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HARMONIC
MATERIALS
OF
MODERN MUSIC
HARMONIC
MATERIALS
OF
MODERN MUSIC
Resources of the Tempered Scale
Ilowar3™™lfansoir
DIRECTOR
EASTMAN SCHOOL OF MUSIC
UNIVERSITY OF ROCHESTER
New York
APPLETON-CENTURY-CROFTS, Inc.
n.
Copyright © 1960 by
APPLETON-CENTURY-CROFTS, INC.
610-1
All rights reserved. This hook, or parts
thereof, must not he reproduced in any
form without permission of the publisher.
Library of Congress Card Number: 58-8138
PRINTED IN THE UNITED STATES OF AMERICA
MUSIC LIBRARY.
'v\t:
H'^
To my dear wife, Peggie,
who loves music but does not
entirely approve of the twelve-tone scale,
this book is affectionately dedicated.
Preface
This volume represents the results of over a quarter-century of
study of the problems of the relationships of tones. The conviction
that there is a need for such a basic text has come from the
author's experience as a teacher of composition, an experience
which has extended over a period of more than thirty-five years.
It has developed in an effort to aid gifted young composers grop-
ing in the vast unchartered maze of harmonic and melodic
possibilities, hunting for a new "lost chord," and searching for an
expressive vocabulary which would reach out into new fields and
at the same time satisfy their own esthetic desires.
How can the young composer be guided in his search for the
far horizons? Historically, the training of the composer has been
largely a matter of apprenticeship and imitation; technic passed
on from master to pupil undergoing, for the most part, gradual
change, expansion, liberation, but, at certain points in history,
radical change and revolution. During the more placid days the
apprenticeship philosophy— which is in effect a study of styles-
was practical and efficient. Today, although still enormously im-
portant to the development of musical understanding, it does not,
hy itself, give the young composer the help he needs. He might,
indeed, learn to write in the styles of Palestrina, Purcell, Bach,
Beethoven, Wagner, Debussy, Schoenberg, and Stravinsky and
still have difficulty in coming to grips with the problem of his
own creative development. He needs a guidance which is more
basic, more concerned with a study of the material of the art and
vn
PREFACE
less with the manner of its use, although the two can never
be separated.
This universality of concept demands, therefore, an approach
which is radical and even revolutionary in its implications. The
author has attempted to present here such a technic in the field
of tonal relationship. Because of the complexity of the task, the
scope of the work is limited to the study of the relationship of
tones in melody or harmony without reference to the highly im-
portant element of rhythm. This is not meant to assign a lesser
importance to the rhythmic element. It rather recognizes the
practical necessity of isolating the problems of tonal relationship
and investigating them with the greatest thoroughness if the
composer is to develop a firm grasp of his tonal vocabulary.
I hope that this volume may serve the composer in much the
same way that a dictionary or thesaurus serves the author. It is
not possible to bring to the definition of musical sound the same
exactness which one may expect in the definition of a word. It is
possible to explain the derivation of a sonority, to analyze its
component parts, and describe its position in the tonal cosmos.
In this wav the young composer may be made more aware of the
whole tonal vocabulary; he mav be made more sensitive to the
subtleties of tone fusion; more conscious of the tonal alchemy by
which a master may, with the addition of one note, transform
and illuminate an entire passage. At the same time, it should
give to the young composer a greater confidence, a surer grasp of
his material and a valid means of self-criticism of the logic and
consistency of his expression.
It would not seem necessary to explain that this is not a
"method" of composition, and yet in these days of systems it
may be wise to emphasize it. The most complete knowledge of
tonal material cannot create a composer any more than the
memorizing of Webster's dictionary can produce a dramatist or
poet. Music is, or should be, a means of communication, a vehicle
for the expression of the inspiration of the composer. Without that
inspiration, without the need to communicate, without— in other
viii
PREFACE
words— the creative spirit itself, the greatest knowledge will avail
nothing. The creative spirit must, however, have a medium in
which to express itself, a vocabulary capable of projecting with
the utmost accuracy and sensitivity those feelings which seek
expression. It is my hope that this volume may assist the young
composer in developing his own vocabulary so that his creative
gift may express itself with that simplicity, clarity, and consistency
which is the mark of all great music.
