficulty, in the resonance of musical sound ? Is it consonant,
or is it dissonant ? But inasmuch as this " natural Seventh "
has no place in our harmonic system, the solution of the many
problems connected with our system of harmony does not
depend on the answer to this question. It is very improbable
that there exists any consensus of opinion among musicians
themselves as to whether this interval (4 : 7) is consonant.
Those who have been accustomed to regard the chord of the
Dominant Seventh as a "fundamental discord" formed by
means of the " natural Seventh," would probably tell us that
this interval is dissonant. Others, again, would be of opinion
that in itself it is a consonance. One thing, however, is
certain : Its employment in harmony and in the art of music
would necessitate a change in our harmonic system.

But how then, it may be objected, explain other upper
partial tones which are higher than the seventh ? Take for
example d", which is the Ninth upper partial of C. Of the
nature of this interval there has never been any division of
opinion among musicians. It has been consistently regarded
as a dissonance, and in the form of the major Ninth (4 : 9),
or major Second (8:9), it has long been actually used in
music. Is this interval derived from the harmonic series, and
if so, why should it have found a place in our harmonic system
while the " natural Seventh " is excluded ? This question
is not difficult to answer. The ear does not regard C, but g,
as the fundamental sound or bass to which d" must be related ;
that is, d" is Fifth (Twelfth) of g. In acting as it does in this
way, the ear alHes itself in a most striking way with the
operations of Nature. For as Rameau points out ^, not only
C gives rise to a certain order of harmonic sounds ; its Fifth,
or Twelfth, g, gives rise to the same order of sounds ;
thus : — _Q „

y II *i2th]]

(y II ^sjiT li

N^ rgHwi II ^3jj

Fundamental.
Fundamental .

See following chapter.

RAMEAU'S GENERATION HARMON IQUE 18 1

For this reason g itself becomes a fundamental sound, and is
the true Fundamental Bass of d" .

Finally, there is little difficulty in disposing of the remarks
of Fetis concerning his alleged discovery of the tritone in tlie
resonance of a metal plate. Why. asks Fetis in effect, does
Rameau not accept this " natural discord," and why does
he not acknowledge this particular metal plate as its source ?
One cannot fail to admire the audacity with which Fetis
advances such a criticism, nor to perceive how little he under-
stood Rameau's theory. It is true that Rameau well nigh
wrecks his theory in treating as he does of " fundamental
discords " formed from added Thirds. But it is not Rameau,
who throughout all his works consistently excluded the
" natural Seventh " from chords, whom we have to thank
for the introduction of " natural discords " into the theory of
harmony, but in the main Fetis himself. It was Fetis
who was one of the first to maintain that the chord of the
Dominant Seventh is derived from the " natural Seventh,"
and that it corresponds with the proportions 4:5:6:7. The
tritone then, according to Fetis. is determined by the pro-
portion 7 : 10. But Rameau gave the tritone quite a
different explanation; it was not a •'•'natural discord." It
is not against Rameau that Fetis should have directed his
criticism, but against himself, and all other theorists who
have indulged in " natural discords."

These theoretical considerations might well have induced
Fetis to exchange his prerogative of "philosophic liberty"
for, at least, an attitude of " philosophic doubt." They may
serve to indicate that Rameau in developing his principles
is on the right track, notwithstanding the contradictions and
even absurdities into which he frequently falls. But even
if these considerations were not present, and we were unable
to evoke in ourselves Rameau's enthusiasm for and faith in
his sonorous body, with its harmonic divisions, it might be
wise to reserve our judgment until we meet with a metaphysical
or psychological theory of harmon}' which does not lead us
into still greater difficulties.

I82

THE THEORY OF HARMONY

CHAPTER Vn.

FURTHER DEVELOPMENT OF RAMEAU'S THEORIES: GENERATION
HARMONIQUE AND DEMONSTRATION DU PRINCIPE DE

VHARMONiE {continued).

Diatonic System (Major Mode)-

Rameau now takes as his fundamental principle of harmony
the resonance of a sonorous body, which in addition to the
fundamental sound causes to be heard also the sounds of the
Twelfth and Seventeenth above, while at the same time it
sets in co-vibration with it — according to Rameau — the sounds
of the Twelfth and Seventeenth below -. —

ZM.nih

-i2t)>-

:z3i

nj,tti —

3p:i7th:

The origin of all harmony, then, is to be found in a single
sound ; all chords, whether consonant or dissonant, the Modes,
Harmonic succession (Fundamental Bass), progressions,
proportions. Cadences, Key-relationship, Temperament even
— all may be traced back to this source. In the Generation
Harmonique, therefore, we find that the first chapter — which
contains the Propositions and Experiences we have already
examined — is entitled: "Origin of Harmony"; Chapter 4,
" Origin of fundamental and harmonic successions from which
are derived geometric })r()gressions " ; Chapters, "Origin of
Consonances and Dissonances"; Chapter 6, "Origin of
the Diatonic System {genre), of Tetrachords, and Systems
ancient and modem," and so on in the case of the majority
of the other chapters. Tliese, and especially the modifica-
tions and further development which Rameau's theoretical

RAMl-.Al'S GHXHRATIOX HARMOMQUH 183

principles undergo, here, as well as in his Demonstration
dii principe dc Vharmoyiie, have now to be noticed. Let us
first of all examine Rameau's explanation of the origin and
nature of our Diatonic System (Major Mode).

A musical sound, begins Rameau, being not simple but
harmonious in its nature, will always represent its harmony.
" The grave and dominating sound, which is generally thought
to be a single sound. ... is always necessarily accompanied
by two other sounds which we will call harmonic. If this
fundamental sound changes its position, it is none the less
accompanied by the sounds of its harmony'. ... It is
necessary, therefore, alwa^-s to consider the sound in its
three-fold aspect.^ " When we hear any sound, we hear
also its harmony, and are preoccupied in favour of its Fifth,
as its most perfect consonance ; consequent!}-, if we do not
intone this Fifth after the sound first given, we intone one of the
sounds of its harmony, which then represents its fundamental
sound" {"qui represent toujours leur son fondamental ") .-
Rameau means as follows : — If, for example, c is the sound
first intoned, we accept this sound as Tonic, and naturalh-
intone after it if not its Fifth g, then one of the harmonic
sounds of this o', that is either d, a tone above c, or b a. semitone
below it. Both d and h represent the harmony of g, the Fifth
of c.

Given then our single sound, how is the mode, the scale,
to be developed from this sound ? "In the sonorous body."
proceeds Rameau, " the only sounds present are the fun-
damental sound, its Octave, Fifth, and major Third ; these
are the only sounds at our disposal, and the only hberty we
have is to take these sounds successi\-ely upwards [harmonic
progression], as well as downwards [arithmetical progression].
But how then ought we to regard the sound which succeeds
the fundamental sound ? Ought we to consider it as a new

1 Dr. Riemann, who in his brief analysis of some points of Rameau's
theory (Geschichte der Musiktheorie, Chap. 2. pp. 454-470) has
done excellent service in drawing attention to the importance, even
for present-day theory, of Rameau's researches, appears to do Rameau
less than justice when he remarks : — " Helmholtz has opened up
quite new perspectives by his conception of klang-rcpresentation .
Theorists may have suspected it, but no one has said [!] that tones ma\-
be regarded as the representatives of Idangs." {Die Xatur der Har-
monik, p. 28.)

« Gin. Harm.. Ch. iS. Art. 2.

1 84 I'HE THEORY Ol- HARMONY

fundamental, or as harmonic ? [representing the fundamental].
This indeed, is the great difficulty. If it is harmonic, there can
be no [fundamental] succession, and we shall be dependent
always on the same fundamental ; then the sound whicli
succeeds that first given must be regarded as a new fun-
damental. . . . This is necessary, since one cannot hear it
apart from the first, except in a new sonorous body, which in
its totahty corresponds to it. For in passing frorn one sound
to another, we pa-s from one sonorous body to another; every
tone of the voice, every pipe, every string are so many different
sonorous bodies, and consequently so many different fun-
damental sounds. . . . From this succession, which we
regard as fundamental, it follows that each of the sounds carries
its particular harmony, . . . consequently from such a
fundamental succession there necessarily results a harmonic
one. For example, when 3 [the Fifth] succeeds i [the Prime],
the harmony of 3 succeeds that of i, and the difference is,
that if the succession of fundamental sounds is determined,
that of the harmonic sounds is arbitrary ; in this respect,
that as each of them represents the fundamental sound,
from which they proceed, the one can be indifferently sub-
stituted for the other. . . . Hence there follows an indispens-
able principle which is that we must be guided only by the
fundamental succession, while on the other hand theterms of
the harmonic or arithmetical proportion should be considered
only as representing their fundamental sound. . . . This
principle, once understood, proves that the only sounds which
can succeed the sound first given are the Octave, Fifth, and,
major Third ; whence the relationships of these intervals
being known, it is quite easy to imagine, in such a case,
progressions determined by each of these relationships." ^

This not very lucid, even in some respects contradictory,
statement of Rameau is important. What he means is
evidently this : — A sound being given, the only sounds which
we have to follow this given sound are those which compose
its harmony. Thus, if c be the given sound, we may take
after it its Octave, c' , Fifth g, or major Third e, but these
only. But as both c and g represent the fundamental sound
c, no progress or movement to a fresh harmony can be made
so long as these sounds are regarded in their harmonic aspect

^ Gen. Hai)ii.. Ch. .|.

RAM i:aus (jhxhra tiox ha RMOSIOUH I 8 5

only, that is, as harmonic conrstituents of the sound c. It
is necessary that the sounds c and g be each regarded as new
fundamental sounds, each of which bears a harmony similar
to that of the sound first given. If we consider the sounds
c-e-g-c', in their harmonic aspect, as composing the harmonj'
of the sound c, we may proceed indifferently from one to
another of these sounds. This succession of sounds is there-
fore arbitrary, and may be said to be a melodic succession.
On the other hand, we may regard the sounds c-e-g-c' as so
many different sonorous bodies, each bearing its own harmony.
In proceeding from one to another of these harmonics, we
make o. fundamental succession. This fundamental succession
brings about a real progression of the harmony ; it is not
moreover like the first, arbitrar}-, but determined. Here
we find a slight contradiction, for the melodic succession as
explained by Rameau is not more, nor less, arbitrary than
the fundamental one.

" Whence," proceeds Rameau, " it follows that the propor-
tion of the Octave being i : J, or i : 2, of the Fifth i : \, or
I : 3, of the major Thircl i : ^, or i : 5, the idea of a duple or
sub-duple, of a triple or sub-triple, and of a quintuple or
sub-quintuple progression immediately presents itself to
us." 1 The duple progression, that to the Octave of the
Fundamental Bass, may be left out of account, as it brings
about no change in the harmony. Of the other two possible
progressions, that to the Fifth and that to the major Third,
which ought we to prefer ? Undoubtedl}- that to the Fifth.
The Fifth is the most perfect consonance after the Octave,
and follows it immediately in the harmonic series. The
fundamental progression (Fundamental Bass) in Fifths will
give us the Diatonic system, the ]\Iajor and INIinor Modes ;
that in Thirds will give us the Chromatic system. It we take

the Fifth-succession c-g (^^~^f) thus :—

(*)

— «

— » n

t'r») '

^—rj-

'■-»

o- ■

F.B.

■^ 8:1:3

^ Gen. Harm., Cli. 4.

1 86 THE THEORY OF HARMONY

we obtain the sounds c-g-c, and g-b-d {a) . These sounds being
approximated in diatonic order, by means of the identity
which we perceive to exist between octave sounds, we
obtain the tetrachord h-c-d-c {b) consisting of the following
degrees : —

b-c, a major (diatonic) Semitone = 15 : 16.
c-d, a major Tone . . . ==8:9.

d-e, a minor Tone . . . . = 9 : 10.

" If these are the smallest degrees which can result from
such a succession, where everything is derived from one
and the same sonorous body, it is necessary to regard them
as the only natural degrees ; we see this ; we feel it. If it
be objected that the minor [chromatic] semitone does not
find a place here, it has to be pointed out that this is not
nearly so natural, as we shall show later." ^ Rameau thinks
that he has here re-discovered the ancient Greek tetrachord
(Dorian tetrachord) which formed the foundation of Greek
theory in regard to their tone-systems. "It is from just
this diatonic order [b^. c^ . d^ . e) that the Greeks formed their
diatonic systems, to which they gave the name of tetrachords ;
its origin is to be found in the fundamental succession in
Fifths. ... It is astonishing that the Ancients have thus
discovered one of the immediate consequences of this principle,
without having perceived the principle itself, without even
having followed it in the proportions which they assigned
to the intervals of their tetrachords.""^

Notwithstanding Rameau's newly-found admiration for
the Greeks, he is nevertheless disposed to censure them, in
that they had discovered a diatonic system without, appar-
ently, being aware that this system really has its origin in the
Fundamental Bass. Zarlino also comes in for a share of
his criticism. " This author," he remarks, " starts with the
harmonic proportion, and consequently derives from it the
Octave, Fifth, and major Third, and indeed nearly all the
consonances. He discovers even the major and minor tone.
But in order to obtain the semitones he is obHged to abandon
liis principle, and can only derive them from the intervals
whicli he has just obtained by his divisions. Such was the
practice of the Ancients, since all say that they derived

1 Gdn. Harm., Ch. (>. * Ibid., Art. 2.

RAMEAUS GENERATION HARMON lOUE 187

the tone from the difference between the Fifth and the
Fourth." ^

But in order to complete the diatonic scale, it is necessary
that a second tctrachord be added to the first. This new
tetrachord is obtained b}' the addition of a new sound to the
Fundamental Bass, which so far has consisted of the
succession from Tonic to Dominant. As in this succession
the fundamental bass proceeded a Fifth upwards, to the
Dominant, the new fundamental sound necessary will be
discovered by allowing the bass to proceed from the Tonic
to the Fifth below — the Subdominant.^ The ascending
progression of the Fundamental Bass has already been
explained ; the descending progression is justified by the
power possessed by the Tonic of exciting co- vibration in the
sound lying a Twelfth (Fifth) below. The first is a harmonic
progression ; the second is an arithmetical one. Thus we
obtain all the notes of t^e diatonic major scale, by means
of a triple progression of the fundamental bass, as

c— (; — D

1:3:9'

Major ]\Iajc)r Minor Major Major Minor Major

h tone. tone. tone. ^ tone. tone. tone. Major tone. .', tone.

(It is necessary, Rameau explains, to take here G as Tonic,
otherwise confusion might result in respect of the terms

of the proportion, as for example, ( \'\ .

It will be observed that a break is made at the sixth degree
of the scale. This is necessary owing to the succession of
three whole-tones which " as one is aware, as one feels,
are not natural, and they are so little natural that they
can never be obtained from a fundamental succession in
Fifths [!]. To obtain them it would be necessary to allow
I to be followed by 9, in which case the third sound (/"#)

' Gen. Har>}!., Preface. - Ibid.. CIi. o, Art. 5.

I 88 THE THEORY OF HARMONY

proves afresh the want of relationship between these
harmonies. For if the harmony of i is 1:3:5, and that
of 9 is 9 : 27 : 45, if we double i and 5, so as to approximate

them to 27 (that is i : 27 = ^ "'^ and 5 : 27 =

' i6 : 27, J /

E — \\

. ^^ ) there is found between 16 : 27 a major Sixth

which is too large by a comma (80 : 81), while the inversion of
this Sixth gives us a minor Third a comma too small. Also
between 20 : 27 we have a Fourth a comma too large ; and
the inversion 27 : 40 gives us a Fifth a comma too small.
Therefore, as the harmonic succession is a necessary conse-
quence of the fundamental one, i and 9 cannot immediately
succeed one another without making the want of relationship
between their harmonies perceptible ; and this is the cause
of the disagreeable effect produced by the third tone." ^

Rameau treats of this also in the Demonst. du Principe de
I' Harmonic (p. 44). although not quite in the same way.
We may represent, he remarks, the triple progression by

^~^-~^' or ^ ~ ^ ' ~ I ^ In the first case, C will represent

3 : 9 :. 27 9 : 27 : 81 ^

the Tonic ; in the second case, G. The Fundamental Bass
may proceed either to its Fifth above, as C-G, or to its Fifth
below, as<3-F. In either case, the succession is perfect : that
is, the intervals thus obtained are of just proportion. But we
cannot have a fundamental succession of the two Dominants

because, as the note ^ is a constituent of the
3 ■■ -^7, 81

harmony of G, by making the two sounds F and G of the
Fundamental Bass succeed one another, we obtain a false

minor Third D-F. "We cannot hear ^nd together

3 27

without being understood since this 81 naturally

sounds along with 27. But from 3 to 81 is the same as from
I to 27, these being the first and fourth terms of the triple
progression, forming between them a minor Third which
is a comma too small. . . . This is evident proof of the
want of relationship between 3 and 27." "It has never
been known why three whole-tones in succession gave an

* Gen. JI arm.. Ch. 5, Art. 5.

RAMEAU'S GENERATION HARMON IQUE 189

unpleasant effect ; they can never result from a funda-
mental succession in Fifths, and the progression which
causes them [as i : 9] gives us, between the harmonic sounds,
intervals proscribed by Nature." ^

Nevertheless, Rameau has an uneasy feeling that all is
not right. " One makes use, however, of these three tones
in succession." 2 Besides, he has set out to prove that the
diatonic Major Mode is not only derived from a natural
principle, but is itself a natural product. "It is from the
resonance of the sonorous body alone that arise all our impres-
sions of harmony and of its most natural succession ; for
it is it alone which contains, which comprises within the
limits of its two Fifths this most natural sequence of sounds
known as the Natural Mode {Mode Nalurel).^ Rameau
is evidently much embarrassed by his inability to make the
notes of the diatonic order, starting with the Tonic and
proceeding upwards by degrees to its Octave, fit his Fun-
damental Bass. He thinks this might be effected by consider-
ing the diatonic major scale to consist of two disjunct tetra-
chords. After the first tetrachord a break would occur,
a point of repose, " a repose by virtue of which that which is
past is forgotten." ^ But the second tetrachord would then
be in a different key from the first. " This repose marks
a change of key, as soon as it occurs, since another sound is
taken as the Principal." That is : —

n *

Tetrachord.

Point of repose.

V »

~ CJ ^^ '^

i^^^^

— -i> —

r-»

i^} —

— .-S*

•^ Tetrachord.

TTiV-

(W— ^.

fr> rj 1

yw^ ,

fr* o< 1

fjt

-& ^

where in the first tetrachord ' is the Principal, or middle

term, that is, the Tonic, but in the second tetrachord

9
He is of opinion that we have not sufficiently profited by

1 Gen. Harm., Ch. o. Art. 5. "^ Ibid.

* Ibid., Ch. 0, Conclusions. ■» Ibid., Ch. 6, Art. 5.

lyo THE THEORY OF HARMONY

the wisdom shown by the Greeks in the disposition of their
tetrachords ; for they made these either disjunct (as above), or

conjunct, beginning with the semitone (that is b-c-d-e-f-g-a) ^ :
"it is only by this last means that one can continue the
diatonic order, whether ascending or descending, without
changing the Mode " [key] -. Here, the three whole-tones
in succession do not occur : —

«;>

-<s>-

o o — /tj r3

But Rameau does not mean to give up the attempt to
prove that the major mode is in reahty a natural product,
and that it is possible to find a Fundamental Bass for it.
This he hopes to achieve by means of Dissonance {double
employment of dissonance). By this means " the diatonic
order can commence with the principal sound, and continue
wdthout interruption up to its octave." ^

1 Kanicau here forgets that the Greeks regarded their tetrachords not
as an ascending, but as a descending succession of tones and semitones.

Thus the Dorian tetrachord of the Greeks, e-f-g-a, commenced, not
with a semitone, but a tone. He also forgets that each of his tetrachords
has " natural," not Pythagorean Thirds.

- Gen. Harm., Ch. 6, Art. 3.

^ Ibid.. Chap. II.

/ I

RAAIEAU'S GENERATION HARMON lOUE 191

Harmonic Dissonance : " Double Employment of
Dissonance " and the Chord of the Added Sixth.

Dissonance in music is, according to Rameau, a necessit\-,
and the cause of this necessity he explains much in the same
way as in the Traite arid Nouveau Systeme. Owing to the
uniformity of the harmony of -the three sounds of the Fun-
damental Bass, it is impossible, Rameau considers, for the
ear to decide which of the three sounds in question must
be regarded as " Principal," or Tonic.

"If the first two fundamental sounds which succeed one
another have nothing distinctive in their harmony, the third
will always be arbitrary ; and, in consequence, the principal
sound, as well as the key, will never be perfectly decided." ^
In making the Dominant and Subdominant harmonies for
this reason dissonant, Rameau's method of procedure is
somewhat different from that which he has followed in his
Traite, and leads to his famous device of the " double employ-
ment {double emploi) of dissonance " by means of the chord
of the Added Sixth. The interval, begins Rameau, which
ought to be added to the Dominant and Subdominant har-
monies in order to render these dissonant, is the minor Third,
because "as we have seen, the minor Third is the smallest
harmonic interval." - Next comes the question as to where
this minor Third should be placed ; and here one or two
considerations claim our attention. First, as no chord can
exceed the extent of an Octave, this Octave " provides us
\\dth a gap {vuide) in which the new sound may be placed "
(thus in the harmony g-h-d' — g' , the " gap " occurs between
d' and g'). Again, the new sound should be one of the notes
of the scale or key in which the original harmony occurs.
" The harmonic sounds of the Mode cannot be altered or
changed \\dth altering the Mode." Further, in adding this
Third, the imperfection of the Subdominant harmony, as
compared with the Dominant harmony, should be kept in
mind : the former arises from the arithmetical (descending)
proportion, but the latter from the harmonic (ascending)
proportion : " the same subordination should exist between
the Subdominant as compared with the Dominant harmony,
as between this and the Tonic." ^ "If the Third which

^ Gen. Harm., Ch. 9. * Ibid. ' Ibid.

J 92 THE THEORY OF HARMONY.

is thus to be added must be derived from the same Mode,
if its diatonic movement, or direction, must conform to that
which the fundamental succession has already determined
for it ; and if its situation as well as its species (genre) ought
to conform to the proportion whence proceeds the fundamental
sound to which it is added, then it must be minor, and must
be added above the Dominant harmony and below that of
the Subdominant ; seeing that the Dominant arises from the
harmonic proportion, in which the minor Third is at the top

(that is I X 1 f 1 whereas the vSubdominant

I, 2i ;ii 45 _ .■;, (i
arises from the arithmetical proportion, where the minor Third

is at the bottom"! (f ^^ f { { {)

The Dominant dissonance, then, will appear as

Min. 3rd Min. 3rd

g — 1) — d — f ', that of the Subdominant as d — / — a — c
36 : 45 : 54 : 64 _ 27 : 32 : 40 : 48

In the first case, the minor Third d-f is added above the
Dominant harmony ; in the second case; below the Sub-
dominant harmony. It will be noticed that in each discord,
the added minor Third has the proportion 27 : 32 ; this is not
a just minor Third, being a comma too small. This however,
Rameau thinks, ought to be regarded as a happy circumstance,
for by this means the dissonant nature of each chord is
impressed all the more strongly on the ear.^ The new sound/,
which is added above the Dominant harmony, is the
fundamental note of the Subdominant chord ; while on the
other hand the new sound d, added below the Subdominant
harmony, is not the fundamental but the Fifth of the
Dominant chord. In this way, the subordinate position
which distinguishes the Subdominant as compared with that
of the Dominant is preserved. This union, so to speak, of
the two extreme terms Dominant and Subdominant, by
means of the added dissonant sound, forces each of the dis-
sonant chords in question to return to the Tonic harmony :
" it connects each with the principal sound in such a way
that they cannot remove themselves from it." -

1 " Les deux memes sons, la und iit (or r^-fa) forment de chaque
.c6te la tierce mineure ajoutee, dont le rapport est meme altere, pour
mieux y faire sentir la dissonance." [Gcyi. Harm., Ch. 9.)

2 Ibid., Ch. 9.

RAMKAU'S GHNHRATION HARMONIQUE 193

What then must we consider to be the fundamental note
of the chord d-f-a-c' ? Is it/, or is it ^ ? According to Rameau,
d is the fundamental note of this chord. This form of the
chord is the most perfect, as it is composed of Thirds, re-
sembhng in this respect the Dominant discord. ^ It has also
a similar (cadential) resolution. " The order of the harmony
which is found above this new fundamental sound, being
like that of the Dominant. . . . obHges us to treat this
sound as a dominant, which must then descend a Fifth."
That is : —

ig=

m-.

-f^-

The chief difficulty, however, is not with the chord in the
position d-f-a-c', but in the position /-a-f'-^'. Already in the
Traite, Rameau had at some length discussed this chord
(chord of the Added Sixth, see pp. 112, 113). We saw how he
attempted to prove that it was possible to consider the chord
f-a-c'-d' in two aspects : first, as an original and fundamental
discord, with fundamental note/; secondly, as the first in-
version of the chord of the Seventh d-f-a-c' with fundamental
note d. In the Generation Harmonique, he makes a fresh
attempt to prove that the chord f-a-c'-d' is an original and
fundamental, as well as a " derived " or inverted chord.

If the Subdominant," he remarks, " receives the new minor
Third below it, note that in accordance with the first order
of the arithmetical proportion, and its necessary subordina-
tion to the harmonic, this Third ma}^ appear as a major

Maj. 6th
Sixth, above the same Subdominant ; since in 5 : 3

f — d — a

in which the lowest sound must be regarded as fundamental,
the major Sixth is direct.'"^ Such is Rameau's new and

2 Gen. Harm., Ch. g.

Ibid.

194 THE THEORY OF HARMONY

extraordinary explanation of the origin of the chord of
the Added Sixth. Here d, the added and dissonant note
of the chord f-a-c'-d' is found to be a constituent of the
proportion 5:3:.!, while c' , the consonant Fifth of the
chord, finds and indeed can find no place.

Rameau's long, confused, and contradictory explanations
of the nature of this chord are all directed towards one obj ect :
he wishes the chord f-a-c'-d' to be considered in a two-fold
aspect — as an original chord, with fundamental note /;
and secondly, as the first inversion of the chord d-f-a-c', with
fundamental note d. Not onl}^ so, he wishes to consider
the chord f-a-c'-d' as an original and a derived chord, at one
and the same time. This is the " double cmploi," which
Rameau regards as " one of the most happy discoveries."
" It is just this chord," he proceeds, " that we stand in need
of in order to carry the diatonic succession up to the
Octave." The manner in which he accomplishes this is
as follows : —

t _ .. S^' -^ - "— jj

^ '^^

4 <"-> ^-> ^^ '^ ~f^

Sir G> =-3 fm\ <5^

t«t

Here, in order to avoid the Subdominant-Dominant
succession, Rameau considers the note e* to form part of the
chord of the Added Sixth c-e-g-a, of which c is the reputed
fundamental note ; he then regards this chord as changing
its aspect ; it is now to be considered as the first inversion of
the chord of the Seventh a-c-e-g, of which a is the fundamental
note. This chord then finds its natural resolution on the
Dominant chord d-f^-a-c, and thus by means of this " double
employment of dissonance " the complete diatonic scale is
made to fit the Fundamental Bass ; or more accurately, the
Fundamental Bass is made to fit the scale. But the Tritone
and the " altered consonances " are still there ! Rameau,
as was to be expected, finds himself totall}^ unable to banish
dissonant intervals from the scale. He makes repeated

t f

RAMEAU'S GENERATION HARMON IQUE 195

attempts, but neither by any system of conjunct or disjunct
tetrachords, nor by any device such as " double employment,"
can he prevent the intrusion of " altered consonances " —
in reality dissonances — into what he rightly or wrongly
considers to be the natural Major Mode. The above scale
he considers to consist of two disjunct tetrachords, g-a-b-c
and d-e-f^-g, but somewhat inconsistently, he wishes the
break or point of disjunction to occur, not after the note c,
the fourth degree, but after d, the fifth degree of the scale ;
after such a break, he remarks, the progression of the
fundamental may be expected to be somewhat arbitrary
in character ! Rameau makes an attempt to justify his
procedure in this respect by a reference to the practice
of composers, who in harmonizing the descending major
scale "change the key, that is, the principal sound, giving
this to the Dominant g, and assigning to this sound a point
of repose "^ : —

_Q-

-<S-

Tv"

-e*-

■^ ^

^^ '^ r^

-<s*-

8 : 9 : 27

Here the sounds of the Fundamental Bass are all comprised
in the three terms 3 : 9 : 27, of which the central tenn 9 must,
according to Rameau, be regarded as Tonic. The term i,
which represents the Subdominant of key G, does not appear
at all ! The complete scale, therefore, would appear to be
in the key of the Dominant, D major. Notwithstanding this
apparent defect, which Rameau does not appear to have
noticed, it must be admitted that this harmonization of the
scale is a vast improvement, from a musical point of view,
on his version of the ascending scale.

In the Demonst. du Principe de VHarmonie, Rameau
makes a still further attempt to adapt his Fundamental Bass

' Gen. Hctnii., Ch. 11.

196 THE THEORY OF HARMONY

to the ascending major scale. He now considers it necessary
to make a modulation to the Dominant kev : —

9 II

/T '^ ''-' "

V L

-et-

/"^N-

(w.

^~\

Njl^

fTj

9 : 27 : 9 : 3 : 9 : 27 : 81 : 27

Here, at the point of repose, G*, Rameau changes

F — c — c;

3:9 : ^7

the Fundamental Bass: for the terms ^ ^ ^' he

substitutes ^ . This G, formerly Dominant, now

9 : 27 : 8i

becomes Tonic, and the second tetrachord appears in the
key of G major. " At sol," observes Rameau, " there
begins a new tetrachord, similar in its proportions to the
first, in which the two tones it contains are taken with
the same faciUty as in that which immediately precedes
it ; this becomes for the ear a new harmonic phrase,
whose connection wdth what precedes it no longer claims
our attention ; in fact in this new phrase the key changes,
and this is e\'ident from the necessit}' to make 8i succeed
27." As for the " altered consonances," matters are
worse than before. For now besides the false minor
Third d-j, we find an altered major Third f-a (64 : 81) a
comma too large, and an altered minor Third a-c (27 : 32)
a comma too small. It is noteworthy that in the Demonstration
Rameau makes no use of " double emplojonent." He indeed
refers to the subject, but dismisses it in a word. Speaking
of the two-fold aspect which the sixth degree of the scale
may assume, namely in the C major scale as Third of /, or
Fifth of d, he remarks : " One should observe in this connec-
tion how the question of " double employment " arises,
since it matters httle to the ear as to whether la should be
related to fa 3 as Third, or to re 81 as Fifth, seeing that it
forms with its Fundamental Bass in each case a consonance
of just proportion."

In all this — the change of kej- which arises in harmonizing

RAMEAU'S GENERATION HARMON IQUE 197

the scale by means of the Fundamental Bass, the impossibility
of making the extreme terms of the triple progression succeed
one another, the " altered consonances " which arise between
the sounds of the scale — Ramcau sees, however, only the
necessity for Temperament ; even more, its origin. He
remarks : — " The mode, in its origin, prescribes temperament
as a necessity, since the diatonic succession cannot reach
its full extent, that is, cannot be extended from a note to its
Octave, without an essential fault, whether as regards the
three tones in succession, the necessity to abandon the fun-
damental succession in Fifths, in order to substitute for it
one, as that between i and 9, which produces false consonances
between its harmonic sounds ; or finally the necessity to
make use of a new fundamental sound at the Fifth of one of
the two extremes in order to extend the diatonic succession
up to the Octave." ^

Such is the extremely important development which
Rameau's theory of the fundamental bass undergoes in the
Generation Harmoniqiie, and which, before we touch on
the question of Temperament, we must examine more
closely.

Examination of Rameau's Views concerning the Origin
AND Nature of the Key-system ; Difficulties in
connection \vith the Subdominant ; Helmholtz's
Theory of the Origin of Scales ; Difficulties
connected with the Tritone, " Double Employ-
ment," " False Intervals " of the Scale.

These investigations of Rameau into the nature of the
Diatonic System, of Dissonance, " double employment,"
etc., are of the utmost importance, containing as they do
the very essence of his fully developed and matured theory
of harmony. Here once again we discover not only Rameau
the musician, with a fineness of ear, with an intuitive percep-
tion of tonal relations, as these find expression in our modem

1 Crcn. Harm., Ch. 7.

198 THE THEORY OF HARMONY

harmonic system, amounting to positive genius, but also
Rameau the by no means clear-headed theorist, who has no
sooner taken up a definite position than he straightway pro-
ceeds to demolish it by a statement of the most flagrantly
contradictory character. Rameau, however, is apparently
less intent on system-building than on the discovery of truth :
and in this he claims our respect, as one of the most honest
of theorists.

Rameau makes the notable pronouncement that the sounds
of the scale, whether of the Major or Minor Mode, have their
origin in a series of harmonic successions determined by a
Fundamental Bass in Fifths, in which a central sound is
taken as Tonic, and a harmonic progression is made to the
Fifth above (the Dominant) and to the Fifth below (the
Subdominant). In this sense, our diatonic system is a Fifth-
system, and arises solely out of the chief harmonies of the key,
those namely of the Tonic, Dominant, and Subdominant.
The influence of this Fundamental Bass of Rameau was
widespread and powerful ; even if it was not fully under-
stood, and its theoretical significance not adequately realised,
it nevertheless came to be regarded as the central point
of his theory, and was held as an article of faith by many
musicians and theorists up to the end of the eighteenth
century. By the beginning of the nineteenth century,
however, it began to be considered as no longer adequate
for the growing needs of composers, or for the explanation
of the many new harmonic combinations and successions
which had, since Rameau's time, been evolved by composers
themselves. Mozart and Beethoven had, it was thought,
given the coiip-de-grdce to the system of the Fundamental
Bass. It soon became almost forgotten, and ever greater
importance began to be attached to that other — unfortunate
— aspect of Rameau's theory, namely, the generation
of chords by means of Added Thirds. In our own day,
however, there has been witnessed the renaissance of the
fundamental bass : its real significance for the theory of
harmony is being more adequately realized (Helmholtz,
Riemann, etc.), while, on the other hand, the generation
of chords by means of added Thirds is falling more and
more into discredit.

The inquiry with which Rameau begins is a pertinent one.
Given the first sound, for example c. what sound is to follow it ?

RAMEAUS GHXERATIOX HARMOMQUH 199

And how is this sound to be determined ?^ In the diatonic
succession of sounds there must clearly be some underlying
principle determining such a succession, and this principle
must be a harmonic one. If not, what other principle is
there ? There is none. It is harmony then which impels us,
after the first sound c, to intone that sound which after the
Octave is most perfectly consonant with it, namely, g, its
Fifth (or at least one of the harmonic sounds of this Fifth,
that is, d or b) or e, its major Third. Of these two sounds
g and e, g is first in the order of generation of the harmonic
sounds. Both are harmonic constituents of the fundamental
sound c, and represent this sound. The succession of sounds
c-e-g may then in a sense be described as a melodic succession.
But each of the sounds e and g may be regarded in another
aspect, namely, as itself a fundamental sound, bearing its
own harmony. In proceeding, then, from one to another of
the sounds c and g there arises a Fundamental Bass in Fifths,
and from this fundamental succession c-e-g — g-b-d we obtain

a tetrachord of the form b-c-d-e. In a similar way, by
means of a fundamental succession between the principal
so^d and that lying a Fifth below (Subdominant) we obtain

the tetrachord e-f-g-a. These two tetrachords when joined
together furnish us, in correct proportions, with all the sounds
of the Diatonic Major Scale.

^ The following passage from the Demonst. du Principe de I'Harmonie
(p. 8. et seq.) shows how strongly Rameau was influenced in his scientific
researches by the philosophic " Methode " of Descartes. Here Rameau
describes how, in order to discover what sound is most naturally
intoned after a given sound, he endeavoured to place himself in the
position of a man totally unacquainted with music ! "I found, in
truth," he remarks, " that there were certain sounds for which my
voice and car appeared to have a predilection [namely, the Fifth and
Fourth] . . . but this predilection appeared to me to be purely a
matter of custom. ... I therefore placed myself, as far as I possibly
could, in the position of a man who had never tried to sing, nor had
even heard music. . . . That done, I searched around me, and in
Nature, for what I could not find in myself. . . . My search was not
a long one. The first sound which fell on my ear was for me as a ray
of light ; I perceived at once that it was not a single but a composite
sound ; there, said I, is the difference between noise and [musical]
sound .... I named the first sound or generator ' fundamoital
sound ' ; its concomitants ' harmonic .sounds ' ; and there I had tliree
things quite distinct, and of natural origin — noise, fiiudamental sounds,
and hatmonic sounds."

;oo

THE THEORY OF HARMONY

Rameau now presents us with what appears to be a regular,
logical, and symmetrical Key-System, arising out of the three

chief harmonies of the key or Mode

b-d

the Tonic harmony occupying the central position, while the
Dominant harmony is represented as that of the upper Fifth,
and the Subdominant harmony that of the lower Fifth. That
is, our diatonic system is a Fifth system. " It is in the Fifth
alone," Rameau tells us, " that the Diatonic System
has its origin." ^ (Rameau, of course, refers here to the
fundamental succession of the bass, above each term of
which the complete harmony. Third as well as Fifth, is
understood.) In Chapter 4. {Gen. Harm.), also, he gives
the following table : —

729 243

la'n - mi^
81 27

sir>
9

■fa
3

la - mi - si -fajff

_i„

« 1

•J 4 A

2 !t

Ut - sol - re

where we find the key-system extending upwards (harmonic
progression) and downwards (arithmetical progression) f^pm
the central sound Ut ; but in which any three terms, of which
the central term is taken as Tonic, will represent the fun-
damental harmonies of the diatonic key system. By means
of the progression of these three fundamental sounds, the
key admits of the clearest possible definition : —

:g=

i^z

-<";>-

-f5>-

-G>-

221

^^—f^

:z3i

(at least, Rameau might quite well have maintained this ;
we have seen, however, that he considers dissonance to be
necessary for the proper definition of the key).

* Gi'n. Harm., Ch. 6.

RAMEAUS GHXHRATfOX HARMOXIQUB 201

From such a Fundamental Bass there follows a whole train
of consequences of the utmost importance for the science of
harmony ; as connection and succession of chords, resolution
of dissonance, modulation, cadences, etc. In the case of the
last, for example, Rameau points out how completely the
different effects produced by the various cadences correspond
with his explanation of their origin. We also find, in a
striking manner, the Tonic determined as the centre of the
whole key-system, the note to which the other notes of
the scale not only are related, but from which they
arise. The Tonic harmony also appears as the central
harmony, towards which all other chords or discords tend
to gravitate.

So clear an exposition does Rameau's theory of the three
chief harmonies of the key, Tonic, Dominant and Sub-
dominant, their tonal functions, determination of Cadence,
etc., appear to furnish of the nature of our diatonic and har-
monic systems, that one feels disposed to accept it without
further examination. But whether or not it be regarded
as necessary for a correct understanding of much in our har-
monic system, it must nevertheless be pointed out that it
has never yet been conclusively established. Rameau does
not succeed in finding for the Fundamental Bass a scientific
or a rational basis. Nor have his successors. This is not
surprising, for the difficulties in the way are by no means Hght.
One or two considerations have to be noticed.

(i) In the three terms which Rameau considers to form the

basis of the diatonic system, for example ^ ', the

1:3:9'

relationship of both Dominants F and G tt)wards the Tonic
C is, Rameau informs us, that of the Fifth ; that is, G is the
Fifth (Twelfth) above C, while F is the Fifth below. But while
G has its origin in the compound tone of C, F has not. F is
a new sound which cannot possibly be discovered among the
upper partial sounds of which C is the prime, or fundamental
note. In order to discover it, Rameau is obliged to have
recourse to a new acoustical phenomenon, namely, co-vibration.
While then G, the Twelfth above C, is a constituent of the
compound tone of C, and sounds along with it, F, the Twelfth
below, co-vibrates with it. The sonorous body corresponding
to this Twelfth below vibrates, Rameau assures us, through-
out its whole length, while at the same time it divides itself

2o:

THE THEORY OF HARMONY

into three equal parts or segments, with two nodes.^ But
here Rameau makes a serious error : for this Twelfth does
not vibrate throughout its whole length. It vibrates only
in segments, which produce a sound corresponding not to
F, the Twelfth below C, but to C itself, that is the Unison.
Therefore the exciting sound C does not produce co-vibration
in F, its Twelfth below. Rameau's explanation of the origin
of the Subdominant is based on a faulty observation of an
acoustical phenomenon. But even if he had succeeded
in proving that this Subdominant does really co- vibrate with
the principal sound, such an explanation would be by no means
free from serious objection. Rameau's fundamental principle
is that of harmonic resonance ; from this everything proceeds.
But while the upper Dominant arises from this principle,
the lower Dominant has to be explained by means of what
Rameau considers to be a quite different, a new and
remarkable acoustical phenomenon, apparently unrelated to
the principle of harmonic resonance.

But in fact the relationship which Rameau perceived to
exist between these two sounds, the Principal and its lower
Dominant, has been almost completely misunderstood, both
by Rameau, and by not a few of his successors. If we take
a principal sound c', its Twelfth (third partial tone) above is
g". In exactly the same way, we find that the Twelfth above
F is c': —

12tli

-GJ-

.12th.

^ " Prenez une viole, ou un violoncello, dont vous accorderez deux
cordes a la douzieme I'line de I'autre : raclez la grave, vous verrez
fremir I'aigue : vous I'entendrez peut-etre meme resonner. . . raclez
ensuite I'aigue, vous verrez non-seulement la grave fremir dans sa
totalite, vous la verrez encore se diviser en trois parties egales, formant
trois ventres de vibrations entre deux noeuds, ou points fixes. Pouy
s' assurer que la corde f rem It dans sa- totalite, pendant qu'elle se divisc
en trois, lorsquc I'aigue est raclee, il faut y effleurer les points fixes
avec rongic, ct on la scntira fremir en ces endroits." {Gen. Harm.,
Ch.i.)

RAMEAU'S GENERATIOX HARMOSIQUE 205

The relationship which g" bears to its fundamental
c', is the same in every respect as that which c' bears to its
fundamental F. But the central sound c' may be considered
as Tonic, while o'" is its upper, and F its lower Dominant. Then
the relationship between Dominant and Tonic is exactly the
same as that between Tonic and Subdominant. It is merely
the same process repeated a Fifth or Twelfth lower. It is,
therefore, obviously as incorrect to describe F as Fifth of c',
that is, considered in a downward direction, as it would be to
describe c' as Fifth of »". On the contrary c' is Fifth of F, just
as g" is Fifth of c'. ^^'e therefore discover the true and simple
explanation of what Rameau and his followers have regarded
as a strange and remarkable acoustical phenomenon. When
the principal sound c' is set in \dbration it causes to co-vibrate
with it, not F its Twelfth below, but that portion of the string
which corresponds to its third upper partial tone, this partial
tone being at the Unison of the principal and exciting sound.
This is easily proved, for not only may this partial tone,
contrary to Rameau's belief, be heard actualh' to sound, but
the string corresponding to the fundamental sound of which
it is a partial ma}' be observed, as Rameau had remarked, to
divide itself into three segments, each of which corresponds
to the sound c'. No doubt Rameau would have been agree-
ably surprised, at least at tirst, had he discovered that the
relationship between Tonic and Subdominant was detennined,
not by a new and unrelated acoustical phenomenon, but,
exactly like the relationship between Dominant and Tonic,
by his first and fundamental principle of harmonic resonance.

But the elucidation of this fact only serves to introduce
fresh problems ; and here we light on the difficulties which
have dogged the steps of every theorist who since Rameau's
time has made use of the arithmetical progression. Rameau
considers F to be the fundamental sound of the Subdominant
harmony Y-a-c. But he does not observe all that this implies.
If the fundamental sound F is to bear a harmony like that of
C and G, then in the harmony ¥-a-c, c must appear as Fifth
of F. That is, the Tonic C appears no longer as the central
and determining note, but is itself a determined note. It is
absolutely necessary for Rameau's explanation of the diatonic
and harmonic system that F should be a determined note, as
Fifth of C. How then is it possible to maintain the exact
opposite and to consider C as Fifth of F ? The Fifth must

204 THE THEORY OF HARMONY

necessarily appear as a determined sound, and the sound
of which it is Fifth must appear as the sound which determines
it, that is, it is its fundamental. If the relationship of a
fundamental sound to its Fifth may be indicated as I = Fun-
damental, II = Fifth, then the relationship of the two

terms of the fundamental bass ^ must be, if C is to be

' • 3
regarded, as Rameau requires, as the determining sound,

as : ■ :; : but the harmony F-a-c appears in the position
, S; and must necessarily do so. With the Dominant

I ir ■^

harmony, the position is quite different. In the harmony
G-b-d, the Tonic C does not appear as Fifth : it does not

appear at all. If we compare the harmonies y ^ t^t

we find g, with its harmony, quite clearly represented as

Fifth of C. If we compare the harmonies '^ ^ S^~^ «

we find c, with its harmony, appearing as Fifth of F.

Further, from the fundamental succession of the harmonies

of G and C, Rameau has generated the tetrachord h-c-d-e.
Proceeding in exactly the same way, he has afterwards

generated the tetrachord e-f-g-a from the fundamental suc-
cession of the two harmonies of C and F. Both tetrachords,
Rameau himself inform us, are generated in exactly the same
way. In each case, then, the fundamental succession is as
Dominant-Tonic, while the Subdominant linds no place.
Thus Rameau is forced to make the key-system appear as

f - a - c - e - g - b - d -^^ which 3 must be regarded as F'ifth of

I 3 9

I, and 9 as Fifth of 3, while J the Subdommant appears

as the determining sound, the foundation of the whole
key-system. But if we concede to Rameau the right to
regard C as determining a harmony in both an ascending
and descending direction, then C appears as the generator
of the two harmonies f-a\^-C and C-e-g. But the harmony
f-a\^-C is not a major but a minor harmony. This minor
harmony however, as Rameau well knows, is not the
harmony of the Subdominant of the Major Mode. While

RyVMEAUS GENERAriON HARMON IQUE 205

then it is impossible to deny the great importance and
theoretical significance of Rameau's conception of a funda-
mental bass founded on the three chief harmonies of the
key-system, we must nevertheless observe that Rameau does
not succeed in finding for it a logical, still less a scientific basis.
And what is true of Rameau is true also of his successors.

Not less important are Rameau's researches in connection
with the origin of the Diatonic Scale. Here we find ourselves
in the very heart of the mysteries of harmonic science, and
confronted by what must still be regarded as unsolved pro-
blems. Rameau is of opinion that the Diatonic Scale owes
its origin to his Fundamental Bass of three terms. In pro-
ceeding from one to another of these sounds, C and G, there
arises a Fundamental Bass in Fifths, and from this fundamental

succession we obtain a tetrachord of the form b-c-d-e. In a
similar way, by means of a fundamental succession between
the principal sound and that lying a Fifth below — the

Subdominant — we obtain a quite analogous tetrachord e-f-g-a.
These two tetrachords, when joined together, furnish us with
all the sounds of the diatonic major scale. Rameau then
compares his tetrachord with the diatonic tetrachord of the
ancient Greeks. He is convinced that he has discovered the
origin not only of the ancient Greek tetrachord, but also of
their system of conjunct tetrachords, and he cannot avoid
dwelling for an instant on the marvellous intuition of the
Greeks, who, without any actual knowledge of the Funda-
mental Bass, but nevertheless unconsciously guided by it,
were able to discover such a tetrachord. But of all comparisons
this, one would think, is the very one which Rameau might
have been expected to avoid most carefully ; for does not the
very fact of the existence of such a tetrachord and such a
diatonic system among the Greeks tend to demoHsh his theory
of the origin of the scale ?

Such, at least, is the view taken by Helmholtz, who
remarks : — "Theorists of our own day who have been bom
and bred in the system of harmonic music have supposed that
they could explain the origin of scales by the assumption that
all melodies arise from thinking of a harmony to them. . . . But
scales existed long before there was any knowledge or experi-
ence of harmony at all. . . . The same remark applies to
Rameau's assumption of an ' understood ' fundamental bass
in the construction of melodies or scales for a single voice. A

2o6 THE THEORY OF HARMONY

modern composer would certainly imagine to himself at once
the fundamental bass to the melody he invents. But how
could that be the case with musicians who had never heard
any harmonic music, and had no idea how to compose any ?
Granted that an artist's genius often unconsciously ' feels
out ' many relations, we should be imputing too much to it
if we asserted that the artist could observe relations of tones
which he had never or very rarely heard, and which were
destined not to be discovered and employed till many centuries
after his time." ^ Helmholtz refers here to the Thirds and
Sixths, which were dissonant for the Greeks and other nations
of antiquity. What then is Helmholtz's explanation as to
how scales first arose ? He explains as follows. A note being
assumed as Tonic, the other notes necessary in order to form
a scale are selected from those sounds which are more or less
closely related to this Tonic. " We find," he remarks, " the
following series of notes related to the Tonic in the first degree,
lying above the fundamental note c, and related to it in the
first degree : —

c c' g f a e e\f

i:i 1-2 2:3 3:4 3:5 4:5 5:6

and the following series in the octave below : —

e C F G Eb Ab A -

1:1 2:1 3:2 4:3 5:3 5:4 6:5

As to the intervals which ought to be selected from the above
series in order to form the scale, this is " a question which
different nations have answered differently according to the
different direction of their taste, and perhaps also according
to the different delicacy of their ear." ^ Helmholtz then
proceeds to show how various scales were formed according
to the principle he has just enunciated. Thus the pentatonic
scale c-d-e-^g-a-^c' is one of the " more irregular forms of the
scale of five tones, in which the major Third g [4 : 5] replaces
the fourth /, which is more nearly related to the tonic c."
Again, one of the most ancient forms of the Greek tetrachord
is explained thus : — " If we assume e — the last tone in the
tetrachord b-e — as a Tonic, its next related tone within the
compass of that tetrachord is c, the major Third below e.

^ Sensations of Tone, Pt. III., Ch. 14. ^ Ibid.

RAMEAU'S GENERATION HARMON IQUE 207

This gives us the ancient enharmonic tetrachord of Olympos —

f*^^ '^ . In this, in many respects the most important

section of his work, the intervals which accordmg to
Helmholtz were selected in order to form these early scales
included not only the major and minor Thirds, but also the
major and minor Sixths, intervals which were dissonant, as
already remarked, for all antiquity. Nevertheless it is
Helmholtz who inquires how scales could be formed from
relations of tones which had either never been heard, or which,
when heard, were rejected ! If it is absurd to suppose that
the diatonic scale owed its origin to Rameau's Fundamental
Bass, it is no less absurd to imagine that scales first arose
according to the principle enunciated by Helmholtz. The
whole question is of the utmost importance for the theory of
harmony. In the meantime, it may be pointed out that
Helmholtz is in substantial agreement with Rameau's theory
of an " understood " Fundamental Bass as applied to our
modern scales. Thus the melodic succession, c-d-e, in which
we find the first three degrees of the major scale, is

c-d d-c
determined by means of the linking sound G : thus : ^-^ ^— ^

G G

It is evidently for Rameau a remarkable circumstance that

the progression from the central term to either of the extreme

terms of his Fundamental Bass furnishes him with a series

of degrees and of intervals of correct proportions. Thus

in the tetrachord b-c-d-e we find not only the major and
minor tone {c-d = 8 : 9, d-e = g: 10, 6-c = i5 : 16), but also the
major Third c-e (4 : 5) the minor Third (5 : 6) and the perfect
Fourth (3:4). It is indeed a noteworthy fact that these
determinations of the various intervals, corresponding to the
necessities of just intonation, were fixed by different theorists
long before Rameau formulated his system of the Fundamental
Bass. Thus in the sixteenth century ZarHno, Sahnas, and
others, recognize the following determinations of the intervals :—

Octave I : 2

Fifth 2 : 3 Fourth 3 : 4

Major third 4 : 5 Minor third 5 : 6

Major tone 8 : 9 Minor tone 9 : 10
Diatonic semitone 15 : 16 Chromatic semitone 24 : 25.

^ Sensations of Tone, Ft. III.. Ch. 14.

2o8

THE THEORY OF HARMONY

But indeed we already meet in Ludovico Fogliano {Musica
theorica, 1529) with the same determinations : —

9
To

1 •■

1 11

But Rameau's pardonable self-congratulation over this fact
disappears before the difficulties with which he soon finds
himself confronted. These arise, not in connection with the
individual tetrachords, but with his attempt to unite them so
as to form a complete scale. For, although each of his tetra-
chords, b^-d-e and 7J-g-a, furnishes him with intervals of

just proportions, no sooner has he joined both tetrachords
together than there arise " altered " consonances, that is,
intervals which are not consonances at all. This scale

h^c-d-e-f-g-a however is not complete, nor does it correspond

to any of our modern scales. Rameau now re-constitutes his

tetrachords, and gives them the form c-d-e-f g-a-h-c ■ From

these disjunct tetrachords he obtains all the sounds necessary
for the complete scale of c major, beginning with the Tonic c,
and proceeding upwards in diatonic succession to the Octave.
Here also false intervals are present. These are intervals
" proscribed by nature," and their presence in the " natural "
Major Mode is evidently for Rameau the occasion of consider-
able perplexity. Still, they are of httle account so long as
they are not perceived by the ear ; and indeed they can never
occur in a fundamental succession of the bass, which is
always perfect. Thus at («) the false intervals d-f (27 : 32)

T3'

m=^

331

-f^h-

/ t

RAMEAU'S GEXERATIOK HARMOXIQUE 209

and a-d (20 : 27) arise in the immediate succession of the
upper parts ; they are found again at {b). but not in immediate
succession.^ There is, therefore, this great difference between
the two cases, that whereas at {a) the false intervals are per-
ceived, at (b) they are not perceived by the ear. This
argument is further developed by Rameau in his remarks on
temperament.

But while Rameau has little difficulty with the first form
of the scale, he is totally unable to find a Fundamental Bass
for the second. Here difficulties crowd thick upon him.
There is the tritone, this " essential fault " of the natural
maj or mode. This can only arise from an immediate succession
of the extreme terms, i and 9, of the Fundamental Bass.
Such a succession however is impossible, and the attempt
to bring it about only results in evident proofs of the want
of relationship between these extreme terms : there is not
only the tritone, but the false intervals which arise in im-
mediate succession. These three whole-tones, as " one feels,
are not natural," and in short they " can never result from
a fundamental succession in Fifths." Here then, one would
think, the whole matter comes to an end. Rameau has set
out to show us that our major scale has been evolved from
his fundamental bass of three terms. He now tells us plainly
that the major scale can never be discovered from such a
bass. It is evident that it is not the presence in the scale
of " altered consonances," or of the tritone, which is the
real cause of Rameau's embarrassment. Instead of exhibiting
such unnecessary sensitiveness with regard to the "intrusion "
of these intervals, Rameau might have deduced from them
important results concerning the origin of harmonic dis-
sonance. The tritone is less an "essential fault " than an
essential part of our scale, and Rameau knows well that
in harmonic music the extreme terms i and 9 may succeed
each other quite freely. His real difficulty is, of course, to
account for such an immediate succession of the two
Dominants. He fails in the Generation Harmonique, and
in the Demonstration he confesses his failure. On the
other hand, Rameau deserves credit in that he perceives,
unhke most other theorists, that some explanation is
necessary.

^ See, however the remarks on this subject in Ch. S (Temperament)
P

210 THE THEORY OF HARMON\'

It is unnecessary to dwell on the contradictions and even
absurdities in which he becomes involved in treating of
Dissonance and the " Double employment of Dissonance."
In generating the Dominant and Subdominant discords, the
added interval should, he thinks, be a minor Third, because
this is the smallest interval used in harmony. He also thinks
that it is quite clear where this added Third should be placed,
for we find a " gap " between the fundamental sound and its
Octave. Thus in the chord g-h-d — g\ the gap occurs between
the sounds d — g' ; in this gap therefore the added third should
be placed. As for the Subdominant chord, Rameau evidently
assumes that the gap, in this case, occurs between c — -/ in
the % position of the chord, thus : c — f-a-c'. In adding
this dissonant sound heloiv the Subdominant chord, Rameau
is evidently quite satisfied that he has proved this chord to
arise from the arithmetical proportion, and that it is only
necessary to extend this proportion further downwards, as he
extends the harmonic proportion upwards in the case of the
dominant harmony. But the Third added to both harmonies
(27 : 32) is not a minor Third. Rameau, however, regards
this as a merit ; it intensifies, he tells us, the dissonant effect
of both discords. So then, as d'Alembert remarked, it is
proportion which enables Rameau to form the discord ; while
it is the lack of proportion which renders the dissonance
perceptible. Rameau does not stay to consider whether the
addition of a true minor Third might not still further increase
the dissonant effect. It is also 'to be noticed that one of the
great advantages, from Rameau's point of view, in adding
the dissonance below the Subdominant harmony is that the
resulting dissonant chord is now similar in form to that of the
Dominant, that is, it is composed of a series of Thirds !

In the Dominant discord, the added dissonant sound is the
Subdominant itself ; in the Subdominant discord, it is the
Fifth of the Dominant. In each case the effect of the added
dissonance, says Rameau, is to compel each discord to proceed
to the Tonic harmony. Whether or not this be true of the
Dominant discord, it is not true of the Subdominant discord.
For if there is a tendency on the part of the Subdominant
harmony /-rt-c to proceed to that of the Tonic, the tendency of
the dissonant chord d-f-a-c is rather to resolve on the Dominant
harmony g-h-d. Rameau himself recognizes this, for he
gives, in one and the same chapter, two absolutely contradictory

RAMEAUS GEXERAriO\ HARMONIQUH 21 r

explanations of the natural tendency of this dissonant chord.
First, by the union of both Dominants in this chord, it is
" compelled to return to the principal sound." Secondly,
" the order of the harmony above this new fundamental sound,
being like that of the Dominant [Dominant Seventh] . . .
obliges us to regard this new fundamental sound as a Dominant ,
for which reason it must descend a Fifth " (that is, resolve
on the Dominant harmony). Rameau's methods reach a
chmax when he makes the dissonance note d, which is added
below the Subdominant harmony f-a-c, the fundamental note
of the chord d-f-a-c. That is, the added and dissonant note
is the fundamental note of the chord ! It is clear that Rameau
has no foundation for his theory of " double employment of
dissonance," in which it is necessary that the discord f-a-c-d
should be regarded now as an original chord, \^dth fundamental
note/, and now as the first inversion of the chord of the Seventh
d-f-a-c, according to circumstances. But there is another
reason why d cannot be the fundamental note of the chord of
the Seventh d-f-a-c, namel}'', because the intervals d-f and d-a
are not hannonic intervals. Rameau does not perceive this
in the Generation Harmoniqiic. In the Demonstration, how-
ever, he sees quite clearh' that if a is Third of / it cannot at
the same time be Fifth of d. In order that a should appear
as Fifth of d, it is necessar}' that a modulation should take
place to the Dominant key. The sound d can then be regarded
as a real fundamental sound : not as Supertonic of key C, but
as Dominant of key G. It is therefore all the more remarkable
that already in the Traite'^ Rameau should insist that the
chord f-a-c I d must be regarded as an " original " discord, in
which a dissonant note is added to the Subdominant
harmony. There is, in short, no other way in which this
chord can be explained, and Rameau's penetration is
nowhere more manifest than in his treatment of it as a
Subdominant discord. Unfortunately he is not consistent,
and he certainly comes to grief when he tries to explain
it as the first inversion of the chord of the Seventh d-f-a-c.
He is impelled towards this by two very good reasons : first,
he had laid dowTi the principle that the Seventh is " the
origin of all the dissonances," and secondly, he had to
discover or invent some means whereby the extreme terms

^ See p. 121.

212 THE THEORY OF HARMONY

of his Fundamental Bass could be brought into immediate
juxtaposition.

The chord of the Added Sixth, according to Fetis, is the
rock on which all theories of harmony have split ; it has
certainly proved one of the greatest stumbling-blocks in the
way of a rational theory of harmony. It is a noteworthy fact
that Fetis and Dr. Day, whose systems are almost diametrically
opposed to one another, nevertheless agree in regarding this
chord not as a Subdominant but as a Dominant discord, of
which the Dominant is the " root," or fundamental note.
Others find no difficulty in explaining the chord as the first
inversion of the chord of the Seventh on the second degree of
the scale, assigning to this second degree the position of fun-
damental note. All, however, agree that not d but c is the
dissonant note of the chord f-a-c-d. The only explanation of
this chord which does not appear to have been popularized
in text-books of harmony is that given by Ranieau, who
explains it as arising from the Subdominant harmony, to
which a dissonant note is added. Theorists, however, have
revolted against such an explanation, owing largely to their
engrained habit of regarding every " original " chord as com-
posed of a series of added Thirds. Hence their inability to
consider the chord f-a-cjd in any other aspect than as the
first inversion of the " original " chord of the Seventh d-f-a-c.
It is a remarkable fact that at the present day, almost two
centuries after the publication of Rameau's Traite de
rHarmome, theorists are reverting to the French master's
explanation of this chord.

But whether or not we accept Rameau's explanation of the
chord of the Added Sixth, we must at any rate reject his
views as to the dual aspect which he supposes it may assume
in connection with the " double employment of dissonance."
Whether, as in the Generation Harmonique, he makes use of
this " double employment " in order to find a bass for the three
whole-tones in succession, or, as in the Demonstration, he
abandons it in favour of a quite open and definite modulation
to the Dominant key, Rameau is obliged to add a fourth term
(27) to the three terms (1:3:9) from which, he has told us,
the whole key system is evolved. That is, the Fundamental
Bass has four, not three terms, the diatonic system is not in
one, but in two keys, and it has its origin in two generators.
Such is the by no means satisfactory result of Rameau's

RAMEAU'S GENERATION HARMON lOUE 213

strenuous endeavours to demonstrate that the key-system has
been evolved from a single generator.

Finally, it has to be pointed out that while Rameau, by his
Fundamental Bass of three terms, puts us in possession of the
three chief harmonies of the major key-system, namely, the
major harmonies on the first, fourth and fifth degrees of the
scale, he is quite tinahle to inform us whence we obtain the
minor harmonies on the third and sixth degrees.^ This,
however, is one of the principal difficulties in the way of the
theory of the Fundamental Bass and of harmony : the
explanation of these minor harmonies is an absolute necessity.
As for the diminished triad on the seventh degree, Rameau is
disposed to consider this as derived from the chord of the
Dominant Seventh.

In treating of this part of his subject, and especially of
" double employment," Rameau is convinced that he is
grappling with the central problems of harmonic science.
One agrees with Rameau, as well as with his remark, which
he makes in the Demonstration r that some of these problems
have never yet been solved. To this we may add that they
still await solution.

1 Sec pp. 99, log.

2 14 THE THEORY OF HARMONY

"Temperament: Its Theory and Practice."

The presence of " altered consonances " in the natural mode
leads Rameau, as we have seen, to conclude that temperament
in music is a necessity, and one moreover prescribed by
nature itself. If Rameau's reasons as to the necessity for
temperament are not convincing, nevertheless his remarks
in dealing with this subject are of considerable theoretical
importance.

In Chapter VII. of the Generation Harmoniqne [Origin du
Temperament, sa theorie d sa pratique) Rameau brings forward
several extremely interesting propositions. In Propositions
I. and //., he asks : ' ' How does the ear distinguish the difference
between the major and the minor semitone [that is, the diatonic
and chromatic semitone, as b-c, &-#6], or between the minor
Third and the augmented Second } " The answer given by
Rameau is, that the ear does not perceive any difference
between these intervals on a kej^ed instrument, except by
means of the Fundamental Bass. That is, it is not the slight
difference in proportion between the diatonic and chromatic
semitones which the ear regards as most important, or of
which it takes most account ; for the same notes on the
Clavecin, as b-c, may represent now a diatonic semitone b-c,
and now a chromatic one, as ^-#6. It is the harmonic
significance or meaning of such intervals which the ear regards
as all important, and the shght difference in proportion matters
nothing, or at least very little, to the ear, so long as the
harmonic meaning of the interval is made clear.^

" The ear does not perceive the difference between the
major and the minor semitone, nor between the minor Third
and the augmented Second, except by means of the fundamental
succession, of which the harmony is understood, even if it is
not actually heard. If the fundamental bass proceeds by
Fifths, the ear accepts all the semitones as major, and all the
minor Thirds as such. If the fundamental succession changes,
and in consequence the key changes, the semitone which

^ Gen. Harm., Ch. 7., Prop. III. IV., and V.

RAMEAU'S GENERATION HARMOXIQUE 215

occasions the change will appear to the ear as minor, and the
minor Third as an augmented Second." ^ Rameau illustrates
this as follows : —

-G>-

(a): ^-^

-rr-

II.

:z5=^?

-<^-

22:

—& (S>-

^^^^^^m

B.F.

^^eJ^3^^

i^i

-s^

.cz:

-*?-

Here the passage in the treble clef at I. is repeated at II., the
same keys on the Clavecin being utilised. The various intervals
at II., however, affect the ear in a totally different manner from
those at I., and this is because the intervals at II. appear in a
quite different harmonic aspect. At A, a, and B,b, we find
the same keys on the Clavecin representing now a diatonic
and now a chromatic semitone. At C,c, we find the same two
keys representing first a minor Third, and then an augmented
Second.

Rameau also brings forward a proposition by Huyghens : —
" How does one proceed in singing a passage such as the

following : ^oI — nt — la — re' — soly> Rameau agrees \\ith

'=' 24 : 32 : 27 : 36 : 24
Huyghens that if one smgs this melody in such a
manner as to give to every interval its just proportion,
it will be impossible to sing the last sol in unison with the
first. For, between 27 : 32 we find a minor Third which is a
comma (80 : 81) too small, and if this interval is intoned in its
just proportion as 25 : 30 (=5 : 6), then, the intervals which
follow being also sung in perfect tune, the last sol must appear

Gcii. Hiinu., Cli. 7. Prop. III.. \\ .. nnd V.

2i6 THE THEORY OF HARMONY

a comma lower, as regards pitch, than the first. Nevertheless,
says Rameau, there is no one who does not pique himself on
being able to sing the last 50/ absolutely in unison with the
first. How is this ? " Without doubt, the impression given
by the first sol, as fundamental, and of its harmony, is retained
by the ear up to the end ; consequently it guides the voice,
which itself tempers the consonance in question, or perhaps
all of them, so as to arrive at the unison of the first sound. . . .
Is it not therefore the Fundamental Bass and its harmony
which guides the ear ? There is no doubt of it ; everything
confirms it." ^

In Proposition V. Rameau discusses the question as to how
the voice is guided in the intonation of different intervals
when it is accompanied by one or more instruments. Suppose
that it is accompanied by a Viola. In this instrument the
Fourths (old tuning) are just, consequently the major Third
c-e formed b}- the two middle strings is a comma too large.
If now both voice and viola begin on the note c, as Tonic,
the voice will tune itself to this c, as the principal and fun-
damental sound. But what happens when both voice and
viola proceed to e : for now they do not accord, as the e sung
by the voice is different from the e played on the viola ?
Does the voice, then, accommodate itself to the viola, slavishly'
imitating its intonation, so as to sing every sound in absolute
unison with it ? By no means. The note e is regarded b\^ the
voice and ear as part of the harmony of the fundamental sound.
The voice therefore intones e in its just proportion, regardless
of the intonation of the viola. If, however, the key changes
so that e now becomes Tonic {e minor ?) the voice \\all take
this e in unison with the \aola, while c will be intoned differently
by each.

If we add to this instrument a Violin, tuned in perfect
Fifths, so that the major sixth formed by the first and fourth
strings is a comma too large, and also a Clavecin, in which there
is not a single just Fifth, what happens ? Which of these
instruments must the voice follow ? As each instrument has
its own peculiar intonation, the c of the one never being
absolutely the same as that of the other, and so with an
infinite number of other sounds, the Fifths being just here,
and the Fourths there, while neither Fourths nor Fifths are

^ Gen. Harm.. Ch. 7., Prop. IV.

RAMEAU'S GENERATION HARMON IQUE 217

just on the Clavecin, how is the voice to proceed amidst such
a confusion, — not of imperfect harmony only, but of false
unisons ? One sees very well that, far from being helped by
this confusion of sounds, it is actually in danger of being
completely bewildered by such a fracas, were the ear not
preoccupied in favour of the Fundamental Bass ; it is this
bass, and its harmony, which guides the ear.^ Whether
actually heard, or only understood, it is to this that the voice
tunes itself. It should also be observed that the triple
progression, extended to its twelfth term (c to 6#) gives us
" a Si^ which surpasses the first sound Ut by a Pythagorean
comma" (524288:531441). "Temperament, therefore, is
in the nature of things a necessity, and of the different kinds
of temperament in use, equal temperament is to be preferred
to any other, in so much as it permits transposition or modula-
tion to any and every key."

With the greater part of these remarks of Rameau one
has little difficulty in agreeing. If they do not furnish us
with much material for a rational theory of temperament,
they at least show how it is that temperament is possible, how
it is bearable. They also suggest that temperament may not
be so great an outrage on the susceptibihties of the ear as
Helmholtz and his followers would have us beheve. As a
matter of fact, the ear will much more readily tolerate an
interval slightly out of tune than a faulty harmonic pro-
gression. This, however, does not mean that the interval
in question would not sound greatly better if it were in tune.
Rameau's strongest argument in favour of equal temperament
is that it permits of modulation to every key. This is for
Rameau, however, only a secondary consideration. The real
necessity for temperament he sees in the presence of " altered
consonances " in the natural Major Mode. It was bad enough
that Rameau, having set out to prove that the Major Mode
was a natural product, should find in this mode an " essential
fault " in the presence of the tritone and the false intervals
which arise in the attempt to fit the Fundamental Bass to
this part of the scale, but it was hardly to be supposed that
he should discover in these false intervals a proof of the
necessity for temperament. In deahng with the extremely
interesting proposition of Huyghens, Rameau is evidently of

^ Gen. Harm., Ch. 7. Prop. V.

