HARMONY AND METRE
PRINTED BY
SPOTTISWOODE AND CO., NEW-STREET SQUARE
LONDON
THE NATURE
OF
HARMONY AND METRE
BY
MORITZ HAUPTMAJNN
TRANSLATED AND EDITED
• vi* -& BY
•&*&*
W: E'HEATHCOTE, M.A.
LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE
LONDON
SWAN SONNENSCHEIN & CO.
PATERNOSTER SQUARE
1888
AUTHOR'S PREFACE.
THE knowledge to be acquired by him who is desirous of becoming
a practical musician has been amply set forth in many treatises.
Few have attempted to examine how the laws of music depend
upon principles of the human mind, and how that form of musical
expression which is true and correct must also be that which is
natural to mankind, which is conformed to human reason, and
which is consequently open to universal comprehension. Researches
of this character will, as a rule, find less general acceptance than
if they were directed towards strengthening the powers of execution,
or towards refining the taste and the judgment. The beginner in
music is absorbed in studies of a practical tendency ; the mature
musician is devoted to the exercise of his profession. They rarely
find time, and have little inclination, to reflect upon principles for
the truth of which it seems to them that instinctive feeling is a
sufficient assurance. Occasionally, however, a desire is manifested
for information as to the ground of certain inevitable axioms ; and
we are asked to assign a reason for rules whose validity is un-
questioned, but which for the most part remain without demons-
tration. To such desire the present treatise is designed as the
vi AUTHORS PREFACE
appropriate response. May it meet with the approval of all those
who concur in, or will follow out, its arguments ! It does not con-
tain a practical method of instruction in harmony and metre, but
is an enquiry into the nature of musical and metrical art.
M. HAUPTMANN.
LEIPZIG.
TRANSLATOR'S PREFACE.
THE following translation was undertaken with the view of
rendering more accessible a work that possesses great interest
both for music and metaphysics. For music, because, independ-
ently of the theory involved, it contains a clear and compact
account, written by a skilful musician and experienced teacher,
of the doctrine of harmony and metre ; in which the received
rules are reduced under general principles, while upon particular
points new and unexpected light is frequently cast. For meta-
physics, because it is an application of Hegel's method of philosophy
to a concrete subject, that has, like logic, the peculiar advantage of
standing by itself and of being comprised in a comparatively
narrow compass.
It has been endeavoured to represent the original in as plain
and literal language as possible, carefully avoiding all arbitrary
interpretations. Also it did not seem that anything in the shape
of a commentary or notes accompanying the text would be likely
to be of use. For Hauptmann does not assume in his readers a
knowledge of metaphysics, nor anything beyond technical ac-
quaintance with music. It is conceived, however, that to readers
unversed in metaphysics the method of reasoning may present, at
viii TRANSLATOR'S PREFACE
the outset, some difficulty. The Translator has therefore ventured to
prefix a short introductory essay, which may perhaps serve to give
a general idea of the scope of the work, and of the nature and in-
tention of the arguments employed in it, and at the same time
elucidate certain special instances and expressions. The proof-
sheets have been submitted to the criticism of Mr. H. Keatley
Moore, B.A., B. Mus., who, besides revising in many respects the
musical phraseology throughout, has also kindly made numerous
suggestions and corrections, which have greatly added to the value
of the translation.
114 EBURY STREET, S.W.
June 1888.
CONTENTS.
A SHORT SKETCH OF THE LIFE OF MORITZ HAUPTMANN
INTRODUCTORY ESSAY (by the Translator]
INTRODUCTION
PAGE.
xiii
xv
. xxxv
I. HARMONY.
SOUND . . . . . . . . 3
MAJOR TRIAD ........ 5
MAJOR KEY . . . . . . . 8
MINOR TRIAD . . . . . 14
MINOR KEY . .... . 17
MINOR-MAJOR KEY . . . . . . . .21
DIMINISHED TRIADS . . . . . . ' . . 23
THE KEY-SYSTEM STRETCHING OUT, OR IN TRANSIT, TO DOMINANT
OR SUBDOMINANT . . . . . . . 28
DIMINISHED TRIADS OF THE KEY-SYSTEM IN TRANSIT . . .30
(a) In the major key ... ... 30
(0) „ „ minor key . 31
(c] „ „ minor-major key . . . . . . 32
SCALE OF THE MAJOR KEY . . . . . . -33
SCALE OF THE MINOR KEY . . . . . . . 39
SCALE OF THE MINOR-MAJOR. KEY . ... . .42
Harmonic determination for the melodic succession in the major
scale . . . . 43
Harmonic determination for the melodic succession in the minor scale
and in the minor-major scale . . . . . . 43
CHORD-SUCCESSION . . . . . . . .45
DISSONANCE (SUSPENSION) . . . . . . . 54
CONTENTS
PAGE
SEVENTH CHORD . -55
RESOLUTION OF DISSONANCE . 65
(1) In suspensions . 65
(2) „ chords of the Seventh . 67
PROGRESSION OF PARTS IN SEVENTH HARMONY . .. . 76
SUCCESSION OF SEVENTH CHORDS . . . . 8 1
SEVENTH CHORDS OF THE KEY-SYSTEM PASSING INTO ITSELF . . 88
I. Dominant Seventh chord. . . . . 88
II. Seventh chord upon the Third of the dominant . . . 101
(a] In the major key . ..... 101
(V) „ „ minor key . 104
III. Seventh chord upon the Fifth of the dominant . . . 107
DEGREES OF DISSONANCE . . . . . . . 108
CHROMATIC RESOLUTION OF DISSONANCE . . . . 113
ESSENTIAL DIFFERENCE OF SEVENTH-HARMONY OF THE UNTRANSPOSED
AND OF THE TRANSPOSED SYSTEM WITH RESPECT TO CHORD-
POSITION . . . . . . . . .116
SEVENTH CHORDS WHICH ARISE FROM THE UNION OF THE LIMITS OF
THE EXTENDED KEY-SYSTEM, AND SEVENTH CHORDS CONTAINING
AN AUGMENTED TRIAD . . . . . . I2O
THE AUGMENTED TRIAD AND ITS OCCURRENCE IN THE SEVENTH
CHORD . . . . . . . . 127
CONCERNING THE SO-CALLED CHORDS OF THE NINTH, ELEVENTH, AND
THIRTEENTH. PEDAL . . . . . . 131
SUSPENSION OF THE NINTH ...... 133
PASSING-NOTES . . . . . . . . 135
(a) Diatonic . . . . . . . . 135
(b) Chromatic . . . . . . . . 136
MODULATION ......... 144
ENHARMONIC CHANGE . . . . . . . . 166
CLOSE .......... 175
II. METRE.
METRE AND RHYTHM . . . . . . .189
METRE . . . . . . . . . . 189
I. Two-timed (Octave) . . . . . .189
II. Three-timed (Fifth) . . . . . . . 191
III. Four-timed (Third) . . . . . . .192
CONTENTS xi
PAGE
THE DIFFERENCE BETWEEN TWICE-TWO-TIMED AND FOUR-TIMED
METRE ......... 195
FIVE-TIMED AND SEVEN-TIMED FORMATION AS ARTIFICIAL AND IN-
ORGANIC ......... 196
COMBINED METRE . . . . . . . 2OO
ACCENT .... . . 204
Accent of the member
(a] In the two-timed metre ..... 205
(b] „ „ three-timed metre . . . . . 205
(c] „ „ four-timed metre ..... 205
COMBINED ACCENTS . . . . . . . 206
(a) Twofold, in the three-timed metre ..... 206
(£) Threefold, in the four-timed metre . . . . 207
THE NOTION OF MAJOR AND MINOR IN METRICAL DETERMINATION . 211
ACCENTS PRODUCED FROM THIS DOUBLE DETERMINATION . . .212
(a) In the two-timed metre . . . . . .212
(b] „ „ twice-two-timed metre . . . . . 213
(c] „ „ three-timed metre, referred to the twice-two-timed . .215
(d) „ „ four-timed metre . . . . . 231
RESUME OF ALL ACCENT-DETERMINATIONS IN THE TWO-TIMED, TWICE-
TWO-TIMED, THREE-TIMED, AND FOUR-TIMED METRES . . 237
I. Accents of the two-timed metre ..... 238
II. „ „ twice-two-timed metre . . . . 238
III. „ „ three-timed metre . . . . .238
IV. „ „ four-timed metre . . . . 239
ACCENTS IN COMBINED METRE . . . . . .248
RHYTHM IN METRE . . . . . . . . 253
THE RHYTHMICAL CLOSE . . . . . . .254
FILLING-UP OF THE METRICAL FORM. REST . . . 257
(a) In the two-timed metre ...... 258
(b) „ „ three-timed metre . . . 258
(c) „ „ four-timed metre . . . . . -259
FURTHER COMPARISON OF THE HARMONIC AND METRICAL ELEMENTS . 262
METRICAL CONSTRUCTION INWARDS AND OUTWARDS . . . 267
UNEQUAL-TIMED DIVISION OF THE METRICAL MEMBER . . .271
The metrical determinations compared with the spacial . . . 271
POSITIVE AND NEGATIVE FORM OF THE UNEQUAL-TIMED DIVISION . 278
xii CONTENTS
THE THREE ELEMENTS OF THE UNEQUAL-TIMED DIVISION, CORRE-
SPONDING TO THE THREE METRICAL ELEMENTS OF THE TWO-,
THREE-, AND FOUR-TIMED UNITIES ; AND LIKEWISE TO THE HAR-
MONIC ELEMENTS OF OCTAVE, FIFTH, AND THIRD . . . 280
THE DOTTED MOVEMENT . . . . . . . 285
Analogy in harmonic melodic determination .... 287
Analogy in the determination of space . . . . 288
METRE OF SPEECH. — FOOT. VERSE-MEASURE. DIPODY — TRIPODY —
TETRAPODY. DIMETER — TRIMETER — TETRAMETER . «, . 2QO
RHYTHMICAL MEMBERMENT OF THE FOOT. SPONDEE ; TROCHEE —
IAMBUS ; DACTYL — ANAPAEST . . . . . . 293
DIFFERENCE BETWEEN THE METRICAL DACTYL AND THE RHYTHMICAL,
OR BETWEEN THE SPONDAIC DACTYL AND THE TROCHAIC . 300
THE MARKING OF METRE . . . . . . . 307
CATALECTIC AND ACATALECTIC METRE . . . . -313
Examples both in spoken metre and in musical . . . 313
QUANTITY AND ACCENT . . . . . . .316
Difference between ancient and modern verse . . . . 316
III. METRICAL HARMONY. HARMONIC METRE.
HARMONIC METRICAL DETERMINATION . . . . 325
METRICAL POSITION OF DISSONANCE . . . -331
(a) In the chord of suspension . . . . . 331
($) „ „ Seventh chord ...... 332
(a) In the untransposed key-system . . . . . 332
(/3) „ „ transposed key-system ..... 333
SUMMARY OF THE FOREGOING CHAPTER ON THE METRICAL POSITION
OF DISSONANCE . . . . . . .338
DISSONANCE IN THREE- AND FOUR-TIMED METRE . . . . 339
SYNCOPATION. . . . . . . . .341
HARMONICAL CORRESPONDENCE OF THE SUCCESSION OF LINKED
SEVENTH CHORDS WITH THE METRICALLY SYNCOPATED SERIES . 345
APPENDIX : A SHORT ANALYSIS OF HAUPTMANN'S TREATISE .... 349
A SHORT SKETCH
OF THE
LIFE OF MORITZ HAUPTMANN.
BY PROFESSOR DR. ALFRED SCHONE.
Translated from the Preface to ' Briefe von Moritz Hauptmann an Franz
Hauserj edited by Prof. Dr. Alfred Schone. Leipzig: Breitkopf and
Hartel. 1871.
MORITZ HAUPTMANN was born in Dresden on the i3th of October, 1792.
His father, who was a provincial architect, recognised the exceptional
talents of his son, and developed them by careful and judicious education.
Inclination for music was early manifested by the boy, but up to his iQth
year he was receiving complete technical training as a future architect,
besides making zealous studies in mathematics, natural science, and lan-
guages. Without doubt this familiarity with architecture brought incal-
culable benefit to Hauptmann in his later labours in musical theory, to say
nothing of the delicate understanding for the fine arts which he owed
chiefly to these studies of his youth. At the age of 19 he turned wholly
to music, and in 1811 went to Gotha, where he had instruction from Spohr
in the violin and composition. In the very next year he entered the
Dresden Court band as violinist, and in 1813 held during several months
a like position in the Vienna Theatre orchestra, at that time conversing
much with Carl Maria v. Weber, with Meyerbeer, and with Spohr, who
occupied the post of conductor. Returned to Dresden, he accepted in 1815
an appointment as music master in the house of the Russian princess
Repnin, and in that capacity passed five years in Moscow, Pultowa,
Odessa, and St. Petersburg. After Hauptmann had returned in 1820 to
Dresden, his teacher and friend Spohr, who had meanwhile become con-
ductor in Cassel, engaged him as violin-player in the Electoral band, and
xiv SHORT SKETCH of the LIFE of MORITZ HA UPTMANN
for full twenty years the great man remained in this modest situation. Yet
his name soon became known in wider circles. His two great masses,
several sonatas for violin and piano, violin duets, some sacred choral
pieces (among them his famous * Salve Regina '), an opera ' Matilda,' secular
song-music for one or more voices (as, e.g., the sonnets of Petrarch, 'Amor
timido,' 'Anacreontiche,' and others), and some lesser pianoforte pieces met,
if not with universal and sudden acclaim, yet with marked recognition from
the best and ablest musicians, and gradually gathered round his name a
small, steadily increasing, company of admirers and friends of his music.
Nor did he meet with less acknowledgment as a teacher of musical theory.
The number of his pupils was over 300 ; and, while thus perseveringly
busied in teaching, he was developing that insight of genius into the essence
of musical theory, which he has set forth in his book (' Natur der Har-
monikundder Metrik,' Leipzig, Breitkopf and Hartel, 1853) and in smaller
essays connected with it. Thus gradually he won the reputation of the
most notable theorist and teacher of his time, and when in 1842 the post
of Cantor at the Thomasschule, hallowed for ever by J. Seb. Bach, at
Leipzig fell vacant, through Mendelssohn's influence Hauptmann was
called to this honourable position, and was at the same time appointed
teacher in the Conservatorium, then about to be founded.- His mind
quickly made up, Hauptmann left his quiet sojourn at Cassel, which he
had broken only in 1829 by a journey to Italy and in 1842 by a visit to
Paris. He was accompanied to his new home by his wife, Susette, whom
he had married in 1841 ; she was a daughter of Hummel, Director of the
Academy in Cassel. On the i2th of September, 1842, Hauptmann entered
upon his office in Leipzig. Happy in his wife, whose great talents for music
and fine art were the ornament of his house, and in his three children, in
friendship with a circle of like-minded worthy families, conversing person-
ally and by letter with many of the most eminent men in art and science,
loved and honoured by the daily increasing band of his scholars, full
twenty-five prosperous years of unenfeebled activity were allotted him.
It was not until the end of the year 1866, shortly after a beautiful celebra-
tion of his silver wedding, that a bodily weakness set in, which rapidly
gained ground and made the last days of his life burdensome. On the 3rd
of January, 1868, he closed his tired eyes for ever ; but in the recollections
of his friends his memory will endure as one of the best and most note-
worthy men that Germany has produced.
INTRODUCTORY ESSAY.
BY THE TRANSLATOR.
ALTHOUGH music itself is ancient, yet its modern European form
is recent and of plain origin. The source of modern music is two-
fold : Church and secular ; of which Church music represents the
more ancient phase, though both doubtless were in their beginning
the same. Modern harmony, counterpoint, modulation, all the
formal part of modern music, takes its rise from the Church
chorale. And while Church music served as the mould, secular
music furnished the material ; so that rhythm, and those forms of
music which are characterised by rhythm (e.g. marches and dances),
are rather due to secular music.
Similarly, among forms of composition the fugue belongs
principally to the Church style, and the sonata to the secular ; the
sonata being distinguished by multiplicity of contents, though its
general course of modulation is not different to that of other forms
of music.
Now the science of music is concerned more with the form than
with the materials. Therefore the influence of secular music is
regarded rather as falling in with the tendencies of modern music
than as guiding them.
With the progress of music a system of rules for composing
INTRODUCTORY ESSAY
-was gradually formed. At first harmony was simple, and simple
rules sufficed. Afterwards new effects were discovered, which
made new rules necessary and also modifications of the former
-ones. To take an early example from Hauptmann : at the time
of Palestrina Seventh chords were not used, or very rarely, in vocal
counterpoint ; later it was discovered that under certain restric-
tions they might be freely used with advantage. But at the same
time the employment of this greater dissonance was the reason for
•certain other rules regarding the more delicate leading of voices in
harmony being discarded ; the introduction of the greater contrasts
made the lesser unimportant. The final result was a body of
rules, consistent indeed and not arbitrary, inasmuch as they formed
in the aggregate a system, but still only gathered from experience
and not illuminated or governed by any philosophic or scientific
explanation. Various attempts have been made to supply such an
explanation ; but the doctrine of harmony, though empirical, is
simple, and the explanations mostly added intricacy without
helping practice. Many partial or particular explanations have
been given, that are interesting historically, and have also left their
impression upon the musical system ; but of none of the older
theorists can it be said that he formed a system to supersede all
others, or gave more than a partial glimpse of the central truth
pervading harmony.
Hauptmann's * Nature of Harmony and Metre' is a philo-
sophical explanation of the received laws or principles of the art
of music, aiming at equal simplicity with the laws themselves. It
explains them by showing that the various rules and principles
are derived from one law that pervades the whole ; also that the
INTROD UCTOR Y ESS A Y xvii
gradual development of music historically is the gradual embodi-
ment of that law, and so may be said to be due to it ; also that
every single phenomenon in music is a perfect instance or embodi-
ment of the law.
The treatise is written in the Hegelian philosophy. It may be
compared to the ' Logic ' of Hegel, which it resembles in method
and in plan. In both works a large body of received rules and
principles, recognised as forming a system, but inadequately ex-
plained hitherto, are established upon a philosophical basis and
shown to depend upon a law. Both start from a simple beginning,
from which the more complicated conceptions are shown branching
out. This development (by which is not meant only development
in time or by succession) is seen to follow a uniform law, by the
operation of which the process of development is separated into
stages ; and each stage is marked by the accomplishment or comple-
tion of some particular notion, which then appears as including
within it all previous notions completed in previous stages.
Recognition of the law referred to, which will presently be
enunciated, is characteristic of the Hegelian philosophy. It was
not, however, an invention or discovery of Hegel's. In one form
or another it was known to and stated by many previous philo-
sophers and metaphysicians. And in the form now to be noticed
it has been stated as plainly by Goethe as by Hegel. But it is called
Hegelian as having been made formally the basis of a system by
Hegel, and by the school of which he is the chief representative.
The law is enunciated thus by Hauptmann : ' Unity, with the
opposite of itself, and the removal of the opposite : immediate
unity, which through an element of being at two with itself passes
a
X
xviii INTRO D UCTOR Y ESS A Y
into mediated unity.' Here three stages or epochs in the process
of thought are marked out : a stage of simplicity or unity ; a stage
of division or separation ; and a stage of reconcilement or restored
unity.
In speaking of the process of thought no contrast is intended
between thoughts and things. All things are regarded as thought,
and this triple organisation is attributed to concrete things no less
than to abstract thoughts and notions. Thus for a first idea of the
kind of triplicity that is meant we might instance the seed, the
growing crop, and the harvest ; the beginning, middle, and end ;
the power, the fulcrum, and the weight ; the premise, the argu-
ment, and the conclusion. Now to learn or follow out any of these
is a process ; but when the process* is completed, then the several
elements appear as factors in a higher notion, in which, moreover,
the process is surveyed as a whole. It is not known that a grain
of corn is a seed until it is known that there will be a harvest from
it. Also a conclusion does not satisfy until it is seen to follow
from the premise, and premise, argument, and conclusion to form
one coherent, necessary whole.
To consider more closely the three stages of thought, or, as it
might be said, of our knowledge of things.
I. Anything is said to be ( unity ' when it is perfect and simple
(' in se ipso totum teres atque rotundum J), having no parts, or at
least only such as are discerned to be necessary, not distinguished
as parts, as when, e.g., a picture is viewed without reflecting on the
parts of it or the reasons for their being such and such. Of this
kind are the notions of ' up ' and * down ' before the question arises
of how much up and down ; or when the branch of a tree is said
INTRO D UCTOR Y ESS A Y xix
to be at right angles to its trunk, which further reflection shows to
be quite indefinite. So to say the sun rises in the east might seem
either a definite statement or an indefinite, according as the quarter
of the heavens or the point of the compass is thought of. But the
later notions are not in reality more precise than the earlier, though
they are apt to appear so ; just as a sum is not made more precise
for being stated in pence rather than in pounds. These are notions
gained by reflection, which have not as yet undergone further re-
flection. In language they are also represented by general terms,
as when a ' chord ' is spoken of, meaning a chord in general, which
was once a concrete thing, though now by reflection become
abstract.
Therefore when musical sound, as the Octave, is said to be
unity, we are not to think of it existing then as it does now, and
capable of being distinguished into chords, notes, or scales. We
are going back to a time anterior to all that — to a time when there
was neither chord nor scale, when sound was indeed perceived as
musical, but without further difference. For in the progress of
music the law (viz. of unity, difference, and union) is the form that
generates the notes ; it is the cause of their existence and anterior
to them. Musical pitch, tone, or quality of sound, which depend
upon the triad, must not be thought of as existing before the triad
exists, and still less as contributing to the formation of the triad.
Therefore, in the beginning, musical sound must be thought of as
existing indeed, but without difference ; that is, as if every musical
sound were understood as being the same.
Even the Octave in the triad of Octave, Fifth, and Third is not
strictly the prime unity of all ; for the Fifth and Third are also the
xx INTRODUCTOR Y ESSA Y
prime unity (being modes of it) just as much as the Octave is. But
within the triad the Octave represents the prime unity ; for it is
the prime unity appearing as such within the triad.
The distinction here pointed out arises merely from consider-
ing the higher unity as produced from the prime unity in a
successive process : first Octave, then Fifth, then Third, which is
the union of the two former. This implies a distinction between
the prime unity, that exists first, and the Octave, that exists con-
comitantly with the Fifth and Third. In reality there is no suc-
cessive process ; but the intellect, in considering things as finite,
necessarily assumes one. It follows that when an assumed prime
unity — e.g. the triad — has through an intermediate process given
rise to a higher unity— e.g. the key, a triad with its subdominant and
dominant — then the chords of the subdominant and dominant are
triads no less than the tonic triad, and the notion of triad becomes
an abstract one. It has become abstract to the intellect because
parted from it by an act of reason. For the process of unity,
difference, union represents to the intellect the act of reason.
II. In the second factor, of division, difference, or separation —
the Fifth in music — the immediate unity of the first factor is broken
up ; immediate unity is as it were perceived to be no limit for
thought. It is not lost, but becomes doubtful and unreal ; it is
and is not at the same time. It becomes two, or at two, in itself ;
by which is not meant that it becomes two numerically, two
separate, distinct unities, but that division or twoness appears in
each element and every part of it. Thus twoness appears in
musical sound when it is perceived to have ' sides ' or double
meaning, to be different in different directions. In the completed
INTRODUCTORY ESSAY
musical system this is represented principally by a note and its
Fifth ; but more generally by any two notes that are different.
For the interval of the Fifth is the type of difference ; and thus in
music the Fifth, the type of difference, is present wherever sounds
are considered as different. So in geometry a direction in a plane
is defined principally with reference to two directions that are at
right angles to one another, but more generally to any two direc-
tions.
III. The third factor is that of union or reconciliation. In the
second factor the unity of the first was not lost ; it only became
troubled, that is, contradictory and opposed to itself. And so in
the third factor the opposition is not lost, but reconciled. Fusion
takes place. The contradiction is removed by being made reason-
able. The Fifth is unreal, because in it there is opposite, unre-
conciled meaning. The Third brings back reality, for now the
opposite becomes complementary meaning in a higher notion.
In the completed musical system this is represented principally
by a note and its Third. For the Fifth is the direct opposite of the
note ; but the Third, holding the middle place between the note and
its opposite, is their bond of union. The incompatible natures of
the note and its Fifth are reconciled by a third nature partaking of
both. Hauptmann takes the example of an end and a beginning.
End and beginning are incompatible natures, but are reconciled in
the nature of middle, which partakes of the nature of end and also
of the nature of beginning, or more truly is both end and beginning.
There are, then, beginning, end, and middle, factors of a higher
nature.
But more generally the nature of the Third appears in any note
xxii INTRO D UCTOR Y ESS A Y
that is considered as having relation to another. For one note is
the same as another in so far as it is its Octave, and different to it
in so far as it is its Fifth ; but union of sameness and difference
constitutes the nature of the Third. So a direction in a plane is
defined by considering how far it coincides with a given direction
and how far it is at right angles to it. In music the Octave is
present wherever sounds are considered as the same ; the Fifth is
present wherever they are considered as different, and the Third
wherever they unite sameness with difference. In other words, the
Octave expresses sameness or identity ; the Fifth, difference ; and
the Third, unity of sameness and difference.
A chief hindrance to understanding the Hegelian process comes
from considering the three elements, through which the higher
notion is manifested, as three distinct separate things, whereas they
are more truly sides of the same thing : the higher nption is viewed
on three sides before it can be seen through and viewed as a whole.
For it might be supposed that there are a great number of par-
ticular things or substances, and that these ' combine chemically '
(a comparison sometimes used of the Hegelian process) to form
new notions ; as if out of the great crowd of things a negative
could always be found for any given positive, and then the two
combined. But the true view is that the negative is produced out
of the positive ; or rather, positive and negative appear together in
the prime unity. So that the better comparison is found, e.g., in
the nature of dimensions, where a point or line considered at first
per se is afterwards found to involve or generate the notion of
space, and can then only be recovered out of space by a mental
effort, so as to appear abstracted from it. Here space is not com-
INTRO D UCTOR Y ESS A Y xxiii
bined out of a number of points in space ; but the notion of space
is generated from the notion of point, which in space appears as
abstract position. True that on the piano a note, its Fifth, and its
Third are three distinct sounds, but that is because the way in which
they have come about is left out of sight. Being themselves the
fruit of previous thought, the embodiment of a whole history of
music, the notes of a keyboard now stand ready-made in a system
taken for granted like the alphabet, or the multiplication table in
working arithmetic. They are thus for use in music, but music
must not be founded on them. And so the question is not
merely of combining them ; for they are made so as to combine ;
but of combining them afresh, so that they transcend their nature.
Thus every advance in music brings about advance in the con-
struction of instruments, which are, so to speak, the register of
results already attained in music. For every instrument is the
evidence of its own use ; as the story seems to imply, of the wise
man who having been shown a chess-board and men discovered
their use by meditation.
It remains to speak of the triad, the completed notion, in which
the three elements appear united into one. In Hauptmann's
words, the unity which at first was simple and immediate has
become a mediated unity. To explain this let us consider the
notion of musical sound. This may be defined as sound capable
of being used as a means of expression, It is the first glimpse of
music appearing in sound. Now at first musical sound is a simple
notion ; all musical sounds are the same. Afterwards there may
be musical sounds that are musical and musical sounds that are
not musical, e.g. concords and discords. So that the triad, the law
xxiv INTROD UCTOR Y ESS A Y
of musical sounds, is to music what musical sound is to sound
generally ; for in it musical sound is doubly musical : as if we should
represent musical sound by x> and the triad by x'x, the x of x.
Thus musical sound is reflected in itself; that is, musical sound
becomes intelligible, is understood, by means of musical sound, or,
as Hauptmann says, is mediated by itself. And this process of
doubling upon itself, or specialisation, takes place continually in
all parts of music, making it into an organic whole.
We have, then : first, musical sound appearing in sound ; next,
musical sound separating into concord and discord ; lastly, the
reconciliation between concord and discord, in view of which
discord is more properly termed dissonance. Now the opposition
between consonance and dissonance is analogous to that between
musical sound and sound generally — noise. And what answers then
to the resolution of dissonance is the whole development of music,
or, as it is sometimes said, of the musical idea. There is no special
charm in music not common to all sound ; but the discovery of the
charm is reached through music. Music is, therefore, in Spohr's
phrase, the consecration of sound ; it constitutes the demonstration
that all sound is musical. The advance of music is marked by the
gradual transformation of discord into dissonance. The meaning
of discord is found, and it becomes the complement, the other side,
of consonance. And in this notion is explained the term so often
used by Hauptmann of the meaning of a sound or combination of
sounds. For every sound employed in a particular way has to
justify its existence ; it has to contain in itself the reason for its
being such as it is. Its existence involves a doubt or riddle and
its solution. De Quincey speaks somewhat similarly when he de-
INTROD UCTOR Y ESS A Y xxv
.scribes ' the questions ' in music ' asked and answered in a deep
musical sense.'
Thus far has been considered the triad in music as the embodi-
ment of a certain law. This, in its abstract statement, is claimed
for the expression of the general process of thought : it is the
shape taken by all thought, just as syllogism is the shape taken by
all argument. Now, as in argument it is not usual to put forward
the form of the syllogism, so neither does the form of thought lie
nakedly upon the surface. Thought laid open to the intellect takes
of itself this form ; but the laying open is a kind of dissection, and
rests upon a fiction. And so we see in Hegel that the abstract
form of the law is not much appealed to ; the natural process of
thought is followed, and takes of itself the proper form. In
Hauptmann the form of the law is made more prominent, and we
are constantly shown the same process repeated in new material.
The reason for this is partly that the symmetrical construction of
the musical system had to be accounted for. It was known that
every chord, every scale was based upon a fundamental note, and
that modulations followed the same intervals that the notes of
melody do ; besides many other symmetrical relations, which,
natural as they may now seem, were yet discovered only gradually
and with difficulty. The outline of the system being, therefore,
known, it remained for Hauptmann to show the connexion of the
parts.
The object of knowledge, as Hauptmann tells us, is recognition
of the particular in the universal and of the universal in the parti-
cular. Here a distinction may be made between universal and
abstract. The latter is ordinarily used as a term in the common
INTRODUCTORY ESSAY
logic. In logical abstraction things are compared, and the quali-
ties in which they differ are rejected. Thus the higher conceptions
are always emptier and emptier, and the highest abstraction is
nearest nothing. The term universal in Hegelian metaphysics has
the opposite meaning to this. There the higher notion is such as
to unite and include the opposite qualities of things ; it is fuller in
contents than the lower or particular, and the particular is regarded
as a limitation of the universal. In the common logic the parti-
cular = the general + a difference. In Hegelian logic the parti-
cular = the general under a limitation. There is a very real dis-
tinction whether the abstract is thought of as emptier than the
concrete, the universal than the particular, or whether as fuller of
contents. The views are, in fact, opposite, and their reconcilement
is found in that way of thinking so often brought forward by
Hauptmann in the course of his work, when he says that the
universal must be thought of in the particular, and the particular
in the universal ; which leads ultimately to abolition of the anti-
thesis between universal and particular.
The following is meant as an illustration : The modulation of
a piece of music may be represented in a series of chords. These
are considered as the groundwork of the composition ; as if a
sketch or outline of the whole were laid down, and afterwards filled
in, just as variations may be considered as filling out or giving
greater detail or finish to a more simple air. The chords are in a
way the abstract or general form of the piece : they certainly can-
not be said to be the general notion or idea of the piece ; still they
may be taken as symbolising it. Now in one view the chords are
simpler, less important than the piece, because the details are taken
INTROD UCTOR Y ESS A Y xxvii
away ; to the finished picture they are the sketch or outline, and
may be filled up too in various ways. Herein the distinction is
apparent. If we conceive the general form as emptier, then it
may be filled up in different ways indifferently ; but then that is
because it is conceived inadequately. To conceive it adequately is
to conceive it as capable of generating not merely this particular,
but also an infinite number of other particulars. Now writing
down the chords that underlie a piece of music is a more or less
mechanical process. But to attain to the conception of the true
form underlying a piece of music is to see it identical with an
infinity of other pieces, and to know the general form as something
of infinitely greater dignity and fulness than the particular piece.
But, again, to know adequately the particular piece is also to know
the general form in it. Then the general form appears in an
individual shape, in which the other individual shapes are latent
but effective ; as when a solo occurs in an orchestra and the other
instruments produce as much effect by being missed as when actu-
ally sounding. It certainly seems paradoxical to say that a simple
succession of chords, such as a hymn-tune, has not less fulness and
complexity than a long and complicated piece. But if develop-
ment can take place towards without — that is, in the apparently
complicated structure — so it must also take place towards within —
that is, in the meaning, the expression of the single notes.1 No
doubt the progress in both directions is correlative. To contrive
a large structure of musical notes is also to see more meaning
1 Thus in beginning algebra the first result of putting letters for numbers seems
merely the substitution of vagueness for precision ; and not until later is perceived the
increase of power that comes from using and comprehending the general values.
xxviii INTRO D UCTOR Y ESS A Y
in the single elements ; and if the outwardly more complicated
form be chosen, this may be merely from the necessity of building,
so to speak, from the ground, the general level of appreciation of
the time, however deep the foundation might be laid. For where
thoughts are to be communicated, over-simplicity fails equally with
over-complexity. Thus in speech a meaning might have to be con-
veyed in phrases, for which in other circumstances words would have
sufficed. And if there is no merit in intricacy, there is also none
in simplicity : the value of each is the same ; the only question is
of means of expression. Though it might be said that outward
complexity rather facilitates execution, as lending mechanical aid ;
but here the gain is apparent rather than real.
There is therefore a notion of progress both extensive and in-
tensive. It aims at knowing more difficult things, but also at
knowing easy things better. That is to say, it must be directed
towards things past and things remote as well as towards things
present and at hand. For to the mind as well as to the eye things
appear simple by reason of remoteness. And it is not enough to
understand a thing by its elements, which are the most remote
parts in it, and most buried in the past. In proportion as anything
is better understood there must be corresponding revivification of
its elements ; as complete command of an instrument may be
shown in touching a single note.
It therefore follows that the explanation which starts from a
simple beginning ought not to be regarded as mere development
from a fixed base. Every enlargement of the system is attended
fay corresponding intensification of the germ from which the system
springs ; and if the first or parent notion be called most abstract,
INTROD UCTOR Y ESS A Y xxix
then the notions developed out of it may be said to form its better
understanding. Hence comes Hauptmann's caution against sup-
posing that the first notion or the first law can be from the begin-
ning understood and then the later notions deduced from it If
the first notions were adequately understood then the rest would
be self-evident ; as it is said that the propositions of Euclid were
self-evident to Newton. But in the process of learning the con-
clusions react upon the premises ; and the growth of the argument
makes it strike deeper root continually.
What has here been said principally with reference to notions
whose connexion is demonstrated, will partly apply also to notions in
which a connexion is merely shown by way of analogy. For instance,,
when Hauptmann finds in space an analogy to time and then
speaks of the ' time of space/ he does not mean that time is a
mode of space, nor any actual time in space, but only that there is
something in space (a mode or property of space) that is to space
what time is to the universe, that contains real space and time. The
' time of space ' is a property of space ; actual time is not so. So
* space of time ' is not space, but a particular kind or mode of time.
Though, on the other hand, if by time we mean the real notion, the
idea of time, it might be said that ' time of space ' is the real
notion, the idea of time, appearing in space, just as ' time ' is the real
notion appearing in the phenomenal world or universe, and that in
this sense the two are identical. An easier instance is found in
considering the bass part of a piece of music. It is a doctrine of
Hauptmann's that the bass part is earlier than the parts that lie
above, that it exists before them and must be supposed that they
may exist. On the other hand thorough bass is primarily regarded
xxx INTRODUCTORY ESSAY
as the art of making a bass or accompaniment to a given air, and
in this way the bass comes last into existence ; also historically the
air is prior to the bass, which involves harmony. Hauptmann's
meaning, however, is seen when we recognise that the earlier forms
of music, the choral and the simple arpeggio often repeated, have
later found their proper place in the bass part. Of these early
forms the meaning belongs rather to the past than to the present ;
now they are hardly to be recognised as ( tunes ; ' and in a piece of
music they gravitate to that part of it which is representative of or
symbolises past time. Here, therefore, that which in development is
latest, in the completed notion symbolises the earliest. So in poetry
Night is said to follow Day, and yet Night is representative of the
ancient time, the survivals or superstitions of which are often found
placed in it. Not that the bass part in its simplicity is equivalent
to the ancient music ; compared to that it is as a fossil to the
living animal. For modern music accomplishes by means of a
multiplicity of parts and many adjuncts no more than ancient
music accomplished. And the bass no longer lives as a separate
thing ; the life having partly gone out of it, it now forms the
standard, the skeleton for the rest. Thus in the bass we have, not
ancient music, but only so much of it (that is, of the spirit of it)
as agrees with the parts of music that are more peculiarly modern.
There is an analogy in history ; for past manners and customs
survive as a basis, retaining only so much of their former meaning
and reasons as agrees with the present time. And earth itself,
which serves as the general foundation, is also made out of the
perished forms of things past.
In estimating the value of Hauptmann's own explanation
INTRODUCTORY ESSAY xxxi
something may be said of the arithmetical theories which he re-
garded as unsatisfactory.
The system of music is wonderfully symmetrical. Yet its deve-
lopment has never been ruled by considerations of outward sym-
metry, but only by the feeling or intuition of what is right in
music. Sound having been made the vehicle of expressing ideas,
a symmetrical construction has resulted : symmetry has uncon-
sciously been attained. Now, it was very early observed that the
notes of the common chord stood somehow connected with simple
numbers. It was therefore natural to suppose that the symmetry
of numbers and the symmetry of music stood also in some close
connexion, so that numbers might be traced in music and music
in numbers. The arithmetical explanations were therefore attempts
at showing likeness between systems that had developed independ-
ently. It was assumed that music depended somehow upon
numbers ; it was left out of sight that the likeness might be due to
some common law of growth. The central fact was taken, that
the notes of the common chord correspond with the numbers 3, 4, 5 ;
whence it was sought to show analogy between all notes and all
numbers. Here was what Bacon calls an ' anticipation of thought,'
a hasty induction springing from an insufficient basis. It is not
enough to say that the numbers 3, 4, 5 do in fact ' correspond
with ' the notes of the chord ; the fact should have been shown
rooted in some higher notion. The ratios 3:4:5 have mean-
ing in numbers ; so have the notes C — e — G in music ; to show
correspondence between 3:4:5 and C — e — G without showing
identity of meaning is as if the same combination of letters with-
out regard to the sense were shown to exist in different languages
xxxii INTROD UCTOR Y ESS A Y
in order to found a theory of the derivation of one from the
other.
Now in Hauptmann's work the whole development is traced
inside music. Though illustrations and analogies are made use of,,
yet nothing is founded on them. The system is as self-supporting
as, e.g., that of geometry ; it is a simple unfolding of the musical
notion. But the principle of development is not peculiar to music ;
it is the same everywhere. If, then, numbers combine, following
the same laws as notes, analogies necessarily arise. For the course
of development is logic operating upon the initial unit, the o from
which the science in question springs. Thus the explanation of
music goes back to the time when there was no music, only sound.
Sound is the element in which the universe of music is generated.
And numbers combining also into a universe by the same law,
there will be a likeness of correspondence between the two, but not
likeness of identity ; because, though the two universes are the
same, yet they subsist in different elements. As in arithmetic
3 = 3 in the abstract, but in the concrete 3 and 3 may be of different
denominations.
Hence it is that numbers and symbols generally may be taken
to represent musical sounds. For while a single note may be
represented arbitrarily by any mark whatever, more perfect sym-
bolisation requires that there shall be a system of symbols con-
nected by a principle of generation to represent a system of notes
also connected by a principle of generation. The systems corre-
spond as wholes, and also in their single parts, and there is life in
the symbols not less than in the things symbolised. Parallel with
the history of the development of music we may expect to find a
INTRO D UCTOR Y ESS A Y xxxiii
history of the development of its written characters, both of inde-
pendent growth, yet representing one the other.
Now as music stands to its written characters, so it will also
stand to other arts and sciences. Every notion that arises in
music will have its analogies in other sciences, and then the question
arises of interpretation : how far it will be possible to pass from one
to another, e.g. from algebra to music, applying a knowledge of the
general structure of any one science to interpret the general struc-
ture of another, as if in a kind of comparative anatomy of sciences.
With the view of music as a pure science it is evident that music
no more needs to be explained by mathematics than geometry
does. Nevertheless results have followed the application of geo-
metry to physics, and of algebra to geometry ; and mathematical
analysis applied to music, and not merely to its physical basis,
might be equally fruitful. But the ultimate end is unity, and unity
may be reached in two ways, either by rinding that anyone science,
e.g. mathematics, includes all others under it, or else that the
ultimate highest notions of any one science are the same in all.
Now the knowledge of any particular branch of music deeply
entered upon leads to that general knowledge of music in which
all the branches converge and coincide. And so it might be
supposed that the general knowledge of music followed as a pure
science might ultimately lead to a point whence it should be seen
to be identical with the general notion of science. And, indeed,
Hauptmann's work is directed towards no less than this ; provided
that what in the book is explanation can be translated into actual
experience.
W. E. H.
AUTHOR'S INTRODUCTION.
IT WAS ALWAYS THE CUSTOM to begin text-books of Thorough
Bass and Composition with an acoustical chapter. In it the rela-
tions of the intervals are set out in known manner by the number of
the vibrations or length of the strings : the ratio of the Octave as
I : 2 ; of the Fifth, 2 : 3 ; of the Fourth, 3 : 4 ; of the major Third,
4 : 5 ; of the minor Third, 5 : 6 ; of the major Second, 8 : 9 and
9 : 10 ; of the minor Second, 15 : 16.
In the ratio of the vibrations the larger number belongs to the
higher note of the interval ; in the ratio of the lengths of strings
the higher note is denoted by the smaller number.
Most theorists then find in the numbers I, 3, and 5, in their
doubles, powers and reciprocal products, the determination of all
harmonic relations of notes.
Some seek it in the progressive arithmetical series from i to 1 6,
and place the notes under the members of the series, thus : '
i 2 3 4 5 6 7 8 9 10 ii 12 13 14 15 i6~*
ccgcegbt>cde f g abfrb c
The notes set against the numbers 7, n, 13, and 14 certainly
do not correspond to the true intonation ; b\> appears too flat, /too
sharp, a too flat again.
This necessitates modification of the degrees in question ; they
must be raised and lowered, whereby occasion is taken for speak-
b2
xxxvi INTRODUCTION
ing pf the difference between a natural system of notes and an
artificial one, as of the difference between a savage condition and a
civilised.
Many authors have believed that they must continue further
the above series and assign to the new numbers which enter, and
are not already designated from the series 1-16, the chromatic
notes intermediate to the diatonic. Then in the former series they
allot to the number II the note/J instead of/, so as to be able to
claim the latter for the number 21, and/J for the number 22 :
1.6 17 18 19 20 21 22 23 24 25.-.
c c# d d# e f fj gb g g#
Excepting the note £$, which in this series is determined by
the number 25, not one of the determinations here corresponds to
the true ratio of these chromatic degrees.
Of the theory which seeks to trace the reason of all harmony
in the so-called partial tones heard at the same time, we have only
to remark that even if, when a note is struck as Root, its twelfth and
the tenth of its Octave are the notes which make themselves heard
most distinctly as sounding with it, still the other notes of the
series — the series, namely, as continued ad inf. — are just as much to
be called partial tones, and must be included in the notion of the
notes heard at the same time. Indeed the degrees determined by
the numbers 7 and 9 can often be heard sounding quite clearly.
Besides, even if the consonance of the notes corresponding to
the first four numbers is alone considered, as being that which is
most distinctly heard, still this yields only the harmony of the
major triad. The minor triad can indeed be discovered in the con-
tinued series ; it occurs first in the combination of the numbers 10,
12, and 15, as e, g, b. Since, however, it here does not proceed
directly from the unity first assumed as Root, and also would com-
pel the omission of the intermediate numbers n, 13, and 14, it has
INTRODUCTION xxxvii
been believed according to this system that there is less justifica-
tion for regarding the minor triad as a natural product and of equal
rank with the major triad, and the minor triad has been called
* artificial ' in contrast to the ' natural ' major triad.
If, then, we may disregard this partial-tone theory, so also does
the theory previously represented, according to which it is thought
that the key to harmony is found in the continued arithmetical
series, reveal even in this first assumption decided untruth and disa-
greement with the structure of what is musically natural.
Better capable of being maintained is the view that all our har-
monic determinations are produced from the numbers I, 3, and 5,
their doubles, powers, and reciprocal products. This assumption
contains nothing that contradicts reality, but has in no way led to
a further explanation of harmony.
In the numerical series resulting from these conditions :
i 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25---
ccgcegcdeg be d e gg#
the inharmonic notes of the arithmetical series are certainly ex-
cluded. But it produces those harmonic notes only which lie on
the upper, dominant side ; the Fifth below can never appear in it.
Further, it does not at all afford any rigidly definite determination
of the triad, nor yet has any striking distinction been pointed out
for consonant and dissonant intervals. For if such distinction is
to be grounded merely upon the more or less ' simplicity ' or ' com-
prehensibility ' of the relations of sound, as has so often been said
and repeated, it needs only the smallest perception to discover that
the difference between consonance and dissonance is not one merely
of degree. We hear the notes of the ratios 2 : 3, 4 : 5, 5 : 6, as
consonant intervals, as agreement of the several pairs of sounds ;
but the notes of the ratios 8 : 9, 9 : 10, 15 : 16, as intervals de-
cidedly not consonant, not in agreement, which cannot persist in
xxxviii INTRO D UCTION
sounding together, as the first-named intervals can. Thus the
question cannot be only of more or less comprehensibility, of a no
more than quantitative distinction between the ratios of these
intervals ; a qualitative distinction must be traced.
Now where these first determinations are yet to seek, we cer-
tainly cannot expect a theoretical establishment of harmony in the
wider sense, an establishment of the laws governing the connexion
and succession of chords, from such data only as the acoustic
ratios.
And so we see that the introductory chapter on acoustics in the
text-books is always entirely left behind in the subsequent doctrine
of chords or harmony. That chapter is prefixed as a beginning to
the book ; its contents, however, can in no way count as an intro-
duction to the doctrine, as a principle from which the subsequent
matter is developed in a natural course. Neither the truth nor
the falsehood of the acoustical presuppositions has any further
influence upon the doctrine itself ; although in view of the untruth
and half-truth of these presuppositions this can only redound to the
advantage of the doctrine.
It will always, however, be an indefensible position, that a
doctrine of this kind has two beginnings : one left behind, given
up, and one carried on.
To take up the neglected beginning and present it in a sense
such that it may be a real beginning, leading up to where the
practical teaching of harmony and composition begins, and that as
a real beginning effective in every further formation it may therein
be, and be seen to be, but a development or further ramification
of itself — this is the aim of the present attempt.
The contents of this book do not run counter essentially to any
practical method of composition, so far as its teaching is right.
But still less should they run counter to that, which to sound human
perception seem* musically sound and natural ; which, if not always
INTRODUCTION xxxix
and everywhere in the rules of the text-books, we at least meet
with always and everywhere in sound compositions.
All that has already been theoretically demonstrated and ex-
perimentally verified of the physical doctrine of sound and intervals
will here be assumed as known. We shall also assume acquaintance
with the general field of practical music, as a whole and in its par-
ticular parts : practical knowledge of harmony and metre in all
elements of their outward manifestation, as also knowledge of the
usual technical terms for all the objects entering upon these fields.
For our intention is not to instruct in these things upon the lines
of their outward occurrence or their use in art, or with a view to
these. For this purpose there is no lack of more or less good and
thorough works of every sort and kind. Rather it is our intention
to seek a natural establishment of the laws governing harmony and
metre, the principle from which the manifold expansions in all
directions issue determined from within, and developing are shaped
into the phenomena known to us and again addressing us inwardly.
This shaping principle must in every element of its operation
always be, and remain, the same in itself. In the broadest relations
of the expanded musical work, so far as it is one whole, as in the
narrowest particular, the smallest member of it, in all elements of
its harmonic-melodic, as also of its metrical-rhythmical existence,
there will always be only the one law to be traced for its right and
intelligible construction. Again, this law cannot be exclusively
musical, but it is rather the wholly universal law of construction,
which operates everywhere, in that operation of it which attains
to musical, i.e. harmonic-melodic, metrical-rhythmical, manifesta-
tion.
Music is universally intelligible in its expression. It is not for
the musician only ; it is for the common perception of mankind.
Moreover music is not of radically different quality in popular
tunes and in fugues of Bach, or symphonies of Beethoven. The
xl INTRODUCTION
contents of the complicated work of art may make it difficult to
be understood, but the means of expression are always the same,
and singly are intelligible universally. Through them the greatest,
as well as the smallest, piece of music speaks to us, is imparted to
us, in a language whose words and grammar we are not first obliged
to learn. 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.
In no other kind of perception are the first elements of expression
given and apprehended with such mathematical determinateness
as in the acoustical. A practised eye is needed to judge the
correctness of optical determinations and relations ; of the acousti-
cal, every sound ear is an unerring judge. To pronounce upon the
purity of musical intervals requires no technical skill ; the feeling
for it is born in us, is given in the nature of humanly reasonable
existence.
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. The rules of musical phrase carried back to their
essential meaning are only the rules for what is in general commonly
intelligible, and in this meaning may be comprehended by every-
one, since they appeal in him only to that which is known to all.
The notion of an artificial system of notes is a thoroughly
worthless one. Musicians were not able to determine intervals and
invent a system of notes, any more than grammarians to invent
the words of the language in which they speak, and the construc-
tions they use in explaining constructions. They speak with the
language, which the general sense of mankind makes. Now as
INTRODUCTION xli
speech does not consist in placing words together, but in setting
them asunder, which in the thought are one, so also musical expres-
sion, which in succession and simultaneous sound is set asunder in
notes, is only one in the contents of the musical thought which is to be
uttered : its single elements are only members of an organic unity.
Of conventional determinations for chords, for the arrangement of
a key or scale, of arbitrary alterations, raisings and lowerings of
the naturally given degrees, although such phrases are often
employed by otherwise intelligent people, there can be no mention
when we proceed rationally.
That which does not rest upon determination universal and
everywhere valid, cannot be everywhere and universally understood.
That which is musically right, correct, addresses us as being
humanly intelligible.
That which is faulty does not address us as the expression for
something faulty, but it addresses us in fact not at all ; it finds no
sympathetic resonance in our interior. We cannot understand it,
for it has no intelligible sense, If incorrectness could be the ex-
pression for what is faulty, for what is bad or ugly, then it would
not have to be excluded from the means of aesthetic representation.
But a painter would never think of carrying out an artistic con-
ception by intentional wrong drawing, no more can a musician
apply what is incorrect to the purpose of delineating characteristic-
ally ; as the story is told of a composer, who thought that the
words, * There is none among us that doeth good,' were nicely
expressed by a row of parallel Fifths. Here it is only the composer
who does not do good ; every Fifth by itself does quite what it
ought.
Rightness or correctness of phrase is the condition under which
generally a sense can be uttered.
This Tightness, i.e. reasonableness of the shape taken by music,
has for its law of formation Unity, with the opposite of itself and
xlii INTRODUCTION
the removal of the opposite : — immediate unity, which through a
moment of being at two with itself passes into mediated unity.
There must always be the repetition of this process on that which
is assumed as immediate unity or given as the result of a pre-
vious process. Thus the unity of sound correlated with itself gives
rise to the triad, and the unity of the triad correlated with itself to
the key. But sound itself is also already such a unity, that has
gone out of and been correlated with itself; as all that is real
always contains, or is, being within itself and being outside itself
as one.
To try to set the full notion of this shaping process at once in
a clear light would be labour in vain. But in the course of the
following investigations it will, we dare to hope, be made out in its
working more and more clearly, and be established as the ruling
principle, as the essence of every intelligible formation, and at the
same time as the right understanding of the same.
At the beginning the reader may perhaps wish and expect
that things, which at first are simply enunciated, should be dwelt
upon more minutely, and proofs brought to support them ; as, e.g.,
when it is said .' there are three intervals : Octave, Fifth, and Third,'
or when it is said of these intervals that ' they are unchangeable '
— without further justification or explanation, why only three inter-
vals are named, and why just these three, and why they are called
unchangeable ; for we know of more intervals, and of many changes
in them. So too the meaning involved in the acoustical ratios of
these intervals is at first only just stated as shortly as possible ; and
a minuter explanation at the first entrance of this subject might be
all the more desired, as this way of thinking is, in the theory of
music, not one already known, and may seem difficult on first being
approached.
But further progress, with the expansion of the material, never
ceases bringing up occasion of returning back upon these first deter-
INTRODUCTION xliil
minations, and of explaining them by tracing them in their effects.
Besides, the sense, which is hardest to grasp in the greater sim-
plicity of the phenomenon, of its own accord lays itself open to
easier understanding in the subsequent unfolding of the principle
at first tightly packed in the germ.
We cannot deduce the law of a progression from one single
member, but only from the succession of members. If we know
the progression, then the single member may be known to us also
as having come into existence in the series, and carrying in itself
the conditions of its coming to be. So is it also with this law of
ours, which rules in music. It is depicted for clear recognition
only in the series of the functions, in which it attains to reality.
Afterwards, the single element of its working, seen in the series of
effects, from which the whole arises, will also become of easier com-
prehension.
For the first step it will only be requisite to acquire an inward
conception of the notion of the formative process in its wholeness,
in the unity of its three elements, with which we become acquainted
in their first utterance as the intervals of the Octave, Fifth, and Third.
This notion is and remains everywhere the same, in every for-
mation and transformation. It is the notion, that something,
which at first subsists for intuition in immediate totality (Octave),
parts from itself into its own opposite (Fifth), and that then this
opposite is in its turn abolished, to let the whole be produced
again as one with its opposite (Third), as a whole correlated in
itself.
Going into the universal sense of this notion, we shall soon be
obliged to grant, that it no less than comprehends in itself the ele-
ments altogether of all knowing, and that anything further for
knowledge is not conceivable ; — just as simultaneous sound admits
no consonance beyond the intervals of the Octave, Fifth, and Third,
which further consideration may show us to be related to the
xliv INTRODUCTION
notions of feeling, understanding, and ( felt understanding,' i.e. as
feeling, intellect, and reason.
It may be well to say a word beforehand in justification of the
way of representing notes by letters, which has been adopted here
as serviceable. For at first sight it certainly seems as if the usual
notation, which recalls our knowledge of music, and which we are
accustomed to read as the equivalent sound, must convey more.
Nevertheless it is useless for our purpose.
Written notes distinguish the degrees said to be enharmonically
'•different, e.g. c and £$, but they do not distinguish degrees differ-
ent in the well-known ratio 80 : 81. They make no difference in
the symbol for the Third of a Root and for its fourth Fifth, e.g. for
.e as the Third of C, and E in the series of Fifths C—G—D—A—E.
How essential this distinction is, and important to the notion
of the system of harmony, and how necessary it is also in the
notation, will be made clear by the contents of the following book.
There a Third-note is denoted by a small letter, a Root- or
Fifth-note by a capital ; e.g. the major triad of C in the first posi-
tion is C — e — G ; its second position is e — G — C\ its third posi-
tion is G — C — e.
It will want only a little practice for the distinction brought
out by this notation between Third-notes and Fifth-notes to be
perceived, both by the eye and also intuitively, in the meaning
it has for harmony.
That which is contained as harmonic determination in the three
intervals of Octave, Fifth, and Third and their mutual relations,
that, in its abstract meaning, we see taking shape in metrical deter-
mination as two-, three-, and four-membered time-unity.
So too the opposition in the musical notion of major and minor,
upon which we cannot now enter further even by way of allusion,
is repeated in metrical determination as metre which begins with
the arsis and metre which begins with the thesis, and as trochaic
INTRODUCTION xl\r
rhythm and iambic rhythm ; for the three elements of determination
must reappear in every manner, rhythmically as well as metrically.
The metrical-rhythmical shaping process will then have to be
combined with the harmonic-melodic. In this a determination of
the one does not necessarily call for the corresponding determina-
tion of the other ; for the same succession of harmony can assume
very different metrical shape, and the same metrical arrangement
can be embodied in harmony very differently. Only in the ele-
ment of dissonance a closer relation enters between the metrical
determination and the harmonic.
Now the diversity of shape must be infinite, first in each sphere
by itself, the harmonic as well as the metrical, and next in the com-
bination of the two spheres. Therefore it must not be expected
that a theoretical explanation is about to be given of every possible
particular phenomenon. But when the general notion has laid
open the course, upon the whole, of the train of construction, then
it will be easy to obtain by it a solution for every single case in its
particular occurrence. Our aim is principally explanation showing
the general in the particular, and the part only in relation to its
whole. And our examination of the particular can go no further
than is requisite for the explanation of the general in it, but leaves
it, when the determination for its kind has been found, to special
and practical treatment of a different scope, which we are not
now to engage in.
So too a last ending of the doctrine in itself is not possible. Its
end is the notion, gathered up, of the whole ; in which the notion
lies extended, while it is also contained concentrated in every single
part of it. However far the doctrine is continued, however far off
its end is put, it always remains unending. Of its nature it must
remain so, if it is to unfold and show organic working and weaving
in living growth.
As organic doctrine has no end, so also it has no determinate
xlvi INTROD UCTION
beginning. Both are to be looked for everywhere, and to be found
nowhere ; for what is outwardly most outward, or inwardly most
inward, is in itself only one and the same thing. Thus a theory of
the objects in the field of music, such as begins here with examin-
ing the phenomenon of sound, might just as well start from a
metrical manifestation of the notion, or from the last rhythmical, the
trochaic dactyl, so as in progression to arrive last at that which is
.here treated first, the phenomenon of sound. In the organic notion
•every beginning is also an end, and just for this reason the notion
is finite-infinite, because in it every end is also a beginning : the
germ is only contained in the fruit, and the fruit can only have
come from the germ. So metre teaches us, that the close falls
always upon a metrical first element, that the end must always be
a beginning again.
We must distinguish this manner of theoretical contemplation
from the theory which bears immediately upon practice : the theory
of harmonic and metrical shape in itself from the theory of the art
of composition.
For the active business of art, theoretical knowledge and under-
standing of the inner finite-infinite unity, or of the substantial
essence of the phenomenon with its intellectually distinguished ele-
ments, are not a necessary requirement ; as science in general is
not necessary to art and its flourishing.
Consciousness of theory in the act of poetical production, which
is rooted in feeling, and creates and forms in inward delight, is not
even conceivable.
Not abstract theory alone, but also theory of art is excluded
from consciousness in that act.
A work of art both in music and painting is called a ' composi-
tion.' The artist composes, puts together, notes or colours. After
an inward image he composes an outward one to agree with it,
which is able by its effect to call up the original again in our
INTRODUCTION xlvii
interior. The choice of notes and colours is guided and determined
by the inward image, that the total effect may correspond to it as
closely as possible. The artist may not be asked to account for the
nature of his means of representing, or of the inward image itself ;
but if this is felt as a harmonious whole, then only by tones of
sound or colour harmoniously joined can it be represented outwardly
and communicated to us through the senses. To the inward thing
thought only an outward thing thought can correspond, and for
this the individual must be compacted and bound up into a whole,
just as it would have been produced from that whole. Only as
having come from unity can anything again become unity, and only
as unity can anything speak to us as feeling and thought.
A person ignorant of music is able, though the keyboard is
strange to him, to pick out on it the notes of a chord or melody,
whichever he has in his mind, without in the least knowing the
meaning of the notes in harmony. A musician knows notes and
chords, understands their meaning in harmony, knows rules for
harmony and melody, metre and rhythm, for musical form in every
sense, but it is nothing of all this that guides him in poetical pro-
duction, and makes him find the right expression for his thought.
With him, just as with the ignorant person picking out his chord
or his melody upon the notes of the piano, it is the desire of making
the outward representation in agreement with something felt in-
wardly, so as to be the very thing itself.
Knowledge of art-theory may help technical acquirement, and
generally' endow the artist with that thorough education which
renders him a master ; in actual production it has no immediate
share. At least the artist will not turn to Knowledge, until imme-
diate Power leaves him, until the right will no longer come to him
unsought, and he is obliged to seek clearness as to his own unclear-
ness.
Those are not the happiest moments of producing, nor yield the
xlviii INTRODUCTION
happiest results ; they take their turn however, driving the unin-
structed to despair of success, the instructed to reflection and con-
scious contrivance of his end.
Here too technical knowledge stands nearer to practice, bears
more immediately upon it, than general knowledge or knowledge
of the general : the rule is consulted sooner than the law. Yet
knowledge of the law is as able to lend clearness and certainty to
technical knowledge, as that to help actual practice.
It is hoped that to scientific knowledge in the field of music the
present treatise may be an incitement and a beginning.
I.
HARMONY
B
SOUND.
1. WHERE sound is to be produced, there is required (i) an
elastic, stretched, uniform material, (2) and trembling or vibrating
movement thereof. The parts of the body moved are then alter-
nately in and out of their state of uniform cohesion. The instant
of transition into this state of equality or inner unity is that, which
by the sense of hearing is perceived as sound. It is the coming
to be of the being which subsists absolutely during rest, and which
is alternately abolished and restored in the elastic movement.
2. Not being in self, or dead persistence in rest, nor yet being
out of self in the motion, is sounding ; but coming to self.
3. Sound is only an element of transition from arising to passing
away of the state of unity. Quickly succeeding repetitions of this
element make the sound appear continuous.
4. We distinguish high and low sounds, and it is known, that
the difference of height and depth stands in relation with the
quickness of the vibrations. But greater quickness, or a greater
number of vibrations following in a given time, cannot be the true
cause of greater height in the sound, if, as stated above, the sound is
contained in one element of a single vibration, and only repeated
in the succeeding ones. For repetition more or less quickly of the
same thing does not change it.
5. Determinate pitch of sound is rather the manifestation of a
determinate degree of tension present in the elastic material. And
we can regard the tension as an effect of force fixed in a resistance,
B 2
HARMONY
which is expressed in sounding as greater, in relation to the re-
sistance, in the higher sound, and less in the deeper.
6. The same force in a quantitatively different resistance, or
quantitatively different force in the same resistance, will equally
produce difference of pitch. For pitch expresses only the relation
of the two conditions combined : of the force as active, and the
resistance or mass as passive. Thus the sound of a stretched string
is raised either by shortening the string or by increasing the weight
which stretches it. And since these conditions are quantitative, this
can be done in determinable degrees and proportions.
7. Sound exists as a phenomenon through a material means ;
to its production there is requisite a body specially conditioned,
and elastic vibratory movement of that body. But sound in its
essence is not contained in the material as an utterance of qualita-
tive attribute. What we perceive as the phenomenon of sound is
only the coming into being of the abstract inner form of unity in
the material, of equality recovering in the elastic movement from
inequality. So too the determination of pitch is not contained
in quantities of force or mass determined in themselves, but only
in the abstract relation in which these factors stand to one
another.
8. For the relations of sound and their harmonic meaning the
particular way in which the different degrees of pitch are reached,
makes no difference. It may be done by increasing the force or
by diminishing the mass : either by stretching a string with a
heavier weight, or by shortening the string stretched with the same
weight. It is known that for double tension of a string there is
wanted, not double weight, but quadruple, sc. in the duplicate ratio,
and for triple tension nine times the weight ; but the half of the
string, in which, as in every single part, the whole of the stretching
force is effective, contains in proportion twice as great tension as
the whole does, and the third part thrice as great, which is ex-
SOUND
pressed in the sound.1 Consequently the quantitative determinations
of sound are most simply considered in differences of quantity of
sounding material at a constant tension. For to obtain them
expressed in differences of the stretching force, we must use
magnitudes which are squares and roots.
But it will soon appear, that the harmonic determinations of
sound do not at all consist of complicated numerical relations,
and that even the few numbers required impart definite musical
character to the corresponding sounds in virtue, not so much of their
numerical, as of a more general signification.
9. A sound of definite pitch we shall call a note, and relations
of notes intervals.
MAJOR TRIAD.
10. There are three intervals directly intelligible :
I. Octave.
II. Fifth.
III. Third (major).
They are unchangeable.
I. The Octave : the interval in which the half of a sounding
quantity makes itself heard against the whole of the Root, or funda-
mental note, is, in acoustic determination, the expression for the
notion of identity, unity and equality with self. The half determines
an equal to itself as other half.