Since this text differs radically from conventional texts on "har-
mony," it may be helpful to point out the basic differences
together with the reason for those diflFerences.
Traditional theory, based on the harmonic technics of the
seventeenth, eighteenth, and nineteenth centuries, has distinct
limitations when applied to the music of the twentieth— or even
the late nineteenth— century. Although traditional harmonic
theory recognizes the twelve-tone equally tempered scale as an
underlying basis, its fundamental scales are actually the seven-
tone major and minor scales; and the only chords which it admits
are those consisting of superimposed thirds within these scales
together with their "chromatic" alterations. The many other com-
binations of tones that occur in traditional music are accounted
for as modifications of these chords by means of "non-harmonic"
tones, and no further attempt is made to analyze or classify
these combinations.
This means that traditional harmony systematizes only a very
small proportion of all the possibilities of the twelve-tones and
leaves all the rest in a state of chaos. In contemporary music, on
the other hand, many other scales are used, in addition to the
major and minor scales, and intervals other than thirds are used
in constructing chords.
I have, therefore, attempted to analyze all of the possibilities
of the twelve-tone scale as comprehensively and as thoroughly as
traditional harmony has analyzed the much smaller number of
chords it covers. This vast and bewildering mass of material is
classified and thus reduced to comprehensible and logical order
IX
PREFACE
chiefly by four devices: interval analysis, projection, involution,
and complementary scales.
Interval analysis is explained in Chapter 2 and applied through-
out. All interval relationship is reduced to six basic categories : the
perfect fifth, the minor second, the major second, the minor third,
the major third, and the tritone, each— except the tritone— con-
sidered in both its relationship above and below the initial tone.
This implies a radical departure from the classic theories of inter-
vals, their terminology, and their use in chord and scale construc-
tion. Most of Western music has for centuries been based on the
perfect-fifth category. Important as this relationship has been, it
should not be assumed that music based on other relationships
cannot be equally valid, as I believe the examples will show.
Projection means the construction of scales or chords by any
logical and consistent process of addition and repetition. Several
types of projection are employed in different sections of the book.
If a series of specified intervals, arranged in a definite ascending
order, is compared with a similar series arranged in descending
order, it is found that there is a clear structural relationship
between them. The second series is referred to here as the
involution of the first. (The term inversion would seem to be more
accurate, since the process is literally the "turning upsidedown"
of the original chord or scale. It was felt, however, that confusion
might result because of the traditional use of the term inversion. )
The relation of any sonority and its involution is discussed in
Chapter 3, and extensively employed later on.
Complementary scales refer to the relationship between any
series of tones selected from the twelve-tones and the other tones
which are omitted from the series. They are discussed in Parts V
and VI. This theory, which is perhaps the most important— and
also the most radical— contribution of the text, is based on the
fact that every combination of tones, from two-tone to six-tone,
has its complementary scale composed of similar proportions of
the same intervals. If consistency of harmonic-melodic expression
is important in musical creation, this theory should bear the most
PREFACE
intensive study, for it sets up a basis for the logical expansion of
tonal ideas once the germinating concept has been decided upon
in the mind of the composer.
The chart at the end of the text presents graphically the relation-
ship of all of the combinations possible in the twelve-tone system,
from two-tone intervals to their complementary ten-tone scales.
I must reiterate my passionate plea that this text not be con-
sidered a "method" nor a "system." It is, rather, a compendium
of harmonic-melodic material. Since it is inclusive of all of the
basic relationships within the twelve-tones, it is hardly likely that
any composer would in his lifetime use all, or even a large part,
of the material studied. Each composer will, rather, use only
those portions which appeal to his own esthetic taste and which
contribute to his own creative needs. Complexity is no guarantee
of excellence, and a smaller and simpler vocabulary used with
sensitivity and conviction may produce the greatest music.