2 I

THE THEORY OF HARMONY

opinion that if the interval

la — lit

be sung in tune,

1 • • / ^

the pitch will flatten. One must however maintain, on the
contrary, that this could only happen if the interval were not
sung in tune. If the interval be sung in tune, and according
to its correct proportions, the last key-note will be absolutely
in unison with the first. Rameau assumes this interval
to he a minor Third, of " altered " proportions. It is,
however, not a minor Third, and its proportions are correct.
In this case also, Rameau should have remarked, the harmonic
succession guides the ear. Thus in the following passage
where, between d-f, we find an interval of the proportion
27:32:—

-f5»-

-Gh-

the voice makes no attempt to convert this interval
into a minor Third, nor indeed is the singer greatly concerned
as to what proportion of interval he forms between the two
sounds in question. What he is concerned about is that
his first sound should form a perfect Octave with the bass
note /, and his second sound a perfect Fifth with g. If he
does this he will form an interval which cannot be regarded
as a minor Third. If he does not do this, he will sing out of
tune. The voice, then, makes no attempt to perform such a
difficult feat as to " temper " the interval d-f. Rameau
does not succeed in making out a case as to the logical necessity
for temperament. Speaking generally, the difficulties in the
way of just intonation for keyed instruments are practical,
not theoretical. The reverse is true of equal or any other
system of temperament.

RAMEAUS GENERA TIOX HA RMOMOUE 2 r 9

CHAPTER VIII.

RAMEAUS cAvAji AT/OX HARMONIQUR AND DEMOXSTKATIOX
nU PRINCIPE DE L'IIARM0NIE.~{C0ntiniied.)

The Minor Harmony.

In the Generation Rarmoniqiic, we find that Rameau has
radically altered his views respecting the origin of the Minor
Harmony. He no longer considers the minor harmony to
be derived from the same principle as the major. If the
major harmony has its source in the phenomena of harmonic
resonance, the minor harmony, on the other hand, has its
source in what is, for Rameau. a new acoustical phenomenon.
This is the power possessed by any given sound of exciting
co-vibration in the sounds of the Twelfth and Seventeenth
below it : —

12th.

17th.

These sounds when heard simultaneously furnish us
with the minor harmony. In the Generation Harmonique,
Rameau imagines that the sounds of the Twelfth and
Seventeenth below actually vibrate, that is, that the sonorous
bodies to which they correspond vibrate throughout their
whole length. In the Demonstration, however, he perceives
liis error. The sonorous bodies in question do not vibrate
in their totahty, but only in segments : the first (12th) in
three, and the second (i/th) in five segments. Each of these
segments corresponds to the Unison of the exciting sound C
That is, the sounds of the Twelfth and Seventeenth below

220 THE THEOR\' OF HARMONY

do not co-vibrate at all, but only the Twelfth (third partial
tone) of this Twelfth, and the Seventeenth (fifth partial
tone) of this Seventeenth. This radically alters the situation
for Rameau. Nevertheless he adheres, at least in the
first part of his Demonstration, to his theory as to the origin
of the minor harmony given in the Generation Harmoniqiie.

In the Demonstration he thus succinctly states his case
for the minor harmony : — " If one tunes with the principal
sound other sonorous bodies which are in the same proportion
to it as the sounds which it produces [by its resonance],
namely, the Twelfth and Seventeenth above, and the Twelfth
and Seventeenth below, it will cause them all to vibrate :
with this difference, that the former will vibrate throughout
their whole length, whereas the latter ^vill divide themselves
into parts, which correspond to the Unison of the principal
sound. . . . These experiences are equally sensible to the
ear, to the eye, and to the touch. From this power of co-
vibration which the principal sound exercises on its multiples

17th

arise these proportions : — ,^ ^^ ' which, reduced to their
^ ^ I a?- fa- lit

I 2 th
smallest terms and applied to string-lengths, give : —

5th
fa — /a|? — lit" ^ That is, the principal sound Uf, or C, is

Min. 3rd. Maj. 3rd.

considered to be the generator, or at least the determining
sound, of both the major and the minor harmonies ; the
first resulting from its resonance, the second from its power
to produce co-vibration in the Twelfth and Seventeenth

5 3 - i ^
below, thus : — a7 — J — C — g — c

Min. harm. Maj. harm.

The principal and central sound C appears therefore to
determine a harmony in both directions ; a major harmony
upwards, corresponding to the harmonic proportion, and a

^ Dhnonst., pp. 21, 22.

RAMEAUS GENERATION HARMONIQUE 221

minor harmony dowTiwards, corresponding to the arithmetical
proportion. But the principal sound C must not therefore,
according to Rameau, be regarded as the fundamental
sound of both harmonies. While then in the case of the
major harmony C-e-g the fundamental sound is C, in the
case of the minor harmony f-a\}-C the fundamental note is
not C, but /. Kameau's explanation of this is as follows : —
" As in the resonance of a sonorous body it is only the sounds
which correspond to the harmonic proportion which strike
the ear, this [proportion] is the only one by which we ought
to be guided ; consequently everything ought to be sub-
ordinated to it. Thus since the grave and predominating
sound of a sonorous body is always, in the judgment of
the ear, the fundamental sound, it is necessary to suppose
that the same will be the case in the arithmetical proportion." ^

Therefore, " in the harmonic proportion [major harmony]
it is the major Third which is directly related to the funda-
mental sound ; whereas in the arithmetical proportion [minor
harmony] it is the minor Third which is thus related." ~

The proportions of both major and minor harmonies may
be expressed by the same numbers : the proportions of the
major harmony := 4 : 5 : 6 ; those of the minor harmony are
expressed by the same numbers in inverted order = 6:5:4.
Rameau remarks that his use of these proportions is not
arbitrary ; he derives them from a natural principle, namel}^
the physical properties of the sonorous body itself. The minor
harmony, however, Rameau considers to be less perfect than
the major. The major harmony is the " direct product of
Nature," whereas the minor harmony is only " indicated by
Nature," and is, in a sense, the result of Art. In all questions
relating to harmonic succession, modal relationship, etc., it
ought therefore to be subordinated to the major harmony, and
be regulated by it. For the same reason the lowest note of
the minor harmony should be regarded as the fundamental
note. " The principal sound Ut," sa5^s Rameau, " which,
through the direct operation of Nature, produces the Major
Mode, indicates at the same time to Art the means of forming
a Minor Mode. This difference between the work of Nature
itself, and that which Nature is content merely to indicate,
is well marked, in that this principal sound Ut itself produces

1 Gen. Harm., Ch. 3. 2 j^i^

232 THE THEORY OE HARiMONV

the major harmony directly b}- its resonance, whereas it
only causes a certain vibration or tremor (fremissemeni) in
those foreign bodies which are related to it in the arithmetical
proportion {genre mineur). . . . But this indication having
been given, Nature then reassumes her rights ; she desires,
and we cannot avoid this conclusion . . . that the generator,
as the originator (fondateur) of all harmony and harmonic
succession, should be here the law-giver."^

So then, Rameau, after having strenuously opposed, in
his Traits, Zarlino's explanation of the minor harmony,
finally adopts it in the Generation Harmoniqiie.

We have seen that ZarUno, like Rameau, considered the
minor harmony to be somewhat less perfect than the major.
The minor harmony is somewhat mournful in character
{mesta) .- while the major is bright and lively {allegra). Once
more we find Rameau taking up the position that because a
thing is natural, it must necessarily also be perfect and
beautiful ; while on the other hand that which is the result
of art, the product of man himself, is necessarily imperfect.
Rameau should have explained more fully in what sense he
understands the term " natural," and also why it is that he
considers that the introduction of the human element is
necessarily bound to result in something imperfect. But
although he may not adopt the best method of proving his
conclusions, it by no means follows that these conclusions are
wrong in themselves. His opinion as to the comparative
inferiority or imperfection of the minor harmony accords not
only with that of Zarlino, but of Helmholtz and other theorists.
He perceives quite clearly what is an undoubted natural fact,
namely, that the harmony which results from the primary
constituents of a compound musical tone is, and can only be,
a major harmony. In this sense, the harmon}- of Nature is
a major harmony-, and it can never be a minor one. Such a
statement does not differ essentially from that of Zarlino :
for if we place, for example, above the note c a major Third e,
and a perfect Fifth, g, we find, as Zarlino had said, that " these
consonances are in their natural places " : they are both con-
stituents of the compound tone of c : whereas, if we substitute
for the major Third e, the minor Third e\^, which gives the
minor harmony c-e\f-g, we find that e\} is a foreign sound, and

* Ddmoiisf., pp. 62-O4 (Du Mode Mineur).

RAMEAU'S GENERATIOX HARMONIQUE 223

has no part in the compound tohe of c. It is, in a sense, a
contradiction of nature ; it stands, and must stand, in
perpetual contradiction with the sound e, which is, indeed,
actually present in, and cannot be separated from, the com-
pound tone of the fundamental c.

Rameau now perceives that while the major harmony
corresponds to the harmonic progression of numbers i, -|, \, |,
!, 1 the minor harmony corresponds to the arithmetical
progression ;}, %, t,, t, H. tV- Or. while the major harmony
may be represented by the proportions 4:5:6, the proportions
of the minor harmony correspond to the same numbers in
inverted order, thus, 6:5:4. He, however, expressly
disclaims attaching any special significance to these numbers
and proportions. In themselves they determine nothing,
but are themselves determined by the physical properties of
the sonorous body. Here perhaps Rameau, after his some-
what reckless use of proportions in the Traite, has become
over-cautious, and errs in the other direction. For it is
something to have it definitely established that the major
and the minor harmonies correspond to these proportions :
and a maker of musical instruments, if it were part of his
duties to manufacture major and minor harmonies, and not
only individual strings, pipes, tubes, etc., would make use
of these proportions, without inquiring very closely as to
whether his procedure were theoretically defensible or not.
But when it is discovered that the same proportions which,
appHed to a sonorous body, or several sonorous bodies, produce
the major harmony, produce also in inverted order the minor
harmony, we are presented with a fact which may not only be
of service to a maker of musical instruments, but which may
and does influence, to a very considerable extent, the whole
theory of harmony. In so far as the question is one of
proportion, the minor harmony must be regarded as an inverted
major harmony.

Nevertheless, from a theoretical as well as a physical point
of view, the question bristles with difficulties. One of these
is, which of the three different sounds which compose the
minor harmony is to be regarded as the fundamental note ?
Rameau proceeds here in a quite arbitrary way. He determines
the lowest note of the harmony as the fundamental note ;
but can give no better reason for this than that the minor
harmony must " conform to the law " laid down by the major

224 THE THEORY OF HARMONY

harmony. Nevertheless, we have seen that he generates

both harmonies f.a\)-(2,-e-g from a single sound. It is enough
for Rameau that he feels, as does every musician, that the
fundamental note of the minor harmony is the lowest note
of the chord. Further, Rameau overlooks the important
fact that other sonorous bodies than those of the Twelfth and
Seventeenth below the principal sound may be, and are,
affected by the resonance of the principal sound. So that the
same difficulty occurs with the minor as with the major
harmony.^ But, as we have already seen, these multiples
of the principal sound are not excited into co-vibration at all,
but only such of their upper partial tones as correspond to the
Unison of the exciting sound. It is Rameau's ultimate
recognition of this fact which induces him, in the latter part
of the Demonstration, to propose an essentially different
explanation of the origin of the minor harmony. This he treats
of in connection with the relationship between the Major and
Minor Modes.

The Minor Mode,

Rameau considers that the Minor ^lode should, in ever3'thing
except its origin, conform to the rules laid down for the
Major Mode. It must therefore be subordinated to the
harmonic proportion, and although the minor harmony has
been generated dowTiwards (arithmetical proportion) the
lowest note of the harmony must nevertheless be regarded
as the fundamental note. One reason which Rameau advances
for this is that " the ear so decides," — a very good reason, but
not a scientific one. Also the Minor Mode, like the major,
must be considered to be determined by a Fundamental Bass
of three terms, and must likewise submit to the operation
of " double employment." As, in the Major Mode, each term
of the Fundamental Bass has above it a major harmony, it
might be expected that in the Minor Mode each of the
fundamental sounds should have a minor harmony. This
does actually occur, but only in the descending form of the
scale, and this form Rameau describes as the ' ' primitive order

^ See pp. 158-164.

RAMEAU'S GENERAriON HARMON IQUE 225

of the mode. In the ascending form of the scale, a minor
harmony cannot be placed on each of the sounds of the
Fundamental Bass. " We will suppose at present that each
of the fundamental sounds of the new mode bears a minor
Third, in this order of proportion 10 : 12 : 15, without con-
cerning ourselves whether this order is found at 27 (the
Subdominant) or any other number, whatever it may be,
especially as temperament, which we have seen to be a necessity,
enables us to dispense with this inquiry." ^

But as a leading-note is necessary in the minor no less
than in the major mode, " in consequence of the close {repos)
it announces on the Tonic, which follows it," the Dominant
must have a major harmony. Also, in the ascending scale,
the sixth degree must be raised a semitone : this is necessary
" in order to procure a diatonic succession, for without it there
would arise an interval of a tone and a half, while the largest
diatonic interval is only a tone." The only form of the minor
scale investigated, therefore, is that of our melodic minor
scale, ascending and descending. The Fundamental Bass
of this scale is as follows ; — ■

32:

cz:

^ie^

-S>-

-&»-

-rj-

-&•-

22:

-<5^

-Tzr

22:

-^w~

-o-

32:

. B.F. 81

243

729

243

:c5i

-Gh-

^> O-

m^=^

32:

123:

-<s*-

-e>-

243

729

243

81 : 243 : 81

In the descending scale we find that at d^ the fundamental
bass has a minor harmony, while at the note h it has a major
one. The reason for this, Rameau tells us, is that " as in the

* Gin. Harm., Ch. 12.

2 26 THE THEORY OF HARMONY

proportions 8i : 243 : 729 the central term 243 appears as
Tonic of the mode, the harmony must be minor, while im-
mediately after 729 it reassumes its major harmony in order to

announce the real principal sound ( \ which follows it."

Still more extraordinary is his explanation of the diminished
Fifth, which appears at the note ctj. "At 729 we find,"
he remarks, " a false Fifth, instead of the perfect Fifth, which
rightly belongs to the fundamental ; but note well that it
[729] always represents in the mode the harmony of the
Subdominant [that is, by virtue of " double employment "]
whose minor Third naturally forms [when placed above /#]
this false Fifth ; then as the Dominant 243, which follows it,
has no longer the character of principal sound, but reassumes
its owm character as Dominant, 729 is obliged to conform to
the original rules of this mode, since the diatonic succession
offers no further obstacle " [!].^

In this connection w^e are now better able to understand
Rameau's remark that the necessity for temperament enables
us to dispense with too close an inquiry as to the nature of
the harmony which, in the minor mode, each of the funda-
mental sounds ought to bear. So highly does he think of
temperament that it apparently reconciles him to the dis-
crepancy of an entire semitone, so that a diminished Fifth
may take the place of a perfect Fifth. One can also well
appreciate the force of his remark, towards the close of his
examination of the minor mode : — " This minor mode has
many pecuHarities which should not be overlooked ; they
are due to the imperfection of its origin." - No doubt the
minor mode has some pecuharities ; but these are not so
peculiar as Rameau's methods of dealing with them.

In the Demonstration Rameau devotes considerable space
to the further investigation of the Minor Mode. " I have
not thought it proper to pass over in silence the Minor IMode,"
he remarks, " as has been done by all the authors who have
treated of the theory of music." He endeavours to prove
that in making the Minor Mode conform to the Major he is
not proceeding arbitrarily. For this subordination of the
minor to the major harmon}^ and mode is indicated by
Nature herself : for the sonorous body, in causing its multiples

1 GSn. Harm.. Ch. 12. 2 Ibid.

RAMEAU'S GENERATION HARMONIQUE 227

(i2th and lyth below) to co-vibrate, does not make them
vibrate throughout their whole extent, but only in segments,
each of which corresponds to the Unison of the exciting
sound. Thus Nature only indicates the possibility of the
minor harmony. ^ Rameau, in the Demonstration, omits the
descending form of the melodic minor scale, with its Funda-
mental Bass ; but instead he gives another scale, ascending,
beginning with the leading-note, thus : —

-e>-

22:

^Q ^

-& O -n

o o :=^

Eii

B.F. 45 : 15 : 45 : 15 : 5 : 15

(Here e 45 is the major Third of c-g, which is the Tonic of
the major scales given in the Demonstration.) He is not,
however, disposed to admit this as a possible form of the
minor scale. " The succession from fa to sol^ [/-5#] is not
diatonic, nor is it natural to the voice ; in order to remedy
this defect, and at the same time add to the beauty of the
melody, it is necessary to raise /« a'semitone. This, however,
is a matter simply of melody, and the harmony does not
suffer." 2

On the other hand, the Dominant must always have a
major harmony. Indeed, Rameau is of opinion that if

1 " On ne peut done supposer la resonnance des multiples dans
leur totalite, pour en former un tout harmonieux, qu'en s'ecartant
des premieres loix de la nature : si d'un cote elle indique la possibilite
de ce tout harmonieux, par la proportion qui se forme d'elle-meme
entre le corps sonore & ses multiples consideres dans leur totalite :
de I'autre elle prouve que ce n'est pas la sa premiere intention, puisqu'
elle force ces multiples a se deviser, de maniere que leur resonnance,
dans cette disposition actuelle, ne peut rendre que les Unissons, comme
je viens de le dire ; mais ne suf&t-il pas de trouver dans cette proportion
{'indication de I'accord parfait qu'on en peut former ? La nature
n'offre rien d'inutile, & nous voyons le plus sou vent qu 'elle se contente
de donner a I'Art simple iudications, qui le mettent sur les voyes."
(Demonst., pp. 65, 66.)

2 Demonst., pp. 77, 78.

228

THE THEORY OF HARMONY

so/tj {g[\) is substituted for soljf^, then the mode becomes
major. He remarks : — " If we descend in the Minor Mode,
thus : La, sol, fa, mi [a-g-f-e, A minor], etc., we modulate
to the Major Mode, from which the minor is derived ; for
all the diatonic difference between these two modes consists
in the so/iq or soljf^ ; not that in practice one has not the
art of preserving the impression of the Minor Mode with
so/ti ; but this is effected by the help of a dissonance, which
cannot be avoided." The dissonance referred to here is
the diminished Fifth, which occurs at the term 729 in the
melodic minor scale {see p. 226).

But in order to preserve the impression of the Minor Mode
in such a case, Rameau concludes that the most satisfactory
way is to abolish the so/t; altogether. " There is, then, only
one means of preserving in descending the impression of
the Minor Mode, namely to exclude sol\] from the harmony,
and to use it simply as a melodic ornament pour le goiU de
chant) as well as may be." ^

But it cannot be said that there is any difficulty in
the following passages in regarding the g\\ as a real
harmonic note ; it is certainly not a passing or auxiHary
note, nor indeed, any kind of ornamental melodic note: —

M * ib) , _^

^E&

=^=^

-i±

:g=

-o-

~ry-

-o-

321

-Gh-

Both passages are throughout in A minor. It would be
rash, however, to conclude from this that our minor key-
system has been evolved, not from harmony, but from
melody, that it has as its real basis an old Church mode (the
,Eohan) and that the other forms of the minor scale are
merely "chromatic alterations" of this old mode. On the
other hand Rameau, as is proved by his remarks concerning
the gfcj, is unable to find a Fundamental Bass for such a
passage, and is forced to admit it.

Such then is Rameau's explanation of the Minor Mode,

1 D6monst., p. 77.

RAMEAU'S GENERATION HARMONIQUE 229

If, in treating of the Major Mode, he has encountered serious
difficulties, he now finds himself in a truly desperate case.
The harmony which the Fundamental Bass should bear
may be major, or minor, or e\'en diminished — notwithstanding
his express refusal, even in the Traite, to accept this last
chord as fundamental — according to circumstances. One and
the same fundamental sound may even at one time bear a
major harmon}-, and at another time a minor one. As in the
major, one kind of bass is necessary for the ascending form
of the scale, and another for descending ; it is, like the
major, in two keys, and it is necessary also to make use of
the weak device of " double employment." The net result,
however, of Rameau's investigations is that we are left
without any form of the minor scale whatever. For as the
Subdominant must have a minor harmony d-f-a, /# cannot
be regarded as an essential note of the scale : for the same
reason the " harmonic " form of the scale must be rejected,
for there arises the augmented second f-g^, which is not a
diatonic interval. Finally, g^ has no place in the minor
scale, but can only occur in that of the relative major. But
if Rameau finds himself baffled by the difficulties of the
Minor Mode, theorists since his time have fared little better.

Further Development of Rameau's Views Respecting
THE Minor Harmony : Relationship of the Major
AND Minor Modes.

The relationship between the Major and Minor Modes is,
Rameau considers, to be explained in the same way as scale-
or key-relationship in general. " It has already been observed
that, unless dissonance is made use of, two terms of this
[triple] progression being given, the third is arbitrary : thus

•^, ^ , being given, the third term may be either "^ or f*^-
sol -re -^ ut la '

so that in this order ^ ^,^ > sol mav be considered as

lit -sol -re ,

Principal, or Tonic : whereas in ^ ^, "^yg' is the Principal."^

sol -re -la,

^ Ge)i. Harm., Ch. 13.

230 THE THEORY OF HARMONY

Between the two principal sounds sol and re, therefore, a
close relationship exists. If we compare the two modes, or
keys, represented respectively by the fundamental sounds
1:3:9 and 3 : 9 : 27, it will be observed that 3 and 9, with
the harmonies they bear, occur in both keys ; " whence it
follows that these terms, representing, as they do, funda-
mental sounds, the more there are sounds in common between
these two modes, the more closely will they be related to
each other." ^

This is the explanation of the close relationship which exists
between the Major and Minor Modes. In the descending
(Melodic) minor scale we find exactly the same sounds as in
the relative major. It is true that the relationship between
the Tonic of a major key and that of its relative minor is
that of a minor Third, which is not so perfect as a Fifth-
relationship, but this is compensated for by the large number
of sounds they possess in common. " The great number of
harmonic sounds common between these two modes . . .
removes the defect as respects the relationship of their
fundamental sounds." The transition from a Major to its
relative Minor Mode is effected by means of the Fundamental
Bass descending a minor Third. " This relationship of the
major to the minor mode introduces a fundamental succession
in Thirds." One also observes that " the Dominant and
Subdominant being obliged to conform in their harmony to
the nature of the mode from which they derive their origin,
it follows that every mode [key] which is the Fifth of another
should be of the same species [genre, that is, major or minor] ;
whereas, on the other hand, because of the relationship of
the major with the minor mode, every mode which is a
Third from another ought to be of a different species." Thus,
if C major be taken as the central key, the two keys F and G,
each of which has a Fifth relationship with C, ought to be
major : while the keys of E and A, which are a Third above
and below C, ought to be minor.

Rameau's explanation of the relationship between the
Major and the Minor Modes has been repeated in countless
text-books of harmony up to the present day. Such an
explanation, no doubt, has served to a considerable extent
a practical purpose. But if it be the case that the degree of

* Gen. Harm., Ch. 13.

RAMEAU'S GENERATION HARMON IQUE 231

relationship between two scales or keys is determined by the
number of sounds they possess in common, then how is it
that, taking C major as a central key, E major with four
sharps is more closely related to C than D major, with only
two sharps : while A^ major, with four flats, is more closely
related to C major than Bj? major, with two flats ?

New Theory of the Minor Harmony and Minor Mode :
Anticipation of Helmholtz's Theory of the Minor
Harmony.

In the latter part of his Demonstration, Rameau devotes
considerable space to the further investigation of this relation-
ship, as well as of the minor harmony and mode. It would
appear that he was not completely satisfied with the position
in which matters had been left in the Generation Harmoniqiic.
Besides, he was no doubt somewhat puzzled bv the defective
relationship existing between the Tonics of two modes other-
wise so closely related to each other as a major mode and
its relative minor. He had himself suggested that such a
relationship was at least as close as a Fifth relationship, as
that of a Dominant or Subdominant. If the origin of both
the major and minor harmonies was to be found in a single
sound, C for example, then this C must form the central point
of both harmonies, with its major Third and perfect Fifth
below ^s well as above it : and the relationship of these

harmonies must appear as /-a?-C-e-g, where/, as fundamental
note of the minor harmony f-a\f-c, must be regarded as the
Tonic of / minor. But Rameau was too good a musician
to attempt to represent / minor as the relative minor of
C major. Nevertheless these considerations must have
frequently presented themselves to him.

Further, he has now become alive to the fact that the
multiples of the principal sound (Twelth and Seventeenth
below) do not vibrate in their totality. He therefore examines
afresh the acoustical phenomenon on which his theory of the
minor harmony and Minor ]\Iode is based, and now actually tells
us that it is impossible to derive the minor harmony from the
co-vibration of the multiples of the Twelfth and Seventeenth !

232 THE THEORY OF HARMONY

Such a proceeding could only be justified if these multiples,
instead of vibrating in segments, vibrated throughout their
whole length, and instead of reproducing the fundamental
tone of the exciting sound (the unison) produced their own
fundamental tone, corresponding to the entire length of the
string. 1

Rameau, then, expressly states that the co-vibration of the
multiples has no basis in fact, and that therefore it cannot
serve as the basis of the minor harmony. But let it be
supposed, he proceeds, that these multiples did co-vibrate in
their totality, would not this he a manifest contradiction of
Nature, and of the principles which she has already established
in the harmonic resonance of the sonorous body ? The
multiples, however, do not co-vibrate, but only those segments
which correspond to the unison of the exciting sound, and
the utmost that can be deduced from such a fact is that
Nature here indicates the possibility of the formation of the
minor harmony. ^ So much then is clear ; the minor harmony
cannot have its origin in a phenomenon which does not exist.
and which, if it did exist, would be a contradiction of the first
principles of Nature : the minor harmony must arise from
some other source.

But what other source is there ? There is but one ; the
harmonic resonance of the sonorous body. " What," he
asks, " does Nature indicate ? She indicates that the
principle which she has once for all established shall, and
must, dominate everywhere, and that everything — harmon\%
mode, melody, etc., must be related and subordinated to
The generator of the major harmon}^ ^-^-" rnust, therefore,

^ " Pour former un accord parfait oil le genre mineur ait lieu, il
faut supposer que les multiples resonnent & qu'ils resonnent dans
leur totalite, au lieu qu'en suivant I'experience que j'ai rapportee, ils
ne font que fremir, et se divisent " . . . . etc. — {Demonst, p. 64.)

2 ' ' On ne peut done supposer la resonnance des multiples dans leur
totalite, pour en former un tout harmonieux, qu'en s'ecartant des
premieres loix de la nature : fi d'un cote elle indique la possibilite de
ce tout harmonieux, par la proportion qui se forme d'elle-meme entre
le corps sonore & ses multiples consideres dans leur totalite, de
I'autre elle prouve que ce n'est pas la sa premiere intention." —
{Ibid., p. 65.)

' " Ce que pretend la nature ? Elle veut que le principe qu'elle a une
fois etabli, donne par tout la loi, que tout s'y rapporte, tout lui soit
soumis, tout lui soit subordonne, harmonie, melodic, ordre, mode,
genre, effet, tout enfin." — -(Ibid., p. 67.)

RAMEAU'S GENHRATION HARMON IQUH 233

also be the generator of the minor harmony. h'or the
reasons ah'eady given, this minor liarmony cannot consist
of the sounds /•«[7-c. But there is still another reason. If C
is the determining sound, the generator, of the minor harmony
f-a\}-C, how can / possibly be regarded as the fundamental
note ? This difficulty has at length been realised bv
Rameau. Nevertheless he is convinced that /, the lowest
sound of the harmony, is in reality the fundamental note.
But then, the harmon3^ which nature places above every
fundamental note is a major harmony : in this harmony
the Third is major. How is the minor Third to be derived ?
this minor Third which determines the Minor Mode just as
truly as the major Third determines the Major Mode.

Rameau now concludes that the minor harmony determined
by the principal sound C is not f-a\}-C, but a-C-e. Here the
difficulty as to the generation of the minor Third disappears,
for the minor Third is represented by C, the principal sound
itself ! The sound e, which is Third of C, is also Fifth of a.
This sound a must therefore be also regarded as a generator.
" The Mode " (major mode), remarks Rameau, " has already
been established : it is beyond our power to change it in
any way. . . . But it is possible to vary it by the new
genre in question [minor mode]. . . . This variety is deter-
mined by the quahty of the Third which appears above the
fundamental sound, or generator. This generator has already
determined its [major] mode, by means of its major Third,
which sounds along with it ; it equally determines a new
mode by forming a minor Third, while still retaining its
character as the principle [or generator]. . . . This admits
of the most positive proof. The major Third alone is actually
generated from the fundamental sound : never a minor
Third, which, nevertheless, we suppose to be related to
this fundamental sound. It is then necessarily this minor
Third itself which is the cause of the difference of effect
between major and minor." ^

^ " Cette variete va devenir la cause des differens effets entre les
Modes, qui en seront susceptibles. Elle existe dans la tierce directe
du gen^rateur. Ce generateur a deja determine le genre de son mode,
par sa tierce majeure, qu'il fait resonner, il va pareillement determiner
celui d'un nouveau mode, en formant, lui-meme, une tierce miiienre
directe, sans cesser d'etre principe. Je dis, sans cesser d'etre principe,
parce que, dans ce cas, le produit, ou cense tel, est la seule cause de
i'effet : la preuve en est certaine. {Dcuionst., p. 69.)

234 THE THEORY OF HARMONY

Rameau here says that the fundamental sound generates
a major Third, but can never generate a minor one ;
in other words, the klang which constitutes an inherent
property of the sonorous body is always major. This
fundamental sound " can never be the cause of a direct
minor Third " which is supposed to be directly related
to the fundamental sound. ^ Thus in the minor harmony
a-c-e, c, the minor Third above a, cannot be related
directly to a. In the words of Helmholtz, c must be
regarded as a foreign tone which has no part in the a
klang.

When we remember the position taken up by Rameau in
the Generation Harmonique, and in the first part of the
Demonstration itself in respect of the generation of the
minor harmony, this new departure must appear extra-
ordinary. But still more extraordinary is to follow. For
in the minor harmony, a-c-e, Rameau regards a as a funda-
mental and generator, not of the " foreign " tone c, but of
the fifth e ; while, as respects the tone c, he explains this
exactly as we shall see Helmholtz does : both c and e he
considers to be constituents of the c klang. From this point
of view the sound a must appear as a sound added to the
c klang, for it is not a constituent of this klang. " Thus,"
he proceeds, " the ear indicates in this case the method
of procedure of the original generator ut [c] ; it chooses for
itself a fundamental sound, which becomes subordinate
to it, and to which it gives all that is necessary in order to
make it appear as a generator. In forming the minor Third
of this new fundamental sound, which must be the sound
la [a], the generator ut gives k) it its major Third mi [e] for
Fifth ; Fifth which, as we have alread\' seen, constitutes the
harmony and determines the proportion on which depends
all fundamental succession of the mode. Thus this new
fundamental sound, which may therefore be regarded as
generator of its mode [!], is a subordinate one : it is

' " La seule tierce majeure directe resonne avec le son fondamental :
il est consequemment la cause de son effet : consequemment encore
il ne pent plus I'etre d'une tierce mineure directe qu'on hii suppose ;
ce sera done necessairement de cette tierce mineure meme, que
naitra la difference de I'effet entre elle ct la majeure." — (Ddmoiist.,
p. 70.)

RAMEAU'S GENERATION HARMONIOUE

-J3

forced to submit in every case to the law of the first
generator." ^

Rameau, then, is the first to present us with this im-
portant theory of the minor harmony, and the Minor Mode.
It is not only an entirely new theory, but one which is
directly opposed to that of the Generation Harmonique, and
the first part of the Demonstration. The considerations
which have influenced Rameau in his remarkable change of
front are not difficult to find, nor to understand. With
regard to the minor harmony, Rameau has from the first,
even in his Traite, maintained that the lowest note of the
harmon}^ is the fundamental note, and he is still of this
opinion when, abandoning his earher explanation of the
minor hannony, he derives it from the arithmetical proportion
determined by the co-\abration of the multiples of the
Twelfth and Seventeenth. But he has evidently seen that
the difficulties, especially that connected with the funda-
mental note, are too great. He therefore abandons this
theory of the origin of the minor harmony, and seeks
to explain it in the same way as the major harmony ;
that is, as arising from the harmonic proportion,
from the upper partial tone series. But here a new
difficulty presents itself, in that the sound c of the minor
harmony a-c-e is not a constituent of the compound tone of
a. Rameau, however, regards c as itself a generator. The
rriinor harmony therefore has two generators. The sounds
c-e are constituents of the compound tone of c, while this
same sound c " gives to a all that is necessar\- in order to
make it appear as a generator," namely, its major third e,
which then appears as fifth oi a. In short, to use the language

1 " Aussi I'oreille indique-t-elle clairement les operations du principe
generateur Ut dans cette circonstance : il s'y choisit, lui-meme, un
son fondamental, qui lui devdent subordonne, & comme propre,
& auquel il distribue tout ce dont il a besoin pour paroitre comme
generateur. En formant la tierce mineure, de ce nouveau son fonda-
mental, qu'on juge bien devoir etre le son la, le principe Ut lui donne
■encore sa Tierce majeur mi pour Qiiinte, Quinte qui, comme on le
S9ait a present, constitue I'harmonie, & ordonne de la proportion
sur laquelle doit rouler toute la succession fondamentale du
Mode : ainsi ce nouveau son fondamental. qu'on pent regarder, pour
lors, comme generateur de son Mode, ne Test plus que par subordin-
ation : il est force d'ysuivre, en tout point, la loidu premier generateur."
— (Demonst., pp. 71, 72.)

236 THE THEORY OF HARMONY

of Helmholtz, the sounds c-e are constituents of the com-
pound tone of c ; while a-e are constituents of the compound
tone of a.

This is exactly the position taken up by Helmholtz in his
explanation of the minor harmony ; except that Rameau,
unlike Helmholtz, gives one of the generators of this harmony
the predominance over the other. It is not surprising that
this new explanation of the minor harmony should have
been imperfectly understood by the French academicians,
who supposed it to have reference merely to the relationship
existing between the Major and Minor modes. Rameau
himself could have been under no such delusion, for he had
started with the express statement that the co-vibration of
the multiples, having no basis in fact, could not possibly be
the source of the minor harmony. The real significance
of Rameau's statements was, however, ultimately recognized
by D'Alembert, who had himself been mainly responsible
for the report dealing with Rameau's theory inscribed in the
records of the Academy. While in the first edition of his
work, Elements dc Mu^ique, siiivant les Principes de M.
Rameau, D'Alembert had considered the co- vibration of the
segments of the multiples to form the proper physical basis
of the minor harmony, in the new edition he abandons this
view and explains the minor harmony a-c-e as liaving a
two-fold foundation : e is Third of c and Fifth oi a. It is
surprising, however, that these facts should have escaped
the attention of Rameau's commentators, especially of
Dr. Riemann, who, in his Geschichte der MtisiUheorie and
other works, demonstrates the superiority of Rameau's
claims as a theorist, as compared with Helmholtz, in that
he explains the minor harmony as arising from the " under-
tone series," whereas Helmholtz, on the contrary, will have
nothing to do with any real or supposed series of " under-
tones." 1

It is remarkable that Rameau should present us, and in one
and the same work, with just those two theories of the minor
harmony, the respective merits of which in our own day
have occasioned so much controversy and divided theorists
into two opposite camps. Whichever view we take there
are difficulties. Rameau found himself obliged to abandon

1 See pp. 387-390.

RAMEAU'S GENERATION HARMONIQUE 237

his explanation of the minor harmony as arising from the
arithmetical proportion because, to say nothing of the serious
difficulty in connection with the fundamental note, he had
discovered that the supposed series of " undertones " was
a mere chimera ; in reality it was only the upper partial tone
series over again. But in his new theory the difficulties are
even greater. For here we find not one fundamental note, but
two ; the note regarded by Rameau himself, and by the vast
majority of musicians since his time, as the real fundamental
note of the minor harmony appears as an added and foreign
tone, derived from no one knows where. Unfortunate^ for
Rameau, who has set out to demonstrate to us that everything
in harmony is derived from a single generator, the further
he proceeds the more difficult does it appear for him to avoid
deriving most things in harmony from two generators. This
ruUng idea of Rameau, that everything in harmony- is evolved
from a single sound, is truly a splendid conception. Every-
thing has its source in Unity, and cannot be properly under-
stood apart from this Unity. But it is an idea which, if
realizable, is certainly not realized by Rameau.

The minor harmony, no less than the major, impresses
the ear and the mind as a harmonic unity ; Rameau's
explanation of it as arising from two generators makes of it
\drtually a dissonance ; even the most mechanical of musical
theorists would look askance at a proposal to apply the
" double-root " theor}' to a consonant chord. Finally,
Rameau does not observe that in abandoning the arithmetical

proportion, (--vritjimedcal^ h^rmmiicx j^^ demohshes his theory

of the Subdominant. It is left without a foundation. Yet
we must believe that in making use of the arithmetical
proportion he was influenced quite as much by the
necessity to find a theoretical foundation for the Sub-
dominant, as to provide an adequate explanation of the
minor harmony. It is not, therefore, surprising to find
that in his last important work on harmony Rameau
seeks for the Subdominant a fresh explanation, and derives
it from the sounds of the harmonic series (see pp. 265-266),

238

THE THEORY OF HARMONY

The Chromatic Genus : Origin of the Chromatic Scale.

Thus far, remarks Rameau, we have not spoken of the
Fundamental Bass in Thirds, nor have we had occasion to do
so (as it finds no place in the diatonic system), " except to
demonstrate the connection between the major and the minor
modes." " It is from the fundamental bass in Thirds that
the chromatic genus derives its origin." ^ If the Fundamental
Bass proceeds a major or minor Third above or below a given
fundamental sound, there arises a new kind of semitone,
namely, the chromatic semitone of the proportion 24 : 25,
thus : —

(6)

:*g

te=t;

22:

iin

:ig

-s>-

fri\'

((*;.

«0

prj

Nw^ rj

U*'^

rj '

rj ^

This semitone, which is called minor or chromatic, because
it is a quarter of a tone (125 : 128) less than the major or
diatonic semitone, "although a natural one, is not nearly so
natural as the latter, and this is proved by experience." It
is much more difficult to sing ; indeed, few musicians can
intone this chromatic semitone accurately, especially in
descending. The Fundamental Bass in Thirds introduces
a change of key, and should be used only for this purpose.
" This chromatic semitone is never used except to change the
key, a change which bewilders {deronte) the ear." ^ But the
ear is supported by the Fundamental Bass and its harmony,
without which it would hardly be possible to intone this
chromatic semitone with even a tolerable degree of accuracy.
" What assists the musician ... is, that he unconsciously
supports himself by means of the fundamental sound of the
new key into which this semitone leads him ; otherwise he
would find himself as much embarrassed as the merest
novice."

1 Gen. Harm., Ch. 14.

2 Ibid.

RAMEAU'S GENERATION HARMONIOUE 239

Rameau however subsequently modifies this statement :
the singer, he says, may help liimself in such a case not only
by means of the fundamental bass, but by means of the
intervals of the diatonic scale. " Note well then, that in
order to form this chromatic semitone one is aided without
being conscious of it either by the natural intervals, as the
tone or the diatonic semitone, or of the fundamental succession
which occasions it."

Thus, in order to intone the chromatic semitone c-cifi^.

jC21

one ma}- first ascend to d, the whole-tone above c, and
immediately thereafter descend to c# by means of the
diatonic semitone d-c^} But here Rameau gives away his
case : for the chromatic semitone c-c^ (key of C major) thus
obtained, is not of the proportion 24 : 25. It is a larger
interval of the proportion 128 : 135, and is the result of the
difference between the whole-tone c-d (8 : 9) and the diatonic
semitone c^-d (15 : 16). But such an interval cannot result
from a Fundamental Bass which descends a minor Third
from c to a.

While then it is true that such a bass gives rise to a
chromatic semitone of the proportion 24 : 25, it does not
follow that our chromatic scale is to be explained in this way.
Rameau, however, although he does not attempt to formulate
a systematic Fundamental Bass for the chromatic, as he has
already done for the diatonic scale, nevertheless implies-
that not only the chromatic semitone, but also the chromatic
scale, has its origin in a Fundamental Bass in Thirds. He
cannot well avoid doing so. P'or, having shown how the
species of chromatic semitone in question arises, he has to
explain what is to be done \\dth it, that is, how it is employed
in harmony. On this point he is quite definite. It is
never used except to change the key. Thus " if one intones
the passage, tit-re-mi-fa-fa^, the /a# cannot be intoned
without considerable difticulty " ; one reason for which is that

^ Geii. Harm., Ch. 14, .\rt. i.

240

THE THEORY OF HARMONY

" at this point the key changes," ^ that is, a modulation is
effected to G major : thus : —

-o~

22:

-Gi-

-o-

-♦"S*-

B.F.

Rameau, however, does not appear to have observed that the
g thus obtained is not the Fifth of c. For this it would be
necessary that the interval which succeeds /-/;{}: (24 : 25) should
be, not the diatonic semitone 15 : 16, but one of the proportion
25 : 27, which is the difference between the chromatic semitone
and the major tone f-g (8 : g). But Rameau has already
demonstrated, and in the most convincing way, that the only
kind of semitone which can arise from the Dominant-Tonic
succession is of the proportion 15 : 16. In the same way,
if the chromatic scale be extended a little further,

-&-

:z2i

~rj-

j:21

izz:

B.F.

it will be found that, if the chromatic semitones be
determined as of the proportion 24 : 25, and the diatonic semi-
tones of the proportion 15 : 16, g cannot be considered as
perfect Fifth, nor a as maj or Sixth of c ; that is, as the fifth and
sixth degrees of the scale of C major : nor can their leading-notes
/# and g^ belong to the chromatic scale of which the diatonic
scale of C major forms the basis. In short, Rameau does not
observe that in addition to the chromatic semitone 24 : 25,
which is the difference between the diatonic semitone and the
minor tone 9 : 10, there is the other and larger chromatic

Demonst., pp. 90, gi.

RAMEAU'S GENERATION HARMONIQUE 241

semitone 128 : 135, representing the difference between the
diatonic semitone and the major tone 8 : 9, and that, of the
five whole-tones of the diatonic major scale, only two are minor.
Nor does he give any adequate explanation of cases such
as the following : —

(b)

rj rtfrj-

-<&»-

^1^

-^W'

-<s>-

-<^^

^-:it^K-*

-*3-

kgE

_Q_

-C2_

where, as he recognizes, the Fundamental Bass remains
stationary. In the first case, the harmony and key are and
remain throughout those of C major. Try as he might,
Rameau could not invent any Fundamental Bass in Thirds
to fit such a passage. For every chromatic note is understood
and intoned as a leading note to the diatonic note which
immediately follows it. Such notes Helmholtz calls " inter-
calated " or transitional notes, of "no harmonic or modulational
significance." So also with the notes e\^ and d% at (6) and (c),
which are frequently described as chromatic passing-notes.
Such notes, then, would appear to be melodically, not
harmonically, determined. Harmony would therefore appear
to have its origin in melody : a doctrine which was, for
Rameau, anathema, for he quite rightly perceived that it
destroyed the very foundation of his system. Rameau,
however, might have objected that if the chromatic notes at
{a) are in reality leading-notes to the diatonic degrees of the
scale which immediately follow, then these notes must have
not a melodic but a harmonic determination. For if, for
example, j% be correctly intoned as leading-note to g, just
as h the seventh of the c major scale is intoned as leading-
note to c, then the interval thus formed must be of the propor-
tion 15 : 16, and whether /# be correctly intoned or not it will
nevertheless be understood as bearing the same relationship
to g as 5 has towards c, that is, it is its leading-note. But the
interval 15 : 16, as Rameau has shown, has a harmonic
determination. How could such an interval be melodically,
that is arbitrarily, determined ? Again, Rameau might have

242 THE THEORY OF HARMONY

objected that if the chromatic notes cj d^, etc., must be
regarded as melodically determined, so also must be the
diatonic notes d\:\ and/fc|, for they also are transitional notes,
which do not belong to the harmony c-e-g. Might it not even
be possible to explain the chord c-e-g itself, not as a harmonic
unity, but as a fortuitous combination of melodic notes ?

As we have seen, Rameau regarded the chromatic notes of the
scale, not as essential elements, or as representing an extension
of the key-s3^stem, but as a means for effecting a change of key.
He lived in an age when " chromatic discords " were much
less in evidence than they are at the present day. For
Rameau, all chords were diatonic in nature and origin.
Thus the chord /#-fl-c-t[7 could not occur in C major, but must
be regarded as the chord of the Dominant Ninth in g minor.

Although the chord of the Augmented Sixth was known
and practised in his time (Heinichen gives examples of all
three forms of the chord) Rameau avoids entering into any
explanation of this chord. Such a chord as the German form
of the chord of the Augmented Sixth, for example, f-a-c-d^,
occurring on the sixth degree of the scale of A minor, must
have been for Rameau pecuHarly embarrassing. It was
impossible for Rameau to explain this chord and its natural
resolution on the Dominant, either by means of double
" employment " or any other device known to him. It is
perhaps for this reason that he avoids the chord for the most
part in his works for the stage, and substitutes for it the
chord of the Diminished Seventh, which occurs, especially in
accompanied Recitative, very frequently. This is accom-
plished by raising the lowest note of the chord a semitone, thus
f^-a-c-d^, which chord is then resolved on the Tonic harmony
of E major or minor.

In short, the only information which Rameau has to offer
with respect to the nature of our chromatic system is that
it has its origin in the Fundamental Bass in Thirds, and that
such a bass determines the chromatic semitone 24 : 25. This
theory of the origin of the chromatic semitone we meet with
again in the work of Moritz Hauptmann. Rameau does
not mention the fact that all the chromatic notes necessary
for the formation of the complete scale of semitones were in
use by the fourteenth century, and that even the Greeks had
the two forms of B — [if ^^'^^ ^'^l-]

RAMEAU'S GENERATION HARMON IQUE 243

The Enharmonic Genus : the Use made of
Quarter-Tones in Modern Music.

In addition to the Cliromatic, there is also an Enharmonic
genus. In the Generation Harmonique Rameau's remarks
on the enharmonic genus are confined to a single chord,
namely, that of the Diminished Seventh. Each of the sounds
which compose this chord may in turn, by means of enhar-
monic change, be regarded as the leading-note of a key :
and the chord may therefore appear as diatonic in as many
keys as there are notes in the chord. This process, whereby
each of the notes of the chord may successively appear as a
leading- note may be explained, Rameau thinks, as a new
kind of " double employment." ^ He draws an analogy
between the chord of the Diminished Seventh, and the
discord — the Added Sixth — on the Subdominant. In the
chord of the Added Sixth either the Subdominant or the
Supertonic may appear as the fundamental note, according to
circumstances ; in the chord of the Diminished Seventh each
of the notes composing it may, in turn, be regarded as the
fundamental note of the chord. Rameau's views respecting
the nature of this chord of the Diminished Seventh have, in
the Generation Harmonique, undergone a radical change. In
the Traits he explained it as a " borrowed " chord, an altered
chord of the Dominant Seventh. Thus the chord g$-b-d-f he
considered to be the first inversion of the chord f-g^-b-d,
where the note / is substituted for e, the real fundamental
note of the chord. He now considers the chord g'j^-h-d-f to
be derived from two fundamental sounds, namely, the
Dominant and Subdominant From the Dominant we
obtain g^ and h, and from the Subdominant d and /.^ The
first two sounds are the Third and Fifth of the Dominant
harmony of A minor, e-g^jj^-b ; while d and / are the funda-
mental note and third of the Subdominant harmony d-f-a,
of the same key. . Rameau is of opinion that the leading-
note is the fundamental note of this chord, although he
does not explain how the Fifth above this fundamental
sound happens to be a diminished one. He states however

1 Gin. Harm., Ch. 14, Art. II. 2 Ibid.

244 THE THEORY OF HARMONY

that " the leading note, although fundamental, owes
this privilege to the Dominant, from which it is derived,"
that is, the Dominant is the real fundamental. This
explanation he no doubt considered to be necessitated by
the fact that the Dominant does not actually appear in
the chord. The original form of the chord, then, being
g^-h-d-f, with the Dominant E as " root," we here find for
the first time the chord of the Diminished Seventh stepping
out to take up the role which it has played in so many text-
books up to the present day, namely, as a chord of the Ninth
with the fundamental note omitted. One of the principal
advantages, according to Rameau, of regarding the chord in
this aspect is that it is " now brought into conformity with
other chords," that is, it now consists of a series of added
Thirds.

The enharmonic change of which this chord is susceptible is
made possible by temperament. If we change the chord
g^-h-d-f into d^-h-d-j, the sounds of g% and d^ " appear, so
far as keyed instruments are concerned, to be the same sound,
but in the nature of the thing there is a difference of a quarter
of a tone," of the proportion 125 : 128, which is the difference
between the major and minor (diatonic and chromatic)
semitones. By means of such an enharmonic change, two
unrelated keys may be made to succeed one another : " this
defect of relationship is replaced by the large number of
sounds in common." ^ Rameau makes a passing reference
to two other species of Enharmonic, namely, the Diatonic
Enharmonic, and the Chromatic Enharmonic, without making
any explanation as to their meaning or use. For their
proper effect, he remarks, there are necessary what he has
so far been unable to find, namely, tractable musicians, who
are willing to entertain some sympathy, and exercise some
patience with difficulties and novel effects to which they have
never been accustomed. Rameau refers here to his experiences
with the musicians of the Parisian Grand Opera. Even
thirteen years later he cannot refrain from again alluding to
the subject. " I am not sure," he remarks, " whether this
genus [the Chromatic-Enharmonic] suits the voice, but it
can at least be realised on instruments, and this I attempted
to effect in an earthquake in my Ballet of the Indes Galantes ;

1 Gtn. Harm., Ch. 14.

RAMEAU'S GENERATION HARMONIQUE 245

but here I was so badly served, and su badly received, that
I found it necessary to substitute for it a more simple kind
of music." ^

But if these new kinds of Enharmonic are not explained
in the Generation Harmoniquc, Rameau, on the other hand,
treats of them at considerable length in his Demonstration.
The Diatonic Enharmonic is explained as follows : " The
alternate succession of a Fifth and a major Third, in which
the triple progression is combined with the quintuple, gives a
composite genus, called Diatonic Enharmonic ; the semitones
which are its products form a whole-tone step which is a
quarter of a tone too large ; thus these semitones, which are
both diatonic, necessarilv introduce the Enharmonic into
the tone which thev form, which makes its performance
difficult for the voice but not impossible": — -

-as>-

'rj> rj

^nz

-»5>-

-?«>-

Here we find at t[7-fl, a diatonic semitone, and another at a-g#.^
Adding these semitones together, we have an interval of the
proportion [^ X { ;"• = H l ;'; . Comparing this with the whole-tone
of the proportion g : 10, thus rTrXlt!-;, we obtain Rameau's
quarter of a tone, that is }4^. This quarter-tone, compared
with unity {M'^) gives us j^^. or nearly 4I, . The interval
g^-h\} is known to theorists as a diminished Third, that is,
a minor Third diminished by a chromatic semitone :
as oXH=iIt- If ^^'6 compare this interval with that
formed by adding two semitones together, we obtain
1 jf2 X|-5-!v=It- The two intervals therefore are not the
same, but differ by a comma.

^ Demonst., p. 95. * Ibid., pp. 93, 94.

' Here B^ in treble clef is J of F, in the bass ; while A is j of F.
Comparing these, we obtain fxj = jg, that is, a diatonic semitone.
Again A, in the bass, is SxJ of B[> in the bass, and = ^^ : while E
is .5X^'^ = f^g, and Gjt is J xfg=2,-:j'', which, compared with A
(ij'tXt= w') is ^6^X7^0 = ^^. A — Gi£, therefore, is also a diatonic
semitone.

246

THE THEORY OF HARMONY

As for what Rameau calls the Chromatic Enharmonic genus,
the procedure is as follows : "If the bass descends a minor
Third, and then rises a major Third, while the harmony above
each sound of the fundamental bass is now major and now
minor, there arises a composite genus called Chromatic
Enharmonic, inasmuch as it gives rise to two minor semitones
in succession which together form an interval a quarter of
a tone less than a whole-tone " : — ^

-?s>-

-e^

4^^ p-

Here, the interval e\}-e^, composed of two chromatic semi-
tones (:jJxH^=Mt) is a quarter of a tone less than a
minor tone (7ftiiXTo = ii^)-

Still another form of the Chromatic Enharmonic, but which
is not so called by Rameau, is where the Fundamental Bass
ascends by an inter\'al equal to the sum of two major
Thirds : —

128

125

-9-S>-

1^^^

->G>-

\^-

B.F.

" If one," he says, " passes from one to the other extreme
of the quintuple proportion i : 5 : 25, there will result the
quarter-tone 125 : 128, which is the difference between the
major and the minor semitones." - " All these new genera,"
Rameau proceeds, " arise from the primar}^ fundamental
successions based on the triple and quintuple proportions,
but the product of these successions has no power as regards

1 Demonst., pp. 94, 95.

- Ibid., p. 91-

RAMEAU'S GENERATION HARMONIQUE 247

expression." Rameau means that the quarter-tone produced
by the genera in question is practically indistinguishable, and
is in itself too small an interval to impress the ear in any
aesthetic sense. Again, " the further we move away from the
primarv fundamental successions, the further we mo\'e away
from the Principal [Tonic centre], and as this Principal can
be no longer understood in its product, such a product has no
harmonic effect or expression ; the Diatonic recalls the triple
proportion, the Chromatic the quintuple proportion, and as
already the latter is less simple than the triple, so the ear
tinds greater difficulty in understanding its product. As for
the Enharmonic, it recalls nothing. It is the product of two
extremes, extremely dissonant with each other, to which
Nature herself has decreed that there should be no immediate
succession : whence it is not astonishing that it cannot be
appreciated by the ear." ^

Nevertheless, although Rameau thinks that the quarter- tone
is too small an interval to be appreciated, or rather under-
stood by the ear, he is of opinion that the effect of it may
be experienced even on a tempered instrument, and that
such effects are owing to the progression of the Fundamental
Bass. " Besides that the quarter-tone is inappreciable, its
expression, if this were possible " {"for example a melodic
series or succession of quarter-tones] " would be^\ilder the
ear rather than assist it ; thus it is banished from our keyed
instruments ; one never even thinks of expressing it on
instruments without keys, where such could be effected by a
ghding of the finger (as on the \dohn) : the same key sers'es

to express the two different sounds ,~jj % , whence it

is evident that if we experience the effect of the quarter-
tone in such a case, this effect can only be caused by the
change of mode [key] occasioned b3' the fundamental suc-
cession. ... Is it, then, possible to doubt that the cause of
such effects exists solely in the greater or lesser closeness of
relationship between the modes of which the Fundamental
Bass is the determining factor } " ~

Rameau, therefore, quite definitely distinguishes three
different species of fundamental bass. : — (i) Fundamental
Bass in fifths (1:3:9): from this arises the Diatonic System ;

1 Ditnonst., pp. 95-97. - Ibid., pp. loi, 102.

348

THE THEORY OF HARMONY

(2) Fundamental Bass in Thirds (1:5; 25) : from this we
obtain the Chromatic System ; (3) The composite Fundamental
Bass formed by combining these two : this gives rise to the
'Enharmonic genus. Rameau's Chromatic and Enharmonic
genera it would be wrong to consider merely as harmonic
curiosities ; on the contrary they are, together with Rameau's
treatment of them, of much importance for the theory of
harmony. It will be noticed however that Rameau treats
of only one kind of Chromatic semitone (24 : 25), and onh^
one kind of quarter- tone (125 : 125). He says nothing of the
Chromatic semitone which arises in the chromatic scale as
the difference between the diatonic semitone and the major
tone (fX-i-f=Tf^), nor of the quarter-tones obtained
b}^ the harmonic division of the semitone (ff and 4}^),
nor that obtained by comparing the sum of two diatonic
semitones wdth the major tone (|5;i}X-^=|i,-f^').

Further, how is Rameau to account for the fact that both
the diatonic and the chromatic semitones, and the quarter-
tone as well, were in actual use among the Greeks ? He is
scornful and entirely sceptical regarding the Greek quarter-
tone. " What," he exclaims, " are we to think of the Ancients,
who were acquainted only with the products of these different
genera, when the effects which the\' attribute to them do not
depend at all on these products, seeing that they — I refer
to the quarter-tone — are inappreciable by the ear ? " ^ Again
Rameau himself, in treating of the chord of the Diminished
Seventh, shows that the quarter-tone may arise, not as the
result of a composite Fundamental Bass, nor of one which
proceeds from one to the other extreme of the quintuple
proportion, but from a Fundamental Bass which ascends
a Third : —

(«)

-Gh'

~rz

(P) , .

:S:

J^MU

|p5:pjj:p^^=:

-iiSr

fcr-^^t

F.B.

^-H-

1 Demonst., p. loi.

RAMEAU'S GENERATION HARMON IQUE 249

At [a] the chord of the Diminished Seventh under-
goes an enharmonic change fe#-«b) whereby there results the
quarter-tone 125 : 128 ; the Fundamental Bass, according to
Rameau, is g^-b. If e and g be regarded as the fundamental
sounds, we have still the minor Third progression of the bass.
In the well-known enharmonic change (gb'/S) ^" ^^e Andante
of Beethoven's C minor Symphony [b) the bass remains
stationary.

Rameau points to several passages in his own works where,
he says, the effect of this quarter-tone is produced, one of
which is the opening Recitative, Act IV., of his opera Dardanits.
The passage is evidently as follows : —

Here Rameau either considers that in the first chord of the
second bar /# undergoes an implied enharmonic change to
^b ; 1 or that the Fundamental Bass proceeds from d to a#,
from one to the other extreme of the quintuple proportion ;
the chord b\}-d-f would then represent the enharmonic
equivalent of the chord fl#-cX-f#. It is certain, however, that
there is no enharmonic change here, whether expressed or
implied, and it is equally certain that Rameau did not " exper-
ience the effect " of a quarter-tone, for there is none. All that
happens is a somewhat abrupt but, since Rameau's day at
least, quite common transition from the Dominant chord of
a minor key to the Tonic chord of its relative major key ; in
this case, from G minor to Bj? major.

In his Noiivelles suites de pieces de clavecin, Rameau
mentions two pieces — L' Enharmoniqnc and La Triomphante,
in which, he says, this quarter-tone occurs, and indeed

1 See Riemann's (p. 402) and Prout's (p. 451) explanations of this
chord.

2;o

THE THEORY OF HARMONY

gives a detailed explanation of the harmonic progressions.
The passages he refers to are these : —

" L'Enluirmonique."

"La Triomphante."

^»

?-i^it?3&E^

Si*?i^Es;

— _t^-

He remarks : " The effect produced in the twelfth bar of L'En-
harmonique wall not perhaps please every one at first ; but
custom will soon overcome repugnance [!] The harmony which
produces this effect is not the result of chance or caprice,
but is authorized by Nature itself. The same effect occurs
in the fifth bar of the second Reprise of La Triomphante,
but the effect here is less surprising, owing to the successive
modulations. The effect arises from the difference of a
quarter-tone found between the cijf and cD^ of the first piece,
and between the h^ and ct] of the second. . . . The impression
we ought to receive does not arise, however, from the interval,
but solely from the modulation."

Here Rameau is of opinion that the " surprising effect "
in the first passage is owing to the enharmonic change
from c^ to ^[7, whereby there arises a quarter-tone. But
this is more than doubtful. The strange effect he speaks of
has really nothing to do with the d!^ of the second bar,
but arises from the progression of the first chord to the
second.' The progression from the second to the third

^ Some theorists would no doubt explain this second chord as
consisting entirely of "non-harmonic" passing-notes, and the chord
itself as possessing no harmonic significance. But this is merely an
easy and convenient means of getting rid of a difficulty. See, in
connection with this subject, pp. 323 and 405.

RAMEAU'S GENERATION HARMON lOUE

251

chord, on the other hand, is quite regular. The essential
harmony of the first chord is undoubtedly a-c^-e ; it is
impossible that f# can represent d\} ; that of the second
chord, according to Rameau himself, is g-h\\-d, and that of
the third c-e-g. If we accept Rameau's statement that
the chord of the diminished Seventh has a Dominant " root,"
these must represent the fundamental harmonies : —

i^.

--^i==i=^

r — *!•—,-?"-

Rameau, of course, does not here view the matter in
this Ught, but considers that every note in each chord
must form an essential constituent of the harmony. His
point of view certainly demands consideration by the
theorist, for even if Rameau is unable adequately to
explain the generation of the chords of the Seventh
and Ninth, nevertheless some explanation is necessary as to
how and why the Seventh and Ninth should be permitted to
intrude themselves into a harmony with which, apparently,
they have nothing to do. The " surprising effect " which
Rameau experiences in the passage in question is owing in
part to the harmonic ambiguity of the chord of the Diminished
Seventh, but chiefly to the immediate succession of two
major harmonies unrelated to each other, and whose " tonal
functions " are but vaguely determined. In addition the
Fundamental Bass descends a whole-tone, a progression which,
be it observed, is expressly forbidden by Rameau. But if
there is no quarter-tone here, in the second example, on the
other hand, the enharmonic change actually occurs. It is
curious that Rameau should have regarded the first passage,
in which there is no enharmonic change, as more strange in
its effect than the second, where there really is such a change.
He certainly describes quite accurately, however, the com-
parative effect on the ear of these two passages, but one finds a
touch of dehcious humour in the way in which Rameau dwells
complacently on the charming and original effect produced by
the first passage, an effect " authorized by Nature," — but
one which, nevertheless, he has himself expressly forbidden !

2^3

THE THEORY OF HARMONY

Other Aspects of Rameau's Theory.

As for the other aspects of Rameau's theory in the works we
are examining, these undergo Uttle or no modification. Only
as respects " Chords by Supposition " Rameau justifies his
procedure in respect of these chordS' — Third or Fifth placed
below the chord of the Seventh — by means of the arithmetical
proportion. " Supposition has its source in one of the sounds
of the arithmetical proportion added below the harmonic
proportion : suspension is only a consequence 6f this." That
Rameau has not yet a clear perception of the mechanism of
suspension is evident from the examples he gives, where
the chord at ia) is described as " a chord by supposition,"
while that at [b) is called a suspension, whereas both
chords in reality contain suspended notes : —

)

(")

_Q_

-C5>-

-&'-

32:

zr2i

-<s*-

231

B.C.

5 etc.

-<s>-

-o-

231

B.F.

Prohibited consecutives — Octaves, Fifths — Rameau dis-
misses with the remark that these need give little trouble
so long as proper attention is paid to the progression of the
Fundamental Bass, and the connection between the harmonies.
The origin of such prohibited consecutives is to be found in
the immediate succession of the two Dominants. " Why
are false relations [of the tritone], two Octaves, Fifths,
and major Thirds in succession forbidden ? You will find

in the fundamental progression of the

( — \} "There is nothing in the

nature of the Octave or the Fifth to displease us when
the two are heard in succession ; these consonances are in

the reason .
two Dominants

^ Gen. Harm., Ch. 6.

RAMEAU'S GENERATION HARMONIOUE

o J

themselves quite pleasing : the bad effect arises when they
represent a fundamental succession wdthout connection . . .
If a good connection be preserved, one need not trouble
much concerning the rest." ^ This does not explain,
however, why the consecutives at {a) should be displeasing
in effect, wliile the same chord successions at {b) sound
quite well. At * there is, besides, a better connection
between the parts than in the example which immediately
follows, in which there even occurs a hidden consecutive
Fifth. Many similar cases constantly occur : —

(«)

(^)

:g:

-frr

~&-

-?2=

(^)

rr\

:?z:

-€^-

-€.»-

* I
-i2.

-Ql

-QL

ZiZiL.

321

'JZH

For the practical use of the Fundamental Bass in composition
the following directions are given : " There are three fun-
damental sounds, Tonic, Dominant, and Subdominant. We
will call the Dominant — Dominant-tonic, every other Dominant
simply Dominant."

" Only the Tonic bears the perfect or natural chord : the
Seventh is added to the Dominants, and the Sixth to the
Subdominants."

' ' There is only one Tonic in every key or mode ; and only
one Subdominant ; every other note of the Fundamental
Bass is a Dominant."

" We may pass from one Tonic to another [that is, modulate]
by every kind of consonant inter\-al : Third, Fourth, Fifth
or Sixth."

" In addition, the Tonic may descend a Fifth to its Sub-
dominant ; ascend a Fifth or a Third to the Dominant-
Tonic, or fall a Fifth, Tliird, or Seventh to a simple Dominant."
[That is, the Fundamental Bass may ascend or descend by
any perfect, major or minor interval belonging to the key ;
the single exception being that of the bass descending a Second ;
such a step is prohibited.]

1 Gen. Harm., Ch. 19.

254

THE THEORY OF HARMONY

" The Tonic [harmony] alone is free in its progression : if
it proceeds to another Tonic, this will possess the same
privilege, otherwise it can only proceed to a Dominant, or a
Subdominant, each of which has a determined progression." ^

In composition, then, in any mode, the method of procedure
should be as follows : " Starting with the Tonic chord, you
may proceed wherever you wish ; whether to the Dominant,
the Subdominant, to the Third below, or the Second above,
and then give to these notes the succession which has already
been determined for them : until finally the Tonic is reached." ^

This abstract of rules for the progression of the Fundamental
Bass does not differ, it will be observed, in any essential
respect from the rules already given in the Traite and the
Notweatt Sysfeme. What is most remarkable is that Rameau,
who has accounted only for the three chief harmonies of the
key, should think it unnecessary to explain whence he has
derived the other harmonies, for example those on the Third
and Sixth degrees of the scale ; and that while the Fundamental
Bass, which properly belongs to the diatonic key system, is a
Fundamental Bass in Fifths, he nevertheless permits it, within
a single key or mode, to ascend or descend by any otherinterval,
the only exception being the descent of a Second. The whole
question is one, as we shall see later, which causes Rameau
much embarrassment.

It is, then, manifest that Rameau is unable to account
for manj^ of the simplest chord successions. In the following
example (a) the chords at * * must be explained as Tonics,
as any addition of a dissonance to their harmony would
destroy their character, and the nature of the chord succession.
This simple passage, therefore, would appear to be in three
different keys. At {b) and (c) we find the forbidden descent of
a Second in the Fundamental Bass. The effect, however, is
good, indeed excellent : —

(b)

(c)

^S=

—c^-

.s-i--^-

-&-

-&*-

-G>-

'^^d^

-e^-

:S=g:

'zq:

-&-

jO.

-€^*~

32;

-S>-

22:

-SI-

^ Gen. Harm., Ch. 18., Art. i. {Dela Composition.) ^ /ti^_

^ss

CHAPTER IX.

OTHER THEORETICAL WORKS OF RAMEAU: NOUVELLES
REFLEXIOXS SUR LF. PRINCIPE SOXORE, ETC.

Of other theoretical works of Rameau, less important
than the Generation Harmoniqiie and Demonstration, but
nevertheless deserving of notice, there may be mentioned the
following : —

(i.) Dissertation siir Ics differentes methodes d'accompagne-
ment pour le clavecin ou pour I'orgue, avec le plan d'lme
nouvelle methode etablie stir une mecanique des doigts que
fournit la succession fondamentale de I'harmonie, et a I'aide
de laquelle on peui devenir savant compositeur et habile
accompagnateiir , mime sans savoir lire la musique. (Paris,

1732.)

(2.) Code de mustqiie pratique ou Methodes pour apprendre
la musique, meme a des aveugles, pour former la voix et I'oreille,
pour la position de la main avec une mechanique des doigts
sur le clavecin et sur I'orgue, pour V accompagnement sur tons
les instruments qui en sont susceptibles, et pour le prelude, avec
de nouvelles reflexions sur le principe sonore. (Paris, 1760.)

(3.) Nouvelles Reflexions sur la Demonstration du principe
de I liarmonie, servant de base a tout I' art musical. (Paris,

1752.)

(4.) Reflexions de M. Rameau sur la maniere de former la
voix, d'apprendre la musique, et sur nos factdties pour les arts
d'exercice. {Mercure de France, Oct., 1752.)

(5.) Extrait d'une reponse de M. Rameau a M. Eider sur
I'identite des octaves, d'oii restdtent des verites d'autatit plus
curietises qu'elles n'ont pas encore ete soupgonnes. (Paris,

I753-)

(6.) Observations sur notre instinct pour la musique et

son principe. (Paris, 1754.)

256

THE THEORY OF HARMONY

(7.) Erreurs sur la musique dans I'Encyclopedie. (Paris,

1755.)

(8.) Suite des Erreurs sur la musique dans I'Encyclopedie.
(Paris, 1756.)

(g.) Reponse de M. Rameau a MM. les Editetirs de
I'Encyclopedie sur leur dernier avertissement. (Paris, 1757.)

(10.) Lettre aiix philosophes, concernant le corps sonore
et la sympathie des tons. (Paris, 1762.)

In the work first mentioned Rameau applies the theoretical
principles which he has developed in his Traite towards the
simplification of the method of accompaniment. " The chief
difficulties in accompaniment," he remarks, " have always
been : (i) The method of fingering employed ; (2) the ndes,
and the methods that have so far been given to us " I Rameau's
" method," in which the rules are few and simple, is distin-
guished by a reduction of the numerous possible harmonic
combinations to a few simple primary harmonies, by the
use of harmonic inversion, and of the Fundamental Bass. He
directs that " the bass be played with the left hand, while the
harmony proper to it is executed with the right." If all notes
common to two successive chords be retained in the fingering,
and in all other cases, movement by step be preferred to
that by leap, the fingering will give very little trouble. For
example : —

i^Q-t-g;

-<s>-

:g=

-&-

Hi— l-h

rpn

-tS»-

'^'^-

-&'■

i:jt

i*=

^6W

-rjr

-s*-

'■rr=-^

KZt

_0_

'Gt-

6
5

6
6

This is far enough removed from the contemporary ItaHan
school of figured-bass playing, the dominant characteristic of
which was the breaking up of the harmony by means of
graceful contrapuntal figuration (Scarlatti, Durante, Porpora,
etc.), but beyond all question it represents the best possible
method of obtaining a strictly logical and connected series
of chord successions, and it is surprising that such a method
of figured bass should not have been adopted in more
elementary text-books of harmony.

RAMEAU'S CODE DE MUSIOUE

257

Code de Miisiquc pratique.

This work, which is much larger than the Dissertation,
treats not only of accompaniment at the clavecin and organ,
but comprises also a school of composition, as well as a
" method " for forming the voice and ear ; here also Rameau
indulges in some interesting reflections on the purely aesthetic
and expressive side of his art.