11. The Fifth: the interval in which a sounding quantity of
two-thirds is heard against the Root as whole, contains acoustically
the determination that something is divided within itself, and thereby
1 The tension is measured by the energy of the stretched string ; in the half or third
part of the divided string the tension, or energy, per unit of length remains constant.—
TRANS.
HARMONY
the notion of duality and inner opposition. As the half places
outside itself an equal to itself, so the quantity of two third-parts ,
heard with the whole, determines the third third-part ; a quantity to
which that actually given appears a thing doubled, or in opposition
with itself.
III. The Third : the interval in which a sounding quantity of
four-fifths is heard with the whole of the root. Here the quantity
determined is the fifth fifth-part, of which that given is the quad-
ruple, 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
multiplier, is contained the notion of identification of opposites :
of duality as unity.
11. The Octave is the expression for unity ; the Fifth expresses
duality or separation ; the Third, unity of duality or union. The Third
is the union of Octave and Fifth.
Before union separation must exist, and before separation unity
The Third fills out the emptiness of the Fifth, for it contains
the separated duality of that interval bound up into unity.
1 2. With the three intervals here named the major triad is known
to be given. But if the determinations of Fifth and Third take place
upon a Root, then the Octave is no longer of essential importance ;
for the Root must in itself answer to the notion of definite unity, if
upon it the Fifth, as interval of duality, and the Third, as interval of
union, are to be determined. Therefore the conditions of the notion
of consonance are completely fulfilled in the combined sound of
Root, Fifth and Third.
1 3. In the notion of the unity of the three elements of the triad
there is contained in brief all determination which underlies the
understanding, not only of chords as the simultaneous union of
notes, but also of melodic progression and succession of chords, and
also, as will be shown later, the requirements of laws of metre and
MAJOR TRIAD
rhythm. Every note of a musical phrase is Octave, Fifth or Third ;
every chord in union with others, and every rhythmical metrical
element, has its intelligible meaning in the notion of the three fore-
going determinations. They must, however, be comprehended as
being of a nature wholly universal, and not merely as intervals of
notes. Rather the determinate character of the latter is itself given
by the universal meaning of the triad notion, whose contents here
with quantitative determinations in the element of sound attain to
sensible intelligible expression as the chord.
14. Of the meaning of unity and opposition we have to say, that
under unity is to be understood being one with self, without distinc-
tion ; under opposition, being different to self. The sense of opposi-
tion that is to be comprehended here, is, not that something is
different to something else, but that it opposes itself as other to
itself. The first is only a difference, but not opposition ; intellectua
opposition can only proceed from identity.
15. We can regard an object in its immediate wholeness, and
comprehend the notion of this wholeness ; this is the unity of the
Octave. We can then regard the object distinguishing ; e.g. form
from contents. Now the intellectual opposition is not at once found
in the fact, that the form is distinguished from the contents. But
when to the form with its contents we oppose, as other determina-
tion, the contents with their form, then the same object appears in
the distinction under opposite determinations, or as opposed to
itself. This is the duality of the Fifth.
But in this opposition reality is suspended ; for that is not con-
tained in the separation of the two determinations, but only in their
united simultaneous existence. When that which is opposed to self
in the determination by distinction, is taken at once and in one, this
corresponds to the notion of real being.
For the phenomenon of sound this is expressed in the Third
which makes heard the separate united. In it duality has become
HARMONY
unity, not in the sense of immediateness, which the Octave offers,
but in the union of the opposites conceivable in it : derived, organic
or real unity, such as is felt in the triad, as against the immediate
wholeness of the Octave and the separated opposition of the Fifth.
1 6. That a construction of fundamental intervals going further
than that now laid down is impossible, is clear theoretically from
the nature of the notion. For all possibility of determination must
necessarily be exhausted, when anything has been traced and recog-
nised (I.) in its totality as a whole, (II.) in its separated opposites,
(III.) in the union of the opposites into a whole. But it is also
confirmed practically ; because not only does the triad not allow of
more consonant notes being added to it, but also, generally, any
note in relation to another can only be understood as meaning one
of three intervals of the triad. This will appear later in the con-
struction of the scale.
MAJOR KEY.
17. As soon as the triad in its three elements has been shaped
into a membered whole, it has again become unity, and passes
entire into the meaning of Octave. This must then split up anew
into its Fifth, and in its Third be restored again to concrete unity of
a higher order.
1 8. The Fifth-notion for the Octave unity of the triad again con-
sists in its splitting up within itself, or coming into opposite deter-
mination to itself. This is fulfilled by means of two other triads,
that of the subdominant and that of the dominant, of which the first
contains the Root of the given triad as Fifth, while the other contains
its Fifth as Root. In this way the triad first assumed comes into
opposition or contradiction with itself. For it has become dominant
chord itself in the first position, and subdominant in the other, and
MAJOR KEY
thus changed in itself from independent Octave unity into meaning
Fifth duality.
19. The Third-notion, uniting, or removing the contradiction,
then causes the opposite determinations, in which the triad is parted
from itself, to be taken up into it both at once, and the passive
' being a dominant ' to fuse with the active * having a dominant ; '
so that the two unities, which make the triad two, are placed outside
it as a duality, of which it is itself the unity : unity of a triad of triads.
20. The finished notion of this organic figuration, this triad of
higher order, whose Fifth is found in the separation of the sub-
dominant chords, and its uniting Third in the chord of the tonic, as
correlated and correlating, determined and determining, we call a
Key. It contains the elements of triad construction quite in the
same sense as the triad itself does ; it is only the triad appearing
in a higher rank.
21. Not to weary with too abstract conceptions, what has
hitherto been said may be made evident in the following way of
representing it.
Let the triad with reference to the inner succession of its deter-
minations be denoted by :
let I— II signify the Fifth ; III, the Third, as union of I— II.
If we denote, now and afterwards, the Root and Fifth by capital
letters and the Third by small ones, e.g.
I— III— II
C e G,
then the Octave unity, the original independence, of the chord
C—e — G is removed in the notion of key, because its Root C appears
in the chord of the subdominant, F— a — C, as Fifth, and its Fifth G
in the chord of the dominant, G— b — Z>, as Root.
I0 HARMONY
I-III-II I-III-II
FaCeG CeGbD
I-III-II I — III-II
This is the Fifth-notion in the key, its splitting into two, which
shows outwardly in the disconnected chords of the subdominant
and dominant, F—a—C, G—b—D, but essentially consists in the
contradiction of the double meaning of the unit chord, C — e — G.
To take at once, or conjointly, that which in the Fifth-notion is set
asunder, answers, here as well as in the chord, to the notion of the
uniting Third. There the Third-meaning does not lie in the separate
note, which forms the interval of a Third with the Root, in the note
e of the triad C — e — G, but in the removal in it and through it of
the opposition of Root and Fifth. And so here in the key, what we
have to think of as answering to the notion of Third is not the
uniting triad separately, but the union itself.
First the given triad is unity, Octave ; then through its two domi-
nant chords it falls apart within itself into opposition, • duality, and
becomes Fifth ; lastly it is restored as uniting Third element in the
correlation of the other two, as higher unity, as unification or unity
of union.
I
I - III - - II
C e G
I II
I— HI— II I — III — II
FaCeG CeGbD
I — III — II I — III — II
I III II
I — HI — II I — III — II
F a C e G b D
I — III — II
MAJOR KEY ii
22. To understand such a scheme rightly, let it be observed
once for all, that by the symbol I — II is expressed, not a first and
second, but the standing apart of opposite determinations, and by
III, not a third or triple, but the coming together of the same. The
organic property of a membered whole can never be represented
exhaustively, either by symbols and numbers or by words ; it can
only be spiritually indicated to intellectual feeling, i.e. reason, that
meets it halfway, and has the power of reproducing alive the living
thought conjured into symbols, numbers, and words. For if in
things surpassing utterance we would cleave only to the literal
meaning, contradiction and doubt would rise everywhere, but never
the living sense. The notion of union in the sense of the Third is
an infinite. The acoustical ' twice two ' of the interval of the Third
contains duality, or separation of unity, in its ' twice ' of the multi-
plier, just as much as it contains unity, or union of duality, in its
' two ' of the multiplicand. Were the last, union, alone contained,
then its other, separation, would be wanting ; union would still have
its opposite outside of itself, and would thus be again only a one-
sided determination. This of itself would be against the notion of
the Third, which does not exclude opposition, but includes it. Now
because this notion has to unite both union and separation, it can
only be fulfilled in endlessly continued passage into contrary and
comprehension of all opposites. Thus it must be conceived as an
infinite process, and consequently as the notion of eternal becoming,
living, or being real. This is Nature, who, produced as duality from
the prime unity, and busied continually in making her opposites be
absorbed into one another, is live being itself and reality.
23. The effect of Octave, Fifth and Third is determined for our
perception quite as unambiguously as are the quantitative rela-
tions from which they proceed. It behoves us therefore to conceive
the relations, which are communicated to us sensibly through the
medium of sound, in their mental meaning, as we have tried to do
12 HARMONY
above ; but the result of the trial must, in the fundamental meaning
of explanation, always be again tested by feeling the effect that
these intervals have upon us. For where what is thought contra-
dicts what is felt, there it can only be untrue. If by theoretical
explanation the Octave were found as the expression for a manifold,
the Fifth as the expression for union, or the Third as the expression
for separation, such a theory must at once be decisively refuted by
the impressions that these intervals excite in us. But that the
Octave should strike our feeling as unity, the Fifth as separation,
hollow emptiness, the Third in the Fifth as a satisfying perfect
contentment, the very meaning correspondingly found for the ratios,
may itself supply another such contenting Third between felt and
thought.
24. In the chord the determinations of Fifth and Third are taken
upon one and the same unity ; therefore there is nothing to prevent
its intervals from being simultaneous. They are elements of a
single existence. But the advance to the key begins with the con-
tradiction of this singleness, because the reciprocal relation of
Root and Fifth is removed by the dominant chords. Whereby the
quiescence of the chord changes to motion, and the simultaneous
becomes successive ; because for simultaneousness it is a contradic-
tion for the Fifth of a Root to be Root of a Fifth ; a contradiction
for simultaneousness, which we learn later to be the essence of disson-
ance, but which in the opposite of simultaneousness, succession, is
none, because it is resolved by the Root becoming Fifth, or, con-
trariwise, the Fifth Root. Thus the key can be set out harmo-
nically only in a succession of chords.
25. The notion of the triad determines first the intervals to
form the chord, and next the chords to form the key. Similarly
it may take the key as Octave unity, and proceed with it to Fifth-
and Third-determination in the same sense as in chord and key
construction.
MAJOR KEY 13
26. The key arose, when the given triad, after coming into opposi-
tion with itself by the subdominant and dominant chords, compre-
hended in itself the opposition as unity, and thereby became tonic.
27. Opposition or Fifth-meaning for the key, which as yet
subsists in absolute unity, is found in its taking on one or the
other dominant meaning through subdominant and dominant
keys ; that is, in its becoming, as a key, a dominant to its sub-
dominant and subdominant to its dominant.
28. The two opposite determinations attain unity by determin-
ing becoming determined ; that is, by the middle key passing from
the determination of being dominant to one key or the other into
that of having one and the other key as dominants. Taking them
together thus again answers to Third unity of the three keys. The
middle key is shown as tonic, or middle of a system of keys, where-
by to its inner determination there is added its outer one of being
principal between secondary keys ; just as the chord, when deter-
mined in itself, could only by secondary chords reach the determi-
nation of being the principal chord in the key.
29. This triad of keys has a link, or element of relationship, in
the tonic triad of the middle key, which appears in it as tonic
chord, in the subdominant key as dominant chord, and in the
dominant key as subdominant chord :
I III II
Bt>d FaCe G b D f# A
I— III— II I— III— II I— III— II
I_III—II I— III— II
30. The linking of chords started in the single key, may be
continued in both directions without end. Now each triad, as it
occurs in successive order, is necessarily determined as middle to
two secondary triads, just as happens in key-union with subdo-
minant and dominant. Thus the keys too appear linked endlessly
I4 HARMONY
to one another. But to a higher unit notion than that of the key
itself, it can never come; no more, indeed, than the triad can
receive any addition in itself. For the latter contains the complete
development of the triad notion inwards, and the key contains it
outwards ; the triad as simultaneity at rest, as chord, the key as
simultaneity in motion, as chord succession. Besides, the last for-
mation does not go beyond the notion of the key ; it only confirms
it, as being one key determined among others. To a determina-
tion of keys going further than that of the two dominants there
would be wanting the direct reference to the unity originally taken.
And things distinguished must necessarily have something in com-
mon, if one is to be able to gather them up into a notion, or to
pass continuously and intelligibly from one to the other. For the
understanding of change, or passage in general, can only be con-
tained in change taking place upon something that remains : not
in another being other or different to one, but in one itself becoming
other.
MINOR TRIAD.
31. The determinations of the intervals of the triad have been
hitherto taken as starting from a positive unity, a Root, to which
the Fifth and Third are 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, then the opposite meaning will lie in a
note being Fifth and Third. Having is an active state, being a passive
one. The unity, to which the two determinations are referred in
the second meaning, is passive : in opposition to the having of the
first idea we find the second, being had. The first is expressed in
the major triad, the second in the minor.
In the latter the relation of (major) Third holds between the
MINOR TRIAD
middle and upper notes, and therefore the two intervals of the chord
are conjoined, not in the Boot, but in the note of the Fifth. 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 last the
common element for both determinations is contained in the note
of the Fifth ; therefore that note, being doubly determined, may be
negatively considered as doubly determining, or as the negative
unity of the chord. Therefore the symbol II — III — I seems not
unsuitable for the minor chord.
32. In the natural infinite series of notes, written by the ratios
of vibration :
I 2 3 45 678 9 10 ii 12 13 14 15 16 17 18 •••
CCGCeG.CDe.G. . b C . D •-.
we find the major triad first occurring under the numbers 4:5:6,
as C — e — G, the minor triad under the numbers 10 : 12 : 15, as
e — G — b. If the series were carried on further, we should see every
member of it that answers to a multiple of 4, bearing the major
triad, and every multiple of 5 that is divisible by 2, bearing the
minor triad, in the same proportion as the first ones above. E.g.
G—b—D as 12:15:18=4:5: 6, b—D—f% as 30 : 36 : 45
= 10 : 12 : 15.
33. The three members of the proportion in the minor triad*
10 : 12 : 15, can be reduced to smaller numbers, if we separate the
two ratios 10 : 12 and 12:15 from one another ; for then they can
be expressed singly by 5 : 6 and 4 : 5. These ratios remain the
same, if we substitute the expressions - : - and - : - ; for 5 ; 6 is
as - : - and 4:5 as - : -. But by the last notation the proportion
10 : 12 : 15 has been expressed in smaller numbers - \ -
. ' - !•=.;—
I6 HARMONY
a common middle term found ; and the proportion g : - : -> or
abbreviated — , may now be taken for the minor triad.
6:5:4
In this expression we get the numbers again, but in con-
trary order, of the proportion of the major triad, which may be
denoted by 4 : 5 : ^ Also the two may be expressed as positive
and negative powers, for there is
io :
e
34. Thus the essential meaning of the minor triad must come
to light, let the expression be of what kind it may, if only it is
taken back to its essential contents. And with this we gladly
leave symbolisation by numbers, which may indeed afford an
interesting play of combinations, but offers no nearer opening
towards the nature of things. It does not make the notion
easier ; rather it can only represent it veiled. For the notion is
contained in determinations far simpler and more direct, those
general terms of unity, its becoming two, and the identifying of both
as union.
35. The minor triad, as an an verted )maj or triad, must, in its
meaning of being considered to originate from a negative unity,
consist of a construction backwards. Referred to the unity C, the
major triad is
I — II
C e G
I— III
MINOR TRIAD 17
The minor triad of the same unity C as negative, that is, as Fifth
determining Koot and Third, is
II - I
F at> C
III— I,
which is the same as if we put
F a[> C
I II
I-III.
In the major triad the unity is the positive which determines ;
in the minor triad it is the positive which is determined.
36. The minor triad thus being of passive nature, and having
its starting-point above (not its most real starting-point, yet that
which is determined as unity), and forming from it downwards, there
is expressed in it, not upward driving force, but downward drawing
weight, dependence in the literal, as well as in the figurative sense of
the word. We therefore find in the minor chord the expression for
mourning, the hanging boughs of the weeping willow as contrasted
with the aspiring arbor vitae.
MINOR KEY.
37. The system of the major key contains the minor triad in a
secondary meaning, that is to say, in the middle of each pair of
major triads : (i) of the subdominant chord and the chord of the
tonic, (2) of the chord of the tonic and the dominant chord.
The 'simultaneous existence of two triads with a note in common
of itself makes a contradiction ; because then opposite meanings in
two directions are attributed to the note at once, which it can only
receive successively.
C
l8 HARMONY
I-III-II I-III-H
F a C e G b D
I— III— II
But the contradiction, which would lie here in C or G, is called
forth only by the extremities of the two chord dualities, in C by
p .£, and in G by C D ; it is not contained in their middles,
a—C—e and e—G—b, because a e and e b as Fifth-deter-
minations, and C— e and G—b as Third-determinations, find their
unity in e and b, passive it is true, but not self-contradictory.
38. Thus there is a motive for linking the minor triads to-
gether in just the same way as we found for the major triads :
...d F a C e G b---
II— III— I
H_III_I II— III— I
But if we tried to gather up a triad of minor triads into chord
union, there would still be nothing at all answering to the notion
of a minor key. Such a series of minor chords would always seem
a mere result of the series of major chords. It can never come to
have independent value, because there the positive unity for the
minor chord is wanting. The minor key, like the major, can only
make its determination of effect in issuing from the positive triad
notion. Therefore the minor chord, as a denial of the major, must
begin by really premising the thing itself, of which it is a negation ;
for a thing, to be real, cannot issue from negation without positive
premise. The element of negation may, however, be taken as
principal determination ; that is here as tonic, middle of a key system,
whose dominant will then, be a major chord, the premised positive,
and its subdominant a minor chord. For in the negative generation,
where the triad determination originates in the Fifth, the minor chord
is the beginning of a series of minor triads continued without limit
MINOR KEY
towards the subdominant side ; just as the positive, where the triad
determination issues from the root, is continued towards the do-
minant side in an infinite major series.
39. In I_III_H
G b D
there is given the positive triad notion for the unity G ; in
II— HI— I
C et> G
the negative triad notion for the same unity G.
In I_IH_II
C et> G b D
II_HI_I
both determinations are contained joined ; and in
II— HI— I I— III— II
F ab C et> G b D
II_IH_I
the second determination, the negative of the first positive one, is
placed as tonic, or principal element of a key, whose contents
accordingly are the minor triad of C, with the minor triad of F for
its subdominant chord, and the major triad of G for its dominant
chord. In this formation we recognise the key of C minor in its
natural and self-determining conditions.
40. Here the process of the formation is shown unfolded in
time ; but, like that of the major key, it is only the concrete
expression of a fixed thought. In the system of the major key
the thought is, that I changes into II ; in the system of the minor
key, that + 1 changes into — I. Both originate from the positive
unity ; but there the notion of the change is that positive one be-
comes positive other ; here it is that positive one becomes negative
one. The former contains the opposition of being and becoming,
the latter the opposition of being and not being. The former is life
c 2
20 HARMONY
carried onwards in another, the latter is solitude and narrowing
down to self.
41. The major key will pass into other keys. The minor key
is isolated, without the power of passage into others. With the
major notion a system of keys could be marked out, containing a
principal key, as middle, with its secondary keys ; and afterwards
each secondary key could in turn appear as principal with se-
condary keys, without contradicting the conditions belonging to
the first as a key. But the notion of separation, out of which the
minor key proceeds, is in principle against the notion of unity
belonging to the major system. Secondary minor keys would make
the fundamental conditions upon which the principal minor key
rests, to be no longer of effect, and thus would abolish the princi-
pal key itself.
II_IH_I I-JII— II
Key of C minor : F at> C efc) G b D
II— HI— I
II— III— I I— III— II
Key of F minor : B|? d[> F a[? C e G
II— III--I
ii— in— i i— m_ii
Key of G minor : C e[> G bt> D f # A
n_ni— i
The key of F minor, as subdominant of the given key of C
minor, contradicts by its dominant chord, C—e—G, that which is
here taken as the principal thing, the negation, C— e\>—G, of the
positive unity, G—b—D ; the key of G minor, as dominant of the
given key, contradicts by G—b\>—D the positive unity itself.
42. The series of chords belonging to the notion of the major
key continues in like form without limit, as a chain of major triads ;
MINOR KEY 21
the series belonging to the notion of the minor key starts from an
element of contradiction, and forms a chain of major triads in one
direction, and a chain of minor triads in the other.
A. Series for the Major Key.
II I-III-II I-III-II I-III-II I-III-II I
I-III-II I-III-II I-III-II I-III-II I-III-II
••-Abe Ebg Bb d F a C e G b D f # A c# E g# B-.
B. Series for the Minor Key.
I-III-II I-III-II
I-III-II. I-III-II I
•••Ab cb Eb gb Bb db F ab C eb G b D f# A c# E g# B—
II-III-I II-III-I II-III-I
I II-III-I II-III-I
MINOR-MAJOR KEY.
43. In the minor key the negative element, the negation of the
positive, or major, triad, which is assumed first, is determined to be
the principal thing, the middle or tonic. But we may also conceive
the notion of the key-system, so that it shall contain the negation,
the minor triad, as essential determination, yet not give it promi-
nence as principal element, i.e. not place it in the middle of the
system. Then the positive, or major, triad represents the middle,
and its negation, the minor triad, occupies the place of subdominant
chord. For the dominant chord there results, by continuing the
positive series, evidently a major triad.
By this there is formed a key-system, which contains in essence
22 HARMONY
and effect the major and minor notions joined. We get then those
harmonies of the major key, in which the minor Sixth asserts itself.
If in the series above for the notion of the minor key we put
the positive triad G — b — D middle, the system takes the following
shape :
II— III— I I— III— II
C et> G b D f# A
I_IH_II
Although it is unusual for the minor-major key to be formally
made the basis of a piece of music, yet it occurs used in the course
of one not rarely ; oftener in the sentimental style of modern music
than in the older. Wherever the diminished chord of the Seventh is
resolved into the major triad as tonic, there this key is present ; in
fact it is then contained in its whole compass in the notes of the two
chords. Similarly, so far as its principal contents, in the plagal
close from the minor triad of the subdominant to the major triad
of the tonic. This key has the diminished triad upon the second
degree, the augmented triad and augmented chord of the Sixth * in
common with the minor key ; only here the chords are not referred
to a minor triad as tonic.
44. When we speak here of the diminished chord of the Seventh,
of the augmented chord of the Sixth, also of other intervals besides
those named at first and explained, that is because we assume
practical knowledge of these chords and intervals, as to their effect
and outer properties. Their relation to the notion of the key could
not up to now be explained, for we have been speaking of consonant
formations alone. From the very beginning only three directly in-
telligible intervals have been named, and it was said of them that
they are unchangeable— cannot, that is, be sharpened or flattened.]
The explanation of the notions which are expressed by the relations
1 See pars. 60-62, and the beginning of par. 236.
MINOR-MAJOR KEY 23
in sound of these intervals, must bring the proof of what we say :
namely, that anything else than one of the elements, which appear in
the notion of a note as Octave, Fifth and Third, but are universally
elements of the notion for all intellectually felt, i.e. reasonable
knowledge, is in itself nothing that can be known directly. There-
fore a minor Third referred to a Root has no more claim to be re-
garded as a direct interval, than a diminished or augmented Fifth has ;
or than have Seconds, Fourths, Sixths and Sevenths with all their
different properties.
Now it would be very uncomfortable and roundabout always to
describe people by their relationship, or by the degree of their
descent from the first human pair, and we prefer calling them by
their Christian or surnames. So here, for shorter description, it
will often be good to use as names the terms 'minor' Third, 'dimi-
nished ' and ' augmented ' Fifth, and others, which describe the in-
tervals outwardly. And as at any rate the expressions * Third,'
'Fifth' and 'Octave' are already taken from numbers of degrees of
the scale, so, when we are only concerned to describe outward
distance, other, indirect intervals may also be named upon the
same system.
DIMINISHED TRIADS.
45. In the linked series of keys, the major key can pass into
either of the secondary keys related to it by the tonic triad, viz.
those of the subdominant and dominant, by the tonic triad itself
taking on dominant meaning in the one case, subdominant in the
other. But for the notion of succession this is a twofold, opposite
determination, and answers to Fifth-meaning. It is a motion
diverging outwards ; and with it, if we regard the rest of the key
in its limits as answering to unity or Octave-meaning, there must be
24 HARMONY
found a motion converging inwards, a passage into self, answering
to Third-meaning.
We can picture the idea of something passing into self by
thinking of a finite straight line bent into a circle with its begin-
ning and end united : finite as infinite, or infinite in finite.
Absolute finiteness would be suggested by the limited line;
absolute infinity by the line running on without limit. The first is
the limited key without passage into itself; the other is its progress
into the keys linked in a chain without limit, each newly arisen
dominant becoming in its turn a tonic.
46. As an effect of sound, the notion of the key passing into
itself is expressed in the chords which contain the union of the
Fifth of the dominant with the Root of the subdominant : the so-
called diminished triads. Now the combination of sound in these
chords rests upon a double basis, upon the dominant and sub-
dominant ; they must therefore always be dissonant.
The notion of dissonance cannot yet be entered upon more
nearly ; only it may be observed in passing, that the expressions
sometimes used in Germany of ' well-sounding ' and ' ill-sounding '
for * consonant ' and ' dissonant ' must be held quite inappropriate.
On the other hand the verbal sense of the latter terms contains a
perfect description : the character of consonance is determined
sounding together in the harmony, and of dissonance determined
sounding apart. A consonance may sound ill in a place where a
dissonance is needed, and where a dissonance sounds well.
The Third and Fifth of the dominant triad can unite with the
root of the subdominant triad to form a diminished triad ; so can the
Fifth of the former with the Root and Third of the latter. E.g. in the
key of C major, b—D\F, D\F—a ; in the key of C minor, as also
in the minor-major key with the same name, b — D\F, DjF—a\> ;
chords which, because they include the limits of the key, have the
property of closing it up into itself. The tendency of such chords,
DIMINISHED TRIADS 25
the reason for their arising, and their mental meaning, we shall
afterwards see ; here they are only to be regarded in themselves as
combinations of sound.
47. The chord upon the Fifth of the dominant of the major key,
D\F — a, must not be confounded with the minor triad, d — F — a ;
which, transgressing the lower limit of the system of the key of C
major, is formed from the Third of B\> with Root and Third of the
major triad of F. And in general notes of the same name distin-
guished by capital and small letters in the notation which we use
here for chords, must not be taken to be the same. The mechani-
cal structure of our keyed instruments with its enforced equal tem-
perament ignores this distinction, equally with the so-called enhar-
monic difference. The ordinary musical notation, too, while it has
a difference of symbol for notes enharmonically different, does
not distinguish notes different in the other meaning. It has
only one sign for the Third of the scale of C major, and for the
Second of the scale of D major, supposing the latter to have the
second degree of the scale of C major as basis : that is, it has the
same sign for e and E. Therefore it may well be, that, from want
of care in practical study, musicians themselves are often unaware
of the difference, although when it comes to the question as to
which of the two meanings is to take effect, instinct will always
make it be perceived clearly enough.
48. What temperament does for instruments with fixed tones,
equally distributing these differences wherever they occur, can
have no influence upon the essence and meaning of the intervals.
The tempered Fifth is not meant to be heard as a flattened Fifth,
nor the Third, which is in the temperament too sharp, as a
sharpened Third ; the intervals are meant to stand for true. Singers
do not temper ; as we shall see in the construction of scales, they
have nothing to determine their intonation but the Fifth and Third,
and they try to take their intervals perfectly true to them. The basis
26 HARMONY
of temperament is certainly nothing else than the using of one
and the same note in several meanings ; whereby there is confused
not only the enharmonic difference, e.g. b$ — £"(125 : 128), but also
that other which exists between the major Third and the fourth
Fifth of a Root (80 : 81).
49. Thus we find the Third e under the number 5 in the natural
series (par. 32). For E, as fourth Fifth from C, we get (34) the
number 8 1. And if e be raised to the corresponding octave (5 x 24),
there is found for it the number 80, different therefore to that for E
as Fifth. But how great or small the difference is does not matter
so much as that there is a difference, and that in the number 81, as
a power of 3, Fifth-generation may be recognised, but in 80, a pro-
duct of 5 into a power of 2, Third-generation.
50. Where intonation is free, not fixed, there is never any
reason for not making the intervals keep perfectly true. For inside
a key, in the compass of three united triads, notes of the same name
with different meanings do not occur ; a key does not even contain
two chromatically different notes. And enharmonically different
notes lie in their real nature so far apart, that it is not possible for
them to meet together in harmony.
51. If the dissonant triad, which has the Third of the dominant
for Root (e.g. in the key of C major, b—D\F} is named diminished,
then we can use the same term for the triad upon the Fifth of the
dominant, D\F—a. For by what has gone before, D - a is no more
a Fifth than b - F is. Both chords have a duality of basis ; the
subdominant and dominant : F and G. So in the minor key with
the triads b—D\F, D \F-cb.
52. Thus the major system
dim. minor dim.
major
DIMINISHED TRIADS 27
contains three major, two minor, and two different diminished
triads.
The minor system
dim. major &ug- major
^^b / F
minor dim.
contains only one major triad of the first order, that of the dominant.
A second is found as intermediate chord between the two minor
triads of the subdominant and tonic. Further there are contained
in it two diminished triads, on the Third and Fifth of the dominant,
made up of notes of the two dominants, as in the major system ;
and lastly the so-called augmented triad, upon the minor Third
of the tonic, a chord which expresses most harshly the twofoldness
of its nature. Thus in the minor key there are three different dis-
sonant triads ; for the two diminished triads contained in it are
not of like structure, any more than those upon the same places in
the major key. Both rest upon the double basis of subdominant
and dominant, but differ between themselves in taking more or
less from one or other of the triads of the two bases : b — D\F>
DlF—a\>.
53. But in the augmented triad dp — G — b the middle note,
G, is in itself decided duality ; it is determined differently in two
directions, as positive Root and negative at the same time :
III -I
+ 1 III
In the diminished triads the dissonance consists in two notes
not being unity ; in the augmented triad it is contained in the
inner duality of one note.
2g HARMONY
THE KEY -SYSTEM STRETCHING OUT, OR IN
TRANSIT, TO DOMINANT OR SUB-DOMINANT.
The Triads joining the Limits, or Diminished Triads,
of this System.
54. All the triad harmonies have now been pointed out which
are found either inside the limits of the major and minor key
systems, or at the meeting of the limits. There still remain to be
mentioned the triads which arise from joining the limits, when the
key system is shifted on through one member of the triad series (A
and B, par. 42) in the subdominant or dominant direction, when it
encroaches, that is, on one side or the other. The system is not
thereby enlarged ; it cannot be enlarged, for what it gains upon
one side it must lose again upon the other, and so keep, as what
its notion includes, contents of no more than three adjacent triad
formations. But besides, by such shifting to the next member of
the series on one side or the other, the existing key is not yet re-
moved ; for one dominant determination still remains. Suppose
the step taken to the subdominant side, the Third of the dominant
remains ; or to the dominant side, the Third of the subdominant
remains. Either of these still prevents the tonic triad from giving
up its determination as principal chord.
55. Such a shifting must not, however, be regarded as a mere
mechanical treatment of the fixed progression of chords ; it can
only rest upon a mental inner foundation. Besides, the progressive
series of fixed, determinate chords has not, strictly speaking, its
counterpart in reality ; it is a means of depicting simultaneously
something that in reality developed successively.
56. If in the key of C major the note/ft, Third of the dominant
Fifth, enters, then in this there is at once expressed an inclination
THE KEY-SYSTEM STRETCHING OUT 29
towards the dominant side, a desire of making the dominant chord
take tonic meaning. But just in measure as this is attained, the lean-
ing to the subdominant side must have lessened ; in the same degree
as the dominant side comes forward, the subdominant side must
recede : the centre of gravity of the equilibrium between the two will
turn to the side towards which the key receives a preponderance.
Supposing the centre of gravity in the system
F — a — C — e — G — b — D
to consist of the tonic Third, as middle of the middle chord, binding
element of the tonic triad binding the dominant triads ; and sup-
posing that the note e is now equally inclined to move towards F
and towards D ; then, when the note /# enters the system, i.e.
when the Third of the dominant of the key of G major is touched,.
as in
a— C— e— G— b— D— f#,
the centre of gravity occupies no longer its former place, but is
situated in that element of the tonic triad which belongs to the
triad the Third of whose dominant has appeared, namely, in the
tonic Fifth as Root of the dominant triad. But the G here does not
enter in full tonic meaning. For with the entrance of the Third of
the dominant Fifth, /Jf, the key has only given up F, the Root of the
subdominant triad, but not the minor triad of a formed from its
Third a and the tonic Third-interval C — e ; and this triad because of
its Root a does not belong to the key of G major. The minor triad
of e is now the triad of reference, and its Third G the middle of the
system both by outward position and by inward meaning. Before,
the middle note e was in equal degree urged towards the limit notes
F and D ; now it is G, that can be determined to move towards a
or to wards /Jf. Upon the entrance of the Fifth of the dominant triad
of the key of G major, the subdominant triad of the key of C major
is wholly given up, because the note A excludes the subdominant
30 HARMONY
Third a. Then the tonic major triad of C will have become sub-
dominant chord, and the middle of the system will lie in the Third
b of the G major triad, now become tonic. The same process would
result in reversed order, supposing the tendency turned towards the
subdominant side. With the entrance of the Third of the major triad
on B\> the chord d — F — a would take subdominant meaning, and
e — G — b dominant meaning ; and C, as middle of the middle triad
a — C—ey would be determined as the middle of the system. By
the entrance of the Root B\> the key of F major would be fully esta-
blished, because then the tonic C major triad would itself have be-
come dominant triad ; and then the middle of the system would be
settled in a.
DIMINISHED TRIADS OF THE KEY-SYSTEM IN
TRANSIT.
(a) In the Major Key.
57- To learn what chords arise from the sounding together of
the limit notes, when the system reaches out to one side or the other,
we now go back to the two series A and B (par. 42), and begin
with the march of the major system towards the dominant side,
whereby the key of C major takes up the Third of the dominant
Fifth, /#, and leaves out the Root of the subdominant, F. The
chords of the joined limits will then be: D—f$\a and fft/a—C'
different in nature and effect from D-f%-A and/#-^/^, the
chords which would be found in the key of G major.
If the system of notes is shifted through one member towards the
subdominant side, then dt the Third of the major triad on /fc comes
forward, while D, as Fifth of G, is at the same time shut out. The
DIMINISHED TRIADS OF KEY-SYSTEM IN TRANSIT 31
combinations joining the limits are now the chords G — b\d and
b\d — F ; to be distinguished from G — b — D and b — D/F, as con-
tained within the limits untransgressed of the C major system, and
formed by joining them.
58. The reception of the Third note, which lies below the
system, ought certainly by parity of reason to let the key continue ;
for the reception of the Third which lies above, does not make it
cease. But the change itself, the difference between D and d, can-
not be brought out in the same way as that between F and /"#.
The Root B\) must have entered before d can be shown decisively as
not D. But with B\> the key of F is determined, and that of C
made to cease. Therefore, because the note gained by the move
cannot be determined but by the note lying underneath that, the
chords belonging here must be referred no longer to the given key,
but to its subdominant. Thus b\d — F and G — b\D no longer
belong to the key of C major, but are seen to be produced by the
stretching out of the F major system, again towards the domin-
ant side. Then the chords G — b\D and b\d — F have the same
relation to the key of F major, which D — f$\a andf^/a — C have
to the key of C major, and can no longer be regarded as derived
from the latter key.
(b) In the Minor Key.
59. The minor key-system, from reasons which lie in its dif-
ferent nature to the major key, can suffer shifting to the sub-
dominant side only under very narrowing circumstances. The
reception of a member of the subdominant series would be an
attack upon the positive premise, that from which the generation
of the key has proceeded ; it would rob the dominant chord of its
Fifth, and the first chord to appear on the dominant side would
then be the augmented triad, a chord of most marked duality.
Therefore the triads G — b — dp, b — d\> — F, which arise in the C
minor system by the move towards the subdominant side, will
32 HARMONY
always attach themselves rather to the F minor key in the move
towards the dominant side.
60. By shifting to the dominant side, there are found, following
the former process, two chords containing an interval of diminished
Third. E.g. in the series B, supposing the outlying note, /ft above
the Fifth of the dominant, received into the C minor system, and F
as Root of the subdominant chord thereby shut out, then the chords
of the joined limits are : D—f$\dv,f%\a\>—C. From these com-
binations the so-called chord of the augmented Sixth is derived, which
indeed makes its leading note strongly perceived as the Third of
the Fifth of a dominant chord.
6 1. Therefore, in the minor key as well as in the major, the
only triads joining limits, which are of real use, besides those
belonging to the closed system, are the ones that can be produced
by taking in the nearest member on the dominant side.
In the key of C major : D—f$\a,f$\a—C.
In the key of C minor : D—f$\a\>,f$\a\>—C.
The particular conditions governing the position of the inter-
vals of the two last will be found later on. Every harmonic com-
bination, whatever the shape it takes outwardly, can be produced
only from inner determinations ; and, to conceive a chord theoretically,
it must be looked upon, never as an aggregate of notes, to which
sharps and flats may be applied at pleasure, but always as an
element of development in the notion of organic reality.
(c] In the Minor-Major Key.
62. The minor-major key is in its subdominant and dominant
chords of like structure with the minor, and, when continued further
in both directions, must also lead to like — on the subdominant side
to minor triads, on the dominant side to major. Therefore, for join-
DIMINISHED TRIADS OF KEY-SYSTEM IN TRANSIT 33
ing the limits of its system, either stretching out or closed, it can
only contain the same chords as the system of the minor key ; for
in them the dominant chords alone have share.
SCALE OF THE MAJOR KEY.
63. The ancient, now somewhat antiquated, dispute or doubt,
whether harmony or melody has precedence in music and must be
taken to have arisen earlier, keeps about equal pace with that
other, whether the chicken comes first, or the egg. That practical
music had historically to begin with melody, one-part song, it is safe
to assume ; but it is also certain that all melodic intervals are only
harmonic determinations, and that these neither are, nor can be,
other than what we have pointed out above. Even a child singing
has in its unconscious feeling nothing for determining the intervals
of its artless song, but the Octave, Fifth, and Third ; every note of a
melody is one of these three intervals to a unity that connects the
melodic notes.
64. First we can think of the melodic principle abstractly, as
what moves ; opposite to it the harmonic principle as what fixes,
The former, also, as the tendency to go out of a subsisting state,
but with no further determinations in itself; these it gets from
the harmonic elements.
65. If we imagine a sound gradually rising from the tonic of the
major key-system, and if we regard its starting-point as the first
degree, then its second degree, as a harmonic-melodic determina-
tion, will be found in the Fifth of the dominant, which is the Second
of the tonic ; the third, in the Third of the tonic ; the fourth, in the
Root of the subdominant, as Fourth of the tonic ; the fifth, in the
Fifth of the tonic ; the sixth, in the Third of the subdominant, as
D
34 HARMONY
Sixth of the tonic ; the seventh, in the Third of the dominant ; the
eighth, in the Octave of the tonic itself. This is the series in which
the ascending motion of a sound in itself undetermined meets on
its way the intervals of the key, and by them is determined into
Degrees.
66. The scale makes the harmonic intervals appear in its
degrees in an order that with each new element of the succession
contradicts the notion of simultaneousness. The second degree
belongs to a different triad to the first, the third to a different
one td the second, and so on. But it is just this that corre-
sponds to the essential meaning of the notion of succession, which
requires a one-after-the-other — i.e. after one, another. But for the
one-after-the-other to be a real connected succession, there must be,
besides its difference, also a unity, a common, binding element ;
which, if the transition be pictured as happening in space of time,
as being the end of one, is made also the beginning of the other.
67. For the first progression of a Second in the scale of C major,
from C to D, the connecting unity is contained in the note G. G is
at first the Fifth determined from C, and then becomes the Root de-
termining D. The melodic progression here is in fact intelligible
only as an expression for the transformation which goes on in G,
out of one meaning into the opposite one. In the next progres-
sion of a Second, from D to e, G passes out of Root-meaning back
into Fifth. The step e-F is determined in like manner upon
the Root C. It is the same with the steps F-G, and G-a\ in
these progressions C changes between Root- and fifth-meaning.
But from the sixth degree to the seventh, from a to b, in so far as
the two notes are contained in the key as Thirds of the subdominant
and dominant chords, such a connecting note to explain the passage
is not to be found. For the triads of the subdominant and dominant
are disjunct ; they have no common element by whose transforma-
tion the step a--b could be given. Therefore between these two
SCALE OF THE MAJOR KEY 35
notes, referred to these two chords, there may be felt a division,
which makes the passage difficult ; for it is in fact not to be called
a passage, but rather a leap. The distance between these two
notes, in their quality of Thirds of the subdominant and dominant
triads, seems to be greater than that of the previous steps of a
Second ; and yet it is equal to the distance between the first
and second, or between the fourth and fifth degrees : described by
the ratio of vibrations it is 8 : 9. But these ratios of numbers
throw no light on the meaning of the intervals. We cannot pitch
the Seconds C» D and F»G by the ratio 8 : 9, nor the Seconds
D'-e and G-a by the ratio 9: 10, nor yet e-'Fand b-C\>y
15 : 1 6. Indifferent to the measure of the outward distance, be
it greater or smaller, we get them determined only through change
in the meaning of a connecting member. And so too the step from
the sixth degree to the seventh can be yielded as intelligible suc-
cession only by means of some such mediation.
68. Here too a mediation is found ; not indeed in the uncon-
nected principal chords, but in the chords of secondary order,
namely, in the two conjunct minor triads of the system, which
have the Third of the tonic for a common note : a is Root to e, and e
determines b as Fifth. Therefore the succession a • - b is made
possible by the change of e out of the meaning of Fifth into that of
Root. The last step b - • C is referred to the same note ; e then
returning to Fifth- meaning. The last passage might indeed also be
given through the note G ; here, however, for the succession of the
three last notes the first meaning is the one of principal account.
69. Thus the whole scale is formed : in its first, second, and third
degrees, on the Fifth ; in its fourth, fifth, and sixth, on the Root : in
its six th, seventh, and eighth, on the Third of the chord of the tonic ;
each of these three elements of the principal triad strikes out of
Fifth -meaning into that of Root and then back again to Fifth-
meaning as at first.
HARMONY
70. The discontinuity which has always been felt between the
two notes of the sixth and seventh degrees of the scale, supposing
them to be referred to the subdominant and dominant chords, and
the obstacle thereby opposed to their following one upon the other,
was the reason why in the earlier solmisation it was not attempted to
group together more than the first six notes of the scale, being those
which afford a continuous succession. Then further progress was
gained by letting the so-called hexachord, with its well-known
syllables ut re mi fa sol la, begin with either the Root, or the Fifth
below, or the Fifth above, of the key :
15 : 16 8:9 8 : 9 : 10
8 : 9 : 10 8 : 9 : 10 15 : 16
CDeFGabCD 6--.
ut re mi fa sol la
ut re mi fa sol la
(A)
ut re mi fa sol la
8 : 9 : 10
ut re mi---
But in this way, supposing the hexachord starts from the Fifth
above, the fifth, sixth, and seventh degrees of the scale of the Octave
appear as first, second, and third of the hexachord, ut re mi ; and
the sixth and seventh are no more in the ratio of 8 : 9, but in the
ratio of 9 : 10 like D--e. The sixth note of the original scale is
SCALE OF THE MAJOR KEY 37
thus no longer a the Third of the subdominant, but A the Fifth of
D ; whereby the scale reaches out into the territory of the key of G
major. But if the hexachord begins with the Fifth below, then the
sixth and seventh degrees of the scale of the Octave become third
and fourth of the hexachord, and the progression mi • -fa is then
that of a minor Second a-BV, whereby again a new key is
touched, that of the subdominant, F major. Therefore, while the
sixth degree of the scale of the Octave is pitched as a, when it means
la or mi of the hexachord, and as A when it means re ; so the
seventh degree varies, accordingly as it gets the meaning of third or
fourth, la or mi, of the hexachord scale, between b and B t>, between
B durus and B mollis. The first, agreeably to its * hardness,' was
drawn square, b , B quadratum, a character related to, and meaning
the same as, "|~[" and t| ; from the last of which probably has come the
h introduced in the German notation only, and standing in the
succession of notes quite out of alphabetical order. It is seen that
the discontinuous juxtaposition of the Third of the subdominant
and the Third of the dominant, which is found interrupting the
progression in the Octave scale, could not occur in the hexachord
system, nor could singers have been encouraged to attempt it.
71. Through the connexion explained above, which takes place
by the minor triads of the key, it is indeed made possible for these
notes to succeed one another. But the meaning which they have
as intervals of the minor chords is only a secondary one. And
here their principal meaning as Thirds of the subdominant and do-
minant triads will all the more count, because the sixth degree
following upon the fifth enters with Third-meaning already. It might
seem that a determination for the passage from this point to the
Third of the dominant could be found in the diminished triad of
the seventh degree ; but then we have only to remind ourselves that
this chord is itself one of twofoldness or division, and that the name
of triad is given it, not as meaning a concrete unity, but only as to
HARMONY
a combination of three notes. A real connexion for the succession
of those two degrees is only given by the Third of the tonic.
72. The descending scale is determined by the same conditions
of succession as the ascending, and contains accordingly the same
series of notes in reversed order. If in the ascending scale we must
take force, manifested in the rising pitch, to be that moving or
melodic principle of direction which is by the harmonic elements
determined into degrees ; so now it is weight, drawing downwards
and deepening, to which is due the formation of the melodic series
in the reversed direction.
73. By the expression melodic, in the meaning which is here in-
tended, there will always be understood successive onward motion
of sound tending upwards or downwards. In melodic succession,
even of intervals that are harmonically simultaneous, the voice has
to go over all that lies between in its harmonic elements, in order
to reach the more remote interval. The progression F»b as Fourth
in the key of C major, the so-called Tritone, contains the same
difficulty of passage, as that from the sixth degree to the seventh,
where these are taken as Thirds of the subdominant and the do-
minant ; although here there is no change of chord, because both
notes belong to the chord b — D\F. But the change is contained
in the melodic passage, which can only take place through the inter-
mediate space with all its harmonic determinations ; here therefore
through F- - G • - a • • b, where the division between a - • b stands again
in the way, as unmelodic. The same notes b:-F, as diminished Fifth,
offer no hindrance to melodic succession ; because the passage
&-• C- D-e-F is continuous in all intermediate elements. All
augmented intervals will be found for this reason unmelodic, but in
their inversion as diminished intervals they will be melodic, i.e.
continuous.
SCALE OF THE MINOR KEY 3g
SCALE OF THE MINOR KEY.
74. The scale of the major key is a successive presentation of
the harmonic determinations of the major key-system, in which
it is completely contained. Each melodic degree is determined
by a harmonic element out of the system closed off in itself.
75. The minor scale up to its sixth degree can be formed quite
in the same way as the major, because in it, too, the first three
degrees are made continuous by the Fifth, and the following three
by the Root. Now in the major scale the sixth and seventh
degrees were at first shown divided ; and it was only in virtue of a
subordinate connexion of chords that a succession of the two
degrees was made possible. So too in the minor scale we come
upon the same division in the same place ; but here we are not
offered the same means for a union, even for one of subordi-
nate meaning, as in the system of the major key. In the major
system the Third of the subdominant can be formed into continuous
succession with the Third of the dominant by means of the Third of
the tonic, which stands to the former in the relation of Fifth, to the
latter in the relation of Root. But in the minor system this inter-
mediate member is not present so as to form a connecting link,
because the minor Third of the tonic does not stand to the major
Third of the dominant in the relation of Fifth. Rather, the aug-
mented Fifth between the two notes expresses most marked sepa-
rateness : determination of a positive Root as negative simultaneously.
Thus a melodic connexion of these two degrees is in no way
granted in the minor system. It is impossible to pass in continu-
ous progression from the sixth degree, as minor Third of the sub-
dominant triad, to the seventh, as major Third of the dominant
triad. That stands in melodic connexion only with the fifth
degree ; this only with the eighth.
40 HARMONY
76. If the seventh degree, the Third of the dominant, is to be
reached, and further progress in general made possible, then the
fifth degree must be followed by a note other than the sixth of the
key. This must be one lying outside the system and connecting
the fifth and seventh degrees, and can be no other than the Fifth of
the Fifth of the dominant chord ; which as sixth degree forms the
passage to the seventh of the key, because now the fifth, sixth, and
seventh are given by transformation of the Fifth of the dominant
triad.
77. In the C minor key-system the melodic succession can
move on through C- D -e^-- F •> G in unimpeded connexion;
the first three degrees being made upon the dominant G, the last
three upon the tonic C, as in the major system. But if after the
fifth degree G, we take a\>, which follows still based upon C, as
sixth, then from this point return to G is alone possible, but not
advance to b. For the triad G — b — D, to which b belongs as its
Third, is not connected with the triad F—a\> — C, whose minor Third
a\> has entered as sixth degree, by any common note through which
the passage could be made intelligible. The connecting link between
G and b can only be determined by the Fifth of the dominant, Z>,
whose Fifth A provides the passage from G to b ; and consequently
the note A, lying out of the system though it does, will take its
place in the scale as sixth degree, after which the seventh and
eighth follow in unimpeded succession.
78. Now if in the ascending minor scale progress was impeded
from the sixth degree to the seventh, then in the descending scale
there will also be no connexion found between the seventh and
sixth degrees. As there the minor Sixth could not form the passage
to the major Seventh, so here the major Seventh cannot lead into
the minor Sixth. The Octave, however, finds a note to conduct it
to the minor Sixth, again outside the system, but this time upon the
subdominant side. While in ascending the Fifth of the dominant
SCALE OF THE MINOR KEY 41
had to become Root, in descending the Root of the subdominant
must become Fifth ; the former change provided the intermediate
step to the major Seventh, the latter change provides the step to
the minor Sixth degree.
79. In the C minor system the melodic progression ascending
from the fifth degree was found in the succession :
G
the first three notes as determinations upon the Fifth of the do-
minant, the two last upon the dominant itself. Here the passage
from G to b, which a\> did not furnish, had to be formed by another
middle member, A. Descending, a continuous passage has to be
found from C to a\>, which is not possible with b. Therefore C and
a\> are referred to the subdominant triad, upon the Root of which,
Fy Fifth-determination passes then by means of B\>. Thus the
descending succession will be :
C
the three first notes determined upon the Root of the subdo-
minant chord, the two last upon the tonic.
The whole scale of C minor, ascending and descending, consists
accordingly of the successions :
C-D.-eb--F-.G-- A- -b-.Q
G D F G
(ascending) (descending).
80. That here or elsewhere there can be no mention of degrees
42 HARMONY
arbitrarily sharpened or flattened, need not be said or repeated after
getting thus far. Again, it lies in the notion of the key-system, that
the major Sixth of the ascending minor scale cannot be major Third
of the subdominant triad, nor the minor Seventh of the descending
minor scale minor Third of the dominant chord ; for both are by the
organisation of the system impossible, they contradict its funda-
mental conditions.
8 1. This account of the construction of the minor scale in its
three last degrees has been compressed as much as was possible,
and yet has proved lengthy. But the thing itself has only been
given in strict necessity, as the course of degrees formed in the
nearest possible connexion. The gap of the major key is linked by
the middle of the system ; in the minor key it is linked by the two
ends. In this linked succession; the minor key again puts forth its
divided nature ; while in the linked succession of the major key
there is expressed the nature of unity.
SCALE OF THE MINOR-MAJOR KEY.
82. The scale of the minor-major key ascending will move like
the major scale through the tonic major Third up to the Fifth ; its
progress beyond will be that of the minor scale. It has no major
Third on the subdominant, and therefore in its last degrees requires
the same connexion by the Fifth of the dominant. And in descend-
ing, as with the minor scale, its passage can only be made continuous
by means of the subdominant Root.
The major scale being formed of the series of notes
SCALE OF THE MINOR-MAJOR KEY
43
then the minor-major scale compared with it in
has A for its sixth degree ascending, B\> for its seventh descend -
ing.
83. The following is a representation of the melodic succession
of the scales according to their harmonic determinations, which may
serve for a general view of the exposition above given :
Harmonic Determination for the Melodic Succession in the Major
Scale.
I— III— II
I— III— II
F a C e G b D
Harmonic Determination for the Melodic Succession in the Minor and
in the Minor-Major Scales.
II_IH_I I— III— II
II I II— III— I I II
Bb (db) F ab C eb G b D (f#) A
(I— HI— II)
84. The minor key has sometimes been called an ' artificial '
one, in opposition to the major, in that case called * natural.' In
44 HARMONY
the first place it is difficult to see what can have been meant by
this expression used to describe a system so directly rooted in feel-
ing, and one in which so many popular songs move. But secondly,
the system of the major key is no more naturally given than that of
the minor key is artificially made. Both are forms humanly ani-
mate and self-generating, i.e. reasonable being and coming-to-be in
sound and determinations of sound ; something higher than
' naturally given ' or * artificially made.'
85. Nature gives determinate notes in a series, which indeed
includes the elements of the triad among its members, but not in the
sense of a determination complete in itself, in which sense alone
it can have musical value for us. We must come to the infinite
progression of the natural series of notes having already in our
mind the notion of the chord, if we want to find out the members
in it which belong to the triad. But again the progression soon
goes beyond what belongs to the chord and has intelligible
meaning in harmony. Now if not even the triad is given in the
natural series as particular determination, much less is the system
of the key so given. For even by its material contents, because
it contains an element formed backwards (the subdominant chord),
the key-system cannot be given in a series which naturally is
formed only forwards. The arithmetical note-progression starting
from C, even if continued to infinity, will never generate the note Ft
nor its Third a. These are no more possible for it than are e\> and
tf t>, Thirds of the minor key.
CHORD SUCCESSION 45
CHORD SUCCESSION.
86. In the scale we considered a sound rising from below up-
wards, whose progress, in itself unbroken, is divided into degrees at
the points where it meets the harmonic elements of the key ; and
we have shown how this is done, both in the major key and in the
minor, preserving continuity of succession. The elements of the
chords were taken as determining the degrees ; but the order in
which they succeed one another was given by the assumed direction,
ascending or descending, of the moving sound. In chord succession,
which consists of a simultaneous advance of several parts, other
conditions of melodic movement will enter. We now get a Harmony
of successions as a Succession of harmonies, and thereby again oppo-
sites made into one, the notion in its essentiality of all that is real :
that is, we have the higher Third-notion of real harmony, whose
Fifth-notion had to deal with the opposites separated ; for previously
we have only had chords determined in themselves, and melodic
progression determined in itself as the scale.
87. The succession of two triads is again only intelligible in so
far as both can be referred to a common element which changes
meaning during the passage.
88. Two triads can be different : (a) in one note ; (fr) in two notes ;
(c) in all three notes. Starting from the middle of the major system,
from the tonic triad, the triads which differ from the first in one
note will be the two minor chords of the key ; in two notes, the
subdominant and dominant chords ; in three notes, the two dimi-
nished or limit-joining chords.
In passing from the tonic to one of the minor tfiads, of the
three parts which form the chord only one will have to move
melodically, while the other two remain, changing the harmonic
meaning of their notes.
46
HARMONY
The passage from the tonic to the subdominant or to the domin-
ant triad makes two parts move melodically ; the third part re-
mains, receiving a new harmonic meaning.
In the passage from the tonic into one of the diminished triads,
all three parts move; and of them one must spring through a
harmonic interval to a note serving to connect the chords, the other
two receive the melodic progression of a Second.
The first and second of these kinds of progression connecting
chords are self-evident, so far as is now necessary, being those
which lie nearest to hand. The third requires explanation.
89. Two triads lying wholly outside each other (such, namely,
as have no common connecting note whose transformation into
another meaning might give the understanding of the passage),
require to be mediated by that triad, lying between the two, of
which the first of the two unconnected triads contains two notes,
and the other one note. And the passage from the first into the
second cannot take place otherwise than in so far as the first
has already this preponderance of community with the inter-
mediate triad, and may therefore be put for it. Or, the pro-
gression from the first of the unconnected triads to the second
is the same as it would be from the mediating triad to the
second.
90. In the system of the C major key
D / F— a— C— e— G— b— D / F
the diminished triads D\F—a and d—D/Fare separated from the
triad of the tonic C— * — G, and therefore the passage from the
latter to either of the former is only possible by the intervention
of a connecting link. But the tonic triad contains two notes of
each of the two minor triads; and again, the minor triads, each in
its own direction, are joined to the corresponding diminished triads
by one common note :
CHORD SUCCESSION 47
C— e— G C— e— G
a— C—e e— G— b
D/F— a b— D/F,
and the passage from the triad C — e—G to DjF — a must here be
taken to be equivalent to the passage from a — C—e to D\F—a,
and the passage from C — e — G to b — D/F equivalent to that from
e—G—b to b—D\F.
91. Thus the three kinds of harmonic melodic triad progression
-within the C major key, starting from the chord of the tonic, will be :
I. To the triads with two common notes, the two minor chords :
From C — e — G to a — C — ey
in the position C — e — a ( J ,
from C—e — G to e — G — b,
in the position b — e — G ( J.
II. To the triads with one common note, the subdominant and
dominant chords :
From C — e — G to F — a — C,
in the position C — F — a I J,
from C — e — G to G — b — D,
in the position b — D — G fj.
III. To the Thirds without common note, the two diminished
triads :
From C—e — G to D — F — a, as if
(*-C-e)
from a — C — e to D — F — a, therefore
in the position a — D — F ( J.
48 HARMONY
From C—e—G to b—D—F, as if
(e-G-b)
from e — G — b to b — D — Ft therefore
in the position D — F— b fj.
92. A second chord, or chord of succession, in every kind of
mediated progression, supposing the first to have appeared in
primary triad form, will assume a position of its intervals different
to the primary one. It will be either a chord of the Sixth-and-Third,
or of the Sixth-and -Fourth ; for its position is not independent, but
conditioned by the succession.
93. If the triad progression is to be carried on further from
these secondary positions of the chords, and if the triad next fol-
lowing is related in two notes or in one note, then the melodic pro-
gression of the parts is self-evident. For the portion common to
the two chords remains in its place, and the different portion can be
reached by progression ascending or descending through a Second.
But if the following triad be disjunct, then the secondary chord
must itself first be referred to some primary chord related to
the new chord to be taken ; and the progression from the secondary
to the new chord can only take place as if from that primary chord,
which the secondary is considered to follow.
94. Now a secondary chord can always be derived from two
different primary chords ; first from that which has the lowest note
of the secondary as its Root, and next from that which has the
highest note of the secondary as its Fifth. E.g. the chord of the
Sixth C—e — a can have arisen either from the triad C— e — G, or from
the triad D\F—a ; the chord of the Sixth-and-Fourth b — e — G from
the triad b—D\F, or from the triad C—e—G. The triad which is
to follow, and which is by hypothesis disjunct from the secondary
chord, will in each case decide which of the two derivations is to
be taken.
95. In the third kind of the above progressions, from the tonic
CHORD SUCCESSION 49
triad to the diminished triads, from C — e — G to D\F — a and
b — D\F (by which for the first there ensues the position a — D — Fy
for the second the position D — F — £), it is true that the position
of Sixth-and- Fourth a — D — F has as a fact been produced from
the primary a — C — e, and the position of Sixth-and-Third
D — F — b from e — G — b. But these derivations are not in them-
selves determined by the secondary chord forms, which can
equally be referred in a — D — F to the primary triad b — DfFt and
in D — F — b to the primary triad D\F — a. For the passages from
these triads taken as primary bring out the same secondary posi-
tions of the two chords as we found for them from a — C — e and
e—G—b.
96. The existence of this double derivation of every secondary
position of a chord furnishes the mediation for the progression from
it to the disjunct triads on either side.
97. The disjunct triads on each side ofD/F — a are now C — e — G
and e — G — b\ those on each side of b — D[F are a — C — e and
C — e — G. From the position a — D — F, mediated through a — C — £,
there ensues for the C major triad the Six- Four position G — C — e,
and for the e minor triad, mediated through b — D\Fy the Six-Four
position b — e — G. But the last triad can also be mediated by
a — C — e> whereupon the Six-Three position G — b — e is obtained
for the same chord. From the position D — F — b, mediated through
DjF — a, is produced the a minor triad in the Six-Three position,
and mediated through e — G — b, the C major triad also in the Six-
Three position e — G — C. Here too the a minor triad can also be
mediated through e — G — b, and receives then the Six-Four position
98. We see that the diminished triads brought from their
secondary position back to the triad of the tonic, from which they
came, can neither of them lead again to the primary position of
that chord ; and manifestly a triad in primary (or root-) position can
E
50 HARMONY
never be followed by a conjunct triad also in root-position. And
if two triads can never follow immediately one upon the other in
primary form (which would in fact contradict the notion of following),
but in every case from a primary chord proceeds a secondary, and
from a secondary a primary or another secondary, then not only is
it impossible for two parallel Fifths to follow one upon the other in
mediated progression, but also the succession of so-called hidden
Fifths, the progression of two parts by similar motion to the Fifth,
cannot occur in a strictly mediated connexion of chords.
99. The prohibition of Fifths, which causes such perplexity to
the beginner not yet clear in harmony and to the amateur, and so
often turns their finest inventions to water, is unnecessary for the
master of harmonic phrase. Given right feeling of what progression
is, and parallel Fifths are self-excluded. Where there is a parallel
Fifth, hidden ever so carefully, the meaning will always sound
through, that here is a second triad trying to make itself again
beginning against a first which is placed beginning. ' This selfishness
of the chord destroys the unity of the phrase. It is ^forbidden to
write consecutive Fifths and Octaves ; with equal right, since both
are of bad effect. But the cause of the bad effect is not the same
in both cases : in the succession of Fifths we miss unity of harmony,
in the succession of Octaves difference of melody. Therefore to
double in Octaves two parts which make no claim to difference is
always permissible ; but to progress in parallel Fifths never, for
unconnected harmonies cannot but be foreign to rational artistic
design. However, this can be said in such strictness only of an
immediate succession of true Fifths, where the parts progress
through a Second and the notes have chord-meaning. Such a
succession does not occur in clear and correct phrase. To admit
its lawfulness when thrust away under many parts is the same as
to defend a lie told under compulsion.
100. The succession of chords, as presented above, is still con-
CHORD SUCCESSION
fined to the linking of harmonies, and exhibits the triads merely in
the abstract sense of following one upon another, according as one
arises out of another. But every chord, the position of whose
intervals has been conditioned by a preceding chord, must, when
present, also put in its own claim to a dignity of independence, a firm
footing for itself. This it gets by the Root as basis or bass, and
may also have it in the Third placed as lowest note ; because the
Third, comprehending in its essential meaning both Root and Fifth,
contains the former, although in combination. But a triad-harmony,
in which the Fifth is lowest or bass note, has not this independence.
For the Fifth is just the decided opposite of the Root, and, placed
in the bass, will therefore mark the chord as decidedly not having a
footing of its own.
101. A succession of chords which, starting from the triad, is
continued in three parts, will therefore need a fourth part to serve
as basis for the chords ; that so a foundation for the independent
presence of each of the members of the succession may be provided,
wherever such foundation is not contained in the position of the
chord necessitated by the succession. But now, after that the
notion of succession has been received, there can be no mention
made of providing single chords which lack foundation with bass
notes having no connexion between themselves. That would offend
against the notion of succession, which admits of nothing isolated.
Rather this part, while having its own relation to the others, must
also in itself answer to the conditions of correct progression.
102. It has been said that, besides the Root, the Third of the triad
can also serve as lowest or bass note to a chord, but not the Fifth, as
being the exact opposite of the Root. Therefore those of the chord-
connexions shown above, in which the Six-Three position appears,
but not the Six-Four position, do not necessarily need foundation
upon a fourth part. They already form in three-part harmony a
phrase in which each chord can maintain itself in the shape which
E 2
52 HARMONY
it takes in the progression. Therefore the succession C—e—G
...2)—F—b"-e—G—C is admissible without a fourth part But
not the succession C—e— G—a— D— F—G— C— e ; because the
second and third chords contain as lowest part the Fifth, which
is not suited for bass. Here a fourth part is required to add the
Root or Third underneath, that the chords may be made to tread
firmly. To avoid like progression with parts already present, it
will take as lowest the course C"D--e\ and the phrase of this
succession is therefore in four parts : C — C — e — G- --D — a — D — F
...e—G—C—e.