Although this text was written primarily for the composer, my
colleagues have felt that it would be useful as a guide to the
analysis of contemporary music. If it is used by the student of
theory rather than by the composer, I would suggest a different
mode of procedure, namely, that the student study carefully Parts
I and II, Chapters I to 16, without undertaking the creative
exercises— although if there is sufficient time the creative exercises
will enlighten and inform the theorist as well as the composer.
During the first part of this study he should try to find in the
works of contemporary composers examples of the various hexad
formations discussed. He will not find them in great abundance,
since contemporary composers have not written compositions
primarily to illustrate the hexad formations of this text! However,
when he masters the theory of complementary scales, he will have
at his disposal an analytical technic which will enable him to
analyze factually any passage or phrase written in the twelve-tone
equally tempered scale.
H. H.
Rochester, New York
XI
Acknowledgments
The author wishes to acknowledge his deep debt of gratitude
to Professor Herbert Inch of Hunter College for his many help-
ful suggestions and for his meticulous reading of a difficult manu-
script, and to his colleagues of the Eastman School of Music
faculty, Wayne Barlow, Allen Irvine McHose, Charles Riker,
and Robert Sutton, for valuable criticism. His appreciation is
also extended to Clarence Hall for the duplication of the chart,
to Carl A. Rosenthal for his painstaking reproduction of the
examples, and to Mary Louise Creegan and Janice Daggett for
their devoted help in the preparation of the manuscript.
His warm thanks go to the various music publishers for their
generous permission to quote from copyrighted works and
finally and especially to Appleton-Century-Crofts for their
co-operation and for their great patience.
Finally, my devoted thanks go to my hundreds of composition
students who have borne with me so loyally all these many years.
H. H.
Contents
Preface
vu
1. Equal Temperament 1
2. The Analysis of Intervals 7
3. The Theory of Involution 17
Part I. THE SIX BASIC TONAL SERIES
4. Projection of the Perfect Fifth 27
5. Harmonic-Melodic Material of the Perfect-Fifth Hexad 40
6. Modal Modulation 56
7. Key Modulation 60
8. Projection of the Minor Second 65
9. Projection of the Major Second 77
10. Projection of the Major Second Beyond the Six-Tone Series 90
11. Projection of the Minor Third 97
12. Involution of the Six-Tone Minor-Third Projection 110
13. Projection of the Minor Third Beyond the Six-Tone Series 118
14. Projection of the Major Third 123
15. Projection of the Major Third Beyond the Six-Tone Series 132
16. Recapitulation of the Triad Forms 136
17. Projection of the Tritone 139
18. Projection of the Perfect-Fifth-Tritone Series Beyond
Six Tones 148
19. The pmn-Tritone Projection 151
20. Involution of the pmn-Tritone Projection 158
21. Recapitulation of the Tetrad Forms 161
xiii
CONTENTS
Part II. CONSTRUCTION OF HEXADS
BY THE SUPERPOSITION OF TRIAD FORMS
22. Projection of the Triad pmn 167
23. Projection of the Triad pns 172
24. Projection of the Triad pmd 177
25. Projection of the Triad mnd 182
26. Projection of the Triad nsd 187
Part III. SIX-TONE SCALES FORMED BY THE
SIMULTANEOUS PROJECTION OF TWO INTERVALS
27. Simultaneous Projection of the Minor Third and Perfect Fifth 195
28. Simultaneous Projection of the Minor Third and Major Third 200
29. Simultaneous Projection of the Minor Third and Major
Second 204
30. Simultaneous Projection of the Minor Third and Minor
Second 207
31. Simultaneous Projection of the Perfect Fifth and Major Third 211
32. Simultaneous Projection of the Major Third and Minor
Second 215
33. Simultaneous Projection of the Perfect Fifth and Minor
Second 219
Part IV. PROJECTION BY INVOLUTION AND AT
FOREIGN INTERVALS
34. Projection by Involution 225
35. Major-Second Hexads with Foreign Tone 232
36. Projection of Triads at Foreign Intervals 236
37. Recapitulation of Pentad Forms 241
Part V. THE THEORY OF COMPLEMENTARY SONORITIES
38. The Complementary Hexad 249