Although Rameau still adheres to his theor}' of " chords
by supposition," he is now much more disposed than formerly
to consider these chords as resulting from the mechanism of
Suspension. This is noteworthy. Thus the " chords by
supposition " of the Ninth and the Eleventh are explained
as being formed from the suspension of the Octave by the
Ninth (9-8), and of the Third by the Fourth (4:3).^ Not
only so : the chord of the Tonic may be suspended — retarded
— by all the notes of the chord of the Dominant Seventh, and
in the Minor Mode by the chord of the Added Sixth. " The
Fifth may not only be suspended by the Sixth, but the Third
may be suspended by the Fourth and the Second [Ninth] ;
that is to say, the Tonic chord may be suspended by the complete
chord i£)hich precedes it : whether this be the chord of the
Dominant Seventh, or, in the Minor, the chord of the Added
Sixth." It should be noted that the chapter in which Rameau
makes this statement is entitled Notes d'ornement ou de
goiit. 2 So then, in the following passage, Rameau no longer
considers the chord g-b-d-f in the second bar to represent
the really essential harmony, for it merely serves to retard
the notes of the Tonic chord : —

^^-

^zr.

zcn

There can be Httle doubt but that Rameau's theory, in
respect of these chords, undergoes a change for the better ;
but he does not observe that he completely demolishes his

^ Cide de Musique,
suspensions."
' J bid, Ch.11.

XXI® Le9on : " Des suppositions et des

258

THE THEORY OF HARMONY

own theory of Supposition, whereby the really essential chord
ought to be regarded as that of the Dominant Seventh, the
note C being added below, as a " supernumerary sound."

With regard to dissonant chords, not only does the chord of
the Dominant Seventh require no preparation, but also
" none of the dissonances of which the leading note forms a
part [Dominant discords!] require to be prepared."^ As
for the chord of the Added Sixth, the Sixth does not require
preparation; "the Fifth, on the contrary, must always be
prepared." Rameau does not explain how it comes about
that the Sixth, the dissonant note of this chord, requires no
preparation, while the Fifth, which is consonant, must be
prepared.

Once more Rameau touches on the vexed question as to
which degrees of the scale may bear a consonant harmony.
In the Traiie he is, at first, in no doubt whatever that the only
consonant chord in a key is the Tonic chord. Every other
chord is dissonant, and whether the dissonance be actually
present in the chord or not, it must nevertheless be understood
as forming an essential part of the chord. Thus the Dominant
chord g-h-d always represents the chord of the Dominant
Seventh g-b-d-f, and the Subdominant chord f-a-c, the chord
of the Added Sixth f-a-c-d. For this reason the Dominant
and Subdominant chords have a strong tendency towards the
Tonic chord, and their resolution on this chord serves to
heighten that effect of repose which properly belongs to it
only. But in treating of the Dominant and Subdominant
Cadences Rameau finds himself obliged to modify this state-
ment, for now the Dominant and Subdominant chords are
themselves chords of repose (a), and this must necessarily
be the case if they are to produce a proper cadential effect.
So with other chords (6) : —

(«)

-<^-

-^>-

(i)

rr\

~JZl.

-f^-

^ZXy-

-€J—

^ Code de Mtisique, Ch. 7. Art. 17.

RAMEAU'S CODE DH MUSIQUH 259

These chords must be regarded as in themselves consonant,
for any dissonance added to their harmony would destroy
tlieir character as chords of repose. For the moment,
therefore, Rameau concedes that a consonant harmony
may appear on the Dominant and Subdominant, as well
as on the Tonic. Nevertheless, we find him very soon
afterwards repeating his former statement that the only
consonant chord is that of the Tonic, and that wherever
such a harmony is found, it must be regarded as a
Tonic harmony. Rameau is here confronted by a very
real difficulty. If the Dominant, or any other chord,
has a decided tendency towards the Tonic harmojiy,
then such a chord is one inducing motion, unrest ; how
then can such a chord produce an effect of repose ?
Must it not be regarded as no longer Dominant,
but Tonic, and bringing about a change of key ?
The problem is one which evidently completely baffles
Rameau.

Let it be observed that Rameau, in insisting as he does in
all his most important theoretical works that the only
consonant harmony in the key is that of the Tonic, is
enunciating a real principle of Tonality. Seeing that he
has this principle already within his grasp, why, it may be
asked, did he not make greater use of it ? He might
have demonstrated that the notes of the Tonic chord are
the only notes of rest in the scale, and that the other
notes are notes of unrest, each of which tends strongly
to proceed to that note of rest whicli lies nearest to it.
Thus in the scale of C major, the notes of rest are
c-e-g-c' : —

^^< ' '1^^

and the notes of unrest d-f-a-h. Of these latter, the notes /and
b, which are respectively distant only a semitone from the notes
e and c, are the notes of greatest unrest. But Rameau quite
rightly avoids such a course. For if the note b, the leading
note, is the note of greatest unrest in the whole scale, how then
can it be a note of rest, as it actually is in the Dominant

26o

THE THEORY OF HARMONY

Cadence ? At {a) the note b, the note of greatest unrest, appears
as a note of rest : while the Tonic c, the note of greatest
rest in the scale, and its Third e, appear as notes of
unrest : —

*7-

®=

(a)

-^-

i2I

(/')

HS»-

T=B:

-s>-

-<s>-

-^j-

:g=

izz

-G^

Again, if the fourth degree of the scale is a note of
great unrest, how can its harmony produce an effect of
repose, as in the Tonic-Subdominant Cadence ? If it be
considered that the effect of repose which may be produced
by the Dominant harmony is owing to the fact that the
Dominant itself is a note of the Tonic chord, and therefore
a note of rest, then the presence of two notes of the Tonic
harmony in a chord ought to increase still further the effect
of repose * : —

* C major.

This, however, is by no means the case. It would
seem, therefore, that the notes of rest in the scale, quite apart
from the question of dissonance and its resolution, may appear
as notes of unrest, and the notes of unrest as notes of rest,
according to circumstances. Not only so ; it results from the
Tonic, Dominant, and Subdominant Cadences that all the
notes of the scale may be regarded as notes of rest. Rameau
no doubt perceived that he was unable to derive much help
from this source. Nor did it enable him to account for the
tendency of the Subdominant or the Supertonic triad towards
the harmony of the Dominant {b).

In the Code de Musiq-ite Rameau emphasizes what he had
already demonstrated in the Traiie, that from the Perfect
Cadence are derived all the rules of harmonic progression.
All harmonic music is but a series of Cadences. " The Perfect
Cadence is the origin of the diversity which characterizes

RAiMEAU'S NOW. REFL. SUR LA DEMONST- 261

harmony. One inverts this Cadence, interrupts it, imitates
it, avoids it, and by such means variety in harmony is brought
about." ^ In this, which is almost his last important work
on the subject of harmony, Rameau makes no further attempt
to account for the triads on the second, third, and sixth degrees
of the scale.

Nouvelles Reflexions sur la Demonstration du principe de

I'harmonie, etc.

Neither in the Demonstration nor in the Nouvelles Re-
flexions does Rameau devote much space to his theory
of " double employment." In the latter work we find merely
a passing reference to the subject. " If the Greeks," he
remarks, " were ignorant of the origin of their diatonic
tetrachord, they were at least well inspired, for this tetra-
chord composed of the sounds si-nt-re'-mi gives, in the
diatonic order, both the Perfect Cadence and the limits of
the mode of which nt is the generator ; whereas in adding
fa and la, these sounds may be regarded as belonging to either
the Dominant or the Subdominant keys ! {en lieu qu'en y
ajoiitant fa et la, les Modes de la dominante & de la soiis-
dominante pourroient toiijoiirs Ic disputer a celui de leur
generateur). If la is employed as fifth of re, it must be
regarded as being in the key of the Dominant ; ... if, on
the other hand, it is employed as third oi fa, then the Sub-
dominant will be arbitrary " — (that is, will have no tendency
towards the Tonic chord). "This two-fold origin of the
sound la, where, in the same key, it may appear as fifth
of re, or third oi fa, is the cause of double-employment." ^

If Rameau is unable to show how the scale is derived from
three fundamental sounds, but considers it to be necessary
to add a fourth for this purpose, the addition of the fourth
fundamental sound has at least, he states, this merit, that by
this means we obtain the fourth proportional necessary
for the " rule of three " in geometry. In his later writings
he is preoccupied with the idea that a close relationship exists
between musical proportions and other arts and sciences,
especially Architecture. He has been confirmed in his views
on this subject by " M. Briseux, the architect, who intends

1 Code de Musique, Ch. 7, Art. 15.

- Nouvelles Reflexions sur la Demonstration, etc., pp. 26, 27.

262 IHE THEORY OF HARMONY

shortly to publish a treatise, in which he is to demonstrate,
among other things, that the beautiful edifices of the ancient
Greeks and Romans . . . were constructed according to the
proportions derived from Music. This justifies the view which
I have long held, that in Music there is unquestionably to be
found the principle of all the arts of taste, ... it is from the
regular division of the string in its several parts that arise
the proportions, each in its order of pre-eminence, or of
subordination . . . wherein it is to be remarked that division
precedes multiplication, a fact which ought to guide one with
greater certainty than has hitherto been possible towards
establishing the basis of a most noble and sublime philosophy. " ^

We find also a reference to Sir Isaac Newton and his " scale
of colours." " If M. Newton had been acquainted with this
principle, would he have selected a diatonic system, a system
simply of products, one which is full of error, in order to
compare it to colours ? Would he not, first of all, have inquired
whether each of these colours ought not to be considered as
forming a bass, a generator, whereby the colours form them-
selves into pleasing groups ? . . . Let us not be deceived :
the Arts, known as Arts of taste, are less arbitrary than their
title would seem to imply. It is impossible at the present
day not to recognize that they are based on principles,
principles so much the more certain and immutable inasmuch
as they are given by Nature ; principles the knowledge
of which enlightens talent and regulates the imagination,
and ignorance of which is a source of the absurdities of men
of mediocre talent, and the blunders of men of genius." ^

Although Rameau seems here to reprove .Sir Isaac Newton
for having failed to perceive what was the true foundation of
a theory of colours, namely, the principle of harmony, never-
theless, in the analogy he draws between the physical
properties of sound and light he is by no means talking at
random. A close analogy exists, closer indeed than the
scientists of Rameau's time were able to suspect. (Light —
a vibration of the ether ; sound — a vibration of the air ; wave-
theory of light, as well as of sound.) Of especial significance
for Rameau must have been the fact that the various colours
of the rainbow existed in white light, just as partial tones
existed in the compound musical sound.

^ Notn.'eUes Rt'flexions stir la Dhnottsfraiion, etc., pp. 49-51. * Ihid.

KAMEAUS NOUV. RHFL. SUR LA DEMONST. 263

Ramcau subsequently enters into an investigation concerning
the natural sounds of the horn and trumpet. He finds himself
unable to show how it is that we receive the major scale — of
the minor scale he takes no notice — directly from Nature.
On the contrary, he demonstrates that of all the sixteen
natural sounds produced by the horn or other instruments, the
only sounds of the scale we are able to accept are those which
together make up the harmonies of the Tonic and Dominant.
The other sounds are false ! notwithstanding that they are
given by Nature. Thus if C be the fundamental sound, the
Tonic, we are able to select certain sounds from the first
sixteen natural sounds, so as to form the following scale
c-d-e-g — b-c' : —

-r-^

All these sounds are in tune : the sounds / and a are certainly
also to be found, but they are so much out of tune that it is
impossible to include them in the above scale. ^

All that Rameau deduces from this circumstance is that
" as Nature has given us at first only those sounds of the
Mode " which correspond to the harmonies of " the generator
and its Dominant," therefore the succession of harmonies,
Dominant-Tonic, is more natural than that of Subdominant-
Tonic. This is confirmed by the fact that inexperienced
persons find it much easier to sing the fundamental bass
id-sol or sol-id, than id-fa, or fa-id ; also by the fact that
the timpani of the orchestra are tuned to the Tonic and
Dominant of the key.^

^ NouveUcs Rijlexions stir la Dc'nionsiradoii, efc, pp. 78-81.
- /hid., pp. 77-So.

264 THE THEORY OF HARMONY

Nouvelles Reflexions stir le Principe sonore.

The Nouvelles Reflexions sur le Principe sonore, appended
to the Code de Musique pratique, is in reaHty an inde-
pendent treatise, and of considerable theoretical importance.
This treatise furnishes fresh proof that the ideas of Rameau
on the subject of harmony were in a state of constant flux,
and his theories in a never-ceasing process of development,
even up to the end of his life. In the physical properties
of the sonorous body Rameau now sees not only the
principle of harmony, as well as of other arts ; it is a universal
principle, the principle of the cosmos itself. He remarks :
" There is one principle underlying all things ; this is a truth
which has presented itself to the minds of all thinkers, but
the more intimate knowledge of which has been attained
by no one. Convinced of the necessity of this universal
principle, the earliest philosophers sought for it in Music.
Pythagoras, following the Egyptians, saw the laws of harmony
in the movements of the planets ; Plato made it the governing
principle of the soul ; Aristotle, his disciple, after saying
that Music is a thing celestial and divine, adds that one
finds in it the explanation {raison) of the world-system.
In short, impressed by the agreement, the wonderful harmon}^
to be observed in their relationship to each other of the
different parts which compose the Universe, these thinkers
considered that the explanation of such a harmonious
relationship must necessarily be found in Music, in which
alone proportions exist ; that is, in the case of every other
sense but that of hearing, these do not present themselves
directly, but only in a symbolic aspect {elles n'en sont, a
proprement parler, que V image). But unfortunately the
system which these great men adopted, far from bringing
them nearer the object of their researches, only served to
remove them further from it. I venture even to assert
that the phenomenon of the sonorous body was absolutely
unknown to them." ^

Rameau subsequently develops his ideas on this subject
in the section entitled " Consequences des Reflexions precedentes
pour I'origine des Sciences," in which he gives a hypothetical
account of the musical impressions likely to have been

^ Nouvelles Rvjlexions siir le Principe sonore. Introduction.

RAMEAUS NOUVELLHS REFLEXIONS. Etc. 265

received by the first man, Adam, when he found himself
placed in a world which was unfamiliar to him. Adam, he
thinks, could hardly fail to observe that musical sound was
not simple, but compound ; after the discovery of tliis fact,
it was but a step to the recognition of the identity of Octaves,
and the discovery of the triple geometrical progression
(i : 3 : 9). From the triple progression would arise the
diatonic tetrachord. These achievements of Adam were
undoubtedly familiar to Jubal, " ipse fitit pater canentium
ciihara et organo." At the time of the Deluge, Noah must
have been able to save amongst his other effects the instru-
ments of Jubal, the tetrachord, and the triple progression.
During the building of the tower of Babel, and the subsequent
dispersal of the peoples, it would seem to be highly probable
that the triple progression was appropriated by the Chinese,
while the tetrachord found its way into Egypt, where
Pythagoras gained his knowledge of it. To the Chinese, on
the other hand, the tetrachord was unknown, but from the
triple progression they formed the pentatonic scale, which
naturally arises from a fundamental bass in Fifths, thus : — '

sol — la — ut — re' — 7ni
F.B. ut — ta — ut — sol — ut

3 — 1 — 3 — 9—3

In all this, which sounds so highly improbable to us, the
time in which Rameau wTote must be taken into account.

More important for our purpose is the new theory of the
" natural mode " now proposed b}- Rameau. This " natural
mode " (major scale) he now^ explains as arising from the
natural sounds of the harmonic series extended to the 45 th
term. By a process of selection we obtain from this series
all the sounds necessary for the formation of the major
scale. Rameau has already pointed out that the fundamental
sound being regarded as Tonic, the only sounds which are
" in tune " of all the first 16 harmonics of the horn or trumpet,
are those which correspond to the harmonics of the funda-
mental sound and of its fifth. That is, Nature presents us
in these harmonic sounds with the harmonies of the Tonic
and Dominant only : the Subdominant can never be discovered
among these harmonics, not even if they are extended to
infinity. Rameau gets over this difficulty b}^ a means
which, if it is not convincing, at least extorts admiration
because of its insjenuitv. He makes the third harmonic

266 THE THEORY OF HARMONY

sound, instead of the first, the determining note of the mode,
or Tonic. Thus, if C be the fundamental sound, we obtain
from the first i6 harmonic sounds the harmonies c-e-g and
g-h-d ; if now we consider this second liarmony to represent
the Tonic harmony, the first harmony will represent that of
the Subdominant. In proceeding thus we are deprived,
it is true, of the Dominant harmony, but this may be
discovered by a further extension of the harmonic series :
the Dominant harmony d-fj^-a, will then be represented
by the numbers 9 : 27 : 45.

We are now in possession of the three harmonies necessary
for the formation of the key-system : —

'C^^e^G^^^D^Jp^a

I 3 9

in which the central harmony, represented by the central
term (3) must be regarded as the determining, that is, the
Tonic harmony : that of the Subdominant (i) is its antecedent,
while that of the Dominant (9) is its consequent. It is true

that the sounds and -^^ cannot be produced on the

27 45 ^

instruments in question ; " but this is not the fault of Nature,

nor that of the instrument ; it is owing rather to our limited

faculties that we cannot produce on these instruments the

sounds of the ^'7 and -4V,."

As for the minor harmony, this is found among the first

16 sounds of the harmonic series ; thus : — ^

■^ 10 : 12 : 15

Rameau proceeds in thorough fashion to develop all this,

and apply it to the explanation of the Major and Minor

Modes, and of their relationship. He has first to explain

whence he derives the liberty of making the Fifth (G) of the

fundamental sound (C) the determining note, or Tonic ;

for in his previous works — even if, in his fundamental bass

of three terms 1:3:9, he has consistently assigned to 3,

the Dominant, the function of central term, or Tonic — he

has given cogent reasons why this privilege should be accorded

to the fundamental sound itself : this sound represents

Unity, by which all the other sounds are determined. The

explanation is as follows : The sonorous body, in vibrating,

^ Noiivellcs Ecjlexions siir le Principe sonore, pp. 198-204.

RAMEAU'S NOUVHLI.es REFLEXIONS, Etc. 267

causes to be heard distinctly among its harmonic sounds
only those of the 12th and 17th. The Octave and 15th also
form a part of the resonance of the sonorous body, but the
ear cannot appreciate them with the same facility : " it is
only with the greatest attention that they can be dis-
tinguished." " They blend together in such a manner with
their generator, that they become one with it ; they become,
in consequence, the Principal itself." ^ In the geometrical
progression i : J : I, the middle term -^, then, represents
the Principal i, and is indeed this Principal itself. Extra-
ordinary as it may seem, Rameau considers this to be a
sufficient reason for making the middle term of the geometrical
progression 1:3:9 (i : ^5 : ?,) the determining sound or
Tonic {ordonnatetir) of the mode. " In short," he remarks,
" the Principal, the generator, in producing Octaves in either
direction [i : |- : ^ or i : 2 : 4], from which arises for the first
time a geometrical proportion, gives us, at the same time,
by means of 3 and 5, other geometrical proportions, namely,
1:3:9, and 1:5: 25, in which the middle term, 3 or 5,
predominates, in the same way as the term 2 of the first
geometrical progression 1:2:4." Thus we obtain the

C - G - D
major system

As for the Minor Mode, and its relationship with the
major mode, Rameau proceeds thus : — " While the sound \
[G] is that which determines all harmonic and melodic
succession, we must not leave out of account the sound i [E] :
if ^ is the cause of what is most perfect in such a succession,
1 adds variety to it. Further, this ^ selects \ as the deter-
mining sound of its Mode [the Minor Mode], and not only
prescribes its progression, but also its harmony. For

12:15:18
example, if sol [G], of which the harmony is sol — si — re'

<r—h—d

determines the Major Mode, it is mi [E] which dcter-

10 : 12 : 15
mines the Minor ^lode with this harmony mi — sol — si

c —g—b
where we find that mi is subordinated to sol, which how-
ever is the sole cause of the difference between the two

^ NottveUes Rt-flexions stir Ic Priucipe sonore, pp. 194-195.

268 THE THEORY OF HARMONY

modes, a difference which consists in the quality of the
Third. At the same time this sol gives its Third ' ^. to

the sound mi, in order to constitute its harmon\', by the
formation of the Fifth ^ . ' ^ -? . The same subordination

Dll - SI

is further to be observed between the extremes of each
proportion, that is to say, that the antecedent of the Major
Mode [that is c-e-g\ lends its Octave and Third to that of
the Minor Mode [that is a-c-e\ ; the same is true of the
consequent [^-/#-«], only the consequent of the Minor Mode
ought to have a major harmony whenever it precedes its
middle term." ^ These remarks should be compared with
the passages already quoted from the " Demoyistration," "^
in which this theory of the formation of the Minor Mode
appears for the first time.

Thus Rameau, in what is practically his last pronouncement
on the subject of harmony, brings forward a theory of the
generation of the Major Mode which, in its most essential
features, differs from that of his previous works. It is a
new theor}^ in which all the sounds of the major scale are
derived from the sounds of the harmonic series. While
Rameau, in his Generafion Harmonique, finds it necessar\-,

1 " En cedant a son 4- la direction de toute la marche harmonique
& melodieuse, ne croyons pas que le principe ait oublie son * : & si
le J produit ce qu'il y a de plus parfait dans cette marche, non seulement
le I y ajoute des varietes qui I'embellissent, mais ce -J le choisit encore
pour ordonner de son Mode renverse, en le revetissant de tous ses
droits, jusqu'a lui prescrire sa proportion triple, & a former son
harmonie de la sienne propre. Si sol, par exemple, dont I'harmonie est

-j , ■ ■? ■ J y ordonne du Mode majeur, c'est pour lors mi qui ordonne

du mineur avec cette harmonie I n^j.^ol-si I o^^/ ^^ s^^roge aux
droits de son legislateur, qui neanmoins s'y conserve celui d'etre
la seule cause de la difference des effets qu'on eprouve entre les deux
Modes : difference qui consiste dans le genre de la tierce, dont il occupe

pour lors la place, outre qu'il livTe encore sa tierce ? a ce meme mi,

pour constituer son harmonie, en formant sa quinte -^ -'•

'^ ■^ nu - SI

" La meme subrogation s'observe, de plus, entre les extremes de
chaciue proportion, c'est-a-dire que I'antecedent de Mode majeure
piete son octave & sa .tierce a cclui du Mode mineur." (Noiwelles
Reflexions sur le Principe sonore.)

^ Ibid., pp. 257-264.

RAMEAU'S NOUVELLES REFLEXIONS, Etc. 269

in utilizing the harmonic sounds for the purposes of his
theory, to call a halt at the number 6, he now presses into
his service harmonic sounds such as the ._}f and 4^. That
Rameau should at the end of his life subject his theory
to such a new and strange development seems at first
sight to be a remarkable circumstance. Seeing that, when
he published the work we are now examining, the master
was yy years of age, and presumably therefore no longer
enjoying the full exercise of his powers, the thought suggests
itself that it is rather to the Generation Haymonique and
the Demonslration that we must look for his mature
and fully-developed theory of harmony. But there are
several circumstances connected with this last development
of Rameau's theory which mihtate against the view that
age had dimmed his insight or impaired his intellectual
vigour. He had discovered that he had made a serious
error with regard to the nature of the acoustical phenomenon
which he had considered to constitute the physical basis
"of the minor harmony. The principal sound had not, as
he had imagined in his Generation Harmoiiqite^ the
power to excite co-vibration in its multiples of the 12th and
17th major, but only in those portions of the string which
corresponded to the Unison of the exciting sound. It is
quite evident that this discovery had caused Rameau serious
misgiving. The minor harmony was left without a physical
basis. And not the minor harmony only, but the Sub-
dominant as well.^ Rameau then turns afresh to the
harmonic series, with the increasing conviction that in it
alone is to be found the explanation of the secrets of harmony.

Hence the new theory as to the origin of the Minor Mode
which he proposes in the latter part of the Demonstration.
He there remarks : " What does Nature indicate ? She
indicates that the principle [harmonic resonance] which she
has once for all established shall, and ought to, dominate
everywhere ; that everything ought to be related to it,
subordinated to it — harmony, melody, mode."

It is not surprising, then, to find that in the work we are
now examining, Rameau not only restates his views which
he had already advanced in the latter part of the Demonstra-
tion as to the origin of the Minor Mode, but demonstrates

1 See pp. 231-237.

2 70 THE THEORY OF HARMONY

that the minor harmony itself ^ ^. ' ' ', ' '^ ? arises from

nit - sol - SI

the sounds of the harmonic series. Here sol, the Tonic
of the major system, has as its major Third the sound si,
but this sound is also Fifth of mi ; sol then appears as minor
Third of the minor harmony. That the minor harmony
should be found among the sounds of the harmonic series
in just such a position as this, is evidently for Rameau a
remarkable proof of the correctness of his new theory of the
origin of the minor mode.^ Further, not only the minor
harmony arises from the sounds of the harmonic series,
but, indeed, all the sounds necessary for the formation of the
Minor Mode.^ At the same time, it must be pointed out
that in the minor we have again the proportions of the major
harmony, but in inverted order. In this sense, the minor
harmony is an inverted major harmony. Such an inversion,
however, is contrary to the natural order. ^

But notwithstanding this reference to the arithmetical
proportion, Rameau's views as to the origin of the minor
harmony and the Minor IMode are in no wise different from
those which he had advanced in the latter part of his
Demonstration. It is not in the co- vibration of the multiples
that we discover the proper physical basis of the minor har-
mony. This phenomenon merely indicates the possibility of the
formation of such a harmony. How this harmony is actually
formed, Rameau has already explained. Quite as remarkable
as Rameau's discovery of the minor harmony and the minor
mode among the sounds of the harmonic series, is his discoverv

1 " Dans ces memes instruments, I'accord de la proportion arith-
metique, renversee de I'harmonique, s'entend entre les sons ." ," ^

"171% SGL St

ou les octaves du + & du J^ sont a lo & a 12, ou ce -J- forme le tierce
mineure du ^, & ou y^, tierce de ce -J-, constitue I'harmonie du i,
dont il est quinte. Ainsi I'oreille & la raison y concourent egalement
pour nous convaincre, et sur le renversement entre ces deux propor-
tions, d'oii suit celle du Mode majeur en mineur, et sur I'agreable
effet que nous en eprouvons. Tout I'ordre diatonique du mineur
s'entendroit memes dans les aliquotes des corps sonore en question,
si Ton avoit la faculte d'en pouvoir tirer les sons." {Nouvelles
Reflexions sur le Principe sonore, p. 203.)

2 " La proportion harmoniquc. form^e des sous-multiples i : 3^ : i;,
se dinatuve totalement dans les nndtiples 1:3:5, car die se renverse
pour lors en proportion arithmetique d'ou resulte le changement,"
etc. — /bid., p. 195-

RAMEAU'S NOUVELLHS REFLEXIONS, Etc. 27 1

of the Subdominant in this same series. In liis Nouvelles
Reflexions sur la Demonstration, etc., Rameau had discovered
that of the three fundamental harmonics which together
constituted the major key-S3/stem those of the Tonic and
Dominant existed among the first 16 sounds of the harmonic
series. Not only the major harmony, then, existed in Nature,
but part of the major key- system itself, and this the most
important part. Further, Nature herself indicated the
relationship between these two fundamental harmonies :
for the third harmonic sound was not only Fifth of the
fundamental sound, but was itself a fundamental, giving
rise to its own series of harmonic sounds. But with regard
to the Subdominant harmony, Rameau was, of course, quite
unable to find for it a similar explanation. He therefore
concluded that " as Nature has given us at first only those
sounds of tlie Mode which correspond to the harmonies of

Tonic. Dominant,
the generator and its Dominant" as c — e — g — h — d

I 3 . .

therefore the succession of harmonies Dominant-Tonic is
more natural than that of Subdominant-Tonic.

But now Rameau, by the adroit addition of a third
term not downwards, but upwards, finds himself able,
as he imagines, to derive all the sounds necessary for
the major key system from the harmonic series, thus

Sub-dom. Tonic. Dom.

c — e — j^ — h — d — -/# — a. Nevertheless, he is not much

^ 'V ■ '

1 J 9

further forward. He is totally unable to explain how c,

the Tonic, has acquired the character of Subdominant,

and the Dominant g, that of Tonic. The Tonic g, then, is

no longer the fundamental sound, the "Principal"; this

privilege is accorded to the Subdominant : and the sound

in which, a's Rameau has assured us, harmony, the Modes,

chord-succession, modulation, etc., have their origin, is not

the Tonic, but the Subdominant. And yet there is Httle

doubt but that Rameau still understands the Tonic as the

" fundamental sound," which represents Unity, and "in which

all the other sounds have their origin."

It is not surprising to find that Rameau's ideas on the

subject of the origin of dissonance, or of dissonant chords,

2 72 THE THEORY OF HARMONY

also undergo a remarkable development. Here we find
nothing less than the Haiiptmann theory of the formation of
chords of the Seventh, by means of the conjunction of triads
closely related to each other by notes which they possess
in common ! Rameau's method of effecting this conjunction
is certainly an extraordinary one. If, he says, we add a

fourth proportional to this major triad '^ : 15 : i8 y^

g - b - d

as to this minor one '° ' ^ " ' ' ^ that is, before the antecedent

e - g - b

of the one, and after the consequent of the other, so that

they are conjoined, we shall have 10:12:15:18 j^ ^ similar

e - g - b - d'
way, if we combine 8:10:12 ^^-^j^ 10: 12: 15 ^^^ obtain

c - e - g e - g - b

the chord of the Seventh 8:10:12:15 p^^^ ^^^ first chord

c - e - g - b

of the Seventh there arises the interval ^^ " '^, which, by

e - d

inversion, gives us the minor tone ^ ' ' ° ; and from the second

a - e

chord of the Seventh, the interval ^ ' \5 the inversion of

c • b '

which is the major semitone '5 : 16 1

b - c '

1 Soit efEectivement ajoutee une quatrieme proportionelle geometrique
.a cette proportion harmonique ["^'^^ ' ^^, en meme-temps qu'^ cette

arithmetique jq : 12 : 15' c'est-a-dire, avant I'anttcedant de I'une &
apres le consequent de I'autre, oh elles se confondent pour lors,
nous aurons " ^^ j ^^^ _" p^ ' J"^, qui donnent une septieme de "" a ^"g,

dont le ton mineur ^^ " ^ est renverse. Assemblons cette meme

proportion arithm6tique avec cette autre harmonique 'g " "" " *° , une

pareille proportionelle, dans un ordrc oppose au precedent, ou les deux

proportions se confondront cgalcment, fournira dans "J ' "" ' *°^ " **

° 8 - 10 - 12 - 15

une nouvelle septieme de ^J a f^, dont le demi-ton maieur ** ' "f

015 ^ 15 : 16

•est renvers6. {Nouvelles Kijlexions sur le Ih-incipe sonore, p. 207.)

RAMEAUS XOUVELLES REFLEXIONS, ETC. 273

As for the chord of the Dominant Seventh, this is formed
in a different manner. It arises from the conjunction of
the extreme terms of the triple proportion. Such a union
in a single chord of the limits of the key-system brings
about the absolute determination of the mode, or key.'^
This is exactly Hauptmann's position with respect to this
chord : it represents the closing of the key-system, and the
clear definition of the Tonic harmony as central harmony.

In forming, as he does, the chord of the Seventh by means
of the addition of a fourth proportional, Rameau proceeds
in much the same way as in the generation of the " natural "
major mode, in which also the addition of a fourth term of
the Fundamental Bass was necessary. It would therefore
appear, he remarks, that there is no reason why the chord of
the Seventh — whether of the form e-g-b-d or c-e-g-b — should
not be considered to be as natural as the major mode itself.
From these two chords we have obtained the minor tone
d-e, and the diatonic semitone b-c. From the chord of the
Dominant Seventh we obtain the major tone (8:9). This
chord must also be regarded as a natural product, seeing
that it is formed by the union of the extreme terms of the
triple proportion. Dissonance, then, is the product of
nature itself, and it also has its source in the one and only
principle of harmony — the harmonic resonance of the sonorous
body ! Such is the opinion now expressed by Rameau.
" How," he asks, " can the dissonances [the tones and semi-
tones] which form the basis of the older systems of music, be
considered to be the work of Art ? Since Nature reveals her-
self harmonically only in the resonance of the sonorous body,
how is it possible to derive these dissonances from another
source ? What blindness ! If I have gone astray on this
point in my first two works, have I not corrected myself in
my later writings ? I at least conjectured what I was unable
fully to understand — the fault of not deriving from the
principle all the consequences of which it was susceptible." -

^ " On la voit cette dissonance se former entre les extremes d'une
proportion triple : on ne la voit possible d'ailleurs que dans I'harmonie
du consequent, a laquelle se joint I'antecedant, pour lui servir de
septieme & s'linir, pay ce moyen, avec lui pour rentrer ensemble dans
I'harmonie de leur terme ■moyen, ou cet antecedant prepare I'oreille a
recevoir le sentiment du genre dont le mode annonce doit etre suscep-
tible." (Nouvelles Reflexions sur le Principe sonore, p. 210.)

* Ibid.,-p-p. 206, 207.

274 THE THEORY OF HARMONY

Rameau, therefore, has now several ways of accounting
for the origin of the tones and semitones of the scale : they
arise from the fifth progression of the Fundamental Bass ;
from the addition of a fourth proportional to the major or
miner harmony, as well as from the conjunction of the extreme
terms of the triple proportion.

Extrait d'linc reponse de M. Rameau a M. Euler sur
Videntite des octaves, etc.

In this brochure Rameau endeavours to prove that the
celebrated mathematician Leonard Euler, in his work Tentamen
novae theoriae miisicac (Petrograd, 1729), had arrived
at wrong conclusions in respect of the nature of the octave.
Euler had taken as the basis of his theory of music the
principle enunciated by Descartes, Leibniz, and other philoso-
phers and mathematicians, that musical sounds are related to
each other, are consonant, and pleasing in their effect, in
so far as their ratios are simple and admit of being easily
understood. Thus the Unison is the most perfect of all
the intervals in this respect, that it gives us the impression
of the most perfect order or harmony, because the vibrations
of the two sounds which produce this impression appear to
the mind like a succession of points in perfect corre-
spondence: thus: — ^ '-Unison. The ratios of the

Octave, and of the double and triple Octave, produce also
in us the impression of order, but not, like the Unison, of
identity, for in the Octave the vibrations of the higher sound
are twice as numerous as those of the lower sound, those
of the double Octave four times as numerous, and so on :

for example, ^ XOctave. Here the vibrations

made by the higher sound of the Octave are to those of the
lower in the proportion of two to one. In this manner,
Euler determines the different degrees of harmonious relation-
ship of different intervals. The Unison is in the first degree
of relationship : while the second and third degrees of
relationship are assigned to the double and triple Octave
respectively.

RAMEAU AND EULER

- / y

It is not difficult to understand how these conclusions of
Euler did not suit Rameau. But the latter, in his " reply "
only demonstrated how badly he was equipped, chiefly
through lack of the necessary scientific training, for entering
the lists against such men as Euler. Rameau thinks that
he furnishes a proof of the identity of octave sounds when he
points to the fact that when men and women sing the same
melody, they appear to sing the same sounds. He also states
that the identity of Octaves is indicated by Nature, because,
while in the resonance of the sonorous body the 12th and 17th
are easily distinguishable, the Octave and 15th cannot be so
distinguished. He says : "It should be remarked that
the Octave cannot be distinguished in any sonorous body
capable of being plucked, struck, or affected by vibrations
of the air {qu'elle ne se distingue jamais dans aucun corps
sonore pince, frappc, ou emu par le vent)^ while on the other
hand the 12th and 17th can be easily distinguished. The
Octave changes in no way the nature of a sound, but, Kke
the Unison, merely strengthens it ; adding to it, however,
greater brilhancy. ... Is it not surprising that the J and 1
[12th and 17th] should be heard so distinctly, while the | and
\, the Octaves, are so to speak mute ? Yet these Octaves
really sound not less but more powerfully than the 12th and
17th. . . . and the reason why they cannot be distinguished
is that they blend or coalesce so perfectly with the funda-
mental sound, which is that of the total sonorous body. . . .
Hence we speak of the representation of a sound by its
Octaves : in short, of the Identity of Octaves."

Rameau does not give any exact information as to the
nature of the sonorous body in which the Octaves sound
more powerfully than the 12th and 17th and j^et cannot be
distinguished. Nevertheless, his meaning is fairly clear.
Octave sounds, although in the resonance of the sonorous
body they are by no means " nmte," but easily distinguishable,
nevertheless unite or blend with the fundamental sound more
perfectly than any other sound. But it does not follow
that Octave sounds are identical : and Euler was quite
justified in regarding the Octave as an interval distinct from
the Unison. Rameau, on the other hand, rightly perceived
that the resemblance between Octave sounds was so close
that for all the practical purposes of harmony the one may
be said to represent the other.

2 76 THE THEORY OF HARMONY

In the Lettre aux philosophes, concernant le corps sofiore
et la sympathie des tons {Memoir es de Trevoux, 1762), which
is his last communication on the subject of harmony, Rameau
pursues his reflections on the sonorous principle. It con-
tains however nothing new, but merely recapitulates the
principles with which we are already familiar. " Harmony is
the gift of nature. The sonorous body vibrates and produces,
besides its own sound, other sounds, from which arise two
proportions : one geometrical, determined b}- the octaves ;
the other harmonic, and determined by the harmonics of
the 12th and 17th." The harmonic proportion determines
harmony ; the geometrical proportion determines its suc-
cession. The remainder of the " letter " deals with the
particular methods of instruction advocated b}^ Rameau in his
practical works treating of accompaniment and composition.

Contemporary Criticism of Rameau 's Doctrines :
Rameau and the " Encyclopedists."

As might be expected, the theories of Rameau did not
escape criticism, even in his life-time. At first Rameau
had the support of the philosophes, the " Enc3^clopaedists,"
including d'Alembert, who^ as is known, was the author of
the little work entitled Elements de Miisique theorique et
pratique suivant les principes de M. Rameau (Paris, 1752),
generally described as a concise and lucid exposition of
Rameau's theory of harmony. Rameau did not fail to
thank d'Alembert for the service he had thus rendered him :
" M. d'Alembert had done him the service of adding, to the
solidity of his principles of harmony, a simplicity of which
he indeed felt that they were capable, but which he himself
had not been able to impart to them " (Letter to the Editor
of the Mercure de France, May, 1752). Of d'Alembert's
w^ork, J. J. Rousseau remarks, in his Dictionnaire de Musique
(Amsterdam, 1772) : " Those who desire to see the system
of M. Rameau, which in his various writings is so obscure
and diffuse, explained with a simpUcity and clearness of
which one could scarcely have imagined it to be susceptible,

should have recourse to the Elements de Musique of

RAMEAU AND THE " ENCVCLOI'.EDISTS " 277

M. d'Alembert." ^ On the contrary, it may be affirmed that
those who desire to acquire a knowledge of Rameau's many-
sided theory and researches in the science of harmony need
not expect to gain this by the perusal of d'Alembert's work.
Its very lucidity and conciseness constitute from this, point
of view its principal defects. D'Alembert has pursued an
eclectic method ; his principal endeavour has been to weld
Rameau's theories into a logical system — a somewhat difficult
task ; he selects, but he also eliminates, and that to a
serious extent.

Towards the end of his life, however, Rameau was un-
fortunate enough to incur the disfavour of the " Encyclo-
paedists," through causes which appear to have been poHtical
rather than personal. The result was that in several articles
deahng wdth music which appeared in the French Encyclo-
pcBdia,^ Rameau found his theory of harmony assailed.
To the articles in question, which were generally attributed
to d'Alembert, but which more probably were the work of
Rousseau, Rameau repHed with Erreurs sur la Musique
dans I'Encyclopedie (1755) and Suite des Erreiirs sur la
Musique (1756).

Against Rameau's theor}- of fundamental chords, and
especially of " fundamental discords," Rousseau urged the
follomng objections : M. Rameau requires the harmony, at
least theoretically, to be full and complete. The result is
that a great many of his dissonant chords are insupportable
when all the notes are present. " The Italians on the con-
trary, care Httle for noise. A Third, a Sixth, skilfully used,
even a simple Unison, when needed, pleases them better
than all our fracas." But indeed M. Rameau, in the
majority of his dissonant chords, does actually find it
necessary, in order to render them supportable, to omit
some of their sounds. The sound which has to be omitted
is sometimes the Fifth. But according to M. Rameau,
this Fifth is the support, the buttress of the harmony ; how
then can it be omitted ? Again, M. Rameau does not inform
one " where to take the dissonance, for he permits three
kinds of harmonic successions : that by consonant chords
only ; that by dissonant chords only ; and that in which
both consonant and dissonant chords are intenvoven v\dth

^ Art. Systeme.

^ Encyclopedic 011 Dictionnairc raisonnc dcs Sciences, dcs Arts, et
des Metiers, par une Societe de Gens de Lettres. (Paris. 1751-80.)

278 THE THEORY OF HARMONY

each other." Rousseau further objects that " M. Rameau
has pretended that Melody arises from Harmony."
M. Rameau himself, however, ascribes different effects to
the interval of the Third as compared with the Fifth ; further.
Accent and Rhythm, on which music depends for so much of
its charm, do not owe their origin to harmony. To much of
this criticism Rameau cannot well find a satisfactory
answer.

On other points Rousseau shows himself less discerning,
as for example when he remarks : "It appears, then,
necessary to suppose that every dissonance should be resolved
downwards ; if there are any which resolve upwards,
M. Rameau's instructions appear to be insufficient " ; to
which Rameau has no difficulty in replying that in his
theoretical works he repeatedly lays stress on the fact that
there are two kinds of Dissonance, the major, which resolves
upwards, and the minor, which resolves downwards, and
that he has explained in the clearest possible manner how
both arise, and how they should be treated. Not infrequently,
one observes that neither Rousseau nor Rameau quite
understands the real nature of the subject he is discussing ;
as for example where the former expresses the opinion that
" chords by supposition " are as susceptible of inversion as
other chords ; and where the latter docs not observe that
the chord of which he speaks — the chord of the Eleventh — is
nothing but a simple 4-3 Suspension.

The Suite des Erreurs sur la Musique dans I'Ency-
dopedie is wholly taken up with an attack on Rousseau's
article " Enharmonique," written for the Encydopcedia,
and is mainly concerned with Greek theory. Rousseau
had remarked : "As modern authors [Rameau] have ex-
pressed themselves somewhat vaguely on this subject, we
consider it necessary to explain matters here a little more
clearly." How, asks Rameau, has Rousseau done this ?
" Simply by copying, word for word, the article deahng
with the subject in the Generation Harnionique " I Rameau,
however, does not think much of Greek theorists, who
regarded Thirds and vSixths as dissonances. He cannot
understand the marvellous effects attributed to Greek music,
as such effects could not be produced without the use of
Thirds and Sixths ! But we have seen that Rameau does not
show to advantage as an authority on Greek musical theory.

RAMEAU AND 'I'HK ENCYCLOIM'-DISTS " 279

On the whole, liowever, Rameau had little ground for
complaint with regard to the articles on Music which appeared
in the Encyclopedia. They followed, in the main, the
theoretical principles he had already laid down in his various
works, and demonstrated to a marked degree that whether
or not Rameau's theory of harmony was to be regarded as
an adequate and well-considered system, there was at least
no other system worthy of being placed beside it. Thus
in the article " Scale " (Gamme) d'Alembert — or Rousseau —
not only accepts Rameau's explanation of the scale as arising
from a Fundamental Bass of three terms (fundamental bass
in Fifths), but also his theory of the " double employment
of dissonance," the chord of the Added Sixth, the two-fold
aspect which the sixth degree of the major scale may
assume, and so on. Occasionally the writer of the articles
considers it necessary to supplement Rameau's theory in
some respects, and on such occasions generally comes to
grief. For example, in the article entitled " Fondamental"
he takes upon himself to explain the chord of the Augmented
Sixth, of the form f-a-h-d^. " This chord," he remarks,
" is not in reality a chord of the Sixth ; for from fa to rej^
[f-dj(f\ there is really a Seventh [!] It is only custom which
makes us persist in giving to this chord the name of augmented
Sixth." The writer here considers that e]^ may be substituted
at pleasure for d^ without in any way altering the tonal
significance of the chord, and, like not a few other theorists,
is of opinion that temperament simplifies and reconciles all
things ! He should have imitated the wise example of
Rameau, and avoided this chord as carefully as possible.

Again, while he agrees with Rameau that the chord of
the diminished Seventh, for example g^-h-d-f, has a Dominant
" root," he feels constrained to add that " this chord is
wrongly called a chord of the Seventh, for from sol^ to fa
[^#-/] there is only a Sixth." He also considers it his duty
to awaken musicians in general to the actual possibilities of
harmony. " I am afraid," he remarks, " that the majority
of musicians, some blinded by custom, others prejudiced
in favour of certain systems, have not derived from harmony
all that they might have done, and have excluded numerous
chords which are capable of producing a good effect. To
mention only a few of these, how is it that one never uses
in. harmony the chords ut-mi-sol^-ut, and ut-mi-sol^-si ;

28o THE THEORY OF HARMONY

the first chord contains no dissonance, while the second
chord contains but one " ! He admits that the first chord
sounds somewhat harsh , but cannot discover the reason for
this. He considers it to consist of major thirds added
together, and asks : " How is it that harmonies which
when heard separately please us, when heard together
sound harsh ? I confess I do not know, and I believe this
is the best answer " !

The other chords to which he calls the attention of musicians
are the follomng : —

c-e\f-g-b c-e\)-g^-l\f

c-e\}'g\)-c c-e-g-a\^

c-c\}-g\f-b c-e-gj^-ci

c-e-g^-b\) c-^-gPci

c-e\^-gj^-c c-e-g'^-b

c-e\^-gi-b c-e-g\)-a\)

Here the list ceases ; a few other chords might have been
added on the same principle ; no doubt at this point the
ingenuity of the author became exhausted. Rameau might
well have asked whether all this represented musical science,
or whether it was not really some new game, perhaps suitable
for a kindergarten ; and what was to be done with a musical
theorist who was unable to discover 2lXvw dissonance in the
chord c-e-gi^-c !

Already, in 1753, in his Lettre sur la Mnsiqiie Frangaise,
Rousseau, in comparing French with Itahan music, to the
detriment of the former, and especialty of Rameau, who by
this time had become recognized as one of the greatest
composers of his age, had suggested that the most important
factor in music and musical expression was Melody, and
not Harmony, and that instead of Melod}' arising from
Harmony it would be more accurate to say that Harmony had
its origin in Melody. This was, for Rameau, an abominable
heresy, and in his Observations sur noire Instinct pour
la Musique, published in the following year, he subjects
Rousseau to sharp criticism. The effect of music, Rameau
begins, depends not so much on the transitions from grave
to acute, from piano to forte, from slow to lively : these
are feeble means. Harmon}^ is the sole basis of music, and
the caiise of its greatest effect. He then proceeds to demon-
strate, by means of the arguments famiUar to us, that Melodj''

RAMEAU AND THE ENCYCLOP.KDISTS" 28r

arises from Harmony, and also remarks : "If we sing a
melodic passage as, c-d-e-f-g, we shall find that the small
degrees of the scale are suggested by the consonances to which
they pass. After singing the whole-tone c-d, one naturally
ascends another whole-tone ; because this gives us the conson-
ance of the major third [c-e). Next, a semitone will be taken ;
one could not naturally sing a whole-tone, because this would
give the augmented Fourth— a harsh dissonance. After
this semitone, we next ascend a whole-tone, so as to arrive
at the perfect Fifth. All this proves that Melody is based
on the harmon}' of the sonorous body."

Rameau examines a passage from a work by Lull}', and
points to the different aesthetic effect produced by a transition
to the Dominant, as compared with that to the Subdominant.
He remarks that the effect of Lully's melody depends almost
entirely on the harmon}-, and that the effect would remain
even if the melody were made to fall wiiere it now rises,
and vice versa. He subsequently analyses a Recitative from
Lully's Armide [del! qui petit m'arreter !), and shows
that although it contains no accidentals, there is nevertheless
much chromatic effect, that is, implied chromaticism result-
ing from the progression of the Fundamental Bass. Rameau's
remarks are extremely interesting, and to a large extent
also convincing. He speaks for example of Lully's masterty
use of an interrupted Cadence where, although the sense of
the words in Armide's Recitative is finished, Armide herself
is not. The latter part of Rameau's brochure is taken up
Math a very keen attack on Rousseau, concerning the article
Lully, written by the latter for the Encyclopaedia.

The Rdponse de M. Rameau a MM. les Editeurs de
VEncyclopedie, etc., concerns a notice which d'Alembert
had prefixed to Volume VL of the Encyclopaedia in which
he defends Rousseau from the attacks made upon him by
Rameau. D'Alembert had remarked of Rousseau that " he
joined to much knowledge of and taste in Music the talent
of thinking and expressing himself with clearness and precision
{nettete), a talent which musicians do not always possess."
He reproached Rameau for having said that geometry was
based on Music, and that in short the principle of Music
influenced equally the other arts ; that a " clavecin ocnlaire,
which would illustrate the analogy between harmony
and colours, would meet with general approbation." etc.

282 THE THEORY OF HARMONY

Rameau, in his reply, denies that he had asserted that geometry
is based on Music. But he adds later : " I believe, however,
that it would be much more easy to prove the possibiUty
rather than the singularity of it." He then proceeds to
restate the main points of his theory. He owes, he remarks,
all his discoveries in music to the observation of the laws
of Nature, as manifested in the sonorous body. " This is
a whole divided into an infinitude of parts . . . from which
there result, in the same instant- — root, tree, branches,
proportions, division, addition, multiplication, squares, cubes,
etc. " !

In the Memoires de Trevoux, of August, 1735, appeared
an article Des Nouvelles Experiences d'Optique et d'Acoustique,
by the Jesuit R. Pere Castel, in which an attempt was made
to belittle Rameau's theoretical achievements, and to prove
that he had merely developed somewhat the discoveries
of his predecessors. Castel claimed that Anathasius Kircher
{Musurgia tmivcrsalis sive ars magna consoni et dissoni,
etc., 1656), had discovered the Fundamental Bass before
Rameau. " Kircher teaches that a true bass should proceed
by a 4th, a 5th or an 8th {Quarta vox Basis, vulgo Bassus

. . ita dictus, quod in eum, tanquam in Basin, onmes inclinent
voces. . . . Gaudet intervallis gravioribus, grandioribusque,
Quarta, Qtdnta, ei Octava : in natura rerum respondet
telluri)." Castel proceeds to argue that Rameau, in admitting
three fundamental sounds in each key- — on the first, the fifth,
and even the fourth degrees of the scale— loses sight of
the unity of Nature ; that he contradicts his principles in
making the Subdominant the bass of a chord which is not
fundamental — the chord of the Added Sixth. In the chord
C — g-b-d-f, Castel argues that the sound c is not merely the
fundamental sound by " Supposition," but that it, and not
g, is the real fundamental. He refers to Musettes in support
of his contention : in these Miisettes one finds the Tonic
sustained throughout (Tonic Pedal). It is through the
Tonic that one understands the other degrees of the scale,
and this is proved by the fact that these degrees have received
names which indicate their relationship to the Tonic.

Rameau's reply appeared in the following year {Lettre
au R. P. Castel, au sujet de qttelques nouvelles rSfiexions sur
la musiqiic — Memoires de Trevoux, July, 1736). He takes
Castel to task for his somewhat belated discovery of the real

RAMEAU AND THE "ENCVCL0P.'?^:DISTS" 283

significance of Kircher's reference to the harmonic bass. " This
is not the time, Rev. Father ... to expound the proper
interpretation which ought to be given to the rules you quote
from this author." The language of Kircher, he remarks,
is merely an echo of a passage from Zarlino, which is
quoted in the first chapter of Book II. of the Traite de
I'harmonie. Kircher gave no determined progression to his
bass ; it could receive indifferently the perfect chord, the

chords of 6 or , or of the 2nd, 7th or 9th. He made no

distinction between the Fundamental Bass and the Basso
Continuo, citing, as fundamental, chords which were
"derived" (inversions), and again, as "derived," chords of
whose fundamental he was ignorant. Kircher, in short, had
no acquaintance \\ith the principle of harmonic inversion.

As for Castel's contention that the fundamental of such
a chord as C — g-b-d-f, is c and not g, Rameeiu replies that
this no doubt is an ingenious theory, namely, that the
sonorous body is the foundation of all the sounds of the mode ;
but if Castcf admits, as he does, that the fourth degree of
the scale is incommensurable ; if it is not found as an aUquot
part of this same body, and if it is the same with regard to
the minor Third, the\'^ixth, etc., Rameau then fails to see
how it can serve as the foundation of all the sounds of the
mode. The union of c \vith g-b-d-f arises from another
principle, which, however, is only a consequence of the first.
We hear c-e-g in the resonance of the sonorous body, and
it is from these sounds, again, that b-d-f-a arise. But all
these sounds cannot be heard together. Rameau adds that
he will deal with the points raised by the R. Pere in
his work the Generation Harmoniqne, which he is about to
publish.

In a pamphlet entitled Nouvelle Decouverte du Principe
de I'harmonie, avec un Examen de ce que M. Rameau a
piihlie sous le tifrc de ''Demonstration de ce Principe,"
by " M. Esteve, de la Societe Royale des Sciences de Mont-
pelUer " (Paris, 1752), Rameau's theories as to the physical
nature of musical sound are subjected to some criticism.
" M. Rameau has said that every sound which is composite,
which includes within itself several other sounds [harmonics]
is a musical sound ; but that every sound which is simple
produces on the ear the effect of noise." M. Rameau, however,

!84

THE THEORY OF HARMONY

avails himself of three harmonics onlj-, and does not mention
the others. But if i\I. Rameau makes use of harmonic sounds
for his principles of harmony, he ought to make use of them
all : for he is not at liberty to select a few and neglect the
others.

Especially noteworthy is the fact that M. Esteve here
brings forward a new theory of Consonance. He does not
agree with Descartes, who has said that the agreeable sensa-
tion we experience from consonance is owing to the fact that
the soul takes pleasure in simple relations. " If the soul,"
he remarks, " distinguishes consonance from dissonance only
when it is attentive to compare sounds (as 1:2 = Octave ;
2:3 = Fifth, etc.), then why is it not conscious of this
operation ' ' ? This sounds like a passage from Helmholtz. But
indeed, Esteve presents us with a theory- of Consonance that
is nothing more nor less than the theory of conso7iance propounded
by Helmholtz. Consonances are such, and therefore pleasing
to the ear, because their harmonics are reinforced ; that is,
consonance is determined by the coincidence of the upper
partial tones — or of some of these tones — of the two sounds
forming the consonance. With dissonance, on the other
hand, there is no such reinforcing or coincidence of the
harmonics : instead, these clash with one another. M. Esteve
then examines the varying degrees of consonance of the
different intervals, determined by the coincidence, or other-
wise, of the upper partial tones. He gives the following
table :—

Table des Harmoniques des Consonances.

Funda-

mental . .

sol

mi , soZ 1 2 i ut

\N i sol 1

Octave . .

sol

Fifth

sol

Fourth . .

lit

a I

a ut

The imperfect consonances, as well as the dissonances, he
compares in a similar way. Esteve refers to M. Sauveur
" who has proposed another principle of harmony " ; the
essence of which is that dissonance fields its physical explanation
in the presence of beats, while consonance is marked by the

RAMEAU AND THE " ENCYCL0P.'!^:DI3TS ' 285

absence of such beats ! ^ The words used by M. Sauveur,
and quoted by Esteve, are : " In pursuing this idea, we find
that the chords in which beats are not heard are Consonances ;
and that those chords in which the beats are strongly felt,
are Dissonances ; and that when a chord is a dissonance in
a certain octave, and a consonance in another [!] the reason
is that it heats in the one and not in the other." Here we find
in the middle of the i8th century the whole Helmholtzian
theory of Consonance completely developed.

^ Cf. also Rameau's remarks on this subject, p. 157. Rameau,
however, does not regard beats as the explanation of, but merely as
incidental to, the phenomena of Consonance and Dissonance.

2 86 THE THEORY OF HARMONY

PART III.
CHAPTER X.

DEVELOPMENT OF THE THEORY OF HARMONY FROM THE
TIME OF RA:\IEAU UP TO THE PRESENT DAY.

Tartini's Trattato di Musica.

The theories of Rameau, notwithstanding much opposition
and criticism, gained a widespread influence, even during
his life-time. His Traite de I harmonic Mas translated into
several languages. Even Rousseau, in his Dictionnaire de
Musique, found himself obliged to adopt, for the articles
deahng with the subject of harmony, the theoretical principles
of Rameau. Rousseau however could not refrain, even in
his dictionary, from making a shghting allusion to the man
whose principles he was willing enough to borrow. " I
have treated," he says, " the part dealing wath harmony
according to the system of the Fundamental Bass, although
this system, imperfect and defective as it is in so many
respects, is not based in my opinion either on Nature or
truth. . . . Still, it is a system. It is the first, and the only
one up to that of M. Tartini, in which an attempt has been
made by means of definite principles to connect the innumer-
able isolated and arbitrary rules which made of the Art of
Harmony a task for the memory, rather than a matter for
the reason. The system of M. Tartini, although in my
opinion superior, is not yet generally known, and does not
enjoy, at least in France, the same authority as that of
M. Rameau. ... I have therefore thought fit to defer to
the nation for which I write, and to prefer its opinion to my
own [!] as to the real foundation of the theory of harmony."
{Preface.)
Thus Jean Jacques, the apostle of truth !

TARTINl'S TRATTATO DI MUSICA. 287

To trace the influence of Rameau on his successors is to
trace the history and development of the theory of harmony
from the middle of the eighteenth century up to our own day.
In order to accomphsh this, however, in any adequate manner,
a volume — perhaps more than one — would be necessary.
Nevertheless, some mention must be made of the most
important developments which the theory of harmony has
undergone since the time of Rameau.

One of the most remarkable works ever written on the
subject of harmony is that of Giuseppe Tartini, the celebrated
Italian violinist and composer, namely, Trattato di Musica
secondo la vera scienza dell'armonia, published in 1754.
Tartini, like Rameau, takes as the starting-point of his theory
of harmony the acoustical phenomena resulting from the
resonance of a sounding body of musical character. The
first portion of his work strongly recalls the Propositions
and Experiences of the first part of Rameau's Generation
Harmonique. Thus Chapter I. is entitled, De Fenomeni
Arnionici, lora natura, e significazionc .

After referring to the sounds produced by such sonorous
bodies as those of the stretched string of the monochord or
cembalo, the marine trumpet, the orchestral trumpet and
horn, organ pipes, etc., Tartini remarks : — " The stretched
string of the monochord, which in itself ought to produce
a single sound, has clearly three sounds," namely, the
fundamental sound, 12th and 17th. " The marine trumpet,
the orchestral trumpet and horn, exhibit the same uniform
phenomena : it is physically impossible for these instruments
to produce other sounds than those of the harmonic series,
corresponding to the fraction i, I, \, \, !, etc." ^

Tartini then examines the nature of the vibration of the

different segments into which a string of the monochord may

be divided. Suppose that the string A-B is divided into

1 i 1 jL

two equal parts at the point C. - — ^ i, 'i — ^ The

A E D C F B

vibrations of A-C will pass with equal velocity into C-B,
which is equal to A-C : will return from the point B through
B-C into C-A ; then back again from A-C to C-B ; and this
will continue as long as the string vibrates. If the string be
divided into three equal parts (as at D and F) then the

^ Trattato di Musica, Ch. i.

2 88 THE THEORY OF HARMONY

vibrations of A-D will pass into D-F, thence into F-B, and
back again, and so on ad infinitum as long as the string vibrates.
Similarly with the division into four equal parts, as at E.
It is physically impossible that any segment which is
incommensurable with the string in its totality can form any
part of its tone, because it will interfere with and finally
destroy the vibrations of the other segments which are
commensurable with the prime tone produced by the entire
string. 1 Then the sounds of instruments such as the marine
trumpet being physically impossible unless these sounds
be in the harmonic series, in this sense, and from the point
of view of harmonic unity, they are true physical monads.-
In this respect also the name of Aliquot part signifies nothing ;
the name of Unity everything.

One now begins to perceive the nature of the conception
which has inspired Tartini's theory. Throughout his whole
work there is the most direct internal evidence that he has
studied and assimilated not only the theories of his country-
man ZarUno, but especially those of Rameau : in particular,
the Traite de I'harmonie and the Generation Harmonique.
In the Traite, almost the first express declaration of Rameau
is that "Unity is the principle of harmony/' that is, the
consonances proceed from Unity as from their source. In
the Generation Harmonique Rameau's first task is to prove
that harmony has its origin in the resonance of the sonorous
body. But while Rameau makes it his principal endeavour
to demonstrate that sound is in its nature not simple
but complex, not uniform but multiform, Tartini's object
is to prove that harmony presents us, not with a diversity,
but a uniformity ; all must resolve itself into Unity :
all is Unity. Rameau has said that musical sound is not
one but three ; Tartini demonstrates that the sounds of
harmony (harmonic series, fundamental note, 12th and 17th) ,
in themselves real harmonic monads, are not three, but one.

But in doing this, Tartini does not set himself in opposition
to the principle of Rameau. He accepts it, and regards the
two principles, that of Unity breaking itself up into a series
of harmonic monads, and that of these monads resolving

1 Trattato di Musica, Ch. i, pp. 11, 12.

'■* " E in tal scnso, e rispetto le unita armoniche, sono vei-e monadi
fisiche." (Ibid., Ch. i, p. 12.)

TARTINl'S TRATTATO DJ MUSICA

!89

themselves into Unity, as complementary principles, of equal
importance and of equal significance for the theory of harmony.
For him indeed they are one and the same. " Therefore,"
he remarks, " the harmonic system reduces diversity to
uniformity, multiplicity to unity ; and, given a simple Unity,
this divides itself harmonically. Then the harmonic system
must, in every respect, be regarded as Unity; rather the
hannonic system resolves itself into Unity, as into its principle.
This is a legitimate consequence, and is physically demon-
strable ; it is, indeed, independent of the human will " {e perb
affatio independent e dall'arhitrio nmano. — Trattato di Musica,
Cap. I, p. 13.)

Tartini then proceeds to demonstrate the existence of a
remarkable acoustical phenomenon which, he considers,
confirms in a striking manner the truth of his theory, namely,
the " combination tones." " One has discovered," he says,
" a new harmonic phenomenon, which proves in a wonderful
way the same thing, and indeed much more." If, he points
out, two sounds of just intonation be sounded clearly and
loudly together, there will result a third sound, lower in
pitch than the other two, and which will be the fundamental ^
sound of the harmonic series of which the first two sounds
form an integral part : —

-«5>-

etc.

-rrr

Resultant sound.

It is from the principle of Unity and from this phenomenon
of " the third sound " {il terzo suono) that Tartini develops
his theory of harmony.

* Tartini, however, does not here say fundamental, but octave of the
fundamental, corresponding^ to the term ^-, and in the examples he
gives of the resultant "third sound," places it an octave too high
This mistake he afterwards corrected.

igo

THE THEORY OF HARMONY

This third sound is considered by Tartini to be nothing
more nor less than the Fundamental Bass [basso fondament ale)
of the harmony ; and this term he uses constantly through-
out liis work. He gives the following example, and points to
the fact that the resultant sounds form the true fundamental
bass of the harmonic succession : —

* ig: -^ -g-

In the acoustical phenomena of the " third sound " we
have therefore a physical demonstration and proof of the
correctness of the theory of harmonic inversion and of the
Fundamental Bass : —

{b) (c)

''/)

22:

-&'

-<s>-

Resultant sounds.

At (a) the harmony is the major harmony of C, fundamental
position ; at (b) we have the first, and at (c) the second
inversion of the same chord. All three chords, however,
have the same " third sound." The fundamental sound
or bass of all three is therefore C. Tartini is of opinion
that the resultant sound of the minor Sixth is the same as
that of the major Third, of which the minor Sixth is the
inversion. The minor Sixth, however, as at {d) has, for
resultant sound, g : a fact which recalls, in a striking way.

TARTINI'S TRATTATO DI MUSICA 291

Zarlino's explanation of this interval as " composite,"
consisting of a minor Third and a perfect Fourth. Tartini also
makes the mistake of imagining that the Octave produces no
resultant sound. At the same time, he is aware that the
"third sound " may result from inharmonic as well as harmonic
intervals. Thus, if d" be sustained on the vioHn, while g", the
fourth above, be gradually approximated to /" so that
several inharmonic intervals must result, the " third sound"
will be found to descend gradually a major sixth, from
g to b\} :—

-G>-

--^-

Resultant sounds.

The point of principal importance, however, is that any two
consecutive sounds of the harmonic series will when sounded
together produce the same resultant tone, this tone invariably
corresponding to the octave of the fundamental tone.
Observe carefully then, says Tartini, that we find this octave,
that is \, established as the physical root or origin of the
harmonic system.^

This settled, Tartini proceeds to develop his system in a
remarkable fashion. Seeing that the sonorous body in
vibrating divides itself into an infinite series of harmonic
sounds ; seeing that any two consecutive sounds of this
series invariably produce, in turn, the same " third sound,"
music must therefore be regarded as a physical science.
Again, as the series of sounds which naturally arise from
the resonance of the sonorous body corresponds to the harmonic
progression i,\,\, I, ',, \, \, etc., which progression must be
regarded, at least theoreticaUy, as continued to infinity, it
is evident that this series of natural harmonic sounds is

1 " Intanto per mezzo di tal fenomcno resta fisicamente stabilita la
unita costante in infinito in A , come radice fisica del sistema arraonico."
( Trattato di Miisica, Cap. i .)

292 THE THEORY OF HARMONY

mathematically determined. The same is true of the
" third sound." Music therefore must be regarded as a
physico-mathematical science. It is necessary to consider
it in both these aspects. Acoustical phenomena are in them-
selves mere isolated facts, without connection ; while
mathematical or geometrical demonstrations may have no
connection with the subject of music. Some connection
must be established between them. For the deduction of
definite principles of musical science geometrical demonstra-
tions are therefore necessary, but only such as can be derived
from the physical facts themselves.

The straight line, divided harmonically, easily lends itself
as a representation of the sonorous body and of the diverse
elements constituting harmony ; but the Unity into which
these diverse elements resolve themselves — how can this
be represented ? For this another kind of geometrical figure
is necessary, which Tartini concludes can only be the circle.
Further, as the straight line must be regarded as antecedent,
both mathematically and physically, to the curve, and as
the circle is itself impossible without the supposition of the
straight fine, the circle must be regarded as inscribed in a
square. The sonorous body will represent the diameter of
the circle. The radius of the circle, therefore, which is half
of the diameter, is half of the sonorous body, that is ^, which
Tartini has demonstrated is the physical root of the harmonic
system. It is unnecessary to follow Tartini into the abstruse
calculations into which he now plunges, especially those in
which he attempts to prove, unsuccessfully, that the system
of harmony arising from the senario is a harmonic system
complete in itself, and that the complex of consonances must
be regarded as being terminated by the number 6. It is
deplorable that Tartini, one of the most gifted of theorists
and musicians, and who intellectually at least was Rameau's
superior, should have taken as the foundation of his system
and of his geometrical demonstrations what was in reaUty
nothing more than the faulty observation of an acoustical
phenomenon. For Tartini places the resultant tone an octave
too high ; it corresponds not to the sound produced by the
half of the sonorous body, but to that produced by its whole
length. Not the half of the string but the whole, that is, the
fundamental itself, as Tartini might have suspected, is
the "physical root," in Tartini's sense of the term, of the

TARTINIS TRATTATO DI MUSIC A 293

harmonic system. The entire string, therefore, should
represent the radius of the circle of which the diameter is
twice the radius, that is, twice the length of the sonorous
body ! — a result which would have considerably embarrassed
Tartini.

Although Tartini is an original and independent thinker,
the main conclusions at which he arrives bear a striking-
resemblance to the theoretical principles formulated by
Rameau. It is perhaps owing to Fetis, who has given a
critique— very inadequate — of Tartini 's theory of harmonv
{Esquisse de I'harmonie), as well as to Rousseau {Art. Sysieme,
Diet, de Miisique), that Tartini's' theory has frequentl}- been
represented as the antithesis of that of Rameau. It ma\-
be asserted however, on the contrary, that in the Trattaio
di Miisica we find a notable attempt to demonstrate, on
scientific and mathematical principles, the correctness of
the theoretical conclusions which Rameau had alread\-
endeavoured to estabhsh.

For Tartini, as for Rameau, the harmonic division of the
sonorous body is the principle of harmonic generation.
From this we obtain the major harmon}'. The minor
harmony, which corresponds to the arithmetical division, is
an inverted major harmony. These constitute the sole
positive harmonic unities of the musical system. If J,
that is the " third sound," and the octave of the fundamental,
is the " physical root," the Fifth is the determining constituent
of the hamionic s^-stem. If the Octave be represented b\-
the ratio 12 : 6 (= 2 : i) its harmonic and arithmetical division
\\-ill be represented respectively by the numbers 8:9. The
product of these two numbers, Tartini points out, is as the
product of the two terms 12 : 6, which here represent the
proportion of the octave. This is the mathematical result.
But the physical result is the same. For the terms 8 : 9
are successive terms of the harmonic series, and if sounded
together there will result the " third sound " which is
represented by J, and equally by the duple proportion {ragion
dnpla) 12 : 6. These are facts which need not be too closely
examined ; it is sufficient to show that without doubt
one of the main objects of Tartini in his geometrical demon-
strations is to prove that the arithmetical as icell as the harmonic
division of the Octave, and also of the Fifth, is a necessity of
the harmonic system. The harmonic division of the Octave

294 THE THEORY OF HARMONY

and Fifth causes no difficulty, for Nature herself divides
these intervals harmonically, thus : —

iS*-4 - —

::qi

-(S>-1

But with the arithmetical division of the Octave and Fifth,
this is not the case ; such a division would appear to be a
contradiction of Nature.

Rameau's difficulties are then also those of Tartini,
namely, the explanation of the Subdominant, and the origin
of the minor harmony. Like Rameau, Tartini considers
that only the major harmony {sistema armonico) is given
directly by Nature. This is proved by the " horrible effect "
of the resultant tones produced by the minor harmony : —

-<^ Gt &-

terzi suoni.

Although it is true that all harmonic music is based on the
two diverse genera of major and minor, and although the
minor harmony, corresponding as it does to the arithmetical
proportion, would appear to arise from a principle which is
the opposite of that of the harmonic, it nevertheless is im-
possible to consider that one and the same musical system
arises from two diverse principles ; this would be absurd,
and contrary to the very idea of a system. ^ The minor

1 " E benche si confessi, che I'arnionia di terza minore, come dedotta
dalla divisione aritmetica, sia quasi prcsa in prestito dalla scienza
aritmetica : e si confessi, chc il sistema armonico sia per natura
I'unico, e per cccellenza il primo, nulladimeno vi e il debito in chi
si propone di formare un sistema imivcrsale di abbracciare i due generi
diversi del sistema, e ridurli ad un genere solo, che sia I'universale.
Altrimenti nello stesso sistema vi saranno due principi diversi, il che
e assurdo, e si oppone alia vera idea di sistema." — (Trattato di Musica,
p. 66.)