103. The strict phrase of successive harmony, even when we
regard it in a succession only of triads, for the present neglecting
the four-part chord of the Seventh, is thus essentially four-part.
It is a union of four melodic series, of which three are given by the
triads passing into one another, while the fourth provides with a
basis the chords not based in the passage.
104. In formal self-determination such as this, by which a
succession of chords may grow only under bound necessity, shooting
out one might even say like a mineral crystallisation, without any
freedom or choice, there would indeed be offered a very cramped
material for musical composition. Its productions in these fetters
would be like the Egyptian sculptures, of which the proportions
were prescribed with such strict precision, that two statues of equal
height, finished by different sculptors, had also to be exactly the
same in all their parts. But what is here shown is only the very
directest and nearest union of chords, as it would be formed
obeying the inner law of succession alone, without the intervention
of any other determination whatsoever. The organism being first
framed according to law, afterwards admits of a freer, nay, of the
freest movement of its limbs inside the regularity. But now it is
the regularity that we are principally concerned with, to find it out
and observe what in the very first place it demands. Its formation
CHORD SUCCESSION 53
under other conditions will be understood the easier when we know
the direct requirements.
105. Here the passage into conjunct and disjunct triads has
been considered starting only from the triad of the tonic. But
in the continued series each subordinate triad too may appear in
primary form, as, e.g., in the series C — e — G- • • C — e — a- • • C — F — a- • •
- - -F — a — C, and so on, where the triads D\F — a, e — G — b, F — a — C
appear in the first position, as well as the tonic triad C — e — G, from
which the series starts ; therefore each subordinate triad may as
primary also become the starting-point, and the passages into the
other triads will then be formed, in the analogy of the relationship,
quite like those which start from the triad of the tonic : the passage
from D\F—a to F—a—C like that from C—e—G to e—G—b,
from D\F — a to e — G — b like that from C — e — G to D\F — a, and
so on.
1 06. In the minor key, supposing the chord-union goes on in-
side the system and may not, as with the scales, reach out beyond it,
between the Third of the subdominant and the Third of the do-
minant the progression will always be met by the impediment of the
melodically discontinuous augmented Second. It cannot be gone
round, but must be overleapt, and stamps the nature of the system in
itself conceived in inner disunion. The passage from C — F — a\> to
the triad of the seventh degree b — D\F can only lead to the posi-
tion D — F—b. Proceeding from C— F — a\> to the primary position
b — DIP would avoid the step of the augmented Second ; but
this form of chord-succession contains no inner union ; as the
hidden Fifths of the outside parts from C -- a\> to b -- F prove.
54 HARMONY
DISSONANCE.
107. Dissonance is melodic succession sounded simultaneously.
If in the C major key the note C should be followed by the notes
e, Fy G, or a, then we do not call it a melodic succession in the sense
here intended ; because each of these different notes forms with C
a triad interval in direct or inverted position, and therefore has to
it essentially harmonic meaning. Only the Second then, ascending
or descending, can be counted an essentially melodic interval, and
that in its meaning of succession, as we have seen it in the scale.
The Second, both as simultaneous sound and as succession, is not
a directly intelligible interval. By the ratios 8:9, 9 : 10, and
15 : 1 6 the feeling has no determination given to it for pitching
any one of these distances ; any more than we can find the intona-
tion for many ratios lying between, as 6 : 7, 7 : 8, ro : u, n : 12,
12 : 13, 14 : 15. But it has already been made evident in the
scale that the ratios of these outward distances do not at all come
into question in determining Seconds, and that this determination
is solely and wholly brought about by the transformation in mean-
ing of a third note.
1 08. The progression from the first degree of the scale to the
second is determined upon the dominant, which passes out of Fifth-
meaning into Root-meaning. If then both degrees are heard at
once, or if the first continues sounding against the second when
that has entered, the harmonic meaning of the Second-interval with
regard to the dominant will be : that the dominant is simultaneously
Root and Fifth. This is a contradiction if the double meaning is
taken as persistent. It may, however, be contained in the note as a
passing meaning, supposing that in passing from one meaning to
the other the note gives up the first, not simultaneously with the
actual passage, but later. Consequently dissonance requires a time
DISSONANCE
55
precedent and a time subsequent for the justification of its exist-
ence, namely, a precedent time of preparation and a subsequent
time of resolution.
109. In the explanation just given, dissonance has not yet
appeared in the meaning of Seventh in a chord of the Seventh.
Rather we recognise here the so-called suspension. Nevertheless
the determinations which are universally valid for all cases of dis-
sonance in simultaneous sounds are already contained therein : that
(1) a dissonance can only be produced from a succession, and that
(2) wherever a dissonance occurs, the understanding of the dis-
sonant interval is to be found, not in the immediate relation of the
two notes which sound dissonant to one another, but in an element
lying outside of them, which by their simultaneous sounding is
determined to twoness.
CHORD OF THE SEVENTH.
no. The chord of the Seventh is the sounding together of two
triads joined by a common interval. It is formed by the passage
from one to the other, so that the first persists along with the
second.
in. The triads lying nearest the principal triad and joined to
it by two notes are the minor triads of the key. The passage out
of the C major triad into the a minor triad, keeping the first on
with the second, gives the chord of the Seventh a — C — e — G in
the position of Six-Five-Three, C— e— G— a. The passage from
the C major triad into the e minor triad under the same condi-
tions gives the chord of the Seventh C — e — G — b in the posi-
tion of Six-Four-Two, b—C—e—G. For there G goes to <z, here C
goes to b.
HARMONY
Here we have the sounding together of each of two successions,
where chord-progression is meant ; where we had it before (par.
108), it was in the meaning of mere melodic note-succession. For
instance, in the sounding together of the Second C—D, the sense of
the dissonance is, that G is determined at once as Fifth and Root.
But here it is, that the middle interval of the chord of the Seventh
(C—e in the chord a — C — e — G, and e — G in the chord C — e — G — b)
has the double determination of belonging at one time to different
triads, and, as of course follows, to each triad in other meaning.
;por £ — e is Root and Third in the C major triad, and Third and
Fifth in the a minor triad ; e — G is Third and Fifth in the C major
triad, and Root and Third in the e minor triad.
112. In the dissonance of the suspension the twoness of meaning
is contained in a doubly determined note ; in the dissonance of the
chord of the Seventh in a doubly determined interval. In the chord
of suspension G — C — D the note G stands in its double meaning
in contradiction with itself; and in the chord of the Seventh a — C—
e — G the middle interval C — e contradicts itself in its different
determination with respect to the two joined triads, and similarly
the interval e — G in the chord of the Seventh C — e — G-^-b.
113. Generally, then, to the notion of dissonance, as being in
itself an opposition, there must be again attributed the meaning of
Fifth, after the universal sense of the interval ; the consonance
which forms the preparation for it has the meaning of Octave ; and
the consonance re-established after resolution the meaning of Third.
Thus harmony gains with dissonance its perfect notion of conson-
ance ; for without dissonance consonance remains fixed in the
immediacy of Octave unity, and cannot reach recognition of itself
in the notion of the Third.
114. But before we speak of the resolution of dissonance, we
have to consider whether chords of the Seventh can result also from
other kinds of triad-progression.
CHORD OF THE SEVENTH 57
115. We have seen that chords of the Seventh are formed when
the progression from the tonic triad to one or other of the adjacent
minor triads is taken sounding all together, and similar chords
will be produced if, starting from any other triad of the key, the
passage from it into the next adjacent triads above or below is taken
sounding all at once. Thus the joined succession of a — C — e -••
F — a — C appears as the chord of the Seventh F — a — C — e in the
form a — C — e — F ; F — a — C joined with D\F — a, as the chord of
\jthe Seventh D\F — a — C in the form F — a — C — D ; or e — G — b
joined with G — b — D, as the chord of the Seventh e — G — b — D
in the form D — e — G — b ; G — b — D joined with b — D\F, as
thte chord of the Seventh G— b — D\F in the form F—G—b—D ;
ana so on.
116. Only those triads which have a harmonic unity, i.e. a
common interval, can be taken together at one time : therefore only
two triads which are related in two notes. For the passage into
the nearest is the only immediately intelligible progression. The
passage from C — e — G to F — a — Cy which leads to the position
C — F — #, is a compounded one, and consists of the progressions
C — e — G--C — e — a-~C — F — a. Both progressions can happen at
once, but the second cannot happen before the first or without the
first (C — e — G--C — F — G\ as the first can before the second
or without the second. Similarly with the succession from
C — e — G to G — b — D which (in b — D — G) is compounded of the
successions C — e — G--b — e — G--b — D — G. There the passage
must lead through the a minor triad, here through the e minor triad.
If we wished to think of an immediate passage from the triad of
the tonic to the subdpminant or dominant triad and to take it all
together in a single chord, then the first — to the subdominant—
would be heard in C—e — F — G — a sounding all at once, and the
other — to the dominant — in C — e — G — b — D. The former contains
the union of .F — a — C — e— G in the form of the immediate succession
58 HARMONY
of the two triads C—e—G and F—a—C', the latter in like manner
the union of C — e — G — b — D as immediate succession of the triads
C e G and G — b — D. The untruth of such a process is at
once expressed as discordance in the combinations C — e — F — G — a
and b—C—D—e—G.
The way in which such groups as F — a — C — e — G and
C—e — G — b — D can have intelligible meaning, under particular
conditions as to the position of their intervals, as so-called
'chords of the Ninth,' does not belong here.
117. The passage from the triad of the tonic to a dominant or
subdominant chord takes place through the intermediate minor
triad : from C—e — G to F — a — C through a — C — e ; therefore in the
progression C — e — G--C — e — a~-C — F — a. Here the first step in
the voices is £••#, and the second is e- F. F makes G impossible
in the union of triads, but not e, for £ belongs to the triad a — C — e,
which is conjunct with F — a — C. Therefore the. passage from
C — e — G to F — a — C can be shown, consistently with right progres-
sion, only in the chord C — e — F — a formed by -taking together the
last members of the succession C — e—a and C — F — a. Similarly
the passage from the triad of the tonic to that of the dominant,
from C—e — G to G — b — D, mediated by the minor triad e — G — b,
in the chord b — D — e — G.
1 1 8. We see that passages even into distant-lying triads,
when shaped into harmony of the Seventh, can only take triads
together which are closely joined. The passage from C — e — G
to F — a — Ogives for its chord of the Seventh the union of the
triads a — C—e and F—a — £*, and the passage from C — e — G to
G — b — D for its chord of the Seventh the union of the triads
e—G—b and G—b—D.
119. It may have been perceived from the way in which har-
monies of the Seventh arise, so far as we have yet gone, that one
note cannot by ascending move on to another, that lasts, so as to
CHORD OF THE SEVENTH 59
be dissonant with it, namely, as if the Seventh should enter to
the Root in ascending motion ; but that the Seventh issues in
descending motion from the same note that must become Root of
the chord of the Seventh. For when the Seventh enters later than
the Root — that is, when the upper triad is joined to the lower—
the progression which is given by the natural succession happens
so that the Root of the lower triad moves downwards into the
Fifth of the upper (which forms the Seventh) as into the
degree melodically nearest. But when the dissonance is pro-
duced by ascending motion, i.e. when the lower triad follows the
upper, then the note, which enters in dissonance with the note
that lasts, can only be Root of the lower triad, and the Root also
of the chord of the Seventh. Therefore when the empirical rule
says that the Seventh, when not prepared, shall only be struck
after the Root or its Octave, that is quite an organic requirement.
The chords of the Seventh in which the diminished triads share
are excepted from this rule. We shall see that the reason of
the exception may be apprehended as easily as the reason of the
rule.
1 20. Generally there is no rule, which has not its reason in
some law of organism. But the rule does not trouble itself to
show the reason of what it orders, is often, indeed, unconscious of it;
and since it has in eye only the outward show and not the essence
of the thing, so for every differing side of the phenomenon, it is itself
different. But organic law is the soul, the inward living unity
itself : it does not receive its determinations after the outward show ;
rather it produces them.
121. Triad-progression in itself we found to be of three kinds
or grades of relationship ; it must now be shown to be triple with
respect to combinations into chords of the Seventh. '
A triad can pass
(i) Into another triad lying next it, i.e. joined to it by two
6o HARMONY
common notes ; e.g. the tonic triad into one of the two minor
triads :
C— e— G-.-C— e— a, C— e— G»-b— e— G.
(2) Into another joined to it by one common note ; the tonic
triad into the dominant or subdominant triad :
C— e— G- - -C— F— a, C— e— G • • -b— D— G.
(3) Into one wholly separate ; the tonic triad into one of the
two diminished triads :
C— e— G- • -a— D— F, C— e— G- • -D— F— b.
How the first two kinds of passage behave with respect to
union into chords of the Seventh has been shown above. Now it
remains to examine the third kind : the formation of chords of the
Seventh belonging to the passage from one triad into another
which is disjunct from the first.
122. Starting from the triad of the tonic, the first kind of pro-
gression leads to one or other of the minor triads, and joins the
tonic triad and the minor triad into a chord of the Seventh. The
second kind leads through the minor triad to the dominant (or sub-
dominant) triad, and binds this intermediate minor triad and the do-
minant (or subdominant) triad together into a chord of the Seventh.
The third kind, progressing in like manner, takes its way through
the minor and dominant (or subdominant) triads to the diminished,
triad, and this last joined to the dominant (or subdominant) triad
is heard as a chord of the Seventh. For simultaneous chord-union,
as we have seen in passages of the second kind, only takes place
between the two last triads related by a Third.
Therefore the passage of C—e—G to DjF—a will produce the
chord of the Seventh DjF—a—C, and the passage of C—e—G to
t>—DjFthe chord of the Seventh G—b—D\F, both in a position
of their intervals answering the conditions of succession.
CHORD OF THE SEVENTH 61
If we consider the mechanism of this formation it is
(1) Towards the subdominant side :
C— e— G . . . C— e— a . . . C— F— a . . . D/F— a = C— D— F— a.
V — -y- ^
(2) Towards the dominant side :
C— e— G . . . b— e— G . . . b— D— G . . . b— D/F = b— D— F— G.
123. For triad-succession in itself, apart from its relation to the
formation of chords of the Seventh, there can be no mention of
taking this triple contracted advance for the link between two
disjunct triads ; for the fourth triad contains no longer a note of the
first, by which it could be seen that the formation of the fourth
triad is transformation from the first. Disjunct triads can only be
joined to follow immediately upon one another when the triad we
start from is replaced by another related both to it and to the uncon-
nected triad. In the passage from C — e — G to D\F — a the triad
a — C — e linked the succession, and in the passage from C — e — G
to b — D\F the triad e — G — b. In this way Seventh-construction
going to the diminished triad of the subdominant side
C— e— G
(a— C— e) . . . a— C— F . . . a— D— F
S - Y ^
gives the position a — C — D — F '; and to the diminished triad of the
dominant side
C— e— G
(e— G— b) . . . D— G— b . . . D— F— b
the position D — F — G — b. For there the note a from the substi-
tuted a minor triad, here the note b from the substituted e minor
triad, supplies the link for the last member.
The above succession, which unites the third and fourth triads
of the series into a chord of the Seventh, also contains in it one
note of the initial triad. We see, however, that in this advance
62 HARMONY
the first and last triads come to stand side by side in the primary
position, for we get towards the subdominant side :
C— e— G . . . C— D— F— a ;
and towards the dominant side :
C— e— G . . . b— D— F— G.
As applied to the particular cases denoted here, i.e. in the
passage from the tonic triad to the upper or lower diminished triad,
there is nothing felt wrong in the progressions. But suppose
we wanted to make the formation universal, and to apply it to
Seventh-construction starting from any other triads, taken primary,
of the key. For with the first two grades of the progression this
did happen, and was allowed by sounding right. E.g.
(1) Towards the subdominant side :
D/F— a.. . D— e— G— b,
e — G — b . . . e — F — a — C,
F— a— C . . . F— G— b— D, &c.
(2) Towards the dominant side :
D/F— a . . . C — e— G— a,
e— G— b . . . D— F— a— b,
F— a— C . . . e— G— b-C, &c.
Thus the progressions carried towards the subdominant side seem
to sound right ; but of those towards the dominant side only the
first, leading from the tonic triad to the upper diminished triad,
remains fit for use unconditionally ; all the others on that side
have something that goes against the grain. Here we again meet
the practical rule already touched on : that in chords of the Seventh
the Seventh must be prepared ; and the exception to the rule : that
in the chord of the dominant Seventh the Seventh may enter free
CHORD OF THE SEVENTH 63
when the Root is held. The reason for this exception, and why
progressions where the triads stand side by side in primary posi-
tion (just as they do in those towards the dominant side) may
be used towards the subdominant side without producing the
effect of consecutive Fifths, cannot be yet explained. We must first
examine the resolution of dissonance with its essential conditions,
and the nature of those chords of the Seventh which contain union
of limits, above all of the so-called chord of the dominant Seventh.
Till now this last has only appeared as a chord among other
chords in the series of harmonies of the Seventh. But it is, as we
know from experience, strongly distinguished by its peculiar cha-
racter from all other chords of the Seventh.
124. First we shall once more place together in a general view
the three kinds of triad-progression, with reference to the Seventh
harmonies thence arising, in the order previously adopted for chord-
succession in itself, without simultaneous combination.
I. Chord of the Seventh, produced by the passage from one triad
into another joined to the first by two notes.
(a) From the tonic triad to the minor triad of the subdominant
side, i.e. from C — e — G to a — C — e :
C— e— G . . . C— e— a = C— e— G— a.
\^ /
v —
(b) From the tonic triad to the minor triad of the dominant
side, i.e. from C — e — G to e — G — b :
C— e— G . . b— e— G = b— C— e— G.
II. Chord of the Seventh produced by the passage from one
triad into another joined to the first by one note.
0) From the tonic triad to the subdominant triad, i.e. from
C—e—G to F—a—C :
C— e— G . . C— e— a . . C— F— a = C— e— F— a.
HARMONY
(b) From the tonic triad to the dominant triad, i.e. from C — e — G
to G—b—D :
C— e— G . . . b— e— G . . . b— D— G = b— D— e— G.
III. Chord of the Seventh produced by the passage from one
triad into another not joined to the first.
(A) Linked by the intermediate triad.
(a) From the tonic triad to the diminished triad of the
subdominant side, i.e. from C — e — G to D\F — a :
C-e— G
(a— C— e) . a— C— F . . . a— D— F = a— C— D— F.
N ' ^ ^x
(b) From the tonic triad to the diminished triad of the
dominant side, i.e. from C — e — G to b — D\F\
C— e— G
(e— G— b) . . . D— G— b . . . D— F— b = D— F— G— b.
(B) In the succession of triads without substitution of a linking
chord.
(a) From the tonic triad to the diminished triad of the
subdominant side, i.e. from C — e — G to D\F — a :
C— e— G . . . C— e— a . . . C— F— a . . . D— F— a = C— D— F— a.
^ Y ^
(b) From the tonic triad to the diminished triad of the
dominant side, i.e. from C — e — G to b — D\F\
C— e— G . . . b— e— G . . . b— D— G . . . b— D— F = b— D— F— G.
RESOLUTION OF DISSONANCE 65
RESOLUTION OF DISSONANCE.
(i) In Suspensions.
125. The resolution of the dissonance of suspension consists in
the removal of the double meaning, which by the dissonance of two
notes to one another is produced in a third note, and the substitution
of a simple one in its stead.
In the dissonance first adduced (par. 108), C D, the relation-
ship of the two notes is contained in the note G. But here by C and D
sounding together G is determined to be at once Root and Fifth :
I— II
C G D
I— II
This double sense neither allows the interval C G to coalesce
by the Third e into a triad, nor the Fifth G— D to be united by the
Third b. But union may follow at once, as soon as either hindering
note is removed. If C gives way, then the union of the Fifth
G D occurs in b ; if D gives way, then the Fifth C G unites
in e. The entrance of the Third b is here a natural consequence of
the removal of the Root C, just as the entrance of the Third e is
a natural consequence of the removal of the Fifth D ; b unites the
Fifth G D, e unites the Fifth C G But neither union could
happen in the presence at the same time of C and D ; because G
meaning Root is contradicted by C, and meaning Fifth by D.
Therefore when C passes melodically to b, the interruption of the
unity of G D is removed, and the union of the Fifth at the
same time effected. Similarly when D moves to £,. G enters into an
uninterrupted relation with C, and is joined with it into a whole.
126. But of the two resolutions the first, with the progression
F
66 HARMONY
C-Z>,is the one principally required here, and for this reason : G
in the preparation by C begins with meaning Fifth ; but as element
of a succession it must next become something else than Fifth, and
so must become Root. Therefore C must proceed to b, and not D
to ey that the dissonance may be satisfactorily resolved and the
required unity effected.
127. This is the dissonance of suspension, in which the note
which is the link and is differently determined by the two dissonant
notes itself takes part in the combined sound, or when not actually
present can be added mentally. In the above example G is Fifth
in the preparation, Fifth and Root in the dissonance, Root in the re-
solution. It passes from one simple meaning through double
meaning into the other simple meaning.
128. The combined sound D — e expresses the arrested passage
of G from Root-meaning into Fifth-meaning, with a tendency
to decide for the latter by C — e. The combined sounds F — G and
G — a, as passages determined upon the Root C, will 'in like manner
lead to the resolutions e — G and F — a. The Second e — F finds its
resolution D — F by b as link of the dissonance ; the Second a — by
the resolution G — b by e ; and the Second b — C, the resolution a — C
by^.
129. The dissonance D — e may, however, be also referred to a
linking a, G — a to D, b — C to e, always according to the sense in
which a linked progression is contained in the dissonant interval,
and according as it is really intended. The resolution will happen
always in like form ; but the resolved note will thereafter differ
in its chord-meaning ; it will be respectively the Root or the Third
of the triad of resolution. So, e.g., the dissonant interval D — e re-
ferred to G as link will lead to the C major triad, and the resolving
C will have Root-meaning. But if the same dissonance D— e is re-
ferred to a as linking note, then the resolution leads to the minor
triad on a, and the resolving C is Third.
RESOLUTION OF DISSONANCE 67
A note resolving downwards can arrive at Fifth-meaning only
if the other note of the dissonance moves upwards at the same
time.
(2) In Chords of the Seventh.
130. In the dissonance of suspension the dissonant chord may
be taken as already essentially that which results after resolution ;
except that it contains a jarring element to be purged out. With
the chord of the Seventh it is otherwise. That consists of a com-
bination of two triads, which cannot pass into consonance by the
advance of one part alone.
131. In the chord of the Seventh the note linking the disso-
nance, which appeared in the suspension and was determined to be
at once Root and Fifth, is not as yet present ; it has to be sought
out.
132. Here, too, the linking note must be Root to one of the dis-
sonant notes and Fifth to the other.
In the chord of the Seventh it must enter instead of the middle
ambiguously determined Third-interval, and the resolution will then
follow upon it and by it, as with suspensions. For by this link of
the dissonance entering instead of the inner Third-interval, the
chord of the Seventh has in fact become a chord of suspension.
133. The resolution of the dissonant interval in the chord of the
Seventh can happen simultaneously with the entrance of the linking
note, or it can follow later. The latter is the treatment, where the
Seventh keeps on as a suspension before resolving. Here we have
the process in detail, while in the immediate resolution of the chord
of the Seventh we have it contracted.
134. E.g. in the chord of the Seventh e — G — b — D, as twofold
chord made up of the triads e—G—b and G—b—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,
F 2
68 HARMONY
D as Root (in the sense in which D a counts as a Fifth in the
key of C major). 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 can substantiate either Fifth-meaning
against D or Root-meaning against e, both of which meanings are now
contained in it at the same time. Therefore either e will go on to F,
or D to C\ and from e — a — D there will arise either F — a — D or
e — a — C. It was found above that in the chord of suspension the
last resolution, where the linking note receives Root-meaning, is the
one principally required ; and that is now too the case. In the dis-
sonance D — e the e has more power to last on and push D from its
place, than D has to hold out against e and force it to advance. But
in the chord of the Seventh the demand for the Seventh to proceed
downwards is less pressing than when the suspension is required to
be resolved in that motion. According to the foregoing explanation
of the preparation and resolution of the dissonance of suspension the
necessity for such progression lies principally in the note which is the
link and doubly determined in the dissonance, requiring a different
meaning in the resolution to that which it had in the prepara-
tion. Now, as link in the chord of the Seventh, it has the oppo-
site determinations immediately upon its entrance ; for when the
chord of the Seventh e — G — b — D is replaced by the chord of suspen-
sion e — a — D, the linking note a in the latter acquires the double
meaning at one and the same instant. Therefore it may decide
subsequently as well for one as for the other.
135. As the one motion or the other, ascent of the Root or
descent of the Seventh, leads alike to unity or determinateness, so
if the two parts are mutually repelled, if e goes to F and D to C
simultaneously, that will also work a resolution. Then a comes
out of the doubt of being Fifth, or Root to the certainty, in F — a — C,
of becoming Third.
RESOLUTION OF DISSONANCE 69
136. But the resolution of the dissonance can, and for the most
part will, happen simultaneously with the entrance of the linking
note ; so that in the above example the triads of resolution
e — a — C, F — a — D, and F — a — C will follow the chord of the
Seventh c — G — b — D immediately, without pausing upon the half-
way chord of suspension e — a — D.
137. This resolution of the chord of the Seventh, in which the
direct opposition of Root and Fifth is established in a note entering
to link the dissonance, and then in it decides for one or the other
simple meaning or for the meaning of Third, we may regard as the
principal form ; namely, because, by the entrance of the note which
links the dissonance and takes the place of the middle interval, the
triad twoness is removed and the consonance can enter unhindered.
138. When it was said above (par. 131) that in the chord of the
Seventh the two dissonant notes as yet wanted connexion, what
was meant was the want of that antithetical relation, which is
present in the chord of suspension, and in the chord of the Seventh
after the connecting note has entered with its simultaneous Root-
and Fifth-meaning. An antithetical relation, but not of strong
opposition, between the dissonant notes of the chord of the Seventh
may, however, be found already in the chord itself. Not that of a
note being determined as at one time Root and Fifth, but of its
being determined as at one time Root and Third, or Fifth and Third.
This determination is already contained in the two notes of the
middle interval referred to the dissonant extremes. In the chord
of the Seventh e — G — b — -Dy chosen above as example, G stands in
a consonant relation to e as well as to D, but to each of the two
notes in a different chord-meaning. So b stands consonantly to e as
well as to D, and again is differently determined to the two notes.
G is Third of the triad e—G—b, and Root of the triad G—b—D,
but it must become Fifth of the triad C — e — G if resolution is to be
effected by means of it ; for the dissonance e — G D can only
7o HARMONY
be resolved into e—G—C. On the other hand, b is Third of the
triad G—b—D and Fifth of the triad e— G— ft but must become Root
of the diminished triad b—DjF for the resolution to be determined
upon it. For, again, e b— D can only reach resolution in
F—b—D.
139. Here, since the dissonant interval can be referred either to
one or to the other of the middle notes of the chord, we see that the
double determination is indeed doubly present. For the lower note of
the middle interval is Third of the lower triad and Root of the upper,
while the upper note is Fifth of the lower triad and Third of the upper.
Now because the resolution can only be effected with respect
to one of the two middle notes, the other double meaning remains
unresolved. The other note in which the double meaning is still
contained has, however, as good a right to have it remedied, as the
one for which that has been done. But instead of receiving satis-
faction it is compelled to move forwards ; unless, indeed, it will
persist as new dissonance in the consonance which has followed
the resolution of the one note. To this, however, not having been
properly cared for in the state of dissonance, it will show greater
inclination than to moving onwards. Thus the resolution of the
chord e — G — b — D upon the Third G, easily produces a new chord
of the Seventh e — G — b — C instead of the triad e — G — C ; and the
resolution of the same chord upon the Fifth b, easily produces the
chord of the Seventh F — G — b — D instead of the triad F — b — D.
In fact by mere reference of the dissonance to one or the other
middle member of the chord of the Seventh the triad twoness is not
yet removed. It was removed in the other resolution of the chord
of the Seventh previously shown; and there, as in the chord of sus-
pension, a more decided restoration of consonance followed.
140. Thus in the second manner of resolution one of the middle
parts of the chord of the Seventh is held, and the progression is de-
termined upon it, while the other, if it does not endure and become
new dissonance, has to proceed to one and the same note with the
RESOLUTION OF DISSONANCE 71
part resolved. Besides this we have yet to mention a third kind,
which proceeds in respect, not of one of the two middle notes, but of
both at once, as an interval which remains and changes meaning ; be-
cause from twofold meaning, to which in the dissonance it is deter-
mined, it arrives at simple by melodic advance of the dissonant parts.
141. This is that resolution in which the interval of the minor
Seventh is by divergent progression of both the extreme parts en-
larged to the Octave, with which the persistent middle interval must
stand in consonant relation : e.g.
G—b—D—F . . ./#— b—D—f%
<or G—b\>—D\>—F . . .
This resolution can therefore only occur for chords with a minor
Seventh, in which the diminished triad takes part ; because only
these can fulfil the conditions of the resolution. It must be observed
besides, that here the resolution leads into another key, because it
can only happen by chromatic progression of one of the two parts.
Now chromatically different notes never lie inside the same key.
142. In the first example above the sense of the resolution is,
that the middle interval b — D, which by G and F is determined to
have a double chord-meaning, receives a simple meaning by the
progression of the two mutually dissonant voices to the Octave
f$—f%- In the second example the case is similar. The middle
interval B\> — d\>, which in g — B\> — d\> — F has double meaning, in
G\> — b\> — D\> — G\> is not different as it is an interval, but only as it
means part of a chord. B\>\D\> would be different as an interval from
B\> — 4> or b\> — Z>[>, as will be understood from what was said earlier.
143. This kind of resolution is less in use than any other.
(For the succession of chords derived from the b minor key-system
stretching out to the dominant side, namely g — B — d — e#~-
F$—B—d—F%, cannot be confounded with the succession
G—b—D—F' • /#—£—£>—/#.) Yet it was necessary to adduce
it, that all possible forms might be surveyed together. And now
72 HARMONY
it results, that the dissonant interval may pass in its resolution
into each of the three intervals of the triad.
(1) Into the Third; when one of two parts dissonant as a
Second moves in resolving away from the other : the Seventh moving
downwards or the Root upwards.
(2) Into the Fifth ; when both parts dissonant as a Seventh
move in resolving towards each other : the Seventh moving down-
wards and the Root upwards.
(3) Into the Octave ; when both parts dissonant as a Seventh
move in resolving away from one another : the Seventh moving
chromatically upwards and the Root diatonically downwards, or
the Root chromatically downwards and the Seventh diatonically
/
upwards.
144. Thus every kind of melodic progression (not by springs)
in the dissonant parts, which leads them away from or towards one
another into one of the three triad intervals, contains the possibility
of a resolution of the chord of the Seventh, and the resolution can t>e
brought about :
(1) by a new note,
(2) by one of the two middle notes,
(3) by both.
145. The general scheme of the resolution of dissonance, as
hitherto discussed, can be tabulated in the following form. But
herein we take no count of difference in the triads combined,
namely as they are major, minor, or diminished ; we denote only
the combination itself. That two triads of the same kind can
never be shown united, is self-evident from the organic connexion
of chords and from the nature of the key-system.
The diminished triads are also counted as organic chord-
formations. The chords of the Seventh G—d—D/F, b—D\F—ay
D/ F — a — Cy though printed with the elements from the dominant
and subdominant chords separated by a line of division, are none
the less grounded as combinations of triads. The chord G—b—D\F
RESOLUTION OF DISSONANCE
73
cannot have organic meaning as a union of the dominant triad
with the subdominant Root, nor the chord D\F — a — C as a union
of the dominant Fifth with the subdominant triad. Only things
of like kind can be united. With the triad only the triad can enter
into union, but not the single chord-element, the solitary note. The
first of the two chords of the Seventh contains the union of the triad
G — b — D with the triad b — D\F, the second contains the union
of the triad D\F — a with the chord F — a — C. At the same time
the interval of disunion D\F still has its meaning, and will always
distinguish the chords of the Seventh in which it takes part essen-
:ially from the rest. But the particular attribute does not shut
them out from the general determination, which, as chords of the
Seventh, they have in common with the others.
146. The dissonance as suspension is in general notation either
For a, resolution may happen in two different ways ; namely,
the double meaning in -j may be determined to I or to II.
For by only one kind of resolution is possible ; that is, for j
to give up the I ; because the Third (III) contained in the combi-
nation has already pronounced for the meaning II, not being able
to progress melodically without coming into dissonance with the
middle note. Thus the resolutions for a will be :
HI -- 1— II I— II -- III,
the resolution for b :
III— II
I -- II
III—II— I.
74 HARMONY
147. The first form, "" ' Sives the s°-called chord of the
Fifth and Fourth, in which the Fourth is contained as suspen-
sion of the Third (Resolution a). The reason why the resolution
^, in which the Fifth must be considered as a suspension of the
Sixth, is not normally effective, has been mentioned earlier (par.
126).
The second form, j L '"_ _H> gives the suspension above the
Boot : the Seventh as suspension of the Sixth in the chord of Six-
Three. A suspension over the Fifth as dissonance will not be found
until the chord of the Seventh ; in the triad, as Sixth, it is neither
dissonant against the Root nor against the Third ; the Sixth added
to the Root and the Third only forms a transposed triad, of which it is
itself the Root, the Third being Fifth and the bass Third. But in the
chord of the Seventh a suspended Sixth is in fact again a suspended
Fourth in the upper triad.
148. The dissonance of the chord of the Seventh may be ex-
pressed generally in the form
(I— III— II
The resolutions of the chord of the Seventh we have seen to be
of two essentially different kinds. The first is, that instead of the
inner interval of the chord there enters a new note mediating in itself
between the two dissonant notes ; resolution then follows as in the
chord of suspension :
!n
I —III— II
I __ III— II
-I
II — I
RESOLUTION OF DISSONANCE 75
The second is when the mediation of the dissonance is found in
the contents of the chord of the Seventh itself.
A. In one (a) or else in the other (/3) of the two middle notes :
I— III— II jI_HI_H
I— III— II I— III— II
Jl— III | I II
1 I— -II I III— II
With a the chord again acquires the meaning of a suspension ; the
resolution here is :
I III
I II
with IB the resolution is :
I II
III— II
II 1— III
B. If the resolution is to be determined in respect of both
middle notes at once, taken as an interval, so that this middle in-
terval persists and becomes consonant in the chord of resolution,
then that again may happen in two ways, according to the quality
of the triads combined in the chord of the Seventh :
I— III— II (I— III— II
<•> i-m-ii
II 1— III II I III— II 1
In (a) the Seventh moves chromatically upwards, in (/3) the Root
chromatically downwards. Here the middle interval in the chord
of the Seventh has the double meaning i ; in the first reso-
lution (a) it decides for I— III, in the second (0) for III— II.
The last kind of resolution of the chord of the Seventh requires
chromatic progression, and thereby brings about a change of key.
76 HARMONY
Yet other ways of resolution will have to be cited, in which chro-
matic changes enter ; but of these the fitting place is not found till
afterwards. For they give rise to a union of chords of the Seventh
following immediately one upon the other ; and that is a succession
which ought previously to be considered in itself, as coming or-
ganically into existence.
PROGRESSION OF PARTS IN SEVENTH HARMONY.
149. The progression of the Seventh in giving rise to harmony
of the Seventh cannot be other than it is in triad-succession.
For harmony of the Seventh is really nothing else than such suc-
cession gathered up into a chord ; and, as we have already seen, it
can only contain two conjunct or rather two overlapping triads.
In the passage from C — e — G to G — b — D, the note b to be under-
stood must mean advanced C, and D advanced e. D cannot be
derived from C. That would express immediate passage from
C- — G to G D, such as occurs in scale construction. But chord
passage from C — e — G to G — b — D can only happen by means of
e — G — b\ as C — e — G--b — e — G--b — D — G\ and the harmony of
the Seventh produced from it can only contain the succession
b — e — G--b — D — G united as b — D — e — G, in the same position
and with the same progression of the parts as in the triad-suc-
cession.
1 50. Accordingly, when the Seventh enters to the Root already
present, i.e. when the triad which lies uppermost is joined on to
the triad next underneath, it can only issue from the Root of the
lower triad ; for no other part can melodically lead to the Fifth
of the upper triad ; and there can be no directly intelligible pro-
gression except that conceived melodically. That is, the Seventh
PROGRESSION OF PARTS IN SEVENTH HARMONY 77
must issue as a Second from the Root of the lower triad, and thus
form the Fifth of the upper triad, or vice versd ; because the change
of chord consists in this difference alone. Where other progressions
happen, or where the note gone over enters in another meaning in
the new chord, there, in fact, combined successions are present : such
that in them has taken place a double progression, a twofold change.
So, e.g., in the succession C — e — G--G — b — D in the position
b — D — G, where the root C has first advanced to the Fifth b of the
triad e — G — b, and then the Root e of this latter to the Fifth D of
the triad G — b — D. In Seventh-formation, then, the first and
second or the second and third triads in the row must be joined, as
b— C— e— G or b—D—e—G.
Wherefore, if the unprepared Seventh can only enter descending
from the Root itself or its Octave, because the connexion of the triads
produces it in this manner alone, it follows that for the Seventh to
move ascending on to the Root must always be against the natural
order of passage.
151. Let us now consider the opposite succession, that towards
the subdominant side, i.e*. leading from an upper triad to a
lower. Here the Fifth of the first triad advances to the Root of
the second joined closest to it. In the triad succession from
G — b — D to e — G — £, which leads to the position G — b — e, D has
advanced to e. The Seventh-harmony of this succession appears
in the position G — b — D — e. Here the note added to the initial
triad is Root of the chord of the Seventh. In the succession towards
the dominant side it was its Seventh, as such the Fifth of the upper
triad, and then by its nature a second, something that makes no
beginning and that can only enter in succession to something gone
before. Therefore the unprepared Seventh will not appear other-
wise than struck after the Root or its Octave, and for this very
reason, be it said in passing, preferably too upon that part of the
bar which corresponds metrically to it, the second or so-called
78 HARMONY
' weak ' part. On the other hand the Root of the chord of the
Seventh, which is as well Boot of the lower triad, has the nature of
a beginning ; it is in essence a first, something that can precede
something else. The unprepared Seventh struck afterwards as in the
succession C—e — G--b — C — e — G will always be heard as a part
that has moved onwards ; the Root entering to the prepared Seventh
produces rather the effect of a fresh part added. The chord of
the Seventh with prepared Seventh finds moreover its appropriate
place upon the first, so-called ' strong,' metrical member of the bar.
152. So too, taken quite generally, to any note held maybe
struck the next lying over it, but not the next lying under. The
former is always a positive, a first, a Root ; the latter a relative, a
second, a Fifth. Therefore the form ~~~TP"^ttr is under all cir-
cumstances a right one ; the opposite, < ^\v, is admissible
only under particular conditions. The simultaneous sound of two
contiguous notes when the lower follows the higher can only appear
in the form j^^p", particular occurrences excepted.
153. From this it becomes manifest, why out of the successions
above (par. 123) constructed into chords of the Seventh, from one
triad into another not joined to it, those which lead to the sub-
dominant side alone sound right ; but those to the dominant side,
with the exception of the first (C — e — G--b — D — F — G\ cross-
grained, disjointed, and * Fifthy.' The succession from the triad
C — e — G to the triad D IF — a, which there comes out as a chord of
the Seventh in the form C — D — F — #,does not sound Fifthy, because
there exists no necessity for deriving the noteZ> from C or for hearing
it as a C that has gone forward; as Root of a chord of the Seventh it
can be just as well derived from the e of the C major triad, supposing
that a natural progression should introduce it by this road. But
taking the triad D\F—a first and letting C—e—G follow, whereby
PROGRESSION OF PARTS IN SEVENTH HARMONY 79
the chord of the Seventh is produced in the position C — e — G — a,
then in that case the Seventh G must either have arisen from the ad-
vance of a, according to which the triads D\F—a and C — e — G
would stand next one another in Fifth-position, unlinked, and
therefore not as a succession ; or else in C — e — G — a the note
G will be heard as an F moved onwards, that is, as a Seventh not
derived from the Root of the a minor triad, but entering disjointedly ;
which, as has been shown, is inorganic, and must also sound wrong.
Therefore, for all cases, Seventh formations going to disjunct
triads lying beneath, i.e. in the subdominant direction, as
C — e — G-..C— D— F-a
D/F— a-.-D— e— G— b
e — G — b---e — F — a — C, and so on,
in this form sound right ; those going to disjunct triads lying
above, i.e. in the dominant direction, as
C— e— G-- b— D— F— G
b_D/F...a — C— e— F
a — C — e • • -G — b — D — e, and so on,
with the exception of the first, C — e — G~-b — D — F — G, sound
Fifthy or disjointed ; in a word, wrong.
154. We have found that for the passing Seventh melodic
motion descending from the Root alone accords with natural suc-
cession, and not motion ascending on to it ; but that with the pre-
pared Seventh the Root can enter either way. The explanation of
those successions, where the Root of the lower triad cannot be
derived from the Fifth of the upper, still remains to be given as re-
gards their inner meaning. We mean those which, like C — e — G-~
C — D — F — a, contain an apparent juxtaposition of two primary
triads, yet do not give the effect of consecutive Fifths. We have
indeed found, that here the note D, in respect of the construction
8o HARMONY
of the dissonance, may be as well derived from the Third e as from
the Root C. But it has been also said, that the progression of
parts in harmony of the Seventh can be no other than that in
triad succession. Now if we make the Seventh- chord arise in the
passage to the disjunct triad, as the Seventh-chords arose in the
passage to conjunct triads, and therefore the above chord
C—D — F— a in the succession
C— e— G . . . C— e— a . . . C— F— a . . . D— F— a = C— D— F— a,
then the note D is by no means produced melodically from e. There
is no possibility at all that it should have been so produced, because
the harmony of the Seventh arises only from the union of the two
last triads of the triple succession, C — F — a and D\F — a ; but with
the triad C — F — a the note £, Third of the C major triad, which in
C — e — a lasted on as Fifth of the a minor triad, is removed. Even if
we would derive the Seventh-harmony C — D — F — a from the a
minor triad of the succession in the manner of chord-unions of the
second degree of affinity (par. 91), C must still progress to D, e to F.
Now apparent necessity for leading e to D, intentional working to-
wards a set end, cannot be brought in here ; because we are not now
speaking of fine-art construction, but only of natural formations
self-produced, without approach of individually determined will.
Therefore the succession C — e — G--C — D — F — a, if justifying
itself to the ear, must be explained from some other grouping than
the one denoted above : C — e — G< • • C — e — a- • - C — F — a- - -D — F — a
— — — - ^-
= C — D — F — a\ for that contains progression of Fifths C — G--
D — a between the first chord and the last.
SUCCESSION OF SEVENTH CHORDS 81
SUCCESSION OF SEVENTH CHORDS.
155. In the second kind of resolution of the Seventh (par. 138)
there is contained the germ of a continued series of joined Seventh
chords. That resolution was effected by the intervention of one of
the two notes of the middle interval of the chord. E.g. in the
Seventh chord e — G — b — D the dissonance can be found as a
double meaning either in G, as
I - II
e G D
III— II
or in b, as
I-III
e b D
I- -II
and the resolution will in the first case be
e G C
III-II — I
and in the other
F b D
In the first case the progression leads towards the subdominant
side, in the second towards the dominant ; for in the first the Third
of the lower triad, and in the second the Third of the upper triad, is
the note which determines the resolution. But because in neither
of these two determinations is the whole of the middle interval
accounted for in the resolution, but only one or the other of its
notes, therefore, as was earlier noticed, the slighted note is more
inclined to stop in its place than to move forward to the restora-
tion of consonance. It has no interest in the resolution. But so a
G
82 HARMONY
new dissonance arises, because the note which moves to its resolu-
tion comes into dissonance with the note which stays, and that as
Root or as Seventh of a new Seventh chord. For with the Seventh
chord above, e — G — b — Dy taking the G as link, the resolution
e — G — b — C follows, and, taking the b as link, the resolution
F — G — b — D ; supposing that in the first case b, in the second G,
stays in its place. With this, one Seventh-harmony has passed
into another. If this succession be continued further in the same
way, then, starting from the tonic triad, the following two series
will arise : —
(1) Towards the subdominant side, when in the Seventh chords
the lower note of the middle interval determines the resolution :
C— e— G ... C— e— G— a ••• C— e— F— a ••• C— D— F— a •••
b— D— F— a---
(2) Towards the dominant side, when in the Seventh chords the
upper note of the middle interval determines the resolution :
C— e— G ..- b— C— e— G ••• b— D— e— G ••. b— D— F— G •••
b— D— F— a--.
In the first series may now be found that which by triad pro-
gression alone could not be produced, continuous succession from
the first member to the fourth, from C — e — G to C — D — F — a ; in
which the Root of the triad D\F—a is produced, not by the
ascent of C, but by the descent of e, which as Seventh of F had to
resolve to D.
1 56. This series contains in each of its members a construction
of dissonance following correctly from the member next preceding.
The other, on the contrary, is at once manifested to be inad-
missible in the succession from the second member to the third,
because there the Seventh moves ascending on to the Root ; now
if not prepared the Seventh can only proceed from the Root.
Therefore if the succession C— e — G--b — D — F — G commends
itself as perfectly right to the ear, its construction is not to be
SUCCESSION OF SEVENTH CHORDS 83
explained out of the second of the above series in the same way
as the construction of the succession C — e — G--C — D — F— a is
explained out of the first, in which every member stands in right
succession with that which goes before as well as with that which
follows. Here the process of construction must be otherwise de-
rived. We shall return to it in considering the chord of the
dominant Seventh.
157. Every succession from a triad to the disjunct triad upon
the dominant side, except only that which leads from the tonic
to the upper diminished triad, will also at once bring out the same
incongruity in the construction of the dissonance as that just found
in the example given ; as, e.g., D\F — a- • • C—e — G — a, e — G — b- • •
D — F — a — b, F — a — C--e — G — b — C, and so on. We always
have to choose between hearing two triads moved side by side in
primary position, and finding the Seventh ascend on to the Root.
But the one and the other are alike against the nature of continu-
ous progression.
158. The passage C — e — G--C — D — F — a, in the subdominant
series above, stands in intelligible succession by its intermediate
members C — e — G — a and C — e — F — a. Now this succession
leads through two Seventh chords preceding C — D — F — a, and so
joins the third Seventh chord in sequence with the first. Thus a
series of Seventh chords progressing through the first, third, fifth,
seventh members of the one above, must also be linked intelligibly.
For from the third to the fifth, from the fifth to the seventh, there
is repeated only the same relation of the first to the third.
159. The first series puts together in Seventh-harmonies a pro-
gression of triads related in the Third. In the second, which every
time passes by a member of the first without stopping, we get as
Seventh-succession a progression of triads related in the Fifth. The
same follows if we progress through the second, fourth, sixth,
eighth members. A third series, led through the first, fourth,
G 2
84 HARMONY
seventh, tenth members, contains in the succession from one
member to another a contraction of three progressions ; and can
therefore, after the first step starting from the triad, no longer
regularly appear to follow correctly. For such a progression brings
with it the simultaneous advance of three parts by a Second,
namely, continual passage into disjunct triads. This, it is true,
may be correctly managed by means of intermediate triads (par.
90) ; but the course of the parts thereby necessitated comes into
contradiction with that which the Seventh to be resolved demands
here. It does so quite decidedly in the succession of the fourth,
fifth, and sixth members of the series, as well as in those which
correspond periodically with that place, where triads in primary
position twice stand next one another. Therefore in this series
those successions will alone seem right to the ear which answer to
linked progression of disjunct triads and at the same time fulfil the
requirements of dissonance.
1 60. The first series of joined Seventh-harmonies, progressing
towards the subdominant side in triads related in the Third, is :
4-
.. e— D— F— a ...
The second, progressing in triads related in the Fifth :
5- 6. 7. 8.
— C— e— G ... a— C— D— F ... G— b— D— F .. G— b— C— e ...
a7 D Gr
SUCCESSION OF SEVENTH CHORDS 85
The third, progressing in triads without direct relationship :
i. 2. 3. 4.
C— e— G .-. C— D— F— a ... b— D— e— G ... a— C— e— F-.-
C D7 e; FX
5. 6. 7. 8.
G— b— D— F ... G— a— C— e ... F— a— b— D ••• e— G— b— C •••
G; a/ b°7 Cx
the last offending against triad continuity in the successions 4-5
and 5-6.
161. To complete our view of the whole subject, the corre-
sponding three series of joined Seventh-harmonies towards the
dominant side shall also find place here ; the stubborn succession of
the same has already been discussed.
The first series, progressing in triads related in the Third, is :
— G ••• b— D— e— G ••• b— D— F— G •••
5. 6. 7. 8.
b— D— F— a ••• C— D— F— a ••• C— e— F— a ••• C— e— G— a •••
b°7 D°7 FX a;
The second, progressing in triads related in the Fifth :
i. 2. 3. 4-
C— e— G -.. b— D— e— G ••• b— D— F— a ••• C— e— F— a •••
C e7 b°7 FX
5. 6. 7- 8.
C— e— G— b .•• D— F— G— b ••• D— F— a— C •.• e— G— a— C •••
Cx G7 D°7 a7
86 HARMONY
The third,
progressing in triads without direct relationship :
I.
C_e— G ..-
C
2.
b— D— F— G .•
G7
3-
. C— e— F— a ••
4-
• D— e— G— b .-.
e7
5-
D— F— a— C
6.
... e— G— b— C
7-
... F— a— b— D
8.
... G— a— C— e ...
D°7 Cx b7 a;
The first two series are marked faulty by the entrance ascending
of the Seventh. In the third, to this must be added the inconse-
quent progression of the disjunct triads, which in the progression
towards the subdominant side was enough to prevent the series
from subsisting without periodic interruption.
162. Thus altogether a sequence of Seventh-harmonies can be
framed only to the subdominant side, because in that direction
alone are the requirements of organically lawful succession corre-
sponded to. And here in the first and second series it may be
continued uninterruptedly, but in the third it is interrupted by
Fifth successions, which set in periodically.
163. But a sequence in Seventh-harmonies towards the dominant
side is self-contradictory. This series is really an inverted one.
In the opposite direction, if the progression were from the third to
the second, from the fourth to the third member, it would seem
quite consequent ; because then it becomes exactly a subdominant
series, or a series in which the Third of the lower triad is taken as
linking the dissonance. (In the dominant series it is the Third
of the upper triad upon which the resolution follows.) So too the
progression here from the fourth to the second, from the fifth to
the third member, and so forth, would follow quite correctly.
164. Seventh-formation, when carried towards the dominant
side, can only consist of mere triad-union, and not of linked Seventh
chords, as found practicable in the direction towards the subdominant
SUCCESSION OF SEVENTH CHORDS 87
side. Therefore upon the dominant side, only the passages into triads
related in the Third and in the Fifth can be gathered up into a
Seventh chord, but not the passage into a disjunct triad. Towards
the subdominant side that also is a possible progression, linked,
namely, by Seventh chords.
165. The possibility of framing dissonance towards the disjunct
triads of the dominant side is not thereby unconditionally denied ;
only the combination cannot be reached by the linking drawn
directly from this series. The succession a — C — e--G — b — D — et
which out of the primary position of the first triad is bad, will
seem perfectly right out of the Six-Three position of the same,
C — e — a • - - b — D — e — G ; because here the chords C — e — a and
b — D — G stand in right succession to one another, and the
Seventh D is produced from the*?. Similarly the Six-Four position
of the first triad would afford a good progression.
1 66. Now if the succession C — e — G--b — D — F — G is found
right to feeling, and yet cannot be linked by the series C — e — G--
b—C—e—G--b—D—e—G--b—D—F—G (as the succession
C — e — G ••• C — D — F — a is linked by the series C — e — G -•
C — e — G — a--C — e — F — a--C — D — F — a); and if further in this
subdominant series every passage to a disjunct triad, as D\F — a--
D—e—G—b, e—G—b - • - e—F—a—C, F—a—C • • • F—G—b—D,
is right equally with the first, but in the dominant series, except
the first, every other, as D\F — a ••• C — e — G — a, e — G — b •••
D — F — a — b, F — a — C--e — G — b — C, seems wrong; then the
goodness of the succession C— e — G-~b—D—F—G must by all
means have its reason in something else than the linking derived
from these series. Now simple triad-linking will bring about the
union of the minor triad with the dominant triad. But from the
Seventh chord hence arising b— D — e — G to the union of the
dominant and the diminished triads in b — D—F — G there is re-
quired a progression of the e to F ; and this makes the Seventh
88 HARMONY
enter ascending to the Root. This is what we hear in the succes-
sion C—e—G-'b—D—F—G, where the entrance ascending of the
Seventh does not strike the mind as being contrary to smooth
progression. This smoothness is founded in the nature of the
chord of the dominant Seventh, which results from the union of these
two triads. We have now to consider this more closely, as the first
of the three chords of the Seventh in which the limits of the key-
system appear joined.
SEVENTH CHORDS OF THE KEY-SYSTEM
PASSING INTO ITSELF.
I. Dominant Seventh Chord.
167. We have found the essence of harmonic dissonance in a
contradiction, a double determination, that the sounding together
of two dissonant notes produces in a third note. It is the simul-
taneous determination of a note to be Root and Fifth, Root and
Third, or Third and Fifth. But decided opposition is only contained
in the first determination. For the Third according to its notion
unites Root and Fifth in itself, and therefore is not the opposite of
one or other of these two triad elements in their separation, but
rather only the opposite of the separation itself. Thus a completely
satisfactory resolution of the Seventh chord could not be reached
through one of the notes of the chord, which have only the im-
perfect opposition of the Third with one of the other two elements ;
but a fresh note had to enter instead of the middle interval, and
then in it the two dissonant notes call forth the double determina-
tion of Root and Fifth.
1 68. This double determination of strong opposition is situated
in the very origin of the key itself, when its dominant and subdo-
minant are sounded at once ; for then the tonic is at once Fifth
of the subdominant and Root of the dominant :
SEVENTH CHORDS OF THE KEY-SYSTEM 89
I II
F C G
I II
This is the Fifth-relation of the two notes, which they have as
regards the Root, and it splits the Root into opposite meanings.
Third-relation (connexion in a chord) the dissonant notes find in
the simultaneous union of the dominant triad with the diminished
triad of the dominant side, G — b — D b — D — F, as dominant
Seventh chord,
I— HI— n
G b D F
I— III— II
which for its resolution postulates that former Fifth-relation of the
notes F and G in the Root C.
169. This Seventh chord has the same importance in the key as
dissonance which the tonic triad possesses as consonance. The
former refers unambiguously to the latter ; for it is precisely the
Root of the tonic triad itself, which is here set at two in itself by
the two dissonant notes, and its unity restored by the resolution.
Therefore it is that this Seventh chord leads to the perfect cadence.
170. Characteristic of the dominant Seventh chord, as also of the
Seventh chord on the Third of the dominant, to be hereafter more
minutely discussed, is the interval of the diminished Fifth, which
in this harmony is contained between the Third of the dominant
chord and the Root of the subdominant chord. If in the key of C
major or C minor the notes b and F sound together, then b will
want to move to C, F to e (in the minor key to 4>)- Between b and
F no triad unity is present ; therefore to bring one about by the
nearest way is what is wanted. But each note seeks to make itself
felt, and so the note F draws b up to C, and the note b drags F
down to e. In the minor key b will draw F to e\>, because b and
4> stand by the augmented triad dp — G — b in nearer chord-relation
9o HARMONY
than the notes b and F taken out of the separated dominant and
subdominant triads. For even if the limits of the key-system are
outwardly united in its passage into itself, yet the inward separation,
the twoness of basis, will always prevent such a union from producing
a triad unity, such as we have comprehended in the notion of the
major or minor triad. The diminished triad D\F — # contains indeed
in F — a the interval of Third, but not in D a an interval of Fifth ;
the diminished triad b — D\F contains neither Fifth nor Third in-
terval. Similarly in the two diminished triads of the minor key
D\F — a\>, b — DjFy neither interval is contained ; and, in respect of
the Fifth, the difference in our notation of capital and small letters
has already brought this prominently before the eyes. It is true
that in the so-called augmented triad
III— I
e[> G b
I— III
Fifth-relation of the outside notes is also not present, but then both
are bound to the middle note in Third-relation. On the other hand
the diminished triads contain only one direct relation between two
notes, D\F — a, b — D\F, while the third stands in triad-relation to
neither of the other two. Thus the dissonant augmented triad may
still lay claim to a meaning of unity as against the diminished, and
the interval b e\>, against b F, count as an approach towards
unity, proportionably to the nature of the minor key.
171. Although the splitting in two of the principal triad is
most strongly manifested in the Third of the dominant sounded
together with the Root of the subdominant triad, yet it is also
contained (and with the same meaning) in the Fifth of the dominant
sounded together with the Root of the subdominant. Only the
dissonance of these two notes is less plainly to be felt, because we
may be tempted to confuse the Fifth of the dominant with the Third
SEVENTH CHORDS OF THE KEY-SYSTEM 91
below of the subdominant (e.g. in the C major key D with d] ;
hence the chord D/F — a, where not determined by the context,
may be easily taken for the d minor triad d — F — a> and in itself
appears less like a diminished chord than the chord b — D\F does.
In the minor key there is not room for this confusion, because
there the difference of D\F — a\> and d\> — F — a\>, which is correla-
tive to that in the major key of D\F — a and d — F — a, is suffi-
ciently distinct to the ear.
172. In the combined sound of the Third and Fifth of the do-
minant with the subdominant Root and its Third, in b — D\F — tf, the
limits of the key-system have come together. In this union
b — D\F — a we must think of the key-system as turned about upon
itself. The boundaries placed united as middle make the middle come
out divided as boundaries.
F a C e G b D
(e) G— b— D/F— a— C (e)
Here the middle b — D\F — a as Seventh chord refers its dis-
sonance b -- a to the middle of the system, i.e. to £, the Third of
the tonic triad, which from meaning union as Third
is by b and a brought into contradiction by meaning at once Root
and Fifth.
I -- II
a e b
I -- II
173. The same applies to the minor key-system with regard to
the Seventh chord b—D/F—a\> and e\>, the Third of the tonic :
92 HARMONY
F at> C eb G b D
(eb) G— b— D/F— ab— C (eb).
174. Reference to the Root 61 in the dissonance G F is a pro-
perty of the dominant Seventh chord G — b — D\F, in the major
as well as in the minor key ; wherefore this chord, and not that
just named, may rightly lay claim to the meaning of principal Seventh-
harmony. Then the Seventh chord which stands opposite to the
principal one, namely, D\F — a — C or D\F — #b — C, relates to the
Fifth of the tonic triad. These three Seventh chords, standing in
their dissonance in antithetical relation to the notes of the tonic
triad, all contain the interval of the joined limits, D\F \ they belong
to the transposed key-system. The rest of the Seventh chords
relating to notes outside the tonic triad are contained in the
untransposed system.
175. The whole system of Seventh-harmonies is :
(A) In the major key.
b— D/F— a
e
G— b— D/F D/F— a— C
C G
e— G— b— D F— a— C— e
a b
C— e— G— b a— C— e— G
F D
(B) In the minor key.
b— D/F— ab
eb
G— b— D/F D/F— ab— C
C G
eb— G— b— D F- ab— C— eb
ab b
C— eb— G— b ab— C— eb— G
F D
SEVENTH CHORDS OF THE KEY-SYSTEM 93
176. Up to this we have considered the Seventh chord as merely
a union of two overlapping triads, without heeding the particular
quality of the triads so combined. But now that too must be in-
vestigated, and the meaning which it has for the character of the
Seventh-harmony brought out.
177. The Seventh-harmonies of the untransposed key-system :
II-III-I II-III-I II-III-I II-III-I
FaCe, aCeG, CeGb, eGbD
I-III-II I-III-II I-III-II I-III-II
each consist of the union of a major and a minor triad. The two
notes forming the middle interval of the Seventh-harmony are con-
tained organically in both triads, with different determination in
each ; so that in the first and third Seventh chords they stand thus :
II— III
III— II
a C
e G
and in the second and fourth thus :
j III— I
I I— III
C e
G b
Here a combination of two real triads appears as harmonic
union, dissonant in its outside members, which last do not bear to
one another the relation of either of the three triad-intervals. This
is the only dissonance contained in the combined sound.
178. But the Seventh chords which contain both the extreme
notes of the untransposed system, D and F, together,
G— b— D/F, b— D/F— a, D/F— a— C,
and thereby define the transposed system, are of other nature
94 HARMONY
and quality than those of the untransposed system. Here the
dissonance does not so flatly consist in ambiguity of the middle
interval, nor its manifestation merely in the sounding together
of the outside notes of the Seventh chord. Indeed it is altogether
untrue that in these chords there is found union of real triads.
We may be able to comprehend Seventh-harmony in general
under the notion only of triad-twoness, and the dissonant triads
may have, like the consonant, organic existence in that notion ; yet
still they have it but as dissonant chords, that are of their nature
cleft, and not self-rounded in their component parts. In the first
of the three Seventh chords above, G — b — £>/F, which con-
sists of the union of the triads G — b — D and b — D\F, the di-
minished triad b — D / F will surrender the component part of the
G major triad, b — D, wholly to the latter triad on being joined
with it : the Seventh chord G — b — D/Fis heard as the dominant
chord G — b — D together with the subdominant Root F. The
second Seventh chord b — D\F — a, made up of the diminished
triads b — D\F and D\F — a, makes itself heard as the sound of
the Third and Fifth of the dominant together with Root and Third
of the subdominant. The third Seventh chord, made up of the
triads D\F—a and F — a — C, makes itself heard only as the union
of the subdominant triad F — a — C with D, the Fifth of the dominant.
1 79. In the dominant Seventh-harmony G — b — D/F, the Seventh
stands apart, and out of all real triad-relation to the chord. In
the Seventh-harmony on the Fifth of the dominant, D\F — a — C
(D/F—a\> — 61), the Root stands similarly separate and out of triad-
relation. In the Seventh-harmony on the Third of the dominant,
b — D\F — a (b — D\F — a\>), the division is there in the middle of
the chord ; it falls asunder into two parts, of which the lower be-
longs to the dominant triad, the upper to the subdominant. Indeed
all three Seventh chords take their contents from the dominant and
subdominant triads alone, and are only distinguished by their taking
SEVENTH CHORDS OF THE KEY-SYSTEM 95
more or less from the one or the other; the first has for contents
the dominant triad complete and the Root of the subdominant
triad, the second has the subdominant triad complete and the Fifth
of the dominant triad, the third has the Third and Fifth of the
dominant and the Root and Third of the subdominant.
1 80. Now since the Seventh of the dominant chord is not related
to any note of the triad lying underneath it, and since further it is
decidedly a Root (of the subdominant chord), that is, a primary, of
independent value, and not a Fifth such as are the Sevenths in the
untransposed system, that is, a secondary getting its derivation from
a primary, or Root — therefore this interval may enter to the triad
ascending and descending alike, altogether freely, just as the Root
of any Seventh chord might enter in ascending or descending motion
or as a new part added to the triad. To the triad G — b — D as
dominant chord, apart from any particular kind of melodic derivation,
F the Root of the subdominant chord may enter, just as e the Root
of the e minor triad related to it in the Third might do. And yet
to G — b — D as tonic triad, /#, the Fifth of the minor triad on b>
cannot otherwise be added than as springing out of the Root GY
just as to the triad G\B\> — d the Fifth F of the B\> therein con-
tained can as Seventh only be derived out of the Root G.
1 8 1. Thus the tonic triad, as well as any other that contains the
Fifth of the tonic triad, may be followed by the dominant Seventh
chord, and the Seventh may enter to the Root in as good succes-
sion ascending as descending. And thus the passage C — e — G--
b — D — F — G) which the ear allows to be right, while others
like it, e.g. D\F— a •- C— e— G— a, e—G—b-.D—F—a—b,
sound wrong, is really only a progression from C — e — G to^ — D — G,
and the Seventh F accompanying the last chord is derived ascending
from the^ : a progression that cannot occur with other passages of
this form, because then the Seventh would want to be introduced
descending, whereby parallel Fifths arise with the lowest part.