39. The Hexad "Quartets" 254
xiv
CONTENTS
Part VI. COMPLEMENTARY SCALES
40. Expansion of the Complementary-Scale Theory 263
4L Projection of the Six Basic Series with Their Com-
plementary Sonorities 274
42. Projection of the Triad Forms with Their Complementary
Sonorities 285
43. The pmn-Tritone Projection with Its Complementary
Sonorities 294
44. Projection of Two Similar Intervals at a Foreign Interval
with Complementary Sonorities 298
45. Simultaneous Projection of Intervals with Their
Complementary Sonorities 303
46. Projection by Involution with Complementary Sonorities 314
47. The "Maverick" Sonority 331
48. Vertical Projection by Involution and Complementary
Relationship 335
49. Relationship of Tones in Equal Temperament 346
50. Translation of Symbolism into Sound 356
Appendix: Symmetrical Twelve-Tone Forms 373
Index 377
Chart: The Projection and Interrelation of Sonorities in
Equal Temperament inside back cover
XV
HARMONIC
MATERIALS
OF
MODERN MUSIC
1
Equal Temperament
Since the subject of our study is the analysis and relationship
of all of the possible sonorities contained in the twelve tones of
the equally tempered chromatic scale, in both their melodic and
harmonic implications, our first task is to explain the reasons for
basing our study upon that scale. There are two primary reasons.
The first is that a study confined to equal temperament is, al-
though complex, a -finite study, whereas a study of the theo-
retical possibilities within just intonation would be infinite.
A simple example will illustrate this point. If we construct a
major third, E, above C, and superimpose a second major third,
G#, above E, we produce the sonority C-E-G#i Now if we
superimpose yet another major third above the GJj:, we reach the
tone B#. In equal temperament, however, B# is the enharmonic
equivalent of C, and the four-tone sonority C-E-G#-B# is actually
the three tones C-E-Gfl: with the lower tone, C, duplicated at the
octave. In just intonation, on the contrary, B# would not be
the equivalent of C. A projection of major thirds above C in
just intonation would therefore approach infinity.
The second reason is a- corollary of the first. Because the
pitches possible in just intonation approach infinity, just
intonation is not a practical possibility for keyboard instru-
ments or for keyed and valve instruments of the woodwind and
brass families. Just intonation would be possible for stringed
instruments, voices, and one brass instrument, the slide trom-
bone. However, since much of our music is concerted, using all
HARMONIC MATERIALS OF MODERN MUSIC
o£ these resources simultaneously, and since it is unlikely that
keyboard, keyed, and valve instruments will be done away with,
at least within the generation of living composers, the system
of equal temperament is the logical basis for our study.
Another advantage of equal temperament is the greater
simplicity possible in the symbolism of the pitches involved.
Because enharmonic equivalents indicate the same pitch, it is
possible to concentrate upon the sound of the sonority rather
than upon the complexity of its spelling.
Referring again to the example already cited, if we were to
continue to superimpose major thirds in just intonation we
would soon find ourselves involved in endless complexity. The
major third above BJj: would become D double-sharp; the major
third above D double-sharp would become F triple-sharp; the
next major third, A triple-sharp; and so on. In equal tempera-
ment, after the first three tones have been notated— C-E-Gjj:— the
G# is considered the equivalent of Aj^ and the succeeding major
thirds become C-E-Gfl:-C, merely octave duplicates of the
first three.
Example 1-1
Pure Temperament Equal Temperament
"%! ] ip")
This point of view has the advantage of freeing the composer
from certain inhibiting preoccupations with academic symboliza-
tion as such. For the composer, the important matter is the
sound of the notes, not their "spelling." For example, the sonority
G-B-D-F sounds like a dominant seventh chord whether it is
spelled G-B-D-F, G-B-D-E#, G-B-CX-E#, G-Cb-C-:^-F, or in
some other manner.