TARTINIS TRATTATO Dl MUSIC A 295

harmony cannot be regarded as foreign or accidental to
the musical system ; on the contrary, the minor harmony
arises from the same principle as the major, and is inseparable
from it.^

Tartini demonstrates as follows : —

-^gr— -^gr

-W<5-

:&■:

terzi snotii.

Here it is certain that all the intervals in the upper stave
are harmonic intervals and deri\-ed from the harmonic series.
It is equally certain that the ' ' third sounds ' ' which respectively
arise are the physico-harmonic roots of these intervals. All
then is harmonic, and derived from the harmonic svstem.
Nevertheless it will be observed that the resultant sounds
are in arithmetical progression, and produce the minor hanuony
f-a\f-c. The arithmetical s\'stem therefore is the inseparable
consequence of the harmonic s\-stem. Such facts would
appear to indicate, at least, that the minor is an inverted
major harmony. B\' no means however can they be regarded
as furnishing an adequate explanation of the origin of the
minor harmony. If it is true that the intervals in the upper
stave are harmonic, and that they succeed one another in
the order determined by the harmonic series, it is equally
certain that they do not belong to one and the same harmonic
system, but are derived from different svstems. They
are related to different fundamentals, and are strictly speaking
in different keys. Thus the Fifth is related to c as its fun-
damental, or haiTnonic centre, the Fourth to F, the major
Third to C, the minor Third to Aj?. But Tartini's object
is to demonstrate that the minor s^-stem arises from one

■• " Che rarmonia cU terza minore si c presa in prestito dalla scienza
aritmetica e sia quasi straniera, e accidentale alia musica, cio nego
assolutamente : c per lo contrario dico, chc il sistema deH'armonia di
terza minore non solo e inseparabile dal sistema deH'armonia di terza
maggiore, ma auzi e lo stcsso idevtico sistema." — {Traitato di Musna,
p. 68.)

296 THE THEORY OF HARMONY

and the same harmonic system, and not from a series of
intervals derived from different harmonic systems.

Further, from such a series of intervals Tartini might
deduce all that he requires, and much more, without any
necessity to have recourse to the resultant tones. For here
we find, not only the harmonic division of the octave c'-g'-c",
but also the arithmetical division c'-f'-c" ; and not only the
harmonic, but also the arithmetical division of the Fifth,
thus •.—c'-e'-g'—c'-e\f'-g'.

Tartini puts the matter in another way. Let C, the
fundamental note of the harmonic series, = 60. The next
five sounds of this series will be represented respectively
by the numbers 30 : 20 : 15 : 12 : 10, of which the complements
are 30 : 40 : 45 : 48 : 50, represented respectively by the
notes c : G : F" : E : E[7 :—

i3-o- — —
20-0

130 rsfi — — •

^"-&- -s>- -&- -©- "tg-

Tartini here considers the lowest sound to represent the
Fundamental Bass of all the harmony heard above it.^
The sounds of the harmonic series which arise successively
above it determine not only the major harmony and
the major system, but also the minor harmony and the
minor system ; for here we find not only the Octave
arithmetically divided (C-F-c'), but also the Fifth (C-Ej?-^').
Once more, therefore, it is evident that the minor harmony-
results as the necessary consequence of the major. But
Tartini cannot possibly consider the sound C to be the Funda-
mental Bass of the Fourth C-F, or of the minor Third C-EJ7,
for he has already demonstrated that the resultant and funda-
mental sound of the Fourth C-F, is not C, but F ; and that
the fundamental of the minor Third C-Ejj, is not C, but A]?.

Further, if the sixth harmonic sound ^^, corresponds to \ of
the whole string represented by ^ , then its complement ^P

1 " Ma accio meglio s'intenda tuttocio praticamente, si supponga
C solfaut 60 Basso fondamentale di tutta I'armonia, come lo e in
fatto." — [Trattato di Musica, p. 70.)

TARTlNrs TRATTATO 1)1 MUSICA 297

corresponds to ;'; of the string. But this sound cannot
be produced by the string, for it is not an ahquot part :
Tartini has himself demonstrated in the most convincing
way that such a sound is " physically impossible."

Nevertheless, Tartini 's researches as to the origin of the
minor system are important and valuable. He does not,
like Rameau, relate the minor harmony to the co-vibration
of the multiples, but regards it as arising from the same
principle as the major, which was the view taken by Rameau
himself in the latter part of his Demonstration and
Reflexions sur Ic principe sonore. The minor system is
then related to the major system, and is inseparably connected
with it. Also, it is important to observe, Tartini considers
the lowest note of the minor harmony to be the fundamental
note. Thus, in the minor harmon}/ c-t'j;-^, c is the " principal
bass " ; it is the generator of all the notes of the harmony
heard above it. But tliis harmony has a secondary bass,
namely e\}, for this e\^ is the bass of the harmony of the major
third g, which determines the major system.^ This is also
the view taken by Rameau in his Demonstration. But there
is this curious difference between the results arrived at by
these two distinguished theorists. While Rameau considers
that the minor harmonj- generated by the fundamental
note C — for, as we have seen both theorists regard the minor

1 " Posto E lafa (secondo la propria natura di mezzo aritmetico
della sesquialtera, o sia quinta gia altrove dimostrato) a confronto di
C solfaut gravissimo, come Basso fondamentale costante, e a confronto
dello stesso G solreut dell' esempio : —

:l2o:

si trova E lal'a seconda basse di armonia di terza minore, di cui e-
prima base C solfaut Basso costante. Dunque resta dimostrato, che
E lata include in se stesso le due armonie di terza maggiore. c di terza
minore." — (Trattato di A/usica, pp. no, in.)

298 THE THEORY OF HARMONY

and the major system as proceeding from one and the same
fundamental note — to be a-C-c, Tartini considers it to be
C-e\}-g. Nevertheless, Tartini does not consider C minor
to be the relative minor key of C major. This conclusion
however would appear to be forced upon him, even if he
insists, as he does, that C minor is the relative minor of £[7
major. Further, in making o' to be a doubly determined
note, that is, Fifth of C and major third of e^, the minor
harmony appears to arise from two fundamental sounds,
and two generators. This does not help us to understand
how the minor, like the major harmony, impresses us as a
harmonic unity.

These are for Tartini the central problems of the science
of harmony. All his demonstrations are in the main directed
towards one object, namely, to prove that the Octave is the
" physical root " of the harmonic s^^stem, and that the
arithmetical as well as the harmonic division of the Octave
and of the Fifth, is a necessity which arises from the nature
of the harmonic series itself. From this two-fold division
of the Octave we obtain all that is necessary for the formation
of the diatonic system, the scale and harmonic succession.
From the two-fold di\dsion of the Fifth there arise the two
diverse harmonic genera — the major and minor harmonies.
Like Rameau, therefore, Tartini's fundamental bass is a
Fundamental Bass in Fifths, and consists of three terms.
Tonic, Dominant, and Subdominant. Each of these sounds
may bear the complete major harmony {sestupla armonica) ;
these three harmonies are the principal harmonies of the
Mode, and from them we obtain all the sounds necessary
for the formation of the major scale. The Fifth, then, is
that portion of the harmony which determines the harmonic
system and the nature of the Fundamental Bass.^ " The
scale therefore proceeds from the harmony, and not the
harmony from the scale." ^

Tartini distinguishes three principal Cadences : (i) the
Harmonic (Dominant-Tonic) ; (2) the Arithmetical (Subdomi-
nant-Tonic), and (3) a " mixed cadence " {Cadenza-Mista =
Subdominant-Dominant). It follows from his method of
dividing the Octave arithmetically as well as harmonically
that he finds no difficulty in allowing the immediate succession

1 Delia Scala, Cap. 4., p. 98. « /^/^.

TARTINIS TRATTATO Dl MUSIC A 299

of the two Dominants. Tlius the fundamental bass to the
ascending scale is as follows : —

Ife — 5—

- rj

— 0

r^ ^

-O,

ft -

—^

<5> -^

/Ti

.•>-^

Cadenza armonica. C. arithmetica. C. mista.

But this only in the ascending scale (which naturally
ascends, rather than descends, being derived from the
ascending harmonic series). The immediate succession of
the two Dominants in ascending is good, because in
this case we ha\-e the progression from the imperfect
(arithmetical) to the perfect (harmonic). But the reverse
succession is faulty, for here we find the progression from the
perfect to the imperfect. Nevertheless, the three terms of
the fundamental bass may still serve as the basis of
the descending scale, by the mediation of the " natural
Seventh " ! {Fa enarmonico) : —

-& — ff-v

-*^-

\V /

''g^' — rs r-j r> r> p-

In Chapter 5 {De Modi, 0 siano Tiioni mnsicali antichi e
moderni) Tartini enters into an examination. of this natural
Seventh, the conditions under which it might be employed,
\vith many references to the enharmonic system of the Greeks.
This Seventh is consonant, being derived from the harmonic
series ; it is for this reason, he thinks, that the Dominant
Seventh, which so closely resembles the " natural " Seventh,
does not require to be prepared. It is, then, not an " un-
prepared discord," but a consonant chord.

W'liile the harmonic and arithmetical progressions are
consonant, the geometrical progression is dissonant. In the
geometrical progression is found the origin of dissonances.

300 THE THEORY OF HARMONY

and of dissonant chords. " The nature of geometrical
quantity is substantially opposed to that of the harmonic
and arithmetical proportions . . . because the latter are
based on an infinite series of diverse ratios, while the former
is based on an infinite series of similar ratios." It would
appear, then, that a chord composed entirely of Octaves
must be dissonant, because it arises from the geometrical
progression 1:2:4, etc. Tartini however will not grant
this, although he remarks that Octaves are consonant
" more through custom, than reason " {piii per uso, che
per ragione).

We therefore find the following rule {il qiiinto Canone
miisicale) : — " Ever}- chord is dissonant which contains two
similar intervals of different species, except the octave." ^
Therefore two Fifths, two Fourths, two major or minor
Thirds will produce dissonant combinations, thus (a) : —

— — G-\ ^ — Q-) —

-sy-

On the other hand, the chord at (6) is consonant, because
both Fifths are of similar species, that is, they belong to
the same harmonic series. Of two intervals of different
species which form a dissonant chord, that interval will
be consonant which is in its right place in the harmonic
series, while the other will be dissonant. Thus in the chord
of the Dominant Seventh g-h-d-f we find two minor Thirds
h-d, and d-f ; the first is consonant, because it is in its natural
place in the harmonic series of which g is the fundamental,
but the second is dissonant, because it does not belong to
this series. Therefore / is and remains the dissonant note,
no matter what forms the chord ma}- assume.

Again, in the chord of the Added Sixth — not so called
however by Tartini — it is the Sixth which is the dissonance.
The chord should be understood thus : —

ti^

* Trattaio di A/nsica. p. 74.

TARTINI'S TRATTATO DI MUSICA 301

Here there are two Fourths of different species : the first
Fourth g-c', is consonant, because it belongs to the harmonic
series oi which c is the fundamental ; it is the second
Fourth e'-a', wliich is dissonant. In the chord c-e-g-a,
it is a, therefore, which is the dissonant note.

It is evident that Tartini's theory of dissonance is not
one which can lead to any satisfactory result. Nor is he
able to draw any effective distinction between consonance
and dissonance. He appears here to be at the mercy of
his system. Besides, in the chord just mentioned, which
is a Subdominant discord, Tartini imagines that there
are two perfect Fourths. Here however he errs ; for if
the first Fourth g-c' (=3:4) is perfect, the second e'-a'
(= 20 : 27) is not. So also with the chord of the Dominant
Seventh, g-b-d-f, in w^hich he considers there are two minor
Thirds b-d, and d-f. But d-f (= 27 : 32) is not a minor Third.

Tartini's seventh " rule " is that " there can be no dissonant
chord which is not based on a consonant one." This follows
from the fifth rule, and also from his explanation of the
major and minor harmonies as containing in themselves
the sole positive and constitutive elements of harmonic
composition. Thus in the chord c-g-d' both Fifths c-g
and g-d' are harmonic, although together they form a dis-
sonance. This " rule " is of extreme theoretical importance.
But Tartini does not observe that it cannot apply to the
two most characteristic discords of the harmonic system.
For, in the chord of the Dominant Seventh g-b-d-f the Third
d-f (27-32) is not a harmonic interval. It is in itself dis-
sonant, and cannot therefore be derived from a consonant
chord. ' Likewise with the chord of the Added Sixth. In
Tartini's system we meet again with difficulties similar to
those with which the works of Rameau have already made
us famihar.

Tartini's work, however, is that of a superior intellect.
It is a reasoned, logical, and closely- welded system, based
on philosophic and scientific principles the like of which
we do not again meet with until we come to Moritz
Hauptmann's Harmonik und Mdrik.

Although Tartini is generally regarded as the first to
discover the combination tones — he had asserted that as early
as 1717 he had made use of them for the purpose of teaching
pure intonation on the viohn to his pupils — it is certain

302 THE THEORY OF HARMONY

that other musicians had discovered them independently.
/. A. Serve of Geneva, and Romieii of MontpelUer, had given
accounts of these tones before Tartini's pubHcation of the
Trattato di Mnsica. Serre is the author of a not unimportant
work on harmony, Essais siir les Principes de I'harmonie
(1752), in which he has to a certain extent anticipated Tartini's
treatment of the combination tones as a basic principle of
the science of harmony. In other respects he adopts, in
large part, the principles of Rameau. Serre also wrote
Reflexions stir la supposition d'un troisieme mode en
musiqiie {Mercure de France, 1742), in which he criticizes
the theory of a pure minor mode advanced by C. H. Blainville
{Essai sur un troisieme mode^ 1751)- Blainville demon-
strated that the pure minor mode was exactly the reverse
of the major mode ; it is to be regarded not in an ascending
but a descending aspect, in which case the order of tones
and semitones is exactty that of the major mode. This

Major=c-d-e-f-g-a-b-c'

Pure Minor=e'-d'-c'-b-a-g-f-e

theory of a " pure minor mode " has in our own day gained
considerable prominence, principallv through the writings
of Dr. Riemann.

G. A. SORGE.

A work of considerable theoretical importance is that b}-
Georg Andreas Sorge, entitled Vorgemach der Musikalischen
Komposition," etc. (1745-1747). In this work, pubhshed
nine yezxs before Tartini's Trattato di Music a, Sorge
demonstrates his acquaintance with the phenomenon of the
combination tones. ^ In the Preface to the first part of his
work, Sorge puts the question. Why do we prefer this succes-
sion of sounds c-d-e-f-g-a-b-c' , rather than c-d-e-f-g-a-l\}-c' , or
c-d-e-fi^-g-a-h-c' ? Because, he answers, the sounds in the
first order are the most closely related to the perfect (major)
harmonic triad. The first, third, and fifth sounds are

^ " Ja so gar zwey Flutes donees geben, wenn man c nnd a rein zusammen
hldset, noch den dritten Klang, nemlich eirt f." — (Vo)i dem naturlichen
Znsammenhang der Consonantien. Ch. 5.)

G. A. SORGE (1703-1778)

j":>

derived from the major triad c-e-g, and the other four sounds
are related in the closest way to the three sounds of this
triad ; for d is Fifth oi g ; f is the Fifth below c ; while a
and h are respectively the lower and upper Fifths of the
Mediant 0. It is thus we obtain the major scale ; each
sound of the major triad C-e-g, requires a Fifth both above
and below it ; thus the Fifths above and below c are g and / ;
those above and below e are b and a ; while those above and
below g are d and c. On the other hand the b\^, in the other
order of sounds, can boast of no such close relationship
with the three essential sounds, but is the lower Fifth of the
lower Fifth of c. So also /#, in the third scale, is the upper
Fifth of the upper Fifth of e. Likewise in the Minor Mode :
the order d-c-V^-a-g-f-c-d, arises out of the Trias minus pcrfecta
d-f-a. c is upper Fifth of/, and h\f is its lower Fifth ; while
g is lower Fifth of d, and c upper Fifth of a.

Generation of Chords.

Sorge repeats this statement later (Ch. 11, p. 28)^ and
proceeds : — " Of these seven degrees, the three which make
up the determining {herrschenden) triad c-e-g, are the essential
sounds." He then distinguishes the following "triads,"

"• Dr. Riemann (Geschichte der Mnsiktheone, p. 442) quotes the state-
ment here referred to, to which he gives the following illustration : —

a - h - d
C - e - g
f -a - V

which however is not given by Sorge : and makes it appear, indeed
explicitly asserts, that Sorge in this passage recognises that the
major scale is composed of the elements of the three major chords,
that is. Tonic, Dominant, and Subdominant. In such a case,
then, h must be regarded as the Third of g, and a as the Third
of /. But this is to contradict Sorge himself, who says nothing
of a Third-relationship. The complete passage in Sorge 's work
(p. 28) referred to by Dr. Riemann, is as follows : — " Wir konnen
atick sagen. . . . denn ein jeder Theil dieser Triadis verlanget eine
reine Quint unter und tiber sich. Da hat nun Sonus infimus c, f unter
sich, and g als partem triadis, iiber sich : Sonus niedius e hat a unfer-
nnd b iiber sich : Sonus supremus hat c als partem triadis unter- und d
iiber sich, woraus denn unsere Klang Folge des Modi masculini oder
perfecti entstehet, nemlich c, d, e, f, g, a, b, c."

304

THE THEORY OF HARMONY

to the examination of which the whole first section of his work
is devoted : —

(i) The triade harmonica perfecta (major harmonic
triad).

(2) The triade harmonica minus perfecta (minor

harmonic triad).

(3) The triade deficiente (diminished triad) .

(4) The triade superflna (augmented triad).

(5) The triade manca (defective triad as d^-f-a, or

b-d^-f.

The major harmonic triad Sorge considers, hke Rameau,
to be derived from the senary division of the monochord, as
well as from the resonance of sounding bodies, as the viola,
cello, trumpet, horn, organ pipes, etc. The numbers i, 2, 3, 4
5, 6, 8, " form a band which links the consonances together."

The minor harmonic triad cannot be represented by smaller
numbers than 10 : 12 : 15. The minor triad then is not
so perfect as the major : for the proportions 4:5:6, which
represent the major triad, are much nearer to Unity than
10 : 12 : 15. " The trumpet gives this triad perfectl3^ pure,
at the sounds e-g-b." {Vorgemach der Altts. Komp., Ch. 7.)

The diminished triad is, strangely enough, treated by Sorge
as quasi-consonant {!) and he justifies his introduction of it
as an independent harmony {Hauptaccord) by a reference
to the Kleine General-bass Schule of Mattheson, who says of
this chord that " it has all the characteristics of a
consonance " ! The trumpet gives the diminished triad
e-g-l\f = 5:6:7. But this 7 is " too flat." The real propor-
tions are 45 : 54 : 64. {Ibid., Ch. 8.)

The augmented triad is found only in the Minor Mode ;

w'hile the " triade manca"' represents the " fundamental "

position of the chord of the Augmented Sixth. Sorge,

then, discovers a triad, which is either major, minor,

diminished, or_ augmented, on every degree of the major

and minor scales : —

Major. Minor. Diminished.

Major -Jf-—
scale. C^ —

JQI

-<S>-

Minor ^
scale. ^

W^^-

—^rj-

Minor.

Major.

Diminished. Augmented.

G. A. SORGE (1703-1778)

j'-'D

In the following sections of his work he treats of the
inversions of these triads ; and disagrees with Heinichen,
who says that the 4 chord is dissonant. " No consonant
chord," he remarks, " can become, by inversion, a dissonant
chord." {Vorgemach der Miis. Komp., Ch, 4, Sect. II.)

In treating of dissonant chords, Sorge devotes a chapter
to the question, " Which is the first dissonance ? " (" Unter.
suchung welches die erste Dissonantz sei"). He answers
that Nature points the way here : for the " natural Seventh "
can be clearly distinguished in the resonance of the trumpet,
horn, 16 and 32 feet organ pipes, the marine trumpet, etc.
Although this minor Seventh, which has the proportion
4 : 7, is a little too flat, this is merely a proof of the necessity
for temperament. In any case, Nature clearly shows that
the minor Seventh is the first dissonance. " Nature,"
comments Sorge, " is the best guide in all Arts and Sciences " ;
it must therefore have appeared all the more strange to him
that Nature should have made the minor Seventh " a little
too flat."

We have therefore five different kinds of Seventh chords,
obtained by adding a minor Seventh above each of the five
triads already treated of. But other chords of the Seventh
may be obtained by adding a major Seventh above the
major triad (as c-e-g-b), above the minor triad {a-c-e-gj(f.)
and above the augmented triad (c-e-g^-b). Sorge does
not say whether or not he has heard this major Seventh in
the resonance of strings or organ pipes ; or whether he
derives it from the natural sounds of the trumpet.

He distinguishes two chords of the Ninth. One is the
chord of the minor Ninth on the Dominant of the Minor Mode ;
the other is the chord of the major Ninth on the Dominant
of the Major Mode. The first chord is really the complete
form of the chord of the Diminished Seventh. " It cannot
be asserted," he remarks, " of this diminished Seventh chord,
that it is based on the diminished triad. It has as its real
foundation the major triad, on which there is built up the
chord e-gj^-b-d-f, by the addition of a minor Seventh and a
minor Ninth. If now e as the fundamental note {Grund-
klang) be taken away, there remains the chord of the
Diminished Seventh " (p. 346). Such, it will be remembered,
was the explanation of this chord given by Rameau, in the
Generation Harmoniqiie.

3o6 THE THEORY OF HARMONY

Several of these dissonant chords may be taken without
preparation, namely, the chord of the Dominant Seventh,
and the Dominant major and minor chords of the Ninth ;
also, the chord of the Seventh on the leading note of the Major
Mode, b-d-f-a, and the chord of the Seventh based on the
" defective " triad d^-f-a-c, or b-d^ — f-a. All other dissonant
chords owe their origin to the mechanism of Suspension
{gebnndene Septimen-accorden) ,^ or arise from passing-notes
[in Transitu).

Sorge makes the noteworthy statement that all chords of
the Seventh, including those chords with the " natural "
Seventh, really owe their origin to a simple passing-note,
of the form 8-7. " The real foundation of all these chords
is the passing Seventh (durchgehende Septime), for instead

7 8 7

of C-G-C-G, we may substitute C- G-C-G" (p. 362). Here
Sorge presents us with a new theory of the origin of dissonant
chords. If he means, as apparently he does, that the chord
of the Seventh has an accidental, that is, a non-harmonic
origin, he does not observe that he contradicts what he has
already said with regard to the natural origin of the minor
Seventh.

Although Sorge does not appear to have been wholly
unacquainted with Rameau's theories, he does not treat
of the Fundamental Bass, nor of " Chords by Supposition."
He quotes a certain chord of the Eleventh from a work by
Telemann, namely — g-b-d'-f'-a'-c", of which he character-
istically remarks :— " Telemann here presents to us a sort of
harmonic tower {Thurm), above which, like a star, we find
the Eleventh, c"." Nor does he trouble himself greatty as to
the origin of the Minor Mode. As in the Major Mode, the
essential notes are those of the Tonic chord. The two modes
are related because of the large number of sounds they
possess in common. The Major Mode, he remarks, might
say to the Minor : " Thou art bone of my bone, and flesh
of my flesh " : — which recalls Tartini's explanation of the
minor harmony, and the Minor Mode, as the " necessary
consequence " of the major.

G. A. SORGE (i 703-1 778) 307

Chord of the Dominant Seventh.

That Sorge regards the chord of the Dominant Seventh
as an " essential discord," that he derives it from the natural
sounds of the trumpet, and that he allows it to be taken with-
out preparation, is considered by Fetis to be an event of
epoch-making importance for the theory of harmony. He
remarks : — " Let this point be carefully noted, for here we
have arrived at one of the most important facts in the history
of harmony : it is the second epoch of the genuine discoveries
which have been made in this science, and the glory of this
discovery belongs to the humble organist of Lobenstein,
ignored by all musical historians up to this day. For the
first time, he has established the fact that there is a dissonant
chord which exists by itself, apart from any modification
of another harmony, and he states that this chord is absolutely
different from other dissonant harmonies. . . . Even if
Sorge has been led astray by the semblance of regularity
presented by the different chords of the Seventh, he has
nevertheless grasped the fundamental character of the chord
of the Dominant Seventh, and of modem tonahty. In
this, he deserves to take rank in the history of harmonic
science immediately after Rameau, who has first perceived
the foundations of this science, and estabhshed them in his
theory of the inversion of chords." ^

Fetis, at least, deserves credit for drawing attention to
the merits of the " Vorgemach," which is in reaHty an
important theoretical work. Fetis however is wrong in his
facts. Sorge is not the first who has said that the chord of
the Dominant Seventh may be taken without preparation.
Rameau, in more than one of his works, permits this not
only in respect of the chord of the Dominant Seventh, but
of any Dominant discord. Again Sorge makes use of the
" natural Seventh " not only for the major, but for the minor
and even the diminished triad, as h-d-f-a, and d-f-a-c. Fetis
considers the theoretical importance which he — not wholly
without reason — attaches to the chord of the Dominant
Seventh to consist in the fact that it is the sole " natural "
dissonant chord, and that, being dissonant, and its resolution
on the Tonic harmony being its most natural resolution, it

^ Esquisse de I'histoirg de I'harmonie.

3o8 THE THEORY OF HARMONY

thus determines our modern tonality. There is no doubt
at least that the distinguishing characteristic of the Dominant
Seventh chord, especially as compared with the Tonic chord,
on which it " resolves," is exactly its quality of dissonance.
Sorge, however, thinks that the " natural chord of the
Seventh " should be regarded as the lirst or principal of
all the dissonant chords, because it sounds almost as well as
a consonance. He calls it an " almost consonant dissonance,"
and imagines that the good effect which this chord produces on
his ear is a sufficient explanation of its theoretical importance.^
In this respect Sorge shows much less sagacity than Rameau.
Rameau refused to consider the chord of the Dominant
Seventh as being derived from the natural seventh harmonic
sound ; and further says that if the Third he adds above the
Dominant harmony in order to form this chord is not of the
correct proportion, this defect of proportion, at any rate,
accentuates the dissonant character of the chord. Compared
with the theory of Rameau, Sorge's generation of the chord
of the Dominant Seventh represents not an advance, but a
retrograde step. For here begins the theory of " essential "
and " natural discords." If, as Fetis thinks, Sorge's theory
of the " natural chord of the Seventh " is an epoch-making
event, it is principally so only in this sense, that it has led
to some extraordinary results in the theory of harmony.

F. W. Marpurg.

Sorge found in Friedrich Wilhelm Marpurg (1718-1795)
a determined, and, owing to the enormous influence he
wielded in Germany and outside of it as a writer and critic,
a formidable opponent of his theory. The influence of
Rameau had extended to Germany, and the theories of
the now famous French musician did not fail to excite the
attention of Marpurg. In 1757 Marpurg pubUshed System-
atische Einleitung in die musikalische Setzknnst nach den
Lehrsdtzen des Herrn Rameau, which was mainly a translation

^ " Dieser Septenarius obey vereiniget sich mil denen vorhergehenden
Zahlen i, 2, 3, 4, 5, 6, und verursachet keine widrige Tremores [heats ?]
wie wohl andere Dissonantzen thun : weswegen diese fast consonirende
Dissonantz vor die alley leidlichste passiret," ( V orgemach derMus. Komp.,

P- 34I-)

F. \V. MARPURG (17 18-1795) 309

of d'Alembert's Elements de Musique ; and in 1755-58 his
Handhtich bei dem Generalbasse und der Composition, in
which he proclaimed himself to be a follower of Rameau.
It was against the faults contained in this latter work that
Sorge directed his criticisms in his Compendium Harmonicum,
Oder kurzer Begriff der Lehre von der Harmonie (1760).
Marpurg repUed in the same year with Herrn Georg Andreas
S or gen's Anleitimg ziim Generalbass, etc., and continued
his attacks in his Kritische Beitrdge ziir Musik}- In these
long, acrimonious, frequently amusing, but always informative
discussions, Marpurg makes his theoretical position even
more clear than in his Handbiich. " I have taken the hberty,"
he remarks, " of making known, not only in Germany, but
still further afield, the system of Rameau. ... As every one
was now able to compare Sorge's system with that of Rameau,
Herr Sorge was clever enough to see that the comparison
was not to his advantage." Hence his attacks on " Herr
Rameau and myself, his unworthy disciple."

Marpurg, however, considered that Rameau's system was
defective in man}' respects. A complete system, he remarks, ~\
must comprise all possible tones, intervals, and chords, in
so far as these are not contradicted in practice. " They must
be of such a character as to conform to the demands of
practice, as well as of pure speculation. Such a [complete]
system is based on the scale of one and twenty sounds, these
l5dng.between its two " termini " (the Octave) : (such a scale
Marpurg considers to be derived from the constituent sounds
of a central key, and its five most closely related keys) and
the different chords of two, three, or more notes compounded i
of these tones furnish all possible intervals and chords." ^ J
This extraordinary pronouncement shows how Httle Marpurg
appears to have really grasped and understood the principles
of Rameau. Rameau insists everywhere in his works that
it is harmony which produces the scale, and not the scale,
harmony. Marpurg imagines that he adheres to this principle,
even if he develops it a little, when he says : — " The intervals
arise, like the tones, ascending and descending, by collecting
together the sounding and co-vibrating Fifths [and Thirds]
of the fundamental notes {c-e-g). One compares with

1 Vol. V. (" Untersuchung der Sorgischen Lehre von der Entstehung
der dissonirenden Siitze ").
- Krit. Beitrdge, Sect. I.

310 THE THEORY OF HARMONY

the harmony c-e-g the sounding Fifth g-b-d, and afterwards
the co-vibrating Fifth f-a-c. One takes again the sounds
g-h-d, and f-a-c [!] and finds in the same way d-f^-a, arising
from the first, and h^-d-f, arising from the second. These
are first compared with c-e-g, g-b-d, and f-a-c, and then
with one another. One proceeds in this way through the
whole table of relationships [die ganze verwatii ^hafts-
tabelle der Dreiklange) of the triads, and finds all the pv.- sible
intervals."^ Such then is the programme of " the combined
Rameau-Marpurg system." Happily we are left only to
imagine what Rameau would have thought and said of it.
"" After this revised and improved version of the manner in
which the scale — the " chromatic-enharmonic scale," consist-
ing of one and twenty notes, — is developed from harmony,
Marpurg now proceeds to show us how harmony (and all
kinds of possible chords, consonant and dissonant) is developed
from the scale. He actually begins by asking the question —
" How do we get chords in music ? " " We have," he says,
" now got tones and intervals. How do we get chords in
music ? In the same way as we get tones and intervals. By
means of the connection of tones with one another we have
obtained intervals : We must now connect the intervals
with one another in order to obtain chords." - The importance
to be attached to each interval as respects its harmonic
significance, is decided by ]\Iarpurg in the followdng extra-
ordinary fashion: — "The quahty of an interval is determined
according as its ratio approximates to, or is remote from. Unity.
Such a distinction, however, is of value only in theory, in
the science of temperament ; but not in practice, in which
the rank of an interval is decided through the frequency of
its species. We must therefore investigate how often each
interval occurs [that is, in the scale of 21 notes], and if we
find that the Augmented Second occurs more frequently than
the Diminished Third, we must conclude that the former
is more necessary than the latter [!], and if we find that two
intervals of different species occur the same number of times,
this is a sign that both are of equal rank in practice. . . ."
" I shall here briefly indicate how often each kind of interval
appears in the complete scale of 21 degrees, which we make
use of for the 12 major and 12 minor keys."

^ Krit. Beitrage, Sect. II. " The Combined Rameau-Marpurg System."
2 Ibid.

F. W. MARPURG (17 18-1 795) 311

Develofment of the Added-third Theory of
Chord-generation.

Marpurg then finds that among the intervals of all sorts,
— perfect, major, minor, diminished, and augmented, — which
he enumerates, the major Third occurs only 17 times, while
the minor Third occurs 18 times. He will not, however,
abide by his own conclusions. " That the minor Third
occurs oftener than the major Third, is not in the least
derogatory to the superiority of the major triad as it is
established by Nature " [!].

The species of interval of which several ought to be
compounded together in order to form chords is, according to
Marpurg, the Third. " Let us now go back' to the two triads
given to us by Nature [major and minor harmonies] and
consider their outward form. We find that, apart from the
difference of the Thirds, each consists of a Third and a Fifth.
A Third and a Fifth above a fundamental note means that
we have a chord arranged in Thirds. How, then, ought the
intervals to be connected wdth one another ? By means of
Thirds." In this way, " by means of the imitation of Nature,
we discover many varieties of chords built up by means of
Thirds."!

Marpurg's ideas concerning the operations of Nature in the
domain of harmony are further manifested in his explanation
of what he calls " fantastic " or mixed triads, as h-d^-f,
d^-f-a, etc. " It is a question," he remarks, " which of
these mixed triads, namely b-d^-f, and &-g^-b\f, likewise
d^-f-a and gj^-b\}-d, ought to have the preference, seeing that
they occur in the diatonic-chromatic scale an equal number
of times. This question cannot be determined until we have
decided what is the origin of the fundamental sounds
obtained from the progression founded on fifths. Now, as
the fundamental sounds b and e exist in Nature sooner than
the fundamental sounds d^ and g^, so, quite naturally, the
major diminished triad [b-d^-f] ought to be preferred to the
doubly diminished triad " [djjl^-f-d].^

Such is Marpurg's idea of a theory of harmony which
" conformed to the demands of practice " ; and it is an

1 Krit. Beitrage, Sect. II. 2 /j^^.

312 THE THEORY OF HARMONY

undoubted fact that there were many, even in this country,
who considered that Marpurg, as a " practical theorist,"
was far in advance of Rameau.

In his Handbuch bei dem Generalbass, Marpurg distin-
guishes the following fundamental chords, which he divides
into two classes. The fundamental chords of the first order
comprise the different species of triad, and the various kinds
of chords of the Seventh. (By a fundamental chord Marpurg
vmderstands aU non-inverted chords, that is, all chords
arranged in Thirds). "There are not more than three
fundamental chords of the first order, namely : —

(i) The Consonant harmonic triad. (Major or Minor
as c-e-g, or a-c-e.)

(2) The Dissonant harmonic triad. (Diminished or

Augmented as, b-d-f, or c-e-g^.)

(3) The Chord of the Seventh, consisting of 3rd, 5th

and 7th {a.s g-b-d-f, c-e-g-b, etc.). The triads arise
by means of the addition of intervals ; thus
the triad consists of two 3rds added together :
the chord of the Seventh of three 3rds."

By fundamental chords of the second order, Marpurg
imderstands " chords by supposition." These are : —

(i) The Chord of the Ninth, obtained by placing
a note a 3rd below the fundamental sound of
a chord of the Seventh, as E — g^-b-d-f.

(2) The Chord of the Eleventh, obtained by placing

a note a 5th below, as C — g^-b-d-f.

(3) The Chord of the Thirteenth, obtained by placing

a note a 7th below, as A — gi^-b-d-f.

Marpurg, of course, does not confine himself to the single
chord g^-b-d-f, in order to form " chords by supposition,"
but makes use of other chords of the Seventh for this
purpose, as b-d-f-a, f-a-c-e, d-f-a-c, etc.

Of several other varieties of chords investigated by
Marpurg, mention may be made of what he calls the " mixed
dissonant harmonic triad." Although IMarpurg tells us that
" the dissonant triad owes its origin to an alteration of the

F. W. MARPURG (17 18-1795) 3i

j» :>

3rd or 5th," ^ — which is not in accordance with his theory
of the generation of chords by means of the compounding of
intervals selected from the chromatic-enharmonic scale of
twenty-one notes ; nor an explanation of the diminished triad
on the leading note — he nevertheless explains the mixed
dissonant triad as one which belongs to two keys. " Thus,
in key C, the other notes of the most nearly related
scales G, F, etc., may enter, so as to form the chromatic scale
c-c#-^-^#-etc. ... If I may for a moment be permitted
to glance into the hidden depths of Nature [an allusion
to Sorge], there exist the following mixed triads " : — ^

(i) The " hard Diminished Triad," b-dj^-f, mostly used
in 1 position (f-b-djl^). With the 7th added
(b-dilf-f-a) we obtain the chord :^ ( French 6th
f-a-b-di^).

(2) The " doubly Diminished Triad," as d^-f-a. The

chord of the Augmented 6th {f-a-d^) is derived
from this triad. With the diminished 7th
added above the triad, we obtain the chord
of the Augmented I (German 6th f-a-c-d^) .

(3) The Triad arising from the Augmented 3rd and

pure Fifth, as b])-d^-f {\)

(4) The Triad formed from the Augmented 3rd and

Augmented 5th, as b^-d^-fi^ (!)

Marpurg goes on to describe several other " chords "
belonging to this class, but perhaps the above are here
sufficient.

It is important to note the development which the theory
of " chords by supposition " undergoes at the hands of
Marpurg. The chord of the Ninth presents in its formation
an unbroken series of Thirds ; not so the chords of the
Eleventh and Thirteenth. Marpurg, however, exerts himself
to remedy this defect. " In the chord of the Eleventh,"
he says, " we must remember that between the fundamental
note and the 5th below [as C — g-b-d-f'\dithirdvim'S,i be inserted,
[as C-e-g'b-d-f] in order that the chord may be properly under-
stood. This six-part chord, however, is of little use in its
complete state." ^ Of the use of this chord in three-part

1 Handbuch, p. 48. ^ Ji^i^^^ p. 43, 3 Jbjd,^ pp. 74^ 75.

314 THE THEORY OF HARMONY

writing, Marpurg gives this example : —

:g=

221

-G>-

::=g;

_c^_

:g=

zcs:

^r^

±;

in which he discovers a chord of the Eleventh at * : whereas,
in reality, there is notliing more serious than one or two
innocent passing-notes.

Of the chord of the Thirteenth, he remarks : — " The chord of
the 13th arises when, to a chord of the 7th, a 7th is added
below, as A — gjl^-b-d-f. It must be remembered that between
the fundamental note [^#] and the 7th [Aj, two Thirds must be
supposed, in order that the chord may be properly understood.
The chord in its complete form, A-ci(^-e-g^-b-d-f, cannot be
used " ! So then Marpurg, having obtained his Thirds,
finds himself obliged to take them away again. Otherwise,
one might say, the chord cannot be " properly understood."
It is especiaUy to the " combined Rameau-Marpurg system "
that we owe the " chords " of the " Ninth," " Eleventh,"
and " Thirteenth."

Chord of the Diminished Seventh.

It is impossible to avoid referring to Sorge's criticism of
Marpurg's theory of the chord of the Ninth : for there
is little doubt that Marpurg's new development of the
Rameau theory of " chords by supposition " was accelerated
by the criticism to which he was subjected by Sorge. The
passage of arms between the two theorists — for Marpurg
was not slow to reply — is amusing as well as instructive.
In the tenth chapter of his Compendium harmonicum Sorge
asks the question — How does the chord of the Ninth
arise ? — and remarks : — " Is it by means of a Third crawhng
under the chord of the 7th, according to the teaching of
Rameau and ]\Iarpurg ? By no means ! That would be a
bad foundation for the free, as well as the suspended Ninth.

F. W. MARPURG (i 718-1795) 315

The free unsuspended Ninth rises above the chord of the
Dominant Seventh and ornaments, hke a beautiful gilded
dome, the harmonic edifice. Its foundation is the chord
of the Seventh, a sure foundation. No use is made of
supposition, or composition (INIarpurg had thrown out
the suggestion in his Handbuch that the chord of the
Ninth e — g^-b-d-f, was compounded of the two chords
of the Seventh e-gj^-b-d and g^-b-d-f) for to make use of
' supposition ' is as if one were first to build his house in
the air, and then proceed to lay the foundation of it !
This is what is done by Rameau and Marpurg. This chord
of the Ninth is the real foundation of the chord of the
Diminished Seventh g^-b-d-f, and of the minor chord of
the Seventh b-d-f-a, or f^-a-c-e (this is a development of
Sorge's theory ; in the Vorgcmach he explains the chord
b-d-f-a, as a chord of the Seventh based on the diminished
triad b-d-f), and of all the chords arising therefrom by in-
version ; hence all these chords, and their inversions, can be
taken without preparation. Only it has to be noted that
frequently the true fundamental note yields up its authority
in favour of the Third of the chord." That is, Sorge permits
the fundamental note of the chord e-g^-b-d-f to be omitted,
and the chord to assume the form g^-b-d-f, a chord of the
Diminished Seventh.

Sorge's insight into the real nature of the chords of which
he treats is evident. Unfortunately, his concluding sentence
presented a weak point which was immediately perceived
lay ]\Iarpurg. In his KritiscJie Beitrdge Marpurg replies :
" i\Iy dear Herr Sorge, what happens when your true founda-
tion of the chord of the 9th ' yields up its authority in favour
of the 3rd ' ? Does it not remind you of a house from which
the foundation has been taken away, and which is left to
swing in the air ? Will it not then fall to pieces ? Only,
this is your affair, not mine ; and I must allow you to prop
up your house in the best way you can. But chords are not
houses. A chord may be placed on its head [inverted] ;
and one may remove one or more sounds from a chord,
but it would be impossible to remove a story from a house." ^
Here Sorge finds himself caught.

Marpurg, then, considers g'^-b-d-f to be a fundamental

1 KriL Beitrdge, Sect. VI.

3i6 THE THEORY OF HARMONY

chord, with fundamental note gi(!^, which is absurd. Sorge
also considers it to be a fundamental chord, but with funda-
mental note e, which is omitted. If the chord be regarded
as a chord of the Ninth, Sorge's ^dew of the matter is the
more reasonable. There are theorists, however, who hold
that the chord g^-b-d-f represents the first inversion of the
chord of the minor Ninth. This is a new theory of inversion,
and one by no means contemplated by Rameau, who held,
quite rightly, that it was the Octave which made inversion
possible, and that no chord could be inverted which exceeded
the compass of an Octave. The omission of a note from a
chord does not bring about the inversion of the chord. But
again, if the chord g^-b-d-f represents a chord of the Ninth in
fundamental position, what is the first inversion of the chord?
It is extremely doubtful if Marpurg reall}^ understood the
theoretical principles of Rameau, whom he professed to
foUow. The " combined Rameau-Marpurg system " is,
at any rate, a monstrous distortion of these principles. And
yet Marpurg was a man of wide erudition, of great and un-
doubted talent, not only as a writer and critic on musical
subjects, but in many respects as a theorist also. His
influence as a theorist was far-reaching — his Handbuch was
translated into at least two other languages — and there
is little doubt that it extended to this country. This can
only be regretted ; for it did not tend to the advancement of
the science or practice of harmonj'.

317

PART III.
CHAPTER XL

OTHER THEORISTS OF THE END OF THE EIGHTEENTH AND
BEGINNING OF THE NINETEENTH CENTURIES — KIRN-
BERGER, FETIS, ETC.

J. P. KiRNBERGER.

According to Dr. Riemann {Geschichte der Musiktheorie,
p. 476, et seq.), it is not Marpurg we have to thank for the
wide dissemination of the theory which considers all possible
chords to be formed from a series of Thirds added together,
but J oh. Phil. Kirnberger (1721-1783). This is a curious
opinion ; for the distinguishing feature of Kirnberger's
works, and that which marks them out from almost all
similar works of his own time, and of later times, is that no
attempt is made to formulate any theory of chord generation,
whether by means of acoustical phenomena, or by adding
Thirds to one another. Kirnberger rejects all chords of
the "Ninth," " Eleventh," and " Thirteenth," and recognizes
as " real " harmonic combinations nothing but the simple
triad and chord of the Seventh. In his principal theoretical
work Die Kunst des reinen Satzes in der Musik^ published
'^774~79' he simply states (p. 26) of the consonant triad that
it consists of a fundamental note (Grundton), a Third, and
a Fifth ; to which there may be added the Octave ; while
the chord of the Seventh consists of a Third, a Fifth, and a
Seventh ; or, more accurately, of a Seventh (not a Third !)
added above the triad (p. 60).

Kirnberger's works, indeed, represent a reaction against
the inconsequences of the Rameau-Marpurg system, and an
attempt to bring back harmonic theory to the paths of sanity
and commonsense. It would appear that the work Die
wahren Grundsdtze zum Gebrauch der Harmonie (1773) was
written expressly with this object. In the Preface to

31 8 THE THEORY OF HARMONY

this work Kirnberger remarks : — " Rameau has filled this
theory [of harmony] with so many absurdities as to cause
one fairly to wonder how such extravagances could ever have
found acceptance among us Germans. . . . Those who are
acquainted with Rameau's theory will, in the course of this
work, soon perceive in what respects his theory and my own
differ from each other, and which it is that explains most
simply and most naturally the origin and treatment of
chords." It was less against Rameau, however, than against
Marpurg that Kirnberger's criticism was most probably
directed.

But although Kirnberger ostensibly rejects Rameau's
principles, nevertheless several of the theoretical considera-
tions he brings forward differ in little or nothing from those
advanced by the French theorist ; and here indeed Kirnberger
more faithfully represents the teaching of Rameau than does
Marpurg. For example, in Die Kunst des reinen Satzes,
Part II., he gives this as the first and most simple method of
harmonizing the major scale {die erste und einfachste Art des
harmonischen Basses) : —

-o~

-e>-

-<s>-

:z3:

~-CJ . -QI

"O"

F.B.

He distinguishes, also, the ascending leading-note (Rameau's
" major dissonance ") from the descending one (" minor
dissonance"). Both the seventh and the fourth degrees
of the scale, Kirnberger states, are leading-notes ; but they
are not of the same character. " The leading-note, which
is the third of the Dominant chord, produces the greatest
unrest in the hearer if the following chord [Tonic chord] be
omitted, even when no dissonant interval forms part of the
Dominant triad." '^ The leading-note on the fourth degree
of the scale "is of quite a different character; when it
forms a part of the harmony of the Dominant, it appears as

1 Grundsatze des Generalbass, Sect. II., p. 43.

J. p. KIRNBERGER (1721-1783) 319

a real or essential {wesentliche) dissonance " (p. 43). In
the case of this note, then, it appears that it is dissonance to
which it owes its leading quality, whereas the first leading-
note retains its leading quaUty whether it forms part of a
dissonant chord or not. As for the cause of the dissonant
or leading effect of the seventh degree of the scale, Kirnberger
says further : — " Every interval smaller than a minor Third
is a dissonance ; as now, b is only a minor Second from c,
then the two sounds must be dissonant with one another." ^
Evidently this explanation is not complete. Otherwise, /,
the fourth degree of the scale, which is only a minor Second
from e, ought to have as pronounced a leading quality as h.
This, however, is not the case. Further, Kirnberger does
not investigate the circumstances under which both the
fourth and seventh degrees of the scale may produce the
effect, not of unrest, but its opposite, rest, as in the Tonic-
Subdominant and Tonic-Dominant Cadences.

Kirnberger, then, distinguishes {a) the ascending leading-
note ; {b) the descending leading-note ; and (c) both leading-
notes combined. 2 These remarks of Kirnberger cannot
have been without influence on Fetis and his theory of
" Tonahty."

Like Rameau, also, Kirnberger knows only two chords —
the triad, and the chord of the Seventh. " The whole of
harmony," he remarks, " consists of two chords only, in
which all other chords have their origin." ^ "These are: —
(i) The consonant triad, which may be Major, Minor or
Diminished [!^ {a). (2) The dissonant, "essential" chord
of the Seventh, which is of four kinds : consisting either of
a minor 7th with perfect 5th and major or minor 3rd {b : c), or,
\\dth diminished 5th and minor 3rd {d), or of major 7th with
perfect 5th and major 3rd [e) " : —

jQl

^S»-

zSzz

'-&

-<Si-

Kirnberger therefore places a triad, as well as a chord of
the Seventh, on each degree of the major scale. All these

^ Grundsdtze des Generalbass, Sect. II., p. 43. 2 i^i^^

' Die wahren Grundsdtze zum Gebrauch der Harmonie.

320 THE THEORY OF HARMONY

chords of the Seventh he describes as " essential." All how-
ever are not equally perfect. " Of these ground-chords the
first, that is the major triad, is the most perfect ; the diminished
triad on the contrary is the most imperfect consonant ground
chord. The chord of the minor 7th with perfect 5th and
major Third (chord of the Dominant Seventh) is most perfect,
and the chord of the major 7th the most imperfect dissonant
ground chord." It is noteworthy that Kirnberger regards
the perfection or imperfection of the chords of the Seventh
as determined by their nearness to, or remoteness from, the
Tonic harmony.

He says : — " The proof of this is as follows. The first
chord of the 7th [^g-h-d-f] is the most perfect, because it leads
directly to the Tonic chord. . . . and brings about a complete

close, ■ The second chord of the 7th \a-c-e-g\ is less

perfect, because it does not lead immediately to the Tonic
triad, but must first proceed to its Dominant, that is
A — D— G.
7 7 The third chord (b-d-f-a) leads to a Minor

B— E— A

Cadence : 7 7 The fourth chord [c-e-g-h] is less adapted

than any of the others to bring about a state of rest,

C-F#-B-E,

77 7 and is the most imperfect of all."

In the resolutions of these discords, Kirnberger exhibits a
curious comphance with the requirements of the Fundamental
Bass of Rameau ; for he might quite correctly have given other
resolutions to some of these chords of the Seventh. Besides,
in the case of the last three chords, they do not reach the
Tonic chord of C at all, nor can they be regarded even as

belonging to this key. Thus the second chord and

■D

the third are Rameau's Subdominant Discords (3rd

added below Subdominant harmony) in the keys of G major,

and A minor respectively ; while the fourth chord is

in the key of E minor, j

J. p. KIRNBERGER (1721-1783)

All these chords of the Seventh Kimberger describes as real
or essential {wesentliche) dissonances. All other dissonant
combinations are accidental '(zufdllige) or non-essential;
more strictly, all other dissonant chords arise by means of
the retardation of the real or essential harmonic notes of the
chord, which retardations take the place of the real harmony
notes. Such are the notes marked* in the following illus-
tration : —

fe±

-o-

-G>-

jC2Z

-rpr

.CL.

:qi

-<s^-

-e>-

Such an "unreal" dissonant note "is most dissonant
against that note in the place of which it stands, and it finds
its complete resolution in the ground chord itself. The essential
dissonance [the Seventh] on the contrary, is not dissonant
because it takes the place of a consonance, but because, being
added to the consonant intervals [of the triad], it destroys
the consonant harmony of the triad, or at least renders it
very imperfect. Therefore it cannot resolve on the same bass
note, for it does not represent another tone belonging to the
harmony of this note, but makes absolutely necessary the
succession of another harmony for its resolution." ^ This
statement represents a notable achievement in the science
of harmony, and brings to light a principle which the practice
of composers, and the course of harmonic development since
Kirnberger's time, have made increasingly important.

Both kinds of dissonance may occur in a single chord, for
exsLvaple , gjf^-b-d-f ; in this chord, Kimberger regards the note
/ as a non-essential or unreal dissonance ; while d is the
essential dissonance, being the Seventh of the chord of the
Dominant Seventh e-gi^-b-d. All this may be clearly perceived
in the resolution of the chord ; /first falls to e, the harmony
note whose place it occupies ; in doing so it merely resolves
on its own ground-chord e-g^-b. The note d, however,
cannot resolve thus : for this, a change of harmony is

1 Die wahren Grmidsdtze, etc., Sect. VI.

122

THE THEORY OF HARMONY

necessary. Although the distinction Kimberger makes here
is a real one, it is doubtful whether, in making use of the
term " essential " to distinguish the dissonant chord of the
Seventh from other dissonant combinations, Kimberger
exactly described the nature of the chord of the Seventh.
For at bottom the dissonance of the Seventh is not more
" essential " than any other dissonance. It is evident that
Kimberger is by no means prepared to concede that the only
really essential chords in music, that is, the only chords which
in themselves possess harmonic significance, are the major
and minor harmonies. We have seen that he considers the
diminished triad {b-d-f) to be a consonant chord.

Kimberger would therefore appear, almost in spite of
himself, to have given a considerable impetus to the theory
of the " essential discord," of which so much has been made
in this country. But in our own day this term has come
to mean almost exactly the opposite of what Kimberger
intended. Thus the following are said to be " essential
discords " ; whereas, according to Kirnberger's teaching,
they are " accidental " or " non-essential " discords : —

z^z

z^-

-<S-

-G>-

ZZZi.

jC21

:qi

Like Sorge, Kimberger sees in the " passing Seventh,"
that is, in the Seventh taken as a passing-note (which frequently
occurs in compositions by the Church composers even before
the time of Palestrina) the real origin of the chord of the
Seventh.^ Noteworthy also is his explanation of the chord
of the Augmented Sixth. " The augmented Sixth," he
remarks, " is purely a melodic ornamentation carried over into
harmony, and, as it takes the place of the major Sixth . . .
it neither brings about a change in the ground harmony, nor,
still less, does it form in itself a distinct ground chord, as

^ Die Kunst des reinen Satzes, Part I., p. 30.

J. p. KIRNBERGER (1721-1783) 323

some have wrongly taught."^ The explanation of these two
chords, then, is as follows : —

-<s>-

:p2=

;zc2:

I ,s»-

J 'j.

-<5> — ■Se'

:c2:

-Q.

:c2i

'.-.'^

_C2_

:qi

One has little difficulty in agreeing with Kirnberger that the
chord of the Augmented Sixth cannot be regarded as a
" ground chord," nor in recognizing the importance for the
theory of harmony of his explanation of the origin of these
chords. At the same time, Kirnberger goes too far and too
fast if he considers, as he appears to do, that the / in the
first chord and dj^ in the second have a purely melodic
but no harmonic significance. His attitude in respect of
these chords is not consistent. For although the origin of
both dissonances is the same, he considers the Dominant
Seventh as an " essential " dissonance, but the Augmented
Sixth as non-essential.

Kirnberger also distinguishes two forms of the l chord,
which represents the second inversion of the major or minor
harmony. Heinichen and Mattheson had considered it to
be a dissonant chord. Rameau denied this to be the case,
seeing that it represented a consonant harmony ; while
writers of this time, even Sorge, generally devoted considerable
space in their works to the discussion of the question as to
whether the Fourth was a consonance or a dissonance.
Kirnberger recognizes a consonant form of the *l chord,
which represents a consonant harmony, but also a dissonant
form, in which the 4th and 6th retard the 3rd and 5th. Here
again, Kirnberger manifests his admirably clear perception of
harmonic and tonal relationships.

So then, our author concludes, the whole edifice of harmony
is built up from two simple ground chords — the triad and the
chord of the Seventh. Only by such principles as he has laid

1 Die wahren Grundscitze, etc., Sect. XV.

324 THE THEORY OF HARMONY

down can the difficulties of harmony be solved and made
intelligible : — " On the other hand, all music which cannot be
traced back according to these fundamental principles to a
natural succession of the two ground chords is incompre-
hensible " {unverstdndlich).^ This is a daring corollary ; but
one nevertheless which deserves consideration. Further,
he remarks, his theory of harmony is simpler and more
true to the facts than that of Rameau. " Many have been
persuaded by French writers that we have Rameau to thank
for this simple theory of harmony. . . . But Rameau has
not at all conceived in his theory the real simphcity and purity
of harmony, as he actually sometimes regards passing-notes
as fundamental notes, on which he bases his chord of the
Added Sixth, which he considers to be a ground chord." ^
Kirnberger considers the Sixth in this chord to be merely a
passing-note.

On the other hand, Rameau might have replied that
Kirnberger not only accepts the scale without any attempt
to explain it, but considers himself at hberty to place not
only a triad but a chord of the Seventh on each degree of
this scale, without appearing to observe that it is necessary
to explain whence these chords are derived. Besides, they
exist as isolated entities, and apparently \vithout any harmonic
connection between them. Nevertheless, this harmonic
connection constitutes one of the chief problems of harmonic
science, and the theory of harmony which makes no serious
attempt to account for it is a superficial theory. Further,
that Kirnberger makes harmony for the most part to depend on
melody. That is, harmony is melodically determined. But
Kirnberger is unable to formulate any fundamental principles
of melody.

After Kimberger's criticism of Rameau, it is curious to
note his explanation of the ground or fundamental bass
rising a Second. He says : — ' ' It often appears that the ground
bass proceeds by the step of a second, when in reahty this
is not the case. In the following passage {a) it appears that
there are simply triads, and the bass of this passage appears
to be the ground bass. . . . But the second chord carries
here, in addition, the 6th [!] [no 6th is present in the chord in

^ Die wahren Grimdsdtze, etc., Sect. XXIII.
2 Ibid., Supplement.

J. p. KIRNBERGER (i 721-1783)

question], and is therefore not a ground chord but a
chord "[!]:—!

J-3

E^S

-<&>-

-&^

(^)

-s. ^

Se^

2d:

-<s»-

321

E3:

Also, Kirnberger tells us that the actual Fundamental
Bass of the passage at (b) is to be understood as at (c).
Here, then, we find Rameau's "double employment" in
full operation ! This, surely, is one of the most curious facts
in the whole history of the theory of harmony. Kirnberger,
the empiricist, who has explained Rameau's chord of the
Added Sixth as arising simply from a passing-note, resuscitates
Rameau's discredited theory of " double employment "
in order to account for the immediate succession of both
Dominants !

Other Theorists of the End of the Eighteenth and
Beginning of the Nineteenth Centuries.

By the end of the eighteenth century, the theory of Rameau
had begun to lose ground, even in France. Thus N . E. Framery,
in referring to it (Art. Accord in his Encyclopedic methodique,
1791), remarks: — "Rameau is the inventor of 'double
etnployment ' which, after being long a subject of ridicule,
has now become forgotten. To-day this chord [Added Sixth]
is no longer regarded as a fundamental chord, at least in
practice, and the best authors only make use of it as an
inversion of the chord of the Seventh." It is not surprising
that the influence of the Fundamental Bass began so soon
• to diminish. Musicians, who for the most part failed to
grasp its real theoretical significance, had regarded it mainly
as a guide to composition. But Rameau's directions for

^ Die ivahren Gntndsdtze, etc., Sect. XXII.

326 THE THEORY OF HARMONY

the use of the Fundamental Bass were, to say the least,
ambiguous. He had never been able to give any adequate
explanation of the secondary triads of the key, nor to say
with certainty whether or not the ascent or descent of the
Fundamental Bass by the interval of a Third brought about
a modulation. It need not be wondered at, therefore, that
the system of Rameau was soon forsaken for new " Practical
schools of composition " which made hght of the difficulties
that had perplexed the great theorist, and saw no theoretical
problems whatever in the way of the immediate succession
of both Dominants, or of the " ground-bass " rising or
falling a Third.

Further, the practice of composers, the new and strange
chords they employed, the novelty of their harmonic succes-
sions, which appeared to outrage all the rules which Rameau
had laid down for the use of the Fundamental Bass, bewildered
even the few who still swore fidelity to it. Nevertheless, the
influence of Rameau persisted in other directions, and chiefly
along two main fines, namely, the derivation of the scale
and of chords from the sounds of the harmonic series, and
the formation of chords by means of superadded Thirds.
As a rule both methods were combined. Only exceptionally
does one meet also with systems in which there is a definite
abandonment of Rameau's principles, especially his use of
acoustical phenomena. Of such works on harmony, which
appeared during the latter part of the eighteenth and beginning
of the nineteenth centuries, there may be mentioned the
following : —

P. J. RoussiER (Abbe) — Traiie des accords et de leur
succession, selon le systeme de la basse fondamentale (1764) ;
Observations sur differ ents points de V harmonic (1765).

In the Preface to the first work, Roussier explains that his
desire has been to write not so much a theoretical as a
" practical " work on harmony. " I have thought," he ■
says, " that a treatise on chords, in which all theory was
suppressed, and which really belonged to the art of Accom-
paniment and of Composition, would render the study of
harmony less protracted, and especiafiy less repulsive."
While, then, he follows in his work the system of Rameau,
he nevertheless thinks it necessary to develop it a little.

"It is sufficient that in several chords . . . the grave
sound is in reality the physical generator of the principal

ADDED THIRDS AND FUNDAMENTAL DISCORDS 327

sounds, in order to call fundamental, by extending somewhat
the meaning of this term, and by a sort of analogy, every
other direct chord.althoughits musical harmonics {harmoniques)
are not always in the same proportion nor of the same kind,
as the real harmonics of the grave sound of the chord [!]. But
is it desirable that Nature should leave Art nothing to do ? "^

Roussier distinguishes, like Rameau, a major and a minor
dissonance. " Every 7th should resolve in descending a
degree : every 6th [added 6th] should ascend a degree."
He adds that before the discovery of the Fundamental Bass,
there was great uncertainty as to the proper treatment of
dissonant intervals, such as the tritone, the augmented
Fifth, etc. : " in these intervals the upper note is a leading-
note."

Every note of the major or minor scale may bear a chord
of the Seventh. " The intervals in these chords are selected
from the notes of the scale, or mode, in which they occur."
(It is, then, the scale which determines harmony.)

" Chords may be derived from other fundamental chords
in four different ways : — (i) by Inversion ; (2) by Supposition ;
(3) by Substitution ; (4) by Substitution and Supposition
combined. The only chord derived by this last method is
the chord of the Diminished Seventh." A chord distinguished
by Roussier is b-d^-f-a, which he calls a " Mixed Dominant "
chord {Dominant Mixte). "This chord is neither a Tonic
Dominant, nor a Simple Dominant, but shares the features
of both. It is analogous to the Tonic-Dominant by reason
of its major 3rd, and to the Simple-Dominant by reason of
its diminished Fifth [!]." The fundamental note of this chord
is b, and its inversions are d^-f-a-b, f-a-b-d^, etc. !

In Part III. of his work (" In which some new chords are
proposed "), Roussier proceeds to explain some " new chords."
It will be found, he remarks, that some of these chords are
less hard in effect than the chord of the Augmented Sixth.
One of the " new chords " is d^-f-a-c. " This chord is
fundamental : the diminished Fifth dj^-a is its original minor
dissonance." ^ Inversions of this " fundamental chord are,
f-a-c-d^, a-c-d^-f, and c-d^-f-a. Other new chords are
g^ — d^-f-a-c : g — d^-f-a-c : e — d^-f-a-c, etc. These are chords
by supposition : rf:j|: is the fundamental note of all three.

^ Traite, p. 26. - Ibid., p. 160.

328 THE THEORY OF HARMONY

It cannot be said that, on the whole, Roussier's development
of Rameau's system tended to improve it.

Levens (chapel-master of the cathedral of Bordeaux) —
Abrege des regies de I'harmonie (1743). Levens derives the
scale from the first ten harmonic sounds. The fourth
degree of the scale not being found among the first ten sounds
of the harmonic series, he makes use of the arithmetical series
in order to discover this note.

Balliere (member of the Academy of Sciences of
Rouen) — Theorie de la Musiqiie (1764). For the genera-
tion of chords, Balliere, hke Levens, refuses in his use of the
harmonic sounds to be Umited by the number six : and betters
the system of Levens by making use of the first thirteen sounds
of the harmonic series.

J. F. LiROU — Explication dn systeme de I'harmonie (1785).
In his generation of chords b\^ means of added Thirds,
Lirou makes use of an ascending succession of sounds
c-e-g-b-d-f-a : as well as of a descending succession, c-a-f-d-
b-g-e.

H. F. M. Laxgle — Traits d'harmonie et de modidation
(1797). Construction of chords by means of added Thirds.
Langle postulates : — " There is but one chord, that of the
Third, the combinations of which produce all other chords."

J. J. ]\IoMiGNY — Conrs complete d'harmonie et de com-
position d'apres une theorie neuve et generate de la musique,
basSe sur des principes incontestables puises dans la Nature,
etc. (1806). Momigny derives the complete major scale
from the harmonics oi a single string, which give him, he
informs us, the sounds corresponding to g-a-b-c-d-e-f. But as
this does not represent the correct order of tones and semitones
of the major scale, he regards the string from which these
sounds are supposed to be derived, not as a Tonic, hit as
a Dominant ! The starting point of the natural major scale
is therefore g, the fundamental sound of the string, and the
order of its sounds may quite well be determined as g-a-b-c-d-
e-f, so long as g is regarded as Dominant and c as Tonic.
This theory has its adherents even in the twentieth century.
(See Art. Harmonics, in Grove's Dictionary of Music 1906).

G. L. Chretien — La Musique etudiee comme science
naturelle, etc. (181 1).

ADDED THIRDS AND FUNDAMENTAL DISCORDS 329

Chretien follows, for the most part, the principles of Rameau.
He sees in the resonance of the sonorous body the origin
of harmony. All theories based on divisions of the monochord,
and on geometrical calculations, are false ; harmony cannot
be generated by any such methods, for neither the monochord
nor geometry possesses in itself any principle of chord
generation. Both may be used as a means of verifying the
proportions of intervals, but they can generate no harmony
and no scale. Chretien, unlike Rameau, derives only the
major harmony from the resonance of the sonorous body.
The minor harmony is analogous in its construction to the
major ; and is obtained by arbitrarily lowering the major
Third a semitone.
Of works which appeared in Germany, there may be noted : —

J. F. Daube — Generalbass in drey Accorden (1756).
The three chords of Daube are those which Rameau had
already made familiar, namely (i) the major and minor
harmonies ; (2) the chord of the Seventh on the Dominant ;

(3) the discord f ] on the Subdominant. By means of

these three chords the whole scale may be harmonized ;
and whether in a central key, or in other related keys to
which a modulation may be made, they constitute the sole
harmonic material of a Mode. It may happen that one of
the notes of the chord is chromatically altered, or even that
some other note is substituted for the really essential note
of the harmony (" wenn ein Interval von einem Accorde
weggelassen wird, an dessen Stelle ein anderes hinzukommt ").

All other chords are the result of the anticipation or
retardation of notes of a chord, or arise from passing-
notes, etc.

C. G. ScHROTER — Deutliche Anweisung zum Generalbass
in bestdndiger Verdnderung des uns angeborenen harmonischen
Dreiklangs (1772). For Schroter there is but one independent
and original harmony, namely the Triad, major and minor.
The chord of the Seventh arises by means of the substitution
of the Seventh for the Octave ; all other combinations arise
by means of the retardation, alteration, etc., of notes.

ABBf VoGLER — Tommssenschaft und Tonsetzkimst {1776) ;
Handbuch ziir Harnwnielehre (1802). Vogler makes use
of the harmonic as well as the arithmetical division of a
string, which he extends to the thirty-second term. From

330 THE THEORY OF HARMONY

the sounds obtained by this process he then constructs all the
chords he requires.

J. H. Knecht — Elementarwerk der Harmonic (1792-8).
Knecht was a pupil of Vogler. He distinguishes 3,600
different chords which may be used in the practice of harmony.
Of original chords there are : — 132 chords of the Seventh ;
72 chords of the Ninth ; 72 chords of the Eleventh, and 36
chords of the Thirteenth !

H. C. Koch — Musikalisches Lexikon {1802) . This work of
Koch's is noteworthy in that we find again in use Kirnberger's
terms "essential" {wesentlich) and "non-essential or accidental"
(zufdllig). Koch, however, makes use of these terms to
distinguish the primary from the secondary triads of a key.
Thus the essential triads in C major are c-e-g, g-b-d, and
f-a-c ; while the secondary triads are d-f-a, e-g-b, and a-c-e.
The diminished triad (b-d-f) is the 3rd, 5th and 7th of the
chord of the Dominant Seventh g-b-d-f, and is to be regarded
as an incomplete form of this chord.

In the j\Iinor Mode, the fundamental form of the scale
is a-b-c-d-e-f-g-a. The seventh degree, however, must in
certain cases be raised a semitone, that is, from g to g^.
The essential triads in this mode of A minor are a-c-e, e-g-b ^
d-f-a, while the secondary triads are c-e-g, g-b-d, and f-a-c.
In the major mode, therefore, the essential triads are all
major, and the secondary triads minor ; while in the minor
mode the essential triads are all minor, and the secondary
triads major (?). Like Kimberger, Koch distinguishes a

dissonant as well as a consonant form of the chord.

4
• -G. Weber — Versuch einer geordneten Theorie der Tonsetz-

kunst (1817-21). Weber's ground-harmonies or fundamental

chords are those which a multitude of text-books on

harmony have made familiar. He follows Kimberger in

placing a grotmd-chord — triad, or chord of the Seventh — on

every degree of the major scale. In the Minor Mode,

however, neither the triad (augmented) nor the chord of the

Seventh on the third degree of the scale is to be considered

as a ground-chord, a curious exception, seeing that all other

degrees of the scale have " ground-chords." All other

combinations are the result of passing-notes, suspensions,

or chromatic alteration of one or more of the notes of a

ground-chord. Such arc entitled non-essential discords.

DDED THIRDS AND FUNDAMENTAL DISCORDS 331

F. Schneider — Elementarbuch der Harmonic imd
dsetzkunst (1820). Schneider's work differs little, in its
'-.ential features, from that of Weber.

■^f works by Italian theorists there may be mentioned : —
t . A. Vallotti — Delia scienza teorica c pratica delta
•derna musica (1779). Only the first part of Vallotti's
•ork was published. The exposition of his theory was
completed by his pupil Sahhatini.

L. A. Sabbatini — La vera idea delle mtisicale numeriche
segnature, etc. (1799). Sabbatini was a pupil not only of
VaUotti but of Padre G. Martini. Sabbatini lays down
the principle that the only numbers of significance for
harmony are 1:3:5:8, which correspond to the major
harmony. From this harmony all other chords are evolved :
these arise, either by means of inversion, or by "accidental"
sounds added to the fundamental consonant harmony.^ In
the minor harmony we find the same consonances as in
the major, but in diverse order. Other chords which,
although dissonant in themselves, are nevertheless "consonant
by analogy " [Armonie consonanii per rappresentanza) are the
diminished and the augmented chords {b-d-f-b and c-e-g^-c).
The Minor Mode, like the minor harmony, has its origin
in the Major.