96
HARMONY
182. Now, with regard to the succession C — e — G- • -b — D — F—G,
what determines ey the Third of the tonic, to split asunder and
move simultaneously upwards and downwards, must be the ten-
dency to gather up the key within its boundaries, to characterise
it as a determined and sharply defined whole. The succession
£7 e G-'b — D — G does not so far contain any determina-
tion of the key; so too the succession C — e — G--C — F — a leaves
the key still undetermined. In the* former either C — e — G or
I — D — G may be tonic triad, in the latter C — e — G or C — F — a ;
in that we can take C — e — G for subdominant triad, in this for
dominant triad, quite as well as for tonic. If determinateness of
key is to be expressed, if C — e — G is to be determined as tonic
middle by one other chord, then the movement must not be to either
the subdominant or the dominant side singly ; it must be carried
to both at once. Beginning and end united in time with the middle
in immediate succession, this is the meaning of the double motion
in the Third of the tonic, when it passes to the Root of the sub-
dominant and the Fifth of the dominant at the same time. But in
the chord-succession C — e — G--b — D — F — G this happens. The
progression e-*F takes on a tendency to the subdominant side, the
progression e- D to the dominant side. But the tendency may
be active towards both sides with equal energy, or with pre-
ponderating energy for one side or the other. In progressing from
the tonic triad to the subdominant triad there arises the succession
I II
C— e— G..-C— F— a,
and in progressing to the dominant triad the succession
II I
C— e— Q.-.b— D— G.
SEVENTH CHORDS OF THE KEY-SYSTEM 97
The first is determined upon the Root of the tonic triad, the second
upon its Fifth ; because in the first the former passes out of Root-
meaning into Fifth-meaning, and in the second the latter passes out
of Fifth-meaning into Root-meaning.
183. In these successions the melodic progression happens so
that e passes to F or D, G to a, C to b.
F— a— C— e— G— b— D.
184. If the motion turns to both sides with equal energy, then
the tonic triad itself is quite dissolved in the passage :
F— a . . b— D.
Here C — e — G passes into b — D — F — a.
185. If it presses preponderantly towards the subdominant side,
then the Root of the tonic triad, as Fifth of the subdominant triad,
will remain unmoved ; it becomes the Seventh of the resulting
Seventh chord :
Here C — e — G passes into C — D — F — a.
1 86. If the inclination is preponderantly directed towards the
dominant side, then the Fifth of the tonic triad, as Root of the do-
minant triad, will keep its place ; it becomes the Root of the re-
sulting Seventh chord :
F . . . G— b— D.
Here C — e — G passes into b — D — F — G,
H
98 HARMONY
187. The same process, employed upon the system of the minor
key, gives results shown as follows :
F-ab-C-eb-G-b-D
1 88. These three dissonance-determinations which arise out of
the splitting asunder of the tonic Third and unite the dominant and
subdominant sides, can, as being nearly related, pass into another
easily in any order of arrangement.
189. All possible successions of these three Seventh- harmonies
are contained —
I. For the major key in the series :
b— D/ F— a- . -C— D— F— a- • -b— D— F— G- - -C— D— F— a- • •
1 2 3 2
b— D / F— a • • • b— D— F— G - - • b— D / F— a.
i 3 — 1
SEVENTH CHORDS OF THE KEY-SYSTEM
99
II. For the minor key in the series :
b_D/ F— at>- - -C— D— F— aj> • • -b— D— F— G- • -C— D— F—
1232
b_D/F— ab-.-b— D— F— G---b — D/F— ab.
i 3 I
These chords always contain two notes, D and Ft in common ;
therefore in every passage only one or two parts have to move,
and, where they progress together, in parallel Sixths or Thirds. In
itself, therefore, the progression of the parts cannot be faulty. The
approach of the Seventh ascending to the Root, as it occurs in the
successions I - 2, 3 - 2, and 3 - 1, might, in view of what
has been said about the entrance of the Seventh, arouse theoretical
suspicion. But the effect of these successions, although a little
rough in I - 2 and 3 - 2 of the major key, may count as right ;
and their justification too in theory will result from the nature
of the diminished triads D\F — aandDjF — a\>, as the free entrance
of the dominant Seventh resulted from the nature of the diminished
triad b~D\F.
190. In the successions I - 2 and 3 - 2 in both series we see
b move to C dissonant against D ; and in the succession 3 - 1 we
find G going in the first series to #, in the second to afr, both dissonant
to 6.
191. With the dominant Seventh chord G — b — D\F the excuse
found for the same progression was : that the Seventh of this chord
is a note not joined to the dominant triad, lying underneath it, not
growing out of it, and therefore not claiming derivation from its Root.
192. With the Seventh chord D\F—a—Cfat Root is similarly
a note parted off from the subdominant triad which lies above it ;
it does not enter into inner union with F and a. The note C as-
cending from the b finds in D no hindrance to its uniting with F
H 2
I00 HARMONY
and a into the F major triad, such as a C ascending from B\> would
find in d ; where F and a belong to the triad d—F—a as Third and
Fifth already, and the passage into the other meaning, in C—F—a,
can only be gained by the progression d> • C.
193. Thus the successions b — D\F — a---C — D — F — a and
ft — .2) — F — G--C — D — F — a do not seem discontinuous or faulty
to the ear ; while, shifted into the territory of the F major key,
with the minor triad d — F — a instead of the diminished D\F — a>
as B\>—d—F— a • • • C— d—F—a, B\>—d—F—G - • • C— d—F—a,
they prove inadmissible.
194. Far easier is it to enter into the meaning of these suc-
cessions where they relate to the intervals of the minor key, as in
t—D\F—a\> ••• C— D— F— a\>, b—D—F—G ••• C—D—F—a\>.
Although the dissonant interval, as to its outward structure, Is
the same in these chords as in those of the major key, yet the
effect of the dissonance is far less rough or hard. In this we once
again find it confirmed, that the effect does not lie in the im-
mediate ratio of the dissonant notes themselves, but is produced and
receives its character from other relations. The reason of these
chords being easier to understand lies in this : that the combined
sound of D\F — a\> is heard distinctly as a chord of division, while
D\F — a by its likeness to d — F — a may leave us in doubt as to-
which of the two chords is to count in the Seventh-harmony. Only
in union with £, the Third of the dominant, is D distinctly deter-
mined as Fifth of the dominant ; joined with F — a alone it may
easily take on the meaning consonant to that interval ; that is, it
may change to d, the Third of B\>, and Root of the minor triad
d—F—a.
195. Now the structure of the chords D\F — a — C and
D\F — a\> — C being such as to make the entrance ascending of the
Seventh seem lawful, so the like entrance for the Seventh a in
the chord b — D\F — a, and the Seventh a\> in the chord
SEVENTH CHORDS OF THE KEY-SYSTEM 101
b — D}F — #[?, is, from the nature of these dissonant harmonies and
their origin, not only admissible, but necessary. For as the Thirds
of the subdominant and the dominant (a and b or a\> and b) cannot
melodically pass immediately into one another, so too the Seventh
a or a\> cannot descend from b, but must ascend from G.
The peculiar nature of these chords we shall now proceed to
discuss.
II. Seventh Chord upon the Third of the Dominant.
(a) In the Major Key.
196. In the Seventh chord b — D\F — a, when the Seventh is not
prepared, not only may the notes a and b not occur in the position
of Second, but generally no note of the chord may lie above the
Seventh a, or the harmony becomes of doubtful effect. In the
interval of the Second itself, apart from the way in which the
notes have come together, lies the meaning of a melodic progres-
sion fixed harmonically. Now we know that a and b cannot be
melodically connected otherwise than through £, the Third of the
tonic :
I II
a e b
I II
But this mediation is decidedly contradicted by the combined
sound D\F '; and thus a — b sounding together as a Second con-
tains, with regard to the harmony b — D\F — a, a contradiction,
because D/F, denying the Third, does away with the mediating e,
and makes mediation of the a and b, placed melodically next one
another, impossible. It is not alone the relation in the Second
of the two notes, as contained in the positions D — F — a — £,
F — a — b — D, a — b — D — F, that sounds incorrect, but generally
every position of the chord in which the Seventh is not the
102 HARMONY
highest part. For then the interval between the Seventh and the
next upper note of the chord will contain by implication the
notes of the harmony that lie in between ; just as every so-called
dispersed or open harmony is for theory only a close harmony
continued without being filled up. The interval a D gives the
feeling of a — b — Z>, and the interval a F ofa—& — D — F\ whence
it follows that if the Second a — b in this chord is not good, then
neither the Fourth a D nor the Sixth a F can be so. There-
fore, like the chord-positions above written which contain a — b
as a Second, other combinations in which the Seventh a is not the
highest part, as b F— a D, b—D a F, will in the
sense of this harmony appear unnatural, though perhaps in
smaller degree : and even where such a position is led up to by
artistic treatment, there will always remain something strange
about it.
197. The position of the intervals below the Seventh is subject
to no restricting conditions ; the single requirement is for the
Seventh to be at the top. Like the position b—D—F—a, all
those due to transpositions of the lower parts, as D b F— a
and F b — D a, will also be well-sounding and fit for use,
notwithstanding that the intervals D b and F b include a, the
Third of the subdominant. This leads to a not unimportant
observation, namely that all harmonic form shapes itself from
below upwards, even in transposed chords which do not contain the
Root as lowest part. In the chord D b F— a an inter-
mediate a for the Sixth D b is not expressed till afterwards,
any more than in the chord F b—D a between F and b.
Until the higher later part enters with *, so long those intervals
belong to the chord b—D—F without Seventh. And even when
the a, as Third of F, is added, it is only operative in its place
and upwards, but does not serve for filling up gaps in the
intervals downwards. Thus the interrupted position of the notes
SEVENTH CHORDS OF THE KEY-SYSTEM
D - b - F — a, and similarly any still wider separation or other
transposition of the three lower notes of the chord, will always let the
chord be recognised as b — D — F — a ; the notes will be condensed
upwards into the close position of the chord and understood col-
lectively, but F will have no effect in filling up the gap between
the deeper D and b, nor a in filling up that between F and b.
On the other hand, if D or F were to come above the a,
the deeper b would thrust its Octave in between a and Z>, and b
and D each its Octave between a and Ft to fill up ; and in the posi-
tion b — D - a - F we should listen to an effect partaking, in
the interval a - Fy of that produced by the position a — b — D — F.
And as the latter actually contains in itself the Second a — b, so
the former too will make palpable by implication the subdominant
and dominant Thirds standing side by side discontinuously.
198. As, then, the notes a and b in the position of Second can
only be linked through e, the middle of the tonic triad, and yet
this linking is made impossible in a combination of sound that
contains D\F ; therefore the positions D — F — a — b, F — a — b — D,
a — b — D — F, are in themselves really foreign to the key
I_HI_II I— III— II
I— III— II
F a C e G b D
in its inner transposition to
I— III— II I— III— II
(e) G b D/F a C (e)
If the interval a — b must find a link in e, that can only happen in
a triad system in which the note can be had for linking ; which in
this case is the system of the A minor key.
II__IH_I I— III— II
II— III— I
•D f A c - E g# B
104 HARMONY
The chord is then, not b—DjF—a, but B\D—f—A. The
effect of the latter we get every time that the Seventh is not the
highest part in the Seventh chord b — D\F — a; the chord then
changes into the meaning of B\D—f—A, and is thereby attached
to the A minor key. The positions D — F — a — b, F — a — b — D,
a—b—D—F, are heard as D—f—A—B, f—A—B—D,
A — B — D—f, and their natural resolution is upon the E major
triad. Besides, in the position of Second the unprepared Seventh
resists being moved ascending to the Root — we may take for in-
stance the progression G-a'm the key of C major : D — F — G — b- - -
D — F — a — b ; while on the other hand the similar succession in
A minor, D—f—g$ — B--D—f—A — B, forms quite a proper pas-
sage. But because the chord B — D — -f — A can in harmony of the
C major key by no means be intended, therefore every position of
the chord b—D\F — a, which does not contain the Seventh, at the
top, is a normally unauthorised one. Here we are dealing with
harmonic formation as it must be to be expressed clearly and
plainly under all circumstances ; for under particular conditions of
different quality of voices, or derivation in the actual phrase, or of
the context of the chord, such a transposed position of this chord
may also be of certain and excellent effect.
(b} In the Minor Key.
199. In the minor key the place of the Seventh chord
b—D\F—a is taken by the so-called diminished Seventh chord
b — D\F — a\>. This in its organic structure has the same relation
to the minor system as the former has to the major system. But
the diminished Seventh chord is not liable to uncertainty in the
meaning of its notes. As in the dominant Seventh chord, so
too in the diminished Seventh chord, the peculiar nature is de-
finitely expressed. Naturally there can be no mention here of
so-called enharmonic multiplicity of meaning. In this chord,
SEVENTH CHORDS OF THE KEY-SYSTEM 105
even in a transposed position, as D — F — a\> — b, F — a\> — b — /),
a\> — b — D — Ft the interval of the Second a\) — b cannot give oc-
casion for mistaking the meaning, because, as we have already
seen in . the origin of the minor scale, a melodic relation between
the two notes can in no way be established. They do not require
to be connected. The note a\> can only be derived melodically
from G, and b only from C ; a mediation for the passage of one
into the other is not contained in the minor system, which sets
out from the notion of division, and in its whole essence rests
thereon. The system of the minor key, having the negation of
unity for its principle, in the sounding together of its subdominant
and dominant Thirds contributes to dissonance that which in this
quality is most decided : the diminished Seventh chord.
200. We have seen how Seventh-harmony in general is formed
by melodic progression in a union of triads. But as for the
Seventh chord which in the major as well as in the minor
system contains the Thirds of the subdominant and dominant in
dissonance, there is a hindrance to its production, namely the
harmonic separation of these notes, that prevents them from
passing into one another melodically. To a • • b, as well as to a\> • • b,
the element to link the passage and make it intelligible is wanting.
For although in the major scale a link for the Second a • - b was
found in *, yet with the harmony b — D\F — a this mediation will
not serve, because the tonic Third is taken away by the combined
sound of D/F, and a thing cannot be affirmed and denied at the
same time. But if the progression here, a • • b, is not linked, then
also the Seventh chord b — D\F — a cannot have come from the
triad D\F — a followed by the triad b — D\F. For that succession
cannot happen otherwise than with the progression a • - b, or from
b — D\F to D\F — a with the progression b-a\ the first in the
form D\F—a-"D—F—a—b\ the other, b— D \F--a— b—D—F.
Both successions awake a sentiment for the key of A minor, in which
106 HARMONY
the step A--B may be linked in E without contradiction by the
harmony B\D—f—A. Similarly in the key of C minor the Seventh
chord b — D\F — a]) is not to be derived from a union of passage
between the triads b — D\F and D\F — a\>, which must appear in
the forms b—D\F--a\>—b—D—F, and D\F—a\>--D—F—a\>—b.
Indeed the separation, for melodic progression, of a\> and b is still
more decided than that of a and b ; or rather it is quite absolute,
because every link fails. Therefore, altogether, the Seventh-harmony
b — DjF — a, or b — D\F — a\>, is not to be looked on as a passage,
fixed, from one diminished triad into the other, but as a passage
of the tonic triad into the subdominant and dominant triads at the
same time : C — e — G--b — D\F — a ; C — e\> — G--b — D\F — a\>
(pars. 182-187).
20 1. Were any other Seventh chord than this to be introduced
with Root and Seventh unprepared, the contradiction would lie in
the note that links the dissonance having opposite meanings at the
moment of entering. But that in itself is contrary to the sense of a
reasonable reality ; the substance of which is, that the one meaning
in its passage to the other is contained at the same time with it in the
intermediate element. Being at two is not an element to start from ;
it can only be an element to pass through. But the understanding
of the Seventh chord b—DjF—a, or b—D\F—a\>, does not at all
depend upon the determination of a linking note as being at the
same time One and the Other ; for in the first chord the linking e,
in the other the e\>, is taken away by the combined sound of D\F
as unity. The intervals b a and b dp here are not dissonant
in the meaning of a doubly determined unity, but because of D\Fy
which is twoness taking the place of unity. And as the dissonant
combination DjF'm following upon the tonic Third cannot have
preparation, and does not need it, because in itself it expresses
unambiguously the sense of an intelligible alteration, so too the
Sevenths b a and b (h, which depend upon it, can enter un-
SEVENTH CHORDS OF THE KEY-SYSTEM 107
prepared. The Seventh chord b — DjF — a can follow upon the
tonic major triad C— e — G, and the Seventh chord b — D\F—a\>
upon the minor triad C — e\> — G, without any unpleasant effect,
such as results from every other Seventh chord introduced in this
manner. For instance, the successions D-\F~-a~-C — e — G — b,
e—G—b.-D!F—a—Cy F—a—C—e—G—b—D\ or D\F— a\>—
C—e\>—G—b, e\>—G—b . - - DjF—a\>~C, F—a\>—C - • -
4> — G — b — D, could not be written.
202. Of the restricted position of the intervals of the chord
b — D\F — a, and the reason for the absence of these restrictions,
in the chord b — D\F — a\>, we have already spoken ; and it will not
be necessary further to explain why the Seventh chord b — D\F—ay
in correct progression, can only be produced from a position of the
tonic triad with the Fifth at the top ; although the Seventh chord
b — DjF — a\> may be derived from any position of the tonic triad.
So that we may have the successions e\> — G — C ••• D — F — a\> — b,
G—C—e\>~- a\> — b—D—F, but not e—G—C-- D — F—a—b, nor
G—C—e - • • a—b—D—F.
III. Seventh Chord upon the Fifth of the Dominant.
203. For the Seventh chord DjF — a — C of the major key, the
minor key contains the chord DjF — a\> — C. The former may be
confounded with the Seventh chord d — F — a — C, the dissonant in-
tervals D\F and D a being really inclined to pass into the
consonant ones d — F and d a ; but in the Seventh chord of like
place in the minor key DjF — a\> — C this ambiguity is not present.
On the other hand the latter has an outward resemblance to the
Seventh chord d — FjA\> — c of the Et> major key, answering to the
chord b — DjF — a of the C major key. Position, and the treat-
ment suited to the dissonance, will always make it easy to dis-
tinguish between the chords DjF — a\> — C and d — F/Afr — cy as well
as between the chords DjF — a — C and d — F — a — C.
I08 HARMONY
204. The structure of our keyed instruments it is principally
that leads to the mixing up of such chords, and altogether that
allows the unclearness of the harmonic notion to continue. In the
heart of the thing itself, to determine the whole is also to determine
with certainty each single part. But for the pianoforte the notes
£Jf, /t>, £, may be written to make up a well-sounding triad ;
clearly then a natural and systematic demonstration of harmonic
laws ought not to be looked for on the keyboard. Where even the
enharmonic difference has gone, which in writing and the names of
notes is still preserved, there the diversity of notes called by the
same names (as D and d, &c.) will be yet more certainly overlooked.
205. That the triads joining limits, and similarly the Seventh
chords in which they take part, are the same in the system of the
minor-major key as in the system of the minor key, is already
known to us ; their occurrence therefore in this system needs no
particular discussion. The diminished Seventh chord here relates
to a tonic major triad, while in the minor system it has relation to
a tonic minor triad.
DEGREES OF DISSONANCE.
206. The difference in the dissonant effect both of the Seventh
chord and of the chord of suspension depends principally upon
the melodic relation of the dissonant notes in their position of
Second. The more decidedly they contain the melodic progression
in direct chord-union, the harsher will be the dissonance of the
interval, harmonically separated, whilst simultaneously sounding.
In the scale we found that progression by a Second was throughout
made possible in the same way ; but in the union of triads the pro-
gression was conditioned by simultaneous motion of several parts.
Thus in the given key of C major a single part could move upwards
DEGREES OF DISSONANCE 109
and downwards in the succession C-D--e — F- G- >a- • £•• C;
but in chord-union, where the triads C — e — G and G — b — D may
not follow immediately but only linked in C — e — G • • e — G — b • •
G — b — D, the very first melodic movement is not C- • D but C- • b,
and the second is e • • D ; and even if both parts move at once, C is
still related melodically to b, not to D, just as when they move suc-
cessively, and D to e. But in the succession C — e — G--C — F—ay
which consists of the contracted progression C — e — G--C — e — a-~
C — F — a, e will have advanced to F and G to a. Lastly, in the
direct union e — G — b--e — G — C, bean only progress to C. There-
fore the harmonic melodic degrees in the key of C major are b» Cy
D — e,e- F, G --a] and the steps C- - D, F' • G, a - - £ remain excluded
from the progression of the parts in real triad-unions. These first
become possible in the successions D / F — a - • • F — a — C, G — b — D • • •
b — DjFy b — D\F- • >D\F — a, i.e. in triads of the transposed system,
as : F—a—C" • F—a—D, D\F—a • • • C—F—a ; G—b—D • • •
F—b—D, b— D\F"'b—D—G\ D\F—a--.D—F—b, b—D\F-"
a — D — F ; consequently the necessity for such progressions exists
only outside the real harmonic unity. The steps C—D, F»G,
a-b, placed as Sevenths in the chords D\F — a — C, G — b — D\F,
b — D\F — a, are less harshly dissonant in proportion as they have
less determination to pass melodically into one another in their
position of Second. Melodic separation is most decidedly expressed
in the minor system between the Thirds of the subdominant and of
the dominant, in C minor between dp and b. And the less a relation
of succession is called up in these two notes placed as a Second, so
much the less harsh is their simultaneous sound in the Seventh
chord b—D\F—a\>. Besides this chord (as well as b—D\F—a,
which answers to it in the major system), when derived from the
primary position of the tonic triad, was necessarily produced with
both the diminished triads contained in it, b— Z>/^and DjF—a^
also in the primary position ; its dissonant notes being thus placed
not as a Second but as a Seventh. The other Seventh chords, which
no HARMONY
arise from unions of triads related in the Third, can only be formed
in an inverted position from the tonic in the primary position and
contain the dissonant interval as a Second.
207. The above may stand as the reason for the mildest effect
of dissonance being exerted by the diminished Seventh. The major
Seventh must, on the other hand, be so much the more rough in
dissonance in that its two dissonant notes have to one another the
closest melodic relation, most strongly determining them to come
in succession one after the other, and not to unite in sounding to-
gether at one time. Thus in the key of C major the Seventh chords
F — a — C — e and C — e — G — b are the most dissonant, because they
contain, fixed in simultaneous sound, the progressions e • • F and
b • • C, which are strongly determined as melodic by triad-union.
208. Less dissonant than these Seventh chords and more dis-
sonant than those previously named will be found the Seventh chords
a — C — e — G and e — G — b — D. They contain in G — a and D — e
as dissonant interval a progression belonging to direct triad-union,
but not one melodically urgent to the same degree as the Seconds
e — F and b— C, and in such less degree these Seventh chords too
will be less harshly dissonant.
209. The degrees of dissonance of the Second are presented
in the following order of ratios, advancing from the less degree
of harshness to the greater : —
b 64 : 75
b)
D = 8:9
D
9 : 10
e:F|
•c - I5:i6
b
DEGREES OF DISSONANCE MI
210. What has now been said of the dissonance of the Seventh
chord may be applied to the dissonance of suspension as well.
That in both cases the difference of effect must depend, not upon
the kind of dissonant interval alone, but upon the whole nature of the
chord, it will hardly be necessary to observe. But it would require a
special treatise upon suspensions and Seventh chords, if an explana-
tion were to be given of all the characteristic peculiarities of disso-
nance. A few remarks only in this respect may still find place here.
First there is the particular quality of the combination of triads
contained in the Seventh chords. In the major key they can only be
formed from a major and a minor triad (C — e — G — b, F — a — C — e),
a minor and a major triad (a — C — e — G, e — G — b — D\ a major
and a diminished triad (G — b — D\F\ a diminished and a major
triad (D\F — a — C\ and from two diminished triads (b — D\F — a).
To these the minor key adds further by its augmented triad
(e\) — G — b) the Seventh chords of which that forms part
(C — 4> — G — b, e\> — G — b — D), as well as the Seventh chords pro-
duced from the union of the major and minor limits (b — DjF — a\>,
D\F — a\> — C\ and the chords hereafter to be considered which
arise from the union of the limits of the extended system. But
besides this difference in the conditions of combination, which must
impart different degrees of dissonance even to chords which have
outwardly equal distance of Seventh, there is also the melodic rela-
tion of the dissonant notes to the notes adjacent on the outside
of the simultaneously sounding interval of a Second, to influence
the effect of the dissonance. A nearer melodic relationship to those
neighbouring notes, because it makes easier the step to resolution,
will also make the dissonance seem less harsh than when there is
a wider separation between them.
Thus the dissonance of the Seventh chord of the tonic,
C — e — G — b, is harsher than that of the Seventh chord of the
subdominant, F — a — C — e ; although the two Seventh chords in
II2 HARMONY
themselves are quite of like structure ; and the Seventh chord of
the tonic in the minor key, C— e\>— G— b, is harshest of all. For
the progression to resolution in the first is b - • a, 9:8, and in the
second e D, 10 : 9 ; while in the third it would have to be b> -a\>>
75 : 64, which for melody is quite discontinuous. On this account
the last chord cannot possibly be resolved inside the key and
with a descending Seventh. Again, similar Seventh chords, as
F—a—C—e, C—e—G—b, or a—C—e—G, e—G—b—D, are
also of different effect by leading on resolution to different kinds of
triads :
F— a— C— e • • • F— b— D, C— e— G— b • • • C— F— a ;
FX b° GX F
a— C — e— G- • -a— D— F, e— G— b— D- • -e— a— C ;
a7 D° e7 a
so that besides the degree of dissonance, and the progression of
the parts in resolution, this condition of succession will also help
to characterise the Seventh chord.
211. The examination and analysis of a given dissonance, to-
gether with all its attendant circumstances, may be laid down
with perfect distinctness for each distinct individual case. On the
other hand, it would be impossible to establish a general formula or
comprehensive scheme for the occurrence of all possible pheno-
mena. The manifoldness of the formation is infinite, even within
the boundaries of what is determined by law. Manifold as are the
ways in which the Seventh chord can be prepared and resolved even
inside its key, yet the multiple meaning of the chord, its presence in
more keys than one, as well as the modulation that may take place
at the very moment of resolution, endow it with a wealth of possible
developments, branching out so that, even if a classification were at-
tempted, no general mental survey would be afforded. If from a
knowledge of the structure of the human body and of the functions
DEGREES OF DISSONANCE 113
of the muscles we can explain every motion of the individual mem-
bers, that will content us ; we shall not set about finding a formula
of motion for the expression of a series of changing actions.
Only those triads that are most nearly related can pass into
one another, or be developed one from another, in a metamorphosis
of triads according to the same elements of the notion from which
the triad itself was produced. But the manifoldness of possible
development and propagation is inexhaustible, and if we wish to
escape indefiniteness, it will be just as requisite to consider each
particular phenomenon in its own individual existence, and to
allow it some special name, as it is to try always to have the whole
in view, membership in which is the life of the individual, seeing
that the whole is reconstituted a whole only through co-ordination
of its parts. For as the life of the member is in the whole body,
so the life of the whole body is in its members.
CHROMATIC RESOLUTION OF DISSONANCE.
212. As regards the progression of the dissonant notes, resolu-
tion of dissonance consists briefly in this : that by diatonic melodic
motion of one or other of them, or of both, a relation of consonance,
Third, Fifth, or Octave, in the direct or inverted position of the
interval, is reached. Now we have already met with one kind of
resolution, involving chromatic progression of one or other of the
parts. It was that in which the interval of the minor Seventh
passed into the Octave by the diatonic progression of one part and
chromatic progression of the other (par. 140). But any chromatic
alteration that, during the progression to resolution of one note of
the interval, is effected in the other which does not move diatonic-
I
II4 HARMONY
ally will be not inconsistent with the other kinds of resolution.
The dissonance C — D is resolved into C — e, but it can find its reso-
lution in c$ — E just as well ; for the interval c$ — E is, like C — e, a
consonant interval, i.e. subsisting in the triad. Thus too the pro-
gression yjf— A— C—D "- G/A—cff—E sounds right and agreeable.
Similarly the dissonance C — D, instead of going to b — D or
fy — Dy is also able by chromatic progression of the upper note to
pass into B — d$ or B\) — cfy ; and hence we perceive the admissi-
bility and reason of successions such as, e.g., /Jf — A — C — /?•••
F$—A—B—d$, and F—a\>—C\D ••• F—a\>—B\> I D\> or
F—a\> — B\> — d\>. It will not be difficult from this process to
explain the successions written below :
gJ-B/D-f
... G-bb-cJ-E
... fJ-A/C-eb •
a-.-vn;
d-.-vn;
g-VII°7
gJ-B/D-f -
... A-c-dJ-FJ
- afl-CJ/E-g •
a : VII°7
e : VII°7
b : VII°7
c—vn;
fjf : VII°
In the first, the Root is lowered chromatically while the Seventh
descends ; in the second, the Seventh is raised chromatically while
the Root ascends. The two successions are in reality no other than
those which we obtained above in the progressions of harmonies of
the Seventh by taking every alternate member of the descending
and ascending series, the first as :
G— b— D/F ••• G— b— C— e ... F— a— C— e ••• F— a— b— D ...
G— b— D/F ••. a— C— D— F ••• a— C— e— G ... b— D— e— G-»-
The last was seen to be only admissible in the Seventh chords
of the transposed system, because in these alone can the Seventh
move upwards to the Root ; but in the above succession, which
CHROMATIC RESOLUTION OF DISSONANCE 115
progresses only in diminished Seventh chords, it may be used
without interruption.
213. There is something violent or forcible contained in these
chromatic sequences, especially in the second of them ; but that
lies in the continual change of key. A chromatic progression,
as regards harmony, always leads into a new key-system ; and the
above successions lead from one minor key into another — that is,
from one isolated system into another. For, as we saw earlier,
the minor keys are not linked together among themselves in the
same way that the related major keys are.
Then, again, the stiffness of the progression of the parts in these
successions is partly due to the notes which are combined. In the
chord g§ — B\D—f of the first series, g§ in union with D and/
has far more inclination to move to A than to G. Similarly in the
second series, / taken with g J and B would like to progress to E
and not to /Jf. The same constraint appears in the descending
motion in B • • b\), in the ascending in D • • d§, so that in these suc-
cessions of harmony not more than two of the four parts are ever
allowed an unconstrained path : D—f in the first going to c$ — E,
_g§ — B in the second to A — c\ but the other two are obliged to
progress contrary to their free tendency. The reason why the
second of the above successions progresses with still less readiness
than the first we shall find opportunity for discussing when we treat
-of modulation.
I 2
HARMONY
THE ESSENTIAL DIFFERENCE OF SEVENTH-
HARMONY OF THE UNTRANSPOSED AND
OF THE TRANSPOSED SYSTEM WITH RE-
SPECT TO CHORD-POSITION.
214. In the descending series of linked Seventh-harmonies :
C-e-G ...
, C-e-G-a -.. C-e-F-a ••• C-D-F-a •••
6
6
6
5
3
4
3
4
2
b-D-F-a
... b-D-F-G •
,. b-D-e-G •
.. b-C-e-G
7
5
3
6
5
3
6
4
3
6
. 4
2
we see the four different positions or inversions of the chord pro-
duced from one another by the conditions of succession, and con-
sequently justified in their effect.
While, however, the first position of the Seventh chord, which
consists of Third, Fifth, and Seventh ; the second, which consists of
Third, Fifth, and Sixth ; and the fourth, which consists of Second,
Fourth, and Sixth, may be freely used even outside this strict pro-
gression in all cases where there is a suitable preparation of the
dissonant interval, the third position, consisting of Third, Fourth, and
Sixth, does not submit, even with preparation of its dissonance, to
such unconditional usefulness. In the Seventh chords of the
untransposed system the third position produces a feeling of
something being upside down, unsupported, wanting basis.
In this inversion the Fifth of the Seventh chord, i.e. the Fifth of
the lower of the two triads joined in the chord, has become deepest
SEVENTH-HARMONY 117
or bass part ; in it therefore we have the Six-Four position of the
lower triad, a position which, even as the inversion of a triad, can
only be introduced when properly led up to, because the Fifth has,
and in its sound expresses, a meaning opposite to that of the basis
of the chord.
215. This position of the harmony of the Seventh can, however,
enter, even apart from the connexion or derivation above, in the
.Seventh chords in which the limits of the key-system are heard
sounding together : that is to say, in the dominant Seventh chord
and in the Seventh chords upon the Third and Fifth of the domin-
ant ; but in the last with greater clearness only in the minor key,
by reason of the ambiguity to which it is subject in the major key.
Thus the Seventh chords of the transposed system, G — b — D\F>
.b-D\F—a^ b—DIF—a\>, D\F—a\>—C, can appear in the inver-
sions D-F—G—b, F—b-D—a, F—a\>~b-D, a^—C—D—F,
without tying the Fifth placed as bass ; the others from the untrans-
posed system, F—a—C—e, a—C—e—G, C—e-G—b, e—G—b—D,
and F-a\>-C-e\>, a\>-C-e\>-G, C-e\>—G-b, e\>-G-b-D
(the last two in so far as they ought at all to be introduced as har-
monies of the Seventh), can only appear in the Six-Four-Three
position with the bass note tied.
216. But if we ask for what reason the Seventh chords which
belong peculiarly to the transposed system admit of an inversion
that appears unsuitable to the Seventh chords of the untransposed
system except where it has arisen by the conditions of a succession,
the answer must be sought for in the notion of the transposition
itself, (e) G — b — D\F — a — C (e). Because that which is pre-emi-
nently separated is here placed together as middle, while the unity of
the middle is separated and placed asunder as boundaries, therefore
the whole transposed system, in everything that is referred to its
middle or that participates in it, is a system of dissonance. Thus
in this dual nature of the whole, the combined sound of D1 F
n8 HARMONY
must have, and maintain, the meaning of unity ; so that the Seventh
chords G—b—D\F, b—D\F—a, D\F—a—Czxz to be considered
as triads of absolute dissonance, G — b — W, b — 13 — a, 3f — a — CY
in which D and F must, for the meaning of the chord, count not only
as unseparated but also as undistinguished. Accordingly the Fifth
D of the first of the three Seventh chords above is in ^7 at the same
time the Root F ; the Fifth of the second, F, has in itself Root-mean-
ing ; and the Fifth of the third, ay is in JF also Third of F (being for
the effect more clearly not Fifth in the minor Third a\>). But in
these chords the Fifth does not receive the double meaning, as
is the case in the rest of the Seventh-harmonies, through another
triad joined to the first ; rather it has it in the chords themselves,
agreeably to their nature, as a meaning unseparated and undis-
tinguished.
2ij. In the two last of the combined triads of dissonance of the
transposed system, b — W — a and ^7 — a — C, having ascribed Root-
meaning to the ^ of the first and participation in Third-meaning to-
the a of the second, it would seem that we ought now to ascribe
clear Fifth-meaning to the a of the first and to the C of the second.
But the notes a and C do not originate as Fifths in triads of
absolute dissonance ; they are contained in them as real Sevenths,
because this interval of the chord can never be anything but Fifth
of the upper triad. Moreover, in virtue of the Seventh-meaning
of the notes a preparation by tying is assured to them in every
position, and therefore also when they appear as basis ; and then
the Six-Four-Two position of the Seventh chord, like the positions
of Seven-Five-Three and Six-Five, may be used as freely in these
chords of dual nature as in any other.
2 1 8. But the Seventh chord on the Fifth of the dominant will
always in the major key maintain its dual nature with difficulty.
The chord D\F — a — C is too liable to have its meaning changed
into d — F — a — C, which does not annul the unity of e. Therefore
SEVENTH-HARMONY 119
it is only in the minor key, where no such ambiguity is present,
that this Seventh chord is capable of being inverted in the Six-
Four-Three position, (D\F—a\>—C in the position a\>—C—D—F\
or of appearing in other transpositions that contain the relative
Fifth a\> of the chord as bass ; but D\F — a — C cannot appear as
a — C — D — F (apart from the derivation of the note a from b, as
in the succession b — D — F--a — C — D — F, or from its being tied,
as in a — C — e--a — C — D — F).
219. For insight into the manner of chord formation and trans-
formation, it is beyond everything essential that the thought of a
complete and originally determined series of notes should be
entirely dismissed. The chord is not determined by given notes,
but they are themselves produced, i.e. determined harmonically,
through the vital weaving and working of the chord-notion.
The harmonic thought itself, incorporated in these determina-
tions of intervals, is as the soul that forms in them the body
for itself. What distinguishes a Third-note from the Fifth-note
of the same name, d from D, is not the trifling difference of pitch,
but the quite different generation of the two. That D stands with
G, d with #, in the relation of Fifth, that the one note belongs to
the dominant side, the other to the subdominant, is their essential
difference. Similarly a is distinguished from the A which forms the
Fifth to D, and therefore cannot enter into any union with D as a
Fifth. Therefore we do not hear consecutive Fifths in a succession like
(7 — e C G "• J) — F b a, in which the meaning
of the note a in the second chord is clearly expressed. But the
opposite succession J) — F b a ••• {7 — e C ^ we
shall find to be inadmissible, because it contains in the outer
parts, J) a • • £j G, similar motion from the interval
D a to the Fifth C G. The difference between a and
A will also be perceptible if we accompany the first four notes
of the chorale <Ach, Gott und Herr,' £••/£••*••/£, first with
120 HARMONY
rv\
the harmony of the Roots C' ' I & ' ' F' ' I C, and then with the
harmony C"ICr"D"l&' In the last the melody of the
chorale C- - \b - - a - - / G changes its Third-note a into the Fifth-note
A, and the intonation of the note in singing will be otherwise
determined (and sharper) than the intonation of the Third a in
the triad F— a — C.
SEVENTH CHORDS WHICH ARISE FROM THE
UNION OF THE LIMITS OF THE EXTENDED
KEY-SYSTEM, AND SEVENTH CHORDS CON-
TAINING AN AUGMENTED TRIAD.
220. We have earlier spoken of a precession, a shifting on of
the key-system for one member of the chord series (pars. 54-62),
and have seen produced from it towards the dominant side, in the
major key as well as in the minor, intelligible triads by union of
its limits. The system of the key of C major
F— a— C— e— G— b— D
was thereby altered into
a— C— e— G— b— D— fj ;
and from the system of the key of C minor
F— a[>— C— e'b— G— b— D
there arose
ab— C— et>— G— b— D— fj.
221. The triads produced by union of limits from this system
extending towards the dominant side are, in the key of C major :
SEVENTH CHORDS 121
a — C\ and in the key of C minor: Z>—
In the major key the chords D—f$\a, f$\a — C maybe easily
known and distinguished from the chords D—f§ — A, f$ — AjC.
If in a harmonic progression the Third of the Fifth of the dominant,
/"$, is led chromatically from the subdominant Root F, then,
although it is not contained within the compass of the C major
system, this/J will not give the impression of the key of G major,
so long as the subdominant Third a remains joined to it ; as, e.g.,
in the harmonic succession :
a-C-F- • -a-C-f J- • -b-D-G, F-a-D- • -f J- a-D- • -G-b-D.
For this progression of F"f§ does not oblige the Third a to
pass into the Fifth A. Consequently the chords a — C— /$ and
f$ — # — D still participate in the subdominant side of the system
of C major, which therefore still continues.
222. And so too the Seventh chords in which the interval of
the joined limits, /"jf/tf, occurs may be produced naturally without
transformation of the Third note 80 into the Fifth note 81 ; except
the Seventh chord b—D—f^fa (b—D—f$\a\>\ which will be
spoken of later on. In sounding the succession F — a — C — D-'~
/Jf — a — C — D ••- G — C — e the note a in the second chord need not
give up its relation to C of minor Third below, namely 5 : 6.
223. But the chords of like position referred to the minor key,
and to the major key with minor subdominant, D— -f$/a\),
f$\a\> — C, must be considered more particularly, both by them-
selves and also as to the part they take in Seventh-harmony.
224. In the union of the limits of the closed system, both of
the major and minor keys, there arises an interval D\F that does
not correspond to the ratio 5:6 of the minor Third. The
ratio 27 : 32, in which these notes stand to one another, is out of
direct triad-reference. And from the union of the limits of the
122 HARMONY
major system extending towards the dominant side there results,
in the combined sound of/J/tf, exactly the same interval and ratio
27 : 32. But in the joined limits of the minor system extending
towards the dominant side, and of the system of the minor-major
key, which, as regards the dominant and subdominant, is like it, we
obtain the interval of the so-called diminished Third, /Jf/0b- The
ratio of its vibrations is 225 : 256, as we easily find by taking twice
the progression of a leading note ; for
fff: G = 15 : 16
G : at>= 15 : 16
225 : (240) : 256
ftf : (G) : at?
In this combination of sound both notes, /( and a\>, supposing
the question to be of their melodic derivation, can only be referred
to the note G ; for in the minor key/ft cannot be led from the e\>
lying below, nor a\> from the b lying above, because the augmented
Second fails to be mediated as a passage.
225. Nor yet in the extended minor-major key
ab— C— e— G— b— D— fj
can the note/ift have come from £, major Third of the tonic. In
the scale indeed a mediation (in ft) for this step would be established
by the Fifths
I II
e b fj
I II,
just as in the closed system the step from the sixth to the seventh
degree is mediated by the Third of the tonic in the same way. But
in the harmonic progression, in the succession of chords, which is
presented in triads related in the Third and not in the Fifth, the
chord-connexion for this case could only consist in the overlapping:
SEVENTH CHORDS 123
of /Jf/^b — C — e- But here neither C nor a\> affords a mediation
for the passage, the possibility of which lies always in this : that a
note which is in itself a triad element shall take on another chord-
meaning. But neither C with /J, nor e with a\>, has any meaning
of harmonic unity.
226. Now the link for the progression G • • a\> is contained in Cr
and for the progression G • «/Jf in D. Therefore for the double
progression G must at the same time be referred to C as Fifth and
to D as Root. For the combination of the diminished Third (/$/#[?)
to be produced, G ought to be simultaneously opposite. And
it is this contradiction that is expressed in the effect of the dimi-
nished Third as a harmonic interval.
227. We have already become acquainted with one chord, for
whose dissonant interval a natural position could only be found
apart from the direct melodic relation of its notes. This was the
Seventh chord upon the Third of the dominant in the major key,
1) — j) IP — ^ in which the Seventh must always be the highest part,
to prevent its dissonant notes b a from being transformed into
the notes B A of the key of A minor (par. 198). Something
akin to this takes place with the interval of the diminished Third
/";ft/4>. .That stands here with both its notes referred melodically
to the note G ; it seems like a progression of G • • a\> and G - •/$
made at one time. But as such it also contains a contradiction ;
just as the Second a — b, which can melodically be linked only by
the Third of the tonic, does in a harmony of the C major key,
which, because it negatives the Third e, cannot therefore provide
that link. Now here, placing the notes b a as a Seventh, a
direct melodic relation of the notes in the series
C
c
is not raised : b may be regarded as derived from C, and a from G ;
124 HARMONY
and the interval as linked in the Root and Fifth — in the progres-
sion C-b by G, in G-a by C. And similarly with the interval
of the diminished Third /J/#b ; if the same notes be placed as an
augmented Sixth a\> — /J, with reference to a separate derivation.
G • • ab — fj • • G
^"c^ TT
it will likewise be no longer contradictory. For though the melodic
derivation of the two notes is still from G only, yet it is not from the
unison of that note, but from its doubling in the Octave : from one
and another G, of which the lower or earlier is to be referred as
Fifth to C and the higher or later as Root to D. And then the
position which the dissonant notes a\> -ffy are found to occupy, is
not such that they are turned melodically towards each other, but
they are placed out of melodic relation. In the interval of the dimi-
nished Third, /Jf — a\>, we hear the progression from the G which
ought at once to be Fifth and Root ; in the interval of the augmented
Sixth, #b /$, we hear the progression from the G which first was
Fifth and then became Root.
228. Here again the one condition established for the position
of the chord is, that in a combination in which these united
boundary notes take part, they may appear only in the position of
Sixth, and not in the position, either close or extended, of a Third.
229. The Seventh chords of the minor key in which the com-
bination is contained are to be found on the Third and Fifth of the
dominant and on the new note which has entered on the dominant
side.
In the key of C minor they are
b— D— fj/ab, D— fj/ab— C, fj/ab— C— eb
The two last, with the interval of the diminished Third in the position
of Sixth, accost us as well-known harmonies ; the middle one,
D—f$la\> — C, is found also in the minor-major system upon the
SEVENTH CHORDS 125
Fifth of the dominant, and the last, as /J/#b — C — ey upon its Third.
But the first, which seems as well authorised a construction as the
others, nevertheless yields no intelligible chord from any trans-
position of its notes.
230. The combination b — Djf$ — a\) contains in b and /J two
leading notes at once. They are shown to be really such by the fact
that both the one and the other in union with a\) can only move up-
wards : b can only lead to C, and /$ only to G. But b can only be
maintained as leading note in combination with F, and /"Jf only in
combination with C ; the one in the chord b — D\F, the other in
the chord /J/#t> — C- The Seventh chords in which these com-
binations take part are in the one case G — b — DJF, b — D\F — a\>,
and in the other case D — /J/tft> — ^ f$\a\> — C — e\>. Therefore
the Seventh chord b — D— -f$\a\> is self-excluded as containing an
inner contradiction.
The same applies fully to the system of the major key. Here
also union of the limits of the extended system can only give
rise to the Seventh chords D — -f$ja — C and/J/tf — C — e, and not to
the chord b — D—f$/a. In its relation to the key of C major
the note /J is still to be derived only from G ; and in the com-
bination b — D—f$la precisely the same doubleness of leading
note is found as in the chord b—D—f$\a\>. What imparts to
the chord b—D—f$\a an appearance of admissibility, can be
only the opportunity for confounding it with the Seventh chord
b — D—ffy — A, in the key of G major upon the Third of the
tonic.
231. Accordingly of the three Seventh chords in which the
diminished Third takes part, only two are left as really possible
and therefore intelligible : that upon the Third of the dominant
and that upon the note which has entered the system: D— -f$\a\> — C
*ndf$ja\>—C—e\> (or, in the minor-major system, f$\a\>—C—e\
each of them with its diminished Third in the position of Sixth.
I26 HARMONY
The essential dissonance of the first lies in C — D, and of the second
jn £|}__f J (y_fj). But besides that, both contain in the com-
bination yjf — a\> the further dissonance of the joined boundary
notes. There was this already in the chords with the combinations
D\F and/J/tf; but here as /$/#[? it is the more harshly pro-
nounced, in that the expression of a note divided against itself
is more decided.
232. The Seventh chord on the Third of the dominant in the
system of the major key allowed only of a restricted position of
its intervals. This was reduced to the condition of the Seventh
having to be highest part in the chord ; the other intervals might
then be used in all transpositions. The chords with the interval of
the diminished Third require that interval to be in the position of
Sixth, and have their peculiarity brought out most clearly when the
lower note of the interval of Sixth is in the bass. Nevertheless
so long as the interval of separation keeps its position of Sixth,
they will admit of another note of the chord being bass without
becoming unintelligible.
233. Here too it must again be remembered that we are still
speaking only of directly intelligible harmonic constructions, as
presented in the natural order. For, as with the dissonant interval
b a in the Seventh chord on the Third of the dominant of the
major key (b — D\F — #), which can only be used with the Seventh
as highest part, if the effect is not to be ambiguous, and can yet under
special conditions be used in the other position with excellent effect,
in good music ; so also the interval of the diminished Third or
Tenth can be used, in certain particular cases, in its untransposed
form as part of the chords discussed here. In especial we find it
very often used in new and the newest music as a means of pro-
ducing a striking effect.
THE AUGMENTED TRIAD 127
THE A UGMENTED TRIAD AND ITS
OCCURRENCE IN THE SEVENTH CHORD.
234. In the system in extension of the minor key, the so-called
augmented triad stands to the Seventh chord upon the Fifth of the
dominant in a relation of harmonic opposition ; in the key of C
minor, e\>— G — b to D— f$/a\> — C. In the system from the junction
of whose limits this Seventh chord is formed,
ab— C— eb— G— b— D— fj,
the dominant (G) of the key in its meaning of Root, determined at
once positively and negatively, (in G — b and e\> — £)> forms the
middle. As in the closed system
F— a— C— e— G— b— D
the Third of the tonic, e\), has its progression to the limits D and
F, so here the dominant G must progress to the limits /Jf and a\> ;
and thus for the augmented triad the Root e\> can only move to
Dy and the Fifth b only to C. Consequently there results the
relation of succession e\> — G — b ••• D—f$\a\> — C.
235. The parallel succession of two major Thirds, which in the
progression of a major Second, F — a • - G — b, would be disconti-
nuous, is here continuous : the succession G — b •• a\> — C or
•e\>— G •• D— -fjjf being perfectly smooth. The linking takes
place thus: the passage G— & - a\> — C is understood as G—b»
G—C • • a\>—C, and the passage e\>— G - • D— /# as e\>— G - • D— G • •
D— /J ; that is, as a contracted double progression, in which the
succession G — b--a\> — C finds its linking element first in G and
then in C, while the succession e\> — G • • D — /J finds it first in G
128 HARMONY
and then in D. In the progression F — a-G — bt however, con-
sidered as a succession of F— a • • F — b • • G — b, such a linking
element does not exist, because b does not stand to Fin any relation
of unity.
236. In theorganic construction of the minor key the augmented
triad is found upon the Third of the tonic, and in the system of
the minor-major key F — a\> — C—e — G — b — D upon the Third of
the subdominant. In both cases its existence is implied in the
notion of the key. But, besides that, it can also be produced in
two ways by progression. Firstly, by raising chromatically the
Fifth of the major triad, e.g. E\>— g — B\>- • -e\) — G — bt which denotes
a passage from the region of the key of E[> major into that of
the related key of C minor. Secondly, by lowering chromatically
the Root of the minor triad, e.g. E — g — B-~e\) — G — by which
would here express a passage from the region of the key of E
minor into that of the related key of G major with minor sub-
dominant. That such a chromatic progression does not effect
a distinct modulation into the other key is easily perceived. The
key is notwithstanding awaked for the moment in the augmented
triad, which contains precisely that notion of twoness :
-I
+ 1
G,
from which the minor key or the major key with minor subdominant
alone can proceed.
237. The augmented triad forms part of the following har-
monies of the Seventh :—
I. (a) The Seventh chord upon the Root of the minor key, e.g.
in A minor,
A-c-E-gff.
THE AUGMENTED TRIAD 129
(b) The same, as Seventh chord upon the subdominant of the
minor-major key ; e.g. the chord just written, in the key
A-c-E-gS-B-dtf-Ffl.
In the latter indeed its appearance is more easily made possible,
because there £-J, the Seventh dissonant to A, can be resolved on
F J ; not by the linking of chords, for that does not exist in
JF$I A — c — E — -^"J, but by continuous melodic progression in the
system. In the system of A minor, g§ has no melodic progression
to/; accordingly the resolution by means of the ascending Root
can alone be used, which we have seen cannot make good a claim to
principal importance. At the same time by this resolution there
is always given the possibility of this Seventh chord in the minor
key.
II. (a) The Seventh chord upon the Third of the tonic of the
minor key, e.g.
c-E-gft-B.
(b} The same, as Seventh chord upon the Third of the sub-
dominant in the minor-major key ; e.g. the chord just written, in
the key
III. The Seventh chord upon the Third of the tonic of the
minor key in extension towards the subdominant side. E.g. in
respect of the key of A minor,
o-E-g}/bb
from the system
bt>— D— f— A— c— E— gj.
This last chord, because it contains the interval of the diminished
Third g$\b\> in the union of the limits, can only be used when that
interval is inverted as the augmented Sixth.
The Seventh chord which we obtain upon the Fifth of the tonic
K
I3o HARMONY
(that is, on the dominant) of this system, E—g$\b\>— D, has the
same form as that upon the Fifth of the dominant of the minor
key in extension towards the dominant side ; in this case, for
example, the same chord would be given by the D minor system
with the Third added beyond the Fifth of the dominant :
The resolution expected, however, in the latter case is that into
the triad of A major, and not that into the triad of A minor, as
required in the first. This is for the same reason which, as we
previously found, made the extension of the minor key-system to-
wards the subdominant side seem barely admissible (par. 59). Even
with the Seventh chord cited under III., c — E — g$lb\>, the feeling
of the key of A minor is almost entirely absent. We hear the chord
much more as belonging to the key of F major ; that is to say, in
c — E — b\> we seem to have C — e — B\>, and in g J a sharpened Fifth
to the dominant C. For the system
bb_ D— f— A— c— E— g ft
which has no complete dominant triad (though we have recognised
this as the positive element of the minor key), is far more inclined to
put forward its positive triads B\> — d — Ft F — a — C, as its principal
contents than the minor triad A — c — E. And so in this combina-
tion we become theoretically acquainted with a chord which
frequently occurs in practice, the dominant Seventh chord with
chromatically sharpened Fifth, The diminished Third which it
contains subjects it to a restriction of position, by always requiring
to be inverted.
CHORDS OF THE NINTH, ETC.— PEDAL 131
CONCERNING THE SO-CALLED CHORDS OF THE
NINTH, ELEVENTH, AND THIRTEENTH. PEDAL.
238. If it is only the most closely related links of a progression
that can be taken together simultaneously as dissonance, and if
therefore it is only two triads having a common interval that can
unite to form a Seventh chord, then no combination going beyond
the harmony of the Seventh is possible as a union of triads. As
we have seen, the passage from C — e — G to G — b — D cannot be
represented in a chord with the contents C — e — G — b — D, but only
in the notes b — D — e — G ; that is, in the union of the triads
e — G — b and G — b — D. We have similarly seen the passages
into the wholly disjunct triads, e.g. from C — e — G into b — D\F
and D\F — a, always resulting in unions of triads most nearly re-
lated to one another : the first in b — D — F — G, the second in
C — D — F — a. Therefore the so-called chords of the Ninth,
Eleventh, and Thirteenth are self-excluded from the harmony of
dissonance which springs from the union of triads.
239. To resolve the chord G — b — D — a or G — b — D — F — # .let
us make the note a descend to G. That by this no resolution of the
dissonance G — a is effected is plain ; for, considering the combina-
tion G — a in itself and keeping inside the key of C major, that could
only consist in progression to F — a, to G — b, or to F — b. Conse-
quently, in the passage G — b — D — a-~G — b — D — G 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 re-
solution b G is given. A direct harmonic reference between the
outer parts is no more to be pretended in this chord of the Ninth
K 2
I32 . HARMONY
and its resolution than in the series continued in the descending
sequence
— D— a- • - Q—b—D—G* • - Cr—a—C—G^
of dissonance chords and their resolutions corresponding with the
first. The Ninth a, which progresses to the Octave G, is resolved
as Seventh of b, just as in the continued succession £r — a — C — G- - •
(£ — a — C — F the upper G moves to F as Seventh of a and not as
Octave of G. In the last succession we cannot hesitate to recog-
nise a pedal, or organ-point, that is a series of chords under which
is placed a note independent of them, and the first succession cannot
possibly be taken in any other sense ; it is not a combination of two
triads related in the Fifth (which, moreover, not being an immediate
succession, could not coalesce in a chord J, but is an independent
chord of dissonance placed over a pedal note, whose resolution is
determined in the chord itself and not referred to an outside basis.
So also with the other chords going beyond the 'harmony of
the Seventh, which it is thought necessary to build up by a pile of
Thirds. The chords of the Eleventh and Thirteenth are not established
as formations of harmony in this sense. The chord of the Eleventh
begins by excluding the Third ; the chord of the Thirteenth excludes
the Fifth as well. This series of Thirds is in like plight with the
arithmetical progression of notes, if we seek to trace in it the basis
of our harmony. As there the notion of harmony guided us to
select from the infinite series of numbers all that answered to itself
and to reject the rest, so also with the edifice of Thirds there
must be previous knowledge by which that is selected which is
agreeable to the notion of harmony. This mechanical construction
by Thirds does not, however, lead on to infinity, like the progres-
sion of notes in the arithmetical series of numbers. In the eighth
member it coincides again with the starting note, for this is the
Fifteenth or double-Octave of the Root :
CHORDS OF THE NINTH, ETC.— PEDAL 133
G b D f a C e g
i 3 5 7 9 ii 13 IS
Otherwise we should doubtless be told of chords of the Fifteenth,
Seventeenth, and so on, as well as of chords of the Thirteenth.
Given a series containing all the notes of the key, it is certainly
not hard to put together all the chords which can occur in that
key, supposing one is free to make omissions at pleasure. Only
after all no generative principle of harmony will have been demon-
strated. In the newer theoretical works this mode of explanation
has been quite abandoned ; neither is it to be met with in the
oldest works. It belongs to a middle period, and at present is
only occasionally heard from teachers who had their education then.
SUSPENSION OF THE NINTH.
240. Only the dominant and tonic of a key can appear as basis
of a pedal, because these two notes alone admit of a change of
principal chords over them. It follows that not every dissonant
chord in which a note suspended over the bass note of another
part is resolved on the Octave of the bass note, i.e. a Ninth, is to
be regarded as a pedal ; for such a suspension may occur upon
every degree of the scale. Rather the pedal here only shares in
the property by which the deepest note of every harmony allows a
suspension in another part. Thus with suitable preparation we find
the chord e — G — C — F permitting the resolution of F to e, in this
as well as in every other disposition of the parts lying above the
bass note ; not so if e were contained in the chord as an upper or
middle part, and F at the same time as a suspension of e. Here
I34 HARMONY
too the chord of dissonance G — C — F stands independently over
the pedal e, and the resolution of its dissonant interval G — F-~
Q — e takes place without finding that obstacle in the e of the deepest
part as a pedal, which would be presented by the same note e placed
in any other part.
241. The parts of a chord which lie above the bass note have
the effect of a harmonic aggregate set against it. They may be
transposed among themselves, and the chord is not essentially
altered thereby. On the other hand it is of striking difference to-
the effect, which note of the chord is allotted to the bass part ;
whether the chord is built upon its Root, its Third, its Fifth, or its
Seventh ; whether it is triad, chord of the Sixth, or chord of
the Sixth and Fourth ; or in harmony of the Seventh, whether it
is chord of the Seventh, chord of the Sixth and Fifth, chord of the
Fourth and Third, or chord of the Second. So too a repetition above
of the progression of the bass (Octave motion with the bass), is not
admissible, though between the other parts it may occur, as we find
it frequently used in the doubled parts of orchestral and pianoforte
music. The bass note, even when it is not Root of the triad or Seventh
chord, always remains the basis for the position of the chord. To
repeat the progression of the chord in the upper parts, to let the
basis be heard a second time in the middle of the harmony or
at the top of it, like a foundation built upwards into the air, can
only be the expression of something contrary to common sense and
upside down.
242. It would also have equal unfitness if the harmony which
stands over the bass note should contain contradictions in itself,
i.e. unresolvable dissonance ; that is, if a note of that harmony
should be sounded in one part and suspended in another, and so at
once be there and not be there, in the way in which it can be pre-
sent in the bass and suspended in another part as a dissonance to
be resolved against a third part.
SUSPENSION OF THE NINTH 135
243. When the suspension is contained in the bass itself, then
the note upon which it is resolved cannot be allotted to any of
the other parts. Here the basis itself enters into the meaning of
the parts subject to harmonic conditions among themselves ; it is
dissonant with a part lying above it, and has to be resolved against
that. But the note on which the resolution takes place, to be
present simultaneously with the suspension, can only be the deepest
part ; it cannot therefore at the same time occur in the harmony
itself. Besides, such an arrangement would again express the
absurdity of a bass lying above the bass. And indeed wherever
anything sounds bad or incorrect the reason of its unlawfulness
should be sought, not in particular technical conditions, but in its
contradiction of a truth and reality to be conceived as quite
universal.
Here again we cannot now enter upon particular instances of ex-
ception to what has here been enunciated as universal, where that
which has been explained to be unlawful becomes with full right
lawful and capable of being used with excellent effect. For our
purpose it is enough to set down that which is directly and uni-
versally valid.
PASSING-NOTES.
(a) Diatonic.
244. In pedal harmony, chords move independently over a sus-
tained bass note. Similarly, if a chord be held, a part moving in
melody can sound notes other than the intervals of the chord.
These melodic passing-notes are none the less determined throughout
by considerations of harmony, for no other determination of a note
is conceivable. But the determination of the intervals of the melodic
progression is independent of the harmony of the sustained chord.
1 36
HARMONY
Supposing the tonic triad to be sustained, and a part to move
melodically in the diatonic scale, then its degrees are given by the
different triads in the system of the key, just as if each note were
accompanied by the triad to which it belongs in the linked succes-
sion. For no melodic note can receive definiteness otherwise than
as it is conceived as the Root, Third, or Fifth of a triad.
(b) Chromatic.
245. In the same way the chromatic scale moving against the
sustained triad can only be constructed independently of the chord,
through unions in which the chromatic notes find their connected
progression. This takes place in such a way that even the notes
that coincide with the degrees of the sustained triad do not re-
ceive their meaning from that triad. They have the meaning
which comes to them from the chords of the connected progression,
which may coincide with the former meaning, but may also be
different to it.
246. The chromatic scale in the system of the key of C major
is formed in the series :
c-. c$-. d •• d$.. e- > f-.f$- %••%$>• a- -b\>.>b.'C',
or in the series :
c • • d[? • • d • • e\) • • e • • f • • fjf • • g • • a[> • • a • • b[? • • b • • c.
In this way of writing the notes, without the distinction of small
and capital letters, it remains undecided in what character they
appear as chord-intervals. But a note raised chromatically can in the
first instance only have the meaning of the Third of a dominant,
i.e. the leading note of a major or minor key, which forms a close
with the note next above it. These two notes stand to one another
in the unchangeable ratio 15 : 1 6, while the chromatic progression,
following the ratios of the diatonic degrees 8 : 9 and 9 : 10, can
PASSJNG-NOTES 137
also vary between the ratios 128 : 135 and 24 : 25. To the first
ratio, 128 : 135, correspond in the key of C major the chromatic
progressions C-c$, F»f§, B\>-b ; to the other, 24 : 25, the pro-
gressions D • • cfjjp G • >g§.
Accordingly we obtain the first of the chromatic scales written
above in the following meaning :
cjf.-D, dJ-.E, F -. fj, G-.gJf, a-Bb, b .. C
15 : 1 6, 15 : 1 6, (128 : 135) 24 : 25, 15 : 16, 15 : 16,
(128 : 135)24 : 25, 15 : 16, 15 : 16, 15 : i6,(i28 : 135)
C-cft D-dft e .. F, fj .. G, gj- A, Bt>» b,
in which, as is evident, only the Root of the tonic triad keeps its
place in the chord-meaning. The Third e and the Fifth Gt coming
after the leading notes d§ and /$, appear with Root-meaning.
Receiving thus harmonic melodic determination, they acquire in
consequence a self-subsisting existence independent of the sustained
chord ; which is also acquired by every note foreign to the scale, or,
generally, by every so-called passing-note of a part that moves
melodically against a stationary harmony.
247. In the scale ascending by chromatically lowered degrees :
Db--d, Eb-e, F -.fft G-.ab, a-Bb, b-C,
(128 : 135) 24 : 25, (128 : 135) 15 : 16, 15 : 16, 15 : 16,
15 : 1 6, 15 : 1 6, . 15 : 16, 15 : 16, 24 : 25, (128 : 135)
C--db, D--eb, e •• F, f#--G, At>--a, Bb •• b,
the tonic elements C and G are transposed from Root-meaning and
Fifth-meaning into Third-meaning, and appear themselves as leading
notes. The tonic Third £, which in the progression by chromatically
raised degrees acquired Root-meaning, here keeps its Third-meaning.
Either mode of chromatic progression can be used ascending
and descending. It is an erroneous opinion that chromatically
,38 HARMONY
raised degrees belong exclusively to ascending motion, and chro-
matically lowered degrees to descending.
248. In both kinds of the chromatic scale a directly determined
progression is contained only as follows : —
(1) In the relation of a leading note to the note lying above it,
15 : 1 6 ; whether the first is a note proper to the scale, or gained
by chromatic raising, and whether the second is a note proper to
the scale, or chromatically lowered :
c-.Db, cJ.-D, d-Eb, dJ.-E, fJ.-G, g.-Afc gJ-A, a--Bb.
(2) In the chromatic change which transforms the major triad
into the minor, or inversely the minor triad into the major : by
chromatic raising,*? — G — b--e — G$ — b, b — D—ffy'-b — D§— -f$ ;
by chromatic lowering,/'* — a — C- - -F — a\> — C, C — e — G- • • C — 4> — G.