The equally tempered twelve-tone scale may be conveniently
thought of as a circle, and any point on the circumference may
be considered as representing any tone and/or its octave. This
EQUAL TEMPERAMENT
circumference may then be divided into twelve equal parts, each
representing a minor second, or half-step. Or, with equal validity,
each of the twelve parts may represent the interval of a perfect
fifth, since the superposition of twelve perfect fifths also
embraces all of the twelve tones of the chromatic scale— as in the
familiar "key-circle." We shall find the latter diagram particularly
useful. Beginning on C and superimposing twelve minor seconds
or twelve perfect fifths clockwise around the circle, we complete
the circle at BJf, which in equal temperament has the same pitch
as C. Similarly, the pitch names of C# and D^, D# and Ej^, and
so forth, are interchangeable.
Example 1-2
GttlAb)
D« (Eb
MK (Bb)
The term sonority is used in this book to cover the entire field
of tone relationship, whether in terms of melody or of harmony.
When we speak of G-B-D-F, for example, we mean the relation-
ship of those tones used either as tones of a melody or of a
harmony. This may seem to indicate a too easy fusion of melody
and harmony, and yet the problems of tone relationship are
essentially the same. Most listeners would agree that the sonority
in Example l-3a is a dissonant, or "harsh," combination of tones
when sounded together. The same efl^ect of dissonance, however,
persists in our aural memory if the tones are sounded con-
secutively, as in Example l-3b:
HARMONIC MATERIALS OF MODERN MUSIC
Example 1-3
(fl)
i
^
The first problem in the analysis of a sonority is the analysis
of its component parts. A sonority sounds as it does primarily
because of the relative degree of consonance and dissonance of
its elements, the position and order of those elements in relation
to the tones of the harmonic series, the degree of acoustical
clarity in terms of the doubling of tones, timbre of the orchestra-
tion, and the like. It is further affected by the environment in
which the sonority is placed and by the manner in which
experience has conditioned the ears of the listener.
Of these factors, the first would seem to be basic. For example,
the most important aural fact about the familiar sonority of the
dominant seventh is that it contains a greater number of minor
thirds than of any other interval. It contains also the consonances
of the perfect fifth and the major third and the mild dissonances
of the minor seventh and the tritone. This is, so to speak, the
chemical analysis of the sonority.
Example 1-4
f
Minor thirds Perfect fifth Mojor third Minor seventh Tritone
It is of paramount importance to the composer, since the
composer should both love and understand the beauty of sound.
He should "savor" sound as the poet savors words and the
painter form and color. Lacking this sensitivity to sound, the
composer is not a composer at all, even though he may be both
a scholar and a craftsman.
EQUAL TEMPERAMENT
This does not imply a lack of importance of the secondary
analyses already referred to. The historic position of a sonority
in various styles and periods, its function in tonality— where
tonality is implied— and the like are important. Such multiple
analyses strengthen the young composer's grasp of his material,
providing always that they do not obscure the fundamental
analysis of the sound as sound.
Referring again to the sonority G-B-D-F, we should note its
historic position in the counterpoint of the sixteenth century and
its harmonic position in the tonality of the seventeenth,
eighteenth, and nineteenth centuries, but we should first of all
observe its construction, the elements of which it is formed. All
of these analyses are important and contribute to an understand-
ing of harmonic and melodic vocabulary.
As another example of multiple analysis, let us take the familiar
chord C-E-G-B. It contains two perfect fifths, two major thirds,
one minor third, and one major seventh.
Example 1-5
*
Perfect fifths Mojor thirds Minor third Major seventh
It may be considered as the combination of two perfect fifths at
the interval of the major third; two major thirds at the perfect
fifth; or perhaps as the combination of the major triad C-E-G
and the minor triad E-G-B or the triads* C-G-B and C-E-B:
Example 1-6
ofijiij^ijiii
*The word triad is used to mean any three-tone chord.
HARMONIC MATERIALS OF MODERN MUSIC
Historically, it represents one of the important dissonant sonori-
ties of the baroque and classic periods. Its function in tonality
may be as the subdominant or tonic seventh of the major scale,
the mediant or submediant seventh of the "natural" minor scale,
and so forth.
Using the pattern of analysis employed in Examples 1-4, 1-5,
and 1-6, analyze as completely as possible the following sonorities :
i
Example 1-7
4. 5. e. 7.