The influence of the two great Italian theorists, Zarlino
and Tartini, is strongly evident throughout Sabbatini's
work. The rule which is laid down by Sabbatini for the
formation of dissonant chords sounds almost like a passage
from Tartini's Tratiato di Musica. " There is not," he
remarks, " nor can there be, any dissonant chord which is
not based on a consonant chord." ^ He proceeds to develop
this. The only harmonic numbers are i, 3, 5, 8 : these
represent a consonant harmony, and an^' other number,
that is, any other sound of the scale added to this harmony,
will render it dissonant. In whichever part of the scale,

1 A questo fine dico, che Tharmonia ridotta alia sua corda
fondamentale, fra I'intiera serie, fa uso di soli tre o quattro numeri,
che sono i, 3, 5, 8, e con questi soli compone I'inalterabile sue consonante
accordo. Che se poi s'introducono ncH'armonia numeri diversi degli
accennati, vi hanno luogo soltanto o per trasporto di armonia, o come
suoni aggiunti, e accidentali (Cap. i).

- " Non si da, ne puo darsi posizione dissonante, se non fondata
sopra la posizione consonante " (Cap. 4).

THE THEORY OF HARMONY

then, this consonant harmony is placed, all the other sount
of the scale will be dissonant with it (Cap. 4) : —

:iiSi

-1 '^-H-

In this way, by the addition of a dissonant sound to the
consonant harmony, Sabbatini obtains various kinds o)
chords of the Seventh. The chord of the Dominant Seventh
occupies a place by itself. It is more consonant than an^
other chord of the Seventh, the reason being that the ratio
of this Seventh approximates so closely to that of the
" natural Seventh." For this reason the Dominant Seventh
may be taken without preparation. Sabbatini quotes Tartini
as well as Vallotti in support of this view.

In addition to chords of the Seventh, there are also chords
of the Ninth, Eleventh, and Thirteenth. What is remarkable
about these chords is that they are not formed by a process
of adding Thirds one to another : —

(a)

gth.

nth.

13th.

(b)

~JZiZ

— C2gl

-<5^-

-€^-

-S>-

-O-

-<s>-

-o-

m-.

The chord of the Ninth is formed by adding a Ninth
above the consonant harmony (not, therefore, above the chord
of the Seventh) ; the chord of the Eleventh by adding an
Eleventh above the consonant harmony (not above the
chord of the Ninth), and so on. Other combinations however
might arise such as that in which both a Seventh and a
Ninth are added to the consonant harmony (&).

The above list of works on harmony is by no means complete,
and it is impossible here to examine even the most important
of them in any thorough manner. Still, the nature of the
development which the theory of harmony has undergone
from the time of Rameau up to the first years of the nineteenth

DDED THIRDS AND FUNDAMENTAL DISCORDS 333

mtury has at least been indicated. The most strongly
larked features in this development are first, the generation
f the scale by means of selection from the sounds of the
larmonic series, and second, the formation of chords by
neans of adding Thirds together. Knecht, with his 132
fundamental chords of the Seventh, 72 fundamental chords
5f the Ninth, etc., and Vogler, with his 32 sounds of the
.larmonic series, from which he derives even a Chromatic
scale, would appear to represent the redudio ad absurdum of
principles which have at least their origin in Ramcau's theory.
Works such as those of Schneider and Weber on the other
hand are less concerned with the theory than the practice
of harmony. Weber, indeed, declares in his work his entire
disbelief in the possibihty of any theory of harmony which
attempts to furnish an adequate explanation of the harmonic
facts ; the best work on harmony, in his opinion, is that
which takes account of the largest numbers of these facts,
and treats of them in a practical way. This, as Fetis remarks,
is to reduce harmony to the position it occupied at the time
of Heinichen and Mattheson. One may add that it was
just the existence of such a multitude of isolated facts,
apparently wthout connection with each other, which brought
about Rameau's attempt to introduce some order into the
domain of harmonj' .

It is not surprising that at the beginning of the nineteenth
century, amidst such a variety and diversity of systems, much
uncertainty prevailed as to the respective merits of these
systems, as to the proper basis of the theory of harmony, and
as to whether indeed any adequate theory of harmony was
possible. From a work on harmony by C. S. Catel {Traiti
de I'harmonie, Paris, 1801), we learn that in 1801 a conference
of eminent musicians and professors met for the purpose
of approving a system of harmony to serve for purposes
of instruction in the Paris Conservatoire de Mnsique.
Among the members of this conference were Cherubini,
Martini, Gossec, Mehul, etc. Several systems of harmony
were examined, and among them that of Rameau, which still
had its adherents. The treatise of harmony of Catel was,
however, ultimately and unanimously adopted as being at
once the most simple and the most comprehensive. The
adoption of Catel's system by the Paris Conservatoire marks
therefore the definite abandonment in France of Rameau's

334 THE THEORY OF HARMONY

theory of harmony. It is instructive to note what this simisle
and comprehensive system was which the most eminent
professors in France thought worthy to take the place of that
of Rameau.

The basis of Catel's system of chord generation he himself
explains in the following short phrase: — "There exists in
harmony only a single chord, in which all the others are
contained." This is not however the major or minor triad.
What Catel does is to divide a string harmonically by the
first nine numbers, from which sounds thus obtained he
claims to derive the combination c-e-g-b\^-d, or, if it be supposed
that the fundamental sound is g, then g-b-d-f-a. This
Catel calls a chord of the Ninth, and takes no account of the
fact that the sounds / and a are not at all the sounds which
correspond with the fourth and sixth degrees of the C major
scale. This chord, which in practice is called the chord of
the Ninth on the Dominant, contains according to Catel the
following harmonies : —

(i) The Major Triad, g-h-d.

(2) The Minor Triad, d-f-a.

(3) The Diminished Triad, h-d-f.

(4) The chord of the Dominant Seventh, g-h-d-f.

(5) The chord of the Seventh on the leading-note, h-d-f-a.

By means of the extension of the series of harmonic sounds
up to the seventeenth term, Catel discovers the chord of the
Minor Ninth on the Dominant g-b-d-f-a\} ; and the chord of
the Diminished Seventh b-d-f-a\^. All other dissonant
combinations are the result of retardation, anticipation, or
passing-notes ; or of the chromatic alteration of the harmony
notes natural to the " fundamental chords " (so-called by
Catel) enumerated above.

Criticism of such a system, if it can really be called a
system, is needless. Catel's fundamental sound is the Dominant,
which is everywhere known as the Fifth of the Tonic, and
determined by the Tonic. This Dominant nevertheless
forms his starting-point, and centre and foundation of his
system. Several of the sounds he admits without scruple
are utterly alien to any known harmonic system. But if
one may construct from the sounds of the harmonic series
almost any scale that one pleases, so also any one is at liberty
to amuse himself by picking out, from such a series, sounds

F. J. YETIS— THE LAWS OF TONALITY 3.35

which will give him almost any " chord " he desires. Only
it is a decided mistake to label such methods " science," or
" theory of harmony " !

F. J. Fetis.

The name of Fran9ois Joseph Fetis (1784-1871), the distin-
guished Belgian musician, musical historian, and theorist,
has several times been mentioned in connection with Rameau's
theory of harmony. Fetis, as we have seen, altogether
rejects acoustical phenomena as a basis for the theory of
harmony, as well as all harmonic, arithmetical and geometrical
progressions and proportions. The only part of Rameau's
system which he accepts is that of the inversion of chords.
In the by no means adequate analysis of this system which
he has given in his Esqiiisse de I'histoire de I'harmonie and
his Traits de I'harmonie, Fetis, however, does Rameau
a great deal less than justice. For example, of Rameau's
Fundamental Bass he remarks : — " Rameau was too good
a musician not to understand that, having rejected the rules
of succession and of resolution of chords, which were in-
compatible with his system, he was bound to supplement
his theory \vith new rules. He therefore invented his theory
of the fundamental bass."'^ And again : — "The doctrine of
the fundamental bass was, with Rameau, only an accessory,
or one might say a complement, of his system of harmony." ^
We must consider that Fetis did not fully understand Rameau's
theory : for it is difficult to imagine that he would wilfully
misrepresent it.

It is characteristic of Fetis that he considers that all theorists
before his time have been on the wrong track, and that they
have altogether failed to perceive what constituted the real
basis of the theory of harmony. In the Preface to his Traite, in
which he expresses his confidence that he has finally succeeded
in discovering the fundamental law of all music, and all
harmony, he remarks : — " In vain have the most distinguished
men flattered themselves that they had arrived at an adequate
system by other means ... in vain have they called to
their aid mathematical science, acoustical phenomena, the

1 Traits de I'harmonie, p. 206. * Ibid., p. 208.

336 THE THEORY OF HARMONY

distinctive qualities of various aggregations, and all the
resources of which the most daring imagination could
conceive. The history of their endeavours is the history of
their errors." Where, then, ought one really to seek for the
fundamental law of music ? In " Tonality." " The only thing
which no one seems to have dreamed of, was to seek for the
principle of harmony in music itself, that is, in Tonality."

" Tonality," What is it ?

What then is tonahty ? " However simple such a
question may appear," says Fetis, "it is certain that few
musicians could answer it satisfactorily. I say, then, that
tonality resides in the melodic and harmonic affinities of
the sounds of the scale, which determine the successions
and aggregations of these sounds. The composition of
chords, the circumstances which bring about their modifica-
tion, and the laws of their succession, are the necessary result
of this tonality. Change the order of the sounds of the scale,
distribute their intervals differently, and the majority of
the harmonic relationships cease to exist. For example,
attempt to apply our harmony to the major scale of the
Chinese {a), or to the incomplete major scale of the Irish
and of the Scotch Highlanders {h) — our harmonic successions
would become impossible in these tonalities " : —

-<9-

-r^ o ^

>o »-^ «r5-

«>-

^ r, ^ o z^

-r^ '^ — ' - '^ ry-

Fetis does not inform us whether the Chinese scale or
the scale which he regards as that of the Scotch Highlanders
has ever been adopted as the basis of harmonic music.
He suggests that the harmony resulting from such scales
would be quite different from our harmony. This no doubt
is not far from the truth. The only question is, could it be
considered as harmony at all ? For example, the Pythagorean

F. J. FETIS— THE LAWS OF TONALITY 337

Third f-a, in the first scale, is not only a dissonance for our
ears, but has never been known as anything else, by any nation,
in any epoch. On the other hand the Perfect Octave,
Fifth and Fourth found in these scales are the " harmonies "
or consonances used by musicians from the time of Pythagoras
up to our own dsLy. When, then, Fetis speaks of " our
harmony," one naturally inquires what other kind of harmony
is there, or has ever been in existence ? The Conson-
ances of the Octave, Fifth and Fourth, which were known to
and recognized as such by the most ancient peoples possessed
of a musical system, are the same in every respect as the
Perfect Consonances known to and practised by us at the
present daj^ To these we have added the major and minor
Thirds resulting from the harmonic division of the Fifth,
and their inversions.

Fetis proceeds : — " What I describe as Tonality then, is
the Order of melodic and harmonic facts which result from
the arrangement of sounds in our major and minor scales ;
if even one of these sounds were to be placed differently,
tonality would assume another character, and the harmonic
results would be quite different. . . . All then, I repeat, is
necessarily derived from the form of our major and minor
scales, and constitutes what one calls the laws of tonaUty." ^

These remarks have been considered by not a few besides
Fetis to be very profound and to betray a deep insight into
the nature of music and harmony. In reaUty they are very
superficial. Fetis asks us to believe that it is the scale which
determines harmony and harmonic succession, whereas the
reverse is the truth, as every musician knows who is acquainted
with the history and development of the Church Modes.
These Modes, quite different as regards the arrangement and
proportion of sounds from our modem modes, were under
the influence of harmon}^ gradually altered until they assumed
the form of our Major and Minor modes. It would be correct
to say that harmony banished these old modes out of
existence.

Fetis asks us to believe that " our harmony " has arisen
apparently in quite an accidental way, through a chance
combination of two or more sounds, from a scale fashioned
on purely melodic principles, that is by means of measuring

^ Traits de I'harmonie, p. 249.

338 THE THEORY OF HARMONY

off certain intervals so as to form a series of sounds varying
in pitch, but not determined by any harmonic considerations
or consonant relationships between the sounds themselves. ^
How was this scale tuned ? When and where did the scale
which has determined " our harmony " come into existence ?
Fetis cannot tell us. It is a remarkable fact, and one of
theoretical importance, that of all the scales which were in
use throughout Europe before the advent of polyphony, there
was not one which corresponded with our major or minor
scale. How then can Fetis assert that our harmony has been
determined by a scale which had never been in use before
the advent of harmony ? Was it necessary to discover some
new scale suitable for the practice of harmony ? Fetis,
seeing that he considers harmony to be determined by the
scale, can hardly admit that it was necessary for harmony to
discover and to form for itself an entirely new scale. This
however is just what happened.

It may be thought that there was at least one of the Church
modes, the Ionian, which corresponded with our major scale.
This is not the case. According to the Pythagorean system
of intonation of the scale which prevailed not only among the
Greeks but throughout the whole of the Middle Ages, the
Ionian scale presented a series of intervals which made it
quite different from our major scale. Each tetrachord of the
Ionian scale consisted of a succession of two whole-tones of the
proportion 8:9, followed by a small interval of the proportion
243 : 256. There was therefore no interval corresponding
to our minor tone (9 : 10) or diatonic semitone (15 : 16), while
all the Thirds and Sixths were dissonant, and were expressly
described as such. It was not until harmony began to be
used for artistic purposes that the Pythagorean tuning of the
Third began to be called in question. Ultimately this Third,
consisting of two major tones, had to give way to the major
Third of the proportion 4:5, which brought about the
formation of new scales, in which the Thirds and Sixths were
consonant, and the minor tone and diatonic semitone found
a place.

These facts are in themselves sufficient to disprove the
whole theory that " our harmony " has been determined
by the scale. According to this theory, the major Third

1 See remarks on the origin of scales, Preface to TraiU, p. 12.

F. J. FETIS— r/fZi LAWS OF TONALITY 339

(4 : 5) (as well as the minor Third 5 : 6) ought to have been
derived from some existing scale. But there was no scale
in use from which such a Third could have been derived.
What " occult influence," to use the language of Fctis,
could have caused musicians to become dissatisfied with
the Thirds they already possessed, derived from scales
which had been in use for many centuries, and what
could have induced them to substitute for these intervals
other Thirds, derived from no one knew very well where,
the effect of which was to banish, so far as harmonic
music was concerned, these venerable scales entirely out of
existence. Here in truth was a musical revolution ; how
great it is difficult for us adequately to realise. These
ancient scales had their origin in Greek antiquity ; they had
been in use for over 1,200 years ; they had become identified
with the services of the Christian religion. How powerful
must have been the influence which brought about their
decay ! This in^uence was Harmony.

Fetis has even less ground for his assertion that it is the
order of sounds in the scale which determines " the tonahty "
and harmonic succession. In his Traite he quotes a passage
from the beginning of the eight-part Stahat Mater of
Palestrina as an excellent example of music which is in
a different tonality from our own. The conclusion of the
passage is as follows : —

^^=:A=d=;^^

-JZiL

-G»-

^G-

'jCSI

f^ -P^ t?g-

-<5>-

Z2z:c:

^^p^

T^-

-^-

iss

-IS-

-f^-

221

-\}Sr

-QL

-J-- -^ #^-

Q ■;-

hS>-

340 THE THEORY OF HARMONY

There is no doubt as to the " modal " effect of this music.
But it does not arise from the order of sounds of the scale.
Palestrina writes in the Dorian Mode, but he alters it to suit
the requirements of his harmony, using not only B[7, but
C:jf, as well as Ct]. This gives what we may regard as two
scales ; F major and D minor, its relative minor, and he
makes use of both. These are our modem scales. It cannot,
therefore, be the order of sounds in the scale which gives to
the music its pecuhar effect, or which determines Palestrina's
choice of harmonic successions. The effect is o^^•ing to the
nature of the harmonic successions themselves.

Fetis does not investigate the nature of the Minor Scale,
nor does he tell us how it is that while the major scale has
- but one form, the minor scale has three, nor why musicians
constantly " change the order of the sounds " of the minor
scale for themselves. Can it be that such changes are necessi-
tated by harmonic considerations, just as in earlier times the
Church composers changed the Bfc] of the Lydian Mode to
B|7, in order to obtain a better harmony, and raised the seventh
degree of the Dorian and IMixoh-dian ]\Iodes in order to obtain
a true Cadence ?

The " Laws of Tonality."

Fetis proceeds to explain what he calls the " laws of tonaUt}'. "
One of the principal laws of tonality is that certain degrees
of the scale have the character of notes of repose. The repose
which characterizes these notes is not, however, owing to
the arrangement of the sounds of the scale. This is owing to
harmony ! Only those degrees of the scale are notes of repose
which admit of the harmony of the Fifth. "The first, the
fourth and the fifth degrees of the scale are the only notes of
repose; they alone admit of the harmony of the Fifth." ^
Immediately afterwards (p. 23), Fetis tells us that the
" sixth degree also admits of this harmony." ^ It is not,

1 " La tonique, le quatrieme degre, et la dominante, sont les seules
notes de la gamme qui sont susceptibles de prendre le caractere de
repos : elles seules admettent Tharmonie de la quinte." (Traite, p. 22.)

2 " Le caractere de conclusion et de repos attache a cet accord lui
assigne une position sur la tonique, le quatrieme degre, la dominante
et le sixieme degre." {Ibid., p. 23.)

F. J. FETIS— THE LAWS OF TONALITY 341

however, a note of repose like the first, fourth and fifth degrees,
but is only a note of " equivocal repose." Whatever
degree of repose it possesses arises from the fact that
"in the tonality of C major, it represents the Tonic of
A minor." In such a case, one would imagine that this
sound then represented, not the sixth, but the first degree
of a scale. As for the third degree of the scale, " its tonal
character is absolutely antagonistic to every sense of repose."
The same is true of the second and seventh degrees of
the scale.

The reason why it is the harmony of the Fifth which deter-
mines the notes of repose in the scale is that this interval
alone (together with the Octave) " impresses the mind with
a perfect sense of tonalit}-, and at the same time produces
in us the sensation of repose, or of conclusion." ^ The Octave
and Fifth, therefore, are the onl}' intervals of repose. The
Thirds and Sixths do not convey the impression of repose :
for this reason they are called Imperfect Consonances ! ^
As for the Perfect Fourth, this is not in reality a perfect
Consonance, for it does not produce the impression
of repose. However it is not an Imperfect Consonance, Uke the
Thirds and Sixths, but should be described as a " Mixed
Consonance.

Concerning the augmented Fourth and the diminished
Fifth, Fetis gives utterance to the following extraordinary
remarks: — " Up to the present day," he says, " the Fourth
and the diminished Fifth have caused great embarrassment
to theorists. The majority have regarded them as dissonances,
but without being able to den}- that these dissonances are
of quite a different character from those of which we shaU
speak immediately. ... It is remarkable that these inter\^als
define modern tonality by means of the energetic tendencies
of their constituent sounds : the leading-note tending towards
the Tonic, and the fourth degree towards the Third. But
this character, which is eminently tonal, cannot constitute
a state of dissonance ; in reality the augmented Fourth
and the diminished Fifth are employed as consonances in
various harmonic successions."

^ Trait e, p. 7.

- " On leur donne le nom de consonnances iniparfaites, parce qu'elles
ne donnent pas le sentiment de repos." — {Ibid., p. 8.)

342 THE THEORY OF HARMONY

" The augmented Fourth and the diminished Fifth are
therefore consonances ; but consonances of a particular
kind, which I describe as ' Appellative.' " ^

The only " natural dissonance " in the scale is that formed
between the Dominant and the fourth degree. " In the
order of tonal unity, these two sounds alone possess the
faculty of forming a dissonance which can be taken without
preparation, and without a preceding consonance." The
faculty possessed by these sounds of forming a " natural
dissonance" is " the result of the arrangement of the notes
of the scale which, we observe, compose two tetrachords,"^
thus : —

ler Tetrachord. 2e Tetrachord.

-G»-

JdZ

" Disjonction des tetrachords :

choc des limites :

dissonance naturelle."

The only " natural fundamental chords " are the major
and minor Triads and the chord of the Dominant Seventh. ^
All other chords are formed from these, by means of the
alteration, substitution, and retardation of notes.

In all the works which have been written on the subject
of harmony, it would surely be difficult to meet with anything
more inadequate, contradictory, and one may say even
absurd, than Fetis's exposition of his much vaunted principle
of " tonality," and of what he calls " the laws of tonality."
He has defined tonality as " the harmonic and melodic
affinities of the sounds of the scale," resulting from the order
and disposition of their sounds. But he has only a vague

1 " II est remarkable que ces intervalles caracterisent la tonalite
moderne par les tendances energiques de leur deux notes constitutives,
la note sensible, appellant apres elle la tonique, et le quatrieme degre,
suivi en general du troisieme. Or ce caractere, eminemment tonal,
ne peut constituer un etat de dissonance : en realite, la quarte majeure
et la quinte mineure sont employees comme des consonnances dans
plusieurs successions harmoniques. La quarte majeure et la quinte
mineure sont done des consonnances," etc. (Ibid., pp. 8, 9.)

2 Ibid., pp. 17, 18.

^ " II n'y a d'accord naturel fondamental que I'accord parfait, et
celui de septieme de la dominante." — (Ibid., p. 251.)

F. J. FETIS— THE LAWS OF TONALITY 343

notion as to what the " laws of tonality " are, and how the
principle of tonality is to be appUed to the theory of harmony.
He is quite unable to explain the "harmonic affinities"
even of the sounds of the Tonic chord, as c-e-g. The Fifth
c-g is an interval of repose ; but the Third c-e, he tells us, is
an interval which banishes all sense of repose. It would
appear, therefore, that in a Final Cadence the concluding
Tonic chord cannot be regarded as a chord of repose. That is,
so long as this chord is complete. If the Third be omitted,
the chord is one of repose ; but if complete, it is not a chord
of repose. On the other hand, the principle of tonahty puts
us in possession of two new " consonances " — the Augmented
Fourth and the Diminished Fifth.

Again, certain notes of the scale produce in us the sensation
of repose. Fetis exhibits great uncertainty as to what these
notes are. At one time he definitely states that only the
first, fourth, and fifth degrees are notes of repose. At other
times he thinks that the sixth degree should also be included.
But it would appear from Fetis's version of the " rule of the
Octave," that the second degree as well is a note of repose,
for he places the " perfect " chord on this degree : — ^

-Gt-

'JOC

-<s>-

-jCZL

-,&-

-«&»-

"C3"

"O"

etc.

321

~jCil

In the first part of his work, however, he has stated that this
degree is not one of repose, and that the perfect chord placed
on this degree destroys " the character of the tonahty. "^ Only
those notes which admit of the harmony of the Fifth are
notes of repose. But what the Fifth has really to do with
the determination of these notes as notes of repose, it is
difficult to understand. Fetis, besides, has assured us that
the tonality of the scale is determined by the order of its
sounds. ,

As for the " consonance " of the diminished Fifth, Fetis
does not clearly explain the " attractive affinity " of this

1 Trailc, p. 85.

2 Ibid., p. 20.

344

THE THEORY OF HARMONY

consonance. He tells us that in this interval the lower
note should ascend a degree, and the upper note descend a
degree. In this case the fourth degree of the scale, which
is a note of repose, leaves its position of repose, and descends,
that is, presumably, resolves on the third degree, which
degree is " absolutely antagonistic to any sense of repose."
As for the explanation of the " natural " dissonance of the
Dominant Seventh, nothing need be said.

Fetis however gives another and quite a different explana-
tion of the nature of our tonalty. What constitutes our
modern tonaUty is not the order of the sounds of the scale,
nor the repose which characterizes certain of these sounds, but
the " attractive affinity " of the two sounds which form the
diminished Fifth, that is, the fourth and the seventh degrees
of the scale. 1 As Fetis considers this interval to be consonant,
and the fourth degree of the scale a note of repose, it is
impossible to understand why the fourth and seventh degrees
should possess any " attractive affinity " at all. But let us
suppose, what is really the case, that these sounds form a
dissonance with each other. In that case our modem tonality
would be determined, according to Fetis, by the necessity for
resohdng the dissonance existing between the fourth and the
seventh degrees of the scale : —

:cz:

-(S>-

-~,S(-

JZfZ

-G> iS>-

-O-

1 " Le rapport etabli dans ces harmonies entre le quatrieme degre
et le septieme du ton est le principe constitutif de la tonalite moderne :
on le chercherait en vain dans toute la musique composee anterieure-
ment a Monteverde et Marenzio : il n'y existe pas : il n'y pouvait
exister sans aneantir la tonalite du plain-chant. L'attraction de ces
deux notes, la necessite de faire monter le septieme degre pendant que
le quatrieme descend, est le caractere propre de la note sensible. . . .
Toute la tonalite moderne repose done sur cette succession ; —

:8-

inconnue a tous les musiciens jusqu'a la fin du seizieme siecle." — Esqtiisse
de Phist. de Pkarm. Art. Monteverde.

F. T. FETIS— THE LAWS OF TONALITY 345

With regard to the resolutions of the dissonance at (a),
Fetis considers that such a resolution estabhshes C as
the Tonic of C major, and E as its Third. But as both
the sounds which form this dissonance occur in the scale
of A minor, why should C not be regarded as the
Third of the Tonic chord of A minor, and E as the Fifth,
or must such a succession of these sounds of the scale of
A minor be considered to destroy the tonality of A minor,
and estabhsh that of C major ? Besides, this dissonance is
susceptible of other resolutions than the one and only resolu-
tion given by Fetis {b). The resolution in which the sound
F remains stationary would, according to Fetis, appear to
be a better resolution than that in which it descends
a degree, seeing that F, the fourth degree, is a note of
repose.

It cannot therefore be to dissonance and the necessity
for its resolution that we owe our present tonality. This
result might have been expected : for our major key system
admits of the clearest possible definition by means of the
three consonant major triads of the key-system. \\'hen Fetis
speaks of the attractive affinity of the sounds of the diminished
Fifth, he imagines he is dealing \\dth melody only ; in reality,
he is deahng with harmony. What Fetis asks us to beheve
is that the melodic tendencies of two sounds determine the
tonahty of the major scale ; at the same time we are to
consider that it is the tonaUtv, the order of the sounds of
the major scale, which gives to these two sounds their
melodic tendencies !

MOXTEVERDE AND THE ChORD OF THE DOMINANT SEVENTH.

Fetis asserts that the change from the old harmonic art of
the Church composers to that of the present da3^ which
is generally supposed to have been effected about the end of
the sixteenth and beginning of the seventeenth centuries, was
brought about by ^lonteverde's employment, in one of his
madrigals, of the chord of the Dominant Seventh.^ The

^ Esqnisse de I'hisi. de I'harm. — Art. JfofUeve>'de. — Traiti, Book III.,
Ch. 2.

346

THE THEORY OF HARMONY

passage in which this chord occurs is quoted by Fetis,
thus : — -

-o-

gizsi^

ZZSL

;g=

-«s»-

-^ C3-

~c?-

I I

2__L— ?^;

-&—r-&-

-<s»-

-/S>-

-<Si<-

?:::j^

:^22.

.qL j?S>.

-o-

-<!5>-

One need not dwell on the fact that in this passage ^lonteverde
employs harmonic combinations and successions much more
astonishing than that of the chord of the Dominant Seventh ; it
is sufficient to note that Fetis finds in the chord at * and its pro-
gression— to the chord of % — the cause and explanation of that
musical revolution which has brought about oiur modem art
of harmony. Did Fetis, learned historian as he was, really
believe that such a change was brought about in such a
manner ? Did he really consider that while every other
transformation that has been effected in the art of music has
been the result of a slow and gradual development, the
greatest change of all, that from the old to the new world of
music, presented the sole exception to this law of development.
Fetis is by no means certain ; he is quite unable to make up
his mind as to whether it is the chord of the Dominant
Seventh which has determined " our tonality " or whether,
on the contrary, it is '' our tonalit}- " which has determined
for the chord of the Dominant Seventh its harmonic and
theoretical significance.

If the former were really' the case, we should expect to find
]\Ionteverde, his contemporaries, and immediate successors,
employing the chord of the Dominant Seventh at the Tonic
cadences, and especially at the Final Cadence, where above
all places it was necessary clearly to define, and firmly to

F. J. FETIS— r/ZE LAWS OF TONALITY 347

establish, the new tonality. This however is not borne out
by the facts. During the first half of the seventeenth century
scarcely a single composer makes use, for the Final Cadence, of
anything but the consonant Dominant harmony, followed
by that of the Tonic. Even Lully and Alessandro Scarlatti
use very seldom anything more ; although with Scarlatti
at least other chords of the Seventh than that on the Dominant
are frequent enough.

Unfortunately tlie F^etis legend regarding Monteverdc and
the chord of the Dominant Seventh has passed into innumer-
able text-books on harmony and histories of music, and has
become almost an article of faith among musicians. Even
Helmholtz repeats it. It has been considered also that Fetis,
in his remarks on this chord, was referring to the harmonic
progression of the Perfect Cadence. But this is not the case.
Fetis was referring to the melodic tendencies of the fourth
and seventh degrees of the scale, both of which find a place
in the chord of the Dominant Seventh. Fetis quite rightly
recognizes that the chord of the Dominant Seventh is of
theoretical importance. But he utterly fails to find the
true explanation of it, or indeed any reasonable explanation.

Fetis considers that harmony has its roots in melody and
arises from it, although he cannot explain how this is brought
about. Nevertheless, he frequently speaks of the " natural "
major and minor harmonies, and of the natural harmony of
the Dominant Seventh. What exactly Fetis means by this use
of the term " natural " may be ascertained from a statement
he makes in the course of his analysis of the theory of
L. Euler : a statement which must appear extraordinary when
one remembers the ridicule which Fetis constantly pours on
all theorists who make use of acoustical phenomena.

The Chord of the Dominant Seventh
A " Natural Discord."

He quotes Euler's remarks that up to the present time
musicians and theorists have not gone further than the
senary division of the monochord for the generation of the
consonances and of harmony, and proceeds: — "This principle,
which is still that of several theorists and geometricians, has
been rejected by Euler in his Memoir c entitled Hypothesis

348 THE THEORY OF HARMONY

as to the origin of some dissonances commonly accepted in
harmony (1764). This Memoir e aims at the discovery of
the principles of the rational construction of the chords of the
Dominant Seventh, sol-si-re-fa, and of the fifth and sixth,
fa-la-ut-re. After ha\dng remarked that the character oi
the chord sol-si-re-fa consists in the relationship of si,
expressed by the number 45, with fa, represented by the
number 64/ he remarks that this last number undergoes
a modification, owing to the attractive affinity of this
interval ; and he adds that the ear substitutes 63 for 64, so
that all the numbers of the chord are divisible by 9, and in
hstening to the sounds sol-si-re-fa, represented by the numbers
36 : 45 : 54 : 64, the ear really understands 36 : 45 ^ 54 • 63.
which, reduced to their simplest terms, give 4:5:6:7.

Fetis continues : — "It is necessary to do justice to this
great man . . . the philosophy of music owes to him, in the
passage of the Memoire from which I have just quoted, a
truth as irrefragable as it is new. He has been the first to
see that the character of modern music resides in the chord
of the Dominant Seventh, and that its determining ratio
{rapport constitutive) is that of the number 7." ^

Fetis, then, after having ostentatiously rejected all acoustical
phenomena, not only follows Rameau in deriving harmony
from the sounds of the harmonic series, but goes one better,
in making use of the number 7, ^^dth which Rameau would
have nothing to do. He has defined the chord of the Dominant
Seventh as " the only natural dissonant chord," and as
the chord which has determined our tonaUty. It is the only
natural dissonant chord because it alone, of all dissonant
chords, admits of being taken \\dthout preparation. Fetis
considers it necessary to advance some reason for this, and
the explanation he gives is that this chord is derived from
those sounds of the harmonic series represented by the
numbers 4:5:6:7. Yet he knows quite weU that this
" natural 7 " is not the real fourth degree of the scale, and
he constantly ridicules other theorists who make use of it.

Fetis borrows from Sorge his explanation of the origin
of the Dominant Seventh chord. He follows Rameau in
regarding the first, fourth, and fifth degrees of the scale
as the determining notes of the key-system. He is of

1 Esquisse de I'hist. de I'harm. — Art. Eitlet-.

F. J. FETIS— THE LAWS OF TONALITY 349

opinion that all theorists before his time have failed to
discover the true explanation of the theoretical significance
of these sounds. Whether this be so or not, it. is certain
that this problem of " tonahty " is one which completely
baffles Fetis. Fetis invites us to consider that the order of
the sounds of the scale has determined " our harmony " :
that this is so is proved by the fact that it is " our harmony,"
and especially that of the natural Seventh, which has deter-
mined the order of the sounds of the scale !

Chord Relationship and Succession.

According to Fetis, the chief defect of Rameau's system
is that the chords he generates appear as isolated chords,
existing \\ithout inner connection. But chords, he quite
rightly argues, are in harmony more or less closely related
to each other, and one of the principal difficulties of harmony
is to explain the nature of this harmonic relationship and
the laws of harmonic succession. It is in connection with
these difficulties that Fetis has led us to expect the principle
of tonality to be most productive of theoretical results. It
is just here, however, that this principle appears to be most
barren of results. Fetis is quite unable to explain chord
succession, nor has he any adequate explanation to offer
of the nature of harmonic relationship, even such a close
and direct relationship as that existing between a Tonic
and its Dominant, between Tonic and Dominant harmonies,
and between Tonic and Dominant ke\'s. It is impossible
for Fetis to maintain that his principle of " tonality "
affords any adequate explanation of harmonic relationship
or of the principles which lie at the root of harmonic
succession.

He, however, accepts Rameau's theory of harmonic
inversion. Tliis part of Rameau's theory he describes as
" a stroke of genius." But he does not appear to reahse
aU that it implies. He rejects Rameau's theory of the fun-
damental note, but nevertheless considers himself at liberty
to speak of " fundamental " and even " natural " chords,
and to make use of Rameau's theory of inversion for his own
theory of harmony. But how then is Fetis able to determine
that, for example, the Fourth is an inverted Fifth ? May

350 THE THEORY OF HARMONY

not the Fifth be an inverted Fourth ? As he regards /, in
the key of c, not as the Fifth below c but as the Fourth above
it, should not c-f in this case be regarded as a fundamental
interval, and f-c, the Fifth, as its inversion ? For the same
reason, should not the chord f-a-c be regarded as the inversion
of the chord c-f -a ? F'etis no doubt would object that this
is not in accordance with the principles of " tonahty " —
an explanation which might be illuminating if Fetis could
inform us what the principles of " tonality " really are.

But, it might be urged, seeing that the Fifth is a more
perfect consonance than the Fourth, the Fifth ought to be
regarded as the original and fundamental interval, and the
Fourth its inversion. But this does not follow. " Tonality "
is a somewhat hazardous foundation on which to build up
a theory of inverted chords, and a somewhat uncertain means
of determining whether chords are inverted or fundamental.
But without the theory of inverted chords no theory of
harmony is possible.

Again, Fetis looks on the scale as consisting of an ascending
series of sounds. He merely assumes, however, that this
is so in reahty. It is certain that this was not the sense in
which the scale was originally understood. The Greek
conception of the scale was that of a descending series of
sounds. As Fetis is of opinion that all music and harmony
have their origin in scales, he might have been expected
to adopt the view of the Greeks, which is the historically
correct view. At the same time, it would have been necessary
for him to point out that the modern theory and practice
of harmony are based on a misconception as to the real
nature of the scale.

" Altered " and " Chromatically Altered " Chords.

A great part of his Traiie is devoted by F6tis to the explana-
tion of the various ways in which " fundamental " chords
may be modified. He here develops Kirnberger's theory of
the modification of fundamental chords by means of the
prolongation (suspension), substitution, and alteration of notes
of a chord. For example, in the first inversion of the chord
of the Dominant Seventh b-d-f-g, the note a may be substituted

F. J. FETIS— THE LAWS OF TON A LI TV

for g, and the chord may appear as h-d-f-a. Again, the tonic
chord c-e-g may appear as a chord with chromatically altered
Fifth, thus, c-e-g^. Here, of course, the question arises — how
much alteration may a Tonic chord undergo before it ceases
to be a Tonic chord ? May not c-e\}-g^ be also regarded as
an altered Tonic chord ? Fetis himself gives examples of
chords in which as many as three of the original sounds
of the chord are chromatically altered. Thus in the
following : —

Z^~

-G>'-

-e^-

i&-

iraz:

he explains the second chord as derived from the first ; that
is, it represents a chromatically altered form of the harmony
f-a-c-d. One may assert on the contrary that the second
chord does not at all represent the harmony f-a-c-d, but an
altogether different harmony. If Fetis holds that the chromatic
alteration of the first chord does not change the harmonic
meaning and significance of the chord, he is plainly in error.
If, on the other hand, he considers that such an alteration
does actuaUy change the nature of the chord, he has not
considered it necessary to explain the nature of this change.
But as Fetis considers that harmony arises from melody, why
should he not consider the second chord to be an entirely
independent harmony, representing nothing but itself ?
Anything more ill-considered, more inadequate than F6tis's
" metaphysical " theory of harmony based on the principle
of tonality which he himself does not understand, and
is unable to explain, it would be difficult to conceive.

352 THE THEORY OF HARMONY

CHAPTER XII.

HAUPTMANN ; HELMHOLTZ ; OTTINGEN ; RIEMANN, ETC.

MoRiTZ Hauptmann.

Within a few years of the publication of the Traite de
I'harmonie of Fetis, there appeared the remarkable work
by Moritz Hauptmann — Natitr der Harmonik und der
Metrik (1853), undoubtedly one of the most important
and valuable works on harmony which we possess.
Hauptmann's musical insight, sound musical judgment, and
clear discernment of harmonic facts, have been surpassed by
no other theorist. The examination of the various existing
systems of harmony appears to have convinced Hauptmann
of the inadequacy of acoustical phenomena or of mathematical
proportions and progressions as a basis for the theory of
harmony. In the Introduction to his work he remarks : —
" It has always been the custom to begin text-books of
Thorough-bass and Composition with an acoustical chapter,
in which the relations of the intervals were set out by the
number of the vibrations or length of the strings." After
a reference to the famihar process of chord-formation by
means of sounds selected from the harmonic series, and the
necessity for the modification of the natural sounds so obtained,
he proceeds: — "Of the theory which seeks to trace the reason
of all harmony in the so-called partial tones, it need only
be remarked that even if the third and fifth partial tones
are those most distinctly heard, nevertheless the other sounds
of the harmonic series, indeed of the infinite harmonic
series, must equally be regarded as partial tones, and as
constituents of the fundamental or ground-tone ; for example,
the seventh and ninth partial tones may frequently be
quite distinctly heard. . . .

" We may therefore disregard this partial-tone theory, as
well as that other theory which supposes that the key to

M. HAUPTMANN— T//£ NATURE OF HARMONY 353

harmony is to be found in the continued arithmetical series,
a theory which is both untrue to fact and in disagreement
with what is musicaUy natural."

Like Rameau and Tartini, Hauptmann is convinced that
there exists but a single original and fundamental chord,
from which all other chords are derived, namely, the major
harmony. The minor harmony, which is as truly a harmonic
unity as the major, is an inverted major harmony: —

Major triad. Minor triad.

^^EKt

Major Minor Major Minor
3rd. 3rd. 3rd. 3rd.

This being so, it is clear that there are but three intervals
which are "directly intelligible," namely, the Octave, Perfect
Fifth, and Major Third. In the major triad c-e-g, the major
third c-e determines the minor third e-g. The minor third
is not a " directly intelligible " interval. The Octave, Fifth,
and Major Third are the sole positive constitutive elements
of harmony.

If it is from this triad of sounds and of intervals that all
chords are derived, it is from a " triad of triads " — the
Tonic, Dominant, and Subdominant triads — that our key-
system is derived and by means of which it is determined.
In the treatment of dissonant chords, Hauptmann also
discerned a three-fold process : — First, we have the consonant
triad or harmonic unity {Preparation), next, the state of
opposition created by the clashing of the dissonant harmonic
elements {Percussion or Suspension), lastly the removal of
these opposing elements, or reconcihation in a fresh unity
{Resolution) .

These and other similar facts relating to harmony led
Hauptmann to the beUef that the princple from which har-
mony proceeds, which underUes all music and which renders it
universally intelligible, must be a metaphysical principle.
His reflections on the aesthetic side- of his art confirmed him
in this behef. " Although," he remarks, " the contents
of the complicated work of art may make it difficult to be
understood, nevertheless the means of expression are always

354 THE THEORY OF HARMONY

the same, and singly are intelligible universally. . . . The
triad is consonant for the uneducated as well as for the
educated : the dissonance needs to be resolved for the unskilled
as well as for the musician ; discordance is for every ear
something meaningless. . . . That which is musically right,
correct, addresses us as being humanly intelligible. . . . That
which is musically inadmissible is not so because it is against
a rule determined by musicians, but because it is against
a natural law given to musicians from mankind ; because
it is logically untrue and of inward contradiction. A musical
fault is a logical fault, a fault for the general sense of mankind,
and not for a musical sense in particular." These significant
remarks might in themselves be held to prove Hauptmann's
worth as a great theorist and musician.

Hauptmann then concludes, as Rameau also concluded
after his own fashion, that the principle on which music
is based must be a principle which operates everywhere,
in the simplest as well as in the most complicated work of
musical art, and not only in harmony but in melody and
rhythm as well. He therefore, as is known, gives to his theory
of harmony a metaphysical basis, the principle of which he
borrows from Hegel and which he enunciates thus : — " Unity,
with the opposite of itself, and the removal of the opposite,"
or (i) Unity ; (2) Duahty or separation, and (3) Union. It
may be at once remarked that the dialectical method pursued
by Hauptmann, applied as it is for a scientific purpose,
is altogether unsuitable and inadequate. Evidently one
of the principal difficulties of such a method is to determine
exactly the premises from which the inference or conclusion
has to be drawn.

Octave, Fifth, and Major Third the only
" DIRECTLY Intelligible Intervals."

One of Hauptmann's first tasks is to explain the major
harmony ; and it is somewhat surprising to find that no
sooner has he begun the exposition of his theory than he
conducts us into the now familiar region of acoustical
phenomena {Major Triad). In demonstrating that the

M. HAUPTMANN— THE NATURE OF HARMONY 35 5

only " directly intelligible " intervals are the Octave, Fifth,
and Major Third, he proceeds thus : —

The Octave ^ is the expression for Unity ; the Fifth
expresses Diiality or separation (f of a string is heard
against the ground-tone) ; the Third, Unity of Duality
or Union (i of the string is heard against the ground-tone).
The Third is the union of Octave and Fifth.

The unifpng property of the Third, Hauptmann demon-
strates thus : — " The Third: the interval in which a sounding
quantitv of four-fifths is heard with the ground-tone. Here,

^ " The Octave : the interval in which the half of the sounding
quantity makes itself heard against the whole of the ground-tone, is,
in acoustical determination, the expression for the notion of Identity,
Unity, and Equality with self. The half determines an equal to itself
as other half."

" The Fifth : the interval in which a sounding quantity of two-
thirds is heard against the ground-tone as a whole, contains acoustically
the determination that something is divided within itself, and thereby
the notion of duality and inner opposition. As the half places outside
itself an equal to itself, so the quantit\' of t%vo third-parts, heard mth
the whole, determines the third third-part ; a quantity to which that
actually given appears a thing doubled, or in opposition with itself."
Harmony and Metre (Major Triad).

It is unfortunate that Hauptmann should find it necessary for bis
argument to make use of two kinds of acoustical determination for
the Octave, as weU as for the Fifth. He first expresses the Octave as
1 : I, which is the correct acoustical determination. But he finds it
necessar}', in order that the Octave may be understood as Unity, to
give it quite a different determination, namely 1 ; i This, however,
does not express the Octave, but the Unison.

Similarly for the Fifth, which is first expressed as | : -|, but which
represents Duality' only if understood as |- ; 1 This, however, is the
expression for the octave, which, Hauptmann assures us, represents
Identity, Unitj'.

As, according to Hauptmann, Duality is " a thing doubled," then
the Octave must be the constant expression for DuaUty, for the
Octave is acoustically determined as 1:2.

It is evident that Hauptmann, by comparing the true acoustical
determinations of the Octave (i : 2 or 1 ; i) and Fifth (2:3 or ^ ; 3 \
might have arrived at quite different results. He might also have
followed Rameau in deriving the Octave, Fifth, and Third from the
harmonic sounds of the sonorous body, represented by the numbers
1 : 1 : 1. Hauptmann, however, does not accept these natural
determinations. If he did, his argument would fall to pieces. But
it is, in short, impossible to demonstrate the facts of harmonic science
by the Hauptmann system of dialectics.

356 THE THEORY OF HARMONY

the quantity determined is the iifth fifth-part, of which that
given is the quadruple, that is, twice the double. In the
quantitative determination of twice two, since the double is
here taken together as unity in the multiplicand, and at the
same time held apart as duality in the multipher, is contained
the notion of the identification of opposites, of Duahty as
Unity." This is surely the most extraordinary explanation
ever advanced to account for the consonance of the Fifth.
As the Third, in itself, represents a " unity of duality, or
union," therefore it renders the Fifth consonant !

So then, concludes Hauptmann, " the conditions of the
idea or conception of Consonance are completely fulfilled
in the sound combination, Ground-tone, Fifth, Third."
Rather, Hauptmann completely fails to give any adequate
or correct idea as to the real meaning of these intervals for
hannony, of the varying degrees of consonance wliich they
express, and by which they are differentiated. ( (i) Unison ;
(2) Octave; (3) Fifth; the Third = an Imperfect Consonance.)

It is plain that the Hauptmann system of dialectics appUed
to acoustical determinations may be made to produce almost
any result. One can only regret that so much ingenuity
should have been expended on the attempt to prove what
is plainly in entire contradiction with the facts. According
to Hauptmann, the Fifth in itself must be regarded as virtually
a dissonance. It may, however, become a consonance, but
this can only be effected by the mediation of the Third !
Hauptmann might have considered that nearly all ancient
peoples, to whom the Third as a consonance was unknowTi,
nevertheless regarded and described the Fifth as a consonance.
One would naturally have expected Hauptmann to postulate
Hke Zarlino, Rameau, and other theorists, the fundamental
sound, rather than the Octave, as Unity. In that case, however,
the Octave might have appeared as Duality, and the Fifth
as the uniting element.

The principle laid down at the outset by Hauptmann, on
which his whole theory is based, that the Octave, Fifth,
and Major Third are the only " directly intelligible " intervals,
has been hailed especially by German theorists as a notable
and astonishing achievement, which marks a new epoch in
harmonic science. But it was Rameau who, following
Descartes, first clearly showed that the Octave, Fifth and
Major Third are the only consonances employed in music

M. HAUPTMANN— THE NA TURE OF HARMON Y 357

which are directly intelligible in the sense that they alone
arise directly from, and are directly related to, the
fundamental note. The other intervals were " derived "
from these three. Rameau took this as his starting point,
developed from it his theory of Harmonic Inversion, and
rightly insisted that it formed the only possible basis for
such a theory, and consequently for any rational theory of
harmony. Hauptmann's acumen as a theorist is evidenced
by his recognition of the necessity which existed to pro\-e
at the outset that the Octave, Fifth and Major Third are,
in Rameau's language, " fundamental," and not " derived "
intervals. But while Hauptmann fails to prove this fact,
Rameau demonstrates its truth in the most complete and
con\dncing way. According to Hauptmann, the Fifth
represents " duaUty, inner opposition " ; nevertheless, he
considers it to be a " directly intelHgible interval." It is
strange that Hauptmann, who found himself obhged to call
in the aid of acoustical phenomena in order to find a firm basis
for his " metaphysical " theory of harmony, should neverthe-
less have rejected Rameau's method ; but in rejecting it,
he rejects the only means wherebj^ the inter\'als of the
Octave, Fifth, and Major Third can be estabhshed as
" fundamental," or " directly intelligible."

The Key-System.

Hauptmann proceeds : — " In the notion of the unity of
the three elements of the triad there is contained, in brief,
all determination which underhes the understanding, not
only of chords as the simultaneous union of notes, but also
of melodic progression and succession of chords." As already
indicated, Hauptmann finds in the primary Triads of Tonic,
Dominant, and Subdominant (" Unity of a triad of triads ")
the complete means for the determination of Key. In order
to help out his argument, he finds himself obhged to bring
in two new Conceptions, namely, that of Having, and
that of Being {" having " a Dominant, and " being "
a Dominant). After some laborious reasoning, Hauptmann
decides " not to weary the reader with too abstract concep-
tions," and presents to us the two triads which are, he teUs us,
and which we know to be in reality, in opposition with each

3 58 THE THEORY OF HARMONY

other, although the opposition is not that of the Fifth, namely,

the triads of Dominant and Subdominant, -p ^

F — a — C

and ^ 7 p) (Here I = Ground-tone ; n = Fifth; 111 =

Third).

How, then, do these triads arise ? Both are derived from

the Tonic triad = ^ P ' G changes its character as

Fifth, and becomes ground-tone = /- 7 j^ : while C

changes its character as ground-tone, and becomes Fifth

= 2, ^^^ ]J- In the Subdominant triad, therefore, C,
F — a — C.

the Tonic and central note of the whole key-system, appears
as Fifth of F. As Hauptmann reckons intervals upwards,
he evidently does not feel justified in describing F as Fifth
of C. But while the two triads, Dominant and Subdominant,
are certainly in opposition with each other, one looks in
vain for any Fifth connection or rather opposition between
them. Nevertheless, asserts Hauptmann, these two triads
represent Duality, that is, the Fifth. They can only be
reconciled and their opposition removed by the mediation
of the Tonic triad, which then appears as the uniting Third
element.

Another question remains to be decided. Which of the
two triads. Dominant and Subdominant, represents Unity,
and which Duality ? It is the latter which, according to
Hauptmann, represents Unity (I), and the former, DuaUty
(II) ; although how he arrives at this result it is difficult
to discover. The complete key-system, therefore, appears
thus :- I ^ m^ II ^

F— a— C— e— G— b^D ^

This is the formula given by Hauptmann himself. The Sub-
dominant triad appears as the root of the whole key-system.
As for the Tonic triad, this represents not only the original
Unity, but is itself the uniting Third element. This is quite
a different result from that of the original major triad.
But this is not a complete account of the extraordinary

^ Harmony and Metre, " Major Key."

M. HAUPTMANN— r///i NATURE OF HARMONY 359

metamorphosis which the Tonic chord has to undergo before
the key-system can become estabHshed. For first it is Unity = I ;
then it is Duahty = 11 (being Dominant of the Subdominant
triad) ; lastly it is the uniting Third element = III.

Surely no theorist was ever in greater straits than Haupt-
mann in his attempt to explain the key-system by means of
this organic " life-" or " world-process " which he conceives
to be the simple and universally intelligible fundamental
principle of all music.

If Hauptmann would but carry out strictly his own philo-
sophical principle, his course is perfectly plain. If the
original Tonic C-d?-G represent the fundamental Unity,
then G-6-D must represent the Fifth duality, for G is Fifth
of C ; and the triad E-o-f-B will represent the mediating
triad, for E is the Third. So then we get the perfectly
logical system : —

T "^ TT

C— t^— C; ^ ' G—b—l)

or rather, the system which logically results from the strict
carrying out of Hauptmann's philosophical principles. Bu-I
so far as the ear is concerned, if it may be left to the ear to
decide anything relating to a musical system based on
Hegehan metaphysics, there does not appear to be much unity
in this " triad of triads." But one must not blame Hauptmann
for being a better musician than a philosopher !

For Hauptmann, then, as for Rameau, the scale is harmoni-
cally determined ; that is, each note of the scale is derived
from one or the other of the three determining chords. It is
harmony which determines melody and melodic succession.
The melodic passage C-^-G is harmonically determined. Not
less is the succession C-b or C-D, for b is Third, and D is Fifth
of the Dominant triad G-b-B. Hauptmann expressly states : —
" No melodic note can receive definiteness otherwise than
it is conceived as Ground-tone, Third, or Fifth of a triad. " ^ Thus
are determined " the sixth [degree of the scale] as Third of
the Subdominant ; the seventh, as Third of the Dominant ;
the eighth, as Octave of the Tonic." -

1 Harmony and Metre, " Passing-notes."
* Ibid., "Scale of the Major Key."

36o THE THEORY OF HARMONY

But while Hauptmann explains C-T>-e, the first three degrees
of the scale of C major, as determined by a Tonic-Dominant
succession of chords, and e-F-G-a by a Tonic-Subdominant
succession, the sounds a-b-C — the sixth, seventh, and eighth
degrees of the scale — he considers to be determined by a
Submediant-Mediant hannonic succession : —

" Thus the whole scale is formed : in its first, second, and
third degrees, on the Fifth ; in its fourth, fifth and sixth
degrees, on the Ground-tone ; in its sixth, seventh and eighth
degrees, on the Third of the Tonic." ^ Hauptmann's satis-
faction \\dth so symmetrical an arrangement appears to have
caused him to overlook the decided contradiction imphed
in this double determination of the sixth, seventh, and eighth
degrees of the scale. The scale concludes ^ith a minor
harmony ; there is no real close or cadence between leading
note and Tonic, while the latter part of the scale is in the
key of A minor, rather than that of C major. Not three
but five " harmonic unities " are necessary for the determina-
tion of the scale-succession. Hauptmann experiences the
same difficulty as Rameau in this part of the scale — a as
Third of the Subdominant, and b as Third of the Dominant
triad cannot succeed one another. For this it would be
necessary that the two disjunct triads Subdominant-Dominant
should succeed one another immediately. Such a succession,
however, would be unintelHgible, for there is no " common
element," no connection between the two triads.^ It would
appear, then, that for the determination of the sounds of
the major key-system, a " triad of triads " is insufficient.
Other " harmonic unities " than those on the Tonic,
Dominant, and Subdominant are necessary, namely, the
triads on the Mediant and Submediant.

Harmony and Metre, " Scale of the Major Key." '^ Ibid.

M. HAUPTMANN— r//Zi NATURE OF HARMONY 361

Secondary Triads of the Key-System.

But where does Hauptmann discover the two minor triads
which he has been obHged to introduce for the harmonization
of the scale ? He explains the matter thus. Between each
pair of major triads there exists a minor one. Thus, between
Tonic and Dominant triads we find the minor triad on the

Mediant : C — e — G — h — D, while between Subdominant and

Tonic triads we find the minor triad on the Submediant :

F — a — C — 6 — G- Two other triads may be derived from

the scale by a process of joining together the limits of

the key-system, thus: D|F — a — C — <?— G — b — D|F. These

■^ ^^ ^- '

triads are D/F-a and b-D/F. Both are dissonant ; both

" have a duahty of basis," and, properly speaking, they

are not triads at all. Hauptmann is truer to fact in his

treatment of these chords than many of his predecessors :

D — a, as well as b — F, are not perfect, but diminished Fifths ;

and both triads are " diminished triads."

But, one would imagine, the minor triads e-G-b, and
a-C-e, the Mediant and Submediant triads of C major, although
their Fifths are perfect, have nevertheless likewise a " duality
of basis," and should therefore be regarded as dissonant
triads. Thus in the Mediant triad e-G-b, e is Third of the
Tonic triad C-g-G, while G — b represent the Ground-tone and
Third respectively of the Dominant triad G-&-D. But another
explanation is possible for this triad. For e-G may be regarded
as Third and Fifth respectively of the Tonic triad, and b as
Third of the Dominant triad. Similarly for the Submediant
triad a-C-e, which may likewise represent a duality.

Hauptmann, however, does not take this view. Each of
the triads in question he regards as a harmonic unity. In
the triad a-C-e, e, he states,^ is Fifth of a ; and in the triad
e-G-b, b is Fifth of e. So also in the chord succession C-e-G —
C-e-a, we pass from one to another hannonic unity ; such a
succession is " only inteUigible in so far as both can be referred
to a common element which changes meaning during the

^ Harmony and Metre, " Scale of Major Key."

362 THE THEORY OF HARMONY

passage." ^ Here the common element which changes meaning
consists of the sounds C-e, common to both chords. That is, e
changes its meanmg as Third and becomes Fifth, while C
changes its meaning as root, and becomes Third. Why then
does not Hauptmann give to the second chord its proper
notation ? ^

It is remarkable that he should employ the wrong notation
for the Mediant and Submediant triads. Thus the Submediant
triad he designates as a-C-e, although he expressly states that
e is Fifth of a. C, then, is the Third of the chord. These
triads, therefore, should have the notation A-c-E, and E-|;-B.
But in such a case we find five degrees of the scale not only
doubly determined, but with their original meanings entirely
reversed. Hauptmann's difficulty with regard to the notation
of these triads can therefore be understood.

The key-system, Hauptmann points out, may be shifted
slightly upwards without inducing a change of Mode. Suppose,

he remarks, we shift the key-system Y—a—C—s—G—b—D,

a little in an upward direction, thus : a — C — ^ — G — &— D— /#,
we must not imagine that the introduction of the note /#
necessarily implies a modulation to the key of G major.
For here G has not full Tonic meaning : for the first key-
system has only given up F, not a as well. On the other
hand, if we shift the same key-system downwards so as to

include BJ? thus : — Bb— ^— F— «— C— ^— G, we have a real
modulation to F major. Here the chord F-a-C appears as
central Tonic chord. The manner in which Hauptmann
explains a modulation to the Dominant key is, then, apparent.
He does not attempt to show that the relationship between
the two keys can be established only through the mediation
of the key of the Third — the Mediant E major.

^ Harmony and Metre. " Chord Succession."

* As is known, Hauptmann devised a new method of designating
the harmonic triad ; as he justly remarks, theorists have not been
careful enough to distinguish in the notation employed, between the
Third-meaning and Fifth-meaning of a sound : Thus e, the harmonic
Third of the triad C-e-G, is quite a different sound from E, the fourth
Fifth of C. This E is the Pythagorean, and not the true harmonic
Third of C. While e then, has Third-meaning, E has Fifth-meaning,
and this distinction must be carefully observed. For Ground-tone
and Fifth Hauptmann therefore makes use of capital letters, and for
Thirds, small letters.

M. HAUPTMANN— T/f£: NATURE OF HARMONY 365

Origin of Discords : Diminished Triads and
Chord of the " Added Sixth."

In Hauptmann we meet with what, at first sight, appears
to be the " double employment " of the Seventh chord on
the Supertonic. He presents us with the two chords D/F-a-C,
and ^-F-a-C. These chords differ from one another. In
the first chord, D is the Fifth of the Dominant triad G-6-D ;
in the second, d is the Third of 6(7. " The chord on the
Fifth of the Dominant of the major key D/F-a," he remarks,
" must not be confounded with the minor triad djF-a, which,
transgressing the lower limit of the C major key-system, is
formed upon the Third of F>\}, with ground-tone and Third
of the major triad of F." ^

This can mean nothing but that the triad d-F-a cannot
belong to the C major key-system. The chord of the Seventh
on the Supertonic of C major can therefore assume only one
form, namely, DjF-a-C ; the other chord d-F-a-Q must of
necessity be that on the sixth degree of the scale of F major.
There can, therefore, be no " double employment " of the
chord of the Supertonic Seventh in C major, or indeed in
any key.

Hauptmann's explanation of this important chord D/F-a-C
differs from that given by Rameau. It will be remembered
that Rameau considered this chord to be formed by the
addition of a Third below the Subdominant harmony. It is
true that he also explained it as arising from the addition of a
Sixth above the Subdominant harmony, and that he regarded
this chord of the Added Sixth F-a-C/D as an original chord.
In both cases, however, he insisted that the fundamental
harmony was that of the Subdominant, F-a-C Hauptmann,
on the other hand, can comprehend the chord of the Seventh —
all chords of the Seventh — only as a triad-duality. " The
chord of the Seventh is the sounding together of two triads

joined by a common interval " ^ P^ ^ ^ ^ p) ....

" Only those triads which have a harmonic unity, that is a
common interval, can be taken together at one time ; there-
fore only two triads which are related in two notes."

^ Harmony and Metre, "Diminished Triads."
- Ibid., " Chord of the Seventh."

364 THE THEORY OF HARMONY

By means of the joining together of a major with a minor
triad we obtain in C major the following chords of the Seventh :

C-e-G-h, e-G-b-D, F-a-C-e, and a-G-e-G. But there are
three other important chords of the Seventh distinguished

by Hauptmann, namely, D/F-a-C, G-b-DjY, and h-Dj¥-a.
These chords are more important than any other chord of the
Seventh ; for, as they contain the interval D/Fwhich represents
the joining together of the Hmit» of the key-system, they are
of the greatest possible value for defining the key. How, then,
are these three important chords formed ? The first chord
contains the diminished triad D/F-a ; the second, the
diminished triad 6-D/F, while in the third chord we find both
diminished triads. These triads are not harmonic unities ;
Hauptmann has rightly pointed out that they cannot properly
be regarded as triads at all. He now finds it necessary to
contradict his former statement, for he is quite unable to
account for the formation of the three most important Seventh
chords of the key-s\-stem except by explaining these diminished
triads as harmonic unities.

" The diminished triads," he states, " must also be regarded
as organic chord-fomiations. The chords of the Seventh
G-6-D/F, h-DjY-a, D/F-a-C, although the hue of separation
indicates the derivation of their elements from the Dominant
and Subdominant triads, are none the less estabhshed as
combinations of triads. The chord G-&-D/F cannot have
organic meaning as a union of the Dominant triad with the
Subdominant ground-tone, nor the chord D/F-a-C as a union
of the Dominant Fifth with the Subdominant triad. Only
things of hke kind can be united. With the triad only the
triad can enter into union, but not the single chord-element,
the solitary note."i Hauptmann must have been in great
straits when he found himself obliged to explain the
diminished triad b-DIF as an "organic chord-formation"
{organische Accordbildung) and of harmonic meaning
{von gleicher Begriffsgaitung) similar to the major triad
G-&-D. Perhaps it is, but not according to Hauptmann's
system.

Beyond all question, Hauptmann's designation of the

Harmony and Metre, " Resolution of Dissonance."

M. HAUPTMANN— r//£: NATURE OF HARMONY 365

diminished triads as 6-D/F and D-F/a indicates their duahty
of origin ; and this is true also of the two minor triads e-G-b and
a-C-e, where, instead of ground-tone, Third, and Fifth, we
find two Thirds and a ground-tone. His explanation of
the Dominant and Subdominant discords cannot be regarded
as an advance on that of Rameau.

Resolution of Dissonant Chords.

Hauptmann's theory of the resolution of dissonance is
characteristic of his system. The essence of dissonance, he
remarks, is that a note is determined as at once ground-
tone and Fifth. Thus, in the dissonance C-D, it is G which
is determined as simultaneously Ground-tone and Fifth : —

I— II

C — G — D. C may proceed to b, or D to e. In either case

I— II

the dual character of the sound G disappears. In the first
case G is definitely estabhshed as ground-tone ; in the second
case C is ground-tone.

It is thus that the resolution of the chord of the Seventh
is determined. " For example, in the chord of the Seventh
e-G-b-D, which comprises the duality e-G-b and G-&-D, the
notes e and D are as yet without relation to one another.
The required note, which brings about the relation, is here a,
to which e stands as Fifth and D as ground-tone. Thus the
note a must enter instead of the Third-interval G-b, whereby
instead of the chord of the Seventh e-G-b-D there is produced
the chord of suspension e-a-D. And now the linking note a
may be regarded as Fifth of D or ground-tone of e ; both of
which meanings are now contained in it at one and the same
time. Therefore e will either proceed to F, or D to C ; and
from e-a-D there will arise either F-a-D or e-a-C." ^

This theory leads to some curious results. The resolution
of the chord of the Tonic Seventh C-e-G-b, for example, has
to be explained thus : — The dissonance is C — b, Cis Fifth of F,
and b is its ground-tone, etc. As to the dissonance b — /F in
the chord of the Dominant Seventh G-&-D/F, Hauptmann is
at a loss, and can only speak of the " attractive tendency "

^ Harmony and Metre, " Resolution of Dissonance."

366 THE THEORY OF HARMONY

of the interval : h tends towards C, and F towards e. Again,
speaking of the resolution of the chord of the Seventh e-G-6-D,
he remarks : — " Here h is Third of the triad G-6-D, and Fifth
of the triad. <?-G-& ; but must become ground-tone of the diminished
triad b-D/F for the resolution to be determined upon it.
For again, e-b-T> can only reach resolution in F-b-D." ^ These
cannot be regarded as very satisfactory results. Hauptmann
is here at the mercy of his system.

The dissonant Augmented triad has to be explained some-
what differently from the chord of the Seventh. Of the chord
e\f-G-b, which occurs' on the Mediant of the key of C minor,
Hauptmann says that in this chord " the middle note G is
in itself decided duality ; it is determined differently in two
directions at the same time, as positive and negative ground-

eb— G— &"2
tone, thus : — III — I

I— III

The chords of the Augmented Sixth a\f-C-f^ and a]^-F>-f^,
are explained by Hauptmann as arising from the union of
the extremes of the C minor key-system extended in an upward

direction, thus a\}—C—e\^—G—b—D—f^. The original form

of these chords is therefore /J/ajj-C and F)-f$/a\^.

Hauptmann, it will be observed, regards all these dissonant
chords, including all the chords of the Seventh, as having a
" double root," a dual origin. Hauptmann has certainly
reason and logic on his side, and his position here is much
more defensible than that of theorists who derive " diatonic
discords," augmented and diminished triads, and so forth,
from one and the same generator. Rameau, in effect, also
gives to the chord of the Dominant Seventh and the chord
of the Added Sixth a twofold origin, when he explains the first
chord as formed by the addition of the Subdominant to the
Dominant harmony, and the second by the addition of
the Fifth of the Dominant to the Subdominant harmony.

^ Harmony and Metre, " Resolution of Dissonance.
2 Ibid., " The Augmented Triad."

M. HAUPTMANN— r//7i NATURE OF HARMONY 367

The Chromatic Scale.

Hauptmann's theory as to the origin of the chromatic
scale does not differ essentially from that of Rameau. Each
chromatically raised note he considers to be the Third of a
Dominant. " A note raised chromatically," he says, "can,
in the first instance, only have the meaning of the Third of
a Dominant, that is, the leading note of a major or a minor
key, which forms a close with the note next above it." ^
But Hauptmann also distinguishes the ascending chromatic
scale with chromatically lowered degrees as : — C-D\}-d-E\^-
e-F-f^-G-a\}-a-B\^-b-C. In this scale " the Tonic elements C and
G are transposed from ground-tone and Fifth into Third
meaning, and appear themselves as leading-notes. "^ He further
states : — " It is an erroneous opinion that chromatically
raised degrees belong exclusively to ascending motion, and
chromatically lowered degrees to descending."

Most remarkable is the resuscitation by Hauptmann of
the " chord by supposition." This is the chord of the Ninth.
We read : — " In the passage G-b-D-a : G-6-D-^ the lowest
note of the first chord is entirely neglected in the resolution,
and the dissonance b-a is alone taken into account, for which
the resolution b-G is given." ^ That is, Hauptmann, like
Rameau, considers the lowest note of the chord G-b-D-a to
be a " supernumerary sound." One would imagine, on the
contrary, that it is just this sound which determines the re-
solution of a on G. Hauptmann does not consider the chords of
the Eleventh and Thirteenth to be real harmonic formations.

The Minor Harmony.

Hauptmann's explanation of the minor harmony does not
differ essentially from that of Rameau ; that is, he considers
it to be an inverted major harmony. When a triumvirate
of theorists such as Rameau, Tartini, and Hauptmann express
the same opinion respecting the nature of the minor harmony,
the correctness of such an opinion becomes more than a
mere probability. But it is one thing to express an opinion,
and another thing to demonstrate its correctness.

^ Harmony and Metre, " Passing-notes." - Ibid.

■* Ibid., " Chords of the Ninth, Eleventh, and Thirteenth."

368 THE THEORY OF HARMONY

Hauptmann remarks : — " The determinations of the
intervals of the triad have been hitherto taken as starting
from a positive unity, or ground-tone, to which the Fifth
and Third have been referred. They may also be thought
of in an opposite sense. If the first may be expressed by
saying that a note has a Fifth and Third, the opposite meaning
will lie in a note being Fifth and Third. Having is an active
state ; being is a passive one. ... In the major triad C-e-G,
C-G is Fifth, and C-e, Third ; in the minor triad a-C-e,
a-e, is Fifth, and C-e, Third. But in the latter the common
element for both determinations is contained in the note of
the Fifth ; therefore that note, being doubly determined,
may be regarded as doubly determining, in a negative sense ;
or as the negative unity of the chord. Therefore the symbol
II-III-I seems not unsuitable for the minor chord." ^

After referring to the fact that the minor triad appears in
the harmonic series, corresponding to the numbers lo : 12 : 15,
he proceeds: — "The minor triad, as an inverted major
triad, must, in its meaning of being considered to originate
from a negative unity, consist of a construction backwards.

I — II
Referred to the unity C, the major triad is C — e — G : the

I-III
minor triad of the same unity C, that is, as Fifth

II — I
determining ground-tone and Third, is F — a\} — C, which is

III— I

the same as if we put I — II

I — III

Hauptmann, then, is of opinion that ^ and |l mean

the same thing. If the minor harmony must be understood
to " consist of a construction backwards," then its correct

expression is ^~^^~^. But Hauptmann, strangely enough,

is not satisfied with this expression ; although there is nothing
in the Hegelian system of metaphysics which would forbid
the determination of intervals downwards as well as upwards.
The real determination of the minor harmony he considers

1 Harmony and Metre, " Minor Triad." * Ihid.

M. HAUPTMANN.— r//£ NATURE OF HARMONY 369

F— ab— C
to be I — II. That is, he relates the question to the

I— III

acoustical determination of intervals. In such a case the

minor harmony appears as a " duality," that is, it has a

two-fold origin. C is Fifth of F, and Third of aj;.

But, urges Hauptmann, the intervals of the minor harmony
may be thought of as being negatively determined. From this
point of view, the minor harmony appears as a unity. But
exactly what importance or significance the negative determin-
ation of intervals possesses for the theory of harmony
Hauptmann does not make sufficiently clear. When he states
that an interval is negatively determined downwards, he
merely means it is positively determined upwards ; and when
he remarks that the minor harmony originates from a
" negative unity," he merely repeats his explanation of this
harmony as being positively determined upwards from a
" double root." It msLV be that the minor harmony, under-
stood as a harmonic unity, must be regarded as originating
downwards ; the whole difficulty is to explain how such a
construction can possibly arise.

Hauptmann's difficulties in connection with the Minor
key-system, are, as may be imagined, much greater than
those in connection with the Major. Like Rameau, he
explained the major key-system as determined by a " triad
of triads." Like Rameau, also, he found three triads
insufhcient for his purpose, and was obliged to utilise other
triads. But while Rameau had to search outside the
key-system, Hauptmann discovered within the key-system
itself the triads of which he stood in need, namely, those
on the Mediant and Submediant. Hauptmann, of course,
was aware that it was necessary to find some explanation
of these triads. But now, in the minor key-system, we find
at the very outset that the principal sounds of the Tonic,
Dominant, and Subdominant furnish us, not with three triads
only, but five. For on the Dominant, and likewise on the
Subdominant, there occurs not only a minor but also a major

harmony : thus .- ^,__J^_^_^^_^^_^^_^.