This is the progression which we have denoted by the ratio
24 : 25. It is given by a comprehensible determination, in so far
as it consists of an intelligible alteration, a becoming-other of the
same thing : namely, when the relation of Fifth between two
stationary notes passes from the positive to the negative meaning
or the reverse.
249. There is another chromatic progression, marked with the
ratio 128 : 135. This occurs in the scale produced by chromatic
raising between the degrees C*-cj)i, F • -/J, B\> • • b ; and in the scale
by chromatic lowering in the two intervals last named, which appear
there also, and in the interval d\> • • D. This progression is not one
that is in itself comprehensible or directly intelligible. If the <:$ that
follows the C is to be related as leading note to D, the Fifth of the
dominant, as is required, then this c$ may not be referred to the Fifth-
interval a e contained in the system. It may not fitly be con-
sidered as a transformation of the triad of a minor into the triad of A
major. For then ^J could only lead to the ^that lies below the system
and not to the D that is Fifth of the dominant. The essential differ-
PA SSIATG-NO TES
139*
ence between these two notes we need not turn back again to explain.
To get the leading note of D we must take, to the Root C, the
Third not of a but of A \ that is, the Third of the triad of A major
in the chord-series C— e — G — b — D—f$ — A — c$ — E; a progression
that, judged by the notion of intelligible succession, is quite without
possible link. A like relation comes out in the chromatic steps
F • -/J, B\)"b, D\)"d. We have denoted it by 1 28 : 135 because
the Third of the third Fifth (33 x 5 = 135) will compare with the Root
raised into its neighbourhood (27= 128) in this ratio of the numbers
of their vibrations. The step C*-c$ in the chromatic scale of C
major, as well as others answering to the same ratio, thus remains
undetermined in itself. The chromatically raised note can here be
comprehended only in its relation of leading note to the note which
follows it, and as determined from that. On the other hand, the
passages D-d^fa, G'-g$y and all progressions answering to the
ratio 24 : 25 (which is found included within the limits of the Fifth,.
20 : 30, as the difference between the major triad 20 : 25 : 30 =
4:5:6 and the minor triad 20 : 24 : 30 = 10 : 12 : 15), contain
a determination in themselves. On this account they are easier ta
sing in tune than the intervals standing in the ratio 128 : 135, as
may be confirmed by testing attentively the free intonation of the
different chromatic degrees. For in the ratio 128:135 there is present
only the relation of leading note to the note following, but no deter-
mination of the chromatic note with respect to the note started from.
250. The diversified and constantly changing meaning, which
must be assumed by the determining notes in the chromatic pro-
gression, makes it more complicated and for free intonation harder
than the diatonic. In the latter, as was shown above, the succession
of the degrees is determined upon the elements of the tonic triad,,
without making a change of meaning for one and the same degree
necessary ; if we except the step from the sixth to the seventh
degree, in which the sixth appears at first with the meaning of Third
I4o HARMONY
of the subdominant, and then passes into the meaning of Root of
the minor triad upon the Third of the subdominant. But in the
chromatic scale, besides this change of meaning of the intervals, a
change of key also enters with each of the determining notes ; and,
since no three successive degrees are ever contained within the same
key, the inner structure of the whole succession becomes so crowded
in its composition, that it is not surprising that a perfectly true in-
tonation of chromatic progressions should be in many cases un-
attainable by singers not thoroughly grounded in harmony, who
yet may be able to move with certainty in the diatonic scale. Thus
often what is outwardly nearest fails to be taken with certainty,
because the determination for it is neither unmistakably felt nor
intuitively known.
251. The chromatic scale, whether progressing in sharpened or
in flattened notes, contains seven degrees of the ratio 15 : 16, three
of the ratio 128 : 135, and two of the ratio 24 : 25. But the ratio
15 I 1 6 is not a chromatic one, but diatonic. It a'nswers to the
progression of the leading note to the Octave of the Root ; also in
the minor key to the difference of the second and third degrees of
the scale.
Where simply a nearest has to be added, there it will always be
some such small diatonic degree. For the chromatic progression
produces a leading note that takes us further, leading upwards in a
sharpened, downwards in a flattened degree. Certainly in the two
chromatic scales written above we meet with no g\> or #J, no
flattened Fifth or sharpened Sixth, but in that which moves in
sharpened progression there is B\> the seventh degree flattened, and
in that with flattened notes /J the fourth degree sharpened. The
reason is that B\> and/J, and not «jf and g\>, find as chord notes a
determining element in the system of the key of C major. Never-
theless the melodic movement of F-f$--F or b->B\>-b could
not be justified to feeling. In the succession F-f$- G only the
PASSING-NOTES 141
degrees F--G and/Jf--^ are determined in themselves, and in
the succession B\> • • b - • C only the degrees B\> - • C and b • • C ; there-
fore the successions F* -f$ - • F and b - - B\> • - b fail of intelligible
foundation. Here too F can only be related as leading note to a
g\> lying above it, and b to a leading note # J lying underneath
it, and now B\>, which lies below the C major system, and /J,
which lies above it, enter as notes linking the steps F- -g\> • • F and
b*-a$f'-b\ just as before we found the sharpened sixth degree in
the diatonic ascending minor scale, and the flattened seventh
degree in the descending, linked by the boundary notes outside
the closed system.
'252. Thus the chromatic places between .F and G and between
a and b, determined only as/Jf and B\> in the progressive arrange-
ment, yet become g\> and # jf when the first is to join on to F and
jthe second to b. But in all other places of the chromatic scale,
Jvhich already in themselves furnish a double progression by
Sharpened and by flattened diatonic degrees, the relation of the
leading note 15 : 16, and not the chromatic 128 : 135 or 24 : 25,
is always given as the difference of two notes that hang together
or desire to pass into one another by natural inclination. Properly
chromatic degrees can only enter in a motion that tends onwards,
C-.db--C--b-.C-- IcJf.-D ••€[>•• D.-c#..D
253. Manifestly the distance of the diatonic interval 15 : 16 is
greater than that of either of the two chromatic intervals 128 : 135
T42 HARMONY
•and 24 I 25. Yet ^J pitched as leading note in the succession
C- - c% - - D will seem higher than the d\> in the succession C- - d\> • • C\
consequently the chromatic interval C-c$ seems to be greater
than the diatonic C-d\>. In instruments with fixed and therefore
tempered degrees of sound this must certainly depend upon an
acoustical illusion, because they use the same note for c§ and d\>.
Singers and players on instruments with free intonation, however,
will feel the necessity of actually taking the leading-note sharper
than truth, but the minor second (which leads backwards) at a less
distance than the ratio 15 : 16 assigns. There is here an endeavour
to characterise the note in its interval-meaning, to enliven or animate
the intonation. Intonation untempered, mathematically true, would
be musically lifeless, and remain an unsatisfying means of expression.
It would be like rhythm moving strictly in time to the metronome.
The beat of the metronome to living performance seems at one time
•to linger, at another to hurry on, because in its mechanical strictness
it cannot answer the light and shade of an animated rhythm. So
too the intonation of characteristic degrees of sound will not be
bound to mathematically determined pitch, but often deviate from
it, pressing upwards or downwards. It must so deviate, if intona-
tion itself is not to remain something merely determined mechani-
cally, as it is in the fixed degrees of keyed instruments. But such
departure from mathematical purity can only touch the interval of
the Third, the interval which alone is changeable : not in the sense
that it can become larger and smaller, but that by shifting chro-
matically it can pass from its relation to the Root into its relation
to the Fifth, whereby the major triad is transformed into the
minor ; e.g.
C — e— G. .. C— eb— G
I— III III — I
This passage, as well as the difference in general of the one and
the other determination, whenever it receives a characteristic mean-
PASSING-NOTES 143
ing, tends to acquire emphasis by intensified expression ; and
this happens in the sharpened pitch of the major Third and in the
flattened pitch of the minor. None but the Third-meaning can
give ocl^sion for altering the mathematically true intonation. The
interval of Fifth, as an invariable, must be pitched always with
perfect purity. Similarly, nothing but the strain of a transition
can bring about the alteration of the Third. Whenever the chord
stands independent and at rest, then that interval too is pitched
according to its acoustical determination.
254. The explanation of the chromatic progressions has kept
us somewhat longer than the occasion seemed to require. But the
opportunity has occurred of noticing the difference of the chromatic
relations 24 : 25 and 128 : 135, the direct meaning of the first and
the indirect meaning of the second ; a difference that should be (but
is not always) observed both in theory and in practice. The singer
who pitches his voice, not by white keys and black ones, but by
harmonic determinations alone, will not be able to take the ap-
parently nearest, the chromatic degree, with certainty, if he be
without feeling of the harmonic meaning of the note. In dia-
tonic progression the linking of the notes is so simple, and up
to the step from the sixth degree to the seventh -so unambiguous,
that a passage presents no difficulty. But in chromatic progression
the conditions are complicated and often change, and in many places
a singer may be in doubt whether he has to take a progression in
the meaning of the ratio 15 : 1 6 or of 24 : 25 or of 128 : 135 ; so
:hat here the intonation is less assured and is exposed to the
vacillations which chromatic song-music so often experiences in
performance. But the blame is not always to be put upon the
singer. Far oftener it belongs to the composer, who should require
from the singer ' nothing unintelligible, nothing unintelligently.'
144 HARMONY
MODULATION.
255. By changed meaning of the note is determined a new
interval ; and by changed meaning of the interval, a new chord.
Similarly the changed meaning of the chord will determine a new
key.
But the meaning of the note cannot be expressed in the note by
itself, nor until the note sounds together with another note in the
interval; nor is the meaning of the interval known until it sounds
together with another interval in the chord. So too the meaning
of the chord cannot receive its determinateness in the chord by
itself, nor until it is placed with other chords in the key. And thus
it comes round that for objective knowledge the key is what deter-
mines the meaning of the chord, the chord determines the meaning
of the interval, and the interval the meaning of the note.
256. The triad of C major, which is tonic chord in the key of C
major, is dominant chord in the keys of F major and F minor ; sub-
dominant chord in the key of G major ; and chord of the sixth
degree or Third of the subdominant minor triad in the key of E
minor. And so universally ; each of the three major triads of the
major key, and each of the two major triads of the minor key, may
take part in three major keys and two minor keys. Also each
of the two minor triads of the minor key, and each of the two
minor triads of the major key, may be parts of two minor and two
major keys.
257. According to this, the major triad has fivefold meaning
with respect to the three major and two minor keys to which it can
belong ; the minor triad has fourfold meaning with respect to the
two minor and two major keys in which it takes part.
258. The diminished triad on the seventh degree forms an
element of relationship only between the major and minor (and
MODULATION 145
also the minor-major) keys with the same name. The diminished
triad on the second degree of the minor key is only contained
again in the minor-major key with the same name. In the major
key the diminished triad on the second degree, as is shown by its
structure, can find place only in one system.
259. In the manifold meaning of the chord lies the possibility
of passage from one key into another. But this possibility is not
matured until the changed meaning has been put clearly forward
by succession and combination of sounds, and until the new key to
be occupied has been marked out in that whereby it is distinguished
from the first. Thus the passage from the key of C major to the
key of G major,
F— a— C— e— G— b— D
C— e— G— b— D— fj— A,
linked by the triads C — e — G — b — Dy can only be effected by
the appearance of the Third of the dominant of the latter key,
that is, f$ ; the passage from the key of C major to the key of
F major,
F— a— C— e— G— b— D
B[>— d— F— a— C— e— G,
linked by the triads F — a — C — e — G, only by introducing the sub-
dominant Root B\> ; the passage from C major to A minor,
F— a— C— e — G — b — D
D— f— A— c— E— gj— B,
linked by the triads F — a — C — e, is only possible by sounding gfy ,
the Third of the dominant of the new key ; and the passage from
C major to E minor,
F— a— C— e— G— b— D
A— c— E— g— B— dj— FJ ,
linked by the triads a — C — e — G — b, only by d$.
L
i46 HARMONY
260. Accordingly, if we wish to make the dominant triad of the
key of C major, G — b — D, which is also tonic triad of the key of
G major, pass from the first meaning to the second, it can only be
done by uniting the major triad on G with the major triad on D
by placing these chords together :
C : I V
C-e-G ••• b-D-G ••• A-D-fJf
G : i — v
In like manner the subdominant triad of the key of C major,
F — & — C, will come to stand in the meaning of tonic triad of the
key of F major, if we bring the subdominant triad of the latter key
into union with it :
c : i iv
C-e-G ... C-F-a — d-F-B[>
F : i iv
The minor triad on e contained in the key of C major as minor
triad of the dominant side becomes tonic triad of the key of E
minor when placed with the dominant chord of that key :
C : I in
C-e-G ... b-e-G •.• B-dJ-FJ
e : i v
The minor triad on a, which lies on the subdominant side, be-
comes tonic triad of the key of A minor when placed with the
dominant chord of that key :
C:T vi
C-e-G ... C-e-a ... B-E-gJ
a : i v
261. It must, however, be admitted that by these successions no
MODULATION 147
modulation into the new key has been effected such as to satisfy
and land us safely in it. Although a chord has entered from its
domain, yet the key itself still remains not definitely marked off in
its boundaries. Scarcely more is felt than that the side or the
direction is opened, towards which the modulation is about to turn.
There is no firm settlement in a new key.
262. Among the harmonies of the Seventh we have seen one
which is quite suited to determine a key in its principal elements, be-
cause its essential contents are the dominant and the subdominant
in its dissonance and the Tonic in its resolution, thus comprehending
in itself and the following chord the principal parts of the whole
system.
This is the dominant Seventh chord, also named the principal
Seventh harmony. As it has the property, in its dissonance, of
causing the limits of the key-system to be heard in union, and, in
its resolution, of establishing the tonic triad as middle of the system,
it will, when introduced in natural connexion, announce the new
key in a decided manner and allow it to enter with certainty.
263. Thus, take the two first passages above, C—e—G •••
b-D-G ••• A-D-f$ and C-e-G ••• C-F-a ••• d-F-B\>, of
which the first contains the transformation of the dominant chord,
and the second the transformation of the subdominant chord into
the tonic, whereby in the former a modulation into the dominant,
in the latter into the subdominant key is stirred but not accom-
plished ; the passage could have been made more decided by the
dominant Seventh harmonies of the two keys.
264. In considering the harmonies of the Seventh, we found
that the chord of the dominant Seventh, which consists of the
dominant triad united with the diminished triad upon the Third of
the dominant, is taken in right connexion if made to issue either from
the tonic or from the dominant or subdominant triad (par. 1 8 1). Ac-
cordingly the tonic triad C — e — G, taken as subdominant triad of the
L 2
i48 HARMONY
key of G major, or as dominant triad of the key of F major, may
be followed immediately by the dominant Seventh chord of the one
key or of the other. In this way, after resolution of the chord, the
new key will not only be introduced, but also will be at once ex^
hibited in its whole compass and firmly settled. Then the modula-
tion is formed on the following plan :
(A) Towards the dominant side :
c : i
C-e-G ••• C-D-fJt-A -•• b-D-G
G : IV V; I
(B) Towards the subdominant side :
C : i
C-e— G ... B^-C-e-G ••• a-C-F
F : v v7 i
The tonic triad gained, of the new key, can in its turn be transposed
into either dominant or subdominant meaning ; and then by the
same modulation it will either lead back into the original key, or
else one degree — that is, one key — further in the direction of the first
modulation.
A. (i) Leading back :
c : i c : v V; i
C-e-G-.-C-D-ffl-A -.. b-D-G ••• b-D-F-G ••• C-e-G
G:IV - — v7 i
(2) Leading onwards :
c : i
C-e-G ... C-D-fJf-A •" b-D-G ••• A-cJf-E-G ••• A-D-fjf
G:IV v7 i
D : iv v7 i
MODULATION 149
B. (i) Leading back :
c : i c : iv v7 — i
C-e-G ••• Bb-C-e-G ••• a-C-F ... G-b-D-F .... G-C-e
F : v v7 — i
(2) Leading onwards :
c : i
C-e-G ••• B[>-C-e-G ••• a-C-F ••• a-C-E^-F ••• B^-D-F
F : v v7 i
Bb:v v7 i
265. The modulations A (2) and B (2), if further continued in the
same way, would lead into the keys which from time to time are
the most nearly related in the two directions : the first, which takes
the tonic triad in subdominant meaning, towards the dominant
side ; the other, which takes it as dominant, towards the subdo-
minant side.
266. But we have to distinguish two kinds of modulation to
remoter keys : one, that advances from one key to the other, alto-
gether leaving the seat of the first and settling in the region of the
second ; the other, that transforms the key to another within its own
boundaries.
267. Suppose we modulate, upon the plan given above, into a
key of the third degree of relationship, from C major to A major or
E[? major; then we have progressed to the new key in one or other
direction in the chain of triads :
E A
C
The connexion with the original key here consists in the uninter-
,50 HARMONY
rupted succession of related keys which has led from the first key
to the last. But now the notes c, g, d in the key of Ej? major are
not the same as those of like name in the key of C major, C, G, D \
nor do the notes A,E, B in the key of A major correspond to those
of like name in the key of C major, a, e, b. The note F, Fifth of the
dominant in the key of E[? major, and subdominant in the key of
C major, is common to the two keys, and similarly the note D>
Fifth of the dominant in the key of C major and subdominant in
the key of A major, belongs in common to the two keys ; yet upon
this a relationship of key cannot be founded. For the key is a
union of triads, and therefore for it the triad alone, and not the
single note, can be an organic element of relationship. This the
single note can be only for the interval, and the interval only for
the chord.
268. But, according to this requirement of inner relationship —
namely, for notes with the same name in two different keys to be
the same with changed chord-meaning — we should be able by the
chain of triads to arrive at no more than the keys that lie nearest,
i.e. the keys of the dominant and subdominant. Even there the
Fifth of the dominant (A) of the dominant key is shown to be
different from the Third of the subdominant (a] of the original key;
and in like manner the Third of the subdominant (d) of the sub-
dominant key is different from the Fifth of the dominant (D) of the
original key. But the demand for identity cannot extend to these
notes, because they are produced by the entrance of a new Fifth-
determination. The notes a and D in the system of the key of C
major do not stand in true Fifth-relation ; consequently the Fifth
D A of the key of G major cannot want to keep a, the Third
of the subdominant of the C major system. No more can D, the
Fifth of the dominant in C major, want to be taken for d, the Third
of the subdominant in F major ; because that must stand in Fifth-
relation to ay Third of the subdominant in C major, and must
MODULATION 151
therefore be a different note from D. According to this the difference
between a and A, or between D and dy should determine the modu-
lation from C major to G major, or from C major to F major, as well
as the difference between F and /$ in the first case, or between
b and B\> in the second. But this difference of the Third-note from
the Root- or Fifth-note with the same name, as it is ignored in our
manner of writing music, so also in practical use is too little for a
change of chord to be made clear by it. Unless /J is added, a
will not pass into the meaning of A ; and, unless B\> is added, D
will not pass into the meaning of d.
269. It is not merely through the greater quantity of like
material that the keys of the Fifth above and below are most
nearly related to the key assumed as tonic. They are much more
essentially related in the sense that the tonic triad of the tonic key
is contained as subdominant triad in the dominant key, and as
dominant triad in the subdominant key ; and similarly the tonic
triads in the dominant and subdominant keys are respectively the
dominant and the subdominant triads in the tonic key ; and that
therefore the relationship, the opposition of difference and equality,
is here to be found in the principal element, the tonic triad itself.
270. With the two keys which follow in the progressive series,
B[> major and D major, the key of C major is related only in one
subdominant or dominant chord : with the key of B|? by the major
triad of F ; with the key of D by the major triad of G. In the tonic
triad, the principal element, these keys remain strange to one another;
principally, therefore, they are not related.
In the keys which follow next, which are more remote by a
Fifth, even this subordinate relationship dies out, and the mutual
reference of such keys, if considered only in this series, ceases
entirely. The plan of modulation introduced above can, it is true,
lead onwards to the remotest keys ; only when even the third key is
reached we are in a wholly strange region, out of all inner con-
nexion with the first.
,52 HARMONY
271. To this plan of modulation stands opposed that other
kind, which does not consist in progressing to another key through
the intermediate keys, but in taking what is common to the two
keys to be united, and transposing it from the meaning which it
has in the first into the meaning which belongs to it in the second.
Thus the new key springs right out of the middle of the first.
272. Key-relationship must be sought principally in the tonic
triad. For the dominant and subdominant keys it consists in this,
that the Root of that triad can become Fifth, or its Fifth the Root, of
the new tonic triad. Hence a relationship may also be contained in
making the Root or Fifth of the tonic triad the Third, or its Third the
Root or Fifth, of a new tonic triad. In this way reappear elements
of relationship constituted by the tonic triad with remoter keys,
which elements, if the keys were reached by progressive modulation,
would have already left us at the third step. For —
(1) taking the Root in Third-meaning :
I
F — a — C — e^-G — b— D,
III
Dt>— f— At>— c— Eb— g — B[>,
we have the identity of the notes F C G and/ c -g
in the keys of C major and A[? major ;
(2) taking the Third in Root-meaning :
III
F — a — C— e — G — b— D,
I
MODULATION 153
we have the identity of the notes a -- e - b and A
in the keys of C major and E major ;
(3) taking the Fifth in Third-meaning :
II
F — a— C— e—G— b — D,
III
At?-c— Eb— g— Bb-d— F,
we have the identity of the notes C -- G -- D and c -- gr-
in the keys of C major and E|? major ;
(4) taking the Third in Fifth-meaning :
F— a— C — e — G — b— D,
II
we have the identity of the notes a - e - b and A - E - B
in the keys of C major and A major.
273. According to this there now enters a nearer mutual refer-
ence between keys of the third and fourth degrees of relationship
than is afforded by those of the second degree of relationship. We
saw that the latter, being, as to their tonic triads, without anything
in common, were in so far to be regarded as, in the principal sense,
unrelated.
274. But if the relationship between the keys is to consist of
the notes with the same name, the modulation must also be of such a
kind that the notes in the.two keys to be united do in fact remain
the same, and that the keys which follow one upon the other do
154
HARMONY
really find their unity in these notes, as something that endures,
though in changed meaning. That, e.g., in the modulation from
C major to Ab major we may see the identity of C and c (C being
here opposed to itself as Root and Third) really preserved, and not>
as happens in the passage through the series in succession of the
united keys :
C-e-G ... Bb-C-e-G ... a-C-F ... a-C-Eb-F ... Bb-d-F ... Ab-Bb-d-F ... g-Bb-Eb ... g-Bb-Db-Eb ... Ab-c-Et>
C:i
F:v v, i
Bb:v v7 i
Eb:v v, i
Ab : v - v, i
obtain c the Third of the tonic triad of Ab major as a note differing
by 80 : 8 1 from the Root of the tonic triad of C major.
275. If instead of the passage just written we hear the following :
C-e-G ... Bb-C-e-G ... ab-C-F ... Ab-Db-f ... Ab-Bb-Db-f ... g-Bb-Db-Eb... Ab-c-Eb
Ab : vi
which might certainly be very much contracted without loss of
clearness, the identity of the note C in the first chord and c in the
last is easily perceptible. The Root of the key of C major has here
become Third of the tonic triad of Ab major, the key of C major
has passed into the key of Ab major, without quitting and annulling
the intervals which can remain common to both keys and in which
their inner relationship is contained. If the passage to the key of
Eb major, related in this manner to the key of C major, is made by
the following series :
C-e-G • • • Bb-C-e-G • • • ab~ C-F - • - Ab~Bb-d-F • • • g-
* *
then in progressing from the third chord to the fourth there will
MODULATION
be perceived an alteration in the note F, the passing of the Third-
meaning into Fifth-meaning. For the transformation of the keys in
the case before us is this :
F — a— C — e— G — b — D
F — at?— C — e — G
(f )— Ab— c— E[>— g— Bt>— d— F.
Here the permanent, binding element is the Fifth- interval, C - G>
which belongs in the first key to the tonic triad, in the second to
the dominant triad, and in the last, where it exchanges the positive
for the negative meaning, to the minor triad on the subdominant
side. But upon this enduring element the rest of the intervals of
the keys are formed, following the assumption that the E|? major
triad shall contain as its Third the Fifth of the C major triad ; and
thus the Fifth of the dominant (F) of the key of E\> major is a
different note from the Root of the key of F minor, and therefore also
from the subdominant Root of the key of C major. Certainly, by
what was said earlier about the extension of the key-system, the
transposed chord might also stand in the course of modulation
as A t> — B\> — d—f with the Root of the key of F minor without
losing its value as a dominant Seventh, which is sufficiently ex-
pressed in A\> — J3\> — d. But the note/" will be hard to maintain as
not- Fifth against B\> — d ; it will always tend to shift into Fifth-
meaning (from 80 to 81).
276. These processes of modulation begin by taking the tonic
triad of a major key as dominant triad of a minor key. By this
turn of meaning we immediately gain a region lying towards the
subdominant side, that in process of successive modulation could
not be reached but by progression through three keys. There-
fore the more distant keys that lie on this side are in general
easily accessible in this way. For if by the triad of C major we
are already landed in the key of F minor, then the passage into
156 HARMONY
the keys related to it may be effected in closest union by dominant
Seventh chords linked and linking. Here, however, the question
was only of modulation to the A[? and Eb major keys.
277. As the keys of A|? and Et> major contain a relationship
with the key of C major in the Root and Fifth of its tonic triad, so
we found the keys of E major and A major related with it in the
Third. In the former cases the Root or Fifth of the tonic triad of
one key is turned into meaning Third of the tonic triad of another.
In the latter cases, inversely, the tonic Third of one key is turned
into meaning Root or Fifth of the tonic triad of another. Modu-
lation into the remoter keys of the subdominant side results when
the dominant chord of a related major key is referred to the minor
key of the same name. It will lead to the remoter keys of the
dominant side if we refer the dominant chord of a related minor key
to the major key of the same name.
Thus the tonic triad of E major may follow the dominant
chord of the key of E minor ; or the tonic triad of A major the
dominant chord of the key of A minor :
C— e— G, C— e— a, B— d#— FJf/A, B— E— gj
c : i vi
e : iv — v7
E : v7 i
C— e— G, a— d— F, g#— B— D— E, A— cj— E
c : i
F : v vi
a : iv v7
A : v7 i
278. But it is to be observed that modulation to the keys on
the dominant side is always less easy than to those on the subdo-
minant side. We remark this even in the modulations which are
nearest, to the adjacent dominant and subdominant keys.
MODULATION 157
First the keys may be considered as members of a chain in the
never-ending triad-propagation, and afterwards each single key as
subsisting for itself. In the former case each key is a uniting
middle member for two others adjacent to it on the two sides : in
the latter the key does not pass into another ; therefore it passes
into not-another, i.e. into itself. We found that this passing into
itself is expressed in the chords which contain the boundaries of the
single key-system united. The separated key has its centre of
gravity in the middle. In the chain of keys each key rests upon
the one that has gone before it. But this is the key of the sub-
dominant ; for positive production, as an effect of force, is directed,
towards the dominant side, upwards. It raises a secondary, the
Fifth, into a primary, the Root ; it does not lower the primary into a
secondary. The dominant key is one that has to be produced out
of the tonic ; it requires productive energy, needs an effort to make
it come out of the tonic. On the other hand the subdominant is a
key that precedes the tonic ; to it, as a thing that has already
been present and determined, the modulation easily descends.
Therefore too a key is far less endangered in its tonic quality by
the dominant than by the subdominant. For if the modulation
turns so easily towards the subdominant side, it will yet more
readily return from the dominant to the tonic, and be able to re-
establish it as principal key after an excursion into the dominant.
On the other hand, a modulation into the subdominant at once
throws upon it the tonic character, which it requires new exertion
to restore to the true tonic.
After more than a short stay in the subdominant key it will be
almost a necessity to again touch the dominant ; so that, returning
from it to the tonic, the latter may be felt quite in its tonic
meaning.
Thus, our regular modulatory form of pieces of music in the
major key, which pass, in the middle of their course, into the do-
I58 HARMONY
minant, is exactly that which fits with reason and nature. It is,
speaking generally, going on, which from first to last cannot be
going back, and therefore cannot lead to the subdominant. Retreat-
ing from the dominant to the tonic is going home. In the minor
key the modulation leads regularly, not to the Fifth, but to the re-
lated major key. The minor key has no going onward ; it is shut
up in itself, and must first get rid in the major key of the fetters
which hamper it, before it can gain freedom and outside alliance.
279. By the foregoing considerations, we see generally, that every
key which, compared with another, contains chromatically raised
notes, will be more exalted, tenser ; and a key that is distinguished
from another by chromatically lowered notes will seem depressed,
quieter, less tense. Moreover, in this alone is to be found the
much talked of character of the keys. That certainly exists ; but
it can only be relative, and not absolute for any single key, be-
cause each particular key by itself rests in its organisation quite on
the same conditions as every other. And there is no absolute
pitch ; therefore no determination for the character of the keys can
lie in that. A song in the key of C major is perfectly identical
with the same song in Dt> major, if the latter be pitched to the
same height as the former ; for in its essence the one key is per-
fectly identical with the other. The characteristic determination
lies in their relation to one another, sc. that the key of D|? major
has the Root of the key of C major for its leading note, and the
subdominant Root of the key of C for the Third of its tonic ;
and that by the transformation of the Root-notes into Third-
meaning all other elements of the key of C are chromatically
lowered and turn towards the subdominant side, towards a region
from whose standpoint the key of C major must itself appear ex-
alted and tense. But the exact character of the difference between
the keys of D[? major and C major will also be shown in the key of
£> major taken with Cjf major — and similarly in E[> major taken
MODULATION 159
with D major — and none of them can pretend to any positive cha-
racter. In orchestral performance single keys can indeed take a
peculiar colouring in the wind and string instruments ; but this,
depending only upon mechanical structure and special acoustical
conditions in the different instruments, and not being founded in
the nature of the keys themselves, cannot here be considered as
essential. In pure vocal music there will be no desire to ascribe
a particular character to single keys. There what is characteristic
is to be found solely in their placing with other keys, in 'the bear-
ings of their relationships, and in how far these are brought out in
the modulation.
280. Succession of keys, like that of chords or single notes, can
, never happen otherwise than continuously. A key may be followed
by the remotest possible, but only in so far as the chord from
which the modulation to the new key takes place is either already
present in it too, or at least belongs to a key most nearly related
to the new one.
281. The passage from the tonic triad of C major to the D[?
major triad is only rendered intelligible by taking the former to
mean the dominant chord of the key of F minor, whereby the D[?
major triad enters as chord on the sixth degree of that key.
282. Supposing the tonic C major triad to be followed by the
B major triad, then in the former there is a change of meaning ; it
becomes the triad on the Third of the subdominant in E minor,
in which the second of the three chords named is the dominant
But we know that the latter (the B major triad) regarded as simply
a major triad can have four other meanings besides this. Here it
belongs to the key of E minor. It is also contained in the keys of
E major, B major, F$ major, and DJ minor ; in the first as V, in the
second as I, in the third as IV, in the fourth as VI. And just as the
tonic triad of C major was determined to be the triad upon the sixth
degree in E minor, in order that the B major triad might followi
160 HARMONY
so now the B major triad, which, following upon the former triad,
enters as dominant of E minor, may in its turn exchange this deter-
mination for another, and afterwards progress agreeably to the newly
chosen determination.
283. Similarly in the succession first named of the major triads
on C and Dt>, the latter, besides that of / : VI, has also at disposal
the meanings of D[? : I, G\> : V, g\> : V, At> : I V ; so that what
follows further may be referred to the keys of F minor, D[? major,
Gt> major, G\> minor, or A[? major.
284. Thus to the initial chord, if it be major, a fivefold indica-
tion may be attributed ; and if it be minor, a fourfold. A diminished
triad has the various determinations pointed out above. Also the
same Seventh chords appear in different keys with changed meaning,
and thereby lead to different modulations.
Thus, even with the condition that the two chords in imme-
diate succession shall belong to the same key, infinitely manifold
change of key is still possible.
285. But also modulation may take place by chords belonging
not to the same, but to very nearly related keys. Supposing
the C major triad to be followed immediately by the triad of E
major, of A major, or of Et> major, then in these successions the
two chords standing next one another are not contained in one
key ; as in all cases where chromatic progression happens, there the
territory of another key is entered upon. In the first succession,
(7 — e — G---B — E — g-jf, the Third of the tonic triad of C major
becomes Root of the dominant chord of A minor. In the second,
C — e — G'-cfy — E — A, it becomes Fifth of the dominant chord of
D minor. In the third, C — e — G--B\> — E\> — g, first the positive
state of the Fifth in C — e — G has passed into the negative C — e\> — G,
and then the negative state of the Third e\) — G has taken on
positive meaning, E\> — g. In the second element of the compound
succession C — e — G> • • C—e\> — G- - -B\> — E\>— g, namely in C — 4> — Gy
no key is as yet determined for the minor triad on C ; whether it
MODULATION 161
belongs to the key of C minor, G minor, E[? major, or A[> major,
remains as yet undecided. Nevertheless the progression e»e\>,
which presses towards D, and would seem to lead probably to
b — D — G, speaks most for the first meaning and least for the last.
The reversed succession C — 4> — G ••• C — e — G would make the
second chord appear pretty clearly as the dominant triad of F minor.
286. By the feeling alone it may easily be perceived that the
first of the chromatic progressions above, e — G--E — £•$, is far
smoother, more pliant than the other, C — e- • -c§ — E. In the first^Jf
is determined as Third by the e already present ; in the other,
C — €•••€§ — E, no determination for ^J is given by e. The first
progression is decidedly the chromatic difference of 24 : 25 ; in the
second it is undecided whether ^J is to be referred as leading note
to d or to D ; whether the step shall enter in the ratio 24 : 25 or
128 : 135 (par. 250). It is true that the ratio 24 : 25 in itself no
more contains a determination for the intonation of the chromatic
progression than does 128 : 135. But the former expresses the
difference between the major Third and the minor, and the intona-
tion of the major Third is directly determined and grasped with
certainty ; while the other progression, 128 : 135, must first assume
the note to which the chromatic note shall stand as leading note,
and the leading note itself finds no determination in the notes of
the given interval. Hence the Fifth of the major triad is always
easier to raise chromatically than the Root, and therefore the
passage
C— e— G ..- B— D— E— gj — A— c— E— A
proves smoother than the passage
C-e— G ... cjf— E-G— A ... D-f— A.
It may also be remarked that the note E in the last example,
which here ought to remain identical with the e and maintain with
G the interval of 5 : 6, is by the chromatically ascending £$ impelled
to become rather sharper, namely to accord with the E of the series
M
102 HARMONY
of Fifths, C G D A E, which in the dominant
Seventh chord A — c§ — E\G forms with G the interval of 27 : 32.
The triad C — e — G here in itself contains no determination for con-
sidering it as the dominant of the key of F major (which for the pas-
sage to D minor would be requisite) ; therefore the chromatic pro-
gression C ^J will be referred only to the D contained in the
system of the key of C major, whereby the Third of the major triad
on A, as it stands in the series of Fifths, is taken. This, being too
high for the a and e given in the key, must raise them to A and E,
to bring about the consonance in A — rj — E.
If we insert the major triad on F between the two first chords,
and instead of the succession C — e — G ••• rj — e — G — A •••
D — f — A take the succession C — e — G •- C — F — a •«•
c$—E—G—A •" D—f—A^hen the Seventh chord A—c$—E\G
sounds smooth and perfectly pure ; because now in the Third of the
major triad on F there is given a determination for the step
C ••- c$. But then we shall have obtained exactly that first suc-
cession, in which the Fifth progresses chromatically.
So too modulation to the key of the dominant is not to be
effected by chromatic progression of the subdominant Root of the
original key to the Third of the dominant of the new key, e.g.
F— a— C ... fj— A— C— D
c : iv G : v7
but by taking the dominant Seventh chord of the latter after the
tonic or dominant triad of the original key : e.g.
C— e— G ... C— D-fJf— A
c : i
G : iv v7
or :
C— e— G ... b— D— G ... A— C— D— f#
c : i v
G : i v7
MODULATION 163
The first modulation would always, in the interval f$—A, have to
alter the a of the key of C major into the A which is Fifth of the
dominant in the key of G major. On the other hand, in the two
last examples the note may at once acquire that meaning.
It is in such differences, outwardly small but inwardly import-
ant, that the reason must be sought why so many modulations,
natural in appearance and into near keys, nevertheless keep some-
thing of harshness and constraint, and in vocal music refuse to be
brought to satisfactory purity; while often others leading into the
most distant keys turn out tractable and easy in performance.
287. The inner relationship of keys, it has already been said,
can principally be determined only upon the tonic triad itself;
which either in entirety, or in Ro.ot and Fifth, Root and Third, or
Third and Fifth, or lastly in only one of the three chord-elements,
discovers a relationship with another key. Thus its positive be-
comes relative, or a relative in it takes positive meaning in the other
key ; or lastly, a relative in the chord takes a new relative meaning in
the new key. But this last is a less essential determination of re-
lationship ; since for 'another' to become 'other' can have but an
indirect meaning for the ( one.' Therefore those relationships of keys,
which indeed are connected with the tonic triad, but in which the
change of meaning is from one secondary element into another —
Fifth transformed to Third, or Third to Fifth — are of a less
degree than those in which the primary element passes into the
meaning of a secondary either in a positive or a negative sense, or a
secondary changes into positive or negative primary meaning.
Of the relationships founded upon change of meaning in single
elements, the keys of A major, Et> major, and E minor have a less
direct bearing upon the key of C major than have the keys of E
major, A|? major, and A minor. In the first set the Third e is
taken as Fifth,
M 2
I64 HARMONY
III
C— e— G
A-cfi-E
II
the Fifth G as Third,
II
C P Cr
V^x C VJT
Eb-g-Bb
III
the Third as negative Fifth, and the Fifth as negative Third,
III II
C — e — G
e — G — b
II III
But in the second set the Third appears as Root, the Root as.
Third, the Root and Third as negatively Third and Root :
III
I
I III
C— e— G
C — e— G
C — e— G
E-gJ-B
Ab— c— Eb
a— C— e
I
I
III I
In the first the transformation acts on a relative and leads to a re-
lative. In the second it is positive changing into relative or else
relative into positive ; whereby in each case there is always a
positive actually present, either as determining or as determined.
288. The relationships in the minor key will be determined,
not on the triad, its positive premise, but only on the negation of
that triad, that is, on the tonic minor triad itself ; for here the nega-
tion is what is principally meant. But where the negation has not
the principal meaning, where the major triad is tonic chord and the
minor triad only subdominant, as in the system of the minor-major
key, there the relationships will be determined only upon the
MODULATION 165
tonic major triad. By change of meaning in the tonic minor triad
of the key of C minor there result the relationships —
II III I
C_eb— G
I III
Eb-g-Bb
III II
Ab— c— Eb
II
G— bb— D
I
F— ab — C
of the keys of Eb major, Ab major, G minor, and F minor. The
key of G major is not related to the key of C minor. The G major
triad is in itself already positive and of primary determination.
But when it is confirmed as such, then the notion of the key of C
minor is taken away ; for the essential content of that notion is,
that this triad presupposed positive is taken as negative. To the
key of C minor, the key of G major with minor Sixth, which we
term minor-major, alone can appear related ; as also to the latter,
taken as principal key, the key of C minor is in its turn related,
which, however, does not stand in relationship to the key of G major.
To every key-system the opposite system with like name will
always stand in near relationship ; the minor key to the major
with like name, the minor-major to the minor or major key with
like name, and so too inversely. For here the transformation acts
upon the tonic interval of Fifth, which passes from positive to nega-
tive or from negative to positive meaning, but in both determina-
tions always contains at once the positive of the one and of the
other.
I— II
II— I.
,66 HARMONY
ENHARMONIC CHANGE.
289. Those passages which are founded upon so-called en-
harmonic change we can here only mention in passing, for they
belong to the tempered, not to the pure note-system. So far as they
may be possibly referred to the latter, they have already been
included in what has preceded. Mostly they depend upon the
diminished Seventh being identified with the major Sixth, or the
augmented Second with the minor Third, e.g.
b— D— F— ab = B--D— f— g J= B— d— eft— G#=cb— d— F— Ab
c : vu°7 a : vn; f J : vn°7 eb : vn°7
On this assumption every chord of the diminished Seventh lands us.
in four different, widely separated, keys. Three such chords, each
looking four different ways, may be set out :
b_D/F— ab, f#— A/C— eb, ctf— E/G— b;
whereby, if the proper Seventh chord be taken, modulation to the
whole twelve keys of the tempered Fifth-circle stands open — to the
major keys as well as to the minor, for we know that the diminished
Seventh chord can be referred to a major triad as well as to a
minor, and accordingly that the resolution b— D\F— a\> ••• C—e—G
is found no less frequently than the other, b—D\F—a\> ••• C— e\>— G
(par. 43). Besides these we know too the lawfulness of the resolu-
tions b—D\F—a\> ... C—F—a\> and b—DjF—a\> ••• C—F—a
(par. 212). In the latter the Six-Four position of the resolving
chord may draw after it progression to the triad C — e — G as.
dominant chord, and bring about the close in the key of F major
or minor, which may thus be equally well reached either from the
Seventh chord e— G\B\>— d\> or from b—D\F—a\>.
A diminished Seventh chord in any of its enharmonically
different attitudes may always be derived unprepared from one of
ENHARMONIC CHANGE 167
the three principal triads of any key. Consequently modulation
into any major or minor key that may be desired is easy to ac-
complish by this method.
290. For so far as this way of modulation is believed to be
authorised in assuming as identical, because of outward nearness,
what is inwardly quite different and without relationship, it is as it
were tainted with untruth, and we cannot rank the constructions,
whose explanation has to be sought in such enharmonic changes,
with those which depend upon an organic union. They have not
a natural life, and exist only in the turbid element of the inaccuracy
of tempered intonation. We have already perceived that in true
harmony the difference between like-named Third- and Fifth-notes
is found to be essential, and that these notes may indeed come into
collision in chord-constructions, but can never stand indifferently
for one another. But the difference of the enharmonically different
notes, b$ -- C, c§ -- D\>, according to theory still less allows
of their being identified ; not because the interval of sound is greater,
but that in their organic generation, the primary source of all
note-determinations, there is no possibility at all of confounding
such different degrees. Even if we will not abide by the positive
series of triads
in which the enharmonically different note does not appear until
the ninth member, and if we look for its nearest possible approach
as given by the series
C.eb..G..bb..D..fff..A..cJt..E..gJt..B..dl..FJt
yet even then the inward gap between enharmonically different
degrees still stretches out far beyond anything that can enter into
mutual relation in harmony or melody.
1 68 HARMONY
291. It is not by any means true, however, that all enharmonic
substitutions that occur in the writing of music are to be taken for
changes of meaning in the above sense. More often it happens
that a composer consciously or unconsciously puts one name for the
other to lighten actual performance, for more comfortable fingering,
and sometimes no doubt also from over haste or want of thorough
knowledge of harmony, without intending thereby harmonically to
alter the meaning.
But in vocal music it is never allowable to write an enhar-
monically different note instead of the right one, with the intention
of facilitating intonation. If a progression is impossible to sing with
the right notation, that is because the harmonic link fails. Diffi-
culties in singing are not made easier by wrong notation. The
minor Sixth is in itself an easy interval to sing, the augmented
Fifth a very uncomfortable one. But when the latter is part of
the harmony, a composer dares not write the easier for the harder,
if he will not risk persuading the singer to attempt what is perhaps
impossible.
292. In view of theory it naturally makes the greatest difference,
whether from C major we modulate to the key of Fjf major or to
the key of G|? major ; for the two last stand to the first in exactly
opposite relation : G\> to C as C to F§. Nevertheless in practice it
often happens in pianoforte and orchestral music that one of the
extreme keys is exchanged for the other. This is not always to be
called modulation in the enharmonic way, for the enharmonic change
of meaning can take place in this case either before or after the
modulation. But so it may chance that a piece of music of some
length with such enharmonic changes shall begin in C major, and
close, according to the connexion of notes, in BJ major or ~D\>\> major,
although the writing may show neither of these, but the original key.
Then, however skilfully the whole may be composed in other re-
spects, as regards the key it will always contain an untruth.
ENHARMONIC CHANGE 169
293. Music in performance passes in time before the hearer,
and while it goes on we have sensibly before us only what hangs
immediately together. This makes us overlook many faults in the
form and conduct of a piece of music, which, if the whole were
set out comprehensively or, if we may so say, architecturally to the
inner sense, could not possibly be hidden. Crookedness, want of
symmetry, disproportion, in visible objects that pretend to regularity,
at once meet the healthy eye unpleasantly. Unfitness in modula-
tory arrangement, as well as in metrical relations of phrase, would
be as easily perceived as faults in the immediate succession of
chords, if it were not already in itself a harder task to glance over
a whole of some magnitude in time with its parts, than to review
in its proportions something made up of parts in space. Now
there is such an architecture in music, and it consists principally
in the regular structure, metrical and modulatory, of the piece ;
a requirement so essential that without it a composition has no
pretence to art. For the first impression these conditions seem to
be of less active influence. For we see productions shapeless,
rhapsodical, without intelligent building up of periods, without
organic unity of manifold contents, extort not seldom a brilliant
success. But the works that have been able to keep in lasting
favour have ever been such as, apart from characteristic pecu-
liarities, apart from charm of melody and harmony, preserve order
of rhythm and modulation ; i.e. which wear their beauties set in
the beauty of the whole, in the truth and reasonable conformity to
law of a form in itself artistically valuable.
294. It is no more purposed to give lessons here on the prac-
tical handling of passages of modulation, than in the earlier inves-
tigations of harmonic combinations it was discussed how technically
to apply them. That can by no means be expected in a treatise
that represents chords exclusively in the closest position of their
intervals and in progressions such as issue, without choice or
guidance, merely from the most obvious requirements.
i;o HARMONY
295. If determinations should be here given for the modulatory
organisation of a piece, they could only be quite general. The
particular form is determined by the particular contents ; it is
subject to principles of universal validity only in the broadest out-
lines and in the narrowest detail. Particular in universal (which
is also universal in particular), i.e. individual, constitutes reality ;
of which the concrete existence is apparent to reason, but by the
intellect can only be imperfectly apprehended, i.e. either in abstract
universality or in abstract particularity.
296. That something leaves unity and enters into opposition
with itself, and then that this opposition is done away with and
linked into union, is the notion and explanation of all real coming-
to-be and of all reasonable formation.
The harmonic succession of the triads fj ••• Q ••• Q, or the
three first notes of the melodic scale, C-D-e, which are based
upon that succession, contain in the narrowest compass everything
that normally lies at the bottom even of the broadest formation.
What is here given within the key as chord-succession can but be
repeated in the same sense, when the key itself is taken as the con-
crete element of unity, and the advance of construction made from
it. Thus in the succession above, the secondary triad-element (the
Fifth, G} is by the entrance of another triad changed to primary
meaning ; or, in other words, the change of meaning in the secondary
triad-element produces another triad ; and then the change back of
meaning (G becoming Fifth again, as at first) reproduces the first
triad, which before was absolute, but now is resultant. And when
not an element of the triad but the triad itself as element of the
key experiences a like transformation (i.e. starts as tonic, travels
through dominant meaning, and finally re-enters upon tonic), the
key will similarly be reproduced from its opposite, and derivative,
instead of being, as at first, posited immediately.
297. Although there are pieces that take another course of
ENHARMONIC CHANGE 171
modulation, and that have not their principal division, the finish of
their first part, in the dominant key, yet we may now set aside
every divergent form as abnormal. The decidedly other key, which
forms the opposite into which every piece of music must pass in
the middle, is the key of the dominant ; not the subdominant, for
from the first an onward, not a backward, course should be taken.
With the latter the beginning could only come upon an earlier
beginning, and would then no longer be itself a beginning. If the
modulation leads straight on to the subdominant, then the principal
key appears itself as dominant ; it loses the tonic character. By the
key of the dominant the tonic is not only not endangered ; rather
this is the right key to settle it firmly. The principal key, coming
after the dominant, is at once felt as tonic ; after finishing in the
dominant, the tonic can re-enter with full power.
298. There are small pieces too constructed all in the same
key. They then have their harmonic opposition not in the key
but in the chord ; and what has just been said of the key is in
this case true of the triad.
299. As a composition with modulation can be constructed in
the tonic and dominant keys, but not in the tonic and subdomin-
ant, so one without modulation may be made with the tonic and
dominant triads, but not with the tonic and subdominant, at least not
in a natural manner. There the principal key, here the principal
chord, would through the subdominant take on dominant meaning :
that is, a meaning, not of setting out, but of going on. It would
exchange the character of positive for that of relative.
300. But though the tonic key, as also the tonic triad, has in
the first place to maintain positive quality, yet must not this posi-
tive continue absolute and given immediately. For then the con-
dition of reality would depart, which requires that it should be
derivative as well. Thus the tonic triad must be accompanied
with its dominant and subdominant triads before it can receive full
I72 HARMONY
meaning as at once source and offspring. For the subdominant
chord has only positive meaning, and the dominant only relative ;
but the tonic has relative meaning to the first and positive to the
other : it therefore comprehends in itself both determinations,
being in fact determined on two sides and not, like the others,
only on one. It is like the notion of the present, which between
past and future is itself to the first a future and to the other a past,
and thus at once future and past ; and in this opposition held
united in one it is the only time that exists really. So for the
complete determination of tonic quality it is necessary to touch
also upon the subdominant side. The succession of the major
triads of C and G contains as yet no unmistakable establishment
of the first as tonic, for it might as well be subdominant. But let
the dominant Seventh chord G — b — D\F follow the major triad
of C, and doubt as to the key will no longer be possible. For in
the dominant Seventh chord dominant and subdominant are held
united, and the resolving chord C — e — G demanded in the disson-
ance then enters as decided tonic. A more detailed and spread-
out determination of tonic character would be given by the do-
minant and subdominant triads in succession preceding the close,
or by the Seventh chord upon the Fifth of the dominant followed by
the dominant Seventh chord, e.g. (marked with the Root-harmony)
C-F-G7-C °r C-D°-G7-C.
301. These successions establish the chord as tonic. With
regard to the key, the condition that it must be established as on
both sides derivative seems less pressing than the need felt within
it of hearing the subdominant, as Seventh to the dominant triad.
Nevertheless in a composition carried to any length the modula-
tion would be felt to be wanting in completeness if keys lying below
the principal one were not also brought in ; if only chromatic
sharpenings, and not chromatic flattening^, were found in it. For,
taken generally, it is in this outward difference that the ' One '
ENHARMONIC CHANGE 173
and the ' Other/ the dominant and subdominant sides, must be
shown, in phrase of the major key. But the first part of a piece,
even when of larger compass, cannot in general be determined to
manifoldness of modulation ; because its business is merely the
setting out of certain contents, in particular of a duality consisting
of a principal and a subsidiary phrase, the first in the tonic and the
second in the dominant. So that other keys, and therefore also
those of the subdominant side, especially the subdominant key
itself, cannot find a place until the subsequent working out, when
the positive character of the tonic key has been established by its
dominant.
302. In the universally valid, normal form of musical structure,
so far as manifoldness of key upon the whole prevails there, every-
thing remote in modulation, and especially everything directed to-
wards the subdominant side, falls only into the second part of the
whole ; and there its place is before the re-entrance of the principal
key, in which the principal and subsidiary phrases, which in the
first part were held apart in Fifth-separation, now come together
and are united tonically.
303. Although the principles which rule the arrangement of mo-
dulation are quite general and may be applied to every musical form,
yet here, when we speak of a succession of principal and subsidiary
phrases, of their Fifth-separation in the first part and tonic union
in the second, we have principally in view that conception of a
musical composition known as Sonata-form. This consists essen-
tially of homophonic phrase with divided periods. It is opposite
to the Fugue-form, which, woven polyphonically, less admits of a
division of periods, or similarly of an abstract determination of the
succession of modulation. And, generally, in the fugue, for
reasons not now to be explained, no richness of modulation can be
developed.
304. Exceptional arrangements of modulation could be caused
174 HARMONY
only if the principal key, instead of passing into the dominant,
sought out one of the other relationships, such as we discussed
earlier ; always excluding those of the subdominant. Thus in
Beethoven sometimes in major phrases we see the principal key turn
to a major or minor key related in the Third, and the first section
finished in that key.
305. The minor key has its principal relationship in the major
key of its tonic Third, which passes from negative Third-meaning
into positive Root-meaning, while the negative Root assumes
positive Third-meaning (A minor, C major). This is the relation-
ship most founded in opposition, and is therefore the one most
universally applied. A relationship of opposition almost as decided
the minor key finds in the major key of the Third of its subdo-
minant, by the negative tonic Third receiving positive Fifth-mean-
ing and the negative Fifth positive Third-meaning (A minor, F
major). This relation too affords a form of modulation well ap-
proved for the first division of a piece. And it is of the first divi-
sion alone that we speak here ; because there only is found a more
or less determinate opposition of related keys and the establish-
ment of a second against a first. For the further carrying out of
the modulation up to the re-entrance of the principal key no
schematic determination of form can be given. It could be ex-
pressed only as obedience to general principles of modulation, and
in aesthetic conditions, as also in the negative determination that
keys wholly without relationship, which may be touched in passing,
must not attain to being tonically established.
306. The minor phrases of earlier times usually pass from
the tonic into the minor key of the Fifth. This is a relationship
rather grounded upon the structure of the particular ' church modes/
as they are called, than in the nature of our more general musical
system. There was also a special claim to it in the polyphonic
manner of phrase ; where if a theme in a minor key has to be
ENHARMONIC CHANGE 175
carried out, it can be transposed into other minor keys, but not
into a major key without undergoing alteration.
CLOSE.
307. The title of this section may seem to indicate a conclusion
of the doctrine of harmony ; but that is not meant by it. Merely
to gain an insight into the general principles of harmony has been
the aim and object in our path hitherto. The conclusion of the doc-
trine is never attained. The end remains out of reach, if not sought
in this, that in all and everything we come back again to the begin-
ning.
Although, in what has preceded, the principal phenomena of
harmonic combination, of the process for construction and recon-
struction both in simultaneous and in successive sounds, as well as
in successions sounding simultaneously, have been discussed, yet
infinitely more might still be offered of interest for theoretical
knowledge. This infinity cannot be exhausted. Everywhere the
way only can be shown, how it leads onward in every direction,
and where it must be struck for the investigation also of any single
case, any particular instance. And the clue is to be found in
following out that process of production which is illustrated in the
present work, by steadily grasping the three factors of development
in their simple abstract meaning, in their universal essentiality, and
by analysing each composite whole in order to see it built up again
from its parts by their union.
What is here denoted by the title is the musical close, the
cadence.
The very essential metrical conditions which co-operate in the
construction of the cadence must as yet remain unexplained ; here
we are dealing with the harmonic conditions alone.
,76 HARMONY
308. The expression contains the sense of something having to
be brought together. Thus the close really presupposes separation.
Chords that are principally united can form no close when following
one another. The tonic triad of the major key cannot unite with
the two minor triads of the system into a close ; their too near re-
lationship is against such union. In the passage from C — e — G to
a — C — e, or from C — e — G to e — G — b, no decidedly antithetical
change of meaning takes place in the notes which remain the same,
£7 — e or e — G. For the Third to become Root or Fifth is not for
it to pass into its opposite, into what is quite other than itself. That
can consist only in the Root becoming Fifth or the Fifth Root The
Third is in itself already Root and Fifth ; if it becomes Root or Fifth,
it only gives up one or other of the determinations contained united
in it and reverts to the single one ; it finds a decided opposite in
neither of the two. That can only consist in one thing appearing
as another that, before, it decidedly was not. Therefore a change
of meaning sufficient to give the effect of a close cannot take place
on the Third, but must go upon the Root or Fifth. Accordingly
in the first instance they must be triads related in the Fifth that
passing into one another can form a close :
C— e— G • • • b— D— G, G— b— D - - - G— C — e ;
C— e— G ••• C— F — a, F— a— C ••• e — G— C.
309. But, again, wholly disjunct triads can also be united into a
close in so far as they have the link of the triad that lies between,
with which the initial triad is united in two notes. This linking
triad stands to the triad, into which the succession is to lead, in
Fifth-relationship, the relation of the close ; and, since the passage
can take place only from the linking triad placed instead of the
initial triad, this succession also yields the close in Fifth-related
triads. Thus the meaning of the close in the succession G — b — D> - •
a — C — F, which consists directly of the succession b — D\F<-
CLOSE 177
a—C—F, is that the Fifth F of the first triad, b—D\F, re-
ceives Root-meaning in the second, a — C — F. In the succession
F — a — C--D — G — b, which is directly the succession D\F — a->
D — G — bt the meaning of the close is that the Root D of the first
triad, D\F — a, receives Fifth-meaning in the second, D — G — b.
Of the relative triad-value which the chords joining the limits
have for their key, we have already spoken above (par. 145).
In the successions of Fifth-related triads this transformation to
the opposite takes place in the common note itself:
II II II
C— e— G • • • b— D— G, G— b— D • • • G— C— e.
Therefore of the three possible kinds of triad-succession, only
that of Third-related triads remains excluded from those that form
a close. The successions
III II III I
C — e — G • • • C — e — a, C — e — G • • • b — e — G
have no meaning as closes ; they do not bring together what is
essentially divided, decidedly antithetical. The tonic major triad
allows the possibility of a coexisting minor triad, but not of a co-
existing dominant or subdominant triad, or of a diminished triad.
Every other progression of triads than that of triads related
in the Third will form a close, and likewise every resolution of
the Seventh chord. For in the latter the resolving chord will
always be related in the Fifth to one of the two triads united in
the Seventh chord, or else disjunct from it.
310. But we have to distinguish two meanings of the word
close : namely bringing together and concluding. The first is present
wherever triads united in the Root or Fifth, or triads not directly
united, follow one another.
The close as conclusion may also be considered in two ways :
it may close altogether, or it may be such as to let an after-phrase
N
I78 HARMONY
C— e— G- • -b— D— G- • -C— e— G,
be expected. The former kind must lead from a dominant or sub-
dominant chord into the tonic triad in its triad-position, in order
quite to fulfil the sense that the beginning appears as the end, and
that first and last merge in one another. Any other than the funda-
mental position of the tonic triad must have been brought about by
some previous progression ; it cannot be absolute beginning, and
therefore also not closing chord. But it is alone the union with a
dominant or subdominant chord, the change of tonic and dominant
or subdominant triad, that in the close leads back into the funda-
mental position of the tonic triad :
© (!)
G- • -C— F— a- - -C— e— G.
The two diminished triads, which besides the dominant and sub-
dominant triads may be united with the tonic in forming a close,
lead, in consequence of their indirect progression, to a transposed
position of the latter :
(D (a) (D 0
C-e-G C-e-G
(e-G-b> - -D-F-b- - -e-G-C, (a-C-e). • -a-D-F- • -G-C-e ;
the diminished triad on the dominant side to the Six-Three position,
the diminished triad on the subdominant side to the Six-Four position
of the tonic closing chord.
311. But the closing chord does not attain to full tonic esta-
blishment until its relation not only to the dominant or subdo-
minant, but to both of them is brought out :
C— e— G • • - C— F— a • • • C— e— G - - b— D— G • • • C— e— G,
or
C— e— G- • -C— F— a- • -b— D— G- • -C— e— G.
That here the subdominant precedes the dominant, is simply in
CLOSE 179
agreement with the direct order, as the former is the earlier in the
harmonic generation. Wherefore this form of close is the most
universal and ordinary, and found repeatedly in the greatest com-
positions and in the smallest, in the trivial as well as in the most
sublime. For composers of genius have ever least sought originality
in oddness.
312. Nevertheless the form :
C— e— G—b— D— G-.-C— e— G-C— F— a—C— e-G,
in which the relation to the subdominant stands last, is also not
less right. In the older music especially it is frequently used,
where, moreover, it was naturally necessitated and brought about
by the structure of some of the so-called church modes. We call
this kind the Plagal close.
313. Under the notation :
C— e— G .-. b— D— G ..- C— e— G,
C— e— G ... C— F — a ... C— e— G,
might be conceived, not merely the form of a close, but also the
notion of that progress of modulation and its return into itself which
makes up the whole of a composition. Now by the requirements
of modulation the first passage must be, not to the subdominant,
but to the dominant ; and when the tonic has been established from
that side, then the subdominant too will be touched on.
Therefore the general form :
I iv V 1,
which places the subdominant before the dominant, is only good
for the close : not as a scheme of modulation, not as a succession
of keys, but only as a succession of closing chords inside the key.
314. When the dominant chord follows the subdominant :
F— a— C ... D— G— b,
no sustained note which changes its harmonic meaning is present.
N 2
l8o HARMONY
Union is therefore sought to be recovered by the entrance, simul-
taneously with the subdominant chord, of the diminished linking
triad D l F— a, whose Root D becomes Fifth in the dominant chord :
I II
F— a— C— D ..- G— b— D;
or by the Root of the subdominant chord continuing on into the
dominant chord as Fifth of the diminished triad b — D\F :
I II
F— a— C ... D— F— G— b
and forming with it the dominant Seventh chord which leads to a
decided close. In the sounding together of the Third of the domin-
ant and Root of the subdominant is contained the compulsion for
the former to proceed to the tonic Root. Therefore also in the
first form F— a — C — D ••- G — b—D, in which F and a go together
to G, the note F, which now is not present as sustained into the
dominant triad, will readily be added as a later Seventh to the
chord, to drive the leading note upwards ; so that this form of
close even in four-part harmony surrenders the Fifth of the closing
chord more readily than the Seventh of the dominant chord.
315. Already (par. 238) we have noticed the Pedal, though only
to allude to it. Here too a detailed discussion of it cannot be
undertaken. That belongs to the technical lesson-book. But so
far as the pedal bears upon the close, something general may still
be said on it.
316. The pedal can be established on two notes of the key-
system : on the dominant and on the tonic. These two are pivots
upon which a change of principal chords moves. Upon the do-
minant, the dominant triad changes with the tonic triad ; and upon
the tonic, the tonic triad changes with the subdominant triad. Such
a simple change of triad and chord of Six-Four upon the two notes
is indeed b no means what is imagined under the term pedal ;
CLOSE 181
rather the web of harmony over such a sustained note may be most
manifold. But those two chords are the fixed points for the passing
harmonies, which are spun upon the dominant before the close, and
upon the tonic after the close, in the one case postponing it, in the
other to prolong and echo it. The pedal upon the tonic, which can
only be formed after the close, is always to be regarded as an appen-
dix or Coda of the piece or section closed ; and here we have always
to distinguish the end from the close.
The newer music is much more exhaustive in closing, in heap-
ing on appended phrases, than the older. In old music mostly with
the closing chord the piece too is at an end. But in modern music
the close must often be sought a great way before the end. With
the older close which has no coda it is almost always necessary to in-
troduce a ritardando before the end, so as to prepare for leaving off.
Otherwise the piece seems to close abruptly and unsatisfactorily.
317. In melodic progression, when a close is to be brought
about, we see that the parts do not take the course that is suggested
by the conditions of chord-union. The succession G — b — D •••
G — C — e makes the Fifth D of the first triad go to the Third e of
the second, a progression brought about by triad-linking for the
two chords, which are joined only in the Fifth, in this manner :
G— b— D ... G— b— e .-. G— C— e.
But in closing, this part needs to pass, not to the Third of the tonic,
but to its Root, to find there the rest not given in the Third. Now
in essentially melodic phrases it is principally this Fifth of the
dominant that precedes the closing note in the part that carries
the melody. Thus the closes of chorales and popular melodies
have this form in greatly preponderating majority ; the melody
closes with a passage to the tonic, not from the leading note, but
from the Fifth of the dominant. In the harmonic close, besides the
Fifth of the dominant, the Third of the dominant will appear,
1 82 HARMONY
another part, and this also must proceed to the tonic. A third part
will hold the dominant Root unmoved as Fifth of the tonic. Thus
with these progressions the tonic triad will remain without a Third ;.
for if the Fifth of the dominant does not progress to that note, no
other interval of the dominant chord leads there. Hence the poly-
phonic music of the old time, which throughout is more melodically
combined, less a succession of chords than a chord of successions,,
a sounding together of melodies, often closes without tonic Third.