±
fit
i
9.
10.
ft
ji8 ijia^ 1%^
=^
Iia^itftt«^i«sp
The Analysis of Intervals
In order again to reduce a problem of theoretically infinite
proportions to a finite problem, an additional device is suggested.
Let us take as an example the intervallic analysis of the major
triad C-E-G:
Example 2-1
Perfect fifth Major third Minor third
This triad is commonly described in conventional analysis as a
combination of a perfect fifth and a major third above the lowest
or "generating" tone of the triad. It is obvious, however, that this
analysis is incomplete, since it omits the concomitant interval of
the minor third between E and G. This completes the analysis
as long as the triad is in the simple form represented above. If,
however, the chord is present in a form in which there are many
doublings in several octaves, such a complete analysis becomes
more complex.
If we examine the scoring of the final chord in Death and
Transfiguration by Richard Strauss we find a sixteen -tone chord:
Example 2-2
:i
*
*
^m
HARMONIC MATERIALS OF MODERN MUSIC
These sixteen tones combine to form one hundred and twenty
different intervals. The relationship between C and G is repre-
sented not only by the intervals
Example 2-3
eta
■<^ -o- ■«■
a o ^
but also by the intervals
Example 2-4
i
^
5
»^^ f^ »^ *^'*
-o — © — © — o — ^
in which case we commonly call the second relationship the
"inversion" of the first. The same is true of the relation of C to
E and E to G.
However, the composite of all of the tones still gives the
impression of the C major triad in spite of the complexity of
doubling. In other words, the interval C to G performs the same
function in the sonority regardless of the manner of the doubling
of voices.
The similarity of an interval and its inversion may be further
illustrated if one refers again to the arrangement of the twelve-
tone scale in the circle of fifths:
8
THE ANALYSIS OF INTERVALS
Example 2-5
Here it will be seen that C has two perfect-fifth relationships,
C to G and C to F; the one, C to G, proceeding clockwise
(ascending) and the other, C to F, proceeding counterclockwise
(descending). In the same manner, C has two major-second
relationships, C to D and C to B^; two major-sixth relationships,
C to A and C to E^; two major-third relationships, C to E and
C to A\); and two major-seventh relationships, C to B and C to
Dt>. It has only one tritone relationship, C up to F#, or C down
to G\). It will be helpful in ,our analysis if we use only one
symbol to represent both the interval under consideration and
its inversion. This is not meant to imply that the interval and its
inversion are the same, but rather that they perform the same
function in a sonority.
Proceeding on this theory, we shall choose the symbol p to
represent the relationship of the perfect fifth above or below the
first tone, even though when the lower tone of each of the two
intervals is raised an octave the relationship becomes actually
a perfect fourth:
harmonic materials of modern music
Example 2-6
#
Perfect fifth p Perfect Perfect
- fifth fourth
The symbolization is arbitrary, the letter p being chosen because
it connotes the designation "perfect," which apphes to both
intervals.
The major third above or below the given tojie will be desig-
nated by the letter m:
Example 2-7
^
^^
Major third, m
(or minor sixth)
The minor third above or below the given tone will be
represented by the letter n:
Example 2-8
i
B^
Minor third, n
(or major sixth)
the major second above or below, by s:
i
Example 2-9
(i'ji)
t»to tib<^
Major second, s
(or minor seventh)
the dissonant minor second by d:
Example 2-10
Minor second, d
(or mojor seventh)
10
THE ANALYSIS OF INTERVALS
and the tritone by t:
Example 2-11
(bo'i
M
^
*^ Augmented fourth,^
(or diminished fifth)
(Tritone)
The letters pmn, therefore, represent intervals commonly
considered consonant, whereas the letters sdt represent the inter-
vals commonly considered dissonant. The symbol pmn, sdt'*
would therefore represent a sonority which contained one perfect
fifth or its inversion, the perfect fourth; one major third or its
inversion, the minor sixth; one minor third or its inversion, the
major sixth; one major second or its inversion, the minor seventh;
one minor second or its inversion, the major seventh; and one
augmented fourth or its inversion, the diminished fifth; the three
symbols at tiie left of the comma representing consonances, those
at the right representing dissonances. A sonority represented,
for example, by the symbol sd^, indicating a triad composed of
one major second and two minor seconds, would be recognized
as a highly dissonant sound, while the symbol pmn would indicate
a consonant sound.