370 THE THEORY OF HARMONY

These five triads, however, are not sufficient : other two are
necessary in order to explain the melodic succession of the
sounds of the scale. In the ascending scale, " the connecting
link between G and b can only be determined by the Fifth
of the Dominant, D, whose Fifth A provides the passage
from G to b." But this A, Hauptmann tells us, " lies out-
side of the system." In the descending scale, the passage
from the Octave C to the minor Sixth a\} can only be effected
by means of a triad whose fundamental note also lies outside
of the key-system, namely, B\^-d-F. " While in ascending
the Fifth of the dominant had to become Root, in descending
the Root of the subdominant must become Fifth." The
explanation of the ascending and descending forms of the
minor scale is therefore as follows : —

C G C

C_D-eb-F-G-A-&-C C-Bb-«b-G-F-eb-D-C

G (ascending) D F (descending) G

Hauptmann is unable to make up his mind as to which note
of the minor harmony should be described as the fundamental
note. He frequently speaks of the lowest note of the minor
harmony as the " ground-tone " or fundamental note. Never-
theless, he represents the minor key-system as follows : —

II-III-I I-III-II

F—a\f—C—eb—G — b — B
II-III-I

Here we find the Dominant G, represented as I or ground-
tone not only of the Dominant major triad G-6-D, but also
of the Tonic minor triad C-eb-G. The Dominant G is the
" ground-tone " of both triads. If we take G as Tonic the
key-system appears to be left without a Dominant ; and if
G be taken as Dominant, it is left without a Tonic ; that is,
unless we regard G as being at one and the same time Dominant
and Tonic.

M. HAUPTMANN— T//E NATURE OF HARMONY 371

Chord Relationship and Chord-Succession.

Hauptmann's theory of chord succession differs from that
of Ramcau. Rameau relates chord succession to the pro-
gression of the Fundamental Bass. Hauptmann, on the
other hand, considers that chord succession can be explained
only as a hnking together of successive harmonies by means
of sounds which they possess in common. It is this common
element between successive chords which renders chord
succession intelligible. Hauptmann says : — "The succession
of two triads is only intelligible in so far as both can be referred
to a common element which changes meaning during the
passage." ^ The succession C-e-G— &-D/F must therefore
be understood thus : — C — e — G

e—G—b

b — D/F, and similarly in
the case of other disjunct triads.

So also the succession from the Tonic chord to that of the
Dominant Seventh must be understood as : — C-c-G. . . .
b-e-G. . . . b-D-G. . . .b-D-K. . . =- 6-D-F-G; and from the

Tonic chord to that of the Supertonic Seventh, as
C-e-G . . . C-e-a . . . C-F-a . . . D-F-a = C-D-F-a.

Hauptmann, then, considers that a Subdominant-Dominant
harmonic succession can only be effected through the media-
tion of one or more linking triads ; that is, he is unable to
find any explanation of the immediate succession of both
Dominants, a succession which continually occurs in harmony.
But even with regard to the succession Tonic-Dominant and
Tonic-Subdominant Hauptmann remarks : — " The passage
from C-e-G to F-a-C, which leads to the position C-F-a,
is a compounded one, and consists of the progressions C-c-G
.... C-e-a. . . . C-F-a. . . . Similarly with the succession
from C-e-G to G-b-F), which is compounded of the successions
C-e-G. . . . b-e-G. . . . b-B-G." It follows therefore that
the reverse progression, namely, from Dominant to Tonic
harmony, should be understood in a similar way, as : —
G-b-D. . . . G-b-e. . . . G-C-*?. If it be true that it is
community of sounds which determines chord relationship,

^ Harmony and Metre, " Chord-Succession."

372 THE THEORY OF HARMONY

then, beyond all question, those chords which possess two
sounds in common are more closely related than those which
possess only one sound in common. The succession G-b-D
.... G-b-e. . . . G-C-c, must be regarded as being more
" directly intelligible " than the succession G-b-D. . . .
G-C-e. This is the essence of the Hauptmann theory of
chord succession. Unfortunately, it conflicts wdth the facts
as manifested in the Perfect Cadence, and therefore breaks
down at a crucial point. The essence of this Cadence hes
in the direct and immediate succession of Dominant-Tonic
harmonies, which furnishes us with the most " directly intel-
ligible " of all harmonic successions.

H. L. F. Helmholtz. — " Sensations of Tone."

Ten years after the publication of Hauptmann's Harmonik
und Metrik there appeared the well-kno\\Ti work by
Professor H. L. F. Helmholtz — The Sensations of Tone as
a Physiological Basis for the Theory of Music {Lehre
von den Tonempfindtingen als physiologische Gnindlage filr
die Theorie der Musik, 1863). Helmholtz's work is, in
many respects, one of the most important of its kind, and
not least in the respect that its author was one of the most
distinguished physicists of his time, who brought to the
consideration of the theory of music and of harmony not
only considerable musical insight, but also a trained
scientific judgment and accurate scientific methods.

As is kno\\Ti, Helmholtz in the first part of his work
investigates in the most complete way the nature of musical
sound and of sound in general, of the Composition of
Vibratioiis, of Sympathetic Resonance, of Upper Partial
Tones, Quahty of Musical Tones, etc. ; in Part H. he treats of
Combinational Tones, of Beats, of the relationship of both
to the Phenomena of Consonance and Dissonance, of the
Relative Harmoniousness of Intervals and Chords ; while in
Part ni., with which we are most immediately concerned, he
treats more specifically of the theory of harmony, and applies
the results of his previous observations to the consideration
of the origin and development of scales, of key-systems,
chord relationship and chord succession, concords and
discords.

HELMHOLTZ— THE SENSATIONS OF TONE 373

Major Harmony.

The explanation of the major harmony advanced by
Helmholtz does not differ essentially from that of Rameau.
As we have seen, Helmholtz agrees with Rameau that the
natural relations which may be observed to exist in the
resonance of the sonorous body constitute the proper basis
of the theory of harmony. In the major harmony, he states,
all the sounds of which it is composed are constituents of the
compound " klang " of the fundamental sound. This sound
Rameau has quite properly described as the fundamental bass
of the harmony.

Minor Harmony and Chord of the " Added Sixth."

As for the minor harmonj', Heln^ioltz considers this to
have a two-fold origin. Helmholtz is gc^riferalh' supposed to
have been the first to advance this explanation of the minor
harmony. We have seen, however, that this is by no means
the case. It was first proposed by Rameau, adopted by
Serre and D'Alembert, and later, apparently independently,
advanced by Hauptmann. It is important to note that
Helmholtz discovers in the nature of the minor harmony a
proof of the correctness of Rameau's theory of " double
emplojonent."

He says : "In the minor chord c-e\f-g, the ^ is a constituent

of the compound tone of both c and e\f. Neither e\^ nor c
occurs in either of the other two compound tones. Hence
it is clear that g at least is a dependent tone. But on the
other hand this minor chord can be regarded either as a

compound tone of c with an added e\^, or as a compound

tone of e\} with an added c. Both views are entertained at
diftercnt times, but the first is the more usual. If we regard
the chord as the compound tone of c, we find g for its third

partial tone, while the foreign tone e\f occupies the place
of the weak third partial e. But if we regard the chord as

a compound tone of e\f, although the weak fifth partial g
would be properly represented, the stronger third partial,

which ought to be %, is replaced by the foreign tone c.

374 THE THEORY OF HARMONY

Hence, in modern music, we usually find the minor chord

c-e\}-g treated as if its root or fundamental bass were c, so
that the chord appears as a somewhat altered and obscured
compound tone of c. But the chord also occurs in the position

e\f-g-c (or better e]^-g-c), even in the key of Bj? major, as a

substitute for the chord of the subdominant tj?. Rameau
then calls it the chord of the major (added) sixth, and, more ,

correctly than most modem theorists, regards e\^ as its
Fundamental Bass." ^

This is an extraordinary pronouncement from so eminent
a theorist and scientist as Helmholtz. He first explains the
minor haraiony as a duahty ; it has two roots : g is Fifth of c

and major Third of e\^. While the major harmony is a single
klang, the minor harmony is a dual klang. Helmholtz,
however, is aware of the objections which may be urged
against this view. He therefore invites us to consider the

minor harmony c-e\^-g as a " somewhat altered " major
harmony. The minor harmony, then, we ought to regard
as a major harmony somewhat out of tune, the " out-of-
tuneness " being of the extent of a chromatic semitone
24 : 25, which is the difference between e and e^ !

Further, Helmholtz quite mistakes the manner in which
Rameau formulated his theory of " double employment."
This device of Rameau had nothing to do with any supposed
ambiguity of the minor harmony. Rameau did not consider
the two chords d-f-a-c and f-a-c-d to originate with a minor
harmony at all, but \\dth a major one. That is, he did not
regard the fundamental harmony of the chord d-f-a-c to be the
chord d-f-a, nor did he consider that the fundamental harmony
of the chord f-a-c-d was f-a-d (which is merely the inversion
of d-f-a). In both cases Rameau expressly states that the
fundamental harmony is f-a-c, and forms the chord d-f-a-c
hy adding d below the chord f-a-c. This note d, he expressly
states, is the dissonant note of the chord d-f-a-c.

But Helmholtz, who is an apostle of just intonation, was
quite well aware that the triad on the supertonic {d-f-a in
C major, or c-e\^-g in Bj; major) is not a minor consonant
triad at all, but a diminished one. His own words are : —

" The chord d-f-a [Helmholtz's notation] which in the usual
1 Sensations of Tone, Pt. III., Ch. 15.

HELMHOLTZ -7//E SENSATIONS OF TONE 375

musical notation is not distinguished from the minor triad
d-f-a, and may hence be called the false minor triad is, as

Hauptmann has correctly shown, dissonant, and on justly
intoned instruments is very decidedly dissonant." ^ Accord-
ing to just intonation, the minor triad cannot occur on the
supertonic of a major key. Helmholtz, not without reason,
dwells with admiration on Rameau's fine tonal sense. It
was Rameau's fine tonal sense which guided him in his
treatment of the chord d-f-a-c, where, between d and a,
there is not a perfect Fifth, but a diminished one.

Origin of Dissonant Chords.

Helmholtz's views on the formation of dissonant chords,
although not original, are nevertheless remarkable enough
to deserve mention. Chords of the Seventh, consisting of a
major triad with major Seventh, or of a minor triad with
minor Seventh, he considers Like Hauptmann to be formed
from the union of two triads. The chord of the Dominant
Seventh, however, has a different origin. Of this chord he
remarks : — " We must observe that the minor seventh g-f
approaches so nearly to the ratio 4 : 7, which would be
almost exactly represented by g-f, that / may in any case
pass as the seventh partial tone of the compound tone G.
. . . Hence, although the chord of the Dominant Seventh
is dissonant, its dissonant tone so nearly corresponds to the
corresponding partial tone in the compound tone of the
dominant, that the whole chord may be very well regarded
as a representative of that compound tone. For this reason,
doubtless, the seventh of this chord has been set free from
many obligations in the progression of parts to which dissonant
sevenths are otherwise subjected. Thus it is allowed to be
introduced freety, without preparation, which is not the case
for the other sevenths. . . . The chord of the Dominant
Seventh consequently plays the second most important
part in modem music, standing next to the Tonic. It
exactly defines the key, more exactly than the simple triad
g-b-d, or the diminished triad b-d-f. As a dissonant chord

^ Seiuations of Tone, Pt. III., Ch. 17.

376 THE THEORY OF HARMONY

it urgently requires to be resolved on the Tonic chord. . . .
This chord appears to have been discovered by Monte verde." ^

Once more we light upon the theory of the " natural chord
of the Seventh," already mentioned in connection with Sorge
and Fetis. It is remarkable that Helmholtz, after explaining
the consonant minor harmony as the result of a dual klang,
should now ask us to consider the dissonant chord of the
Dominant Seventh as the result of one and the same klang
Further, the most characteristic discord of the key-system is
now a quasi-consonant chord. Speaking of the " natural "
Seventh earlier in his work, Helmholtz had stated that " the
sub-minor Seventh 4 : 7 is very often more harmonious than
the minor Sixth 5:8; in fact, it is always so when the third
partial tone of the note is strong as compared with the
second." ^

But although Helmholtz makes of the chord of the
Dominant Seventh a self-sufficing combination, existing in
and for itself, he nevertheless thinks that it " urgently
requires to be resolved."

He is also of opinion that it " exactly defines the key,"
notwithstanding that all its sounds are the result of a single
klang. But this can only happen if the Seventh of the chord,
which is dissonant with the Dominant, but consonant with
the Tonic, be regarded as the Subdominant itself, as Rameau
asserted it to be. If on the other hand the Seventh be
regarded as a constituent of the compound tone of the
Dominant, we get a note which brings about quite new
relationships, for it bears a quasi-consonant relationship with
the Dominant, but is dissonant, decidedly dissonant, with the
Tonic.

It appears to have escaped the attention of Helmholtz
that the ratio of the augmented Sixth f-d^ (/= fourth degree
of the scale of C major; rfijf = chromatically raised second
degree) approximates more nearly to the ' ratio of the
" natural " Seventh than does the minor seventh g-f=g : 16.
Here are the respective ratios : —

Natural Seventh 4:7 = 128

Augmented Sixth = 128

Minor Seventh (9:16) = 126

224
22^
224

» Sensations of Tone, Pt. III., Ch. 17. - Ibid., Pt. II., Ch. 10.

HELMHOLTZ— THii SENSATIONS OF TONE 377

While the difference between the Augmented Sixth and
natural Seventh is represented by the extremely small
interval 224 : 225, the difference between the minor Seventh
g-fsind the natural Seventh is that of the much larger interval
63 : 64, an interval larger than the syntonic comma (80 : 81).
It would be much more reasonable, therefore, to identify
the chord of the Augmented Sixth with the natural Seventh,
rather than with the chord of the Dominant Seventh, as does
Helmholtz. There is of course no more reason for describing
the seventh partial tone as " a Seventh " than there would
be for describing the fifth degree of the diatonic scale as
" a Third " because it is the third partial tone of the
Tonic.

The chord of the Dominant Major Ninth is explained by
Helmholtz in similar fashion. In the chord b-d-f-a we must
observe, he remarks, " that the two tones / and a approach
very closely to the two next partial tones of the compound
tone of G. Hence the chord of the Ninth g-b-d-f-a may
represent the compound tone of the dominant g, provided
that the similarity be kept clear by the position of the tones ;
g being the lowest and a the highest. . . . This seems to me
to be the simple reason why musicians find it desirable to
make a the highest tone in the chord b-d-f-a." ^

Of the chord of the Diminished Seventh b-d-f-a\^, he states
that "it contains no note which belongs to the compound
tone of any other note in the chord, but the three notes
b-d-f may be regarded as belonging to the compound tone

of g, so that it also presents the appearance of a chord of the
Ninth in the form g-b-d-f-a\^. It therefore imperfectly
represents the compound tone of the dominant, with an
intruded «[?.""

Helmholtz docs not distinguish a chord of the Dominant
Eleventh sls g-b-d-f-a-c ,nov a chord of the Dominant Thirteenth
as g-b-d-f-a-c-e. The c in these chords differs from the real
Tonic by nearly a quarter tone (32 : 33). But apart from
this fact there is an obvious difficulty in explaining c, the
Tonic, as having its source in the compound tone of g, the
Dominant.

1 Sensations of Tone, Pt. III., Ch. 17. ^ 7^j^_

378 THE THEORY OF HARMONY

Chord Relationship and Succession.

Helmholtz identifies himself also with Hauptmann's theory
of chord succession ; he holds, with Hauptmann, that chord
succession is intelligible by virtue of a common element
existing between successive chords. He says : — " Just as
the older homophonic music required the, notes of a melody
to be linked together, so modern music endeavours to link
together the series of chords occurring in a web of harmony."
Again, " When disconnected triads would come together
it is frequently advantageous to transform them into chords
of the Seventh, and thus create a bond between them. ' ' Thus,
in place of f-a-c — g-b-d, we may substitute f-a-c-d — g-b-d.
Helmholtz is not more in a position than Hauptmann to
explain the immediate succession of both Dominants.

Although Helmholtz, in many difficult questions relating
to the theory of harmony, is too often content merely to offer
hints and suggestions rather than venture on any positive
statement, we nevertheless find the following : — " When two
chords have two notes in common they are more closely
related than when they have only one note in common. Thus
c-e-g and a-c-e are more closely related than c-e-g and g-b-d." ^
This is the logical outcome of Helmholtz's theory of chord
succession, as it is of Hauptmann's. It follows, and must
follow, that a Dominant-Tonic succession of harmonies, as
in the Perfect Cadence, is less " directty intelligible " than
a Dominant-Mediant-Tonic succession.

Tonality.

Notwithstanding the deservedly high position as a musical
theorist which Helmholtz occupies in the esteem of musicians,
it would nevertheless be somewhat difficult to state exactly
what original contribution he has made to the theory of
harmony. Dr. Riemann is of opinion that Helmholtz's
greatest contribution to the science of harmony is his principle
of " klang-representation " (Klangvertretung). "Helmholtz,"
he remarks, " has opened up quite new perspectives by his
principle of klang-representation." This honour however
belongs not to Helmholtz, but, as we have seen, to Rameau.^

1 Sensations of Tone, Pt. III., Ch. 15. ^ See p. 183,

HELMHOLTZ— r//£: SENSATIONS OF TONE 379

Besides, Hauptmann had already stated that every note in
our modern harmonic system of music must be regarded as
tlie Fundamental note, Third, or Fifth of a triad. It is much
more in the principle of Tonality that Helmholtz discovers
the ultimate explanation of the art of music, melodic or
harmonic. He explains " as the fundamental principle
for the development of the European tonal system," that
" the whole mass of tones and the connection of harmonies
must stand in a close and always distinctly perceptible relation-
ship to some arbitrarily selected tonic, and that the mass
of tone which forms the whole composition mtist be developed
from this Tonic, and must finally return to it. [Italics
by Helmholtz.] The ancient world developed this
principle in homophonic music ; the modern world in
harmonic music." ^

It may at once be said that this principle of Tonality, as
enunciated by Helmholtz, represents a distinct advance on
that of Fetis. It has frequently been thought to mark a
fresh and important stage in the development of the theory
of harmony. It has even been considered, somewhat too
hastily perhaps, to furnish an adequate solution of some of
the most obscure facts of harmonic science. But the principle
on which Helmholtz here lays stress does not mark a new
conception. It was enunciated quite clearly by Rameau.
It forms the root idea of his whole work as a theorist. It is
the root idea of the numerous theorists who since Rameau's
time have regarded the harmonic series as the principle of
chord generation. It was Rameau who, for the first time,
stated in his Generation Harmonique that all harmony is
developed from the Tonic, and that the Tonic is the centre of
the harmonic system.

Helmholtz's enunciation of the principle of Tonalit}', in
itself admirable, is therefore little more than a statement
of the problems which the theory of harmony has to face.
It was to their solution that Rameau addressed himself in his
numerous works on harmony. We saw that the principal
difficulty was to determine exactly how the " mass of tone "
is " developed from " the tonic.

Helmholtz's views as to the origin of early scales have

^ Sensations of Tone, Pt. III., Ch. 13.

38o THE THEORY OF HARMONY

already been referred to.^ In these scales Helmholtz is of
opinion that each degree of the scale, with the exception of
the Tonic itself, must have been selected on the principle of
its relationship to the Tonic. It is thus, according to
Helmholtz, that the sounds of the scale have been " developed
from " the Tonic. In the case of the Greek tetrachord

e-f-g-a, we must, then, believe that the " relationship "
between g and a (8:9) and between / and a, which was an
interval of the proportion 64 : 81, was " distinctly perceptible "
to the Greeks. It was not, however, the " relationship "
between these sounds, but the want of relationship, which
appears to have most impressed Greek musicians and writers
on music.

The value of Helmholtz's theory as to the origin of early
scales, and as to the manner in which the sounds of the
scale were developed from an " arbitrarily selected Tonic "
on the principle of " close and distinctly perceptible relation-
ship " to this Tonic, may be judged from the following frank
statement : — " Pythagoras constructed the complete diatonic
scale from the following series of Fifths : F-C-G-D-A-E-B.
In his diatonic scale there are but two kinds of small intervals,
the whole-tone, 8 : 9, and the Limma, 243-256. In this
series, if C be taken as Tonic, A would be related to the
Tonic in the third degree, E in the fourth, and B in the fifth
. . . but neither singer nor hearer could possibly discover
in passing from C to E that the latter is the fourth from the
former in the series of Fifths. Even in a relation of the
second degree through Fifths, as of C to D, it is doubtful
whether a hearer can discover the relation of the two tones." ^
In fact, in this scale, no matter which sound be " arbitrarily
selected " as Tonic, not more than two sounds, if we exclude
the Octave, namely, the Fourth and the Fifth, will be found
to bear " a distinctly perceptible relationship to the Tonic."
If, as is most natural, we select F as Tonic, seeing that it
forms the starting point of the series of Fifths, only one
sound, C, the Fifth above, bears such " a distinctly perceptible
relationship."

Helmholtz, therefore, in treating of early scales, prefers
to give them " natural " or just rather than Pythagorean

^ See pp. 205-207.

* Sensations of Tone, Pt. III., Cli. 14.

HELMHOLTZ— THE SENSATIONS OF TONE 381

Thirds and Sixths. But that such scales cannot represent
" early scales " is evident from his own remark, wliich he
makes in another portion of his work, that " all antiquity
refused to accept Thirds as consonances . . . the proper
intonation of Thirds was not discovered in early times,
and the Pythagorean Third, with its ratio of 64 : 81, was
looked upon as the normal form till towards the close of the
Middle Ages." 1

It is just the use made in modern music of these " natural "
Thirds which constitutes a fundamental difference between
our modern scales and early scales. The introduction of
these " natural consonances," as has frequently been insisted
upon throughout the course of this work, marks an event
of the greatest theoretical importance. It led directly to
the decay of the old scales, and made possible our modern
tonal system. In referring to this tremendous change,
however, Helmholtz is content to repeat the legend so
sedulously propagated by Fetis concerning Monteverde's
epoch-making introduction and employment of the chord
of the Dominant Seventh.

Helmholtz is not more successful in his attempts to show
how, in our modern scales, the sounds have been " developed
from " an arbitrarily selected Tonic. He is of opinion that
all the sounds of the scale may be regarded as constituents
of the harmonies of the three sounds. Tonic, Dominant,
and Subdominant. But he is by no means prepared to allow
that these are the only or ultimate determinations of the
sounds of the scale. For example, a, the sixth degree of
the scale of C major, may be determined in three different
ways : — (i) as major Sixth of the Tonic ; (2) as major Third of
the Subdominant ; (3) as perfect Fourth of the Mediant.
Again b, the seventh degree, may be determined as (i) major
Seventh of the Tonic (!) ; (2) Third of the Dominant ; (3)
perfect Fifth of the Mediant. In the same way, Helmholtz
might have proceeded to show that the Dominant, instead
of having a perfectly definite and fixed relationship
to the Tonic, in which, as Rameau stated, it has its one
and only source, has other determinations ; for example,
as perfect Fourth of the Supertonic ; as minor Third
of the Mediant ; as major Second of the Subdominant,

^ Sensations of Tone, Pt. II., Ch. 10.

382 THE THEORY OF HARMONY

or minor Sixth of the leading-note, and so on. Only
how all this enables us to understand better the nature
and origin of our tonal system it is somewhat difficult to
imagine. As for the minor scale, or rather scales, matters
are even worse.

With regard to chords, consonant and dissonant, which
belong to the key-system, how many of these did Helmholtz
reaUy consider he had succeeded in proving to be developed
from an arbitrarily selected Tonic ? Like other theorists
before and after him, Helmholtz has little difficulty in
pointing to the fact that all the sounds of the major harmony,
as that on the Tonic of a major key, are constituents of the
compound tone of the Tonic. But what of the other chords ;
for example, that on the next degree of the major scale :
the diminished triad on the Supertonic, as d-f-a ? Whence
is this triad derived ? We may, of course, explain d as the
major Second of the Tonic C ; / as its perfect Fourth, and
a as its major Sixth, but this does not help matters greatly.
In the case of the minor harmony, as a-c-e, we have seen
that Helmholtz considers c to be a " foreign sound " ; such
a sound, therefore, cannot properly be said to be " developed
from " the Tonic a.

Of the Subdominant, the despair of so many theorists,
Helmholtz treats thus : — " When we pass from C-E-G to
G-B-D, we use a compound tone, G, which is already con-
tained in the first chord. ... It is quite different wdth the
passage from C-E-G to F-A-c. The compound tone F is

not prepared in the first chord, and it has therefore to be
discovered and struck. The justification of this passage,
then, is not complete on the ground of close relationship
between the chords, until it is felt that the chord of F contains
no tones which are not closely related to the Tonic C." ^
Helmholtz evidently considers this to be an adequate explana-
tion of the Subdominant.

Helmholtz is even less successful, as might be expected, in
his attempt to show how the principle of TonaUty determines
chord succession. He cannot explain on the principle of
the " relationship of the mass of tones " to the Tonic,

'^ Sensations of Tone, Pt. III., Ch. 15.

HELMHOLTZ— r//£: SENSATIONS OF TONE 383

some of the simplest of harmonic successions, as, for
example : —

Key C Major. ^^

-Gh-

:?2:

-<s>-

-^>-

-«s>-

The sounds which form the combination at * may be explained
thus :— / is Fourth of the Tonic, while a is its major Sixth ;
c is the Tonic itself. But this does not help us to understand
the progression of this chord to the disconnected triad g-h-d.
But, as we have seen, Helmholtz brings forvvard quite a
different theory of chord successions : those chords tend to
succeed one another which are related by means of one or
more common notes.

Theory of Consonance and Dissonance.

Helmholtz 's theory of Consonance and Dissonance, already
referred to, has been subjected to so much examination and
criticism in other works that it is unnecessary to enter into
the question here. Dr. Carl Stumpf has shown ^ that it
is possible to construct by means of simple tones most
discordant combinations of sounds, which, nevertheless,
produce no beats. He has also pointed to the fact that rapid
intermittent sounds do not necessarily always produce an
unpleasant or irritating, effect on the ear, and has instanced
as proofs of his contention the tremolo of the stringed instru-
ments of the orchestra, and the vibrato and other devices
resorted to by both vocahsts and instrumentalists in order
to obtain a rapid intennittence of the tone. He has remarked
also that Helmholtz distinguishes varpng degrees of dissonance
for the same dissonant interval, according to the position it
occupies in the scale of sounds. For example, the semitone

1 Tonpsychologie, 2 vols., 1883 and 1S90, and Konsonanz und
Dissonanz, 1898.

384 THE THEORY OF HARMONY

h' c" , which produces 33 beats in a second, is pronounced b;
Helmholtz to be an extremely harsh dissonance ; its dissonan
effect is, however, considerably modified by its being takei
an octave higher, in the position h" c'" , with 66 beats ; while
in the position h'" c"", two octaves higher, which produces 132
beats, the roughness of the interval becomes very sensibly
diminished. This is owing to the increased rapidity of
the beats. " The beats of a whole-tone," remarks Helmholtz,
" which in low positions are very distinct and powerful,
are scarcely audible at the upper limit of the thrice-accented
Octave."^ Stumpf, however, is unable to account for the
phenomenon of Consonance on psychological grounds ; it
must have, he thinks, a physiological explanation. ^

The considerations advanced by Stumpf cannot by any
means be held to justify the entire rejection of Helmholtz's
theory of consonance. At the same time such a theory is
plainly inadequate. The explanation of consonance as
arising from the absence, or comparative absence, of beats
is a negative rather than a positive one. But, urges Helmholtz,
such an absence of beats results in a certain smoothness of
effect ; and smoothness is an aesthetical quality. The
difference of effect, however, produced on the mind by the \
major, as compared with the minor harmony, is not accounted
for by describing the major harmony as smoother in its effect
than the minor. The explanation given by Helmholtz of the
sensation produced by a single musical sound does not differ
essentially from his explanation of consonance: — " A musical
tone," he states, " strikes the ear as a perfectly undisturbed,
uniform sound which remains unaltered as long as it exists."
This is the physical explanation of the sensation of musical
sound. But, as Helmholtz shows, the flow of sound resulting
from perfectly simple tones is much smoother, more uniform
than that resulting from musical sounds with well developed

1 Sensations of Tone, Pt. II., Ch. 8.

* " Die Ursache der Verschmelzung ist eine physiologische. . . .
Dafiir sprach ohnedies schon von vornherein der Umstand, dass die-
selbe eine Tatsache der Empfindung, ein den gleichzeitigen Tonquali-
taten immanentes Verhaltnis, und von der Uebung in individuellen
Leben unabhangig ist. Empfindungsverhaltnisse sind aber, wie
Empfindungen selbst, nicht auf weiter zuriickliegende psychische Ursachen
sondern nur auf physische zurUckzufuhren," — Tonpsychologie, Vol. II.,
p. 211.

0TTIN(;EN— DUAL NATURE OF HARMONY 385

upper partial tones, one reason being that beats arise between
the upper partial tones themselves. It would appear then
that such composite musical sounds must be greatly inferior
in respect of the musical sensation they produce in the ear
as compared with simple tones. The opposite, however,
is the case. Simple tones are dull, poor, and comparatively
devoid of musical charm. On the other hand, " musical
tones which are accompanied b\' a moderately loud series
of the lower upper partial tones, up to the sixth upper partial,
are more harmonious and musical ; compared with simple
tones, the}' are rich and splendid." ^

Ottingen and the Origin of the Minor Harmony.

Three years after the appearance of Helmholtz's work,
A. von Ottingen {Harmoniesystem in dualer Entwickelimg,
1866) made a severe attack on Helmholtz's theory of con-
sonance and dissonance. It was especially against the
latter's inadequate treatment of the minor harmony that
Ottingen's criticism was directed. Ottingen was not slow
to point out the inconsistency of Helmholtz in admitting
foreign and added sounds in a klang. He maintains
that consonance and dissonance do not find a completely
adequate explanation in Helmholtz's theory- of the coincidence
or non-coincidence of upper partial tones. He argues that
the clashing of upper partial tones is as marked in the major,
as in the minor harmony {a) : —

-? b3.

"^gs-

-I-

Major.

Minor.

The analogy between both harmonies, in which the minor
is considered as the reverse of the major harmony, is also

^ Sensations of Tone, Pt. I., Ch. 5.

386 THE THEORY OF HARxMONY

shown by the fact that the two strongest secondary tones,
the common partial tone of the major {phonic overtone),
and the chief combination tone of the minor harmony (the
tonic ground-tone) occupy a hke position in respect of both
chords {h).

In the major harmony, all the sounds of which it is composed
find their unity {Einheitheziehung) or central point in the
fundamental or ground-tone {tonic ground-tone) ; in the
minor harmony, on the other hand, the element of union
is found in the first partial tone common to all the three
sounds of the harmony : —

« 5 4

Major. _^_ -?- -it' -*'

(■)

-<s>-

-^-

Minor.

Considered in a major sense, that is, in an upward direction,
the minor harmony is in reality dissonant ; in a downward,
sense, it is consonant. The major harmony, on the other
hand, considered in the first aspect, is consonant, and in
the second, dissonant.

Whether or not Ottingen's conclusions are to be accepted
as finally determining this difficult question of the minor
harmony, they at least deserve the fullest consideration.

It should be observed that Ottingen is very far from
establishing that a complete analogy exists between the two
harmonies. For example, the major harmony is consonant,
not only in an upward sense, in respect of its upper partials,
but in a downward sense, in respect of its combination tones,
while the minor harmony is consonant neither with respect
to its combination tones nor its upper partials. Further,

Ottingen cannot well maintain that there is any real analogy
between the " tonic ground-tone " of the major and what
he describes as the " phonic overtone " of the minor harmony.
He cannot maintain that, while c is the fundamental note of
the major harmony c-e-g, g is the fundamental note of the
minor harmony, c-e\f-g. This g of the minor harmony is a'
determined note, and is shown by Ottingen himself to be
the Fifth of c. So that between c-g of the major harmony,

OTTINGEN— DUAL NATURE OF HARMONY 387

and c-g of the minor harmony, there is absolutely no difference;
c is fundamental and g is Fifth in each case. When, then,
Ottingen shows us that in the minor harmony c-e\}-g, g is

Fifth of c, and Third of e\f, his position does not appear to
be materially different from that of Helmholtz, that is,
he considers the minor harmony to arise from two sources.
At the same time Ottingen makes it increasingly evident
that the only sense in which the minor harmony can be
regarded as a harmonic unity is that of an inverted major
harmony. But how such an inversion is to be brought
about, how it is possible for the ear to conceive a chord,
which is doubly determined, as a harmonic unity, still remains
a mystery.

Dr. H. Riemaxx and the " Uxdertoxe-Series."

In his attack on Helmholtz's theor}-, Ottingen found
supporters in H. Lotze {Geschichte der Aesthetik in Deutsch-
land, 1868) ; Dr. Stumpf, already mentioned ; Hostinskv
{Die Lehre von den niusikalischen Klangen, 1879), and Dr.
Riemann {Musiknlische Logik, 1873 ; Mnsikalische Syntaxis
1877 ; Die Natur der Harmonik^ 1882 ; Geschichte der
Miisiktheorie^ 1898; Musiklexikon, etc.).

Dr Riemann is of opinion that Ottingen has given to
Helmholtz's (Rameau's) conception of the principle of " klang-
representation " an unexampled reach, in that he has rendered
it possible to consider not only the major, but the minor
harmony as a real " klang," represented by a single sound. ^
The great defect of Helmholtz's theory. Dr. Riemann remarks,
is his failure to give an adequate explanation of consonance
and dissonance, and especially of the consonance of the
minor harmony. " The most controvertible chapter of
Helmholtz's work is that treating of consonance and dis-
sonance, which Helmholtz sought to explain on physiological
grounds by means of differences of euphony or harmoniousness.
He finds the cause of dissonance in beats. The major chord
is more free from beats than any other chord, but the minor
harmony is the obscuring {Triihiing) of the physiological

^ Natur der Harmonik, p. 29.

388 THE THEORY OF HARMONY

consonance. Beginning with the complete fusion of the
sounds of a harmony, as represented by the first overtones,
he gives an entire scale of chords, ranging from those of the
most perfect degree of harmoniousness to the harshest dis-
sonances, according to the measure of their beats, so that
neither for the major and minor harmonies, nor for consonance
and dissonance in general, is any distinction made except that
of their varjdng degrees of euphony. This highly unsatis-
factory result has given rise to the most violent opposition." ^
Dr. Riemann's explanation of the major harmony is
essentially the same as that of Rameau. As for the minor

harmony, he agrees with Ottingen that it must be regarded
as determined in a downward direction, but is of opinion
that the latter is not radical enough in his treatment of this
harmony, in that he relates it to the series of overtones.^
Nevertheless, he thinks that in his "phonic" explanation

of the minor harmon}^ Ottingen has succeeded in giving
" a physiological basis to a series of undertones, which for
him is nothing but the series of those tones whereof a certain
note selected as the starting point [a Prifite] is the overtone,"
thus : —

1 2 3 4 5 6

_a. Q • • •—

-«5>-

t«==f

In describing Ottingen's " phonic " overtone as a Prime ^
as he does here, Dr. Riemann of course knows that it is
not actually such. It is an upper partial tone, a determined
and dependent tone. But, he thinks, all that is necessary
for the complete establishment of the minor harmony as the

antithesis of the major, and the gaining of Ottingen's " phonic
overtone " as a real Prime, is the scientific demonstration
of the objective existence of a series of " undertones," in
the same way as the ascending series of sounds, the
overtones or upper partials, have been proved to exist
objectively.

He says: — "As the consonance of the major chord is
explained not only by means of the combination tones, but

^ Natur aer Harmonik, pp., 23, 24.
* Geschichte der Miisiktheorie, p. 499.

DR. RIEMANN— 1 ONAL FUNCTIONS OF CHORDS 389

has its real foundation in the phenomenon of the overtones,
so hkewise for the completely adequate explanation of the
minor consonance there is only necessary the opposite
phenomenon of the undertones. Even if the existence of
such a phenomenon did not admit of positive proof, never-
theless it must be remembered that the minor correlatives
have a subjective existence, in that the major proportions
may be measured downwards as well as upwards. I have
already pointed out that the co-vibration of tones points
the way towards the existence of a series of undertones ;
and the same may be affirmed of such acoustical phenomena
as are furnished by the sounds produced by striking rods,
metal discs, etc. {Klirrtone). If one takes a vibrating tuning-
fork and allows the prongs to touch quite lightly a resonance
box, or if one sets in violent vibration a loosely held metal
plate or disc, there may be heard, instead of the proper tones
of the tuning-fork or plate, the lower octave or twelfth,
even the lower 15th, or 17th, as well as lower undertones.
1 1 is even probable that every tone has not merely a series of
overtones, but also a series of undertones, of the same
proportions, but gradually becoming more feeble as they
recede from the prime tone, and being more difficult to
distinguish, that is, to separate from the klang of the prime
tone, than the overtones." ^

It is impossible to enter here into any detailed examination
of the arguments by which Dr. Riemann, in several of his
works, attempts to prove that the series of undertones has
a real objective existence. This however is really un-
necessary, for it eventually turns out that Dr. Riemann
is quite unable to furnish any scientific proof of the objective
existence of undertones. In the article Untertone, in
his M usiklexikon , he remarks : — " The compiler of this
dictionary has made repeated attempts to demonstrate
the existence of undertones, corresponding to the overtone
series ; in his MusikaUsche Logik he has demonstrated
their objective existence in the ear, and, from various signs,
he thinks himself justified in believing in their objective
existence. In his Katechismus der Miisikwissenschaft (p. 79)
he has shown finally by proof of a scientific character
why, in spite of the commensurability of the vibration forms,

^ Natiiv der Harmonik, pp. 21, 22.

390 THE THEORY OF HARMONY

a tone by summation of its vibrations cannot produce the
undertone series, and that the question may thus be considered
to be finally closed." The conclusion then is that
Dr. Riemann has had little better ground for his theory of
"undertones " than a somewhat too speculative imagination.

After this, one is not surprised to read in the Natur der
Harmonik, a few pages after the author has given an account
of his experiments with metal plates, and of tuning-
forks placed wrongly on their resonators, the following
statement : — " The principle of klang-representation is really
not a matter for physics, nor for physiology, but for
■psychology. The minor as well as the major harmony is a
'fact of experience.'" It is therefore "a scientific fact,
which forms as good a foundation on which to build as
acoustical phenomena." ^

The foundation on which Dr. Riemann wishes to build
is a somewhat insecure one, namely, that the Fifth of the
minor harmony is the fundamental note of this harmony.
It is surprising to find that he assumes as a fact what he is
unable to prove, and that notwithstanding his failure to
demonstrate the objective existence of an undertone series,
he has nevertheless not been deterred from building up,
in his work Harmony Simplified, a complete system on what
he has himself admitted to be incapable of proof.

" Tonal F'unctioxs of Chords."

In Harmony Simplified, or Theory of the Tonal Functions
of Chords {Vereinfachte Harmonielehrc ^ 1893) we have a
notable attempt, by one of the most eminent authorities
on the subject of harmony of the present day, to arrive at a
logical and consistent theory of harmony. Especially note-
worthy is the fact that Dr. Riemann makes a return to
some of the most essential of the principles enunciated by
Rameau as well as by Hauptmann. There are but two
harmonies, he states, which exist in and for themselves,
that is, which are " directly intelligible," namely, the major
and minor harmonies. The major harmony [Overklang]
is determined in an upward direction, corresponding to the

^ Natur der Harnwiiik. p. 29.

DR. RIEMANN— TONAL FUNCTIONS OF CHORDS 391

first six of the ascending series of overtones ; the minor
harmony (Under klang) is determined in a downward
direction, corresponding to the first six of the descending
series of " undertones." In the first harmony, the lowest
note of the chord is the fundamental note ; in the second,
the highest note is the fundamental note.

All other chords must be considered as modifications of
one or the other of these harmonies.

Further, the chords of primary tonal significance within
a key are those of the Tonic, Dominant, and Subdominant.
These three chords, the first of which is taken as harmonic
centre, the second as the harmony of the upper Fifth
(" overtone " series) , and the third as the harmony of the
lower Fifth (" undertone " series) define the key-system.
As for the secondary triads on the second, third, sixth, and
seventh degrees of the scale, these are described as parallel-
klangs, qiiasi-consonances {Scheinkonsonanzen) and in other
ways, or as derived from discords.

The principles on which Dr. Riemann has built his
system are thus briefly stated in the " Introduction " to his
work : —

I. " There are only two kinds of klangs : overklangs and
underklangs. All dissonant chords are to be conceived,
explained, and indicated as modifications of overklangs
and underklangs.

II. "There are only three kinds of tonal functions, namely,
tonic, dominant, and subdominant. In the change of these
functions lies the essence of modulation."

In accepting Rameau's explanation of the generation of
the major harmony, and the nature of the major key-system.
Dr. Riemann adds nothing to the considerations already
advanced by the French theorist. All the sounds of the
major harmony combine so as to form a single klang. The
seventh upper partial tone, as well as others higher in the
harmonic series, cannot form part of such a klang, for such
tones are " out of tune." ^ In the key-system, the upper
Dominant is derived from the overtone series and the lower
Dominant from the undertone series. Rameau however,
it should be remembered, ultimately abandoned his theory
of the existence of a real series of " undertones." '^

^ Harmony Simplified, " Introduction," p. 6. ^ See p. 232.

392 THE THEORY OF HARMONY

" OVERKLAXGS " AND " UXDERKLAXGS " : THE KeY-SYSTEM.

In regarding, however, the Fifth of the minor harmony as
the fundamental note, and especially in the application of
his conception of the " underklang " to the theory of harmony.
Dr. Riemann parts company with Rameau. One or two
results of the appHcation of the " underklang " theory to the
minor key-s\-stem may be noted. In the minor key-system
which Dr. Riemann recognizes as the most representative,

d-f-a e-g^-b

a must be regarded as the fundamental note of the

minor harmony d-f-a ; while e, as fundamental note of

the minor harmony a-c-e, appears as the Tonic. This
note e, however, is also the fundamental note of the

major harmon}^ ^"-g^-b- Is, then, the note e both Tonic and
Dominant at one and the same time ? It results further
that in the Perfect Cadence in the minor mode there is no
real harmonic progression from a Dominant to the Tonic

which determines it. Thus, in the succession e-g^-b — a-c-e,
the note e must. Dr. Riemann considers, be regarded as the
fundamental note of both chords. There is therefore no
real Cadence, but only a species of harmonic oscillation ;
e appears as a sort of pivot or fixed point on which the
harmony may swing from one side to the other. But this
does not at all accord with the nature of the Cadence, in
which, as every musician feels, there is a real movement
and progression of the harmonies.

Again, if the central harmony of this mode is a-c-e, then e,
as the fundamental note of this Tonic harmony, is the Tonic

of the mode. But the major harmony e-g^-b has also e as

its fundamental. The harmony e-gi^-b must therefore properly
be regarded as a Tonic harmon}-. In this mode, therefore,
there is no Dominant harmon}-. Instead, we find a Sub-
dominant harmon\- (d-f-a), and two Tonic harmonies, one of

which [a-c-e] is determined downwards, and the other {e-g^-b)
upwards.

DR. RIEMANN— TONAL FUNCTIONS 01' CHORDS 393

But if it results from Dr. Riemann's theory that in tlic
minor Perfect Cadence the " fundamental bass " remains
stationary, in the common change from a minor mode to
its tonic major, on the other hand, it is necessary to suppose
that there occurs a real movement and succession of the
harmonies, for here the " fundamental bass " descends a

Fifth. Thus in the succession c-e\^-g — c-e-g, the fundamental
note of the first chord, according to Dr. Riemann, is g, and
that of the second c.

With regard to the ascending form of the Melodic Minor
scale, which has/# as well as gj^, matters are no better. For
here : —

a-c-e
d-f^ e-gP^

d is the fundamental note of the Subdominant " overklang,"
while e is the fundamental of the Tonic " underklang." That
is, the Subdominant, in this case, is not a Fifth but a Ninth
below the Tonic.

To Dr. Riemann and others of the post-Helmholtz school
of writers already referred to, who claim that tlie major mode
must be regarded as composed of a system of " overklangs,"
and the minor mode, the antithesis of the major, of a system
of " underklangs," it must be somewhat disconcerting to
discover the presence of " overklangs " in tlie minor mode.
Dr. Riemann, however, is of opinion that tliis defect maybe
remedied by means of the introduction of an " underklang "
in the major mode. An analogy is then perceived to exist
between this minor-major scale, and the harmonic form of the
minor scale : thus : —

Harmonic minor : — , ^ ^ ,

d-f-a e-g$-b

c-e-z

Minor-major : — / , ^ , ,
•* y-«p-c god

In this minor-major scale the fundamental note of the Tonic
harmony is at the same time the fundamental note of the
Subdominant harmony.

394 THE THEORY OF HARMONY

Dr. Riemann, however, agrees that this is not the " pure

major " scale, which is of the form c-d-e-f-g-a-b-c. What
then is the " pure minor " scale ? This is not, as might be ex-
pected, of the form a-b-c-d-e-f-g-a, but of the form e-f-g-a-b-c-d-e.
This descending scale. Dr. Riemann points out, is of exactly
the same form, and consists of the same order of tones
and semitones as the major, but in inverted order. That is,,
it is exactly the reverse of the major scale : —

c-d-e-f-g-a-b-c

e-d-c-b-a-g-f-e.
It is unfortunate that the minor scale which Dr. Riemann
presents to us as the direct antithesis of the major is not our
minor scale at all. Dr. Riemann considers it to represent the
Dorian Mode of the Greeks. This however it does not do. The
Greek Dorian Mode had Pythagorean tuning, with dissonant
Thirds and Sixths. But even if we suppose such a scale to
have originated from a system of consonant "klangs,"
it is impossible to regard it as being consistently generated
downwards, or as composed exclusively of " underklangs " : —

g-b-d a-c-e
Maj. Min.

If Dr. Riemann is bent on discovering a minor key system
which can be consistently regarded as generated downwards,
and as composed exclusively of a system of " underklangs, "^
it is quite possible to find one : —

. . 5th

7_r_ ^ „ 7, fundamental note.

^^ "sth

Here b is the starting point, and fundamental note of the-
mode ; e is its Fifth below ; while a is Fifth below c. The
scale which results from this system of " underklangs " has
therefore the form b-a-g-f-e-d-c-b.

Similarly with regard to the major mode : —

5th,

Fundamental note

DR. RIEMANN— TONAL FUNCTIONS OF CHORDS 393

Of tlie Subclominant liarmony in this mode, that is, J-a-c,
Dr. Riemann himself states that / is the fundamental note :
c, therefore, is its Fifth, and g is F'ifth of c. The note /
represents the starting point of this mode, and now the mode
may be consistently regarded as generated upwards. The
scale, then, with /as starting point, has the iormf-g-a-b-c-d-e-f.

Dr. Riemann however knows well that this will not do ;
but finds it necessary, for the major mode, to have recourse
to the " undertone " series (Subdominant), and for the
minor mode, to the " overtone " series (Dominant).

It is, again, a decidedh^ awkward circumstance that the
minor scale should have three different forms, while the major
has but one. Dr. Riemann does not help us to understand
why this should be. As for the relationship between the
major and minor modes, he does not add anything to the
explanations already advanced by Rameau. He considers
that this relationship is sufficiently explained by the great
number of sounds which a major and its relative minor mode
possess in common. This, so far as it goes, is quite a good
reason ; but plainly it cannot be the only nor indeed the
chief explanation. For if the degree of relationship between
two keys is determined by the sounds they have in common,
then how is it that, for example, E major \\ith four sharps
is more closely related to C major than is D major with only
two sharps ; and similarl}- with other keys ? Another
difficulty is that the relationship between the Tonic of a
major and of its relative minor kev is that of a minor Third.
Dr. Riemann howe\-er strongly holds, with Hauptmann, that
the minor Third is not a " directly inteUigible " interval.
And yet the relationship between the two keys is of the
closest possible kind.

Dr. Riemann appears to be of opinion that by means of his
system of " overklangs " and " underklangs," for the notation
of which he has invented special signs, he has greatly sim-
plified the science of harmony. On the contrary, one may
assert that what with " underklangs," " contra-klangs,"
" contra-fifth klangs," " plain-fifth klangs," etc., he has made
of harmony, especially considered in its didactic aspect, a
subject of quite needless complexity. It is needless, because
in a succession of chords the student does not understand,
for very good reasons, the Fifth of the minor harmony as the
fundamental note. It is needless also, because Dr. Riemann,

396 THE THEORY OF HARMONY

strange as it may appear, in the very work in which he has
developed his system of " under-klangs," " contra-fifth
klangs," etc., himself tells us that in the minor harmony the
loivest note ought to be regarded as the fundamental note.
He makes the following statements, surely the most extra-
ordinary, in the circumstances, which have ever proceeded
from a musical theorist. He says :■ — " The under-klang,
which on account of the peculiar dependence of its notes on
a higher principal note appears to tend downwards, first
receives d, firm basis through the choice of the under-fifth for
its bass note." ^ Consequently, he points out, this " under-
fifth is the best note to double, and the fundamental note,
the Prime, may be omitted " ! It is evident that Dr. Riemann
has an uneasy feeling that all is not right, for at this point
he adds a long note of explanation: — " In order more fully
to explain the somewhat strange-looking fact that in the
' under-klang ' the Fifth [that is, the lowest note of the chord]
forms the fundamental note [!] , we submit the following short
considerations." 2 The passage is too long to quote, but it
is worth reading. Its perusal, and the consideration of all
the facts, make the reader disposed to wonder how it is that
Dr. Riemann does not appear to have a sense of humour.
It is Dr. Riemann who, in the majority of his works, has
insisted that the minor harmony must not be regarded as
generated upwards ; also, that harmony must be understood
as a logical and rational science.

" Characteristic Discords."

The dissonant chords recognized b}- Dr. Riemann
(" characteristic dissonances ") are, in the major mode, the
chord of the Dominant Seventh, and the chord of the Added
Sixth — Rameau's Subdominant discord. Here the method of
procedure is similar to that of Rameau ; and so also the
explanation as to the necessity for adding a dissonant note to
the Dominant and Subdominant chords, namely, that by
such means the real character of these chords is rendered
perfectly clear, and there is no danger of their being mistaken
for Tonic chords. It is by no means certain, however, that

^ Harmony Simplified, Ch. i. - Ibid.

DR. RIEMANN— TONAL FUNCTIONS OF CHORDS 397

the tendency of the Subdominant harmony towards that of
the Tonic is made more decided by the addition of a dissonant
note to its harmony.

In the minor mode, as in the major, we find a Dominant
and a Subdominant discord. In a minor, the Dominant
discord is e-g%-b-d, and the Subdominant discord b'd-f-a.
Here the analogy which Dr. Riemann wishes to maintain
between the major and the minor modes again breaks down.
He fails to show why, in the case of the Subdominant discord
in the major, as f-a-cjd, the interval added above the major
harmony /-fl-c should be a Sixth, while in the case of the same
discord in the minor, as hid-f-a, the interval added below
the minor harmony d-f-a is a Seventh ; especially as the
" function " and meaning of both chords is the same.
There is really no good reason wh\- the Subdominant discord
in the minor should not appear, like that in the major, as a
chord of the Added Sixth : d-f-a/b.

It would be possible however to preserve a strict analogv
in respect of the construction of these two discords, if the
Sixth were added below the minor harmony d-f-a, in the same
way as the Sixth had been added above the major harmony.
Between the chord of the Added Sixth in the major f-a-c'd,
and the chord of the Added Sixth in the minor c d-f-a, there
would then exist a real analogy, as respects the construction
of these chords. Dr. Riemann however does not consider
this alternative.

Instead, he distinguishes a certain chord of the Added
Sixth in the minor mode which, he seems to imagine, is the
counterpart of the chord of the Added Sixth in the major.
This chord is d/e-g-b {a minor). It is however not a Sub-
dominant but a Dominant discord. It is difficult to consider
that any real analogy exists between a Dominant chord of the
Added Sixth in the minor mode, and a Subdominant chord of
the Added Sixth in the major mode. This chord die-g-b
appears to have been introduced by Dr. Riemann merely in
order to impart to his system of " characteristic dissonances "
an appearance of symmetry.

It may also be noticed that the Dominant discord in the
major, as g-b-d/f, and the Subdominant discord in the minor,
as b/d-f-a, are composed of exactly the same intervals. The
former consists of a major Third, perfect Fifth, and minor
Seventh ; the latter consists of the same intervals in

398 THE THEORY OF HARMONY

descending order. A strict analogy exists, therefore, as
respects their construction, between the Suhdominant discord
in the minor mode, and the Dominant discord in the
major mode. Dr. Riemann, however, brings forward a
new species of Subdominant discord in the major, namely,
dlf-a\f-c, which he regards as analogous to the Subdominant
discord in the minor mode bjd-f-a. But this new discord
has not an ascending but a descending construction. It
should properly have been compared with the minor
Dominant discord, which consists of exactly the same
intervals, but taken in ascending order. The major
key-system is now in possession of two Subdominant
discords, while the minor key-system has but one.

But with respect to the new Subdominant discord djf-a^-c.
Dr. Riemann had already plainly stated that " a\} is foreign
to the key of C major, as g^ is foreign to that of a minor [!]...
The contra-klang of the Tonic [/-ab''^] i^ really a plain-fifth
klang of the Tonic- Variant, i.e., of a Tonic of the other klang-
mode ; the F minor chord in C major is really the plain-fifth
klang [Subdominant] of the C minor chord." ^

" Parallel-kl.-\ngs."

Another feature which distinguishes Dr. Riemann 's work
from previous works on harmony, is his theory of what
he describes as " Parallel-klangs." Rameau, although he
had demonstrated that the key-system received its complete
definition by means of the three principal harmonies of Tonic,
Dominant, and Subdominant, had never been able to furnish
any adequate explanation as to the nature and origin of the
secondary triads of the key, with one exception, namely, the
diminished triad on the leading-note, which he had explained
as derived from the chord of the Dominant Seventh, through
■omission of the fundamental note. Dr. Riemann accepts
Rameau's explanation of this chord, and in doing so proves
his superiority to other theorists who have explained it as
an independent chord, and given it a place among even the
primary triads of the key.

1 Harmony Simplified, Ch. i.

DR. RIEMANN— TONAL FUNCTIONS OF CHORDS 399

Dr. Riemann's " parallel-klangs " are the secondary triads
on the second, third, and sixth degrees of the scale. In his
treatment of these chords he makes a notable attempt to
develop and complete Rameau's theory as to the origin of the
various triads of the key-system. He recognizes the necessity
which exists to explain these secondary triads as arising in a
different way from the primary. But unfortunately it is by no
means an easy matter to ascertain what exactly Dr. Riemann
wishes us to believe concerning these secondary triads.
In his Musiklexikon he gives the following terse definition
of "parallel-klangs": — "Parallel-klangs are klangs which
stand to each other in the relationship of tonics of parallel
keys ; for example, C major and A minor ; that is, klangs

which possess a third interval in common : a-c-e-g." If,

5th
tlien, in the major chord c-e-g, which represents the Tonic
chord of C major, we substitute the note a for g, we obtain
the " parallel-klang " c-e-a. This " parallel-klang," there-
fore, must be understood as follows : — The notes c-e represent
the fundamental note and Third of the Tonic chord of
C major, while a is the Tonic of a minor. The chord
therefore is derived from two keys, C major and A minor.

The triad on the second degree of the C major scale would
therefore appear similarly to be derived from the keys of
F major and D minor ; and that on the third degree from the
parallel keys of G major and E minor. It is clear that Dr.
Riemann cannot mean to present this as an adequate ex-
planation of the origin of the secondary triads in question.

This explanation of the secondary triads on the second,
third and sixth degrees of the scale would appear to apply to
the major key-system only; there are obviously serious
difficulties in the way of its application to the minor key-
system. For example, the triad on the second degree of
the minor scale is a diminished triad, while that on the third
degree is augmented.

Dr. Riemann, however, gi\-es another and a quite different
explanation of the " parallel-klang." He explains the Sub-
dominant " parallel " as being derived from the " character-
istic dissonance " on the Subdominant, as f-a-cid, by means
of the omission of the Fifth, C. The other " parallel-klangs,"
that on the sixth degree of the major scale, a-c-e or c-e-a, and
that on the third degree, e-g-b or g-b-e, cannot, however, be

400 THE THEORY OF HARMONY

similarly explained. " These cases have to be explained in a
different way, since for the tonic there can be no characteristic
dissonance, and the dominant klang, with its own character-
istic dissonance, cannot produce any quasi-consonance." ^
But Dr. Riemann fails to discover any adequate explanation
of the " cases which have to be explained in a different way."
It is impossible to understand why he should describe the
triad d-f-a as a " parallel-klang," seeing that it is derived from,
and represents, in incomplete form, the Subdominant discord
f-a-c/d. There is another reason why this triad cannot be
considered as a " klang " : it consists not only of a dissonant
Fifth, d-a (27 : 40), but of a dissonant Third d-f (27 : 32).

But we find still another explanation of these " klangs " as
" leading-tone-change-klangs " {Leittonwechselkldnge). It is
evident that Dr. Riemann has no settled idea as to what his
" parallel-klangs " really are, and what they really stand for.
Still, if we select one out of the various and contradictory ex-
planations which Dr. Riemann has given of these " klangs,"
it is possible to perceive what it is he is principally aiming at.
His theory of " parallel-klangs " is the necessary complement
of his theory of " tonal functions of chords," in which he lays
down the principle that every chord within the key-system
must have either a Tonic, a Dominant, or a Subdominant
significance. When, then, he defines " parallel-klangs " as
' ' klangs which stand to each other in the relationship of tonics

of parallel keys " as a-c-e-g, he is evidently of opinion that he
has demonstrated the possibility of considering both chords
as having a parallel or similar " tonal function." That is,
he wishes us to consider the minor " klang " on the Sub-
mediant in C major — or is it the Tonic in A minor ?^ — as having
the same harmonic significance as the major " klang " c-e-g,
that is, a Tonic significance. Similarly, we must consider the
" klang " on the Supertonic to have a Subdominant, and the
" klang " on the Mediant a Dominant significance, or function.
It is not difficult to discover whence Dr. Riemann has derived
this theory. It was Helmholtz who stated, in his explanation
of the minor harmony, that the minor triad, for example,
a-c-e, may appear in the form c-e-a, in which form it is to be
considered as a C klang, in which the foreign note a takes

^ Harmony Simplified, Ch. i.

DR. RIEMANN— TONAL FUNCTIONS OF CHORDS 401

the place of g. Unfortunately Dr. Riemann, in several of
his works, has made it one of his principal tasks as a theorist
to demonstrate the utter impossibility, even absurdity, of any
such explanation of the minor harmony, which must be
regarded as the antithesis of the major, and as being generated
dowTiwards, not upwards.

The Three Tonal Functions of Chords.

In Rameau's explanation of the diatonic ke3^-system as
determined by the three primary harmonies of the key, we find
the origin of Dr. Riemann 's theorj- of the " tonal functions of
chords." Every chord within the key-system must, according
to Dr. Riemann, represent one or other of the three chief
harmonies of the key. This is an important theoretical con-
ception, and one which Dr. Riemann, in Harmony Simplified,
has made a notable attempt to develop. If such a theory is
really feasible, the result undoubtedly is greatly to simplify
the science of harmony. But the difficulties in the wa}" of its
application as a theoretical principle are not a few.

Dr. Riemann, then, sets himself to demonstrate that every
chord within the key-system has, and must have, either a
Tonic, Dominant or Subdominant function or significance.
For example, the secondary triad on the sixth degree of the
scale of C major, a-c-e, or rather c-e-a, is a Tonic " parallel,"
and has a Tonic significance, because the chord represents the
C major " klang," into which the foreign note a is introduced.
This, as we have seen, is the explanation which Helmholtz
has given of this minor chord. This being the case, Dr.
Riemann is of opinion that the Deceptive Cadence, in which
this chord plays a part, is the result of a Dominant-Tonic
succession of harmonies.

He gives the following examples of the Deceptive Cadence : —
(a) _ (6) {c)

^r\

;q:

-i^-

-o-

--1-

-g^

fJ?^

'JHZ

-G>-

221

-<s»-

-Bz

IQ.

^vzs;

=Tp

402

THE THEORY OF HARMONY

In examples {a) and (b) Dr. Riemann is of opinion that the
ear understands a Dominant-Tonic harmonic succession in the
key of C major, and in example (c) a Subdominant-Tonic
succession in the same key ! If not, his use of the signs D — T
and S — T has no meaning. The manner in which the ear
understands the chord c-e-a to be derived from c-e-g he has
already explained. It is more difficult to understand how
e]}-g-b\^ can represent and be derived from the chord c-e-g.
The process of evolution is as follows : — c-^-^ = the " Tonic
Variant," c-£\f'g = the "Tonic-parallel" of this "Variant"
b\f-e\^-g. Therefore c-e-g == l\}-e\f-g !

In the same way, the following are to be understood as
Dominant-Tonic successions in A minor ! The chord /#-fl-c$
is derived from a-c-e as follows : —

xTp

-u

1=^

a-c-e = the "Tonic Variant," a-cj^-e = the Tonic -parallel of
this " Variant " fl-c#-/#. These are extraordinary results.
In the following harmonic succession {a) : —

^==2=

321

:c3:

321

(D)[Tp]

(D) Tp

Dr. Riemann considers both chords to be in the C major
key. He is evidently much puzzled as to how the first
chord e-g^-b should be denoted. First he marks it as (D)
because " it has a kind of Dominant significance," but
assigns to it also the mark (Tp) {c-e-g = b-e-g = b-e-g^) .

DR. RIEMANN— TONAL FUNCTIONS OF CHORDS 403

But, he remarks, " the g^-gl^ as a cadential step is not
quite logical." In order, then, to obtain for this chord
" real cadential significance " it should be understood
as at (b). Dr. Riemann also suggests the possibility
of explaining this chord succession as arising through
an " elision." That is, a " mediating " chord a-c-e is under-
stood (c) . In one and the same harmonic succession, therefore,
the chord e-gjf^-b may be understood in three different ways :
first, as a " kind of Dominant," next, as a " Tonic-parallel,"
lastly, as a " Dominant-parallel."

Similarly with the succession a\^-c-e\^ — c-e-g, which Dr.
Riemann explains in the following different ways : —

(a)

(c)

-mi

=2^:

5g^

:cz:

-f^-f^-

:^g=tt|

*iS-

=68:

:i2Q:

:ai

JQI

^Sp

Svii'

:Uz3=-^z=:

IQI

He apparently considers that in such a succession a\f-c-d^
may quite well be substituted for a\^-c-e\f, just as e-a\}-b may
be substituted for e-g^-b.

Dr. Riemann gives several other examples of chords (of
which only a few need be quoted) to which he is unable
to ascribe either a Tonic, Dominant, or Subdominant
significance : —

ZCjL

-o-

7-
\'
•2"

Nevertheless he says of them, " as they arc (//msi-consonances,
they share with all such the peculiarity that they may be

404 THE THEORY OF HARMONY

treated as real harmonies." That is, although they " may-
be treated as consonances," they must nevertheless not be
understood as such. But if they are ^was^'-consonances, they
represent at least as real harmonic formations as the " parallel-
klangs " to each of which Dr. Riemann has found it possible
to assign a " tonal function." The real difficulty is, of course,
that Dr. Riemann is unable to account for these chords, or to
explain their " tonal functions," except that they arise from
" leading-tone steps."

A word must also be said with respect to Dr. Riemann 's
theory, or rather theories, a$ to the origin of chords. Of
consonant chords there are but two, the major and the minor
harmonies. Of dissonant chords. Dr. Riemann has stated that
" all dissonant chords are to be explained as modifications of
overklangs and underklangs." He therefore follows Kirn-
berger in a notable attempt to reduce all harmonic formations
to a few simple primary chords, and in making a firm stand
against all theories of " fundamental discords," or of chord
formation by means of added Thirds, he has done a real service
to the science of harmony. But in getting rid of the " added
Third " theory he by no means gets rid of the difficulties
which beset the problem of chord generation. He has made
us acquainted with a number of " characteristic discords "
which cannot properly be said to arise from the modification
of a major or minor harmony. For example, the chord of the
Added Sixth, f-a-c/d, cannot be understood as a modification
or alteration of the major hannony, f-a-c. Nor can the note d
in this chord be regarded as having no harmonic significance, as
a non-harmonic note. Dr. Riemann has stated that the
addition of this dissonant Sixth " renders the meaning " of
the Subdominant harmony " still clearer."

Melodically Altered Chords.

As for all other dissonant formations, he is of opinion that
these have their origin in " melodic figuration " ; that is, they
represent in reahty modifications of " overklangs " and
" underklangs " brought about by means of the introduction of
passing- and auxiliary-notes, etc. Thus the chord * in
example {a) has its simple origin in a chromatic passing-note

DR. RIEMANN— TONAL FUNCTIONS OF CHORDS 405

(b) ; wliile the chord h-d-f-a in example (c) has its origin in a
suspension {d)^: —

(h)

{c)

=^S

{d)

-&-

-«5^

-f3-

-«S>-

-■^

But let us take from among many such chords the familiar
chord h-d-f-a, as in example \e) : —

r>7

Here the note a cannot ver}^ well be explained as arising
from a suspension, or a passing-note. It is true that by
means of distorting the melody a plausible explanation of the
note may be found (/). In any case, Dr. Riemann assumes that
the note a is the note of melodic figuration. He may be right ;
but, theoretically considered, this is a mere assumption.
Why should not the notes d-f-a, rather than h-d-f, represent
the true harmony-notes, and h the note of melodic figuration,
which arises as a passing-note, as at {^ . This might even be
maintained to be the more reasonable view, seeing that the
" parallel-klang " d-j-a is ^'i/asi-consonant, while h-d-f is
decidedly a discord. Still other views are possible. For
example, / and a might be regarded as the actual harmony
notes, while d arises as a passing-note, and h as an auxiliary-
note (A).

But, as we shall see, Dr. Riemann is of opinion that
" harmony has its roots in melody." Hauptmann, not without
reason, has stated that the essence of melodic succession is

Harmony Simplified, Ch. 3.

4o6 THE THEORY OF HARMONY

progression by step. In the chord we are considering, then,
it would be reasonable to assume that the notes of melodic
figuration are those which proceed by step, namely b-d-f,
while the note a, which proceeds by leap, is the harmony note,
representing perhaps the fundamental note of the " parallel-
klang " a-c-e.

But it is really astonishing, in the case of theorists who
claim that harmony arises from melody, to observe with how
little compunction such theorists distort and torture the
melody for which they profess so much regard, and even lop
off a member here and there from a chord, in order to make
it fit the Procrustean bed of some preconceived harmonic
formation.

Another species of discord explained by Dr. Riemann is that
which owes its origin to a " leading-tone step progression " : —

(«) {h) {c) id) {e)

^ — ^-

-g=

-.M^

Z.CfZ

:g=

jci-j:z=^

-3t^

:z5:

^fS-a

'^m^

1 I I

Thus in the chord at [a] (key of C major) the d^ is a leading-
note which tends towards and takes the place of the real
harmony note c. The real origin of the chord is seen at (6).
This is at least an intelligible and even reasonable explanation.
Dr. Riemann, however, does not consider it necessary
to inform us how this note d^, in Hauptmann's language,
" acquires definiteness." It does not even appear that
definiteness of intonation is necessary. Not being harmoni-
cally determined nor possessing any independent harmonic
significance, it is apparently only necessary to intone this
sound as leading-note to c. Consequently neither the singer
nor violinist will be careful to be accurate in his intonation
of d^; indeed it is very improbable that he could be accurate,
if the tuning of d\) is to be determined by means of the
sounds of the chord which appear below it.

We now come to a discord consisting of two leading-notes,
d\) and/, and one harmonic note, g, as in example (c). This
chord finds its explanation at {d). Still another discord is
that composed entirely of leading-notes, and in which there is,
presumably, no harmonic note (<'). There can be no question
as to the dissonant character of this chord : for the notes

DR. RIEMANX— TONAL FUNCTIONS OF CHORIJS 407

f-a\f-d^, taken in free melodic intonation, represent as dis-
cordant a combination as could well be desired. Dr. Riemann,
however, is of opinion that this chord may be considered as
consonant. It is, in fact, not only a consonant chord, but
one of the most important consonant chords used in harmony.
In a similar way. Dr. Riemann might consider that in the
following passage, which is taken from Wagner's Tristan : —

etc.

the chord at * represents a discord composed entirely, with the
exception of the lowest note g^, of chromatic leading-notes.
These notes, therefore, have only a melodic significance.
They introduce no real change of harmony. Throughout the
whole passage, the only actual harmony, whether from a
purely theoretical point of view, or from the effect produced
upon the ear, is that of the Tonic chord g^-b^-d)^. All this
of course is the grossest travesty of the actual facts. The
chord in question does, very decidedly, introduce a fresh
harmony, and brings about a very real harmonic change.

In Haymony Simplified, Dr. Riemann has endeavoured to
develop and establish two main theoretical principles, neither
of which is new, but which are derived from other theorists.
The first is that not only the minor harmony but the minor
key-system must be regarded as the direct antithesis of the
major harmony and major key-system. It can scarcely be
maintained that Dr. Riemann has succeeded in establishing
this part of his theory. On the contrary, anything more
topsy-turvy it would be difficult to imagine, e.xcept it be his
treatment of the " parallel-klangs." First, the minor harmony
arises from the " undertone " series, and is the antithesis of
the major, which arises from the " overtone " series. Secondly
the objective existence of a series of " undertones " cannot
be proved, and therefore the minor harmony can only be
explained as a psychological fact, as a fact of experience.

4o8

THE THEORY OF HARMONY

That the highest and not the lowest note must be regarded as
the fundamental note of the minor harmony results also as a
fact of experience. But thirdly, in practice the lowest note
of this harmony should be regarded as the fundamental note,
while the real fundamental may be omitted without altering the
fundamental position of the chord.