The dominant Seventh, which with us leads to the tonic Third, was
still strange to that time, or at least unusual and of rare occurrence ;
as indeed was Seventh-harmony altogether — the dissonances of
the old style are as a rule suspensions. When the top part has
leading-note progression, an inner part will readily progress from
the Fifth of the dominant to the Third of the tonic. With that
melody in the upper voice the Third is not wanting even in the old
vocal phrase, unless the tenor is obliged to close on the Root
by the Cantus firmus, which in many cases it has to carry. In the
minor key yet another motive comes in for not letting even an
inner part close with the Third. It is that between the Fifth of
the dominant and the minor Third of the tonic there exists leading-
note relation. In the key, e.g., of C minor to the chords Cr-.-C
such an inner part would receive the melody D • • e\>. This taken in
itself expresses, not a close in C minor, but one to E|? major, the
key of the Third. To avoid this the inner part here too goes
better to the Root ; or else instead of the minor it takes the tonic
major Third. Thus in phrases of old music in the minor key we see
the close, when not without the Third, always made with the major
triad. But the reason is not to be sought in the minor triad having
been deemed too little consonant for a closing chord ; it could not in
fact be introduced naturally if the melodic independence of the part
was to be preserved.
318. In the major close the Fifth of the dominant should for
CLOSE 183
melodic determination of the close pass to the tonic ; but, as urged
by chord-union, it would rather progress to the tonic Third.
Therefore it is brought by the two claims into division. It cannot
do both at once ; but it does one after the other. It lets the Third
be heard struck into or out of it before going to the Root. It also
repeats the Third struck afterwards, and with that there arises
the shake upon the Fifth of the dominant, the ornament so long
customary at the end of the old airs and solo pieces of every
kind ; which was therefore not a mere prettiness at pleasure, or
fashion of the time, but given with the kind of close and a natural
condition in it. Although the shake has its origin and proper seat
upon this note, it can nevertheless occur also upon other notes ;
but, in obeying nature, only upon such as admit of a double pro-
gression, upwards and downwards. The upper note of the shake
contains the first ; the so-called turn belonging to the shake contains
the second. The Sixth and Seventh degrees of the minor key-
system allow of the shake only when the relations of passage for
each are discovered outside the limits of the closed minor system :
as we found them from the Octave downwards to the minor Sixth
in the minor Seventh, and from the Fifth upwards to the major
Seventh in the major Sixth. But in a harmony that contains both
degrees united according to their determinations in the system,
these cannot be had recourse to without violence. Thus in the
diminished Seventh chord neither a shake upon its Root nor upon
its Seventh is found natural. To the former the turn is wanting, to
the latter the upper note of the shake ; the one has the augmented
Second below it, the other has it above.
319. To the close from the dominant or subdominant triad into
the tonic is opposed the close from the tonic into the dominant or
subdominant triad. With the former a whole is concluded, or the
principal section of the whole which has established on its own be-
half a key related to the principal one. The latter only marks the
•i84 HARMONY
fore phrase to an after phrase : not a period, only a clause. And
as such a clause need not present a closing chord, it follows that the
dominant or subdominant chord, due regard being had to its deriva-
tion, may there appear in one of its passing shapes. Besides this,
the condition whereby the perfect close could only be led from either
the dominant or the subdominant triad, here lapses, and the dominant
or subdominant triad may issue from any succession that meets the
case of a close. In the perfect close only the two cadences V---I
V---I ; G'-'C, F'~C, could be realised. But for the half-close
upon the dominant and subdominant, besides the two cadences
opposite to the two former, I---V, I---IV ; C'"Gr, C'"F, tnese
stand also at disposal :
II0-- -V, IV---V, vi ..-V; D/F-a ••• b-D-G,
F-a-C .-. D-G-b, a— C— e ... b— D— G, anc
vn... IV, V..-IV, in- »IV; b-D/F ••. a-C-F,
G-b-D • « • a-C-F, e— G— b • • • C— F— a.
Similarly this close may have its derivation from the triads that
unite the limits of the key-system in extension ; into the upper
Fifth f$la—C'-D—G—b, into the under Fifth e— G/£\>---
C—F—a.
320. We have yet to mention that form of close in which the
dominant chord is followed by some other in place of the tonic
triad expected. Such a succession is well known under the name
of False close. Within the key this succession will be subject only
to the principles which make up the general conditions of the
close. It cannot lead to chords related in the Third. Thus, after
withdrawal of the triad on the tonic C—e—G, there remain the
triads a — C — ey F—a — C, D\F — a, into which the dominant triad
£ — b — D can pass so as to meet the conditions of a close : as in
the successions .
V vi V — IV V — 11°
G— b— D-.-e— a— C; G— b— D-.-a— C— F; G— b— D • • • F— a— D ;
CLOSE 185
where we still denote the triad-progression only in close three-part
harmony, and neglect the consideration of one part serving as basis
for the others ; though this, in many cases, will itself take up one
of the progressions governed by the succession.
Besides the false closes which are yielded within the key, a
much larger number will be offered if the closing chord may belong
to another key. Here every way stands open which the arrange-
ment of modulation allows. We can ascribe to the dominant
chord four other meanings in different keys, agreeably to which it
can take the most manifold progressions. These will, however, be
curtailed, both here and also in the false close within the key, when
the chord leading to the close is not merely dominant triad but
dominant Seventh chord ; because then the progression in reso-
lution of the Seventh receives determinations by which many of
the otherwise possible successions are shut out. Instead of them,
with the Seventh chord, there are now successions found suitable for
a close which would not be so with the plain dominant chord : those,
namely, which suit with the upper of the two triads joined in the
Seventh chord. A false close in another key will by preference
fall always upon one of the three principal constituents, tonic,
subdominant or dominant, according to chord-union as issuing
from the dominant or subdominant chord with or without Seventh,
agreeably to the precepts in general of succession. The entrance
of a new chord of the dominant Seventh, if it can be prepared in
the chord preceding, is eminently fitted for determining the new
key ; because then the new leading note stands out clearly in its
quality of Third of the dominant.
II.
METRE
METRE AND RHYTHM.
1. WE shall call the constant measure by which the measure-
ment of time is made — Metre ; the kind of motion in that measure
—Rhythm.
2. The measure, as to outward structure, is found to be a two-,
three-, or four-part unity. For the motion in that measure, it may
in itself be infinitely manifold of shape ; nevertheless as measured
it can be understood only by the determinations that issue from
the metrical notion.
3. And here we shall meet again with the same elements of the
notion, by which the essence of the triad was explained to us :
namely, those of the Octave, Fifth, and Third, taking these intervals
in their abstract meaning, i.e. of unity, opposition, and unified
opposition.
METRE.
I. Two-timed. (OCTAVE,)
4. For the beginning of metrical determination we must take
an interval of time that at first is still undivided. Two successive
audible beats, supposed one second of time apart, may be the
sensible image of such an interval of time.
5. These two beats enclose only one space of time. But with
the two beats we have, not one, but two times determined. With
1 9o METRE
the second beat, marking the end of the enclosed space of time, there
is given the beginning of a second space equal in duration to the first.
At the end of this second space we may expect a new beat, which,
however, cannot happen earlier than at that point of time without
causing an interruption, a curtailment of the time determined for
us by the two beats. What is injured by a later beat happening out
of time is not the actual interval of time bounded by the two original
beats ; for that in itself cannot experience disturbance from some-
thing that does not enter until it has expired. Yet we feel that a
beat happening before the completion of the second space of time
does disturb the metrical determination given by two beats. Con-
sequently what is disturbed is not the enclosed interval of time
simply, but the metrical unity made up of this and the interval
which follows it.
6. A single beat then cannot determine a space' or magnitude
of time. Rather it denotes only a beginning without an end. But
with two beats following one another we obtain a whole determined
in time, of which the space of time enclosed by the two beats is
the half. The first metrical determination is not of a simple inter-
val of time, but of a twofold or repeated one.
7. A simple time is not a metrical unit, and cannot stand as a
metrical whole. A single thing in metrical determination has its
meaning only as part of the whole, as first or second. For the
metrical whole, from its first determination onwards, is an undivided
double, a twin unity.
8. This first determination is to metre that which the Octave is
to the intervals of harmony. The Octave too is in reality only a
half ; and in this meaning it opposes itself to its other self, i.e. the
other half ; and taken together with this other (the reflexion, out-
side, of itself), it ' then ' fulfils the notion of itself as half of a whole.
METRE 191
II. Three-timed, (FIFTH,)
9. As two beats enclose one space of time, determine a second
and join it to the first, so three beats, actually bounding two spaces
of time, cause a third to follow as echo of the second. But this
third part of that which is now to be comprehended as a whole of
three parts does not stand in a relation of equality to the two pre-
ceding parts, but only to the second of them. It arises by echoing
the second, just as we have seen the second arising as an echo of the
first. And thus the second member of the three-part unity gets
the double meaning of being second to a first and first to a second.
But in the latter meaning, because it becomes first to a second, it is
withdrawn from union with the first member, which is left standing
solitary. Separation of the unity enters in the first pair. The twin
unity becomes twoness. This and the contradiction of the double
meaning in the second element is what we have already pointed out
as the essence of the Fifth.
10. It is not as a succession of three members strung together
that the three-part in time is metrically determined and intelligible.
For then as a mere chain of members every other quantity, fivefold,
sevenfold, elevenfold, would be so too. But its metrically intel-
ligible sense is the interlacing of the twoness of the first and second
members as first pair with the second and third as second pair ; a
formation in which the middle member of the three-part whole has
the determination of belonging to both pairs, and, self-opposed, of
being end or beginning.
II. If a second is to be added to a first, then it cannot be
otherwise than equal to the first ; for unequals cannot be counted
I92 METRE
together. In the three-part whole, to the first single part a part
of double magnitude, or to the two first parts comprehended in
unity a single part, would be opposed as other part. In neither
case would the notion of equality in opposition be satisfied. That
which is single can only have another single ; the pair can only have
another pair, to be its other or second. Thus if the three-part
unity is to admit of intelligible partition, the pair made up of the
first and second parts of time can come into opposition only with
the pair made up of the second and third parts.
12. In a succession of three equal elements of time a, b, c\
if the part a, as first, be taken single, then the second b — c, being
double in magnitude and unequal to the first, for this reason can-
not be a second to a ;
a b — c
nor yet if a — b be joined to make a first, can c, being single, be
second of this first.
Only the double times a — b and b — c can here be opposed to one
another as A and B.
A — B
a b — c
III. Four-timed. (THIRD.)
13. A fourth beat happening after completion of the third
space of time now causes a fourth part of time to follow as echo
of the third, which, at first itself preceded, now precedes, and has
become a first with the fourth space as its second.
METRE 193
14. This last metrical formation, being four-membered, is twice-
two-membered, and in this sense is Third. But in the course of its
successive growth — and it is shaped in time, and therefore can
have its nature and reality only in this process of becoming and
having become — it is at the instant of its first determination two-
membered, or Octave ; next it becomes three-membered, or Fifth ;
and lastly four-membered, i.e. twice-two-membered, Third. To the
last determination it cannot attain otherwise than by passing
through the shapes proper to the first two. And thus on reaching
the last it is a successive union, a union in time, of Octave, Fifth, and
Third : the metrical triad.
15. The first beat gives us the Root, as yet undetermined in
duration. The second gives us the Octave, the determinate time ;
the interval of time joined to its copy as metrically determined
unity. With the third we have the copy of the second space of
time, reckoning the second space now as a first ; consequently the
two first times, which belonged together, are now separated ; the
half is withdrawn from its whole, and there is the contradiction in
the second time of being self-opposed, second and first, end and
beginning. This is the metrical Fifth. The fourth beat causes the
copy of the third element to come into existence, and the third,
from being a second, to become a first. Thereby the second,
which in its relation to the third was withdrawn from its union
with the first, is restored to the first, and again becomes one with
it. And now the first and second, being in a state of unity pro-
duced and derivative and no longer merely immediately given,
have themselves become a first, that has for its second that like
double unity of the third and fourth which is its copy. The
whole has become also a part in the notion of the Third.
1 6. It is this inner reconcilement of separation in unity and
unity in separation, the completed negation of every negativing
excluding element, that speaks to us here in metrical determination
o
i94 METRE
as the essence of the triad ; but in combinations of notes as the per-
fection of harmony ; and generally in any guise of phenomenon as
the perfected notion of determinate reality.
17. Now in the processes of metrical formation there is one
thing that must be kept in view as an essential condition to their
right understanding. It is, that the changes happen upon a unity
always one and the same. Otherwise a change, into another, could
have no intelligible sense. Only in so far as a determination is im-
parted to the first metrical element by the later ones, have they
a meaning of unity with it. The unity given undetermined by
the first beat, is determined by the second, splits into twoness
by the third, and passes by the fourth beat from twoness into
unity of twoness. It is the passage from the feeling of the imme-
diate whole, through the intellectual analysing perception of its
members, to the intellectually felt, i.e. reasonable, notion of the
whole in its memberment.
1 8. The four-part, then, as a musical measure -of time is the
metre which is perfectly determined in itself and independent, con-
taining within it all elements of the notion of a membered whole,
and needing no addition to complete its unity. For the unities of
the metrical two-part and three-part taken alone are imperfect in
inner determination of memberment. In the former the element of
separation is wanting ; in the latter the element of reunion. Both
of them need to be repeated in order to find determination as part,
as half, in the notion of unity of a whole of higher order.
TWICE-TWO-TIMED AND FOUR-TIMED METRE 195
THE DIFFERENCE OF TWICE-TWO-TIMED
AND FOUR-TIMED METRE.
19. Two-part time repeated is always easily distinguished from
time essentially four-part : the -J bar repeated, from the | bar.
The first is only opposed as a whole to itself :
I — 2 i — 2
I 2
whereby the second and third times in succeeding are not united.
The union is only between the pairs themselves and between the
members of each pair by itself. But in four-part metre, whose full
notion in fourpartedness is reached after passing though three-part,
I. II. III.
I — 2 I — 2
the third member is not merely a repetition of the first, as begin-
ning in the second part ; it has previously, in three-part time, also
been the successor of the second member, relation to which it gives
up only with the entrance of the fourth member ; i.e. it gives the
second member back again to its first, and causes the two to be
united which at first were one and then separated.
20. Now it is true that the determinations of the metrical for-
mation have their essential bearing upon the first pairs of members
only. Yet the difference of two-part, three-part, and four-part
division, as well as of four-part and two-part repeated, is represented
also in the time-figures just as we have drawn them. Like the body
showing the soul, or the outside of a thing showing the inside, so
the figures show to what degree less or higher the unit-notion first
posited is developed.
Thus the difference between two-part metre repeated and metre
o 2
I96 METRE
essentially four-part, which in outward compass are both alike,
comes out clearly when we consider the two figures standing below,,
and compare them with one another.
Here the eye tells us that the last as against the first is undivided
in the middle, is organically richer determined, and more luxuriantly
twined.
FIVE-TIMED AND SEVEN-TIMED FORMATION
AS ARTIFICIAL AND INORGANIC.
21. The three-part metrical unity consists as to its formal struc-
ture of an overlapping double pair ;
for the twin unity has here half gone out of itself, and taken its
second element anew as first. Also the four-part begins anew pair
with its third member without thereby denying the past existence
of the union between the second and third members,
although that union is set in the background now that the united
whole pairs are opposed. From this one might easily be tempted
to advocate a construction carried on with overlapping pairs in the
manner of the three-membered formation, so as by continued link-
ing together of halves to give rise also to metrical unities of more
than four times.
But we have seen how with the entrance of the fourth mem-
FIVE-TIMED AND SEVEN-TIMED FORMATION
197
her the separation of the first pair, which sprang up in the third
element of time, is annulled. The pair has again become whole,
and therefore can now find its second, or opposite, only in the other
pair, which is set quite outside it. So that such an articulation
by halves can make metrical union only in Three-time. With the
fourth time the determination of the whole in its parts is closed, and
now to produce a further formation the whole must itself enter into
the meaning of the part.
22. It will therefore be self-evident that anything extending
beyond the fourth member, beyond the end of the second pair, can
no longer exercise an influence upon the interior of the first pair,
and therefore too can no longer stand to it, as such, in an organic
relation of unity ; and that therefore a metrical formation going be-
yond the four-part lies outside the notion of unity, and consequently
falls asunder into twoness. Anything metrically five-part cannot
be understood otherwise than as artificially put together out of two-
part and three-part, as 2 + 3 or 3 + 2,
3 3
2 2
Similarly seven-part can only be metrically intelligible as artifici-
ally made up of three-part and four-part ; or else of two-part, three-
part, and two-part ; as 4 + 3, 3 + 4, and 2 + 3 + 2.
3 3
2 2
But such formations are by no means capable of being shaped
into a metrical unity, as were those of two, three, and four parts. Not
I98 METRE
having sprung out of organic determination, they will never seem
more than artificially put together. Here the one is not followed
by another of like quality, i.e. a second, as the first in its opposition.
Instead of this there is another first, a new determination, which
can only make another beginning, and not a succession to what
has gone before.
In chord-union an immediate succession of two Fifths is self-
excluded. In harmony taught rationally no special prohibition of
that progression would be wanted, for between united triads it can
never occur. Rather it marks discontinuous juxtaposition of two
triads in the primary position, two beginnings placed next one
another ; and it is precisely this want of union that comes out so
offensively in consecutive Fifths. So too a metrical formation
placing two- and three-membered unities alternately makes us feel
how rhythmically incongruous is the repeated shock of the new be-
ginning which it causes at every change instead of steady progress.
23. Still a thing irregular in itself may yet form a regularly sym-
metrical whole if it be opposed to itself in a regular pattern. We
see this in the figures of the kaleidoscope, in which the most hetero-
geneous objects thrown together quite at random are shown as a
regular star by repeating them symmetrically about a centre. So
too such metrical formations as the five-part and seven-part may
attain to a degree of admissibility by being received as members,
into a metrical formation of a higher order and repeated ; that is
to say, when the evolution proceeds initially from them as from a
given quality. But even in this use of them the feeling for unity is.
not fully satisfied, and less when the two-part or four-part precedes-
the three-part,
3 3
FIVE-TIMED AND SEVEN-TIMED FORMATION 199
than when it follows it.
3
2 2
434
In the last form, after the termination in two- or four-part of
the first member, it is easier to begin the second member again in
three-part than it is in the first form to join the two- or four-part
beginning on to the three-part end. Even by itself the five- or seven-
part member is produced more readily when the crooked precedes
the straight. The crooked, the three-part, contains the element of
dissonance, which finds its resolution in the straight, the two- or
four-part. Nevertheless such determination is too abstract for
every case of concrete detail to be included in it. Formations of
this kind, which spring out of an evolution, not that progresses
steadily, but only that is steadily interrupted, regularly irregular,
can never reveal a metrically healthy nature ; and they are as little
suited to the continued time-measurement of a whole piece as
diminished and augmented triads for carrying out its harmony.
Attempts to apply composite bars in music are as a rule far more
apt to impress us with the perverse eccentricity of the composer
than with the naturalness of growth, in metrical structure, of the
composition. Besides that such metres cannot hold out in five or
seven parts for long, and usually soon pass again into two-, three-,
or four-part measure ; so as to be resolved in a determination in
itself intelligible, and therein to attain steadiness and quiet progress.
200 METRE
COMBINED METRE.
24. Now though such an artificial putting together of different
metrical formations, i.e. the addition of them, has thus small power
to form a metrical unity ; yet on the other hand their multiplication,
the combination in which something two-, three-, or four-fold is
again taken two, three, or four times, and where the two-, three-, or
four-membered unity becomes itself a member of a two-, three-, or
four-membered unity of higher order, will always result in none but
natural, easily-comprehended metres.
In the multiplication of the quantities of metrical determination,
the quantity of the multiplicand is taken as unity, and in this
quality is taken metrically double, triple, or fourfold together into a
whole. Then every single element of such a combined formation,
as member in a member, has its value with respect to the whole
determined to it by the whole, and stands to every other member
in a determinate reciprocal relation.
25. In a metre composed by addition of straight and transverse,
e.g. in five-part, each single part has metrical organic determination
only as either half of the two-part or a third of the three-part ; it
belongs to the whole not in the same quality : in one member of
the compound metre it is different to what it is in the other. In
the metre arising from multiplication of straight and transverse, in
the six-part made out of twice three or three times two, every
sixth part has its determination as a third of the half or half of the
third of the whole — if we may so express it, as Fifth of the Octave
or Octave of the Fifth of the Root of the six-membered metrical
unity — and in each position it remains the same with respect to the
whole. In the five-part each single part is Octave of the two-part,
Fifth of the three-part ; it is determined differently from two
COMBINED METRE
201
different roots, and remains disparate in itself, an unresolved dis-
sonance.
26. Thus besides the simple two-, three-, and four-part, further
formations, metrically intelligible, may be constructed, by taking
two-, three-, or four-fold as units again in two-, three-, or four-fold,
namely :
2x2, 2x3, 2x4.
3X2, 3x3, 3x4.
4x2, 4x3, 4x4.
2X2
2X4
3X2
4 x 2 £
4x 3
4x4 OC^g^C^g^X
202 METRE
In these forms is necessarily contained everything that as
metrical construction can be comprehended under the notion of
unity. This in no wise limits us from giving wider scope to the
whole of the formation or a more minute articulation to its parts.
For we are quite able either to regard any one of the metrically
combined forms of unity as being in its turn part or member in a
new arrangement of higher order, in simple or combined form ; or
else to think of the part of any whole as being in its turn a whole,,
i.e. a unity capable of being metrically articulated.
27. From what has been said already, it is a self-evident result,,
that metrical articulation does not consist of dividing up a whole
previously given ; nor yet should- the whole be imagined to be a
grouping of unities into a plurality. Metrical formation is always
simply the product sprung out of the evolution of a first time taken
as beginning, and all the manifold construction here issues pri-
marily merely from simple opposition of the thing premised simple,
i.e. from doubling it. In two-membered formation, this opposition
acts productively outwards. The three-membered annuls the pro-
duction ; it denies the determination of the first member in appoint-
ing the second to be itself a first, thereby withdrawing it from the
first, out of which it was produced, and giving rise to the double
meaning in the second of being one and other, second and first in self-
contradiction, or diverse within itself. In this property we encounter
the essence of the Fifth, which already above ('Harm.' par. 113)
has been shown to be also correlative to the notion of dissonance ;
namely, inasmuch as in the harmonic union of the chords of pre-
paration, dissonance, and resolution the middle one also contains the
element of double meaning, of being at two with self, as the Fifth.
28. Thus in the passage from the major triad of C into the
major triad of G, in the dissonance C — D prepared by C and re-
solved into b — D, the note G, to which the dissonance is referred, is
COMBINED METRE 203
Fifth in the preparation, Fifth and Root in the dissonance, Root
in the resolution :
II — I
II — I.
In the passage from the major triad of G into the major triad
of C, in the dissonance D — e prepared by D and resolved into
C — e, the note G, to which the dissonance is referred, is Root in the
preparation, Root and Fifth in the dissonance, Fifth in the resolu-
tion :
I — II
I — II.
In both cases the middle element is oppositely determined in
just the same sense as in the metrical three-membered unity :
which, as we shall shortly see, can also appear in the meaning of
the first case :
29. How in four-time the third element in the evolution annuls
the separation of the first and second members, while in place
of their first immediate oneness it now brings about the derivative
unity of unitedness — all this has been explained in detail in
what has gone before, and nothing more is necessary to be added
here.
30. It is by no means the idea that we ought, in the metre
divided into three parts, to listen for a Fifth of sound, or in the Fifth
204
METRE
for a dissonance according to the special musical notion ; nor yet
in the consonant chord which prepares a dissonance, and in the
metre of two members, for an Octave, nor in the chord of resolution
and in the metre of four members for a Third. But then in such
identifications we have to seize upon and hold fast in proportion-
ately greater generality all that is essential to the notion of these
elements in their qualitatively different manifestations. So we may
also discover the same determinations joined into unity of notion
in subject matter seemingly yet far more remote. E.g., in the
division of the notion of regular extension or of space generally, by
considering its vertical dimension, height, as unity ; its horizontal
bilateral dimension, in every direction opposed to itself, breadth, as
duality ; and both united, as one in other, making a Third in which
every element of duality participates in the unity and is absorbed
in it, as unity of duality, and therefore as the Third, which completes
the notion by being the union of height-unity, or Octave, with
breadth-duality, or Fifth ; that is, as the close of the determination
of space. Therefore, as in the notion of completed space there is
no longer room for a further determination to be added, and as no
further consonance can be joined on to the triad ; so too the
metrical unity cannot extend beyond a fourth element of time with-
out becoming again twoness in itself ; and this we have seen in the
metrical formation of fivefold falling apart into two- and three-fold.
ACCENT.
31. A first element of time, which metrically can only be the first
of a second equal to it, is, in regard to its second, determining ; the
second is determined. A first as against its second has the energy
of beginning, and consequently the metrical accent.
ACCENT 205
ACCENT OF THE MEMBER.
(a) In the Two-timed Metre,
32. In the two-timed metre the first member is accented the
second is without accent :
(b) In the Three-timed Metre,
33. In the three-timed metre, as we regard its meaning up to
now, the first and second members are accented, the last is without
accent.
i — 2
i —
(c) In the Four-timed Metre.
34. The four-timed metre may not and cannot disown that union
of the second and third members, which was constituted by the
third portion of time in determining the second to be a first ; even
though such determination has been given up again with the
entrance of the fourth member. Hence the second member must
here receive another meaning than the one attributed to it in the
two-timed formation. It cannot be 'altogether without accent, as
in two-timed metre repeated ; for what essentially distinguishes
double two-timed metre from four-timed is that the latter has
submitted to the determination of three-part time, and that from
this third stage of its growth it contains the two halves of the
whole, not merely as the first and second to be united, but as a thing
already one in itself and knit together. Thus if the accent in double
2o6 METRE
two-timed metre can exalt only the first and third members, letting
the second as well as the fourth drop altogether, yet in four-timed
it cannot suffer the second member to recede, as against the first
and third, in the degree in which the fourth recedes. It must bring
out the value which,, before the fourth entered, was attributed to the
second in its relation to the third.
35. Here, then, three accented members follow one another ; for
as the first precedes the second as primary in the pair of members,
so the second precedes the third, and the third the fourth.
COMBINED ACCENTS.
36. To this determination of accent, which only touches the
members in the meaning that each has in its pair, must be added,
when pairs are united, another of higher order : that, namely, for
the pairs themselves. Everything that is to be comprehended in
the notion of a succession partaking of unity can have but one
beginning, one first, and not a repeated beginning nor several
beginnings. And so in every order of the formation one member
must be the first, and the member which follows equal to it must
be the other ; and should the formation be carried further, yet still
these two united can but be again a first to a following equal
member.
(a) Twofold, in Three-timed Metre.
37. The only metre without combination of several superposed
orders is the simple two-timed. The three-timed metre contains
COMBINED ACCENTS 207
already a first and second of higher order ; it has as members a
first and second pair of members ; only here they are not yet fully
parted from one another, as in the four-timed. But the second
pair, which begins in the middle of the first pair, has in respect of it
the secondary meaning, just as the second pair has in the double
two-timed metre. There the second pair is without accent as
against the first ; and so too the second member beginning in the
middle of the first, both of the higher order, is without the accent
belonging to this order. Consequently the second third part of the
three-part metre receives only the accent which it gets by being
first time in the pair of members of the lower order. This accent
it has in equal strength with the first third part. But the first
third part bears the accent of the higher order : that of the first of
the pairs ; and this it is that makes the first time of three-part
metre stand out above the second in having the principal emphasis.
Thus in the three-timed metre the first time is strong of the
strong ; the second is weak of the strong and strong of the weak ;
the third is weak of the weak.
(£) Threefold, in Four-timed Metre.
38. The accents of the four-timed metre, in so far as the members
of the formation in their pairs are concerned, have already been
demonstrated : in this meaning the three first times are accented,
the fourth is without accent. Taking the higher order into account
in the three-timed metre, we have found upon the first time the
accent of the first pair combined with the accent of the first member
of the pair ; upon the second, only the accent of the first member of
the unaccented second pair ; and the third time is without accent.
If three-timed advances into four-timed, and if four-timed is to
be conceived as sprung out of three-timed and succeeding to it ;
then, agreeably to the notion, the ' one and other ' of the last form
•can again be sought only in opposing to itself the three-timed.
2o8 METRE
39. As the three-timed contains a first and second of two-time,,
so must the four-timed, succeeding to the three-timed, consist of a
first and second of three-time.
40. It is not this side of the organic structure of the four-timed
metre that stands out in the effect as the principal division of it..
That is rather the opposition, also contained in it, of the first and
second halves of the whole : the twice-two, as in the double two-
timed metre. It is, however, easy to feel how much closer is the
linking of these two members in four-timed metre ; which comes
from the element of three-time and the union of this with itself, and
is in fact the essential distinction between the four-timed formation
and the double two-timed.
41. As regards the accent of higher order in the four-timed
metre, it will result as different from that of the double two-timed.
In the latter it is the second pair that is, as against the first, alto-
gether without accent. Hence the third member only has the
accent of being first in its pair. This in itself is equal to the accent
of the first member in respect of the pair. For the accent of a
member is independent of the meaning of the pair. It is in its order
the same in an accented as in an unaccented pair ; for it depends
merely upon the determination of distinguishing a first above a
second of the same order.
42. The four-timed metre in its derivation from the three-timed
consists of three overlapping pairs,
of which the second is without accent as against the first, but the
COMBINED ACCENTS 209
third must be accented as against the second ; for that which has to
follow a last can but be a new first.
This inverted succession will come up for detailed discussion
later on, and its consideration is therefore deferred to that place.
Of the two three-timed unities which exist interlinked in the
four-timed, the first is the accented one, and the second is without
accent :
i — 2
Again, in the first three-timed unity the first of the pairs joined in it
is accented, and the second is without accent ; in the second, which
begins with an unaccented first pair, the second pair is accented :
i I
2 2
and by this determination the third as well as the first of the over-
lapping pairs receives an accent :
i I
This accent will be increased, in the first pair, by the accent belonging
to the first three-timed unity ; it will thus be exalted above the second
accent of like order ; so that here, as in the twice-two-timed forma-
tion, the first pair is the accented pair, while at the same time full
value is given to the accent which is due to the second half of the
whole as accented pair in the second three-timed unity.
43. These determinations of thrice-two-timed and twice-two-
timed combined in the four-timed metre may be thus summed up :
The first member, besides its accent as member, which it has in like
measure with the second and third, receives an accent as first of the
P
2IO METRE
two-timed, and another accent as first of the three-timed, members.
The second member has only the accent of first in the second un-
accented pair. The third member, besides the accent of the first beat
in the pair, receives that of the pair itself, which is an accented one.
The fourth, because there is nothing left to which it can stand as
a first, remains without accent. Thus the stronger accent of the
first member will exalt it above the third, i.e. the first half of the
whole will rise above the second half, while the accent of the second
member, which it has as member, will make it recognised as a first
of its order whose second falls into the second half of the whole ;
whereby both halves appear in an inwardly joined unity, and not
merely strung together into a whole, as in the two-and-two formation.
This accent of the second member is the characteristic element
for distinguishing the four-timed metre from the twice-two-timed :
e.g. the | bar from the f bar repeated ; the -1/ bar from the £ bar
repeated, and so on.
44. Since no formation beyond the four-timed can afford a
metrical unity, therefore the determination of accent closes here.
Combined metres, as the twice-twofold, the twice-threefold, the
thrice-twofold, and the thrice- threefold, and similarly the combina-
tion of the twofold and threefold with the fourfold, and of the latter
with itself, will in each of the orders which exist interlinked in
them follow the same determinations of accent which would be
valid for them when standing alone. But the accent of the member
will always be absorbed in that of the pair which stands over it,
so that the latter gives the determination for the principal division
into members ; and the accent of the member in combined forma-
tions can only stand out in places where the pair itself has no accent ;
as, for example, in the second member of the three-timed and four-
timed metres.
45. Perhaps this exposition of inward and outward metrical
relation may seem far-fetched and artificial, importing into the matter
COMBINED ACCENTS 211
meanings and subtleties which do not lie in it, to gratify a theory
set up. But if we consider the results produced from this seemingly
too complicated procedure, nothing has been brought out but what
rhythmically squares with our feelings and seems natural, nothing
but what, in the sense of metre, comes naturally ' of itself.' And
this indeed is the sole aim of these investigations : we wish to make
clear in what sort and guise that is made which * makes itself — in
the simplest thing as well as in the most complicated — or which is
artificially made in the way we think natural when using art. The
artist's endeavour can only be to make anything so that it may seem
to have made itself. But, to enable him to accomplish this, the
means for representing his thought must be universally intelligible,
i.e. naturally given. A good musician will no more take pains to
discover new chords and new varieties of accent than a painter will
labour to invent a new shape for man, or to give man's form
another set of members than that which it has received from God.
THE NOTION OF MAJOR AND MINOR IN METRICAL
DETERMINA TION.
46. What in harmony lies at the base of the opposition in the
notions of major and minor, in metre serves to determine the emphasis
of the first or second member of the pair.
In the major triad the element of unity is placed in the Root of
the chord : in C— e— G, C G is Fifth and C— e Third ; both
intervals have their meaning determined by C, and find in C the
agreement of their sound. In the minor triad both intervals are
referred to the note of the Fifth. In a— C— e, a e is Fifth,
C — e Third ; here the note e is unit-element in the chord, and in
it the sounds of the Fifth and Third intervals find agreement.
Since both positive determinations meet in this note, we may also
P2
212 METRE
say that they issue negatively from it ('Harm.' par. 31). In this
sense we have throughout denoted the minor chord as a negative
triad: II — III — I. This elevation of the second element of the
triad, making it a primary and converting the first to a secondary,
will be expressed in metrical sense when the second and not the
first member of the metrical pair receives primary or positive deter-
mination, and the first receives secondary or relative ; i.e. when not
the first but the second member is accented.
Then in the metrical notion of minor, as formerly we saw in the
harmonic notion, the duality of unity will be expressed ; the notion
of major expresses the unity of duality. For the fact of emphasising
the second member marks it out as a positive beginning ; because
an accented element can be nothing but the positive first of a
second, since it is only in that quality that it receives an accent.
And if the metrical dual unity corresponding to the notion of major
contains a first and second as a whole, then the dual unity in the
notion of minor must comprise a second and first. Also the metrical
beginning, the accented, positive member, occurs in the second
half ; thus making it apparent that in the middle of the minor
formation separates have been united.
ACCENTS PRODUCED FROM THIS DOUBLE DE-
TERMINA TION.
(a) In Two-timed Metre.
47. In the metrical positive or major unity a first is followed by
a second :
1 2
in the metrical negative or minor unity a second is followed by a
first :
2 — i
ACCENTS FROM DOUBLE DETERMINATION 213
In the first case the beginning, in the second case the end, stands
out as the principal thing. It is seen that the first form has for its
contents what is sole ; the second, what was separated and is united.
(b) In Twice-two-timed Metre.
48. But now in combined metre, where the whole of a lower
order is contained as part of a higher order, such a pair (either of
one kind or of the other) may become member in a pair of higher
order (again either of one kind or of the other) ; that is to say, the
positive of the lower order in either the positive or the negative of
the higher order, and similarly the negative of the lower order in
either the negative or the positive of the higher order. In this
way, if we place the two-timed as member in the two-timed metre,
there arise four varieties of construction :
A, (a) positive in positive ;
(b} negative in positive ;
B. (a) positive in negative ;
(b) negative in negative.
2
A. (^
I--2 I--2 2-- I -I
The principal accent, because it betokens the element positive
to the highest power, must in such combined formations fall always
upon the accented member of the lower order in the accented mem-
ber of the higher order : in A (a) upon the first time ; in A (b) upon
the second time ; in B (a) upon the third time ; and in B (b) upon
2i4 METRE
the fourth time. The subordinate accent belongs to the accented
member of lower order in the unaccented member of higher order.
This is in fact the accent of the lower order of members, which here
stands out. It falls in A (a) upon the third time ; in A (£) upon the
fourth ; in B (a) upon the first ; in B (b} upon the second.
Musical notation comprehends the metrical two-, three-, or
four-part unity within the compass of a bar. In combined metrical
formations it is the multiplier that determines the principal
division of the bar. The f bar, as being of twice three parts, is
ranked among two-part metres ; the ^/- bar, which is of four times
three parts, is four-part. The beginning of the bar is always deter-
mined by an accented element. But the most highly emphasised
need not always be the beginning of the bar. There are also
metrical formations in which the principal accent falls upon another
member of the bar than the first.
49. The metrical forms considered earlier, before mention was
made of the notion of minor in metre (the succession 2 — i), all of
them begin with the beginning of the bar ; because in a combination
of several orders with none but positive determinations the accents
of all the separate orders must fall upon the first member.
In the metrical notion of major the first and second as positive
unity is musically written :
In the metrical notion of minor the second and first as negative
unity is musically written :
I ' I
This beginning with the unaccented member is called the up beat.
50. The above four metrical formations we should not consider
as four-timed, but only as twice-two-timed, according to the dis-
tinction that has already been made apparent between the two
determinations. For they do but contain two-timed in two-timed,
ACCENTS FROM DOUBLE DETERMINATION 215
without having passed through three-timed into four-timed ; their
fourfoldness is only repetition of the whole pair, whereby no separa-
tion of it is induced. These four formations, then, combined from
the positive and negative meaning in the pair of lower order with
the positive and negative meaning in the pair of higher order, will
be presented in musical metrical notation as follows, the determina-
tion being that the principal accent [ A ] must fall upon the accented
member in the accented pair, and the subordinate accent [ v or • ]
upon the accented member in the unaccented pair :
^. (*)
I 2
«-- " ~^.<^~ ~^*>v
I — 2
(M ^ ^^^- -^
I - - 2 I - - 2
-±-+±±-r-
i 1 1 1
2 I
c^ _>s^ _zxir^_^^— j^>
2 - - I 2 - - I
2 I
(K\ s~~ vf" """>
I - - 2 I - - 2
_^_*4-^-r-
1 1 '1 1
2 - - I 2 - - I
r r n r"
A («)
We find coincidence of the accented elements of both orders (i.e. the
accent of the member in the accented pair), in A (a) upon the first
time, in A (&) upon the second, in B (a) upon the third, and in B
(b) upon the fourth.
(c) In Three-timed Metre, referred to Twice-two-timed.
51. The three-timed metre is already in itself a formation con-
sisting of two orders united. It contains a pair of pairs, the second
of which begins in the middle of the first :
With this it has a double determination of accent : one for the
216 METRE
single members in the pairs, the other for the pairs themselves. The
pairs can here lie next one another, or rather overlapping one another,
either in the positive succession as first and second, or in the nega-
tive as second and first :
I — 2 2 — i
Similarly the members may be related to one another either in
positive succession or in negative :
i — 2 2 — i
12 21
From these different determinations for the members of the two
orders there result again four different kinds of accentuation in the
three-timed metre.
52. But every determination of accent in the three-timed metre
will always have its derivation in some determination of accent in
the twice-two-timed. For the three-timed metre is in fact a con-
traction of the twice-two-timed, or, more properly, it is a twice-two-
timed metre imperfectly spread out.
Thus the twice-two-timed positive determination of accent :
i — 2 i — 2
is presented in three-time in the following involved shape :
I 2
I — 2
I — 2
where the accented member of the second pair coincides with the
ACCENTS FROM DOUBLE DETERMINATION 217
unaccented member of the first. And similarly every other form
of accent in twice-two-timed metre is translated in like sense into
three-timed ; and the combinations of accent placed side by side in
what follows as twice-two-timed and three-timed must mutually
correspond to one another.
A. (a) Positive of the lower order in positive of the higher :
Twice-two-timed. Three-timed.
12 12
I 2 I 2 I — 2
I — 2
TTT
(b) Negative of the lower order in positive of the higher :
Twice-two-timed. Three-timed.
I — 2 I — 2
2 — I 2 — I 2 — I
2 I
+L±. -
I i
B. (a) Positive of the lower order in negative of the higher
Twice-two-timed. Three-timed.
2 I 2 — I
I — 2 1 — 2
218 METRE
(b) Negative of the lower order in negative of the higher
Twice-two-timed. Three-timed.
21 21
2 — I 2 — I 2 — I
2 —
53. The third of these determinations of accent in the three-
timed metre, B (a), has, like the second, A ($), the principal accent
upon its middle member. But it has at its beginning the accented
member of the unaccented pair, and cannot therefore, like the
second, begin with the up beat. Here the first part of the bar
has the subordinate accent, and the second part has the principal
one. This accentuation under proper metrical conditions is
also in practice found to be not unnatural. Yet if the metrical
figure be long-continued, entering as member into a formation
of higher order, the position of the accent upon the second time
of the bar will soon become doubtful in its effect. The
accented member will require to be -heard as first in the bar,
that is, to determine itself as the beginning of it. The accen-
tuation
i i i M i i >| i i
will soon come to be heard as one in which the first member bears
the principal accent, and the member with the up beat the inferior
accent :
I A • i A • I A • I A
-*— .*— +-H* — +—*~^*~
, 'TTTTTTTTT'i i
54. We find here an analogy to what was previously (' Harm. 'par.
38) said of the minor triad in reference to the construction of the
minor key ; namely, that a minor key can never be determined
from a series of minor chords only, in the way in which the
ACCENTS FROM DOUBLE DETERMINATION 219
major key is determined from a series of joined major chords. In
the chain of minor triads
G— bb— D— f— A— c— E— g— B— d— FJf
II I II I
II I II I II I
the positive
g— B[>— d— F— a— C— e— G— b— D— fj
I II I II
I II I II
will always have a tendency to put itself forward so long as the
negative, which is intended for the principal determination, is
deprived of the positive presupposed in it. Since in the minor
triad
II— III— I
A c E
the negative Third c — E is in fact also the positive Third C — e,
while the note c finds its positive Fifth in g ; and since similarly
every note which is a Third in the minor series can be assigned its
positive determinations in the same series, therefore the positive
altogether will come to prevail in the series and will make the series
itself appear a positive one. Only by the major triad on E can the
minor triad on A be determined as a tonic principal element, and only
by the major triad on B can the minor triad on E be so determined.
Hence too the rhythm
-rrr
must be united with one of the other rhythms that accent the first
member of the bar :
-rrr1 -rrr' -rrjf-'
before it can carry on its metrical meaning in a prolonged succes-
sion. We are now speaking only of a prolonged accentuation of
220 METRE
the second element. A single minor triad, e.g. a — C — e, would
neither awake the adjacent major triads F — a — C and C — e — G,
nor lose its independence through them ; nor would a single metrical
unity having the principal accent upon its second member call up
at once the determination of the up beat. It might even per-
sist without ambiguity through several repetitions. Only it does
not admit of being carried on as uninterruptedly as those rhythms
which are accented upon the first member of the beat. Accentuation
of the second time in three-timed metre is characteristic of many
dance rhythms, e.g. of the Mazurka. Yet even here it is not kept
up steadily, but alternates with other forms of accentuation.
55. Besides the four differently determined forms of accent in
the three-timed metre which are derived from those of the twice-
two-timed, there are yet four more to be added, whose determina-
tion is indeed to be referred to a double two-timed metrical forma-
tion, but which can never appear in double two-time, because their
nature is to pass at once into three-time. These are the forms of
accent which arise from union of pairs of opposite kind, and that
within the same metrical order. For hitherto the opposition of
positive and negative determination has been contained only in the
different orders superposed. The positive pair of the superior order
could have for its contents negative pairs of the inferior order ; or
inversely the negative pair of the superior order could be filled out
with positive pairs of the inferior order.
56. But if in the lower order itself a negative pair is set to follow
a positive, or a positive to follow a negative :
I — 2 2 — I 2 — i i — 2
then in the middle of such formations there arises a contradiction
against the condition of succession ; for they set equals after one
another :
2 — 2, i — i.
ACCENTS FROM DOUBLE DETERMINATION 221
Now ' after one another,' agreeably to its notion as well as to its
verbal expression, requires after one, another : after a second, a first ;
after a first, a second. The elements which meet in the middle of
the formation above, 2 — 2, i — i, belong as the same elements to
different pairs ; and in this meaning they certainly have so far their
difference. But in themselves they are alike. They are not one
and another, but in fact one and the same, and will also want to take
up one and the same place. In this sense these formations at once
of their own accord take the shape of three-timed metre :
1 - - 2
2 - - i,
2 - - I
I - - 2 ;
for the second member of the first pair is not different from the first
member of the second pair ; the same metrical element, a relative
or a positive, is presented in both. A succession, a division and
consequent union, could only arise here if these places should contain
relative and positive one after the other.
57. In what way two and three accents of members can follow
in immediate succession, we are taught by the positive three- and
four-timed metres. There the accent-elements get a difference
among themselves through the accentuation of members of higher
orders. But unaccented members cannot present any difference of
accent ; moreover no kind of metrical combination yet considered
shows a succession of several unaccented members. If the twice-
two-timed metrical formations just shown, with opposite pairs, are
wanted expressed as four-membered, it can only be by the first,
I — 2 — 2 — i, taking the metrical shape :
i — 2 2 — i
and the second, 2—1—1—2, the shape :
222 METRE
2 — I I — 2
-rrr;--
Then, in the first case, the second element is relative to the first
and the third relative to the fourth, and at the same time there
is brought in a relation between the second and the third ; for the
second, without detriment to its relativity to the first, is a positive
to the third. In the second form, 2 — I — i — 2, which does no
more than place in succession two positive and differently accented
members, the first rhythmical figure would also be brought to light,
were the metre to be continued ; for the same succession of un-
accented elements enters again, on the boundaries of the members
of higher order :
2—1 — I — 2 2 — I — I — 2
Thus these rhythmically four-membered determinations pass of
their own accord into three-timed metre. But the distinction here
taken in the relative elements which meet together vanishes in the
three-timed form of the metre with opposite pairs, and with it also
the small accent which emphasises the first of the contracted
members as against the second. And there arise for the three-
timed metre four accent-determinations different from those already
given.
A. (a) Positive-negative of the lower order in positive of the
higher :
Twice-two-timed. Three-timed.
I — 2 I — 2
I — 2 2 — I I — 2
2 I
TTTT- _ ,_
-±-i-+-±- I I I
Contracted • \ i
ACCENTS FROM DOUBLE DETERMINATION 223
(b) Negative-positive of the lower order in positive of the
higher :
Twice-two-timed. Three-timed.
12 12
^ ^" ^
I
I — 2
-r-rr
ea : ,— • !— f ~^
• i X
(<?) Positive-negative of the lower orde
higher :
Contracted : f
+ \ \ +
B. (a) Positive-negative of the lower order in negative of the
Twice-two-timed. Three-timed.
I 2 — I
2 — I I — 2
2 — I
V
Contracted : — *
I
(^) Negative-positive of the lower order in negative of the
higher :
Twice-two-timed. Three-timed.
2 — I 2 — I
I — 2
Contracte :
-rtr-
The difference that there is in the twice-two-timed accentuation
of the metres A (£) and B (b) disappears in the three-timed metrical
224 METRE
forms that correspond ; because there the two differently accented
positive elements coalesce into one. For the accent of this metre
it therefore comes to the same thing, whether the order of the
superior members is of positive structure or of negative.
58. Thus for the three-timed metre by determination of the
overlapping of opposite pairs there result kinds of accentuation
that are not contained among the earlier ones; namely, that of
the unaccented member between two differently accented ones,
and that of the accented member between two wholly without
accent Now the latter appears free from difficulty ; we know it as
the triple bar accented only in one element, where the accented
element is preceded by one unaccented element and followed by
another :
But the kind of accentuation that contains the unaccented element
in the middle :
— «•
I I 'I I i I I I 'I I
is far less able to be used without violence to rhythmical feel-
ing. The accented end of the formation will not easily be followed
by an accented beginning ; a shock is felt, as of a beginning repeated.
Here are two primary elements relating to the same secondary, two
positives to the same relative.
In this case too we have to look for, and represent, the relation-
ship of the metrical determination with the harmonic.
A series of major chords :
F— a— C— e— G— b— D
I — II
I _ II I _ II
ACCENTS FROM DOUBLE DETERMINATION 225
is to be identified with the metrical positive construction :
j 2
I - - 2 I - - 2.
A series of minor chords :
D— f— A— c— E— g— B,
II I II I
II I
answers to the metrical negative construction :
2 - - i 2 - - i.
2 - - t
Here a steady, continuous transformation of the relative element
into a positive is always proceeding.
The manner in which a harmonic element can subsist at one
time in double determination of unity is seen in the system of the
minor key, which contains the major triad of the dominant joined
to the tonic minor triad by the Root of the former, so that both
triad-determinations originate from that one note. E.g. in the
system of the key of A minor :
A-c-E-g{-B,
I II
II I
the note E is at once positive Root and negative : it is Root in
two opposite directions.
To this double determination corresponds metrically the form :
2 — i
i 2,
namely, accentuation of the second time without any subordinate
accent upon the first and third ; for the formation contains only
Q
226 METRE
one (differently determined) positive place and two relative, un-
accented places.
In triad-construction an element may occur as at once positive
and negative Root, but never as at once positive and negative
Fifth. And similarly in metrical construction, while the coexistence
of a positive and negative First is allowed, a positive and negative
Second cannot coexist in the same element.
The following is an attempt to establish this metrical principle.
59. We may imagine the metrical notion of major as a relation
of present to future, and the metrical notion of minor as a relation
of present to past.
The present is in both cases the positive (I) ; past and future
are the two relatives (II).
The metrical positive succession, I — 2, joins to-day and to-
morrow ; the metrical negative succession, 2 — i, joins to-day and
yesterday. It goes, not from to-day to yesterday, but from yester-
day to to-day ; beginning, not with the positive,- but with the
relative : it begins with that which is presupposed in the positive
present, i.e. with the up beat ; and it reaches the positive present only
with its second element. That this succession takes for beginning
what positively is not beginning, what to the positive present is a
relative, and that it places the positive itself as relative, is the
negative element in it, its essence as the notion of minor.
In the harmonic notion of minor, which has precisely the same
contents, there can be no expression of a negative by a single in-
terval ; for that remains the same, whether it be called positive or
negative. But when the Third placed inside the interval of Fifth,
instead of being joined on to the Root, is joined on to the Fifth —
issues negatively from it — then the Root may also be regarded as a
negative Fifth to the Fifth as a negative Root.
Similarly with the two beats which in the beginning we assumed
to denote a first metrical determination. If they are of equal
ACCENTS FROM DOUBLE DETERMINATION 227
strength, if one is not accented above the other, it is left un-
decided whether their mutual relation shall be metrically positive
or negative. For in the succession one after the other of two
equally strong beats we can imagine the one meaning
just as well as the other
I ' i
But with the accent, either can be established decidedly in metrical
first, or positive, value.
60. Now the metrical determination :
2 — I
I 2,
is thoroughly natural and consistent. It places in the middle
the to-day, to which a yesterday and a to-morrow relate. As has
already been said, it is in every respect to be identified with the
harmonic determination :
A— c— E— gj— B.
I II
II I
But the other :
— 2
2—1,
places a to-morrow as yesterday ; the first referring to a foregone,
the other to a future to-day. This is the same contradiction already
met with in the Fifth-, or separating, element of key-construction,
228 METRE
when the triad is determined in self-opposition as dominant chord
and as subdominant :
F— a— C— e— G, C— e— G— b— D,
IV IV I
a contradiction which is afterwards resolved in the Third-element,
when the triad determines, instead of being determined, on two sides,
and has therefore become a tonic triad.
The form
expresses a natural relation of time ; as also does the opposite form
2 I
2 I.
The first places to-morrow as to-day to another to-morrow : the day
after to-morrow ; — the second places yesterday as to-day to another
yesterday : the day before yesterday. The former begins with the
positive (to-day), and transforms its relative (to-morrow) again to
positive (to-day), The latter begins with the relative (yesterday),
and makes its positive (to-day) again into relative (yesterday).
In neither determination is any contradiction contained ; the
middle element in both is a positive towards one of the two sides,
to-day to a to-morrow or yesterday.
In the negative-positive succession :
it is to-day to a to-morrow and yesterday.
ACCENTS FROM DOUBLE DETERMINATION 229
In each of these three cases the change of meaning in the
middle element is in intelligible succession.
But in the form
2 I
the middle element, from being future, must immediately become
past. This it cannot do, unless during the passage it becomes
present (as must also the tonic triad in the key-system, when it is to
pass from the dominant meaning, which it has to the subdominant
triad, into the subdominant meaning, which it has to the dominant
triad). The abrupt transformation, or unreal metrical succession, is
the untruth, and therefore too, in this direct sense, the impossibility of
the rhythmical form. It may also be shown to be untrue in harmony,
in its application to the union of triads. For if we try to take the
Fifth of a major triad as Root of a minor triad, i.e. change the posi-
tive Fifth to negative:
C— e— G— bb— D,
I II
II I
the immediate result is an organic impossibility, viz. the minor triad
on G in a key that contains the major triad on C. In the key of F
major the triad on G would be, not G — b\> — D, but GjB\^ — d\
consequently not a minor, but a diminished, triad.
61. The form
I - - 2
2 - - I
contains two positive elements, which give rise in the relative which
lies between them to the contradiction of being past and future.
It is therefore precluded from passing immediately into Three-time ;
because its middle element is not truly a single element. The middle
member can be applied in its relative meaning only to one or other
230 METRE
of the positive members ; either to the first, I — 2 | I, or to the
second, I | 2 — i. Thus either the second positive element or the
first remains solitary. Then in the first case the second positive
element will seek its relation in the future, in the second case the
first positive element will seek a relation out of the past ; but in
both cases twice-two-timed construction will arise, a distinct pair
of pairs, of positive nature or of negative :
1 — 2
2 — i
i — 2 | i is filled out as i — 2 | i — 2 ;
1 — 2
2 I
i I 2 — i is filled out as 2 — i | 2 — i.
Moreover we feel plainly, that the first time and the third being
accented, and the middle time absolutely without accent, the third
cannot at once be followed again by an accented first ; but that an
unaccented time must either, as last, conclude the form, or else, as
first, begin it.
62. In order to link the succession we might also have recourse to
that separation of the middle element into one and another relative,
i — 2
I — 2 — I,
by which a relation of positive and relative is formed within the
middle element itself. But now, in so far as the middle time is a
time separated and united, it is no longer absolutely without accent.
Its first half receives the accent that must fall to it in this lower
order :
whereupon this three-timed rhythm has again returned to the point
ACCENTS FROM DOUBLE DETERMINATION 231
from which it started, and appears as what it really is, contracted
four-timed.
63. That a four-timed metre can sometimes be used with the
accent upon the first and fourth times ought not to be found in
contradiction with the statements just made ; since the explanation
of such rhythms is not derived at all from this source, and will
be given later on in considering syncopation. But moreover, in the
following accentuation :
A I VIA I
TT~ rr~nf!
the third member is by no means wholly without accent. If it were
without accent, it could not occupy a first place in the twice-two-timed
metre. Its accent of lower order is merely overshadowed by the
syncopated accent.
If, then, we proceed presently to rank the metrical formation :
i — 2
2 - - I,
among the three-timed, yet so far as the organic structure and ac-
centuation of it are concerned, it must not be supposed that we
have lost sight of the explanation just given of this metre.
(d} In the Four-timed Metre,
64. The essentially four-timed metre — i.e. that which consists,
not merely of a repetition of two-timed, but of a formation that has
passed through three-time — must also make apparent in its ac-
centuation the conditions through which it has come into existence.
That is, it must be distinguished in its accentuation from the
twice-two-timed.
The twice-two-timed formation consists merely of a pair of
pairs :
(~^ * \r — .. *s
2 32 METRE
If we assume positive nature of memberment throughout, it bears
the accent of the member on the first time and the third, and the
accent of the pair upon the first.
•'• TTTT '
The four-timed formation is something more than a pair of
pairs. Even if we could consider it, in accordance with its mere
extent, as twice-two-timed, still there is an essential difference
between the two in the fact, that in four-timed the second half of
the whole is not a member of opposite quality to the first, or at
least is not so necessarily. On the contrary, those determinations
which are most natural to the metre, viz. those which contain no
unusual accentuations, are derived from the following form.
2
And even if hereafter we shall see the forms
I 2 2
maintaining themselves as admissible, yet we shall observe that from
them arise only such formations as accentuate that part of the bar
which in the natural order is without accent : the so-called weak
part.
65. In the four-timed metre the two halves of the whole do not
stand to one another in immediate succession. The supreme op-
position here is that of a first and second of three-time.
In this twice- three-timed whole, the third two-timed member,
i.e. the second half of the whole, has direct relation only to the
second, and may be related indirectly to the first as I — I, I — 2,
ACCENTS FROM DOUBLE DETERMINATION
2 — I, 2 — 2 ; while in the twice-two-timed metre the two halves can
only stand to one another in the relation I — 2 or 2 — I.
If, then, the succession is the first of those given above :
the second half (the third two-timed member) has, like the first,
the accent of the pair, which therefore falls upon the third member,
as well as upon the first member ; the second member, as first in
the second pair, has, like the first and third, the accent of the member;
but the first member, in addition to the two former accents, has also
the accent of highest order, that of the first of the two overlapping
three-timed parts.
66. If we denote the accent of the three-timed part by 3, that of
the pair by 2, and that of the member by I, without intending by
these numbers to indicate the specific strength of the different
accents ; then upon the first member of the four-timed metre will fall
the accents 3, 2, and I, upon the second the accent i,upon the third
the accents 2 and i, and the fourth will remain without accent. If
we assume that the accent of the lower order is absorbed in the accent
of the higher order, the accentuation of the four-timed metre will be :
Represented in the same way, the three-timed metre has the
accents :
234 METRE
and the twice-two-timed :
2 o i
A like determination results if we represent the accents of every
order each for itself, and take the sum of them upon the respective
places.
In the four-timed metre :
•
-i-i-i-+- = -±-i-±-+-
iii! i i r i
In the three-timed metre :
-1—+—+- — -^— *— *-
111 III
In the twice-two-timed metre :
" A •
rrr = rrr
This is the accent-determination of the four-timed metre, as it re-
sults when the succession in every order is taken as I — 2, i.e. in
positive progression. It must now be examined how these accents
will be presented when the positive of one order combines with
the negative of another, or when the succession in every order
is negative.
67. The two-timed metre admits only of a twofold form of ac-
centuation ; it can consist either of a positive or of a negative suc-
cession of members :
i — 2
68. The three-timed metre follows the twice-two-timed, in so far
as it can be referred to it in its pairs ; and in the first place can
have, like the latter, fourfold form :
ACCENTS FROM DOUBLE DETERMINATION 235
Twice two-timed. Three-timed.
1 — 2 1—2
1 ' • ' • A. (a) '- " • i
I — 2 I — 2
I 2
0)
i i i i
2 — I 2 — I
2 I
B. (a\
i i i i
I — 2
2 — I
B. 0) i 1 • «
I 21 — 2 I — 2
1 — 2
2 — I
(J) ._•_ -.
2 — I
2 — I
69. The four-timed metre, consisting as it does in the highest
order of two overlapping three- membered parts, will contain double
the number of the formations of the three- timed metre ; because
the succession of the two three- membered parts may be either
positive or negative. Hence arise eight different determinations
for the accentuation of the four-timed metre.
70. But to the accent-determinations of the three-timed metre,
besides those which it takes corresponding to the accents of the
twice-two-timed, must be added also those which arise from com-
bination of opposite pairs ; these consist of a positive-negative or
negative-positive succession of the pairs of the lower order combined
with either positive or negative of the pair of higher order, and
form four other determinations, as previously detailed.
These will now be transferred also into the four-timed metre,
and that in double number ; because the succession of the two three-
timed members of the four-timed metre may be either positive
or negative. Hence arise eight new accent-determinations for the
four-timed metre.
236 METRE
But in joining opposite pairs :
I — 2 i — 2 2 — i
2 — I I 2
the double determination of the second member, which distinguishes
the four-timed metre from the twice-two-timed, is cancelled ; and
in accent these two formations fall back into the meaning of the
twice-two-timed, for they can only appear as such :
I — 2 I 2 2 I 2 I
71. Further, in four- time one more determination is to be added,
that in which the interwoven pairs are either of uniformly positive
or of uniformly negative structure throughout.
When all three orders take positive shape, we see the accents
hence decreasing step by step :
If negative throughout, the accents increase step by step :
0123
72. One last possible determination remains : that in which the
pairs follow one another in like succession, but are related oppo-
sitely in their members. Hence there are again produced eight
modes of accentuation different to the previous ones. Nevertheless,
as in the construction last but one discussed, the result of the
accentuation is only as if it belonged to twice-two-time ; for in
this case, as in the former, the second member is deprived of double
determination.
73. That we should obtain sixteen accent-determinations in
the four-timed metre coinciding with those of the twice-two-timed,
while the latter can only show four different forms in all, ought not
to be looked upon as a contradiction. Those sixteen forms may
be like the four in their result, and again may show a fourfold
ACCENTS FROM DOUBLE DETERMINATION 237
difference in the inner conditions which give rise to them. In the
twice-two-timed metre, as in the four-timed with opposite pairs
overlapping, the accents will fall upon the first and third or upon
the second and fourth members, and the intermediate members will
remain without accent ; one of the accented members receives the
principal, the other the inferior accent. In this the likeness of the
two metrical species consists. But the conditions from which these
like external results are produced show in the twice-two-timed
metre a fourfold, and in the four-timed metre a sixteenfold difference.
By negative elements the sixteenfold difference is reduced to four-
fold. But the conditions of the negation here belong no less to the
organic determination of the four-timed metre, than do the con-
ditions which pass positively into the final result. Accordingly the
accent-forms of four-time which agree with the twice-two-timed
must also be drawn out fully in the description below.
RESUME OF ALL ACCENT-DETERMINATIONS IN
THE TWO-TIMED, TWICE-TWO-TIMED, THREE-
TIMED, AND FOUR-TIMED METRES.
74. It appears from the preceding that there result
for the two-timed metre .... two,
„ twice-two-timed .... four,
„ „ three-timed eight,
„ four-timed thirty-two
different determinations of accent. Of the last of these, half the
number coincide with those of the twice-two-timed metre and are
absorbed in them, which leaves sixteen properly belonging to four-
METRE
time. Consequently the accent-forms for the four metres named
stand with regard to their number in the relation of 2 : 4 : 8 : 16,
or 21 : 22 : 23 : 24. In order to obtain a comprehensive and sys-
tematic review of the whole of them, we shall now set them down
in their connexion by metrical and musical notation. Not to in-
terrupt the tabular form of the description, any observations that
may seem required by particular rhythmical examples will find
room at the end, with a reference back to the examples in question.
A.
B.
I. Accents of the Two-timed Metre.
I 2
_
T" =
II. Accents of the Twice-two-timed Metre.
I 2
-~— — w. • •
— • — •• — ••— =
i i i
rr
-rr
2 —
III. Accents of the Three-timed Metre.
I -—2
RESUME OF ALL ACCENT-DETERMINATIONS 239
I — 2
2 I
2 — I
2 — I
2 I
(*)
A. (a-b]
•
* — ^H--
i I '
~ = ~~
- rrr
= r+Tr
A.
IV. Accents of the Four-timed Metre,
i — 2
5 frfT = trfnr
I — 2 — I
I — 2
08)
-rrrr -rrnr
240
METRE
2 — I — 2
I 2
-rrrr ~rrr\
i i i
• I • A •
-r = -rrrr
2 — I
B. (a-b, a)
08)
I — 2 I
2 — I
rr
I 2 —
2 — I
'"*' a) ^-^^T^F T r r r
2 I 2
= -»_i_£-i-i-
I I ! 'I
rrr
r+r-rr
•rrfnr
r+rrr
R&SUM& OF ALL ACCENT-DETERMINATIONS 241
2 — I
I 2 — I
2 — I
frfr - fr+hr
08-)
G3-)
I 2 — I
2 — I
2 — I — 2
2 — I
2 — I — 2
I — 2
I
• •
rrrr
I I
I -- 2
I 2
I — 2
• A V •
-f-r = rrrr
^^£^ rrf-r r+f-rr
I 2
I 2
2 — I
2 -- I
I — 2
rrr = -rrrr
2 -~
2 — I
242
METRE
2 — I
B. (a, a)
I — 2
I — 2
2 — I
-f-
I — 2
I -- 2
2 — I
2 -- I
2 — I
2 — I
"^ -+—+—+—+- =• -*— *-
> i i i i i i
— * — » — *•-
1 I I
(-8)
i^£>^> -p p p p- ' -p p p4
2 — I
2 — I
I — 2
A. f^, a-;
I — 2
I — 2
I — 2
rrr
I 2
I — 2
2 — I
2 — I
fr-h-
I — 2
^is^e^ -T-t-r-r- =
2 — I
2 — I
RESUME OF ALL ACCENT-DETERMINATIONS 243
-rrrr
. to, a-/3) o -o^j
2 — I
03-) - =
2 — I
2 — I
i i
-rrr
2 -- I
The accents of two-time and of twice-two-time in the foregoing
description need no remark. From what has been said before,
they are perfectly intelligible as they stand in the notation.
The eight different kinds of accentuation of the three-timed
metre are in their four first numbers taken from the twice-two-
timed, whence is explained their arrangement with regard to the
chief emphasis and the beginning of the bar.
Of the four last accent-determinations of this metre, which re-
sult from the succession of opposite pairs within the lower order,
we have before spoken particularly.
The thirty-two accent-determinations of the four-timed metre
are comprehended in four divisions or groups, each of which con-
tains eight differently intended kinds of accentuation.
In the two-timed metre there is contained only one metrical
order, in the twice-two-timed and in the three-timed two orders, while
R 2
244 METRE
in the four-timed three orders of metrical pairs are combined;
accordingly, we have denoted the succession of members in the first
(which in the two-timed metre is the only) order by A, B, in the
second order by a, b, and in the third order by a, /3.
The succession is marked as positive by A, a, and a, according
to the order referred to, and as negative by B, b, and ft ; a-b,
a-fi denote positive-negative succession in the second or third
order; b-a, fi-a negative-positive succession.
The second group, consisting of eight differently determined
dispositions of the accent in four-time (p. 240), can, according to
the conditions of formation which operate here, only repeat the
accents of the twice-two-timed metre ; for in this form the second
member has lost all trace of the double meaning attributed to it.
But if these eight accent-forms are to be shown equivalent to the
four of twice-two-time, their number must be reducible one-half.
And we find, accordingly, the accent 2 equal to the accents 4 and 8,
and further the accent 3 equal to the accents 5 and 7 ; consequently
the numbers 4, 5, 7, and 8 fall out as repetitions, and there remain
over the accentuations I, 2, 3, and 6, corresponding to 1,2, 3, and
4 of twice-two-time.
Simil'arly the fourth group of the four-timed accent-forms
(p. 242) being also without double determination of the second
member, is, like the second just spoken of, only twice-two-timed
in accentuation. Here too the resulting accents, 2, 4, 6, are alike,
and also 3, 5, and 7. The difference to be noted, that 2, 3, and
7 carry the secondary accent as simple emphasis of the member,
while in 4, 5, and 6 it is the accent of the pair at those places,
is not one that affects the result ; because in this metre there is
only room for a discrimination of accents into principal and
secondary. Accordingly these eight modes of accentuation also
are reduced to the four accents of twice-two-time.
In the third group of four-timed accents (p. 241), in which the
RESUME OF ALL ACCENT-DETERMINATIONS 245
second two-timed member has its determination changed (in the
preceding formations it remained unaltered), we notice a succession
of accents increasing or decreasing step by step, such as can only
arise under the following conditions : when all orders are positive
in structure,
3210
-* — *> — *•-— *- .
I. I I I
and when all are negative,
0123
-rrrr
This is the most complicated combination in respect of the
determination of accent. Nevertheless the result will not be other-
wise than clear and unambiguous even here, if we consider, both
singly and also in combination, the causes which co-operate in
bringing about the several determinations of accent.
In the formation
as in all others, we have first to consider each order by itself as a
succession of an accented and an unaccented part, and to reckon
as adding to the effect of every member of higher order only that
part of the lower order which fully belongs and is proper to it. Thus
this formation contains in the accented three-timed member
246 METRE
one accented and one unaccented two-timed member. The second
two-timed member has, it is true, an accent ; but only in virtue of
belonging, not to the first accented, but to the second unaccented
three-timed part.
Hence the accent of the second two-timed member, although in
taking the whole construction together
it falls within the compass of the accented three-timed part, never-
theless does not receive triple emphasis like the first, nor can
there be a doubt, which of the two members it should be as-
signed to.
In the second three-timed member (which as such is without
accent) this accent joined with the accent of member ranks only as
double, and with the two remaining members yields the succession
But the first, the accented three-timed part, receives the accentuation
T" — » — *•->
I I
and both three-timed members together
RESUME OF ALL ACCENT-DETERMINATIONS 247
combine to produce the figure
TTT
The member-accent of the second time of the first three-timed
part is the same which in the first time of the second three-timed
part was added to the pair-accent belonging to that member, and
thus gave it a double emphasis. This accent therefore, being
already reckoned in the double emphasis, is not to be added to it
again to make it triple.
In this way each of the accent-determinations here represented
may be accounted for, and justified as brought about naturally on
rhythmical metrical lines.
What we have marked as the beginning of the bar is always an
accented element. But the accents of second and third order, that
cf the pair and that of the member, may also be found as first
time in the bar. For a formation positive in the highest order, the
beginning of the bar can fall only in the first half of the four-timed
metre ; for a formation negative in the highest order, the beginning
of the bar can fall only in the second half. But moreover, an ac-
cented first time must be answered by an accented third time ; for
four-time is always also twice-two-time, its two-timed half will
always be apparent. That the whole is of twice-two-times is its
Third-condition ; just as the three-times of its two overlapping
highest parts constitute its Fifth-condition, and the two-times of its
three overlapping parts its Octave-condition.
Accordingly metrical forms such as « » « » (p. 240, .#,#-#,/?),
« « (p. 242, By a, /3) cannot be distributed in bars thus :
» «
i r
• | A V , A .
r r I r r r r I r r-
even though in this way the beginning of the bar falls, agreeably to
248 METRE
the first condition, in the second half. If the beginning of the bar
were placed thus, the third member, which corresponds metrically
with the first, would be without accent :
1-2 1-2 1-2 1-2
A • . A . V
r r r r r r r r
and so in contradiction to the twice-two-time of the four-timed
metre. The determination of the bar for these two formations can
therefore take place only as denoted in the scheme above. That
is to say, it must take place so that the member-accent of the fourth
element may form the beginning :
V A I •
and --
A member determined in the twice-two-timed metre as un-
accented can through the meaning which it assumes in the four-
timed receive single, double, or triple emphasis. But a member
which in the twice-two-timed metre is accented, even though it be
but singly, cannot in the four-timed become unaccented.
ACCENTS' IN COMBINED METRE.
75. The combined metres, in which any one of the formations
drawn out above with their accent-determinations may be taken up
as member in a formation like it or different to it and of higher order,
are, as may easily be imagined, of the utmost manifoldness of con-
struction ; but still the emphasis must always be subject to the
conditions here exhausted. In structures of greater compass the
chief accent must still fall always upon an element that is in all
orders emphasised or positive. This might have been seen already
in the twice-two-timed metre, which is really a combined one, and
ACCENTS IN COMBINED METRE 249
was here taken in anticipation, partly because of its kinship with
the three-timed, and partly in order that by it might be elucidated
the peculiarity and independence of the four-timed.
76. The reciprocal combination of the two-, three-, and four-timed
formations has been investigated already (( Metre/ par. 25) and
represented metrically. It would be a useless as well as lengthy
undertaking, were we to draw out in detail the arrangement of
accent in these nine forms of combination, as was done for the
simple two-, three-, and four-membered metres ; were we to exhibit
in metrical and musical notation the twice-three-timed and the
twice-two-timed with their 2x8 or 8x2= 1 6 different accent-deter-
minations, the thrice- three-timed with its 8 x 8 = 64, the thrice-four-
timed and the four-times-three-timed with their 8 x 32 or 32 x 8 =
256, and, lastly, the four-times-four-timed with its 32 x 32=1024.
By what has preceded, the accentuation for each individual case of
higher metrical combination may be found without difficulty, even
for one crossed repeatedly with positive and negative determina-
tions, by taking the sum of the accents of every order upon the
element in question, whereby the chief as well as the subordinate
accents will manifest themselves in their proper degrees. So, e.g.,
the thrice-three-timed metre with negative pair of the lowest order,
positive of the second and third, and negative of the highest, will
take the following shape and emphasis :
250
METRE
The same with positive pair of the highest order :
T
T
til II
In the first case the principal accent falls upon the fifth member,
the principal secondary accent upon the second ; in the second case
this is reversed, and the secondary accent falls upon the fifth, the
principal accent upon the second member.
77. Suppose it is required to throw the principal accent upon
an element of time previously determined on, e.g. upon the fourth
member of the thrice-three-timed metre. First the place must be
fixed as being emphasised in all orders :
and then the formation will in the rest of its members be neces
sarily determined as :
which produces this accentuation :
ACCENTS IN COMBINED METRE 251
Here if we assume the highest order negative, and then the first
three-timed member positive :
T
whereby the second acquires positive value in the second pair of
three-timed members, then the principal accent will indeed fall upon
the same fourth member, but cannot form the beginning of the bar.
For there arises for the whole the formation :
which begins with the full bar, and carries the principal accent upon
the second three-timed member :
The first metre rests in its highest order upon the determination :
and therefore in that order begins without accent.
78. Thus all accentuations possible to feeling, manifoldly dif-
ferent as they can be, will always be found rooted in the organic
metrical forms treated of. And on the other hand every accentua-
tion that conflicts with nature is also self-excluded from those
forms. In metre, what harmony has already laid down is but
repeated. There it is on the one hand impossible for systematic
harmonic construction of chords to produce a combination of sound
unfitted for practical use, and incapable of being justified to hear-
ing, and on the other hand every chord perceived as correct in
252 METRE
practice must allow of its derivation, its nature, being traced in the
organic system of harmony.
Certainly in the teaching of harmony one hears notes spoken
of as arbitrarily or accidentally sharpened or flattened. In the two
successions :
G --gj --a G
e - eb -
c - c ..cb-.Bb,
the^-J of the first is called a sharpened Fifth, and the c\> of the
second a flattened Root. But why in the collocation
G •• gj •• a
eb -
C -
does the same sharpening, and in
G -
e —
C .-cb-.Bb
the same flattening, seem something altogether repugnant to feel-
ing and inadmissible, if, as here in both successions, the first chord
indeed stands in intelligible connexion with the last, and if the
progression from one to the other means nothing more than that the
note which progresses is drawn upwards or downwards, being
arbitrarily sharpened or flattened ? But we know that the so-called
augmented triad, which may here be recognised in the middle chord
of both successions, and which has been spoken of in its place
(' Harm.' par. 234), is one that can be systematically accounted for ;
that it exists in a natural system ; and that, when the chord appears,
the system must be able to be present connectedly. The sub-
sequent course of the harmony may either remain in that system or
pass into another.
ACCENTS IN COMBINED METRE 253
In like manner no accent can be an isolated determination,
nor occur in a single portion of time as a solitary element not
standing in an arrangement of accents and not in reciprocal relation
with all the other parts of time in a metrical unity. Each single
accent is always rooted in the metrical system ; in its order it
is conditioned by the whole system, or conditions a whole metri-
cal system present at its entrance or arising with it ; which after-
wards may pass into another related system, from which again new
accents may be determined ; just as in harmony every change of
meaning in a chord, or chromatic alteration of a note of a chord,
is founded upon, or founds, a transformation of the key-system.
RHYTHM IN METRE.
79. The system of accents, their order, and their change, is
that which in the chief sense we shall name rhythmical in metre.
Hitherto this expression has been avoided ; for it was necessary
first to become acquainted with the conditions upon which this order
and this change depend. These must always be metrical deter-
minations, just as the notes of melody must always be parts of
harmonies. For in this meaning rhythmical in opposition to metrical
may justly be compared to melody in opposition to harmony.
As the melodic succession called the scale resting upon harmonic
basis, joins together opposite triads in each element of its pro-
gression :
C •• D -• e •• F .. G •• a •• b •• C
0 G C F C F
a e a
so also the rhythmical formation unites what is metrically opposite,
related, i.e. diverging one from the other : it goes on beyond the
254
METRE
end of the metrical positive unity and holds that and the beginning
of a following one together in a close.
THE RHYTHMICAL CLOSE.
80. The notion of the close is, that something separated becomes
united, that it is closed up, joined together.
Union always presupposes a separation, and separation unity
(* Harm.' par. 11). Thus the close is the contentment of recovered
unity.
By itself the positive metrical pair forms no close. It is one in
itself, and therefore has nothing to unite. Thus the magnet with
its positive and negative poles cannot by itself exhibit any attractive
force. But as the opposite poles of two magnets seek each other,
and as the minus of the one tries to attach itself to the plus of the
other and to close on to it, so we see too that in the metrical nega-
tive form, • • 2 — i • • , the two members cleave together all the more
firmly for being as yet decidedly not metrical unity.
In a continued positive series :
I — 2 I — 2 I — 2 I — 2,
one may easily see that it is the negative series also contained
in it
that with cohesive force couples together the double members of the
first series, effecting a close between them.
8 1 , Thus from the very notion of union it follows that a positive
metrical second member cannot be rhythmical closing element.
The close will at all times fall upon a positive first, to which a
second has gone before ; upon the beginning, not upon the end of
THE RHYTHMICAL CLOSE
255
a metrical positive duality : the last member of a positive pair will
always postulate the first of another following pair to form with it a
close. In the negative metrical formation the last member is last
also rhythmically ; for negative last is in fact positive first.
82. When the rhythmical close coincides with a metrical second
member, that happens in so far as every one of a higher order
within itself is, or can be, one and another again of a lower order.
If in the simple positive two-timed form the rhythmical close falls
upon the second time, then the second half of the first time has
united with the first half of the second time :
for in its quality of two-timed the form cannot with its first and
second give rise within itself to any close, since the close presupposes
being divided, which cannot yet be said here. So that here too
the closing element is in its rhythmical meaning the first of a
second and not the second .of a first.
Only try to have the close fall upon an absolute last, or, since
that is impossible because every unity may always be divided again
into halves, upon a second member that may be accounted small
in comparison with the whole, and the impossibility of considering
such a form as a rhythmically closed one will at once make itself
felt:
|—
I —
On the other hand the smallest second member will always easily
unite with a first of highest order following it into a rhythmically
closed figure :
256 METRE
83. In this sense rhythmical unity is something opposite to
metrical unity. That which in the positive metre is separated and
would fall asunder is by the rhythmical close united and held to-
gether. To be rhythmically united is to be metrically separated,
and to be rhythmically separated is to be metrically united. By
this it is not said that what is metrically at unity has to be divided
rhythmically, and that what is rhythmically at unity has to be
divided metrically : not, that the positive of the one determination
is cancelled by the negative of the other ; but only that the negative
of the one is everywhere covered by the positive of the other.
Thus wherever metrical union exists, and just because it exists,
there a rhythmical division, a section, a caesura, is possible. And
every metrical element that can give effect to the meaning of a
first may, in so far as it can do this, be a rhythmical last, a closing
element.
84. If, then, in the positive two-timed unity :
i -- 2
&
\
the second time is to become the rhythmical closing element, that can
happen only by understanding rhythmical union between the second
half of the first member and the first half of the second member :
i — 2
2 — I
and the second time here, if held on to the end, i.e. given its full
contents, would seem too long, heavy, and dragging. The close
is in fact completed in the figure :
and the last, fourth part, if appended to the closing member, would
THE RHYTHMICAL CLOSE 257
be a useless burden to it, for it stands here, not in the meaning
of second half of the whole, but only in that of first part of the
second half. The close is therefore
not —& — j^— but — & — p— F- ;
as indeed it would naturally be performed in practice.
85. In the rhythmical closing figure just given :
2 — i
i — 2 i — 2
fulfilment of the condition of the close has turned the simple two-
timed formation into a twice-two-timed one. The last member,
however, of the latter, the second time of the second half of the
metrical whole, has not been called into the rhythm. Supposing the
close to fall upon the second half, and that with the second member
of that half no new rhythmical figure is to open, then the place
remains empty, void of contents ; it becomes a metrical rest.
FILLING -UP OF THE METRICAL FORM. REST.
86. The metrical form cannot of itself make manifest the ele-
ments belonging to its determination ; for this it needs contents
to fill it out. From the beginning we have taken audible beats
to represent the metrical sections ; and little as these may be con-
sidered as filling out a space of time in the way in which the sound
of a continuous note would fill it, yet even they must be thought of
as absent from the abstract formal determination. Of itself the
metrical form is still only an empty space of time, a metrically
determined rest, and the membered form is only a rest conceived
as membered.
8
258
METRE
This form, then, in all its different metrical determinations of
memberment, may be filled out with contents either as a whole,
or as divided, either 'in its parts collectively, or in individual parts.
(a) In the Two-timed Metre.
87. The two-timed metre, which has in it the two determinations
of containing once twofold and twice single :
i x 2
2X1,
can be filled out with contents in a fourfold manner: (i) as a
whole, (2) as divided ; as such (a) in both parts, (b) in the first part,
(c) in the second part :
(0
(b) In the Three-timed Metre,
88. The three-timed metre has in it the determinations of con-
taining once threefold, twice twofold, thrice single :
i x 3
2X2
3X1.
It can be filled out with contents in a twelvefold manner.
This takes place here with wider scope of combination than in
FILLING-UP OF THE METRICAL FORM. REST
259
the two-timed metre, but its inward process follows the same plan.
The metrical and musical notation will now suffice to represent
this process clearly enough, without its being necessary to append
further explanations.
IT r'
hr ~G~
(<;) In the Four-timed Metre.
89. The four-timed metre has in it the determinations of con-
taining once fourfold, twice threefold, thrice twofold, and four times
single :
i x 4
2x3
3x2
4x1.
S 2
260
METRE
It affords thirty-two different ways of filling it out with contents
hr -r-
h-r-— - H
\-&— * P-|
i_P_r_., — |
hr — r— i
r
hrr -r
h-r-r-r- -i
-M
-P — /s>-
r
IT-T
r~r
-F — ,<& p—
FILLING-UP OF THE METRICAL FORM. REST 261
|-rr-rr|
l-p — *— » — p— I
! I
-H
H»-F— P— F—
|-»-F
-— S
90. In these filled-out metrical forms there is assumed only
positive determination of accent for all orders ; as may be seen from
the musical notation, in which they begin throughout with the
beginning of the bar.
How they would combine either as wholly or as partially filled
out under all other determinations of accent, and how far the latter
would be able to be discerned in the forms partially filled out, it
would be a vain undertaking to represent particularly. For it
would lead us on into the unlimited, and therefore could not after-
262 METRE
wards afford a general view, which is to be won only in the notion
embracing the conditions that give shape.
The thirty-two different manners of filling out the four-timed
metre combined with the thirty-two different accent-determin-
ations of it yield a result of 1024 different rhythmical metrical
figures ; but this numerical determination gives no insight into
the notion, which, as has already been said more than once, is
everywhere contained, not in numbers, but in simple opposition,
and its removal afterwards : that is to say, in opposing the being and
not being of the opposition itself.
FURTHER COMPARISON OF THE HARMONIC
AND METRICAL ELEMENTS.
91. If the Octave, J, the Fifth, §-, the Third, |, are opposed to
the Root, then the quantities representing the sound, ^, f , £, must
first be compared to the whole which represents the Root :
i=*=f=i;
they must enter into a relation of equality with it :
***-**+*-* *+*-f
The Octave is opposed to the Root in ±J,
the Fifth „ „ ±|,
the Third „ „ ±i ;
for the same that must be added to the quantity of the Octave to
make it equal to that of the Root, must be taken from the quantity
of the Root to make it equal to that of the Octave. That which
produces equality is in the first case +i, in the second — \ ; and con-
sequently is the same, opposed to itself as positive and negative, i.e.
COMPARISON OF HARMONY AND METRE 263
posited and annulled. In the same way we have for the Fifth -f £
and -J ; for the Third + 4- and — £.
With these elements of comparison,
the Octave, £, is found to be simply equivalent to the element in
question (1x^=4); the Fifth, f, to be double of its element (2 x -*-
= f) ; the Third, |, to be fourfold (4 x £=•£), i.e. twice double of
its element. And as in the first interval there is unity ; in the
second, duality set asunder, i.e. doubling ; in the last, duality
simultaneously as doubling and as halving, and therefore in the
latter sense brought under unity, or, we may say, duality simul-
taneously posited and annulled ; so the Fifth in the Octave acts
as the interval that separates, and the Third in the Fifth as the
interval that annuls separation and unites.
92. The same meaning is contained in the metrical determina-
tions of the twofold, threefold, and fourfold metres : in the first the
simple meaning of the Octave, in the second the double meaning
of the Fifth, in the third the fourfold meaning of the Third.
93. But further, in the intervals the positive may be put nega-
tive, so that we think of the determining element in the relations as
determined ; i.e.
for C— C, put C— C,
i : i | : i
„ C-G, „ F-C,
t *
„ C — e, „ at> — C,
i : I f : i
whereby in the Octave f : I, in the Fifth f : i, in the Third f : I,
the lower note appears as a determination from the Unity of the
higher. And so too in the metrical relations the sense of something
264 METRE
4 determined ' will be substituted for 'determining/ if the negative
succession of members (2 — 1} be put instead of the positive (i — 2) :
2 — i
2 — i
94. Without acknowledgment of the opposition of positive or
negative unity in the metrical twofold, of positive or negative
duality in the metrical threefold, and of positive or negative tri-
plicity in the metrical fourfold, the nature of these metrical forma-
tions cannot be comprehended. But the notion of their organic
essence goes past the determination of opposition on to that of the
removal again of it ; inasmuch as it collects the two, three, or
four members into a membered whole and fuses them into a
unity. This unity, moreover, finds further determination in its own
opposition and the negation of it.
95. Here we may meet an objection that might be started against
these comparisons of metrical with harmonic determinations of
relation. For harmony, in the three intervals J, f , -f-, groups together
simple, double, and fourfold in the meaning which we know ; but
metre groups together double, triple, and fourfold. Here, then,
the difference from the twofold to the fourfold, as the twice-
twofold, is not present in metre, at least in the outward structure of
the determination of quantity, in the way in which it is present in
the interval-relations of harmony. For in metre the determinations
that we have described as answering to those of the Fifth and
Third, stand in the relation of 3 : 4, instead of fourfold against
twofold, as in harmony.
But, in comparing the outward differences of the harmonic
and metrical determinations, we must take notice of the nature of
COMPARISON OF HARMONY AND METRE 265
the respective spheres in which they occur, and the manner of
their springing into form in each.
96. The musical relations of the intervals are dynamic : they are
relations of tension. The Octave has double tension to the Root ;
double force in the same inertia or weight ; the Root compared with
the Octave opposes double inertia to the same force. In the Fifth
| of the same inertia, in the Third ^ of the same inertia, is over-
come by an equal force. And apart from the meaning of the ratios
there is in the higher note less heaviness, it is lighter and brighter ;
the lower is dragged down by weight less matched by force, and is
heavier and more sombre. The positive intervals are determined in
the direction of height ; they are force-determinations. The negative
are determined downwards ; these are weight-determinations. The
rational meaning of their relations, simple, double, and quadruple,
is contained in them in the sense already known to us.
97. The metrical determinations are extensive in space of time.
But their rational meaning does not lie in the number which gives the
sum of their successive parts in time. It is contained intensively in
the transformation which such succession produces in a time-unity
posited in the beginning. This time-unity is by a second in time
determined to be first of that second ; by a third it is separated
from its second ; and by a fourth united to its second.
"The sense of these three elements of metrical construction is
completely the same as the sense of the Octave, Fifth, and Third
in harmonic construction. The latter is no more contained in the
numerical ratios by themselves, than the former is contained in the
mere number of the metrical members. A third and a fourth member
have no metrical meaning as being third and fourth, either in the
most complicated or in the simplest metrical combinations ; every-
where we find only a first and a second, a determining element
and a determined, in positive or negative succession. And so too
the fourfold in the quantity of the Third has reasonable acoustical
266 METRE
meaning only as a twice-twofold, and the fourth part only as half
of the half.
98. As anything in metre that extends beyond the fourth
member, i.e. beyond the pair of pairs, no longer exerts influence
within the first pair, and cannot therefore be comprehended
in a metrical unity of an order in which the first pair stands
as such ; so too in harmony that which exceeds the fourfold ceases
to be directly and immediately intelligible as an interval-determin-
ation. In the ratio 4 : $ or f , that of the Third, the complementary
part £ has no meaning in relation to the number 5, but only
to the number I ; for the quantity of the Third, |, is predicated four-
fold, i.e. twice-twofold of the numerator, which alone is what here
determines. Precisely as the complementary ^ of the ratio of the
Fifth, |, determines the quantity as twofold ; and the complementary
^ of the Octave-ratio, J, determines the quantity as simple of its
measure.
In the ratio J the quantity compared with the Complementary
part J would appear threefold, and consequently beyond the
directly intelligible opposition of duality. Similarly in the ratio
f, which to the complementary part is fivefold, we have a quantity
extending beyond the twice-twofold.
99. Such ratios as differ in their numbers by more than unity,
as ^, f , can therefore afford no directly intelligible determination.
For here the element which compares between the whole and the
thing compared, f , -f-, is itself not unity, not a measure but
measured, not determining but determined.
100. Hence there are left for directly intelligible harmonic de-
terminations the ratios i, f , f ; for metrical, the two-, three-, and
fourfold, in their metrical meaning agreeing with the former, as
being those that can be comprehended in a membered whole, and
that determine the whole in its members by opposition and by oppo-
sition of opposition.
METRICAL CONSTRUCTION 267
METRICAL CONSTRUCTION INWARDS AND
OUTWARDS.
101. Every simple metre may become combined by sub-
division of its parts. A part is unity in the order within which it
is subdivided.
In the so-called semibreve bar with motion in semiquavers the
crotchet is, in respect of the whole, a part, half of the half ; in respect
of the semiquaver it is a whole, containing halved halves.
1 02. But division into sixteen parts, which is here comprised in
one bar, may also be comprised in a series of sixteen bars ; and each
of the bars may again be divided into sixteen. It is the same in
the combination of different metrical determinations, the two-,
three-, and four-parted, which readily explain themselves. So that
a thing of six parts, made up of two three-parted or three two-
parted unities, is in its highest order twofold or threefold, in its
second order threefold or twofold, and may be further determined
as a member in larger formations, as well as more minutely articu-
lated in its own members.
103. The division into parts, so far as it is still conceived as
metrical determination, remains always subject to the principles of
metre.
A memberment into 5, 7, n, 13 equal parts is not conceivable.
It is otherwise with the formation which is grouped together and
constructed by augmentation, so that a whole becomes part, or by
diminution, where the part becomes a whole.
If when to one single another single has been joined, to this
twofold then another twofold, to the fourfold a fourfold, to the
eightfold an eightfold, and so on, and if to the sixteenfold, thirty-
twofold, sixty-fourfold, &c., there must always be added, as neces-
sarily following member, the equal of the whole that has preceded,
268 METRE
then (still apart from all aesthetic conditions and only considering
formal admissibility) such a progression would very soon extend
beyond any possibility of being seen through or grasped.
Both in extreme height and in extreme depth sound has a limit
of being audible and determinate. So also the comprehensibility
of metrical relations in both directions, that of diminution and
that of augmentation, has its limits. Now in the first direction
the aggregate of the members is held together by a whole already
formally determined ; if division be carried too far, the clearness of
parts may be endangered, but the whole of its order remains
secure. But the combination which augments must determine
the whole by means of the part, and here the too long lapse of time
demanded by the progression of the members would soon pass
beyond the bounds of a unity able to be reviewed and compre-
hended in beginning and end. In things visible in space, a whole
that stands before the eyes as unity can be reviewed at once in all
its elements. In things audible in time, only one element of the
train is present, which, though it leaves its impression behind it,
is liable to be pushed back into the past and obscured by the
following element, and by others following that, and the more other
elements follow it the more its definiteness diminishes.
Supposing a musical period could be lengthened out intelligibly
to an extent of thirty-two bars, slow movement, yet it would not by
any means require a second equal to it as necessary answer. Even
a first phrase of sixteen or of eight bars need not always be followed
by an after phrase of equal number.
, The two-, three-, and four-part kinds of bar do not admit of being
joined into an agreeable course of rhythm. But in more advanced
metrical forms, two-, three-, and fourfold combinations, explained and
governed by their contents, may very well be brought together in
an aesthetically satisfactory construction. And it ought not to be
regarded as showing a want of the sense for regular construction,
METRICAL CONSTRUCTION 269
or an incapacity for the review or comprehension of a whole
of any considerable size, if it should seem to us that a metrical
formation not strictly to be called regular nevertheless fulfils our
aesthetic requirements. In reality the form is everywhere only the
form of the contents. The artistic work that is richer in contents
and higher of purpose is precisely that which contains such devi-
ations from the absolute transparent regularity of pure metrical
structure, and which can make them approved oftener than we
should be willing to tolerate in productions of lesser rank. So too
the organic structures of nature upon the lower levels show a more
comprehensible regularity of form, appear to follow a stricter law,
than the more highly organised ; in which the richer and more indi-
vidual life passes too into their formal existence, shaping it more
completely : not less by law, but under multiplied conditions.
Only in pieces of music of the smallest compass shall we find
the metrical parts arranged in the simple regularity with which
metre by itself makes them follow, or be produced from, one
another. Thus metrical dance-forms keep up, as a rule, a constant
number of bars, two, four, or eight ; for there regular metrical
division is the first requirement, they being intended to lead the
figures and steps of the dance and lend them metrical support.
Now as already in simple three-timed metre a metrical second
element receives the value of a first, is first second, and then
becomes first ; so it happens too in broader formation that a larger
metrical group may be related in one direction as after phrase and
in the other as fore phrase. This is the same as a change of
meaning of a chord with reference to the key, or modulation : as
when a dominant chord (II) is used as a tonic (I),
I — II
I -II,
and we find ourselves thereby carried from the tonic into the
270
METRE
dominant key. And such a change of meaning in metre, according
as its sense is expressed decidedly or doubtfully, clearly or un-
clearly, will prove easy to understand and correct in effect, or
appear incorrect and a mere mutilation of rhythm.
Should it be attempted to represent throughout a whole large
composition its formal metrical web apart from all reference to the
contents, there would always, even in the most regular of the
classical masters, be found much that is not with clearness metrically
self-evolved ; although with the context it appears easy of compre-
hension, unambiguous, and altogether such that an educated sense
of rhythm cannot perceive in it anything conflicting with good
order. As in all things healthy and natural, theoretical conditions
will not once occur to the mind.
But in many productions of newer and the newest music
deviations from the directly intelligible metrical regularity do not
always imply masterly twisting of the web. More often it is
nothing but a tumult of sound, in which the composer has himself
not arrived at clear metrical perception, and now inflicts the un-
clearness upon us also. The defect in such an artist — if a composer
with this defect can still be called an artist — is the not being able
to comprehend, or not wishing to comprehend, a whole as whole in
its members. It is thus at bottom a defect of proper artistic sense,
which demands, not a piece in isolated fragments, but a body
of healthy coherent members.
Music in its rhythmically moving course cannot do without
metrically regulated support. The rhythmical phrase derives its
meaning in art from metre, in vocal music as well as instru-
mental.
Prose speech is also made up of rhythmical phrases. Recitative
is rhythmical without being metrical. Now as recitative is dis-
tinguished from melody proper, from the metrically periodic com-
position, with which the sphere of musical art in the narrower
METRICAL CONSTRUCTION 271
sense is first entered upon, it is a great error in a composer to
suppose that in setting a text to music he need only follow the
course of its rhythm, and is exempted from conceiving it musi-
cally in metre. Even the words of psalms, being in themselves
unmetrical, can be handled in musical art only in a metrical con-
ception of independent value. For music must always be music in
itself, apart from the words sung, and carry its own determination
of form. Had music no other task than that of emphasising the
words agreeably to their accents and logical import, then the first
things to be thrown aside would be bar- and part-singing. Music
must then be confined to declamation in recitative. For even
measured verse is not spoken by bar ; and it is impossible for
several melodies sung at one time, under the condition in part
phrase of being different, to be in equal measure adapted to the
logical emphasis.
UNEQUAL-TIMED DIVISION OF THE METRICAL
MEMBER.
The Metrical Determinations compared with the Spacial.
104. Architecture has been called frozen music ; in the same way
music might be called fluid architecture. Things of time have a
notion of symmetry like things of space. Architecturally we might
call bilateral breadth the space of space, and height the time of
space. Symmetry is to be found only in the space of space, in the
sides opposite one another of the dimension of breadth. Height
is a progression, is evolution, and cannot oppose like parts to one
another. It has in it no opposition at all ; by itself it is absolute
unity, just as breadth by itself is absolute duality ; then the one in
the other is real determined space.
272 METRE
105. Hitherto in the metrical determinations we have only seen
what may be compared to the space of space : namely, the space of
time ; which, agreeably to that, has in it too its notion of symmetry
meant of time, its things of like form in opposition. To this space
of time must be opposed a time of time : a notion in time answer-
ing to the notion in space of time of space, or height-unity ; just
as the space of time, the metrical determinations up to now, may
be put answering to space of space, the bilateral opposition of
duality, i.e. breadth.
This side of metrical determination we have now to consider in
its notion and in its manifestation. But at the same time it will not
be going out of our way, having already found for it a counterpart in
the notion of space, also to investigate its relation to one aspect of
the notion of harmony. Such an aspect there must be ; for there
could not be musical metre and metrical music, were not the musical
and metrical determinations rooted in the same nature and principles.
1 06. Metrical determinations, as heretofore given, with all their
manifoldness of accentuation, are nevertheless always made up
of parts equal in time. They rest upon the opposition of a first
and second in direct or inverted succession. The single may be
put double or halved ; yet in the double as well as in the halves
only equal is set against equal, never the part against the whole
or the whole against its double. Everywhere the only differ-
ence made is between accented and unaccented of equal quantity.
107. So is horizontal symmetry in ruled space : it demands like
to like on both sides. In this determination of equality, in space
as in time, single produces double, and double fourfold ; for the
same is always put against the same :
Our metrical three-timed formation, too, contains only opposition
DIVISION OF THE METRICAL MEMBER
273
of like determinations, and if we represent it in its notion in space it
must not be drawn as a symmetrical figure, made up of two halves
in themselves unequally divided,
but as a horizontal space-determination with two middles.
This corresponds again to the notion of Fifth, and in an abstract
sense to the dissonance-notion of a double unity, such as might be
pictured architecturally by a building with two porticos or two
main entrances placed side by side.
108. What gives rise to the Gothic pointed arch is a similar
duality of centre ; since the centre of the arch of one side falls
upon the periphery of the other,
in contradistinction to the round arch, which is produced from a
single centre.
The Gothic arch contains in its point, in the middle that has come
to be, the resolution of its dissonance, its Fifth-duality. The round
T
274 METRE
arch cannot show a determinate middle in itself, because it is only
unity and every part in it passes into its other.
109. In all symmetrical determinations, as such, equal will only
enter into union with equal, quantitatively.
But if we now take a rhythmical movement like
s~r
then, as the marking with six quavers shows, a twice-three-times
may certainly be discerned in the metre ; for the three-timed
may appear in the filled-out form (' Metre,' par. 88) of a whole pair
and a single member. But it makes a difference, whether in a
unity we have to consider the whole as the principal determination,
or the parts.
In the figure
occurring in the three-timed metre, the first double time is two-
part brought together ; wholeness is not its first determination ; it
has come originally from the growing together of the parts. The
figure
-JUt ,
from the rhythm above, also admits of being resolved into a three-
membered one :
but if we think of this rhythm in quick movement and many times
repeated, the twopartedness of the first double time seems, not its
original determination, but division of a length originally deter-
mined as undivided ; and the whole figure therefore consists of
the succession of an undivided double and a single, or of a time-
unity and its half.
no. This same determination is also contained already in the
DIVISION OF THE METRICAL MEMBER 275
equal-timed metrical formation, namely in the second or Fifth
element of it : in the three-timed metre. The fundamental metrical
determination is two-timed unity. This in the three-timed metre
becomes doubleness ; it is divided, the halves stand out in it. In
the meaning of equal-timed metre, whole and part here subsist in
one another.
in. The unequal-timed rhythmical formation causes the whole,
on ceasing, to be followed by the half of that whole. It places
one after the other in time, that which the equal-timed metrical con-
tains one in the other (in the meaning of rest) in space. The same
notion lies at the bottom of both, in positive meaning and negative,
posited and annulled. Rhythmical determination is the * not being
at once ' of the * being at once ' in the metrical : the coming to be
of being in time : time in time.
112. In the metrical three-timed there has arisen an extension
by enlargement of the metre. The rhythmical unequal-timed
determination does not admit of being increased in the same way,
for it originates within the metrically determined member. The
member must therefore contain both elements of the determination,
and they must be formed in it successively ; for rhythm is essentially
successive, just as metre is essentially simultaneous.
1 1 3. Certainly, nothing can take shape in time without being
successive. But we have pointed out that there is space in time
and time in time ; and similarly in space, that there is time of space
and space of space, picturing by the latter the stationary and
simultaneous horizontal determination of space, and by the former
the vertical and progressive.
If the notion of a determination of time thought of as space
could not be conceived, then we could form no picture at all of
shape in time. For only one element of what passes is ever really
present; and not until this is taken together with the element
that has gone before and the element that follows after — therefore
T 2
276 METRE
with something that is no longer and with something that is not
yet — can the notion of an image in time be realised.
114. In determination of space symmetrical relations of equality
are natural to the horizontal dimension when considered as base,
and progressive relations, increasing or decreasing, to the vertical
dimension. If we look at a building, it seems to us a construction
which has arisen from below upwards, out of the ground-plan pre-
supposed in its whole breadth. In its horizontal proportions it is
a two-sided equal, symmetrically measured together and at once ; it
is in space. In its vertical proportions it is successive, progressive,
growing : it comes to be in space.
115. In the sphere of determination of sound the same oppo-
sition which in the notion of space is presented by the determina-
tion of horizontal and vertical, that namely of ' being at once ' and
' not being at once,' or of being and coming-to-be, is found again as
the opposition of harmony and melody : that is, of simultaneous
sound and successive, if they are contrasted independently, or of
harmony of melodies and melody, i.e. succession, of harmonies, if
they are considered in combination.
1 1 6. The notion of harmony places Root and Fifth, C G,
sounding together as simultaneity or space-interval. The notion
of melody places Root and Fifth as time-succession or interval. In
the melodic progression, C- • Dy C is Root to G and D is Fifth to G ;
that which therefore was itself at first Fifth, then became Root, and
if its two meanings are taken together forms the unity by which
the difference of the two successive notes C—D becomes intelligible
and by which their combined sound exists as an intelligible dis-
sonance. C and D sounding together in harmony contain the same
contradiction that we should obtain if we placed a whole and a half
in architectural symmetry, i.e. a whole on one side and a half on the
other ; a contradiction that would demand resolution into one or
the other, to one side or to the other in symmetrical equality. On
DIVISION OF THE METRICAL MEMBER 277
the other hand this proportion of 2 : I or I : 2 in the vertical
line is quite suitable as an architectural arrangement.
117. Thus in the notion of space equals are situated as equal in
time, or horizontally symmetrical ; unequals are produced suc-
cessively, or vertically, increasing or decreasing in their propor-
tions ; nevertheless the vertical determination first comes to reality
in and with the horizontal determination, i.e. ascent can only become
real and perceptible by reference to something that ascends.
Further, in the notion of sound the Fifth C -- G as an interval is
simultaneous in sound, and therefore is as it were in space ; while
the Fifth of the Fifth, the Second C-D, has only a relation of
succession, and is therefore an interval in time, yet again, as being
in time, it has the foundation of its intelligibility only in something
which persists (G). And so too the metrically unequal, which we
compare to the ascending in space and to the progressing in melody,
can come to real existence only upon a metrical basis, only within
the equal-timed form of metrical unity. Thus the rhythmical figure
belongs to the unity-determination of time in time, and not to that
of space in time. The latter is double, the former single. It belongs
therefore to the single in double, to the part in the whole, and can
therefore be realised only in repetition.
A | bar is but the half of a metrical unity ; a unity that can be
either of positive form or of negative, i — 2 or 2 — I, whereby the
unequal-timed division begins either with the beginning of the bar,
or on the up beat.
A two-, three-, or four-timed metrical basis is always required,
just as a harmonic basis is necessary to melodic progression.
278 METRE
For the unequal-timed is not in itself an independent metrical
construction in the sense in which the equal-timed is.
1 1 8. A member that carries the unequal-timed metrical deter-
mination cannot at the same time contain the equal-timed. For so
it would itself be a double unity, a metrical one-and-other, and as
such it could not have the unequal-timed articulation, which can
only be formed in a member which is single.
POSITIVE AND NEGATIVE FORM OF THE
UNEQUAL-TIMED DIVISION.
119. The unequal-timed determination having in its rhythmical
meaning to be identified with the metrical three-membered for-
mation, its parts, its long and short, must next be shown to be
correlative to the metrical first and second of the two-membered :
the rhythmical long to the metrical accented member, and the short
to the unaccented. Then all that has been described and con-
trasted as metrically positive and negative and metrically major
and minor will find its corresponding application also in the
unequal-timed division. The metrical equal-timed unity contains a
first and second, an accented member and an unaccented, and places
them one after the other in direct or inverted succession ; in the
unequal-timed division of the member these are replaced by a
long and a short : the accent-determination by a quantity-determin-
ation ; the accented member by the long, the unaccented by the
short.
1 20. The equal-timed positive succession,
THE UNEQUAL-TIMED DIVISION 279
appears in unequal time as a succession of long and short,
| *
the equal-timed negative,
2 I
r
in unequal time as a succession of short and long,
121. The long, as such, has no accent ; for a metrical element
receives the accent only as being first to a second which is equal to
it. Without the condition of equality a succession cannot, in this
meaning, be comprehended. But the long as against the short is
in itself a double : and is, moreover, of decidedly positive nature,
seeing it appears unseparated ; for negative succession would have
separated it. And thus the beginning of the long is accented in
that member by reason of its double and positive nature, but the
short in its quality of single can have no accent.
1 22. According to the two possible accentuations of the metrical
dual unity, in which dual unity alone the unequal-timed division is
determined into a whole, the latter is capable of being emphasised
in four different ways. It can be contained : (Ay a) as positive in
metrical positive, (#) as negative in • metrical positive, (B, a) as
positive in metrical negative, and (b) as negative in metrical
n egative.
B
•
28o METRE
THE THREE ELEMENTS OF THE UNEQUAL-
TIMED DIVISION, CORRESPONDING TO THE
THREE METRICAL ELEMENTS OF THE TWO-,
THREE-, AND FOUR-TIMED UNITIES; AND
LIKEWISE TO THE HARMONIC ELEMENTS OF
OCTAVE, FIFTH, AND THIRD.
123. Since the unequal-timed division can only develope in a
given member of the metrical equal-timed determination, it is
bounded, extensively, by the latter : the unequal-timed construction
can take place only within the member. Now the unequal-timed
division of the metrical member being identified for rhythmical
meaning with three-timed metre, we know that the metrical deter-
mination does not cease with the three-timed formation, the
element of duality or Fifth in metrical meaning, but that it goes
on to reach repose of unity and completion in the four-timed.
Therefore for the unequal-timed memberment there must likewise
still be left an element of determination wherein it shall find the
finish of its process of memberment.
124. The element of the unequal division, in its present aspect,
corresponds with the notion of duality in rhythm. The two-timed
metre contains first and second still only as unity, while the
three-timed calls up decided separation in the pair, and the four-
timed brings about union of the separated pair ; in the unequal-
timed division, the first of these three elements of the notion is
contained in the undividedness of the metrical member, and the
second is contained in its unequal division, according to which
one part has the determination of being whole as against the
other, and this other has the determination of being part as against
the first.
ELEMENTS OF THE UNEQUAL-TIMED DIVISION 281
125. The metrical equal-timed determination now proceeds to
its third and last element essentially as follows : It makes the
whole become part or half ; the membered pair it places as member
in a pair of higher order. The advance in this, as formerly with
the three-timed, is by enlargement in extension : the formation
now claiming twice the space of time occupied by the two-timed.
126. But division of the member cannot be carried beyond its
extent ; it must be completed within it. In the unequal division
the whole, i.e. the long, cannot, as in the equal, become part, i.e.
rhythmical short. For that by which the long could appear short,
or half, would be greater than the member, or more extended than
the space of time within which the determination must take effect.
But the short can appear long by a shorter than it within the
member, i.e. by the half of the short ; consequently the part may
appear as whole. And thus, as the equal-timed metrical formation,
in the completion of its notion, puts the whole as part, and thereby
removes the opposition of one and other; so the unequal-timed
formation, by putting the part as whole, thereby arrives, agreeably
to the nature of its method of construction, at the same completion
of its notion.
127. The short is made to appear long by a part preceding it,
of which it is itself the double. Consequently the second element
of the unequal division is determined as short against the long
which precedes it, and as long against the short which precedes it,
and thus contains united in itself both opposite determinations of
the unequal division.
The rhythmical figure that is produced from this determination
is one well known to us from the seventh symphony of Beethoven :
where it is kept up so consistently in its peculiarity and essential
282 METRE
difference from a real double three-timed bar, that though the bar
is marked, as is customary, with f , yet a division of the half-bar
into three equal parts instead of the two unequal parts, or a com-
bination of the two kinds, never once appears in the whole long
phrase. The first alteration would make a variety in the rhythm,
but the second would change its whole character.
128. It is easy to perceive that in a spirited performance the
middle element in this rhythm does not receive quite the full
value of third part of the first : that it is taken shorter, and in
fact is not connected with the first, but with the third. For the
third member as against the second receives the opposite meaning
to that which it takes as against the first, and therefore compre-
hends within itself the double meaning of short and long, and there
is a tendency to put stress upon the latter meaning by more
sharply marking as short the middle element. In this sense there is
also imparted a proportionate degree of accent to the last element,
in so far as against the immediately preceding member it is a
double ; although against the first it is a half, being thus opposite
in itself, \ and f .
129. With the three elements of the unequal-timed division of
members :
I. _J_,___J_.,_
III. _
this rhythmical determination closes, just as the metrical determina-
tion ceased with the two-, three-, and fourfold, and the harmonic with
Octave, Fifth, and Third. Further division of the member can only
proceed as in equal-timed metre, or by a repetition within a smaller
equal-timed member of the unequal-timed division in the second
or third element of its notion.
ELEMENTS OF THE UNEQUAL-TIMED DIVISION 283
130. As the unequal-timed division of the member is also, in a
sense, equal three-timed metre, being marked f, $, f, y, even
when, as in the phrase from the symphony just given, a real divi-
sion of the half-bar into three equal parts does not occur ; so, on the
other hand, the three-timed metre will also offer a point of view
from which it may be regarded as unequal two-timed, as a succes-
sion of long and short or of short and long. The first pair of
members of the three-timed formation can take effect as an un-
divided unity ; then the single member, the half of the other pair
of members, remains over as the complementary part, and we get
at once a long and a short. Then, further, the short here may
be made to appear long by a half put before it ; so that an
unequal-timed determination, like that of time in time, arises also
in larger measure in the real three-timed metre.
The difference between this metrical figure and the rhythmical
one before noticed will not be overlooked. Wherever a strict
metrical behaviour of the parts to one another remains perceptible,
especially of the long long and short short, there the sharply
accented, elastic nature of the rhythmical formation is missing.
For those two elements cannot enter into direct relation to one
another ; the second is only a relative to the third.
131. A relative should always be referred to its positive alone.
To another positive it stands in no intelligible relation ; moreover it
284 METRE
does not hinder the relation of this other positive to its relative
from being truly presented. In the rhythmical figure
5-W-
the two determinations
— ^ ^— and —i
stand together, and at once,
i— i
in such a manner that no relation between the long long and the
short short comes into question. A direct relation is only found
between the first and third and between the second and third
elements, the third having the double meaning of long short and
short long. Therefore also in the figure
the middle member presents no considerable proportion to the
first — it would be represented here by i : 3 — but the first long has
its duration and metrical meaning, irrespective of the entrance of
the intermediate member, and lasts up to the beginning of the
second principal element of the formation.
So also between two notes that do not form the interval of an
Octave, Fifth, or (major) Third, no direct harmonic relation can
exist. The concord of the minor Third, e.g. e — G, will always
merely point to a third note, C or £, in which the two notes e and
G may then attain to relation in unity, as Fifth and Third of positive
or negative determination.
132. The expression for the negative form of the third element
of the unequal-timed division must likewise be completed within
ELEMENTS OF THE UNEQUAL-TIMED DIVISION 285
the compass of the member. It must also, like the positive, fulfil
the determination of opposite meanings being contained at once
without contradiction. It can therefore be no other than that
represented below — the rhythm of the quail's cry.
As the positive form must at the same time comprehend in it the
negative :
f*
so too in the negative there must at the same time be contained
the positive :
i-4 i-i
r~s , r™2 i
-£-HT— ;-*~T-
i i i- - i
the figure beginning with the short of the up beat, which also takes
to itself the meaning of long.
THE DOTTED MOVEMENT.
133. By these rhythmical figures we are led to the so-called
dotted movement in general ; which is also to be considered as an
independent determination, for it can be produced from metrical
formations of every kind.
It has already been seen in the unequal-timed dotted rhythm
that the little short has no influence upon the effect of the great
long, and that it exerts an influence only upon the intensive quality
of the second principal element, the longer short; which, from being
286 METRE
wholly without accent, it strengthens into being accented in due
proportion. And so generally, to this intermediate element
metrical meaning ought not to be ascribed, at all events not such
as would claim for it a measured portion of time determined or
determinable in duration. It is an element of absolute shortness,
joined on to the next following, relatively long, element.
134. The metrical figure
III!
— •» — •* — •* — W—
has as four-timed the accentuation
but as twice-two-timed the accentuation
Besides the fourth time, which in the former is without accent,
the second now drops its accent as well. Similarly the dotted
rhythm of the following figure :
will make the difference of accent, both in the fourfold of the four-
timed and in the threefold of the three-timed metre, far less
noticeable ; because, on account of the short prefixed to the second,
third, and fourth times, each of these elements has an accent
imparted to it, and thereby an accent-determination of four
equally emphasised elements arises,
— ^-+— i±-+- + ±-+— i— = —4 -- » -- » -- ^~
s * s * s 0 I I I 1
in place of the four-timed,
or of the twice-two-timed.
THE DOTTED MOVEMENT 287
135. As has already been observed, the short in the dotted
movement has no metrical quantity. As soon as quantity can be
recognised in it, and it thereby becomes determinable for duration,
the rhythm loses its character ; this movement requiring a sharp
contraction of the intermediate element, in consequence of which
the following portion of time always appears accented, apart from
its metrical duration and other metrical meaning. In this manner
the accent-determination stands by itself, and does not essentially
change, whether the time following upon such a short is metrically
long or short, emphasised or not emphasised.
Analogy in Harmonic Melodic Determination.
136. This rhythmical division, which has not and cannot have
metrical meaning, because here as well as in the unequal-timed
rhythm it is concerned with the member only as a finished metrical
determination, we may compare to melodic passing notes struck
before or after a chord-note. These indeed have their origin and
existence in harmonic determination alone — the only way of
writing them is as notes thus determined — yet it is not in their
chord-relation that they are used, but, on the exact contrary, as not
harmonic notes, notes not belonging to the subsisting chord.
Having this negative meaning in harmony, they are positive for
melody ; they are notes of union in melody, because they are notes
of separation in harmony.
137. The succession of two chord-notes, e.g. C'*e over the
stationary major triad on C or minor triad on a, is melodic, inas-
much as the notes enter successively. But they might also sound
at once, or the first might be prolonged to sound with the second :
they make a successive harmony.
138. The passage C"D-*e over a stationary triad prevents the
first chord-note of the melody from sounding on into the second ;
288 METRE
it separates their harmonic unity. The third note of this melodic
figure, which in the immediate succession of C-e is merely a
harmonic echo of the first, and has therefore but a secondary
meaning to it, is after the second unharmonic note to be regarded
as a newly entering chord-element, and so gets a primary meaning.
It is exactly like the short of the unequal-timed division appearing
as a new long against the smaller short prefixed to it, i.e. acquiring
a primary meaning without giving up its secondary meaning
against the first long. And as in that case the intermediate
element has reference only to the last and is joined on to it, so too
in melody the passing note is attached closely to the next following.
Between it and the next following note no division can be placed ;
and it is no more possible to close with the passing note than with
the short of the short in the unequal-timed or with a decided
metrical second element in the equal-timed metre.
Analogy in the Determination of Space..
139. If we want to discover in the determination of space some-
thing analogous to the unequal-timed rhythmical division, corre-
sponding to the analogy between the equal-timed metrical division
and the bilateral symmetrical base, it will have to be sought among
dimensions of height and other determinations drawn from the
nature of progression. Now in general the proportions of height
in constructions which shall correspond to a reasonable free
determination of space must not be uniform, but increasing or
decreasing ; and will in the first order be made out of single and
double. And if this principal proportion corresponds in the first
instance to our Fifth-notion of the unequal-timed rhythmical de-
termination, then the Third-notion, which is to show the half also
as whole, the secondary also as primary, the unaccented also as
accented, and to present the separated elements of the opposition
determinately in their state of union as well as in their state of
THE DOTTED MOVEMENT 289
independence, will find a place here also as last determination in
the distribution of spa.ce.
140. First height is unity, or whole. Then it is divided within
itself into an unequal duality, of positive or negative succession.
Next the two immediately contiguous elements of this determina-
tion must be separated, that they may be able to be united ; and
this is effected by the interposition of smaller intermediate mem-
bers, just as" it was effected in the melodic succession of C->e by
means of the passing-note D, and in the unequal-timed rhythmical
form by means of the intermediate member of the smaller short.
The principal proportions lying one above the other might so far
be viewed as produced immediately one from the other, but with
intermediate members interspersed they seem continuous, united, and
knit together just by reason of the separation that they thus have.
However attractive it might otherwise be to pursue further the
principle of the laws of harmony and metre as applied to the
determination, by law, of space, especially with reference to
architectural formation, yet to attempt it here would take us too
far out of our way ; wherefore with these general indications the
subject must rest.
u
290
METRE
METRE OF SPEECH.
FOOT. VERSE-MEASURE.
Dipody— Tripody— Tetrapody. Dimeter— Trimeter— Tetrameter.
141. IN the art of scansion a single member of a verse is named
a foot : the parts of a verse are called after the number of the feet
contained in them, and the whole verse after the number of such parts.
In speaking of verses with six or five feet, of hexameters, penta-
meters, or of five-footed iambics, no more is indicated than the mere
superficial counting up of the members ; the inner structure of the
metrical form is quite left out. For names like these tell us little
more of that than might be learned from reckoning the number of
syllables ; and so they must be regarded merely as names for things
which in their contents and properties are already known to us.
142. But the name of foot for the single member is very well
adapted to the matter, because it brings out the natural necessity for
a pair of such members. For there is no going with one foot : it
wants a pair, and a pair with right foot and left, one to step out
and one to be brought after.
143. Such a pair of feet, or the pair of steps which it is engaged
in taking, corresponds to our first metrical determination, the two-
timed. If we picture the step of the right foot as the stronger, it
will then count for the accented member, and the step of the left foot
brought after for the unaccented. Stepping out with the right foot
FOOT. VERSE-MEASURE 291
will make the accent fall upon the first step, and stepping out with
the left will make it fall upon the second ; the former to be re-
garded as the metrical positive succession, i — 2, the latter as the
negative, 2 — I.
144. This metrical duality of members is commonly called a
dipody. If we wished to substitute the expression * two-foot/ yet
the three-membered unity, the tripody, could not properly be called
a ' three-foot ' member. Walking on three feet is in itself hard to
imagine ; and besides we know that the three-membered metrical
unity is in truth also a formation by pairs — that is to say, pairs of
higher power. Here it would have to be regarded as the function
of a pair of pairs of feet, the second pair commencing with the
second step of the first pair. To pursue further the comparison of
metrical members with the action of these members of our body,
would probably prove generally inappropriate, and might even
become laughable. So, e.g., if we wished to compare a combined
metre, such as the twice-two-timed, to the gait of a father walking
in long slow steps, and holding his little son by the hand, who
must take two steps to one of his father's, so that the child's right
foot treads once with his father's right and once with his left. Yet
the three-membered metre no longer admits of such apportionment
of step between two persons, at any rate not in continued succession,
because the third member brings to a stand the person who first
steps out. And even the last comparison of the twice-two-timed
metre is wanting in inner truth. For every metrical formation,
even the most complicated, ought always to be regarded as one sole
organism with members proceeding from itself ; the conditions of its
memberment may not be apportioned between two or more indivi-
duals ; to make it single, they must have their ground in one and the
same individual.
145. The memberment of speech-metre can in essentials be no
other than that of metre in general, as we have seen it up to now ;
u 2
292 METRE
namely, equal-timed, consisting of opposition of equal members, and
unequal-timed, opposed within the member. But now let us call the
first determination, occurring in what order it may, the metrical,
as pre-eminently such, and the other the rhythmical.
146. The metrical determinations are :
A. Lower Order.
1. The dipody; two-membered.
2. The tripody; three-membered.
3. The tetrapody ; four-membered.
B. Higher Order.
1. The dimeter ; twofold.
2. The trimeter ; threefold.
3. The tetrameter ; fourfold.
From combining the determinations of both orders there arise the
thrice-three formations before demonstrated (' Metre,' par. 26), which
by reference to the former description we may now name for speech-
metre—
I. (a) 2x2. Dipodic dimeter.
(&) 2 x 3. Tripodic dimeter.
(c) 2x4. Tetrapodic dimeter.
II. (a) 3 x 2. Dipodic trimeter.
(P) 3 x 3- Tripodic trimeter.
(c) 3x4. Tetrapodic trimeter.
III. (a) 4 x 2. Dipodic tetrameter.
(#) 4 x 3. Tripodic tetrameter.
(c) 4x4. Tetrapodic tetrameter.
Tetrametric formation in practice occurs only dipodically ; but here
the tripodic and tetrapodic are included for the sake of systematic
completeness.
FOOT. VERSE-MEASURE 293
In general, the system of possible formations may be gathered
into an easier view, if we combine only the two- and threefold of
both orders, for this purpose allowing the four-timed to count as
twice-two-timed. The practical metres, with the exception of the
tetrameter, are contained in these forms.
This combination is fourfold :
A. (a) 2 x 2. Dipodic dimeter.
(b) 2x3. Tripodic dimeter.
B. (a) 3x2. Dipodic trimeter.
(^) 3 x 3- Tripodic trimeter.
147. Every metrical shape will depend upon one of these deter-
minations. But only the outline of its principal division is thus
given. The manner of dividing the member gives the character
and colour to the metre.
148. Here the only rhythmical division of the member that can
be considered is the unequal-timed. The equal-timed would only
repeat the metrical determination in another order, e.g. would make
the two times of the dipody appear twice-two times. But if the
dipody is to remain two-timed, its member must not become two-
timed. So, then, for further enlivenment we must come to that
division of the member which puts not equals as first and second one
after the other, but long and short. In scansion we know this form,
as rhythmical positive, under the name of trochee — ^, and as
rhythmical negative, under the name of iambus ^ — .
149. This rhythmical dual determination, which has been de-
noted as corresponding in its sphere to the Fifth-notion, is preceded
by another element of determination ; and it is also followed by
another. The first, corresponding to the Octave, is that of unity,
that of the undivided member. The last is that of the Third,
corresponding to unified duality, that of the short determined to
long in unequal division.
294
METRE
150. The dipody with its members undivided appears as the
spondee. (— — ).
This takes up a pair of members. It is not a foot like the
trochee or iambus ; it consists of a pair of such feet, which may
become trochees or iambuses.
151. The short determined as long (by means of a smaller short
prefixed to it) produces the dactyl in the single member, the trochee.
The dactyl contains the iambus in the trochee. What was
rhythmically opposed as positive and negative, it has merged one
in the other ; and in this sense has formed the rhythmical Third.
The short of the trochee appears here as at the same time the long
of the iambus.
152. The opposite of the dactyl is the anapaest.
It contains the trochee in the iambus. The determination corre-
sponding to the rhythmical Third-notion with regard to the negative
trochee, that is, to the iambus, consists in this, that the short of
the iambus becomes the long of the trochee.
153. Accordingly the metrical two-timed unity may be rhyth-
mically divided, according to the elements of the unequal-timed
determination : —
(a) In Rhythmically Positive Form.
i. as spondaic
2. — w — w „ trochaic
3. _^o_^o „ dactylic
-dipody
FOOT. VERSE-MEASURE
295
(b) In Rhythmically Negative Form.
_£-* ----- £_.
1. - — as spondaic \ I
2. w — w — „ iambic }- dipody ~J 'rf™ £
3. o^i_o^_ „ anapaestic j
154- In this the difference of positive and negative quality is
put in the rhythmical determination alone ; the metrical is taken
only positive. Hence the spondee is the same in both forms,
sinking.
/
The metrical negative determination would throw the accent upon
the second member of the spondee, rising :
whereby the principal accent of the rhythmically-membered for-
mations must also find its place on the second principal metrical
element.
155. In metrical negative meaning the above rhythmical deter-
minations will be as follows :
(a) Rhythmically Positive.
(b) Rhythmically Negative.
-+—L-1—+ !_V-
296 METRE
156. Of these rhythmical forms with positive and negative
meaning, the first stands in notion inwardly akin to the Octave
and to the metrical two-timed division ; the second to the Fifth
and to the metrical three-timed division ; the third to the Third
and to the metrical four-timed division.
157. Metrical determination by itself requires only equality of
the members in the whole of their duration. Rhythmical determina-
tion is completed within the member. It may be different in the
single members of the metrical unity ; and therefore suffers com-
bination dipodically in 3x3 = 9 ways. These we now draw out in
positive meaning only, metrical and rhythmical :
3. i. /_ro — = -4>.±-
Tripodically the combination happens in (3 x 3) x 3 = 27 ways.
1. I. I.
2. 2. 2. -v w >
3. 3. 3. v^o.^o^
FO O T. VERSE-MEA S URE
297
I. I. 2.
I. 2. 2. •<
^-i « 90 »
I I > I i/
—+ *-j*~ f 1*"
1^1 I/
_W = _^_J
I. I. 3. K ^O =
r
•sen
= — +—+-+—+-*- ^
_±W W — — A_i ^
3* 3' I ^-^ w
^irr^ ~L^;
X i
k I >
= —+S.-*.
2- 3- 3- 1 — w — w^w
o»_MO _^O
k
i*-i"
-r— f
T— f
2. 3 .
f— j-
298 • METRE
158. The tetrapodic form, combining the three rhythmical un-
equal-timed determinations, would again yield thrice the number of
the preceding, i.e. (3X3X3)x3 = 8i different ways of division.
But the tetrapody, the essentially four-timed, differs from the
double dipody, the twice-two-timed, only in determination of
accent, and not in the arrangement of its members. The twice-
two-timed is indeed without an accent-element, which the four-
timed has ; yet the four-timed is always a twice-two-timed as well.
Therefore the 8 1 forms of the tetrapodic rhythmical division need
not be written down. They can only consist of the combinations
two at a time of the two-timed forms, and must consequently con-
tain the ninefold of the two-timed nine times repeated, because
each of the nine two-timed forms is to be combined with itself and
with the eight others.
159. But also the rhythmically divided double dipody will in
its accent-determination take to itself the nature of the tetrapody.
We know that the accentuation of the twice-two-timed metre and
that of the four-timed differ only in the emphasis given to the second
member ; this in the four-timed is accented, but in the twice-two-
timed remains without accent. But in the rhythmical unequal-
timed division there is allotted to every long the accent which falls
to it as double short. Hence in every case the metrical second
element, though as such it is unaccented, receives rhythmically a
proportional emphasis.
The double dipody, which, when undivided, is without accent
upon the second as well as upon the fourth time,
will, when divided, receive rhythmical emphasis upon both elements ;
thereby appearing, as far as the second element is concerned, also
tetrapodically accented :
FOOT. VERSE-MEASURE 299
Thus the rhythmically divided dipodic tetrameter passes easily
into the tetrapodic dimeter, or may be conceived as such.
160. The rhythmical negative form of division of the members,
which begins with the short, would reproduce with iambuses and
anapaests what in the foregoing description of the positive appears
with trochees and dactyls. The trochaic figure
I K I >
is transformed into the iambic ;
/ , > J
and the dactylic
into the anapaestic.