Dr. Riemann's second main theoretical principle is that
every chord within the key-system must have either a Tonic,
a Dominant, or a Subdominant " function " or significance.
He completely fails to prove this, for the very good reason
that there are chords within the key-system which do not
possess a Tonic, Dominant, or Subdominant significance.
Of the existence of such chords he is himself aware, but is
unable to discover any adequate explanation of them. In
order to ascertain the exact value of Riemann's theory of
" tonal functions " and " parallel-klangs," all that is needed,
it might be imagined, is to cast a glance at his designation of
the following chords (in C major) : — ■

-0-

^O"

-r>«^-

:-tLo^:

:g=

= =i2^i

IzJLq:

^-n-p

zfeo:

3Tp

4 :
Dp

rt^Qi

Svii

-C3_

It is an offence not only against the ear but against the
intelligence of the average musician to ask him to believe that
chords such as a\}-c-e\^ and e\f-g-b\^ stand for, or represent, the
Tonic chord c-e-g.

Chord-succession : Basis of the Theory of Harmony.

Dr. Riemann's theory of tonal functions has evidently been
inspired by Rameau's fundamental bass in Fifths. But
although his appreciation of the theoretical significance and
value of Rameau's bass does him credit, he is by no means
prepared to accept its limitations. He is quite prepared to
accept Rameau's fundamental bass, but he is unwilling to be

DR. RIEMANN— TONAL FUNCTIONS OF CHORDS 409

liampered by the difficulties which arise in connection with it.
He sees no difficulty in the way of the immediate succession of
l^oth Dominants, and is even of opinion that Rameau evinced
a quite unnecessary sensitiveness on this point. He is by
no means certain that it is the fundamental bass which deter-
mines the sounds of the scale. He is even less prepared to
state that it is the fundamental bass which determines and
explains chord-succession, although it might have been
imagined that this was one of the principal objects of his
theory of " tonal functions." In the " Introduction " to his
work we read : — " The theory of harmony is that of the
logically rational and technically correct connection of chords
(the simultaneous sounding of several notes of different pitch).
The natural laws for such connection can be indicated with
certainty only if the notes of single chords be regarded, not
as isolated phenomena, but rather as resulting from the motions
of the parts."

If this statement has any meaning for the theory of harmony
then we must consider chord-succession to be determined by
the " melodic tendencies " of the parts. In the perfect
cadence, then, (a) , we must believe that the progression of the
first chord to the second is brought about by the tendency of
/; to proceed to c, of d to e, and so on : —

(a)

(b)

-fS-

"O"

jOL

-«s-

:c5i

'&-

-o-

"O"

-JOT-

'JC21

'jrjr

jCSI

ZZ21

-<5»

In the half-cadence {h) the succession is determined by
the tendency of c to proceed to b, of e to d, and so on, that
is the sounds have now exactly the opposite tendencies !
Is it only in the case of " over-klangs," " under-klangs,"
" contra-klangs," etc., that we must consider the constitutive
harmonic elements of a chord to be " isolated phenomena " ?
Is it by means of the "motions of the parts " that Dr. Riemann
has been enabled to determine the connection between the

4IO THE THEORY OF HARMONY

harmonies in the examples of " parallel-klangs " just given ^
and to assign to each chord its correct tonal function ?

Indeed, one of the puzzles presented by Dr. Riemann's
works on harmony is that of ascertaining what exactly is the
position of their author with respect to the basis of the theory
of harmony. At one time he discovers for it a physical, at
another, a psychological basis. As a rule h& accepts both.
He suggests that harmony has its origin in melody. It is-
not for nothing that Dr. Riemann makes this statement. He
has in view the large number of chords for which he can
find no explanation except that they arise through melodic
figuration. But immediately after this pronouncement we
find him engaged in investigating the " undertone " and
" overtone " series, and numerous species of klangs.

It is probably in order to justify this procedure that he
remarks towards the end of his work : — " Harmony is certainly
the fountain-head from which all music fiows, but the diatonic
scale is the primeval bed, the banks of which the stream may
at times overflow, but into which it is always forced again " '
Ought we to conclude from these remarks that, in the first
place, harmony is derived from the scale, and that, in the
second place, the scale is derived from harmony ?

Dr. Riemann has based his work, as of course he has a
perfect right to do, on principles derived from Rameau^
Kirnberger, Fetis, Hauptmann, and Helmholtz. These
principles, however, frequently mutually opposed to one
another, we find strangely jumbled together. Dr. Riemann
has adopted certain theories without having sufficiently
considered whether, in the first place, they are tenable, or
where, in the second place, they are likely to lead him ;
witness his operations with regard to the " undertone " series,
and the fundamental note of the minor harmony. In Harmony
Simplified we have the latest noteworthy attempt to evolve
a logical harmonic system, by one of the most erudite musicians
and theorists of his day. The whole work is an eloquent
testimony, not only to the enormous difficulties of the subject,
but as to the actual state of harmonic science at the beginning
of the twentieth century. In no previous work of the kind —
not even in Helmholtz- -docs one observe such extraordinary
uncertainty, hesitation, and evasion as to what constitutes-
the fundamental principles, and indeed even the proper
basis of harmony.

ENGLISH THEORISTS— ALFRED DAY 411

CHAPTER Xni.

ENGLISH THEORISTS : DAY, MACFARREX, OUSELEV,

staixer, prout.

Day's Treatise ox Harmoxy.

Of works on harmony by English writers, the first and in
many respects the most important to be mentioned, is the
Treatise on Harmony by Alfred Day, M.D. Dr. Day's treatise
represents a characteristically straightforward attempt to
reduce harmony to its fundamental principles, and to evolve
from such principles a rational theory of harmony. In the
" Preface" to his work he remarks: — "The following work is
the result of immense labour during the leisure time of many
years." The work itself was pubUshed in 1845, only a few
years before his death (1849).

Dr. Day divides his work into two main sections. In the
first he treats of Diatonic or Strict, and in the second of
Chromatic or Free, hannony. In so doing he makes some
remarkable distinctions. In the first section he explains the
major scale as determined by the three principal harmonies of
the key : — " The foundation of the major scale is the common
chord of the tonic, which supplies the first, third, and fifth of
the key ; of the dominant, which supphes the major seventh
and second, and of the subdominant, which supplies the
fourth and sixth. " ^ Likewise, with regard to the sounds of the
minor mode, in which we find a minor harmony on the Tonic,
a major harmony on the Dominant, and a minor hamiony on
the Subdominant. In the second part of his work however

^ Treatise on Harmony, Pt. I., Ch. 2.

THE THEORY OF HARMONY

he gives, as we shall see, quite a different explanation of
both the major and minor scales.

Again, while " Diatonic " harmony allows of such harsh

combinations as the following * : —

it does not permit of such comparatively innocuous harmonic
successions as those of Dominant and Diminished Sevenths
preceded and followed by the Tonic chord {a, b) : —

(«)

:gi

(^)

:?2^

-@-

-o-

-&-

the reason being that in the first case the discords are, or are
said to be, prepared, while in the second they are taken
without preparation. The chord of the Dominant Seventh
therefore, when prepared, belongs to Diatonic harmony ;
when unprepared, to Chromatic harmony.

Not only so, for this chord has two different origins : if
prepared, the dissonant note is derived from the Subdominant ;
if unprepared, the whole chord, dissonant note included, is
generated from the Dominant. These are a few of the curious
distinctions drawTi by Dr. Day between diatonic and chromatic
harmony.

ENGLISH THEORISTS— ALFRED DAY 413

Chromatic Harmony : The Key-system :
Generation of Chords.

In the second part of his work, entitled Chromatic Harmony ^
or Harmony in the Free Style, Dr. Day treats of what he calls
" natural discords." He remarks : — " Diatonic discords
require preparation because they are unnatural ; chromatic
do not, because they may be said to be already prepared hv
nature."'^ He therefore suggests that, for example, the chord
of the Dominant Seventh, when prepared, is an " unnatural
discord " ; the dissonant Seventh is not derived from nature.
On the other hand, when the chord is taken without prepara-
tion, it is a " natural discord " and derived directly from
nature.

Dr. Day's methods of procedure in respect of chord gene-
ration are similar to those with which previous works on
harmony have made us already familiar ; namely, the selection
of certain sounds as " roots," and the building up upon these
roots of chords and discords by means of sounds selected from
tlie harmonic series.

" The harmonics from any given note (without taking the
order in which they arise, but their practical use) are," he
remarks, " major third, perfect fifth, minor seventh, minor or
major [!] ninth, eleventh, and minor or major thirteenth."
He does not suggest that these represent all the sounds of the
harmonic series ; there are, of course, many more. Dr. Day is
evidently in no doubt as to what sounds he requires. He does
not tell us how he has gained this knowledge ; certainly not
by the study of the sounds of the harmonic series.

From the sounds thus derived he obtains a major common
chord, a chord of the minor Seventh, and so on, up to the
chord of the major Thirteenth. These chords may be con-
sidered to arise from the Tonic ; from the Fifth of the Tonic
(Dominant), and from the Fifth of this Fifth {Siipertonic) ; the
reason for this being that " the harmonics in nature rise in the
same manner : first, the harmonics of any given note, then
those of its fifth or dominant, then those of the fifth of that
dominant." But here Dr. Day quite overlooks the existence of
the Third {Seventeenth) of the Tonic, which arises before the

1 Treatise on Harmony, Pt. II., Introduction.

414

THE THEORY OF HARMONY

Fifth {Tiaelfth) of the Dominant. If he is guided as he
professes to be by nature, and is selecting his " roots "
according to the manner in which they arise in nature, then he
must include the major Third of the keynote as a " root "
before he proceeds to the Fifth of the Dominant, which only
arises after this Third.

He informs us, however, that the Tonic, Dominant, and
Supertonic are the three " roots " from which all chords in the
key-system, major or minor, are derived. Conversely, all
chords derived from these " roots " belong to one and the
same ke3^ It should be observed that the chord of the
Eleventh can appear only on the Dominant : —

-g?"

Tonic.

Dominant. Supertonic.

One may observe, also, that although the order of " roots "
is deteiTnined, according to Dr. Day, by the manner in which
they arise in nature, the order of sounds in the chords which
spring from them is not thus determined. In the harmonic
series we find first an Octave, then a Fifth, then a Fourth, and
so on, the intervals gradually becoming smaller. But in Dr.
Day's chords of the Tonic and Supertonic Thirteenths, the
largest interval is at the top. ]\Iust we understand this as
brought about by means of the omission of a Third ? In any
case, this is a defect which has been remedied by some of
Dr. Day's disciples.

The reason why the order of " roots " cannot be continued
beyond the Fifth of the Dominant is that, in the case of
the next Fifth (Fifth of the Supertonic) " that note itself is
not a note of the diatonic scale, being a little too sharp." ^
It is important to observe, then, that Dr. Day makes a sharp
distinction between the sixth degree of the major scale and the
Perfect Fifth above the Supertonic of this scale. Such a
distinction is necessary, and is one made by every theorist of

Treatise on Harmony, Pt. II., Introduction.

ENGLISH THEORISTS— ALFRED DAY 415

importance. The difference between the two sounds is that of
a comma (80 : 81).

We are now in possession of all the sounds of the Diatonic
Scale, major or minor. " The notes of the diatonic major scale
are produced in the following manner : C (tonic) produces G,
its fifth, and E, its major third ; G produces all the rest, as
D its fifth, B its third, F its seventh, and A its major ninth.
The minor scale in a similar manner : £[>, the minor third, is an
arbitrary, not a natural third, of C." ^ But the three " natural "
and fundamental discords contain not only the sounds of the
diatonic, but also of the Chromatic Scale, which, then, ought
to be written thus (C major or minor) : —

—1-

■ 1 ^ 1 \r^

EjEi^=§5i=g=^^^^^^^

The same method of notation should be employed for the
descending chromatic scale.

But not only the major and Ininor (harmonic) scales, and the
various kinds of " natural discords," but also the common
chords which occur in the diatonic scale, are derived from the
same source. Thus the Tonic " root " produces its own
common chord ; " the minor [!] common chord on the major
second of the scale is part of the chord of the minor seventh
and major ninth on the dominant ; the common chord, major
or minor, on the Subdominant, is part of the chord of the
eleventh accompanied with the seventh, and either major or
minor ninth ; the major common chord on the minor sixth
of the scale is part of the chord of the minor thirteenth,
accompanied by the eleventh and minor ninth ; the minor
common chord on the major sixth of the scale is part of the
chord of the major thirteenth, accompanied with the
eleventh and major ninth."- With regard to the common
chord on the major third of the scale, it "is not allowed,
because it appears to belong to another key." As for
chord-succession, a chord will proceed to another chord
derived from the same " root," or from either of the other
two " roots." Much in the same way, a discord will resolve
either on its own " root " or on a chord derived from
another root.

^ Ti-eatise on Harmony, Pt. II., Ch. i. 2 Jjjid^

41 6 THE THEORY OF HARMONY

Day's " Fundamental Bass " ; System of " Roots."

Such then in brief is the system of Dr. Day, which, in some
respects, suggests to us the " simple and comprehensive "
system of Catel, who also derived the various chords of which
he had need from a single chord. But even more striking is
the resemblance to be observed between the principles which
influenced Dr. Day, and those which formed the basis of the
theory of harmony of Rameau. Both agree that all the notes
of the scale are developed from a single sound — the Tonic ;
that all chords must be developed from a single chord (for
Dr. Day does not present us with three different chords, but
with the same chord on different notes of the scale) ; and
further, that the " roots " or fundamental sounds (" funda-
mental bass ") of the key-system are three in number. But
if the principles of both theorists present a striking re-
semblance, the difference between the results obtained is still
more striking. Not only in musical intuition, but in
theoretical acumen. Dr. Da}' proves himself to be much the
inferior of the great Frenchman.

At the outset. Dr. Day lays down a definite principle that,
he says, should guide us in determining which sounds ought
to be accepted or rejected as roots. The Fifth of the
Supertonic cannot be accepted as a " root," because it is
sharper (80 : 81) than the major Sixth of the scale. He also
states that the minor Third Ej?, for example, is not a root
in the key of C, because its minor Ninth F[7 contradicts the
major Third E, the difference between the two intervals
being the enharmonic diesis (125 : 128) ; also that wherever
" this enharmonic diesis takes place it always implies a
change of key." Such being the case, one naturally expects
that the principle which applies in the case of " roots " wdll
apply also to the sounds of the " natural discords "
which arise from these " roots." But if we take the discord
of the Dominant Thirteenth g-b-d-f-a-c-e, which we have
been led to suppose is generated from the " root " g, we
shall find that the majority of its sounds do not belong to the
scale of C major. The sound / is decidedly flatter (63 : 64)
than the fourth degree of the major scale of C ; a is sharper
than the major Sixth (80 : 81) ; c is almost a quarter-tone
(32 : 33) sharper than the Tonic, while e is much flatter

ENGLISH THEORISTS— ALFRED DAY 417

(39 : 40) than the third degree of the scale. According,
therefore, to the principle laid down by Dr. Day himself,
all the sounds f-a-c and e ought to be rejected. Nevertheless,
Dr. Day informs us, from these sounds f-a-c-e we obtain the
harmonic or consonant major harmony f-a-c, and the con-
sonant minor harmony a-c-e. The proportions of the first are
7:9:11, and of the second 9 : 11 : 13. We therefore obtain
harmonic and consonant (!) formations hitherto unknown to
any musical system.

We presume, of course, that these sounds represent re-
spectively the seventh, ninth, ele\'enth, and thirteenth upper
partials of the " root " g. It is true that Dr. Day makes no
absolutely definite statement to this effect. But although
he has said that he has selected the harmonic sounds, not
according " to the order in which they arise, but their prac-
tical use," it is difficult to conceive that he imagines himself
at liberty to select, in an arbitrary way, whichever sounds he
pleases from the harmonic series. If so, the subject is hardly
worth discussing furtlier. To pick out sounds here and there
in such a way is in itself, no doubt, a quite harmless amuse-
ment , but it is decidedly erroneous to dignify such a procedure
by describing it as harmonic science.

It is difficult to understand by what method Dr. Day obtains
the sounds he requires. For example, he is able to present us
not only with a major, but with a minor thirteenth. Nature
provides him with neither, for the thirteenth harmonic sound
is neither a major nor a minor thirteenth, but is, one may say,
between the two. It would appear that Dr. Day considered
this a sufficient reason for making use of both.

But with regard to the sounds f-a-c-e. Dr. Day tells us that
these are sounds of the C major scale. If so, they cannot be
derived from the harmonic series of which g is the prime.
Further, it is impossible to understand how Dr. Day can
describe such a combination as a "natural discord"
generated from its " root " g. Dr. Day professes to be
guided by nature ; but it would seem that it is nature which
requires the guidance of Dr. Day. He has led us to beheve
that he is going to produce certain sounds from the harmonic
series ; he does not produce them, but furnishes us instead
with quite other sounds, which he has obtained from no one
knows where.

But let it be supposed that the sounds comprised in the

41 8 THE THEORY OF HARMONY

Dominant discord g-h-d-f-a-c-e are in reality those of the
C major scale. In that case, it is clear that the sounds of the
Supertonic discord d-f$-a-c-e — b cannot likewise belong to this
scale. For example, a in this discord is sharper (80 : 81) than
the a of the Dominant discord. This a is the (perfect) Fifth
of the Supertonic, the same sound which Dr. Day rejected as
a root because it was not the real sixth degree of the major
scale. It ought to be rejected now. Similarly with regard
to the Tonic discord c-e-g-b\^-d — a.

Some apologists of the Day system, notwithstanding that
Dr. Day himself draws a distinction between sounds which
differ by a comma (80 : 81) have sought to defend Dr. Day's
use of " natural discords " by references to our tempered
scale, in which, with the exception of the Octave, everything
is more or less out of tune. For certain theorists temperament
reconciles all things. It is evident that such theorists have
not contemplated what would be the result if the sounds of
the " natural discords " on the Tonic, Dominant, and Super-
tonic, all of which, Dr. Day has assured us, belong to the
major key-system, were actually placed in tune. What sort
of scale would emerge from such a confusion of sounds ?
Notwithstanding the large number of new sounds and intervals,
hitherto unknown in harmonic music, now in our possession,
the effect of which would be to bring about a complete change
in our harmonic system, and for which a new notation would
require to be in\'ented, we would still be without the sounds
necessary to form a harmonic triad or consonant major
harmony on the Subdominant, or a consonant minor harmony
on any degree of the major scale, and similarly for several
other of the most important chords of the key.

Although some of the combinations which Dr. Day succeeds
in evolving from a single " root " are about as harsh in effect
as any one could well desire, he explains, as is known, the
comparatively mild discord of the Augmented Sixth as
derived from a " double root." Thus, in the case of the
Augmented Sixth £?[?-/#, a\} is the minor Ninth of the " primary
root " g, while /if is the major Third of the " secondar}^ root "
d. For this he has been much criticized, and somewhat
unjustly, for to explain a discord as arising from a " double
root," as Hauptmann did, is much more sensible than to
explain it as arising from a single " root." Dr. Day, however,
is far from identifying himself with Hauptmann's theory of

ENGLISH THICORISTS— ALFRED DAY 419

the dual origin of discords. The chord of the Augmented
Sixth is the only chord he explains in this way, and it is an
explanation which would appear to be forced upon him by
the circumstances of his theory. As it is, he still retains his
conception of the " primary root " as the ultimate source of
both sounds a\^ and /#.

The Minor Harmony and Minor Mode.

As for the Minor harmony and the Minor Mode, Dr. Day
merely touches the fringe of one of the most difficult problems
connected with the subject of harmony. In the case of the
minor harmony c-e\}-g, he tells us that e\^ is an arbitrary sound
(which, apparently, has strayed into a place where it has no
right to be), and in the Tonic discord in the minor mode, he
actually substitutes for the minor Third c-e\^, the major Third
i-clq. In the minor as well as the major mode, then, we find
a major harmony. Other theorists have regarded the minor
as the antithesis of the major mode ; Dr. Dav demonstrates
their identity.

The only form of the Minor Scale which he thinks to be
worthy of consideration is the so-called " Harmonic " form,
as a-b-c-d-e-f-gi/^-a . " Here," he remarks, " no major si.xth
nor minor seventh is to be found ; and, strictly speaking, no
major sixth nor minor seventh should he used. . . . This scale
may not be so easy to some instruments and to voices as the
old minor scale ; therefore, let all those who like it practise
that form of passage, but let them not call it the minor scale." ^
Dr. Day evidently intended this as a warning to composers.
Unfortunately, by the time Dr. Day's Treatise had appeared,
much mischief had already been done by composers such as
Bach, Handel, Haydn, Mozart, Beethoven, Schubert, Chopin,
Mendelssohn, and many others, who not only used other
forms of the minor scale, but even treated chords such as c-e-g
and e-g-b, as if they belonged to A minor. But it is un-
necessary to refer to tl*e practice of the great composers.
Even on purely theoretical grounds. Dr. Day's views with
regard to the minor mode appear inadequate. He makes an
arbitrary statement, which is little more than a mere

^ Treatise on Harmony, Pt. I., Ch. 2.

420

THE THEORY OF HARMONY

expression of opinion. The difficulty with regard to the
ascending and descending Melodic forms of the minor scale
cannot be solved by aboUshing these forms. The question
is a more difficult one than he seems to imagine. At the
same time, he is a better theorist than to adopt the easy-going
explanation of the harmonic form of the minor scale as
arising from a chromatic alteration of an old Church mode.

The Subdominant : the Augmented Triad.

Another original feature of Dr. Day's theory is his treatment
of the Subdominant harmony. We have seen the difficulties
which other theorists have experienced in connection with
the Subdominant. These difficulties do not exist for Dr. Day.
He gets rid of them all by getting rid of the Subdominant
itself. The harmony /-fl-c, in C major, does not, as a matter
of fact, represent the Subdominant harmony : that is,
/ is not the " root " of the chord. It is really a part, and
indeed, the most dissonant part, of the Dominant discord
g-b-d-f-a-c. The Subdominant harmony, therefore, is not a
concord but a discord, and represents the discord of the
Dominant Eleventh. Nevertheless he repeatedly refers to
the " Subdominant " as if, in his theory, such a term had any
meaning, and he even speaks of a modulation to the Sub-
dominant key. If we accept Dr. Day's view, we must regard
the following Cadence: —

(«)

(^)

IC>"

-IS>

---g---

"c?"

11^^

iq:

zai

-o

-c?"

not as a Subdominant-Tonic, buf as a Dominant-Tonic
succession of chords, in which the bass makes a leap from
the Seventh of the " Dominant discord " f-a-c, to the
" root " of the Tonic chord. In the reverse progression {b)
the bass make^ a leap from the " root " of the Tonic chord

ENGLISH THEORISTS— ALFRED DAY * 421

to the Seventh of a Dominant discord. In the Interrupted
Cadence (c) : —

-Gh-

IC2Z

-<5>-

the fundamental bass, the " root," does not move at all ;
both chords being derived from the Dominant, we have no
real succession of harmonies. The second chord is not
consonant, but dissonant, and represents the Ninth, Eleventh,
and Thirteenth of the Dominant discord. Again, in the
following passage {d) : —

0 ^'^-.

1 — ,

TT ^ ^

1 ''

im <^

«.-j

VW X. /TJ

•^ ^

«5

i^\

1
1

1
i

-Gt-

-f^-

1
0

-^-

-,-j. c> ^.

r:t

f-*

— ^^

^^L^ \

C-*

-^ -^ir—

'-^

we must not suppose that any real change of harmony
occurs ; for, except at the final chord, we merely pass from
one portion to another of a Dominant discord.

With regard to the Augmented Triad, Dr. Day takes the
view that this chord represents the " root," Third, and minor
Thirteenth of a Dominant discord. This chord, therefore,
should be written not as at {e) but as at (/) : —

422

THE THEORY OF HARMONY

This being so, it is surprising that he did not explain the
Augmented Sixth chord in a similar way. If Dr. Day's
views as to the proper notation of the augmented triad
are correct, then beyond all question the correct notation
of the chord of the augmented Sixth should be that at {h)
and not that at [g). This chord is now quite easily
explained : it consists of the Seventh, major Ninth,
Eleventh, and minor Thirteenth of the chord of the
Dominant Thirteenth, and resolves quite regularly on the
Tonic " root " (!) . Dr. Day appears to attach much importance
to the fact that c\f is a sharper note (^-^#=24 : 25 ; d-e\f =
15 : 16) than d^, as if a question which he has shown to hinge
on the harmonic determination of a note could be settled by
a reference to melodic intonation. But it is quite easy to
understand why he should give such an explanation of the
augmented triad ; for d^ does not exist in any of the funda-
mental discords in C major. He is in short at the mercy of
his system.

Of other results of this system, one observes that Dr. Day,
while he considers the succession at [a) to be in the key of
G major, explains that at {h) as in C major; —

-o-

221

-O-

This is too fine a distinction. Again, we are to believe that
in the minor harmony, even that on the Tonic of a minor
mode, the Third is an " arbitrary sound."

It cannot be maintained that Dr. Day's system has tended
in this country towards a clearer understanding of the nature
of harmony. It has tended rather to obscure it. His
" natural discords," the majority of whose sounds are foreign
to any known harmonic system, are not derived from Nature,
but are manufactured by Dr. Day himself ; while his treatment
of the Subdominant, of the minor harmony and minor ke}'-
system, are indefensible. How, then, explain the considerable
importance to which Dr. Day's theory has attained, and the

ENGLISH THEORISTS— G. A. MACFARREN 4 = 3

undoubtedly great influence it has exercised on subsequent
English writers on harmony ? It has been thought that his
system of " roots " represents the rachcal defect and even vice
of his theory, and that when liis theory of harmony finally
disappears, the whole system of " roots " must disappear
along with it. This, however, by no means follows. The
real defect of Dr. Day's theory lies not so much in his principle
of " roots " as in the use he makes of these roots, and of the
" fundamental discords " which he builds upon them. One
may even venture to assert that it is just Dr. Day's explanation
of all harmonies within a key as derived from a simple system
of two or three " roots," an explanation in which he allies
himself with Rameau, that explains the influence his theory
has exercised on niusicians.

MacI'ARREn's A\i)Imh\ts or T/m^mox) and Si.\ Lectures

ox //armoxy.

In Sir G. A. Macfarren, Dr. Day found an ardent supporter.
It was the former's great influence as a composer and teacher,
as well as his w^ork Rudiments of Harmony (i860) which
contributed so largely to the wide dissemination of Dr. Day's
theories. In his Six Lectures on Harmony (3rd ed., 1882)
Macfarren remarks : — "My late friend, Alfred Day, commu-
nicated to me his very original and \-ery perspicuous theory of
harmony, by means of which many obscurities in the subject
were cleared that my previous anxious study had vainly sought
to penetrate. . . I am indeed so thoroughly convinced of the
truth of Day's theory, and I have derived such infinite ad-
vantage from its knowledge in my own practical musician-
ship, that I should be dishonest to myself and to my hearers
were I to pretend to teach any other." '

The first part of Macfarren's work, like that of Day, treats
of " The ancient strict or diatonic style," and the second part,
of " The modern style " (Chromatic or free harmony). " The
former style," he remarks, " is conventional, limited, and, so
to speak, dogmatic ; the latter is, in every respect of subject
and treatment, natural and free." This is to do a great deal

^ Six Lectures on Harmonv, Introduction.

424 THE THEORY OF HARMONY

less tlian justice to the ancient " strict or diatonic " style.
But for Macfarren the principal distinction between the
" ancient " and " modern " styles is that while the former
" allows of no unprepared discords, save only passing-notes,"
the latter " accepts the natural generation of discords in place
of their artificial preparation." ^ Macfarren has not much
to say regarding the " natural generation " of concords. But
as in the free style we find concords as well as unprepared
discords, we must assume that he understood this style to
permit of the natural generation of concords as well as of
discords. Of the common chords, or consonant triads avail-
able in the major key, he remarks : — " There are five common
chords available as concords in the major key ; those upon the
keynote, the subdominant, and the dominant are major ;
those upon the 2nd and 6th are minor."- In describing the
triad on the Supertonic of the major key as a consonant chord,
Macfarren overlooks the pecuhar character of this chord, and
the important part it plays in Rameau's theory of the chord
of the Added Sixth. The triad is in fact a diminished
one.

As for the common chords on the Tonic and the Dominant,
these were doubtless considered by ]\Iacfarren to arise respec-
tively from Tonic and Dominant " roots." With regard to the
other triads, namely, those on the 2nd, 4th and 6th degrees
of the major scale, he does not appear to be disposed, like Day,
to explain these as arising from, and as constituent portions of,
" natural discords," for, he tells us, they are concords.
]\Iacfarren is positive that the triad on the Subdominant is a
concord ; and he is no less positive that the Subdominant
itself is, not as Day explains it, the "natural Seventh" of
the Dominant, but a true or perfect Fifth of the Tonic. He
remarks : — " I may recur here to what has already been
advanced as to the faculty of the tutored ear for adjusting the
prevarications of equal temperament ; the 5th of a keynote
and of its dominant, or of a keynote and its subdominant are,
in the scale of nature, perfectly true in intonation as compared
with each other, which is not the fact with any other two dia-
tonic fifths in the same key ; equal temperament gives equal
imperfection to all intervals in all keys, but the ear accepts
for what they should be these exceptionally perfect 5ths in

^ Six Lccitoes 011 Harmoity, Lecture VI. - Etidimcitts of Harmcny, Ch. 4.

ENGLISH THEORISTS— C. A. MACKARREN 425

every key, and hears in them what nature would produce rather
than what is positively sounded." ^

It is not difficult to understand the reason for Macfarren's
liesitation in accepting Day's explanation of such a chord as
that on the Subdominant of a major key, namely, that it
represented the 7th, 9th and nth of the chord of the Dominant
nth : nor is it surprising that he experienced some difficulty
in understanding a triad represented by the proportions
7 : 9 : II as a consonant chord. Similarly with regard to the
chord on the sixth degree. But if Macfarren is not disposed
to accept Day's explanation of these chords, he is unable to
furnish any other explanation. He presents them to us
without telling us whence he gets them, and does not obser\'e
that it is necessary to account for them in some way, and
especially to explain the origin of the important harmony
of the Subdominant.

In demonstrating that the chief and essential characteristic
of " modem " harmony is the use of " natural " or unprepared
discords, Macfarren docs not make quite clear how we should
understand the long passages and even complete compositions
by modem masters in which there are no unprepared discords,
that is, whether we should regard these as belonging to the
" ancient diatonic " or the " modern chromatic " styles of
harmon}'. Nor does he sufficienth- explain why the mere fact
of such discords being, as he alleges, " natural " should justify
their being taken without preparation. Is it because the
" natural " 7th, 9th, nth and 13th, all of which are constituent
parts of the resonance of the prime tone, have a quasi-
consonant character ? But Macfarren himself points to at
least one unprepared discord in use in " ancient " harmonj^
namely, that which we know as the first inversion of the
diminished triad, as d-f-b, where between /' and h we find an
augmented fourth. Of the discord in question Macfarren
remarks : — " This inverted chord with the diminished fifth
was often written by early composers in preference to the
dominant as the penultimate chord in a full close ; the reasons
for the satisfactory effect of which will be best explained when
the true fundamental origin of the chord has been discussed."
Macfarren's explanation of course is that the combination

1 Six Lectures on Harmony, Lecture III. (" The Modem Free or
Chromatic Style ") .

425 THE THEORN' OF HARMONY

d-f-h is a ".natural discord " and an incomplete form of the
chord of the Seventh g-b-d-f. Here, then, in " ancient "
harmony we find a " natural " and " unprepared discord,"
which, according to Macfarren, is no less than that of the
Dominant Seventh itself.

In drawing the distinction he does between the two styles
of harmony, Macfarren not only follows Day, but also Fetis,
who held that the change from " ancient " to " modern "
harmony was effected by means of the introduction into music
of an " unprepared discord " — that of the Dominant Seventh.
Such conceptions have led to much error and confusion in the
domain of the theory of harmony. Although, as we have
just seen, unprepared discords were not altogether excluded
from " ancient " music, it is quite true that a distinguishing
characteristic of modern music is the frequent use of what
have come to be known as " unprepared discords." Musicians
and theorists have perceived this fact, and without probing
the matter further, or inquiring as to whether this really
constituted the essential and fundamental difference between
the two styles of harmony, they have assumed that the change
from " ancient " to " modern " harmony has been effected
by means of unprepared discords. The bold and original
genius, then, who first in harmonic music introduced an
unprepared discord — to him must be ascribed the immortal
honour of having accomplished the vast change from the
ancient to the modem world of harmony. This genius, says
Fetis, was Monteverdc. Macfarren, however, states that it was
not Monteverde, but Jean Mouton, who lived about a century
earlier, and in whose works occurs the unprepared discord of
the Dominant Seventh.^ But before musicians begin disputing
over this matter, it would be wise if the\' first made quite sure
as to whether the great and epoch-making change in question
was really owing to the introduction of an unprepared discord,
or whether, perchance, it was not the slow, gradual, and
consistent development to our present harmonic and key-
system which gave such discords their harmonic significance
and made them artistically possible.

1 " It is common to ascribe the discovery and first employment
of this chord to Monteverde. . . . There are examples of the un-
prepared discord of the dominant seventh, however, in the music of
Jean Mouton, who lived and wrote a century earlier than he." — (Six
Lectures on Harmony, Lecture [II.)

ENGLISH THEQRISTS— (i. A. MACFARKHN 437

Although Macfarren is convinced that his " fundamental '
discords are derived from Nature, he is nevertheless aware that
the " natural " dissonances of the 7th, nth, and 13th have
never actuall}- been used in any system of harmonic music,
" ancient " or " modern." " Although there can be no
question." he remarks. " of the names of these notes, the
universal practice of all singers and players, of all instrument
makers, and of all tuners, is to intonate these notes differently
irom their true harmonic sound." ^ He now actually tells us
that this " universal practice" of musicians has been based
on a complete misapprehension as to the true nature and
intonation of these sounds. " The minor 7th of nature is
somewhat flatter, and the nth somewhat sharper than the
notes rendered in musical performance, which from custoni
the ear accepts as correct, and players on brass instruments,
which naturally sound no notes but their harmonics, are
obliged to have recourse to some artifice for sharpening the
7th and flattening the nth, in order to render these notes
available for combination with the rest of the orchestra."
Still, it does appear strange that in a performance by choir
and orchestra, not only the players on brass and stringed
instruments, but singers as well, should not make use of these
" natural " sounds, w^hen they might easily do so, and should
even take considerable trouble to avoid them.

Macfarren has already remarked, quite justly, that in
liarmonic music a tempered fifth represents to the ear a true
or justly intoned fifth. He thinks that the ear acts in
exactly the same way with regard to the " natural " sounds
of the 7th, nth and 13th. He dwells with admiration on this
" wonderful faculty " possessed by the ear. But in the case
of the fifth, there is nothing really wonderful, for it is out of
tune only to the extent of the twelfth part of a Pythagorean
comma. In the case of the other intervals, however, we find
differences of nearly a quarter of a tone (32 : 33). That the
ear should take no account of this, and that an inter\al
which is out of tune to the extent of nearly a quarter of a tone
should represent to the ear the justly intoned interval,
is certainly a wonderful circumstance : so wonderful,
indeed, that one may be pardoned for indulging in a little
incredulitv.

S/.v Lectu7-es on Harmony, Lecture ^\.

428 THE THEORY OF ^ARMONY

Other theorists have stated that these natural sounds are
" out of tune," and that it is necessary to temper them before
they can be employed in music. Macfarren, however, takes
the opposite view. These sounds are not out of tune ; the
fault lies with our singers and instrumentalists, who never
give, and never have given them, their correct intonation.
In confirmation of this, he remarks : — " That it is an abnormal
condition of the musical sense to tolerate, nay, to look for,
these qualified yths and iiths — that this condition shows us to
be in a state of cultivation, and not a state of Nature — is proved
by an interesting passage in Spohr's Autobiography, wherein
he gives an account of his observations — and the observations
of such a musician compel our respect — of the music of the
Swiss peasantry. Every one of j^ou has heard of their
custom of calling together their cattle by playing on the horn ;
every one is familiar with the term Ranz des V aches that defines
the melodies they play, whose peculiarity results from their
being composed of the harmonic notes of the horn on which
they are played. These notes are sounded without sophistica-
tion in Switzerland, the horn players there having no regard
for the civilised intonation of the orchestra or the drawing-
room. Such of the peasantry as do not plav regard the notes
of the horn as their musical standard, since probably they hear
no other instrument ; and their ear being thus tutored, they
habitually sing their minor 7th so flat and their i ith so sharp that
they would be inadmissible into culti\'ated musical societj'."^

This is a curious passage. Must we infer from it that
because the Swiss horn-players, in simple melodies, habitually
make the minor 7th flat, therefore intonation in our har-
monic music ought to conform to the standard set by the Swiss
horn-players ? The question, however, is less one of intona-
tion, than of the harmonic significance of sounds and chords.
Day and Macfarren do not appear to have attached much
importance to Rameau's explanation of the chord of the
Dominant Seventh, an explanation which was adopted by
Hauptmann. This chord, stated Rameau, was of peculiar
significance in our harmonic music, not because it represented
the " natural " Seventh, but because it comprised within
itself the limits of the key-system, and thus completely defined
the key.

^ Six Leclures on Harmony, Lecture W.

ENGLISH THEORISTS— G. A. MACFARREN 429

But it is evident that IMacfarren's explanations did not
convince even himself, and it is probable also that he had
reflected on what the result would be if his " natural discords "
on the Tonic, Dominant, and Supertonic were actually placed
" in tune," that is, according to the natural intonation of
their sounds, for still treating of the same subject he says : —
" Let us turn from music to the other arts, and we shall find
a like disparity between what Nature gives and that which is
changed by cultivation. Do we not increase the complexity
and diversify the colours of our flowers ? Do we not augment
the nourishment and enrich the taste of our fruits ? . . .Who
would be content with a picture that represented its objects
with the faithfulness of a looking-glass, without the tempera-
ment they receive from the painter's imagination ? "^ This
is to put the matter in quite a different light. Whereas, for-
merly, Nature's intonation of the " natural discords " was the
correct intonation, now Nature is taxed with being " out of
tune." The cultivated ear is obliged to " temper " the sounds
given by Nature. While in the case of the consonances the
cultivation of the ear must be directed towards giving these
consonances their natural intonation (as Fifth = 2:3, Major
3rd = 4:5, etc.), in the case of the " natural discords " the
cultivation of the ear must be directed towards avoiding the
intonation given by Nature ; it is necessary to " temper "
them, some to the extent of nearly a quarter of a tone. The
great importance which attached to the " natural discords "
employed in " modern " harmony was owing to the fact that
we received these discords directly from Nature. Now it
appears that these discords are not derived from Nature at all
in the sense understood by.Macfarren and Day ; and this is
nothing but the bare truth. Had Macfarren not been so
strongly prejudiced in favour of Day's system of " natural
discords " he would not have made such contradictory state-
ments, nor would he have described such chords as the follow-
ing as chords of the 13th* : —

* Six Lectures on Harmony y Lecture VI.

430 THE THEORY OF HARMONY

With regard to the minor key-system, Macfarren has little
to add to the considerations already advanced by Dr. Day.
He is of opinion, however, that the principal chord of the
minor key-system, namely, the common chord on the Tonic,
is properly a minor and not a major harmony ; but he is quite
unable to inform us whence he has obtained this minor har-
mony. With regard to the relationship existing between the
major and the minor modes, he thinks that the belief enter-
tained by musicians that the third degree of the minor mode
is the keynote of its relative major mode, is based on a mis-
conception, anfl that, in fact, it represents little more than a
survival of ancient modal theory. He says : — "The 6th degree
of the major key is the keynote of a minor key, which un-
fortunately is called its relative minor. . . . The relationship
of these keys consists in there being more notes in common
between them than there are between a major key and any
other minor key than its so-called relative ; and the relation-
ship is indicated by the two keys having the same signature.
There is some analogy to the Ecclesiastical system in the
frequent use of the term mode when speaking of these qualities
of major or minor in a key ; it is a remnant indeed of the
Church theory to regard the major mode and its relative minor
mode as modifications of the same scale — a theory which is
opposed to natural truth, and which has consequently some-
times induced harmonic obscuritv in compositions even of the
greatest masters." ^ Macfarren appears here to insist on the
fact that a minor mode and its relative major mode do not
have one and the same Tonic, or keynote, but have each its
own keynote. In this he does quite rightly. But what does
he consider to be the true relative minor of a major mode ?
It is, he tells us, the Tonic minor : — " It must be understood,
then, that the variations of major and minor are modifications
of the one same key, not of the two relative keys." That is,
the real relative minor of C major is C minor. More than
ever, then, the minor mode appears as a modification of its
relative major mode. It is a modification, also, to some
purpose, for in the minor mode we find three sounds, the
minor Third, minor Sixth, and minor Seventh, which are not
in the relative major mode.

Macfarren cannot dismiss in this way the actual relationship

* Six Lectures on Harmony, Introduction.

ENGLISH THEORISTS— F. A. G. OUSELEV 431

existing between the major and minor modes, as, for example,
between A minor and C major, the belief in wliich, as he
himself admits, is " deeply rooted in general acceptance,"^
nor docs he succeed in getting rid of the peculiar difficulties
of a problem which has up to the present baffled every musical
theorist.

• Ouseley's Treatise o\ Harmow.

The Treatise on Harmony (1868) of the Rev. Sir F. A. Gore
Ouseley, from 1855 until his death in 1889 Professor of Music
at Oxford, opens up no fresh ground. We find again the
essential features of the Day theory, although in a modified
form, while the System of the Science of Music of Jt)h. Bernard
Logier (London, 1827) has also, as the author acknowledges,
been laid under contribution. Ouseley states that in his
work he " has aimed throughout at a consistent theory
founded in Nature," and also at " the combination of true
l)hilosophical principles with simpHcity of explanation."

In Chapter 2 he proceeds to explain the generation of
chords. He gives a diagram of the first sixteen " natural
harmonics " of the sound C, assumed as an original " root "
or " generator," and demonstrates that with the exception of
the first six all the others are either octave repetitions of
sounds previously heard, or are out of tune ; of these latter
he remarks : — " These are not only foreign to the key of C,
but are out offline in any key," and italicises this statement.
The choice of harmonic sounds is therefore limited to the
first six ; from these we obtain the Tonic chord, c-e-g.

He then gives the first sixteen harmonic sounds of the note
g, which is the Fifth of C : —

_Q_

—S^-

"25"

Here, extraordinary as it may seem, it is unnecessary
to call a halt at the number six ; in this case we may
proceed as far as the tenth harmonic sound. That is,
in this case we may avail ourselves of sounds which are

' Six Lee/ II res on Harmony^ Introduction.

432 THE THEORY OF HARMONY

not only out of tune, but are "out of tune in any key."
Ouseley remarks : — " Here it will be observed that everj'
note belongs to the key of C till we come to the double
bar ; and, although the note /, marked *, is not perfectly in
tune, yet we can substitute a really true f without at all materi-
ally disturbing our new series of sounds." ^ Such methods
can scarcely be described as consistent \\dth a " philosophical
theory of harmony, founded in Nature."

Ouseley proceeds: — "We obtain, then, the chord g-))-d-f -a,
which is called the ' dominant chord of Nature,' being based
on the fifth of the key." More accurately, this chord should
be described as " the dominant chord of Nature, corrected b}-
Ouseley." Ouseley rejects the natural Seventh with which
Nature presents him, and substitutes for it a " really true "
minor vSeventh. In Chapter 5, however, he has changed his
opinion, and now considers that the " really true " minor
Seventh is the natural seventh harmonic sound, which he has
already rejected. " We may regard," he says, " the ordinary
minor Seventh as a tempered modification of the fundamental
Seventh found among the harmonic sounds of Nature."

But Ouseley brings to our notice another " dominant chord
of Nature," the chord of the Dominant minor Ninth, obtained
by substituting for the Ninth, rtt], the seventeenth harmonic
sound a\}. While atj, he remarks, is perfectly in tune, a\} is
" very nearly in tune." Ouseley therefore does not agree
with Day that this rtlq, the Fifth of the Supertonic, is a comma
(80 : 81) too sharp. /\s for a\^, this sound differs from a true
minor Ninth (i : 2 +15 : 16) of ^ by the interval 255 : 256.
But as this small interval is almost negHgible as compared with
the much larger comma 80 : 81, it would seem that the chord
of the minor Ninth much more truly represents the " dominant
chord of Nature."

Ouseley rejects the chord of the Eleventh, as this Eleventh
is " too sharp," but does not observe that this leaves him
without even Day's Subdominant chord, to say nothing of
the harmony on the sixth degree of the scale. As for the
chord of the Minor Thirteenth, he is unable to say defirytely
whether it is a real chord or not. But he is in no doubt as to
the chord of the Major Thirteenth ; this Thirteenth, he tells
us, is " in perfect tune," ^ and is represented not as one would

^ Treatise on Harmony, Ch. 2. ^ j^ij,^^ Ch, jg_

ENGLISH THHORISTS— F. A. G. OUSELEY 433

suppose by the thirteenth liarmonic sound from the " root,"
but by the 27th harmonic sound. This gives a Sixth of the
proportion 16 : 27. This Sixth Ouseley evidently considers
to be a true major Sixth, and of correct proportion. In
reality it differs from a true major Sixth (3:5) by a
comma, 80 : 81.

Ouseley 's views as to the origin of the diatonic major scale
deserve notice. The Subdominant, he is of opinion, can only
be the principal generator of a new key, that is, the Tonic of
the Subdominant key ; the original Tonic, then, relinquishes
its character as Tonic, and becomes Dominant. By the time
the sixth degree of the scale is reached, it becomes necessary
to return to the original key " by a modulation to the original
tonic " ! Strange to say, Ouseley regards such a scale as
" a true diatonic scale which begins and ends in the same
key."i

With regard to a Subdominant-Dominant succession of
harmonies, we must assume that Ouseley would consider the
lirst chord to be a Tonic chord of the Subdominant key. But
indeed he does not appear to observe that the explanation
of such a succession is necessary.

With regard to the Minor Mode, the only form of the minor
scale which he considers to be deserving of recognition as a
real scale is the " harmonic " form. Of the relationship
between the major and minor modes, he remarks that " this
connection can hardly be said to be of natural origin, inasmuch
as the harmonics of the root of the major key do not give us the
common chord of its relative minor." ^ He proposes a new
explanation of the minor harmony. " If," he says, " we take
the first fifteen sounds of the harmonic series with, for
example, C as the root, we find the minor harmony represented
by the numbers 10, 12, and 15 of this series. But the root of
all the notes in this series is C, not E. C cannot be the root
of the minor triad of E. Therefore the numbers 10 : 12 : 15
do not correctly produce a genuine minor triad "(!).^ He then
extends the harmonic series to the 24th term, and, leaping over
all the intermediate sounds, discovers the minor Third of the
root C at the 19th term of the series. This sound " gives us
the minor third of Nature ... it is almost in tune [!]. . . .
Let us, then, assume the fundamental minor Third of Nature

^ Treatise on Harmony, Ch. 4. ^ Ibid., Ch. 5. * Ibid.

434 THE THEORY OF HARMONY

to be i6 : ig." ^ The correct proportions of the minor harmonv
then, should be i6 : 19 : 24. Needless to say, Ouseley's views
with regard to the minor harmony have not found much
acceptance among musicians. As a " philosophical and con-
sistent theory of harmony " Ousele3^'s performance cannot
compare with that of Dr. Day.

Stainer's Theory of Harmony.

In Sir John Stainer's Theory of Harmony, Founded on the
Tempered Scale (1871), we meet with a type of work very
different from that of Ouseley. Stainer was an original and
independent thinker, and it was consistent neither with his
vigorous personality, impatience with unreality, nor ad-
mirable musicianship, that he should have remained satisfied
with works which were passing current in his time as standard
works on harmony. In the Preface to his work, he criticizes
severely the methods adopted by certain theorists in their
manipulation of the harmonic series. " It is interesting," he
remarks, " to watch the process. . . . From a few natural
harmonics exhibited on a diagram, about a dozen of the
hundreds of chords in use are constructed ; the insufficiency of
the number of the chords being then too apparent. Nature is
taxed with being out of tune, and tempered intervals are
introduced to allow of the construction of some of the most
ordinary chords in music." In discarding, then, the harmonic
series, he points to the impossibility of constructing a rational
theory of harmony on a mathematical basis. He says : —
" When musical mathematicians shall have agreed amongst
themselves upon the exact number of divisions necessary in
the octave, . . . when practical musicians shall have framed
a new system of notation which shall point out to the per-
former the ratio of the note he is to sound to its generator,
when genius shall have used all this new material to the glory
of Art — then it will be time enough to found a Theory of
Harmony on a mathematical basis."

It is, then, on the tempered and not on any mathematicallv
correct scale, supposing that such a scale could be found, that
Stainer proposes to base his new theory of harmony. It is

1 Treatise on Harmony, Ch. 5.

ENGLISH THEORISTS— J. STAINER 435

true, he remarks, that " the tempered scale is out of tune,
and will not bear to have its proportions exhibited to an
audience with better eyes than ears ; but its sounds have
nevertheless been a source of as real pleasure to all great
composers, as of imaginary pain to certain theorists."

We must assume, then, that Harmony has its source in
Melody. " Melody existed before Harmony. ... A Chord,
therefore, is defined as a combination of notes taken from a
scale, or sometimes (but rarely) from two closely-allied scales."
On what principle, then, are melodic notes combined so as to
form chords ? " It is," says Stainer, " simply this : by adding
Thirds together, the Thirds being major or minor according
to their nature in the scale from which they are taken. Until
the interval of a Third," he proceeds, " is allowed to be the
basis of all harmony, no theory of music can possibly be formed
which will be true to facts. The old veneration for the per-
fections of the Fourth and Fifth, hardly yet extinct, helped
to degrade Thirds by calling them imperfect intervals. . . .
If any interval ever deserved to be called perfect, it is the
Third."!

Added Third Generation of Chords :
The Tempered Scale.

" The simplest and most natural way of arranging chords,"
then, " is evidently to begin with the tonic, and to go on adding
thirds from the scale, until the whole of the notes of the scale
are exhausted."^ In this way, starting with the common
chord formed by adding two Thirds together, wc obtain, by
means of the addition of a third Third, a chord of the
Seventh, and so on until we finally arrive at a chord of the
Thirteenth : —

C major.

-^-

^— ^^

-&-

-&'

C minor.

:2-

&^F==^^

-s>-

H^-

-o-

1 Theory of Harmony Founded on the Tempered Scale, Ch. 3. * Ibid.

436 THE THEORY OF HARMONY

Such a series of Thirds, starting from the Tonic, may be
derived not only from a major scale but from a minor scale.
It will be observed that Stainer does not, like Day, consider
himself at hbertj- to place a major triad on the Tonic of the
Minor Mode.

But it is not only on the Tonic that such a structure of Thirds
may be raised. The Dominant also ma}' be selected as a
starting point. Thus from the Dominant of the major scale
there arises the series g-h-d-f-a-c-e, and from the Dominant
of the minor scale the series g-h^-d-f-a^-c-e^. Stainer then
proceeds to show how various chords are derived from these
structures of Tliirds. Such, in brief, are the main hues on
which he draws up his theory.

After Stainer's demonstration of the futility of deriving
harmony from the harmonic series, of the impossibility of
arriving at a clear understanding with regard to what scales
ought to be considered as acoustically coirect, and in general
of founding a theory of harmony on a mathematical basis,
but especially after the inference he has drawn in connection
with his statement that it is the tempered scale from which
has been derived the harmonic material of the great composers,
one is not quite prepared for his frank admission that the
tempered scale is " out of tune." How out of tune ? And
out of tune with what ? Is it out of tune witli the oldest and
original t}-pe of the diatonic scale, the scale of Pythagoras ;
this scale which was in use not only among the Greeks but
throughout the whole of the Middle Ages ? Compared with
this \-enerable scale which, dating from Greek antiquity, was
still in use tiU near the dawn of the Renaissance, our modem
scales are of mushroom growth. If Stainer's \'iews as to the
nature of hannony are correct, it is this scale which we ought
to regard as the real foundation of European music and Euro-
pean harmony. The only drawback to such a view is that this
old scale, with its P\-thagorean tuning and false Thirds, was,
during the development of polyphony, ultimately discarded,
the reason being that it could not produce a true major nor a
true minor harmony. This is surely an extraordinary cir-
cumstance for those who hold that Harmony arises from
]\lelody. What did these musicians want with a true major
or minor harmony ? Where did they acquire the foolish
notion that such a harmony existed, or could possibly exist ?

ENGLISH THEORISTS— J. STAINER 437

And wh}- should the\- not have remained satisfied with the
harmony whicli their scale offered to them ?

Nevertheless this new harmony, what Zarlino called the
" Harmonia perfetta." arrived, whence no one very well knew.
— apparently from the clouds — and brought about a musical
revolution of which we are to-day reaping the fruits, and of
which our whole modern harmonic art is the direct result.
Instead then of the scale determining harmony, we actually
and before our eyes find harmon}^ determining the scale.

But it is not, of course, the Pythagorean scale wdth which
Stalner compares his tempered scale, nor would he propose for
a moment to hold up this scale as the true foundation of
harmon}'. What Stainer means, and knows quite well, is
that with the exception of the Octave every interval in the
tempered scale differs sUghtly from its acoustical and mathe-
maticallv correct determination. For example, the Fifth, as
c-g, is flatter bv the twelfth of a Pythagorean comma (524288 :
531441) — more or less, according to the experience and skill
of the tuner — than the perfect Fifth (2 : 3) which arises in the
harmonic series immediately after the Octave.

The alarming thought then suggests itself — could the tuner,
without this acoustically determined Fifth, possibly present
us with a tempered Fifth ? The answer must be in the
negative. Even the tempered scale then is dependent on
harmony ! Stainer is now in possession of two acoustically
perfect intervals, — the perfect Octave, which not even a
tempered scale can dispense with, and without which the
limits of the scale could not be defined, and the perfect Fifth.
The ratio of this octave is i ; 2, and of this Fifth 2:3. But
although Stainer does not beheve in " generators " or " roots,"
he believes nevertheless, like every theorist who tells us that
harmony arises from melody, in the inversion of intervals
and of chords. The inversion of the perfect Fifth, then, gives
him the perfect Fourth, the ratio of which is 3 : 4. Further,
as the Third, according to Stainer, is the primary and most
perfect constituent of harmon}-, we must include both the
major and minor Thirds. Stainer knows, and says, that
in the tempered scale both these intervals are out of
tune, that is, they stand for or represent true major and
minor Thirds. The ratios of these interv-als are respectively
4 : 5 and 5 : 6. Collecting these results, we find that the
harmonic material of which Stainer is obliged to avail himself

438 THE THEORY OF HARMONY

is all mathematically determined, and that it is all comprised
in the arithmetical series 1:2:3:4:5:6.

This is not a bad beginning for a theorist who has set out
to demonstrate the impossibilit}- of founding any rational
theory of harmony on a mathematical basis. Stainer, indeed,
succeeds in demonstrating the impossibility of dissociating his
theory of harmony, based on the tempered scale, from the
acoustical determination of the intervals which form the
constitutive elements of harmony.

Stainer's idea that the scale is the basis of all music and all
harmony is not new. The same theory had been promulgated
by Fetis and others. F'etis, as we saw, considered that the
fundamental law of all music was Tonality, and that tonality
was the result of the arrangement of the sounds of the
scale. We saw that Fetis experienced considerable difficulty
in deducing definite principles of harmony from the mere
arrangement of sounds of the scale ; he arrived, by some
unknown means, at the discovery that certain sounds in the
scale possessed the character of repose. These sounds
proved to be nothing more nor less than the sounds of
Rameau's fundamental bass in Fifths. Fetis, however, had
to admit other sounds of the scale as being also sounds of
repose : and might also have pointed out that his sounds
of repose became sounds of unrest, or movement, and
conversely his sounds of unrest, sounds of repose, according
to circumstances.

It is interesting, then, to observe what success attends
Stainer's efforts to deduce definite principles of harmony from
the tempered scale. His theory is at least simple and
straightforward. If he is less subtle than Fetis and other
theorists whose works we have been examining, he does not
try to entertain two entirely contradictory and irreconcilable
propositions at one and the same time, nor does he use language
designedly ambiguous, and calculated to provide the theorist
with an avenue for escape should the suggestions he has thrown
out prove to be untenable. Stainer's fundamental principle
of harmony, his principle of principles, is simplicity itself,
namely, that all harmony is the result of Thirds added together.

How does Stainer arrive at such a conception of harmony ?
It is certainly not the result of his study of the tempered scale.
There is nothing in the tempered scale which even suggests
such a principle ; rather, the contrary is the case. F"or, as

ENGLISH THEORISTS— J. STAINER 439

Kamcau maintained, the scale is comprised within the Hmits
of an Octave. If now the scale has anything to teach us
with regard to the formation of chords, it teaches first of all
that no chord must exceed the compass of an Octave. But
of course there is no difficulty in discovering whence Stainer
has derived his fundamental principle of harmony. He has
borrowed it from Day and the other theorists who make use
of the harmonic series for the generation of chords.

The Third the " Basis of all Harmony."

Stainer does not, like Fetis, attach great importance to the
harmonic significance of the Fifth. It is the Third which
possesses real generative power. On this point he is quite
definite. " Until the interval of a third," he has said, " is
allowed to be the basis of all harmony, no theory of music
can possibly be formed which will be true to facts." But no
sooner has Stainer proceeded to form chords by adding Thirds
together than certain facts of the utmost importance arise,
which he seems not to have observed, much less considered.
The first chord he obtains is that of the major harmony c-c-g.
This chord, Stainer says, is formed by adding together the two
Thirds c-e and e-g. Stainer knows that the second Third
ought to be minor, and not major, as c-gj^, because this iff
does not belong to the key of C.

But what Stainer fails to observe is that, having added his
two Thirds together, there results quite another interval,
namel}/, the Perfect Fifth c-g. He does not pause to consider
whether this Perfect Fifth may not possess quite as great a
theoretical significance as the Third ; whether it might not
be more correct to explain the two Thirds as arising from the
Jiarmonic division of the Fifth, rather than the Fifth as arising
from the addition of the Thirds ; and whether it is the scale
which determines the order of sounds and quahties of the
intervals in the harmony c-e-g, or whether, perchance, it might
not be the harmony c-c-g which determines the nature and
character of the scale. Stainer does not speak of the chord
of the Dominant Seventh as " a chord of three Thirds," but
calls it of course a chord of the Se\-cnth. In this chord we
find not only major and minor Thirds, but a diminished and a
perfect Fifth, as well as a minor Seventh.

440 THE THEORY OF HARMONY

Again, why does Stainer select the Dominant, in addition
to the Tonic, as a basis for the formation of chords ? Between
Tonic and Dominant we have the interval of a Fifth, an interval
which, according to Stainer, is of little theoretical significance.
Why then for his second " root" does he select the Fifth,
and pass over the Third, seeing that " the Third is the basis
of all harmony " ? Why also does he build his chords
upwards, rather than do\\Tiwards ? Does he look on the
descending scale as something theoretical^ inadmissible ?
In short Stainer, although he discards the harmonic series,
finds it impossible to get rid of the idea of a fundamental note,
root, or generator. It cHngs to his theory from start to finish ;
and one is not surprised to read, at the " Conclusion " of his
work, the following statement: — " All discords, even of the
most complicated kind, are as it were built round a common
chord," and that " the fundamental note of this common chord
is the note from which the whole chord is said to be derived —
or, its ground-note, or root."

This is a somewhat belated confession ; a statement which
Stainer ought to have placed at the beginning, not at the end
of his work. In what sense does Stainer use the terms ground-
note or root ? Does he consider that in his chord of the
Dominant Thirteenth all the sounds of which it is composed
are " derived from " the Dominant ?

Discords which Arise from the Tempered Scale.

When Stainer comes to apply his principles to examples of
harmony by the great composers, his theory, as might be
expected, completely breaks down. At [a] we find one of
those discords which Stainer describes as being " built round
a coinmon chord " : —

ENGLISH THEORISTS— J. STAINER

441

The common chord, in this case, appears to be tlie
Subdominant Chord in Ej; major, namely (i\^-c-t\}, and
the discord in fact is that of Rameau's " Added Sixtli."
But this, according to Stainer, is quite a wrong explanation.
The common chord round which this discord is built is not the
Subdominant, but the Dominant harmony b\^-d-f, and the
discord itself is " the third inversion of the chord of the
Eleventh of Bj? " ; that is, it is not a Subdominant, but a
Dominant discord. Neither the root nor Third of this chord-
of the Eleventh is present. Such being the case, ought we not
likewise to consider the chord of the Dominant Seventh,
g-h-d-f a.s, in reality, a chord of the Tonic Eleventh, c-e-g-b-d-f,
with the " root " and Third omitted ?

The chord at (6) is described by Stainer as a chord of the
Dominant Thirteenth in C major. In this case a gap occurs
between /, the third Third of the chord, and e, but all that
is necessary is to keep on adding Thirds until we arrive at the
latter sound. Stainer does not describe e as the sixth Third
of the chord, or as the Third of the Tonic, but as the Thirteenth
of the Dominant. It would be more easily understood as the
Sixth of the Dominant ; but an insuperable objection towards
regarding it as a Sixth is that its real origin — which is that
of added Thirds — would be rendered obscure. Nevertheless,
we must beUeve that this chord of the Thirteenth has its
basis in the tempered scale.

The chord of the Augmented Sixth Stainer considers to be
derived " from two scales," therefore from two keys. Thus
the chord at (a) " consists of the minor ninth of the dominant
{g), with the major third, minor seventh, and minor ninth of
the second dominant {d) " ^ : —

(«) {^) *

a^Sg^fefi^PipiE;

JZCj

-Gh-

etc.

This is also Day's explanation of this chord. Both theorists
are evidently quite satisfied that the sounds c and ^(7,

Theory of Harmony, Ch. S.

442 -THE THEORY OF HARMONY

of this chord, cannot possibly be considered as the Eleventh
and Thirteenth of the Dominant g. But it is by no means
clear why aj? should belong to one key, or " root," and
c and e\f to another : especially as all three sounds give us
the major harmony a];-c-e\}. Still another pecuhar formation
is that at (b). This chord, Stainer says, is composed of the
tonic C, its tnajor (!) third E, Aj? the minor ninth of the
dominant G, and F# the major third of the second
dominant D.

As Stainer takes as his starting-point the tempered scale,
he does not consider it necessary to enter into the question of
the nature and origin of the scale. Only, in the chapter on
"' Modulation," he states that the major scale is in two keys.
Thus, in the scale of C major, the lower tetrachord is in F
major, and the upper tetrachord in C major. It results from
the nature of Stainer's theory that the section of his work
treating of Modulation and Chord Succession is quite
inadequate. Like Day, he has no explanation to offer of
the minor harmony and the minor mode, nor does he seem to
recognise their pecuhar difficulties, making use only, for
purposes of chord formation, of the " harmonic " form of
the minor scale.

Prout's " Harmony : Its Theory and Practice."

Finally, mention may be made of Professor Ebenezer Prout's
Harmony: Its Theory and Practice (1889, 20th ed. 1903).
In Prout's work we find the Day theory again in full \-igour.
Briefly stated, Prout's theory is"^as follows : — The selection of
certain sounds of the scale, major or minor, as roots or gene-
rators ; the building up of chords by means of added Thirds ;
and the arbitrary selection for this purpose of sounds from
the harmonic series. It is true that Dr. Day, even if he had
perforce to make use of added Thirds as a principle of harmonic
generation, nevertheless avoided making too definite a state-
ment on this point, and especially as to exactly what sounds
of the harmonic series he considered it necessary to make use
of for the purposes of chord formation. Prout, however, is
much less cautious than the older theorist. He states plainly: —
" E\ery chord is made by placing not fewer than three notes

ENGLISH THEORISTS— E. l^ROUT 443

one above another at an interval of a Third " ; and tells us
also exactly what sounds he selects from the harmonic series.

Starting with a major harmony, as c-c-g, in which c is the
root, Prout has evidently no difficulty in deciding that the
next Third to be added must be a minor Third. The seventh
sound of the harmonic series provides him with a Third of the
proportion 6 : 7. This, however, is not the minor Third
required. Prout therefore substitutes for this seventh har-
monic another and sharper sound, which will give him the
Third required. Nevertheless, curiously enough, Prout
•considers that this new sound {h^) has been derived from the
harmonic series. We have now the chord of the Tonic
Seventh c-e-g-h\f.

" When we come to add another Third above the Seventh,
a choice offers itself. We can either take a minor 3fd (Dj?, the
17th harmonic) or a major 3rd (Db], the 9th harmonic)."^
Prout takes both, and thus obtains a chord of the Tonic Minor
Ninth, c-e-g-b\f-d\f, as well as a major Ninth, c-t'-g-b\^-d\^. The
next Third must be a minor one. The eleventh harmonic is
much too sharp to represent the new sound/ which is required ;
it must therefore be rejected, and we must select in its stead
the 2ist harmonic sound, which is much better in tune. We
have now the chord of the Eleventh c-c-g-b\}-d-f. If now, we
add to this chord a major third, we shall have a chord of the
major Thirteenth ; and, if a minor Third, a chord of the minor
Tliirteenth. The new major Third a cannot be represented
by the 13th harmonic sound of C, because it is much too flat ;
we therefore take, instead, the 27th sound of the harmonic
series of which C is generator, this sound being " much better
in tune." As for the minor Third a\}, this is represented by
the 51st sound of the harmonic series. Prout also describes
this sound as the " 17th harmonic of the dominant," which
of course means nothing, as he is developing his complete
■chord from the Tonic root.

" Having exhausted the available harmonic resources of C,"
he proceeds, " as a fundamental note, we must look elsewhere
for the materials to complete our key." He therefore takes
G, the Dominant, and D, the Supertonic, for " roots," as well
as C, and, following Dr. Day, places on these sounds a chord
similar to that which he has considered to be developed from

^ Harmony: its Theory and Practice, Ch. 3.

444 THE THEORY OF HARMONY

the Tonic. He is now in possession of all the harmonic
material he requires : —

^- -•- •^

50:

# '-J^JC

:si

All this, apparently, we must understand as " the theory
or science of harmony." But anj'thing more unscientific,
more opposed to common-sense even, it would be difficult to
imagine. Such a theory only requires to be stated to convince
any discerning mind of its absurdity. Anyone is at liberty,
if he so desires, to build up huge sound-combinations by means
of adding Thirds to one another ; anyone may, from a har-
monic series extended to the 51st term, pick out whatever
sounds he ma}' please, but why describe this as the science of
harmony ? What Prout does is as follows : — He proposes to-
be guided by Nature, and to derive from the sounds of the
harmonic series the harmonic content of the key-sj'stem. He
does not however take the sounds of the harmonic series as
they arise in Nature, but in a quite arbitrary way selects
those sounds which he considers necessary for his purpose.
But even the sounds thus selected are, it turns out, for the
most part " out of tune." He is therefore obhged to reject
them, and as a matter of fact does not make use of them at all.
Here then, one would imagine we have the end of the whole
matter. Nevertheless, Prout presents to us certain huge
combinations of sounds which he describes as " natural
discords." Most wonderful of all, he is of opinion that he has
derived these monstrous structures from the sounds of the
harmonic series !

Abandonment of the Harmonic Series as the
Basis of Harmony.

Origin of Discords : the Suhdominani.

In 1901 Professor Prout published what is best described
as a new work on harmony. In this work he has considerably
modified his previous theories, and introduced many changes.
" First and foremost among these," he remarks, " is the

ENGLISH THEORISTS.— E. PROUT 445

virtual abandonment of the harmonic series as the basis on
which the system is founded." He is now of opinion that
" the modern major or minor is largely the result of aesthetic,
rather than of scientific considerations." ^ Having abandoned
the harmonic series, Prout, indeed, is quite unable to find
any scientific basis for the theory of harmony. He does not
state whether he considers that " aesthetic considerations "
form a suitable basis for such a theory. It is necessary to
suppose that he has a basis of some sort, for his new work
bears the title, Harmony : its Theory and Practice.

But it does not appear that Prout has any real foundation
for his theory, or that his work can properly be described
as a theory of harmony at all. He makes no serious attempt
to grapple with those important questions which Rameau
rightly considered to constitute the central problems of the
theory of harmony ; such as the nature and origin of the
key-system, of the major scale, of the minor scale (or, more
accurately, scales), of the major harmony, of the minor
harmony, of the generation of discords, of the relationship
between the major and minor modes, of chord succession, etc.
It is not sufficient to state that all these things are based on
assthetical considerations.

Prout altogether discards the " fundamental discords "
of the Tonic and Supertonic, while retaining that on the
Dominant. But of this chord of the Dominant Thirteenth
he can give no adequate explanation. Of the chord of the
Dominant Seventh he remarks : — " We meet here for the
first time with a ' fundamental discord,' that is, a discord
composed of the harmonics of the fundamental tone, or
generator." - But the discords of the Dominant Ninth,
Eleventh, and Thirteenth, do not admit of a similar explana-
tion. " Further investigation and thought," Prout states,
" have convinced the author that the practical objections
to the derivation of the higher discords — the Ninths, Elevenths,
and Thirteenths — from the natural series of upper partials
were far greater than he had realised." ^ While, then, the
discord of the Dominant Seventh is a " natural " and " funda-
mental " discord, those of the Ninth, Eleventh, and Thirteenth
of the Dominant are not. This is not easy to understand.

1 Harmony ; its Theory and Practice, New Edition, Preface.

2 Ibid., Ch. 8. 3 Ibid., Preface.

446 THE THEORY OF HARMONY

Day explained these discords as differing from ever}- other
class of discords in that they did not require preparation ;
and the reason he assigned for this was that they had a
natural origin, and arose from the harmonic series. It is
difficult to understand why Prout should admit the Seventh
as a harmonic sound, and exclude, for example, the Ninth.
The Dominant Ninth requires preparation almost as little
as the Dominant Seventh. Besides, this Ninth is more
" in tune " than the Seventh. While the latter is flatter
than the fourth degree of the major scale by the interval 63 : 64,
the former is only a comma (80 : 81) sharper than the sixth
degree of the scale.

Prout makes a notable advance in his recognition of a
Suhdominant. In his previous work he had as in most
other things followed Day, who, while retaining the name
Subdominant, and recognizing the possibihty of a modulation
to the Subdominant key, had nevertheless explained the
Subdominant as part (Seventh, Ninth, and Eleventh) of a
Dominant discord, and as having a Dominant " root."
Prout now states that there are three Primary Triads in
every key, namely, the Tonic, Dominant, and Subdominant
triads. He goes further in the direction of Rameau's
theory : — " The three primary triads," he says, " absolutely
define the key."^ In explanation of this he remarks: —
" The only notes which make perfect consonances with C
[the Tonic] are the dominant G [a fifth above] and the
subdominant F [a fifth below]. The tonic, dominant, and
subdominant are therefore called the three Primary Notes
of every key."^ This however, by no means follows;
nor does the mere fact of the two Dominants being
perfectly consonant with the Tonic furnish a sufficient
explanation as to why the three primary triads absolutely
define the kev.