161. Moreover rhythmical divisions of positive and negative
meaning will readily unite in the same metre ; as, e.g., the anapaestic
with the dactylic :
the dactylic with the anapaestic :
without imperilling metrical order and unity, for these have nothing
to do with the rhythmical structure of the members. The members
are regarded for metrical determination merely as wholes of time.
METRE
DIFFERENCE BETWEEN THE METRICAL DACTYL
AND THE RHYTHMICAL, OR BETWEEN THE
SPONDAIC DACTYL AND THE TROCHAIC.
162. The third element of the unequal-timed positive rhythmical
division, where the short of the trochee appears also long in respect
of a smaller short prefixed to it,
has been named by us a dactyl. But the rhythmical structure of
this foot, as also the representation of it thereon founded, does not
coincide with what in the science of metre is commonly called a
dactyl and with the way of representing it, namely, as a long
followed by two equal shorts.
If, then, both forms of the dactyl
are to exist side by side, we have in the first place to make the
distinction, that the latter is not to be named a foot in the sense in
which the first is. For that comes from dividing a single member,
while the other plainly embraces, like the spondee, a pair of
members. Let us call the first construction the trochaic or rhyth-
mical, the latter the spondaic or metrical dactyl. The rhythmical
dactyl is formed in the member, it stands for the trochee ; the
metrical dactyl is a form of dipody, and stands for the spondee.
The latter cannot find place in a series of trochees, any more than
in music the figure
METRICAL AND RHYTHMICAL DACTYL 301
can occur in a bar of § or £ without changing the nature of the
time.
Although, then, both forms of the dactyl exist, yet the customary
way of representing metre knows but of one, the dactyl with equal
times :
i — 2
and conjoins it, not only with spondees, but also with trochees.
With the latter, however, the rhythmical dactyl, _^o , alone can be
combined,
while the metrical dactyl finds its place only in the series of spon-
dees.
In the latter the division is all of metrical equal-timed struc-
ture.
163. But then this marking is used (and that not only where
the dactyl is concerned, but also for the other members) when
trochaic rhythm is proper to the metre, as, e.g., in the so-called seolic
and logacedic verses.
We find series such as
w — o — w —
denoted only by
where the first trochee, that which precedes the dactyl, appears as a
spondee, its short as a spondaic long ; while the short of the third
trochee, that which follows the dactyl, remains unchanged in value.
302 METRE
Whereby, if the metre were to be performed as it is denoted, the
following unmetrical formation would be produced.
J_,LiJ_,rL,j_jXJ_.
And if, notwithstanding, this marking should be allowed to represent
the rhythm
then the question arises, how should a spondee stand for the first
trochee, an undivided pair of feet for a divided foot ?
The unlawfulness of putting one for the other is obvious. The
first foot marked with two longs cannot, if a trochaic dactyl follows,
be a real spondee, and the second long cannot be a metrical long.
164. It is well known that in the dipodic trochaic series the
short of every second trochee is written over with a long.
The iambic series has the long marked upon the short of the first
foot of every dipody.
The trochaic dipody forms by itself a metrical whole ; it has its
metrical first and second, and should the second die away in a
weak echo, there it will want to end. If another dipody is to be
produced out of the first, then the first must not end with weakness,
the sound dying away ; on the contrary it must show generative
energy at that place. For this the short of the second trochee
must receive a more generous, a superabundant fulness, swelling
it out and joining it tighter to the following element ; so that the
boundaries of the dipodies are pressed together, and they pass into
one another, and appear joined undividedly into a whole. The
METRICAL AND RHYTHMICAL DACTYL 303
place contains a prosodic long in a metrical short, a fulness of
syllable in a narrowly determined but extensible space of time. If
these places are furnished with logically closing but prosodically
trivial contents, then the series of dipodies is deprived of the tie to
unite and cover the joins.
165. Moreover it is the same reason that requires the rhythmical
or trochaic dactyl to be preceded always by this kind of seeming
spondee, a trochee with more-than-filled short. When trochees
and trochaic dactyls are joined, the dactyl will always require to
keep the positive first place in the dipody.
I — 2
It has the greatest energy of memberment, has more weight in
the whole, is the stronger member, so that the trochee can follow
as its weaker echo. A dactyl cannot, on the other hand, be pro-
duced after a trochee. In this latter succession the formation
would merely tend to assume negative dipodic meaning ;
2 — I
but then the trochee has in fact become second member, and the
dactyl is first in positive meaning :
I - - 2 i - - 2
and the short before the dactyl is exactly that of the trochaic
series, which has to unite two dipodies and therefore pretends to
greater fulness. The difficulty of closing a metre with a dactyl is
readily perceived ; the dactyl always requires one more element to
follow it.
304 METRE
1 66. This must not by any means be taken as insisting that
in metrical series the dactyl should occupy only the first place in
the dipody. Most metres would be found to contradict that con-
dition. But the dactyl cannot be an unaccented member in the series,
as a second trochee can (disregarding the small accent which every
long carries on its beginning).
Thus in the three-timed metre, whenever the second time is
divided as a dactyl, it has rested upon its meaning of being first to
the time which follows :
for the three-timed formation allows this increase of weight upon
the second time as forerunner of the third. In four-timed formation
the third is the important time as against the fourth. Only this
last remains unaccented in four-timed metre, as the third time in
three-timed metre, and the second in two-timed. Accordingly
the dactyl, with the condition of having to be first to a second
member, may always be formed in any member of the metrical
series with the exception of the last, which alone is a decided second.
And so too the prosodically filled out short must precede the dactyl
in every position, inasmuch as the latter begins a dipody, and the
preceding trochee enters to it in metrical secondary meaning.
167. In the scheme previously given of the trochaic dactylic
memberment for the two- and three-timed metres — the four-timed
being considered as twice-two-timed for the purpose in hand, it
was not found necessary to draw it out in its 81 forms — the com-
bination of the three elements of unequal-timed division was shown
to the full number of all possible cases. But in those forms which
METRICAL AND RHYTHMICAL DACTYL 305
close with a dactyl, in which, therefore, a decided second foot of
a dipody has received dactylic division, a continuation is always
felt to be necessary. Otherwise the metre, leaving off so, seems
broken off, suspended. Dactylic memberment always requires to
end with a non-dactylic member or part of a member following the
dactyl.
1 68. Dactylic verses are measured by scansionists monopodically,
i.e. the measure of the verse is named after the direct number of
dactyls, and not reckoned by dactylic dipodies or tripodies, as
trochaic and iambic verses are by dipodies and tripodies. If under
the dactylic form the spondaic dactyl alone is to be understood,
then, inasmuch as the real spondee embraces by itself a whole
dipody, no objection could be made to this measurement. Only it
must seem strange that anapaestic verses are not then measured
likewise by the number of anapaests, but in dipodies like iambics.
169. Whether all dactylic movement in spoken metre does not
at bottom belong to the trochaic rhythm, might at any rate still
be debated. Not that the spoken dactyl is obliged always to move
exactly in the trochaic rhythm
For with equal justice might the strict spondaic rhythm
be deemed to suit all words of dactylic form. It seems, however, as
if rhythmical enlivenment has its first origin in the trochaic element,
to which the spondaic is ordained to form merely the metrical sub-
structure. The metrical equal-timed formation, presented as two-,
three-, and four-timed, lacks the rhythmical tension, the elastic
nature, which first comes into the metre with the unequal-timed,
or animating, distribution of the member ; because then it contains
x
3o6 METRE
the metrical opposition of first and second rhythmically united as
single and double in one, whereby the divergence of metre is at
last wholly negatived.
170. In the metamorphosis of plants, blossom-forming goes on
the principle that the leaves, standing opposite one another on the
stalk, are at the same time gathered round a centre or axis ; from
being separate in opposition, upon this side and that, they are con-
gregated into the circle, into the unity of union. Thus the equal-
timed metrical may be compared to the diametrically separated
position of the leaves ; the unequal-timed rhythmical to the centrally
united. Similarly, trochaic in its abstract meaning may be put as
the melody of metre, and spondaic as its harmony.
171. By the natural rhythm of words the strict metrical quantity
of the spondaic dactyl must nevertheless be subject progressively
to the most manifold modification ; because, without forcing the
metre very harshly, it is not practicable to continue speaking dac-
tylic movement in the time of
Similarly the trochaic dactylic form too must always be ready to
give way freely to the conditions of language. Although metrical
quantity is determined independently in itself, yet in its reciprocal
relation with the spoken contents which fill it out it nevertheless
acquiesces in the rhythmical modifications which are the demands
laid upon it by the latter. And it is the union of both together
that gives the finished picture, metrically ordered and rhythmically
animated, in form and contents correlated and made one.
172. If all dactylic movement is of trochaic nature, there will
then have been obtained an explanation why trochees are rightly
admitted in dactylic series as well as dactyls in trochaic series. On
the other hand, it might then seem that the spondee is wrongly
joined with the dactyl. But in places that may be regarded as
METRICAL AND RHYTHMICAL DACTYL 307
joining dipodies the spondaic form of word does not stand in the
meaning of metrical spondee ; it then represents the trochee with
the accented short, which in trochaic series may stand before every
dactyl, consequently in every place except the last and the last but
one.
173. All that has been said here about the difference between
the spondaic and trochaic form of dactyl, will apply also to the
anapaest, which we must accordingly distinguish into spondaic and
iambic.
ME TRE-MARKING.
174. The customary way of marking verse-metre is wanting in
means to discriminate accurately the fine shades of rhythm ; nor
is it employed to disclose the inner metrical structure of verses.
By the scheme which it presents we are taught only the order of
succession of long and short syllables ; which taken by itself is but
the surface, the outside of the edifice of verse.
We may pass by the circumstance that it makes trochees and
iambuses with accented short equal to the real spondee, which em-
braces a trochaic or iambic dipody, and that it thereby represents
doubtfully the total contents of the metre. But even then single
metrical quantities strung nakedly together will give us no image
of the inner conditions, on which a metrical formation as a whole
rests ; the latter being always, not aggregated, but expanded from
the metrical unity, an unfolding of the metrical notion that underlies
the whole.
175. Thus, e.g., the hexameter is laid before us schematically in
the following shape :
X 2
3o8 METRE
From this series we cannot in the first place see whether the sixfold
of its members consists of a twice-threefold or of a thrice-two-
fold. Nor yet is it evident whether the parts of the highest order
shall be taken in positive succession or negative. Further, in lower
order the pairs of members may be positive or negative, and thus
doubly different, in themselves and in their succession. From all
these different possible determinations, even if we have decided
for one or the other assumption with regard to the highest order —
viz. that the metre is a twice-twofold one, or a thrice- twofold —
there will still result a sixfold difference in the metrical organic
form : in fact any one of the six members may appear principally
emphasised.
We must not confound what here is only doubtful multiplicity
of meaning, nor even bring it into connexion, with what is called
eth rhythmical caesura of the verse : — which receives its determina-
tion from the logical contents, and, as is known, can in the hexameter
enter at sixteen different places, namely after every' single member
of the dactyl. For now we are speaking only of the metrical form in
itself, within which the shaping of the rhythm has afterwards its own
special determinations. In metrical sense the question here, ex-
pressed in empiric musical fashion, is merely whether the sixfold of
the metre is a f bar or a f ; further, whether it begins with the full
bar, with the down beat, or with the up beat ; and in the latter case,
how many members belong to the up beat.
176. The six-membered metre as a twice-threefold,
i — 2
can be shown metrically in the six forms :
I 2, 2
T*-»-0-U
METRE-MARKING 309
but as thrice-twofold,
in the six forms :
I - o —^—
1,
*--— *—*—*
U rLJ LJ
trtr
In these different metrical constructions, the organic conditions
of which are known to us from what has gone before, the rhythmical
determination, as given by the spoken contents according to logical
meaning and independently of the metrical, may still be most
manifold ; for it is only in the accent-determination that it comes
in contact with the metrical.
177. The possible meaning that such a series of members can
assume being thus various, a closer representation is needed before
metrical definiteness can be recognised in it. Knowledge of metre
can come to us only from its practical use ; here from the hexa-
meter itself. Of this we know from experience that it has its
principal section in the middle, consequently that it consists of a
twice-three-timed formation ; i.e. that it is a tripodic dimeter.
Further, the beginning of the second half is perceived to be an
3io METRE
element of especial emphasis. The rhythmical flow is urged
towards this element as to a highest point, and from it to the end
seems to sink again to its own level. Therefore we may assume
the second principal member of the twice-three-timed whole as the
principally accented, emphasised in the highest order, or positive
of that highest order. Thus the principal formation of the metre
is determined as a rising spondaic dipody :
2 —
I.
If the dactylic division be taken to be trochaic dactylic, then
there results for each half of the verse a trochaic tripody :
which in the hexameter is normally manifested positive in the pairs
of members, i.e. emphasised on the first time.
But the dipodies themselves in the first half-verse are related
oppositely to the emphasis of the members ; the second dipody is
the accented one.
2 — i.
In the second half-verse the first dipody is accented
the accent of the first half lies upon the second time, in the second
half it is borne by the first time.
METRE-MARKING
Accordingly the scheme of the whole appears in this shape :
2 — i i — 2.
That dactylic division is not given to every trochee is known.
The last but one will hardly do without it ; on the other hand the
third will the more readily, that in it the principal caesura enters
normally, whereby the short of this trochee unites with the long of
that following, forming an iambic beginning to the second half of
the verse, which contrasts as iambic with the trochaic first half.
178. To the hexameter is joined in elegiac metre the penta-
meter. It is the female hexameter. Like to the first verse in tri-
podic dimetric structure, the pentameter is contrary to it in carrying
the principal weight upon the first half; the separated second half
is a weaker echo of the first. In its principal formation the penta-
meter is a sinking spondaic dipody :
i — 2.
The other relations of memberment are the same in the penta-
meter as in the hexameter, and its scheme accordingly :
2 — i
i — 2 ;
the positive half, which ends in the hexameter, begins in the penta-
meter.
3i2 METRE
1 79. Again, in the union of both verses into a distich the hexa-
meter is itself the first half of the whole, and the pentameter the
second half. Therefore the principal accent of the pentameter is a
secondary one, thrown into the background by the principal accent
of the hexameter ; for the pentameter's first is the first in a second :
2 — I
v A
1 In the Hexameter rises the fountain's silvery column ;
In the Pentameter aye falling in melody back.'
(Coleridge's Translation of Schiller.}
1 80. More might be said about the aesthetic conditions and re-
quirements, as well as about the rhythmical caesuras of this verse ;
but it must be withheld here, where a nearer consideration of this
particular kind of metrical formation was undertaken altogether by
way of example, as illustrating the general principle in a concrete
form supposed familiar to us. To draw up a method of verse, theo-
retical and practical, an elucidation of the customary verse-measures
with their specific peculiarities, again, does not lie within the scope of
these investigations, any more than the preceding part, on Harmony,
was meant to contain a method of thorough bass or instruction in
the practical use of chords. There it was undertaken to investigate
alone the natural laws of harmony and melody, upon which every-
thing that can be made of use in practice is grounded. So here we
have to set out alone in their principles the natural laws of rhythm
and metre, which are the same in metre of music and of speech. We
are dealing only with the rational ground of the phenomena, not
with the phenomena themselves ; these we must dismiss as soon as
the firm basis is found for them. Everywhere threads of connexion
METRE-MARKING 313
are left standing, which would have to be taken up in carrying out
further the particular parts. But if the principle has been pre-
served, then it will be less difficult to follow out the intricacy of
the numerous branches, and to see that the particular things of the
phenomena are organically determined. If considered singly, or
only outwardly placed together, they might easily seem to us
arbitrary formations, which yet they in no wise are or can be.
CATALECTIC AND ACATALECTIC METRE.
Examples both in Spoken Metre and in Musical.
1 8 1. In the foregoing we have marked with a rest the termina-
tion of the two tripodies of the pentameter ; they end with the first
member of the foot and leave the second unfilled :
whereby the two halves of the verse appear separated, the tie which
should unite them is wanting.
In scansion generally a distinction is made between catalectic
and acatalectic metres : verses or parts of verses which leave a
rhythmical member or even a foot unfilled at the end of their
metre, and such as fill their measure quite up.
The verse, e.g., of the ancient drama, the trimeter :
is acatalectic ; its three iambic dipodies are completely filled up.
314 METRE
The newer dramatic verse, the metre of the so-called five-
footed iambics :
\J I W W I W
is of the catalectic kind. It is an iambic dipodic trimeter, like
the former ; but it leaves one or two places empty at its end,
either the last iambic long or the whole last iambus. According
to the notion of musical time these places ought to bear rests before
the beginning of the following verse. That, however, would only be
practicable in cases where a logical caesura, a break or breaking off
in the thought, takes place ; which again precisely in this place ought
not to occur too often, in order that the verses may not be detached
lyrically, but joined into continued oration. Therefore a catalectic
metre in its continuance is kept up more by shading the emphasis
(which in itself has the power of betokening one rhythmical element
against the others in its metrical meaning — i.e. as first, second,
third, or fourth) than by progressive movement in bar.
1 82. Here at any rate is found an essential difference between the
metre of music and that of speech ; for the former is not at liberty
to begin a new bar before the last has quite run out. But also
from the quality of its contents the necessity for its so doing can
never arise ; since, being wholly bound to the metrical determina-
tion, and not subject to other conditions, the contents receive their
shape in time from the metrical determination alone.
183. Yet another difference between spoken metre and musical
consists in the former not offering the larger contrasts in the dura-
tion of its rhythmical elements that the latter does, in joining into
metrical figures members of any length and any shortness. The
CATALECT1C AND ACATALECTIC METRE 315
former combines metrically only single and double time-differ-
ences, modified variously, it is true, in practical use, but not so as
to be capable of determination, and still only in the meaning of the
above proportions.
Of the overlapping of several orders in the arrangement of the
metre itself, in so far as it is twice-threefold, thrice-twofold, and
so on, no mention is here necessary — for spoken metre has this in
common with musical — but only of the differences of quantity
which reach expression in the spoken syllables.
184. Vocal rhythm by itself, apart from metre, is in its shades
of quantity comparable to the melody of speech, in which height and
depth of sound are regarded as lending emphasis to words and
syllables. The latter could hardly be represented in a determina-
tion by harmonic intervals, although it makes the vocal note rise
and fall ; nor would it be easier to establish a determination for the
infinite gradations and transitions in which the rhythm of the parts
of speech approaches the pure metrical forms, coincides with them,
and again deviates from them ; because in measured speech the
rhythm preserves measure in the whole, and also is seemingly at
one with it in the members.
But it would deserve to be called downright absurdity, to let
oneself imagine that a poetically animated delivery must or could
everywhere conform exactly to the mathematically determined
forms of a rigid metrical system, or reflect it in all strictness. The
metrical form is the solid skeleton, the bony framework, round
which the soft parts, which the life inhabits, grow in rounded,
mutually re-entrant forms, and, while they cannot do without the
firm determined support, yet let it appear not at all, or only in
veiled, softened, and apparently self-determining outlines.
316 METRE
QUANTITY AND ACCENT.
Difference between Ancient and Modern Verse.
185. The art of ancient verse has the determinations of vocal
quantity for its formal elements : length and shortness of syllable.
Modern verse substitutes for the long the accented or logically
emphatic syllable, and for the short the unemphatic and unaccented.
The modelling of the former art stands upon its own merits, is not
directly affected by the emotional or mental life of the thing re-
presented. For syllabic quantity is not determined by logical
meaning. The short may be the logically emphatic syllable in a
word, the long may be logically without accent.
So far therefore as the construction of verse follows determina-
tion of quantity, its metrical structure is quite independent of the
sense contained. The form is therefore more self-sufficient ; it can
in itself be of metrical artistic importance. In modern verse, which
seeks its accents in the logical meaning of syllables, the formal
construction has not this self-sufficiency. Here the form is merged
and lost in the contents ; it is, if we compare the modern verse-
metres with the ancient, altogether of less consequence artistically.
Where the latter in their strophes afford an inexhaustible variety
of rhythmical division, our verse in strophes for the most part con-
sists merely of an alternation of rising and falling, measured in two,
three, or four times. The metrical art-element plays so small a part,
and is so much absorbed by the poetical contents, the logical mean-
ing of the words, that now, to prevent the verse from being quite
inartificial, another element is needed besides the metrical : such as,
being again formally self-sufficient, is fit to carry the contents with-
out being their slave. Such are rhyme, assonance, alliteration. These
conditions of sound have no more interest in the contents than
QUANTITY AND ACCENT 317
metre by quantity has. The inner references which creep in some-
times between rhyming words, as when heart and smart, sweet and
greet are made to rhyme, are quite accidental ; it is by no means
the business of rhyme to hunt for them. Rhyme consists in pure and
simple likeness of sound, and is in itself artistic form. So too with
assonance, beginning with the same vowel, and alliteration (in
German Stabreim), beginning with the same consonant, of which
the most elaborate use is found in Scandinavian poetry.
1 86. To verse ruled by quantity we might ascribe more of a
plastic, to modern accented and rhymed verse more of a musical
nature ; or in the former meet rather the principle of form, and in
the latter rather the principle of colour. The accented strophe will
hardly do without rhyme ; to the strophe with quantity rhyme
would be quite an unsuitable, even an unwelcome addition. Rhymed
endings to the ancient strophe might be compared to colour upon
a statue.
Then, again, verse by quantity is compared to polyphony in
music, and accented verse to homophony. As verse by quantity
tries to avoid coincidence of the logical caesura with the metrical,
so too the polyphonic phrase spins a web over its metrical form,
and covers the caesura of one part by the progression of another ;
while the homophonic phrase holds its parts in metrical unison,
and accented metre, especially in verse destined for song, need not
avoid the lyrical caesura, but rather courts it.
Historically we see among the Greeks poetry by quantity and
homophonic music, among the old Italians poetry by accent and
polyphonic music standing together. In our time homophonic
music and accented poetry are the more natural growth ; poly-
phonic music and poetry by quantity belong more to the nursery
of art.
1 87. But where, in the alliance of poetry and music, the latter is
to have full play, there the poetry can only be controlled by accent ;
318 METRE
for it must have the lyrical caesura. The newer attempts to set ancient
poetry to music have never fallen out otherwise than to the detri-
ment of the poetry. The delicate distribution of the ancient metre
is crushed under the weight of our self-sufficient musical forms ; or
else the music, in trying for a more intimate connexion, must make
surrender of its own most special nature, because our song is less
the emphasised word, than the contents of the words set to music
in forms of independent musical value.
1 88. It has been said by a writer upon art, doubtless between jest
and earnest, that in music poetry seems to have but one privilege ; it
may be bad with impunity. Poetic both in contents and expression
it must always be, if it is to be capable of being represented music-
ally. Mattheson pledged himself to set a placard to music. But
the contents of a placard or a bill of fare could not be reached in
the musical expression ; though certainly joy at famous names in
the first and at favourite dishes in the other would admit of being
expressed musically. But to emphasise speech according to its
verbal expression, to tint it in its single elements, cannot be the
task of music, which by its nature has to do the precise opposite.
Music has to express in the language of feeling unitedly, what
intellectual language of words can only put dividedly, successively.
Where the latter speaks of gladness and sorrow, and must name them
separately, first one and then the other, there music can express,
and ought to express, sorrow in gladness, and gladness in sorrow ;
but not to emphasise one word joyfully and the other mournfully.
Herein musical expression leaves the speech of poetry far be.-
hind ; and the music, where not merely declamatory, not merely
lending emphasis to words, will always take rank above the poetry.
The verbal expression can make good no other demand upon the
musical, than that it shall not be injured by unintelligent emphasis
conflicting with the sense ; but not that the music shall enter into
all its particulars and try to express them by notes. For music
QUANTITY AND ACCENT 319
emphasises the complex of feeling contained in the words, and not
the words themselves.
Music may be compared to algebra, speech to arithmetic.
What music contains in a general expression, language can only
express as particular. An algebraical formula shows the factors in
their mutual dependence and operation : the factors and the pro-
duct in one. Arithmetic shows either the factors alone or the pro-
duct alone. Algebra gives the universal meaning for infinitely
many particular values that may be taken. Music is like it in this.
One has often seen the experiment made of expressing the contents
of a piece of instrumental music in words, in a poem. The result
can never be satisfactory. If the algebraical expression makes
a + b=c, and one chooses to replace this by 2 + 3 = 5 with arith-
metical values, this application of the formula is certainly quite a
correct one. But there is an infinite number of other values to be
put for a and b, which yield c as another sum, and where the com-
bination of factors fulfils the purport of the formula just as correctly.
So too the same music might be expounded verbally in the most
different ways, and of none of them could it be said that it was
exhaustive or that it contained the proper and the whole meaning
of the music ; for that is contained with the utmost definiteness only
in the music itself. Music has not an indefinite sense ; it tells the
same tale to everyone ; it speaks to men, and says only what men
feel. But ambiguity comes to light when each in his own way
seeks to comprehend in a particular thought the impression of
feeling that he receives, trying to fix the fluid element of music and
to utter the unutterable.
189. We see that as used in music metrical forms are not followed
with mechanical strictness ; because by conditions of harmony and
melody, as well as of animated performance, they continually suffer
small deviations from exact mathematical definiteness, which yet
never seem like losing the time. But the metre of speech is in the
320 METRE
relative quantities of its members still far more given up to modifi-
cations by the conditions, logical and phonetical, of its contents — the
words that fill it The unequal-timed feet, the trochee and the
iambus, with a sonorous or logically important syllable in their
short, will often pass almost into equal-timed ; the trochaic dactyl1
may, by reason of syllabic contents and emphasis, assume the form
of the spondaic dactyl or even of the tribrach, the form of equal-
timed three-membered division ; thus it may appear as trochee with
metrically divided long :
or even as trochee with metrically divided short :
and yet not give up its meaning of dactyl in rhythmical metrical
determination. In like manner the iambic anapaestic form too will
bend to the quality of its verbal contents and submit to manifold
modification.
190. Not to be confounded with these rhythmical modifications
arising from particular verbal contents, is the rhythm which in itself
progresses only by equal times, such as arises from metrical con-
struction without quantity, but with accent alone. Here the differ-
ence of long and short is in fact not present ; the change consists
only in the succession of emphasised and unemphasised members,
in rising and falling. Our rhymed verses are mostly of this kind.
191. But these are not alone in passing over the difference of
long and short parts of time. Even in verses marked trochaic and
iambic it is only brought out noticeably, where dactylic or ana-
paestic movement accompanies the trochaic and iambic. In pure
trochaic or iambic lines there would be difficulty in continuously
doing justice to rhythmical quantity, by double and single duration
QUANTITY AND ACCENT 321
of time. With the trochaic dactyl especially (and also, but less
decidedly, with the iambic anapaest), if it follows soon after the
beginning of the line, rhythm by quantity may make its appear-
ance, and, being once started, it is then kept up through several
members. Conversely, after a longish series of accented rhythm
the dactyl may easily take the metrical equal-timed form ; and
then the succession is also further continued as equal-timed.
192. Musical metre always makes a much more definite dis-
tinction between equal- and unequal-timed movement. The £ bar
cannot be changed for the \ bar without causing an interruption in
the rhythm. In passing from one to the other a break is always
felt, a change of the prime rhythmical condition.
In speech-metre, where the distribution of members has to be
impressed upon the syllables of words, the rhythm adapts itself on
the whole to the metrical relations ; these, on the other hand, accept
their modifications in particular from the rhythm. Here form and
matter are both of elastic nature ; they spread and contract ac-
cording to the claims which the one enforces upon the other. Too
small a syllable, however, is ill suited for filling the metrical long ;
while too heavy a syllable will resist being crowded into the metrical
unaccented short. But in modern languages the logical accent,
above all, is that which determines the metrical position of syl-
lables ; not only in accented, but also in quantitied metre.
193. In scansion the metrical positive first element is named
the arsis, and the second the thesis. In musical meaning this is
reversed; for the first part of the bar, the so-called 'strong'
time, is called thesis, and the second part, the so-called ' weak '
time, is called arsis. The expression thesis in music points to the
down beat, with which the beginning of the bar is marked ; the ex-
pression arsis in scansion for the same element of time to the lifting
force with which the positive metrical determination begins. This
difference might indeed have been assumed to be already known
322
METRE
to the reader ; but for our purpose it seemed best to avoid these
names altogether, because of their opposite meaning in music and
prosody ; lest the metrical notion, which is the same in both spheres,
should, through these different names applied to the same thing,
be brought into seeming contradiction with itself. Knowledge of
technical names, as well as of the outward appearance of the things
called by the names, has always been assumed in our treatise
hitherto, and so we have used the names as known ; our business
being less with the outward appearance of the things named, than
with their inner entity and connexion in unity.
III.
METRICAL HARMONY.
HARMONIC METRE.
Y 2
HARMONIC METRICAL DETERMINATION,
I. HAVING considered by themselves the process of harmonic
melodic construction in the first place, and the metrical rhythmical
in the second, it now remains to unite the two double factors into
concrete unity as they exist in music, so wrapt up in one another
that every element of harmony must have its meaning also as an
element of melody, and at the same time also as an element of
metre and of rhythm.
But melodic rhythmical does not admit of being gathered up
into an abstract system, or of being developed in the way that
harmonic metrical does. With the former, in the infinite multiplicity
of the possible phenomena, nothing could be discussed except what
is most general or most particular. With the latter, the particular
can be comprehended in the universal, and from the whole can be
deduced the explanation of each single thing.
The following contains only harmonic metrical investigation.
In the notion of a succession of notes or chords an advance in
time is already expressed ; but still it has only the general meaning
of sequence without any metrical determination necessarily being
connected with it.
2. Now the first metrical determination is that of the succession
of a first and second, a positive and relative, an accented and un-
accented :
i — 2.
326 METRICAL HARMONY— HARMONIC METRE
Its opposite is the same in inverted order :
2 — i.
3. Harmony also has its positive and relative in the notion of
succession. We have similarly denoted it by I and II, as the rela-
tion of a dominant or subdominant triad to its tonic triad : I — V,
I — IV, both included generally under the above expression : I — II.
Here again the * other,' the opposite of these successions, is their
inversion : V — I, IV — I, under the general expression : II — I.
4. In uniting the harmonic with the metrical notion — that is to
say, in the harmonic metrical or metrical harmonic notion — we get,,
as in every twice-twofold combination, a fourfold possible relation of
the harmonic determination to the metrical : A (a) harmonic posi-
tive in metrical positive ; (b) harmonic positive in metrical negative ;
B (a) harmonic negative in metrical positive ; (b) harmonic negative
in metrical negative :
A. (a) I— II (b} I— II
I — 2, 2 — I.
B. (a) II— I (b) II— I
I — 2, 2 — I.
5. The positions in which the harmonic positive counts as
metrical negative are not contradictory to rational meaning. Their
sense is, that something that has a relative, harmonically, becomes
something that is a relative, metrically ; that in them a harmonic
active is found as a metrical passive. A chord cannot be at the same
time harmonically positive and harmonically negative : the triad
C — e — G cannot at once be tonic and dominant ; but it can, being
tonic, occupy metrically positive or metrically relative position, just
as a metrically positive element, though it cannot at the same time
HARMONIC METRICAL DETERMINATION
327
be metrically relative, may have for its contents harmonic positive
or relative.
6. Thus succession of consonant harmony is in itself as yet with-
out determination for the metrical position of its successive members.
The same series of consonant chords may take most different shape
metrically, and thereby also become most manifoldly different in
inner meaning. For even the succession of the triads C— e £•••
Q — l — 2}t according as it is placed in metrically positive or nega-
tive order :
C— e— G b— D— G, C— e— G | b— D— G,
1—2 2 i
according therefore as it has its metrically positive, accented element
upon the tonic chord or the dominant, lends expression in the first
case to the notion of major, and in the second to the notion of
minor ; that is, to the notion of independence or to the notion of
dependence, thus with the same harmony expressing opposite
meanings ; and if we continue with further triads in more advanced
metrical formation, in three-timed or four-timed and in combined
metre, we may be led to the greatest diversity of harmonic metrical
meaning.
7. As with the triad, so also with its first inversion, the chord of
the Sixth ; which in itself contains no determination for metrical
position. Among triad forms the Six-Four position alone in many
cases is limited metrically to only one place or the other ; the same
position whose occurrence in harmony was subjected to multiplied
conditions. Thus the Six-Four position of the tonic triad, when
following the subdominant it passes into the dominant chord and
leads to the close, will always require a metrically accented place,
while its resolution into the dominant chord takes a second, un-
accented place. Here it is the succession involved in the nature of
the chord-position, and the condition of the closing element having
328 METRICAL HARMONY— HARMONIC METRE
to be a metrical first, that imparts to the chord its metrical deter-
mination.
8. Otherwise it can only be said generally of groups of triads in
this respect that no kind of metrical succession is unconditionally
impossible in them. For as the successions of the triads :
I — 2 I — 2 2 — I 2 — I
C—G, G— C; C/G, G/C;
I_V V— I I— V V— I
contain united what is metrically and harmonically similar and
also what is metrically and harmonically opposite ; and as the un-
connected dominant and subdominant chords may also succeed one
another in a fourfold harmonic metrical sense, as :
I — 2
I — 2
2 — I
2 — I
F-G,
G-F;
F/G,
G/F;
IV— V
V— IV
IV— V
V— IV
so too every passage from the tonic triad into the conjunct minor
triads :
I 2 I 2 2 I 2 I
r (a a) r . r i (a a) /p .
C~le, e|-C' C/ie, ef/C>
or into the disjunct diminished triads :
I — 2 I — 2 2 — I 2 — I
C-JD; D-|_c C/{D ij}/c
^ D , D ) ' ( D , D j '
may be placed in all the various metrical determinations of succession.
9. In these successions the chord in the position of relative, if
it belongs to the subdominant side, will also seem akin to the sub-
dominant chord ; and if it belongs to the dominant side, akin to the
dominant chord. So that the successions
HARMONIC METRICAL DETERMINATION 329
I— VI ( I— IP
C-a and C-D°
veil the meaning
I— IV
C-F,
and the successions
I— III . , I— vii°
C_e and C-b°
the meaning
I— V
C— G.
Now, since it is principally the Third that in these secondary
chords suggests the subdominant or the dominant, because the
subdominant chord here is touched by a, and the dominant by b \
therefore the minor triad on the subdominant side, because it con-
tains both Root and Third of the tonic triad, is also fit to represent
the tonic triad itself. So much depends here upon the particular
position, upon the prominence of one or another interval of the chord,
that to establish an abstract, universally valid, determination for
the substitution of secondary for principal chords, is not possible.
In concrete cases it will always be easy to perceive, and express, the
meaning.
10. In the scale, whose degrees, as before was shown, are deter-
mined in the major key by the three notes of the tonic triad, a
change always takes place between tonic and dominant or sub-
dominant chords :
C •• D •• e •• F •• G •• a •• b •• C
I— V— I, IV — I — IV
(VI— HI— VI)
I — V — I.
To a tonic chord. I, answers directly the metrical positive, I ;
330
METRICAL HARMONY— HARMONIC METRE
to the chord of the Fifth above or below, II, answers the metrical
relative, 2.
According to this the metrical position of the scale notes up to
the sixth degree will be :
I— V I— IV I— IV
C •• D-.e •• F ..G.. a
I — 2, I — 2, I — 2.
But from the sixth degree onwards it is :
I— v i
i /*"*
a • • D • • w
I — 2, I.
Therefore the sixth degree has metrically relative meaning to the
fifth degree, metrically positive meaning to the seventh degree.
In every sense at this place there will always be found a drag upon
the progression, rhythmical as well as harmonic. Here, if metrical
positive is to coincide with harmonic positive, this sixth degree in
changing its harmonical determination must also receive metrically
twofold determination, first relative and then positive. This can be
done by doubling or halving the metrical value of the place, letting
it be repeated or else conjoining it with the seventh degree :
G — a
a — b
c,
I — 2
r^
I — 2
r^
i
^^_
r
i
1
or
G — a — b
I — 2
I 2
In both ways the sixth degree counts as relative to the fifth degree
and positive to the seventh degree, and the metrical determination
squares fully with the harmonic.
ii. It is now plain that here in the scale, as in the before-
mentioned triad successions, the metrical determination may in every
HARMONIC METRICAL DETERMINATION 331
way come into opposition with the harmonic, in every sense go
chequer with it ; because the scale can only have degrees deter-
mined by the interconnexion of triads. They are determined, how-
ever, not by most nearly related triads connected in the Third,
but only by triads connected in the Fifth, related in the second
degree. The former imply melodic quiescence principally, advance
only secondarily. In the latter melodic advance is the principal
thing ; the harmonic support is secondary, subordinate.
METRICAL POSITION OF DISSONANCE.
1 2. With the entrance of dissonance there also comes in a more
definite appointment of metrical position in the harmonic phrase.
We know dissonance in two principally distinguished kinds :
suspension and Seventh-harmony,
The resolution of the suspension follows, or may follow, wjthout
alteration of the Root-harmony of the chord in which it is contained
dissonantly.
By the resolution of the Seventh chord a new Root-harmony is
necessarily brought about.
(a) In the Chord of Suspension.
13. The passage from one triad into another closely connected
with it does not give rise to a chord of suspension ; this can result
only from passage into a triad that is principally not connected, more
separated than connected. Hence suspension can never arise but
with the appearance of a triad essentially different to the preceding
one, related to it in the Fifth or altogether separate. Moreover, its
resolution is not upon an essentially different harmony, but upon
the Root-harmony of the chord of suspension itself. With this,
332 METRICAL HARMONY— HARMONIC METRE
then, is given the determination that this dissonance must be a
metrical First and its resolution a metrical Second ; that the dis-
sonance must stand upon the accented part of the bar, and the
resolution upon the unaccented part. For with the dissonance a
new harmony has entered, which is not altered in the resolution
and merely draws after it a necessarily following second element.
Dissonance and resolution belong to the same Root-harmony, and
so stand in metrical positive unity as a first and second member.
Every suspension has the metrical first place, and its resolution the
metrical second.
(£) In the Seventh Chord.
(a) In the Untransposed Key- System.
14. The dissonance of the Seventh chord is introduced in two
different ways : it is prepared either in the Seventh or else in the
Root, according as the Seventh chord contains in simultaneous
sound the passage into a triad lying above or into a triad lying below.
Thus from the passage of C — e — G to a — C — e arises the
Seventh-harmony a — C — e — G in the position C — e — G — #, for the
Fifth of the major triad on C has progressed to the Root of the
minor triad on a, and has at the same time remained stationary as
Fifth : the dissonance in C — e — G — a is prepared by the Seventh
G. And this G becomes Seventh by the entrance of the Root a.
15. The opposite passage, from the triad a — C — e to the triad
C — e — G, gives rise, when framed as harmony, to the same Seventh
chord a — C — e — G ; and in the position G — a — C — e, for here a
has progressed to G. But in this succession it is the Fifth of the
new triad that enters dissonantly to the Root of the first.
The succession a — C — e--G — C — e = G — a — C — e determines
no new basis with its second element; but the succession
METRICAL POSITION OF DISSONANCE 333
C — e — G--C — e — a = C — e — G — a does. The Seventh chord of
the last succession, as arising from the entrance of the Root, will
require to have the metrical first place, the accented element. But
in the Seventh chord of the succession a — C — e- • - G — C — e the basis
of the second triad is already contained in the first ; it does not
enter with a new Root, it only brings the Fifth to a Root already
present, and the Fifth is a harmonic secondary or relative. There-
fore metrically it will not take a primary, positive, first, or accented
position, but rather that which corresponds to its harmonic meaning,
i.e. relative or secondary : it will find its place upon the second, un-
accented metrical element. Thus the prepared Seventh stands nor-
mally upon the ' strong ' part of the bar, and the passing Seventh
upon the ' weak ' part.
This harmonic metrical determination is valid for all Seventh-
harmonies which are combined within the untransposed system
from two real triads, one major and one minor, i.e. for all those
in which, as we have earlier seen, the Seventh could not enter
ascending to the Root.
(£) In the Transposed Key -System.
16. There are also the Seventh chords that contain the joined
limits of the system. We were then able to consider the system as
transposed within itself, having its middle divided up and taken for
limits, and its limits joined and taken for middle. Thus the systems
of the C major and C minor keys :
(e) G— b— TT — a — C (e)
— at>— C (e[>)
are characterised by the element W, the Root of the subdominant
sounding with the Fifth of the dominant. In these Seventh chords
the Seventh might move upwards to the Root. Unlike those con-
tained within the untransposed system, they do not consist of a
334 METRICAL HARMONY— HARMONIC METRE
union of two overlapping triads ; they are combined out of the
dominant and subdominant chords, that is, from triads that are
decidedly separate. Also their production, unlike that of the
others, is not governed by conditions of passage, for they are not
necessarily produced only from the succession of the two triads
contained in the Seventh-harmony : in fact, one at least of these is
here a diminished triad and therefore not truly a triad.
17. What was earlier said of the nature of these Seventh-har-
monies need not be repeated in this place, nor how that special
character is contained in them which they have over those of the
untransposed system. We will here lay stress only upon one
characteristic ; that in the Seventh chords of the transposed system
we have elements of like harmonic meaning sounding together to
form the dissonance, while in the Seventh chords of the untrans-
posed system there is always the Fifth of one triad sounding dis-
sonantly with the Root of another.
In the Seventh chords :
I I
G— b— D/F,
III III
b_D/F— a,
II II
D/F— a— C,
there are in the first the Roots of the subdominant and dominant F
and G, in the second the Thirds of the subdominant and dominant a
and b, in the third the Fifths of the subdominant and dominant C
and D, confronting one another and forming dissonance.
The Seventh chords :
I II
F— a— C— e,
_ METRICAL POSITION OF DISSONANCE 335
I II
a— C— e— G,
I II
C— e— G— b,
I II
e— G— b— D,
on the other hand, all of them mean by their dissonance, that the
Fifth of the upper triad stands in contradiction with the Root of
the lower, or rather that sounding together they give rise to the
contradiction in the interval that lies between.
1 8. Now in every case, since the dissonant notes cannot come
together simultaneously, for that would lie beyond all interpreta-
tion, they can only bring about the dissonance by entering succes-
sively, coming one after the other. The notes must enter successively
in the order either of Root and Fifth, or of Fifth and Root. The
metrical equivalent meaning is then plainly found in the order of
first and second, or of second and first : to the Root as Seventh
belongs the first metrical place, and to the Fifth the second :
"'
2 — I.
But those Seventh chords which are contained in the transposed
system oppose as elements of dissonance, not a Root against a
Fifth, but two harmonic elements of like determination : Root
against Root, Fifth against Fifth, Third against Third. Here, then, a
self-evident metrical determination for the position of the disso-
nance-chord, according as it is prepared in the Seventh or in the
Root, is no longer to be found. At least it is not given in the
difference of the dissonant notes according to their harmonic
meaning, as it is in those Seventh chords in which Root and Fifth
336 METRICAL HARMONY— HARMONIC METRE
are dissonant ; for here the Seventh is of like harmonic meaning
with the Root. Both confront one another as the same elements
of the subdominant and dominant chords, and may, at any rate
according to the harmonical meaning which they have with respect
to their own triads, with equal right lay claim to equal places. And
so in fact we see the Seventh chords of the transposed system, i.e.
those which contain its limits joined as middle — in the key of C
major :
G— b~D | F, b— D | F— a, D | F— a— C ;
and in the key of C minor :
G— b— D | F, b— D | F— afc D | F— a[>— C,
with prepared Seventh, occupying the metrical second time with
good, irreproachable effect, and having both their preparation and
their resolution upon metrical firsts.
19. A sequence of prepared Seventh chords with their resolu-
tions into the corresponding triads, where the Seventh chord is
taken upon the second time and the resolution upon the first,
cannot but prove strained and unnatural in the dissonance-har-
monies of the untransposed system. But in the dissonance-har-
monies of the transposed system, nothing faulty is perceived. For
there the metrical arrangement with preparation of the Seventh on
the first time and entrance of the Root on the second is as well
founded as the reverse. Root and Seventh being harmonically of
equal dignity, neither is preferred before the other.
20. The following series :
C | FX - b° | e7 - a | D^-G | CX - F | b;-e | a; - D° | G7 - C,
which with this metrical arrangement contains the dissonance-
chord upon the first time, cannot seem otherwise than unexception-
METRICAL POSITION OF DISSONANCE 337
ably correct in each element. If the same series is placed in
metrically opposite order :
C - FX | b° - e7 | a - D; | G - CX | F - b; | e - a7 | D« - G7 | C,
we then see the dissonances fall upon the second, unaccented time,
and this cannot correspond to the weight with which the Seventh
chords of the untransposed system enter. The Seventh chords
F—a—C—e, a—C—e—G, C—e—G—b, e—G—b—D find their fitting
place only upon the metrical first time.
21. On the other hand the Seventh chords G — b — D\F,
b — D\F — a, D\F — a — C quite readily adapt themselves to the
unaccented place. With the chords G — b — D\F, b — DjF—a
there is unmistakably felt the entrance into the other region of
dissonance, different to that in which the rest of the Seventh
chords have their being. With the Seventh chord DjF—a — C
this is less noticeable ; because, as has already been remarked in
cases where it has occurred, the diminished triad D\F—a may
easily be confounded with the minor triad d — F — A, or the latter
may really be substituted for the former ; and then the chord ap-
pears, like the Seventh chords a — C— e — G and e— G — b — D,
as one of those that must fall upon the metrical first time.
In the minor key no ambiguity is found at this place : the chord
j)IF—a\>—C is in relation to the minor key-system of precisely
the same nature as D/F—a—C in the major ; in its effect, how-
ever, it is of a more decided kind, and is not liable to be confounded
with a Seventh chord of the untransposed system. Hence also it
can be placed metrically second with less hesitation than can the
chord corresponding to it in the major key.
338 METRICAL HARMONY— HARMONIC METRE
SUMMARY OF THE FOREGOING CHAPTER ON
THE METRICAL POSITION OF DISSONANCE.
22. As existing in harmony, every dissonance-chord is at once
a first and a second. It is second in consequence of the previous
preparation of the dissonance, and first in respect of its resolution,
which necessarily follows. But, metrically, prominence is given to
one harmonic relation or to the other, according as the dissonance
is either determined by a Root newly added to the chord, or as it
enters against a Root already present.
In constructing the dissonance of the chord of suspension,
the dissonant element is always metrically first, accented ; for here
the dissonance arises by a new Root entering. Its resolution is
metrically second and unaccented; it ensues without essential
alteration of harmony.
In the construction of the Seventh chord the dissonant element
is metrically first if the dissonance is prepared in the Fifth of the
upper triad, and metrically second if in the Root of the lower
triad. But it may be first or second, if the dissonance is not
between a Root and Fifth, but between chord-elements of equal
order, which then can only be equal elements of the opposed do-
minant and subdominant chords.
Suppose that the harmonic process of dissonance-construction
may again be represented generally under the form
I — II
I - II,
in which the middle doubly determined member is to signify the
chord of dissonance, the first member the chord of preparation,
and the last the chord of resolution. Then the metrical determina-
METRICAL POSITION OF DISSONANCE 339
tion joined with the harmonic will give a twofold result. It must
take as accented either the first harmonic member or the second,
the element of preparation or the element of dissonance ; then the
third member, the element of resolution, is, in consequence of the
determination of the second, at once determined as accented or not
accented.
The harmonic metrical determination is accordingly either :
I-II I-II
I— II or I— II
I— 2 | I--, .-2 | 1—2.
Under the first of these forms appears the so-called passing
Seventh ; and such Seventh-formations as have their origin in the
transposed system will also be adapted to it : pre-eminently the
dominant Seventh chord and the Seventh chord upon the Third of
the dominant ; less unrestrictedly the Seventh chord upon the Fifth
of the dominant in the major key, for reasons previously discussed.
Under the second form the dissonance of suspension is always
represented ; also Seventh chords of the untransposed system
when not prepared in the Root ; while those of the transposed
system may thus appear, and in tied harmony they mostly will.
DISSONANCE IN THREE- AND FOUR-TIMED
METRE.
23. Hitherto, in its bearing upon combined harmony and metre,
the behaviour of first and second time has alone been taken into
consideration. Now two-timed metre is but the beginning of
metrical construction, which goes on to three- and four-timed,
finding in these two elements its development and completion.
z 2
340 METRICAL HARMONY— HARMONIC METRE
By this further formation the accent-determination becomes com-
bined. In the three-timed, as well as in the four-timed metre, the
second time is no longer unaccented ; in the three-timed only the
third, and in the four-timed only the fourth is without accent.
But now the accents appear in different orders. The two-timed
metre contains only one simple order of accents ; but in the three-
timed we find them doubly, and in the four-timed triply superposed.
And a metrical member, which in a higher order is without accent,
but in a lower order is accented, may claim this accent for the
harmonic meaning in its order.
Thus a harmonically accented element, which in the positive
two-timed metre can only coincide with the first time, may in the
three-timed coincide with the first and second times, and in the
four-timed with the first, second, and third.
24. In the three-timed metre the first time is doubly accented,
the second is singly accented, and the third is without accent.
Hence on the last time can stand only the unaccented dissonance,
or passing Seventh, which is prepared in the Root, and also the
dissonance-chords belonging to the union of the dominant and sub-
dominant. But the suspension and the Seventh prepared in the
upper Fifth may occur on the second time just as well as on the
first. The preparation then happens, in the first case upon the
first time, in the second case upon the preceding third time.
25. In the four-timed metre, in which only the fourth time is
wholly without accent, everything holds true of it that was said of
the third time in the three-timed. But what was true of the first
two times of the three-timed metre cannot without restriction be
applied to the first three times of the four-timed. In the four-
timed the first time is triply accented, the second singly, and the
third doubly. Therefore the four- timed metre lays stress principally
upon its first time and its third as accented ; while on the other
hand the second with its single accent, that of the member,
DISSONANCE 341
certainly not seem fitted to carry an element of heavy harmonic
emphasis. On this account the harmonically accented dissonance-
elements, the suspension and the tied Seventh in the real combina-
tion of triads, will fall only to the first and third members of the
four-timed metre ; but the second, feebly accented, member, coming
after the triply accented first, and also the unaccented fourth, follow-
ing the doubly accented third, can only receive the harmonically
unaccented passing Seventh.
In place of the latter the other not necessarily accented dis-
sonance-harmonies may also enter, as is manifest, since they may
even occupy wholly unaccented metrical places.
SYNCOPATION.
26. In the three- and four-timed metres, in closed formation,
the weak member of one pair is covered by the strong member of
another, and is in its turn made prominent by accent ; and such
an arrangement of members in linked positive pairs may also be
carried further and continued in a series.
27. Every series of progressive formation necessarily affords
already a double point of view. A series of major chords contains
in it at the same time a series of minor chords ; and similarly in
the metrically positive series a negative series is simultaneously
contained :
B|>— d — F — a — C — e — G — b — D — fj — A
The metrically separated pairs of the positive series are united
by the pairs of members of the negative series ; and here we must
refer the reader to what has been said in its place about the notion
342
METRICAL HARMONY— HARMONIC METRE
of the close and about the difference between metrical and rhyth-
mical unity.
28. But the linking of members now to be considered is of
another kind. The second member of one pair is here joined
with the first of the pair following, not as having the meaning
that belongs to them already as members in the positive series, i.e.
not as a negative pair, but as a positive pair again.
In music this proceeding is known to us under the name of
syncopation, which joins a metrically second member to the follow-
ing first member in positive undivided unity, and lends an accent
to the unaccented member :
i — 2 i — 2 i — 2,
I — 2 I — 2 I — 2 I — 2
r
29. If, however, a series is to appear syncopated, then the urn
syncopated series must at the same time be present with it ; for
without the normal [series, of which the syncopated forms the
metrical contradiction, the syncopated would itself be shown nor-
mally accented. The above would seem a series commencing with
an up beat.
• _A___ A A? ^_
i i i i i
The syncopated movement will be yielded as such only if the
accented elements of the normal series are marked at the same
time with it.
r =-. ="! > i
SYNCOPATION 343
Consequently, to exhibit syncopation, two parts at least are
required, of which one contains the normal, the other the syncopated
emphasis. But this condition of two-partedness is only absolute
where the syncopation enters undivided, without distinction of a
second element. Otherwise a phrase might have syncopated em-
phasis, and yet in its memberment always allow the normal metrical
structure to show through, as we have before seen many times in
rhythms accented upon the second metrical element.
30. In such syncopations, ligatures, and ties from a second
time-element to a first, the latter cannot have longer duration than
the former ; as is seen to be the natural conclusion when we consider
that both elements here stand to one another in the relation of a
first member to a second, just as in the two-timed metrical unity
by itself. Starting from a metrical beginning, there is nothing to
prevent us from lengthening the duration of the member put first ;
for its duration from the beginning is self-determining, and not
determined, as is that of the second element. But where the latter,
being already determined in its duration, is taken as positive first,
there its relative second cannot be more than equal to it. It may
indeed have shorter duration, because the actual filling up of the
measure need not be complete ; but not longer, because the contents
cannot exceed the measure.
31. Consequently a tie from a shorter to a longer portion of
time :
is on the face of it always something metrically untrue. The synco-
pated first time can only have a second of equal or less duration
united with it ; a longer would be, relatively to the preceding
member, more than single, and being then related within itself as
first and second, it breaks up the inner unity of the tie : it makes
an accent felt.
344
METRICAL HARMONY— HARMONIC METRE
I
I — 2
From this it follows that those forms (included above with the
others) of filling up the four-timed metre, in which an undivided
triple follows the single time,
h-f
and
are not metrically justified, which is why they are instinctively re-
jected. For in them a double containing the third and fourth is
tied to the single second member.
r
r r
32. All such rhythmical arrangements, in places where they
occur (as they can occur) with excellent effect, are intelligible as
something out of the common way, the expression of particularity,
or of that which is not of universal validity ; and may therefore act
as rhythmical passionate excitement or as rhythmical stimulus.
But where such particularity is not intended, or not given in the
structure of the phrase, there they seem mere lawlessness, a diseased
rhythm in. a healthy metre.
33. What has just been discussed is only connected with the
metrical conditions for the treatment of dissonance in the require-
ment by which the tied dissonance in general has a claim to a
metrically accented element. But it is not by any means insisted
upon, that the duration of the chord of preparation shall be equal
to the duration of the chord of dissonance ; only that the tied dis-
sonant note itself must not be longer than the consonant note
which prepares and precedes it. The resolution of the dissonance
SYNCOPATION 34S
may equally well follow later on, after other harmonic intermediate
notes ; for the law of the tie, which requires the length of the pre-
paring note to be equal to that of the note tied to it, is a rhyth-
mical metrical law on its own account : it is the same for consonant
and dissonant harmony, and is no more touched by particular
harmonic conditions than, in its turn, it imposes particular con-
ditions on them.
HARMONICAL CORRESPONDENCE OF THE SUC-
CESSION OF LINKED SEVENTH CHORDS WITH
THE METRICALLY SYNCOPATED SERIES.
34. Now if a syncopated series in union with the normal one
contains in each of its metrical members an accented element —
with alternate normal and syncopated accent — then in this view
every member of such a series should also be able to bear an ac-
cented dissonance-harmony. A succession of linked Seventh
chords, such as we have previously seen formed with three kinds of
passages, in the two first running on, in the third periodically inter-
rupted (* Harmony,' pars. 155-160), answers in harmonic sense to the
metrically syncopated series. Each of the dissonance-elements
immediately following one another is at once a harmonic First and
Second ; in the meaning of First (I) it is dissonance, in the mean-
ing of Second (II) it is resolution and preparation.
35. But the syncopated progressive series is metrically intelli-
gible only as periodic, as numbered in two, three, or four times ; and
the accents of higher orders will prevail in it above the accents
of members emphasised with equal strength. Hence particular
elements of the series are put forward as places of principal accent
and as principally suitable for the tied dissonance. Thus in the
346 METRICAL HARMONY— HARMONIC METRE
three-timed metre the second member as compared with the first, in
the four-timed the third member as compared with the first, or the
second with the third, will seem always of slight metrical weight ;
while, as .compared with the rest, the fourth in the four-timed, or
the third in the three-timed — being wholly without accent in the
formation by itself, and having only the syncopated accent — will
have the weakest emphasis of all, and therefore will be the least fit
to receive heavily emphasised dissonances. To these places will
be allotted by preference one of the Seventh-harmonies that may
stand even upon an unaccented time. Such are the Seventh chords
of the transposed system, and among them pre-eminently that
chord of the dominant Seventh which leads irresistibly to the tonic
close, and next to it the Seventh chord upon the Third of the
dominant.
36. But in the succession of linked Sevenths, in the sense of
syncopation, every metrical place may bear an accented dissonance.
The dissonance of suspension alone cannot acquiesce in removal
from normal into syncopated accent ; for its resolution must neces-
sarily fall upon the normally unaccented place.
37. And here it must be called to mind that the determination
of first and second metrical element, so often named, is repeated in
every order, and no other is conceivable, and therefore everything-
that has been said relating to syncopation and position of dis-
sonances is equally true for all orders. Thus in distribution by
bars the application is the same, whether made to members of the
simplest or of the most complicated uniform partition. The first
member of the two-part division, the first and third of the four-part,
the first, third, fifth, and seventh of the eight-part, and so on, receive
with respect to their order the meaning of a normal First ; the
uneven numbers marking the normal accents, and the even,.
2, 4, 6, 8 ••-, denoting what is without accent or has to be accented
by syncopation.
LINKED SEVENTH CHORDS 347
38. In all organic existence the mutual interaction of opposite
factors has always to be recognised in the notion of unity ; and
here too, when harmonic and metrical determinations are contrasted,
it should not escape us, that in essence both are really but one
and the same thing seen as determined from one side or from the
other, and that in the concrete whole the one ought only to be in-
tellectually distinguished, but not separated from the other. The
prepared chord of dissonance does not seek for a metrical place
that shall be in itself accented ; but itself determines the place on
which it stands as a metrical first or accented. Because, harmonic-
ally, it must have a second to follow it, therefore it must of itself
be*a first in time. Yet, as spoken metre has to unite logical and
metrical accent, and cannot let heavily emphasised syllables fall
upon light times ; so too will musical metre demand that first-
timed and heavily emphasised dissonance shall not be given a
place that should by the natural metrical order be unaccented.
APPENDIX.
A SHORT ANALYSIS OF HAUPTMANWS
TREATISE.
HAUPTMANN'S book is divided into three parts, treating respectively of
Harmony, of Metre, and of Harmony and Metre combined.
I. The first part begins with a short deduction of the triad from
acoustical notions. The triad is shown to be made up of three factors or
elements, whereof two are in their nature antithetical, and the third is such
as to bring about reconciliation of the other two, and to stand as a link
between them, so that the three elements stand together in a unity that
both contains and is made up of them. Also the three elements are not
utterly distinct and as it were disjoined from one another, but connected
organically and fused together.
These are the Root, the Fifth, and the Third ; and if regarded as gene-
rated successively (which yet in reality they are not) the Root is the original
unity that generates or gives rise to the triad ; but with respect to the
acoustical notions the Root — that is, the musical sound — is a derived or
generated unity.
Now the fundamental idea of the philosophy is that every notion — as
key, scale, Seventh chord, resolution, and so on — is made up after this
fashion ; i.e. that it possesses three elements involving an antithesis and a
reconcilement, and that one of the three elements is the Root from which the
other two, and so the whole construction, springs. This Hauptmann regards
as self-evident, and it is the basis of Hegelian metaphysics.
Thus from the triad posited as unity springs the key, a triad of triads ;
and from the key as unity springs the system of modulation, comprising the
tonic, the dominant, and the subdominant keys, or we may say modulation
in general.
350 APPENDIX
Again, the chord is of its nature simultaneous ; but the key can only be
manifested in succession. This antithesis of ' simultaneous ' and ' successive '
is identified with the antithesis of harmony and melody, which are opposed,
though one involves the other. As the chord represents simultaneous
sound, harmony, so the scale, the diatonic succession, represents successive
sound, .melody.
Now if successive sound, i.e. the diatonic interval, be taken as simul-
taneous, this is a contradiction, successive and simultaneous being anti-
thetical. And it is this contradiction that is the essence of dissonance, which
in this light, i.e. as involving a contradiction or unreality, is a /^^-notion.
The other two elements are the chord of preparation, from which springs
the dissonance, and the chord of resolution, which produces reconcile-
ment of dissonance with consonance, and so is the 7%/^-element.
Dissonance is treated in two kinds : the chord of suspension and the
Seventh chord. Both contain succession taken as simultaneous ; but in the
Seventh chord it is a succession of adjacent chords, in the chord of sus-
pension it is a succession only of adjacent notes. The dissonance of the
Seventh chord is the completer notion, and historically is later. It is the
complete antithesis of consonance, and only by its antithesis of dissonance
is the notion of consonance completed.
Another antithesis or opposition that occurs frequently is that of major
and minor. The minor triad is a major triad measured in the opposite
direction, an inverted major triad. Thus the notion of major and minor
in music corresponds to that general one of positive and negative ; as, e.g.,
when a straight line is reckoned positive if measured in one direction, and
negative if measured in the opposite direction. The major or positive is
the primitive notion and is presupposed in the negative or minor, of which
it is the positive premise.
There are also to be noticed two special phases of the key-system.
One is when a key tends to pass into its dominant key and yet not fully
accomplishes the transition ; when it takes as it were but half a step.
Then there is subsisting a key-system intermediate between that of the
original key and that of the dominant. For example, /Jf appearing in the
key of C major does not necessarily indicate a complete modulation into
the key of G major, which the chord D—f$ — A would indicate. In half-
closes upon the dominant it often happens that e.g. in the key of C major
an /sharp occurs without a modulation being effected into the dominant.
A SHORT ANALYSIS OF HAUPTMANWS TREATISE 351
Hauptmann names this the system stretching out or in extension^ and
several chords are to be referred to it.
In the other phase the key-system is regarded as passing, not into
another key-system but into itself, whereby it becomes inverted. The un-
inverted and the inverted states of the key-system are principally dis-
criminated by the Seventh chords that arise in them. The uninverted state
is the primitive one, in which the tonic triad lies between the subdominant
and the dominant, thus :
F— a— C— e— G— b^D.
Here the system is bounded on the two sides by F and D. To express
that the system passes into itself, the boundaries F and D must be brought
together in a chord. But then the system becomes
(e)_G— b— D | F— a— C— (e),
which is named the closed, otherwise the transposed or inverted system.
By closing its ends the system is in fact rendered circular :
/e\
C G
1 I'
a b
XF|D/
i.e. infinite in the Hegelian sense.
II. The second part treats of Metre and Rhythm, of which the first, the
measure, is compared to harmony, while rhythm, the kind of motion in
the measure, is analogous to melody.
The metrical unit is shown to be a two-parted unity. This, as two-
timed metre, is identified with the Octave (or Root) in harmony ; then three-
timed metre, which contains two overlapping metrical units, is identified
with the Fifth, and four-timed, which is the last of the uncompounded metres
and includes the other two, with the Third. The four-timed metre is the
metrical triad.
Next, accent is considered as attaching to the first member of the
metrical dual unit ; and hence are derived the various accentuations
possible in all metres, simple or compounded.
The notion of major and minor-is then shown to have its analogy in
352 APPENDIX
metre ; viz. the metre that begins with its first or accented member is
analogous to the triad that issues or is measured from its Root, while the
metre that begins with its unaccented member (as, e.g., a metre beginning
with an up beat) is analogous to the minor triad that issues downwards
from its Fifth.
The metre is the measure, but rhythm is the filling out of the measure.
The rhythm that fills out a metre may be equal-timed or unequal- timed. A
rhythm is equal-timed when the members of the dual unities that make it
up are equal in duration, as | J J | J J | . . . The equal-timed rhythm is
identified with the Octave.
The unequal-timed rhythm in which the least element is a whole
followed by its half, e.g. J J* , is identified with' the Fifth ; and the unequal-
timed rhythm in which the least element is r"T-j ? such that the last quaver
is the half of the whole n that nas g°ne before, but also the whole to its
half ^ that immediately precedes, and so at once half and whole, is iden-
tified with the Third.
These three elements, the equal-timed and the two unequal-timed
divisions, together constitute the determination of rhythm. The two last
have also their minor forms : j* | J and rj I J •
III. The last part of the book considers the union of Metre and
Harmony ; that is, harmony and melody in concrete combination with
metre and rhythm. In this the few general principles that can be laid
down regard only harmony and metre, for these elements are more fixed
and determinate than melody and rhythm. Thus the metrical-position of
dissonance is discussed, both of suspension and of the Seventh chord.
Also continued accent by syncopation is shown Jo correspond with the
series of linked Seventh chords.
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