^Harmony : its Theory and Practice, New Edition, Ch. 4. * Ibid., Ch. 2.

ENGLISH THEORISTS— E. TROUT 447

Secondary Discords.

Prout, however, still retains his extraordinary theory as
to the origin of what, in liis former work, he describes as
" diatonic discords," and now as " secondary discords."
If above each of the triads which may occur on each degree
of the major scale — the Dominant, however, excepted — we
place a Seventh in accordance with the key-signature, we
obtain all the secondary chords of the Seventh which belong
to a major key : thus : —

rz^€-

:SE

:g:

:g=

:Si

All these chords, according to Prout, are derived from
Dominant discords of the Ninth, Eleventh, or Thirteenth.
Thus the Tonic chord of the Seventh " consists of the
Eleventh, Thirteenth, root, and Third of the Dominant
Thirteenth " ^ ; that on the Subdominant is a " derivative "
of the same Dominant discord (7th, gth, nth and 13th), and
the other chords are explained in a similar way.- But with
regard to the Tonic Seventh c-e-g-b, one would naturalh"
suppose that the harmonic foundation of this chord is the
Tonic chord c-e-g ; and that in the case of the Sub-dominant
Seventh the foundation of the chord is the Siih-dominanf
harmony f-a-c ; especially as these secondary chords
of the Seventh on the first and fourth degrees of the scale
have been formed by the addition of a Seventh above the
Tonic and Subdominant harmonies.

This however, according to Prout, is not the case. The
sounds c-e-g and j-a-c must here be understood as being
derived from the Dominant. It appears, then, that while
the chord c-e-g has as its generator the Tonic, the generator
of the chord obtained by adding a Seventh above the Tonic
harmony is not the Tonic, but the Dominant. And similarly
with the chords f-a-c and f-a-c-e.

But notwithstanding that the secondary chords of the
Seventh have their origin in Dominant discords, we must not,

^ Harmony : its Theory and Practice, New Edition, Ch. 14.
2 Ibid.

44^

THE THEORY OF HARMONY

says Prout, consider them to represent such Dominant discords.
" Notice," he remarks, " that in none of these chords is the
characteristic interval of a fundamental discord — the
diminished fifth between the major third and minor seventh —
to be seen." ^ He even thinks that in practical composition
the origin of these chords should be quite disregarded. " With
all these secondary Sevenths," he says, " the student has not
to concern himself in the least with the relationship of the
various notes of the chord to the dominant, but only with
their relations to each other." This is, doubtless, good advice.
For example, in the following succession of chords {a) : —

(a)

— o

— <s<

tf — *^ —

-o-

fZi

«5>l JJ

fri) • ^''

(W-

<rj

^'i:> r>

we find that the bass note in the first chord which, according
to Prout, is the Eleventh of the Dominant g, leaps a Fourth
upwards to what we must suppose to be its note of
" resolution." Similarly, in the first chord of example (6)
the bass note in the first chord, which is the Thirteenth
of the Dominant, " resolves " by rising a Fourth.

But the most remarkable of all these " Dominant discords "
is that on the Submediant, a-c-e-g. Here the generator g,
the Dominant, is at the top, while the Ninth a is at the
bottom. It is not the Ninth, however, which requires to be
resolved, but the generator itself. The Ninth, on the other
hand, while the generator descends a degree to its note of
resolution, may rise a Fourth, or fall a Fifth. Strange to say,
Professor Prout is not only aware of these circumstances, but
points them out. He remarks : — " In the Chord VI. ^ [a-c-c-^
the root of the chord {a) is the ninth of the Dominant, and
the Dominant (the generator) is the seventh. But it is not
the Ninth which is restricted in its movement by the presence
of the Dominant, as in a chord of the Ninth ; it is the Dominant,
which has now become the seventh of the chord, that is itself
restricted by the presence of the root below." (As is known,

Harmony: its Theory and PracticCy New Edition, Ch. 14.

ENGLISH THEORISTS— E. TROUT 449

Professor Prout draws a distinction between the " root " and
the " generator " of a chord. He describes as " root " the
lowest note of any chord consisting of a series of Thirds.
While therefore a is the " root " of the chord a-c-e-g, the
" generator " is not a but g.)

It is difficult to understand why Prout, in the face of these
facts, and without even being able to furnish any adequate
explanation of the Dominant discords themselves, should go
to such extraordinary lengths to explain the secondary chords
of the Seventh as derived from Dominant discords. Is it
because he is unable to find any explanation of the secondary
triads of the key-system ?

Chord Succession : " Tonality " and the
" Melodic Tendencies " of Sounds.

Prout now finds himself unable to explain even such a
simple succession of chords as that of Dominant followed by
Tonic harmony. In his previous work he had been able to
furnish a quite adequate explanation of this succession,
borrowing from Dr. Day the explanation which the latter
theorist had in his turn derived from Rameau, namely, that
in the Perfect Cadence the Fifth returns to its " root " or
source. »

He adopts, it is true, theoretical ideas from various quarters.
He identifies himself with the somewhat lame, certainly
inadequate explanation of Fetis, of the tendency of the chord
of the Dominant Seventh towards the Tonic chord. ^ It is,
he says, the interval of the diminished Fifth which determines
the resolution of this discord. Prout appears to be of opinion
that the only natural resolution of the diminished Fifth, or of
its inversion, the tritone, is that on the " root " and Third of
the Tonic chord. On the contrary, these intervals may
resolve in various ways. For example, the tritone f-b may
resolve in a perfectly natural way on the perfect Fifth e-b ;
from the point of view of the resolution of dissonance, the
resolution e-b is better than the resolution c-c. For, in the
first case, one note remains while the other moves, while in
the second, both notes move to the notes of resolution. Also,
in the first case, the interval of resolution is a perfect Fifth,

1 See remarks on Fetis's theory, pp. 343-345.
2G

4 50

THE THEORY OF HARMONY

while, in the second case, it is a minor Sixth, an interval
described by Helmholtz as the worst of the consonances.

The theoretical value of Front's newly acquired notions
respecting " Tonality " and the " melodic tendencies " of the
sounds of the scale, we have already ascertained in our exami-
nation of the theory of harmony of Fetis. Prout has nothing
very definite to state with regard to what the melodic tenden-
cies of the sounds of the scale really are. He does, it is true,
make a definite statement to the effect that the general rule
to be observed is that, " two notes forming a diminished
interval have a tendency to approach one another," while, on
the other hand, " two notes forming an augmented interval
have a tendency to diverge." ^ But it is evident that this
" rule " is insufficient. For if we resolve the tritone f-h on
the Octave d-d' , we find that, while the notes of the tritone
diverge, we do not obtain the resolution required by Prout.
His rule requires to be supplemented to the effect that the
notes forming the dissonant interval should, in resolving,
proceed by the step of a tone or semitone. But it is to be
feared that this " rule," if it be a rule, is honoured as frequently
by musicians in the breach as in the observance. At {a) and {h)
(a) {b) (c)

^^SE

-^5'-

-&-

te^

-e>

.c^.

^I=P

_Q_

-G>-

-O.

1^31

-G>-

-<SP-

£:

-fSi-

-f^

we find an augmented interval which does not diverge in
resolution, while at (c) we find a diminished interval which
does not contract. It cannot be contended that there is
anything strained or unnatural in these harmonic successions.
Many others of a similar kind, which are constantly being
used in harmonic music, might be quoted.

Still, one need not seriously object to Prout's " rule "
except that it suffers so many exceptions, and that it does
not inform us how to treat other intervals which are neither
augmented nor diminished, but principally that considered
as a principle of " tonality " it forms such a meagre and

1 Harmony : its Theory and Practice, New Edition, Ch. 8.

ENGLISH THEORISTS— E. PROUT

451

insecure basis for any adequate theory of harmony. But of
course the resolution of dissonance is determined not by any
" melodic tendencies," real or imagined, of the sounds of the
scale, but by harmonic considerations. In the following
well-knowTi passage from the Prelude to Wagner's Tristan,

:ti:

f=r?

^%i^

^ "1

^ 1

we see the process going on under our e\^es. First g^ proceeds
to a, thus forming the famihar chord of the Augmented
Sixth, then fl# proceeds to b, the Fifth of the chord of
the Dominant Seventh e-gj^-b-d. It might be assumed that
\\'agner's harmonies represent nothing more serious than
a simple diatonic succession of chords, modified and
"ornamented" by means of sounds which possess merely a
melodic, but no harmonic significance, i.e., "chromatically
altered " notes, and a few auxiUary and passing-notes. But
even if we eliminate g^f in the second bar, and a^ in the third,
there still remain the chords of the augmented Sixth and
Dominant Seventh. ]\Iust we believe that several of the
sounds in these chords have no real harmonic significance ?

An extremely curious instance of the strange manner in
which Prout jumbles together his new ideas on the melodic
tendencies of sounds with his old theory of fundamental
discords, is his explanation of the major harmony on the
third degree of the major scale, which is followed by the
Tonic harmony. He has evidently some difficult}^ in
understanding how g%, in the first chord, can proceed to g^
in the second. He therefore rewrites the chord as at (6) : —

Tj^-

•" — s>-

iq:

452 THE THEORY OF HARMONY

Here, instead of g#, we find aj?, which as a downward leading-
note may now resolve quite regularly on g. But the chord
at {b), he remarks, is derived from the Dominant Thirteenth :
e is the Thirteenth ; b is the Third, while a\^ is a chromatically
altered Ninth. " The note e belongs to c major, while a\}
is borrowed from c minor." ^ Prout, therefore, like Riemann,
turns what is most decidedly a harmonic triad, or major
harmony, into a discord.

In short, as an exponent of the doctrine that chord
succession is to be explained as resulting from the " melodic
tendencies " of the sounds of the scale, Prout meets with
as little success as those from whom he has borrowed his
ideas. He has cast overboard the harmonic series, he has
no longer the guiding hand of Dr. Day, whose theory in his
previous work he had closely adhered to, and he is now as
it were groping in the dark, and totally unable to formulate
any independent theory of his own. One can scarcely
avoid concluding that Prout, to repeat the remark already
made in connection with the practical works on harmony,
of Weber, Schneider, Albrechtsberger and others, had
abandoned his belief in the possibility of any theory of
harmony which attempts to co-ordinate or systematize the
harmonic facts, and that he had formed the opinion that
the best work on harmony is that which takes account of
the largest number of these facts, and treats of them in a
practical way. But it was just the existence of such a
multitude of isolated facts, apparently \dthout connection
with each other, which was the occasion of Rameau's attempts
to introduce some order and system into the domain of the
theory of harmony.

* Harmony : its Theory and Practice, New Edition, Ch. i8.

453

CHAPTER XIV.

RESUME AND CONCLUSION.

We have now concluded our examination of the most
important works treating of the science of harmony by the
theorists who followed Rameau. The list has not been
complete, but of the works which have been omitted some
cannot properly be described as theories of harmony, while
others are for the most part merely text-books of figured
bass and composition. In our examination of the works
I of Rameau, we found that Rameau derived his fundamental
'principle of harmony from Zarlino and Descartes.'' Both
of these distinguished men had pointed out, as a fact of the
first importance for musical theory, that all the consonances,
all the positive constitutive elements of^ harmony, arose,
not arbitrarily, but according to a certain definite mathe-
matical principle, namely, that of the " senario " or arith-
metical progression 1:2:3:4:5:6. From this " natural "
principle of harmony Rameau developed his theories of the
Fundamental Note in chords, Generation of chords,
Harmonic Ifiversion, the Fundamental Bass, and Chord
Succession : harmony in all its manifestations had, he
contended, its source in this mathematical principle. Later
Rameau became aware of the fact that the major harmon}',
resulting from the union of all the sounds represented by the
proportions of the senario, actually existed in Nature as a
physical fact. Musical sound contained within itself those
natural divisions, and in its resonance the actual sounds,
represented by the proportions of the senario. This Rameau
considered to be not only a wonderful circumstance in itself,
but also a remarkable confirmation of the truth of his
theories.

454 THE THEORY OF HARMONY

We found, liowever, that Rameau had not proceeded
far in the development of his theories before he encountered
serious difficulties. That several of the essential features of
his theory did not wholly satisfy him is proved by the fact
that he frequently changed his views respecting them ; such
were his theories of the origin of the Subdominant, of the
origin of the minor harmony, of the generation of chords,
of the relationship between the major and the minor modes,
of the fundamental bass in Thirds. Other difficulties he
either did not perceive, or did not fully appreciate. We
found that Fetis, Berhoz and others advanced objections
against Rameau's or any other attempt to relate the
theory of harmony to acoustical phenomena, or to discover
for it a physical basis. They pointed out that if Rameau
was justified in considering the resonance of certain sonorous
bodies to constitute the " natural principle " of Consonance,
he was bound to consider also such resonance to be a natural
principle of Dissonance, for even in the sounds of the
harmonic series not only consonances, but dissonances,
are to be found. Further, that if Rameau was justified
in deriving the major harmony from the resonance of strings,
organ pipes, and other similar bodies, he was unable to derive
the minor harmony from the same source ; on the other
hand, he took no account of the many other sonorous bodies
which were in existence, capable of producing " natural
discords " of various kinds. . Berlioz concluded that in respect
of music and harmony, the ear was the sole judge. Musical
intervals and chords were determined not by any natural
acoustical law, but solely according to the impressions they
made on the ear ; while Fetis contended that music had
nothing to do with anything external to man — harmony
existed in and for itself, and in his music-making man enjoyed
and exercised to the full his " philosophic liberty."

The objections of Berlioz and Fetis, however, we found
not to be of the most serious kind. It is quite futile to
assert that consonance, to which we may relate the
phenomenon of beats, has nothing to do with natural
acoustical law. While it is true that in respect of harmony,
of consonance and dissonance, the ear is the principal judge,
it is not true that the ear is free to choose the sound
which it may regard as consonant. For a similar reason,
all the " philosophic liberty " enjoyed by the musician

RESUME AND CONCLUSION 45 5

does not enable liini to invent or create a single new
consonance, any more than it enables him to dispense with
the consonances with which Nature has already provided
him ; he enjoys his " philosophic liberty " only so long as he
conforms to the natural determinations of these consonances;
and the harmonic facts which arise from them and from the
principle of harmony which Rameau observed to reside
in musical sound itself.

Much of the criticism directed against Rameau turned on
his use of the word " natural." Certainly Rameau did not
make sufficiently clear the exact sense in which he made use
of this term. He was content to state that harmony is a
" natural effect " and is " derived directly from Nature."
He might, of course, have pointed out that all motion in
Nature is or tends and strives to become rhythmical, and
therefore harmonious or musical. He might have pointed
to the periodic motions of the heavenly bodies, to the regular
ebb and flow of the tides, to the rhythmic surge of the waves
upon the shore, to the rhythmical bodily movements of men
and animals, to the accents of speech, in prose as well as in
poetry, to daily human activity and intercourse, and social
institutions. It is the lyre of Orpheus which, as the Greeks
finely imagined, charms and sways not man alone, but all
Nature. But it is not by the meaning which Rameau assigns
to the word " natural " that his theory must stand or fall.
The criticism directed against Rameau leaves unaffected his
fundamental principles of harmony, the principles on which
his whole theory is based. It remains true that " harmony
does not arise arbitrarily, but from a definite principle " ;
further, that " this principle of harmony resides in musical
sound itself."

Rameau's fundamental principles stand firm : his theoretical
difficulties and failures were chiefly the direct result, not of
his adherence to, but his departure from, these principles.
It cannot be said that we find, among his successors, any who
have been able to remove these difficulties. In treating of
music and harmony as a physico-mathematical science, and
of the theory of harmony in general, Rameau re\-eals himself
as one of the greatest of musical theorists ; his theoretical
researches are of pre-eminent importance ; indeed, among his
successors, we meet with only a few who appear to have
completely grasped the full significance of his theories. In

456 THE THEORY OF HARMONY

the case of the majority of those who, fohowing Rameau, have
related harmony to acoustical phenomena, the most char-
acteristic feature of their work is undoubtedly the extra-
ordinary development at their hands of the principle of the
generation of chords by means of added Thirds, and their still
more extraordinary manipulation of the harmonic series for
this purpose. It is a characteristic confined not to one country
nor to one school.

In Day's work we find a Fundamental Bass consisting
like Rameau's of three terms, but on different degrees of
the scale, on each of which is placed, not a consonant
harmony, but a gigantic discord. As from his three huge
" natural discords " Day derives all the sounds which
he considers to be comprised in the key-system, to sav
nothing of several others which are not required, and which
have certainly never appeared in any known harmonic system,
Day would seem to have taken the most effective precautions
against being left without an explanation of any sound-
combination which has ever appeared, or is likely to appear,
in music. No sound-combination, it might be imagined,
which could be evolved by the genius of composers but
could be derived from one or the other of his " natural
discords," or failing this, from two, or even all three com-
bined. Unfortunately Day's precautions are unavailing ; he
finds himself unable to account for one of the only two
consonant chords used in music, namely, the minor harmony.
On the Tonic of the minor key-system we find, not a minor
harmony, but a major one. Further, there is no Subdominant
and no Submediant ; no consonant Subdominant harmony,
and no consonant Submediant harmony. It has been
considered that the radical defect of Day's theory lies in
its system of " roots." On the contrary, it is Day's
conception, in which he follows Rameau, of a simple system
of " roots," from which the complete harmonic material
of the key is derived, which explains the influence his theory
has exercised upon musicians. Only, our key and harmonic
systems, including the whole harmonic material utilised by
even the greatest masters, are much more simple than Day
ever imagined. Professor Prout, having closely followed
Day's system, even if, as he considered, he developed it
somewhat, and having exploited the harmonic series for the
purposes of his theory, suggests in the Preface to his new

RESUME AND CONCLUSION 457

work on harmony that the harmonic series has led him astray.
He ftnallv discards Day's system, and practically throws the
harmonic series overboard. In doing so, however, he finds
himself rather more badly off than he was before. He is
quite unable to formulate any independent theory of harmony.
Instead, he borrows theoretical ideas from various quarters.
He considers that much in harmony may be explained by
means of the principle of TonaUty, and of the " melodic
tendencies " of the sounds of the scale. In taking up this
position Prout, however, meets with no better success than
those from whom he has borrowed his ideas.

Against such an absurd manipulation of the sounds of the
harmonic series, and the no less absurd consequences which
follow therefrom, the theoretical works of Kirnberger, Haupt-
mann, Fetis, Stainer, and others, may be regarded in a sense
as a protest and a reaction. Kirnberger, however, finds it

impossible to dispense mth Rameau's principles of a Funda-
mental Note (Grundton) and of Harmonic Inversion ; while
on each degree of the major scale he places not only a triad
but a chord of the Seventh, without considering it necessary
to explain where he obtains these chords, or the hberty to place
them where he does. Kirnberger, after informing us that a
great deal of unnecessary pother has been made over Rameau's
chord of the Added Sixth, which he thinks admits of a quite
simple explanation as arising from a passing-note, nevertheless
avails himself of Rameau's theory of " double employment "
in order to account for a Subdominant-Dominant succession
of harmonies. Stainer, like Kirnberger, is of opinion that
" it is time enough to found a theory of harmony on a mathe-
matical basis . . . when practical musicians shall have framed
a new system of notation which shall point out to the per-
former the ratio of the note he is to sound." Stainer therefore
proposes a theor}^ of harmony based on the tempered scale.
But unfortunately Stainer finds it necessary to point out
that this scale is " out of tune." Like Kirnberger also, Stainer
cannot dispense with a " ground-note " and the inversion of
K chords. But, unhke Kirnberger, he has a principle of chord
generation, which is that of added Thirds. On the Tonic and
.Dominant of both major and minor keys he erects huge

-^1 structures of added Thirds. Stainer does not derive this
principle of chord generation from the tempered scale, but
from Day's theory.

4 58 THE THEORY OF HARMONY

With regard to Hauptmann, it is a matter for lasting regret

that such a musician and theorist, undoubtedly one of the

greatest after Rameau, should, in abandoning the soUd facts

of acoustical science, have imagined that he could discover

a firm basis for his theory in a system of Hegelian metaphysics.

Helmholtz is of opinion that Hauptmann has needlessly buried

his valuable theoretical apperceptions under the abstruse

terminology of metaphysics. But Hauptmann was well

aware that, for a theory of harmony, a basis of some sort was

necessary. He decided against a physical basis, and sought

for his theory a metaphysical one. It is just one of the

principal defects of Hauptmann's system that it has no solid

basis. Hauptmann begins with the important declaration,

on the truth of which he considers so much in his system to

depend, that in music and harmony there are only three

intervals which are " directly intelligible," namely, the

Octave, Fifth, and (major) Third. But this, if true, cannot be

proved by a method of dialectics. Immediately thereafter,

Hauptmann finds it necessary to assume that the two sounds

forming the interval of the Fifth are opposed to one another.

These two sounds, however, which he has already stated to

form a " directly intelligible " interval have, by ah nations

in possession of a musical system, and at all times, been

regarded as directly related to each other in a consonant

relationship. Hauptmann's declaration, supposed by German

theorists to mark an important epoch in the history of the

; theory of harmony, is merely an echo of what had been

A previously stated by Rameau, namely, that the only " directly

'-/ \ derived " intervals are those of the Octave, Fifth and major

Third. Hauptmann must frequently have cast a longing

glance at Rameau's extremely simple and clear method of

demonstrating that these were the only intervals which arose

directly from the fundamental note. It is just the principal

weakness of Hauptmann's theor\^ that it does not appear to

permit of anything in harmony being regarded as " directly

intelligible." Each harmonic fact can only be understood

through the " mediation " of something else. Thus the

sounds forming the Fifth are opposed to one another, and are

^only rendered " intelligible " through the " mediation " of

the Third. Similarly, the Dominant-Tonic succession of

. harmonies (Perfect Cadence) can only be understood as brought

about by the " mediation " of the triad on the Mediant.

RESUME AND CONCLUSION 459

But if there is one thing more than another in harmony which
is " directly intelligible," it is the succession of harmonies
in the Perfect Cadence.

The works of Dr. Riemann, who must be regarded as the
foremost representative, in the domain of musical theory, of
latter day German " culture," present not only an interesting
theoretical but also psychological study. Dr. Riemann's
theoretical methods, as we have seen, are not above criticism.
Even if we leave out of account his — doubtless unintentional
— misquotation of important passages from eminent theorists
whose works nevertheless are but httle known to the average
musician, the fact remains that Riemann is not over-careful
as to the means he adopts to buttress, as he imagines, his own
theories. His treatment of ZarUno is a case in point. Not
only has he widely disseminated statements respecting the
nature of Zarlino's theory which are not borne out, but
actually contradicted by the facts, but he quite fails to grasp
the real significance of ZarUno's theoretical researches, and
his real position in the history of musical theory.

Of his own theory of harmony, he tells us at the conclusion
of his Geschichte der Musiktheohe, that it stands firmly
and solidly on the rock of truth (" der Standpunkt, auf
dem ich stehe, cin felsenfcstes Fundament erhdlt "). This
is somewhat confident language to come from a theorist
who is unable to make up his mind as to what is the foundation,
the fundamental note, of the minor harmony, or as to what
constitutes the proper basis even of the theory of harmony,
and whose principal work on harmony. Harmony Simplified,
is characterized by the most extraordinary uncertainty and
contradiction. The difficulties of the subject, as Rameau
discovered, are great. But it is certain that it is not in
modern German " culture " that we find their solution.
Riemann's first great theoretical principle is embodied in his
theory of the " tonal functions of chords." Every chord in
the key-system, he states, has and must have either a Tonic,
Dominant, or Subdominant " function " or significance. But
Riemann quite fails to demonstrate that this is really the case.
In order to support his theory, he is obliged to introduce an
elaborate system of " parallel-kkngs," of whose origin he is
unable to give any definite explanation, and which, not-
withstanding that they appear to admit of the most extra-
ordinary harmonic metamorphoses, he still considers to possess

460 THE THEORY OF HARMONY

a Tonic, Dominant, or Subdominant " function." But in
spite of the manifold and quite unrecognisable forms which
the parallel-klangs may assume, Riemann presents us with
several chords which he himself admits cannot be explained
either by his theory of parallel-klangs or of " tonal functions."
Such chords have to be understood as arising from " leading -
tone steps." In reviewing Riemann's procedure in respect
of his " parallel klangs " and chords arising from " leading-
itone steps," one can well appreciate the force of the statement
^ j jwhich he makes at the beginning of his work, that harmony
'has its roots in melody. On the other hand, it is doubtless
'his theory of " klang-representation," his generation of the
major harmony from the hannonic series, and of the minor
harmony from the " undertone series " which occasions his
remark at the end of his work, that " harmony is the fountain-
,1 head from which all music flows."

Dr. Riemann's second great theoretical principle is that
not only the minor harmony but also the minor key-system
must be regarded as the direct antithesis of the major harmony
and key-system. Yet at the beginning of his work he tells
the student that in practice he had better consider the
lowest note of the minor harmony to be the fundamental
note. But this fact does not prevent Dr. Riemann from
introducing a bewildering variety of " klangs," " over-klangs,"
" under-klangs," " contra-klangs," etc., into a work
already sufficiently comphcated by an elaborate system
of " parallel-klangs." In his work Harmony Simplified,
Dr. Riemann has made of harmony a subject of quite needless
complexity.

With regard to Fetis, we saw that he considers the funda-
mental principle ol all music to be what he calls Tonalit}-, a
principle however as to whose nature he has himself only a
vague conception, and of which he is unable to furnish any
clear explanation. All music, according to Fetis, has as its
^ basis or source the scale. The nature of music and harmony
is determined by the order or arrangement of sounds in the
scale. Change the order of sounds in the scale, and the nature
N/of the harmony resulting therefrom becomes likewise changed.
Scales are not all of one type, but are of the most varied type.
Take, for example, the Chinese or other similar scales : our
harmony would become impossible in such tonalities. But
Fetis does not inform us whether the Chinese, or the other

RESUME AND CONCLUSION 461

nations to whom he refers, have ever attempted to make their
scales the basis of a harmonic art of music. He inverts things
in a curious fashion. He considers that the harmony resulting
from such scales would be quite different from " our harmony,"
which is no doubt the case so far as the Thirds and Sixths are
concerned, but which is false with regard to the consonances
of the Octave, Fifth, and Fourth found in these scales. These
consonances are the same in every respect as our perfect
consonances. What Fetis ought to have said is that the inevi-
table result of the appUcation of harmony, of the " natural "
Thirds and Sixths, to the scales he mentions, would be to
change the arrangement of the sounds of the scale. This,
however, would make it appear that it is harmony which
determines the sounds of the scale, and not the sounds of the
i scale which determine harmony. Fetis speaks of "our
I harmony." But what other kind of harmony is there ?
There is none, nor has there ever been any other in existence.
;The constituents of " our harmony " are the perfect and the
jimpeifect consonances. The consonances of the Octave, Fifth,
■ and Fourth in use at the present day, are the same in every
respect as the consonances known in the sixth centur}/ B.C.

Fetis is quite unable to inform us where and when the
scale which has determined " our harmony " came into
existence. Of all the scales which were in use throughout
Europe before the advent of polyphony there was not one
which corresponded with our major or minor scale. If
Fetis had carried his researches into the nature, history, and
development of scales, and especially of the Church Modes,
a little further, he would have discovered that these Modes,
quite different as regards the arrangement and proportion
'^ of their sounds from our modern modes, were under the
influence of harmony gradually altered until they assumed
the form of our major and minor modes. It was harmony,
and especially the use of the " natural " Thirds, which
played the greatest part in banishing these old Modes out of
existence. Fetis evidently wishes us to believe that " our
harmony " has arisen through a chance combination of two
or more sounds from a scale fashioned on " purely melodic
principles," so as to fo'm a series of sounds varying in pitch,
and of intervals readily appreciable to the ear. That such
was actually his view is confirmed by his remarks on the
origin of scales in the Preface to his Traite dc Vharmonie.

46:

THE THEORY OF HARMONY

It is when he comes to formulate what he calls the " laws
of tonahty," and attempts to explain chord succession,
that the real barrenness of Fetis's principle of tonality
becomes apparent. What the laws of tonaUty really are,
Fetis has only a vague idea. These laws do not appear
to arise from, or to be connected with, the order of sounds
in the scale, but \\ith the fact that certain sounds in the
\scale have a character of repose. Only those sounds have a
I jchafacter-of repose which admit of the harmony of the Fifth.
What sounds these are, Fetis does not find it easy to deter-
mine. He first postulates the first, fourth, and fifth degrees
of the scale as the sounds of repose, but afterwards finds
himself obhged to admit others, especially that on the sixth
degree. As for the third degree of the major scale, this
is not a sound of repose, although the Fifth above this sound
is a Perfect Fifth. The reason for this, according to Fetis,
is that " its tonal character is absolutely antagonistic to
every sense of repose."

Here, the theory of Fetis appears to be not altogether
.unconnected with Raraeau's Fundamental Bass in Fifths, the
, /jthree terms of which consist of the sounds on t'he'' first,
fourth, and fifth degrees of the scale. Fetis, in fact, perceives
that for the different kinds of Cadence, which is the principal
means used in harmonic music to produce the effect of repose,
these three sounds, with their harmonies, are indispensable.
Fetis knows well, also, that a Cadence may occur not onlv
on the Tonic, but on the Dominant and Subdominant as
well. But it is important to note if, in the Cadence at {a) : —

(«)

(b)

'JOiZ

-<s>-

-e>-

32:

:q:

-o-

\/\

we regard, as we needs must, the Dominant ^ as a note
I of repose, so also must we regard its Third h and Fifth d,
which are sounds of its harmony. Similarl^-, we must
consider all the sounds of the Subdominant harmony f-a-c
at (6) to be sounds of repose. The sounds of repose in the

RESUME AND CONCLUSION 463

scale of C major are therefore c, d, c, f, g, a, b, c' ; a fact
which, however interesting it may be in itself, does not help
us much towards a solution of the problems of harmony.

It is noteworthy that musicians who hold by the principle
of tonality, whatever that may mean, and by the character
of repose attaching to certain sounds in the scale, have not
yet made up their minds as to what sounds these are. Some
consider the third of the major scale, which Fetis pronounced
to be absolutely antagonistic to any sense of repose, to be
actually a sound of repose, while the fourth degree of the
scale, which Fetis considered to be a note of repose, they
describe as a leading-note, a note of unrest, which tends
to fall to the note a semitone below. In a work on ear
training recently published the author considers the first,
third, and fifth sounds of the major scale to be the true
sounds of repose ; the leading-note he describes as the note
of greatest unrest. But he finds himself obliged to add
that in certain circumstances this leading-note may appear
as a note of rest, which is of course the case. To this we
might add that the notes of rest may appear as notes of
unrest : —

Thus c and e, notes of rest, are perceived to be notes-
of unrest, which find rest in h and d, notes of great
unrest ! It may be objected that it is dissonance which
brings about the downward " resolution " of the sounds
c and e. But e is not a dissonant sound ; and why should c,
which forms a perfect Fourth with g, be regarded as dissonant ?
The real explanation, of course, is to be found in Rameau/s
principle of the Fundamental note. The two sounds c and e
move downward in order to form a harmonic triad on ".
In short, the notes of rest in the scale may become notes of
"unrest, and the notes of unrest notes of rest, according to
circumstances. These circumstances are determined by
harmonic, not melodic considerations.

But all this represents only one side of the theory of Fetis.
It has another side. According to Fetis, we must believe not
only that it is the scale which has determined our harmony

464 THE THEORY OF HARMONY

and our tonality, but that it is harmony which has determined
( the sounds of our scale. This has been brought about by the
7 cliord of the Dominant Seventh, which Fetis calls the " natural
Vhord of the Seventh " ; this chord, he says, has its source in
the harmonic series, and is represented by the terms 4:5:6:7
of this series. It is Fetis, however, who in his works on har-
mony has made it his principal object to prove the absurdity
of relating the theory of harmony to acoustical phenomena.
Fetis is not the only musical theorist who has attempted
the impossible task of running two absolutely contradictory
theories side by side. What is surprising is that musicians
should have accepted either of them, much less both. The
widely disseminated doctrine of Fetis that our modern har-
^ monic system has been brought about by the introduction into
] harmonic music of the natural chord of the Dominant Seventh
has become almost an article of faith among musicians.
1 Nothing has tended more to obscure the true nature of harmony
and of our harmonic system. Musicians have not sufficiently
considered whether it might not have been the developments
resulting in our present harmonic system which made the
chord of the Dominant Seventh and, in general, unprepared
discords artistically possible, and gave them harmonic value
and significance.

The most important part of Helmholtz's work. The Sen-
sations of Tone, is undoubtedly that in which he treats of the
physical properties of musical sound. When he approaches
the theory of harmony, it becomes evident that something
more is necessary in dealing with so elusive and subtle a subject
than trained scientific perception and judgment. This
" something "^ — intuition or genius — Rameau possessed in . a-
marked degree. Helmholtz's statements with regard to some
of the most fundamental principles of the science of harmony
.are marked by a curious hesitation and uncertainty. He
considers that consonance is to be explained by means of the
phenomenon of beats, but also suggests that the real explana-
tion of consonance is to be found in Fourier's law. The riddle
of consonance, he states, has been solved by the discovery
that tjiej^ar resolves all complex sounds into pendular oscilla-
\/ [tions, according to the laws of sympathetic vibration. Again,
when he treats of the origin of early scales, he finds himself
obliged to make use of " natural " TJiirds and Sixths, but tells
us in another part of his work that such consonances were

RESUME AND CONCLUSION 465

unknown until the close of the Middle Ages. In one place
he tells us that it is absurd to consider that the second degree
of the scale was determined by an " understood " fundamental
bass at a time when harmony was unknown ; in another place,
that there is no other means by which this second degree can
be determined and accurately intoned. If at one time he
expressly states that the closest relationship existing between
chords is that of the Fifth, at another he insists that the
closest relationship is that of the Third, where the chords have
two notes in common with each other.

Helmholtz's views as to the nature of the minor harmony,
we found, were not original, but were anticipated by Rameau,
followed by d'Alembert and Serre ; while as for his principle
of " klang-representation," considered by Dr. Riemann to be
liis most original contribution to the science of harmony,
Rameau not only understands this principle, and explains it
in the most complete way, but makes use of it for his system
of the Fundamental Bass.

Like Rameau, Helmholtz is of opinion that in the natural
relations to be observed in the resonance of a sonorous body,
we find the proper basis of the theory of harmon}-. But
Helmholtz does not appear to have an}' firml\^ rooted con-
victions on this point, and does not seem disposed, like the
great French theorist, to consider such relations to constitute
the fundamental principle of harmony, and of our harmonic
system. Here again Helmholtz speaks with two voices, for
he finally informs us that it is really in Tonality that we
discover the " fundamental law " of all music, melodic or
harmonic. Further, .tonalitv, in which the principal role
is assigned to an " arbitrarily selected " Tonic, is " not a
natural law, but an sesthetical principle." The theory of
harmony, then, would appear to haVe not a physical, but a
metaphysical (psychological) basis.

' It is largely owing to Helmholtz and Fetis that the doctrine
of tonality has become so prominent at the present day.
Now that the " root " theory, thanks to the extraordinary
exploitation of the harmonic series, is falling so rapidly,
into discredit, tonalitv has become the mystic woi'd which-' ,
is to solve for us all the mysteries of harmonic science// t"'^
TH of all who make use of the term how many could giva
a clear answer to the question — What is tonality ? Helmholtz
is at least able to reply that tonality is the relationship which

2H ' ~ '

466

THE THEORY OF HARMONY

; the nqtes_of the scale bear to the Tonic, and the chords of
Ai tHe'key to the Tonic Chord. That such a relationship exists
' has been known for a few centuries ; but the theory of
harmony begins when the attempt is made to discover
what exactly is the relationship which notes and chords
bear to the Tonic, and to the Tonic chord. It is true
that the vast majority of musical compositions begin and
end on the Tonic chord, and it is quite correct to consider
this as a fact of theoretical importance ; but what goes on
in the middle is also of considerable importance, and it is
here that the principal theoretical difficulties he. That is
not an adequate harmonic analysis of a musical composition
which merely points out an occasional Tonic (or even, in
addition, an occasional Dominant or Subdominant) chord,
which stand like harmonic oases in the midst of stretches
of arid waste from which harmony seems for the time being
to have disappeared. It has been thought sufficient to point
out that everything in harmony gravitates towards the Tonic
chord, but this does not help us to understand how so much
in harmony gravitates away from the Tonic chord. It may
explain why every chord should proceed at once to the
Tonic chord as at (a) : —

7.^1

supposing that chords generally proceeded in this way, but it
cannot explain the extremely simple chord successions in
C major, which follow (6) and which might be multipHed
almost ad infinitum by means of chromatic chords. " What
J call TonaHty," exclaims Fetis, in one of the numerous
attempts he makes to define this term, " i^ the aggregation
, of facts, harmonic and melodic, presented to us in the artistic
works of composers themselves." But, it is important to
note, the theory of harmony begins and does not end here ;
it begins when we set out to discover the principles which
underhe such an " aggregation of facts."

RESUME AND CONCLUSION 467

But Helmholtz, although he expressly declares that the
fundamental law of all music, melodic or harmonic, is
Tonality, which is " not a natural law, but an sesthetical
principle," does not appear to be quite satisfied with this
statement ; he is also of opinion that music and harmony
depend, to a certain extent, on natural acoustical law. This
theor\' appears to be quite a feasible one, and it is needless
to find fault with it so long as it is properly understood that
the creative work of the musician or tone-poet is accomplished
without any conscious dependence on natural or acoustical
law, but solely on aesthetical principles, and that that part
of music which Helmholtz considers, quite righth', to depend
on natural law, for example, the determination of the con-
I '^onances, depends quite as much on aesthetical principles
as any other part. When properly considered, the theory
in question does not appear to have much meaning, and it
is not surprising that hitherto the results of attempts to
explain harmony on aesthetical principles have been dis-
appointinglv meagre, superficial, and inadequate. The
application of such a theory to the simplest harmonic facts
produces some curious results. Thus, while Helmholtz con-
siders the rqajor harmony to be determined by natural law,
such is not his~vtew"of the m'inor'Farmony. It would appear
then, that while we must regard the major harmony as .|
based on a natural law. we must consider the minor harmony '
to arise from an aesthetical principle ; or, as some theorists
■"^ell us, while the major harmonj' is a " natural " harmony,
Mthe minor harmony is an "artificial " one. More stricth',
/ seeing that Helmholtz considers the minor harmony C'e\f-g
//to be a compound tone of c into which the "foreign " sound
e^ is introduced, the sounds c-g of the minor harmony c-e\^'g
arise from a natural law, while the " foreign " sound e\f has
its source in an aesthetical principle. We saw that Professor
Prout considered himself at liberty to select, reject, or even
modify, in the most arbitrary fashion, sounds from the
harmonic series in order to form chords. In this he supports
himself on the authority of Helmholtz, quotes the statement
referred to, and gives it a prominent place at the beginning
of his work. For Professor Prout, the harmonic series^ |\
represents the natural law,~while his selection of sounds from ' ^
this series in order to form chords of the Eleventh, Thirteenth^,
etc., represents the aesthetical principle. Dr. Riemann, ,

468. THE THEORY OF HARMONY

in his Natur der Harmonik, considers the major harmony
to arise from a natural law, but is of opinion that the minor
harmony can only be explained on psychological grounds.
Again, we meet with theorists who derive a consonant
harmony from the first sounds of the harmonic series, but
who explain all dissonant harmonic formations as arising from
"non-harmonic" notes, "chromatic alteration," and so forth.
It is evident that one of the principal drawbacks of such a
theory is that it is too elastic ; it presents too great a
temptation to the theorist, who has httle difficulty in referring
sounds and chords whose natural origin he can easily trace
to a natural law, but all other sounds and chords, that is,
all those of whose origin he is ignorant, to an aesthetical
principle.

Still it is asked, has music really to do with anything
external to ourselves ? Is music not the expression

' of man's inner nature, of his sensations, emotions,

I tideas : is it not, therefore, in the human soul that we must,
discover the true source and explanation of music ? Is not
.music man-made ? It proceeds from man and wiU perish

\ 'with him. Why not accept this fact, brush aside the cobwebs
of mediaeval mysticism, and give up the attempt to explain
music as related in some way to the eternal laws, the Supreme
Intelligences, which guide the stars m their courses ? It is

i true that in present-day musical theory we find a marked
\^^i/ tendency to refer many of its problems to ps^'chology. But it

Iris to be feared that what temperament was to an older genera-
\^' I tion, that psychology is at the present day, namely, a haven
(1 of refuge for the distressed musical theorist. When a musical
theorist tells us that a certain fact admits only of a
psychological explanation, it is more than probable that he
has failed to discover for it any adequate explanation. It
should be remembered that if the difficulties connected
with the science of harmony are great, so likewise are those
connected with the science of psychology, and that if musical
theorists are turning to psychology to help them out of
their difficulties, psychologists themselves on the other
hand are searching in music and harmony, and their effects
on the human organization, for the solution of problems
which confront the science of psychology. Further, that
music is the expression of man's inner nature does not mean
that harmony cannot possibly have a physical basis.

RESUME AND CONCLUSION 469

There is, undoubtedly, a marked tendency at the present
I day to accept the view that all music, melodic or harmonic,
I has its origin in the scale. _. Melody, it is pointed out, existed
Tbefore harmony ; consequently we must regard harmony
■gs having its roots in melody, that is, in the scale. All
melodic and harmonic facts, then, are developed from the
scale on purely ;esthetical principles, and can admit only
of a psychological explanation. But what, then, is the
origin of the scale ? This is a matter with regard to which
much speculation has been indulged in. Helmholtz's
explanations as to the probable origin of early scales are not
convincing. Others are of opinion that the matter admits
I of a quite simple explanation. They point out that whereas
the scales of man consist of a succession of degrees, of intervals
of sound, the scale of nature on the other hand consists of
an unbroken stretch of sound. Such a scale is useless for
artistic purposes, for which a series of definite intervals
is necessary. But nature does not supply us with these
intervals ; strictly speaking, nature furnishes us with no
scale, but only with the raw material from which scales
may be formed.

From the stretch of sound supplied by nature man has
measured off certain intervals easily appreciable by the ear,
and suitable for his artistic needs ; but such a process, it is
evident, admits of a quite simple psychological explanation,
and has nothing to do with mathematics or acoustical
phenomena. It might at first be imagined that scales formed
in this way would consist of a succession of eqiial intervals.
That this is not the case, but that musical scales consist of a
series of intervals of different sizes, and that we find tones
which differ by the extremely minute interval of a comma
(80 : 81), an inter\^al which the unaided ear could not possibly
determine correctly, is no doubt to be explained by the
necessity for variety of artistic material.

Fetis is able to supply us with numerous particulars as

Lto the origin of early scales. The first scales, he states,

/fl consisted for the most part of small intervals of a quarter

/ / of a tone ; these in course of time gave place to scales

" consisting largely of semitones, from which was eventuallj'

developed the diatonic scale, consisting mainly of tones.

' " The interval of a tone in music," he remarks, " can only

be understood as arising from the elimination of a number

470 THE THEORY OF HARMONY

of smaller intervals, notably that of the semitone." ^ Fetis
attempts to support these views by a reference to ancient
Greek scales. The facts, however, so far as we know them,
appear to point to quite the opposite conclusion. The
ancient enharmonic tetrachord of the Greeks {b-c — ^) contained
no quarter-tones ; it was the latey enharmonic tetrachord
which comprised iwo quarter-tones and a Third. Of these
quarter-tones Aristoxenus has said that " no voice could
sing three of them in succession." Further, Boethius, in
his De Musica, states that, according to Nicomachus, the
most ancient method of tuning the lyre was as follows :
c — -f-g — c' , where we find a Fourth and Fifth above c, and a
Fourth and Fifth below c' ; while between / and g there
is an interval of a whole-tone. This whole-tone interval,
it is evident, is determined as the difference between the
Fifth c-g and Fourth c-f. All this is only what might be
expected ; for it is quite natural to suppose that it was the
larger intervals which at first acquired definiteness, and only
subsequently the smaller intervals.

But it is all the more strange that theorists should indulge
in so much speculation regarding the probable origin of musical
scales, including Greek scales, and as to the principle on which
their intervals were selected and determined, seeing that
early writings give us the most definite information on this

/j jmatter. Greek writers on music tell us plainly that the
j jwhole-tone is the difference between the Fourth and the Fifth.
IThe__Greek semitone, on the other hand, represented the
Mdifference between two whole-tones and a Fourth. The Fourth
formed the basis of every species of tetrachord. The Octave
j constituted the limits of the complete Octave scales. The
. y I JGreeks, then, derived their scales by means of a process of
^^ I Mining in Fourths and Fifths, a process not essentially different
from that by which we obtain our scales at the present day.
Without some such method of " tuning " it is difficult to
understand how any musical scale could be formed, much less
perpetuated. These consonances of the Octave, Fifth, and
Fourth, appear to have been known to all ancient peoples
among whom music was cultivated", no matter what form their
scales assumed ; and it is quite impossible to consider that
they were arrived at, among the various nations, and defined

^ Traitf de rHarmonie (Preface) and Hist. Cn'n. de la Mitsique.

RESUME AND CONCLUSION

471

by means of the addition, the accumulation, of small intervals

— the Fourth, for example, as determined by means of the

addition of so many quarter-tones, or so many semitones.

In the case of the Octave, such a view is manifestly absurd.

f As Descartes remarked, " We never hear a musical sound,

witliout our ear being affected at the same time by its Octave."

/// And what is true of the Octave is true also of the Fifth, or

''' Twelfth, of which the inversion is the Fourth.

Rameau stated that we must believe that the fundamental
bass in Fifths was known to the ancient Greeks, or, at least,
that their marvellous intuition had enabled them to discover
its principle, for otherwise they could never have accurately
determined their whole-tone. Such a statement must to
some have appeared bold, to others merely foolish. It is,
of course, impossible to maintain that any system of a funda-
mental bass was known to the Greeks. Nevertheless the
Greek method of deriving the whole-tone led directly to the
principle of the fundamental bass. The Greeks derived their
whole-tone (8 : 9) as at (a) ; we, as at {b) : —

F.B.

The only difference between the two processes is that we, in
possession of a harmonic art of music, have supplied the
fundamental note to both the intervals d-g and c-g. It is a
' remarkable fact that in the Greek method of determining the
'whole-tone we discover the germ of our harmonic system.
Theorists do not appear to have observed this fact ; it is,
ne\-ertheless, a pregnant one for musical theory, and one
which manifests in a striking way the gradual unfolding of
harmonic principles, and the intimate relationship existing
between various stages of musical development.

These early scales were therefore derived from the har-
monies, or consonances, of the Octave, Fifth, and Fourth.
The Octave defined the limits of the complete Octave scale ;

vi.

;?i-^ '

'/■

472 THE thp:ory of harmony

the Fourth formed the basis of the tetrachord ; the whole-
tone was accurately defined as the difference between the
Fourth and the Fifth ; while the semitone was determined,
as the difference between the Fourth and two whole-tones.
These scales, then, had their source in harrnony. It is true
that we meet with theorists who object to such a view, and
who impatiently inquire how scales having their source in
harmony could possibly arise among nations to whom har-
mony was unknown. Such theorists are evidently of opinion
that two sounds blending together in the consonance of the
Fifth or Fourth do not constitute harmony.

The arguments so often met with respecting harmony and
scales — harmony arising from arbitrary melodic combinations,

V from " chromatic alteration " of sounds, and so forth — are
ingenious, but sophistical, and calculated to mislead those
who are unable to give to the subject the necessary patient
investigation and reflection. For example, there is a degree
of truth in the assertion that when the cock crows, the sun
rises. But it would be rash to conclude that the latter
phenomenon is the necessary consequence of the former. It
is frequently pointed out that among the scales of various
nations, including savage races, we meet with examples of
inharmonic or irrational scales, and it is argued that the
mere existence of such scales is sufficient to prove the futiUty
of relating music and harmony to natural acoustical law.
As the scale is not derived from harmony, , then harmony
must be derived from the scale ! It is true that in the rudest
Pjtype of chant, the cadences, the rising and falling of the
V / (voice, furnish a not inconsiderable means of expression. Such
transitions from acute to grave, and from grave to acute,
may be said to constitute a scale, and in this sense all scales
might be said to be originally inharmonic. But the next
and the inevitable step in artistic progress is to turn this
I inharmonic scale into a harmonic one, i.e., a musical scale,
and one of the earliest of musical scales is the so-called,

J pentatonic scale, — arrived at by means of a process of
turning in perfect Fifths, as C-G-D-A-E — a scale which appears
to have been in use amongst nations the most widely
separated from one another. When we have satisfied
ourselves that any given scale is in reaUty representative of
a genuine musical culture, and not merely a sort of musical
toy, it is necessary to exercise some degree of caution before

RESUME AND CONCLUSION 473

pronouncing judgment as to its nature. If it is to be
submitted to a mathematical test, too great care cannot be
taken to ascertain whether its proportions are correct. It is
true that we meet with investigators, whose devotion and
enthusiasm are beyond question, who are able to present us
with scales of the most diverse tN'pes, the proportions of
which are set forth with astonishing mathematical accuracy.
One might submit that these are, if an^-thing, too accurate.
Such accuracy tends to produce some misgi\nng. When it
is remembered that a considerable degree of skill is required
for the accurate, that is, the mathematically exact intonation
of even a consonant interval, where the ear is supported and
guided by harmony, and indeed by Nature, it is e\ident that
the difficulties in the way of the correct intonation (if there
be really such a thing) of an inharmonic or irrational inter\-al
must be enormous, notwithstanding all that has been said
with regard to the sensitiveness and dehcacy of ear of savage
or semi-civilized peoples. Yet even a slight divergence from
the true intonation will seriously affect the mathematical
result: How often does one hear a justl}' intoned scale, t^at
is, a scale in perfect tune? Of the many varieties of the-
'^e-mpered scale, which is it that exhibits the correct propor-
tions? And yet these scales undoubtedl}- represent real
harmonic scales, scales which are in tune, just as most circles
are meant for perfect circles. Not so many years ago it was
the fashion with the interpreters of Oriental music to describe
the Arabic and Persian scale as one consisting of 17
degrees, or of 16 intervals, each interval corresponding to
about a third of a tone. It was customary to point out that
such a scale did not at all agree with Western harmonic
notions, until it was discovered from certain 14th century-
writings of Persian theorists themselves, that the Persian
JC3J£. was a^rrived at by means of a perfectly S3-steniatic
process of tuning in Fifths. In fact, in the folk-music which
has enriched the world we discover tonal relationships much
more delicate and subtle than mere differences of pitch.
Such tonal relationships have their sole and ultimate basis in
consonance. The whole question, in so far as it relates to
our subject, can be cleared up in a word. If the scale is
hannonic in its origin, then it is derived from harmon}-. If,
"on the other hand, the scale is inharmonic in its origin, then
harmonv cannot be derived from an inharmonic scale.

2H*

474

THE THEORY OF HARMONY

Between the second chord at {a) and that at (b) :-

(«) (b)

-<si-

IQZI

-G>-

:2ZZ2I

:2E2:

there is a difference of tonal effect, of " tonahty." We may,
if we choose, explain this difference by stating that in the first
case there is a whole-tone between the second and third
degrees of the scale, whereas in the second case there is a
semitone. Tlje difference of tonal effect is, therefore, owing
tQ^ the different order or arrangement of the sounds of the scale.
But this explanation, if it can properly be regarded as such,
is worse than no explanation, because it only serves to totally
obscure the truth. The true explanation, of course, is a very
simple one, and, one would imagine, almost self-evident,
namely, that while in the first case we have a major harmony,
in the second case we have a minor one. Hence the semitone
between d-e\}. It is necessary to lay stress on this point,
because the arguments of such theorists, ingenious and subtle
though they be, and indeed in great part because of this,
completely bar the way to a proper understanding of the
nature of harmony. It is all the more necessary because
composers are experimenting more largely in old Modes,
while every now and again we are reminded that much in our
modern harmony arises out of, or is based on, a " scale of
whole-tones." From J. S. Bach we have received some noble^
compositions, which are generally described as being in the
Dorian Mode. But as Bach requires for his harmonies a
major as well as a minor Seventh, and a minor as well as.
a major Sixth, he in reality makes use of our minor harmonic
system, and of all the forms of the minor scale. The same
remark applies to later composers who have imitated Bach
in this respect. But in the case of old IModes which are not
merely disguised forms of our major or minor modes, or are
not otherwise altered beyond recognition, it should be re-
membered that the peculiar effect of harmonic successions
within these Modes is not owing primarily to the order of
sounds in the scale in which they occur, but to the peculiar
nature of the harmonic successions and relationships them-
selves.

RESUME

AND CONCLUSION

475

With regard to the "whole-tone scale," it is somewhat difficult
to maintain that it has had a purely melodic origin. It cannot
have been selected for its intrinsic melodic beauty. We
possess, one might say, documentary evidence in the works
of composers themselves that it has been developed from
harmony.

The so-called whole-tone scale at (a) : —

is not in itself inteUigible ; but it acquires " definiteness," or
musical significance if understood harmonically, as at (b). It
might of course be " harmonized " in other ways. In short,
scales, chords, harmonic successions and relationships, are but
different manifestations of one and the same principle. But
it is by no means reassuring, so far as the proper understanding
of harmony at the present day is concerned, not only that we
should be so frequently informed that much of our modern
harmony arises from a " whole-tone scale," but that such a
scale should be described as a whole-tone scale at all. Since
when did such an interval as g^-b\^ become a whole-tone ?
One might as well describe the inter\'al c-gj(f as a consonance,
because g^ is the same note as a\}.

So far, it cannot be said that the application of psychology
to the solution of the problems connected with harmony has
produced any very striking results. Stumpf, in his Ton-
psychologie, gives up the attempt to explain the phenomenon
of Consonance on psychological grounds ; it must have, he
thinks, a physiological explanation of some kind. Even if
psychology had succeeded in solving tlie riddle of Consonance,
it would only be at the beginning of its task, in so far as the
tlieory of harmony is concerned. One may even venture to
suggest that, at the present time, it appears to be much more
hkely that the science of psycholog}^ is to be advanced by the
successful solution of the problems connected with a theory
of harmony based on " natural principles," than that the

476 THE THEORY OF HARMONY

problems connected with harmony shall be soh^ed by means
of the science of psychology. The question as to whether
music, so intimately connected with mental processes, witn
modifications of the human soul, has a physical basis, is one
of the greatest consequence for psychology.

But why, it is objected, persist in regarding harmony as
having its source in " natural principles," as determined by
natural laws ? Have not theorists for generations followed
this road, only to find that it leads nowhere, unless indeed to
a morass of confusion and difficulty ? What, in reality, has
music to do with mathematics or proportions ? Are we,
frankly, really conscious that in hstening to a Third, Fifth,
or other interval, such an interval corresponds to a certain
numerical ratio ? Especially let us not be asked to believe
that the consonances were selected from any other than
purely aesthetical considerations. On the contrary, how many
centuries of experiment, of education of the ear, in the case
of primitive peoples among whom a certain rude type of
musical art was cultivated, how much apparently aimless
wandering from one sound to another may have been necessary
before even these simple intervals were distinguished, and the
relationship existing between their sounds properly recognized
and aesthetically appreciated ? And in general dp we not
find in music, in its nature so impalpable, elusive, subjective,
an art essentially different from all other arts, in that ij:_is
jnanifestly unrelated to objective phenomena.^ to anything"
external to itself ?

Let us concede, then, that in his music-making man
exercises to the full what Fetis terms his " philosophic
liberty," that is, music is man-made and has nothing to
do with anything external to man. This being understood,
there should be noted a few facts of some importance. The
first is, that early peoples, guided by their sense of the
I beautiful, perceived that between certain sounds heard
simultaneously, or in succession, there existed a definite
1 relationship. 'Here at the very outset we light on a fact
of supreme importance. It is not only that this fact forrns
the only possible basis of a rational theory of harmony :
without such relationships it is difficult to conceive how
there could be any art of music, harmonic or melodic, or
anything but a mere aimless wandering from one sound
to another. These relationships, as Tartini pointed out, are

RESUME AND CONCLUSION 4 77

" independent of the human will," that is, they were not
created by man for his artistic needs ; they were suggested,
revealed to him. He is no more able to create than he is to
destroy them. One sometimes hears of " artificial " key-
relationships. But a composer might as well attempt to
cultivate artificial flowers as to discover artificial key or
sound-relationships.
)s Thus were recognized the consonances of the Octave,
hifth, and Fourth, and from these the Greeks derived their
scales. Greek writers themselves state that these consonances
formed the basis of their tonal systems. Long before the
classical period of Greek antiquity we find, among early
peoples, the most extraordinary beliefs respecting the divine
origin of music. Some assert that it has descended from the
gods. Others, like the Egyptians, compare the sounds of
their scale with the heavenly bodies, and name them after
them. To Pythagoras is attributed the discovery that the

consonances were determined respectively by the ratios.
1:2, 2:3, and 3:4. This discovery led directly to the
h^rst solid achievements in the science of harmony.

fl * Further, the diatonic scales of the Greeks passed, under
different names, into the service of the early Church. They < \
were perpetuated throughout the whole of the Middle Ages^^^C!--^
and were in use at the time of the rise of polyphon}'. During
the development of polyphony, the correctness of intonation
of several intervals made use of in composition, and especially
the Thirds, began to be called in question by musicians. J
Eventually the Thirds, in these time-honoured modes, whicli"
had retained their Pythagorean tuning, had to give way to
the " natural " major and minor Thirds. To the intro-
duction of the natural Thirds may be ascribed in great
measure that great artistic development, the nature of which
became apparent after the death of Palestrina. They not
only altered the character of the Modes — which now Zarlino
himself divided into Major and Minor — but were a powerful,
factor in their gradual extinction, or more accurately their]
transformation into our major and minor modes. The/
essential and determining sounds of the major mode were
now those which constituted a major harmony above the
PTnal ; those of the minor mode constituted a minor harmony.

\ A considerable time before the death of Palestrina, then, the
tonal system of European harmonic music had its basis in

47« THE THEORY OF HARMONY

the Perfect and Imperfect Consonances. It was discovered

that the new consonant major and minor Thirds corresponded

respectively to the ratios 4 : 5 and 5 : 6. Taking them in

their order of perfection, the perfect consonances and the

two Thirds were all expressed by the ratios i : 2, 2 : 3, 3 : 4,

,4:5, and 5:6. It was Zarlino who pointed out that the

^_X^ y]|Consonances in question arose according to a quite definite

/] principle — that of the senario, or arithmetical series,

^ % 2, 3, 4> 5. 6.

Starting from the opposite direction, Zarlino arrived at a
similar result. He set himself to classify the great variety
of intervals which constituted the harmonic material of
polyphony, and to determine whether these intervals arose
arbitrarily, or from some definite principle. He first divided
the intervals into two classes, consonant and dissonant.
He showed that the dissonances were not in themselves
intelligible, but intelligible only by virtue of the consonances
which they served to retard, and into which they resolved.
Of the consonances two classes also were to be distinguished,
namely, simple and compound. But the latter had the
same harmonic significance as the former. There remained,
therefore, only the simple forms of the consonances as the
essential and constitutive elements of polyphony. All,
Zarlino stated, had their source in the " senario." The
consonances did not, then, arise arbitrarily, but from a
definite principle, indeed, the simplest and most definite
conceivable — the series of numbers 1:2:3:4:5:6.

Zarlino did not find his path free from difficulties. Although
he maintained, and as we now know quite rightly, that all
the consonances had their source in the " senario," he was
obliged to point out that all did not arise directly from this
source. The two Sixths, major and minor, arose indirectly.
Zarlino explained these intervals as " composite " intervals,
that is, compounded of simple inter\'als which arose directly
from the " senario." It was evident, however, that this
explanation did not entirely satisfy him. Another difficulty
was that in connection with the Fourth. The Fourth,
recognized as it had been from the most ancient times a^ a
consonant interval, was nevertheless perceived to produce
frequently a dissonant effect. It had in fact, as Zarlino
perceived, a dual character. There was something Hefe7
some principle in operation, which Zarlino did not fully

RESUME AND CONCLUSION 479

understand, and which he felt himself unable to fathom.
ZarUno's difficulties were to lead to fresh and important
theoretical results. The principle of the fundamental bass
, was already, and even T^efore the time of Zarlino, making
its influence felt. It had turned the Fourth, of which the
l^wer note impressed the ear as the fundamental, into a
'dissonance.

ZarUno had declared that, the bass was the foundation
•of the harmony, and in doing so he was doubtless merely
•expressing what had already been revealing itself to the
•consciousness of composers. It was a^ statement of much
theoretical significance. But he made the mistake of
imagining that the bass was the foundation of every harmonic
combination heard above it. The mistake was a natural
one, but although it represented a serious theoretical error,
the consequences from a practical point of view were of
no great moment at a time when the harmonies in actual
use were few and simple. But during the seventeenth and
beginning of the eighteenth centuries, as chords became ever
more numerous, figured bass practicians and writers on music
of the time became more and more embarrassed in their
attempts to systematize the new harmonic material, to
reduce it to a rational order. ]\Iany expedients were tried,
but without success. The only possible solution of the
problem was by means of the theor}- of Harmonic In\-ersion.
But the theory of harmonic inversion depended on a principle
Avhich was not reahzed by the figured bass practicians, who
were accustomed to regard the bass note of every chord as
the fundamental note. This principle was brought to light
by Rameau.

ZarUno ha.d_stated t2iat the terms of the senayio had their
origin in unity, but he had not dared even to imagine that
the sounds represented by these terms had their source in the
first or fundamental sound. It was Rameau who-inade this
•statement, and who, startUng though it seemed, demonstrated
its truth. By means of his principle of the fundamental
note or bass, to which is closely related that of harmonic
generation, Rameau was enabled to establish his principle
of harmonic inversion. The theory of the inversion of
chords has been universal!}- accepted. But we find e\'en
■eminent theorists and musicians who, A^'hile accepting and
utiUsing Rameau's theory of inverted chords, have neverthe-

48o THE THEORY OF HARMONY

less considered themselves at liberty to reject his principles
of the fundamental note or bass, and of harmonic generation
(of at least the major harmony) and who in general are
firmly convinced that the theory of harmony has nothing
to do with acoustical phenomena. This raises the question
as to whether Rameau's theory of hannonic inversion,
although it has been universally accepted, has at the same
time been universally understood and its significance reaHsed,
especially the manner in which it affects the whole question
as to whether the theory of harmony has a physical basis.
Such a theory cannot be established by merely pointing to
the fact that a chord and its inversions consist of practically
the same sounds. The difficulty is to determine which is the
" original " chord, and which the chords that are derived
from it. Rameau's theory of harmonic inversion cannot
be dissociated from his principles of a fundamental note,
or bass, and of harmonic generation. It arises from these
principles. If then harmony does not have a physical
basis Rameau's theory of harmonic inversion must be
abandoned. Where shall we turn for something to take its
place ? Certainly not to the works of any of Rameau's
successors, and least of all to the " metaphysical" theories
of Fetis and his disciples. It is a striking testimony to the
value and adequacy of Rameau's theory of harmonic inversion
that in scarcely a single work on harmony which has appeared
since his time has there been even an attempt made to
formulate an independent theory. Yet without an adequate
theory of harmonic inversion there can be no possible theory
of harmony.

But, it may be objected, if Rameau's theories of harmonic
inversion and harmonic generation are so closely connected,
how can the former be regarded as adequate when the latter
is so evidently, so almost absurdly inadequate ? This leads
to a point of considerable theoretical importance. It has
already been pointed out in the course of this work that of all
/ itjie chords used in music one, and one only, can be directly'
/derived from Rameau's principle of liarmonic generation,
namely, the major harmony. This result led Rameau to
turn away from his original principle of harmonic generation,
and adopt another quite opposed to the first, and in itself
quite indefensible, namely, that of added Thirds. It has led
not a few since Rameau's time to reject the harmonic series

RESUME AND CONCLUSION 481

as a totally inadequate basis for the theory of harmony. It
has led theorists like Day, Prout, and others, to exploit the
harmonic series for purposes of chord generation, and to select
from this series sounds which have never formed a constituent
part of any known hannonic system. The only thing which
theorists who have made the harmonic series the principle of
chord generation appear to have omitted to do has been to
abide by the results of their own theory. Having accepted
a fundamental and guiding principle of harmony, they have
nevertheless refused to be guided by it, and have virtually
abandoned it, or, while still professing to do it homage have
vainly attempted to e.xploit it for their own purposes. The
principle of harmony of Zarlino, Descartes, Rameau, Tartini,
furnishes us with but a single chord. But this ought not to
be regarded as a negative result, but as a positive result of the
greatest theoretical signiiicance. It is the one fact of supreme
importance which this principle has to teach us. This has not
yet been realised. Theorists have long enough rejected it
because it did not conform, but was opposed to, their pre-
conceived notions as to what was fit and proper wth regard
to harmony and its theory. There exists in our harmonic
music but a single chord, from which all others are developed.^
But as the sounds of this harmony are contained in the
resonance of musical sound itself, all harmony has its source
in a single musical sound. The development of harmony has
been a more simple and beautiful process than musicians and
theorists have imagined.

In laying the foundations of the science of harmony,
Rameau builded better than he knew. He did not for
example perceive that the minor Sixth, like the Fourth,
quite apart from its position within the key-system, had a dual
aspect, or if he did he was unable to advance any explanation
of this fact, although the explanation lay within his grasp.
Of the dual nature of the Fourth he was quite aware, but
explained it wrongly. The Fourth when consonant, he stated,
represents the inversion of the Fifth ; but vvhen dissonant, it

1 Our examination of various theories of harmony in the course
of this work will have helped the reader to understand, to some extent
at lecist, how this can be the case. The writer may be permitted to
state that he hopes to complete shortly a new and smaller constructive
work on the theory of harmony, the materials for which he has
already prepared.

482 THE THEORY OF HARMONY

1^ represents not a Fourth but an Eleventh, and must be
regarded as the highest sound of what is sometimes called
the " chord of the Eleventh." This explanation has
been advanced by not a few theorists since the time of
Rameau.

Zarlino had discovered that the complex of consonances
comprised in the senario, when sounded simultaneously,
resulted in a " most perfect harmony." This Harmonia
Perfetta, which represented the consummation of Zarliiio.'.S
laBoursHn the classification of the harmonic material of his
tfme,' as well as the labours, from a harmonic point of view,
of the entire polyphonic period which reached its climax in
the sixteenth century, formed the starting-point of a., new
musical epoch. After the publication of his Traite I^ameau
discovered, to his astonishment, that the fundamental prin--
ciple of harmony which he had received from Zarhno actually
existed, so to speak, in the flesh. Rameau's astonishment
that what he and others had recognised to be a rational
necessity should actually exist as a fact in nature, arose in
great measure from his having perceived such a fact for the
first time. In reality, however, it was the natural mani-
festation of a principle which had existed from the beginning
of things. When primitive peoples affirmed that their music
had originally descended from the gods, there were doubtless
philosophers of the time in whom such a belief excited not
reverence, but ridicule. The discovery of Pythagoras that
the harmony of his time had its .source in the series 1:2:3:4
may have given such philosophers food for thought. But
now, as Rameau points out, this principle of harmony reveals
itself in nature as an actually existing fact. In his Nouveau
y ySystenie he remarks that " those who refuse to beheve their^
I ears may at least accept the evidence of their eyes " ; and
also points out that one may in addition convince himself
through the sense of touch, by placing his finger on the nodes
of the vibrating string.

As is known, the Church Modes were divided into two
main classes. Authentic and Plagal. The Authentic mode (a)
was considered to consist of a Fifth and a Fourth : the
Plagal (b) :

^ (^) (ft)

RESUiME AND CONCLUSION 483

of a Fourth and a infth. In making these distinctions,
musicians were undoubtedly guided by their sense of what was
artistically appropriate and beautiful, and not by theoretical
or mathematical considerations. Nevertheless, Glarean ^
pointed out that the division observed in the Authentic mode
was neither more nor less than the harmonic division of the
t)ctave (2:3:4), while that in the Plagal mode arose from
the inversion of this division.

In our Major and Minor Modes, on the other hand, the
determining sounds are the first, third, and fifth of the mode.
Zarlino, going a step further than Glarean, demonstrated
that the determining sounds of the major mode (which con-
stitute a major harmonv) arise from the harmonic division
of the Fifth (4:5: b) (c) :—

{c} (d)

-s>-

-Sf-

those of the minor mode (which constitute a minor harmony) ,
from the inversion of this division (d). Our major and minor
modes have taken the place of the authentic and plagal .
modes of a former epoch. The Octave, the consonance^
arising from its harmonic division and that of the Fifth,!
form the sole constitutive elements of harmonic music. '

In all this a consistent and beautiful development may
easily be traced. From the earliest beginnings of the art of
music, and underlying the whole course of its development,
a single principle may be observed, steadfast and invariable.

It has been argued that the consonant intervals were
selected by man from a large variety of sound-combinations
as those most suitable for his artistic needs ; that they were
arrived at only after long periods of testing and experiment ;
that, in short, the appreciation and recognition of these
intervals as consonant was the result of a long process of
education of the ear. There are certainly grounds both
physiological and psychological for such a view. The history
of the major and minor Thirds would appear to confirm it.
These were arrived at only after long experiment. But it

^ See p. 48.

484 . THE THEORY OF HARMONY

would be a decided mistake to imagine that in his searchings,
wanderings, even blunders, man was left wholly to himself.
He had a guide. In every musical sound that he produced,
the principle of harmony was revealing itself to him. When
at last he discovered the consonances most suitable for his
artistic needs, it was found that they were none other than
those which this natural principle had all along suggested to
him.

These are important considerations for the theory of
harmony, and not for the theory of harmony alone. What,
then, may we infer from them ? We may at least infer that
this natural manifestation, this principle of harmony, has
been and is, to make use of the felicitous expression of Rameau,
" IHz invisible guide of the musician." Long enough have
theorists professed to do it homage, while actually engaged in
vain attempts to exploit it. In so far as the theory of harmony
is concerned, the way may be difficult. Still, it is the way.

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Full text of "The theory of harmony; an inquiry into the natural principles of harmony, with an examination of the chief systems of harmony from Rameau to the present day"

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