The complexity of the analysis will depend, obviously, upon
the number of diflFerent tones present in the sonority. A three-
tone sonority such as C-E-G would contain the three intervals
C to E, C to G, and E to G. A four-tone sonority would contain
3+2+1 or 6 intervals; a five-tone sonority, 4+3+2+1 or 10 in-
tervals, and so on.
Since we are considering all tones in equal temperament, our
task is somewhat simplified. C to D#, for example, represents
the same sound as the interval C to E^i; and since the sound is
" For the sake of uniformity, analyses of sonorities will list the constituent inter-
vals in this order.
11
HARMONIC MATERIALS OF MODERN MUSIC
the same, they would both be represented by the single symbol
n. A table of intervals with their classification would, therefore,
be as follows:
C-G (orG-C),B#-G, C-F^K<,etc. = p
C-E (or E-C), B#-E, C-Fb, BJf-Fb, etc. = m
C-Eb (orEb-C),C-Dif, B#-Eb, etc. = n
C-D (or D-C), Bif-D, C-Ebb, etc. = s
C-Db(orDb-C),C-C#,B#-Db,etc. = d
C-F# (or F#-C), C-Gb, B#-Gb, etc. = t
xo
efc.
Example 2-12
it\^ ^° tf^g'1 '^» r i^i »j|o ^^
SE
t^
i
ife
efc.
etc
£
^^g
bo bo^^'^'
^^
i--*0 fv^
»
m Qgyi
Ed -XT
*
¥^
For example, the augmented triad C-E-G# contains the major
third C to E; the major third E to G#, and the interval C to G^.
Since, however, C to G# sounds like C to Ab, the inversion of
which is Ab to C— also a major third— the designation of the
augmented triad would be three major thirds, or m^. A diagram
of these three notes in equal temperament quickly illustrates the
validity of this analysis. The joining of the three notes C-E-G#
(Ab) forms an equilateral triangle— a triangle having three equal
sides and angles:
Example 2-13
B /-^ ' -\ CJt
G<t (Ab)
12
THE ANALYSIS OF INTERVALS
It is, of course, a figure which has the same form regardless
of which side is used as its base:
Example 2-14
Similarly the augmented triad sounds the same regardless
of which of the three tones is the lowest:
G# • B#(C) E
E G# C
C E G#(Ab)
One final illustration will indicate the value of this technique
of analysis. Let us consider the following complex-looking sonor-
ity in the light of conventional academic analysis:
Example 2-15
f
The chord contains six notes and therefore has 5+4+3+2
+1, or 15 intervals, as follows:
C-D# and Ab-B
C-E and G-B
C-G and E-B
C-Ab and D#-B
augmented seconds
major thirds
perfect fifths
minor sixths
13
HARMONIC MATERIALS OF MODERN MUSIC
C-B = major seventh
D#-E and G-A^ = minor seconds
Djf-G and E-Aj^ = diminished fourths
D#-Ab = double-diminished fifth
E-G = minor third
However, in the new analysis it converts itself into only four
types of intervals, or their inversions, as follows:
3 perfect fifths: C-G, E-B, and Ab-Eb (Dif ).
6 major thirds: C-E, Eb (D^f )-G, E-G# (Ab), G-B, Ab-C,
and B-D#.
3 minor thirds: C-Eb (D#), E-G, and G# (Ab)-B.
3 minor seconds: DJf-E, G-Ab, and B-C.
The description is, therefore, p^m^n^d^.
Example 2-16
i
Perfect fifths
ii
Mojor thirds
o \fv:w^
vObB l^» ^
^ b8(tfo)tlit^^
i
Minor thirds
Minor seconds
b^||o)t8 tfS^^^ Ijl^ ^'^'
=^a=
A diagram will indicate the essential simplicity of the structure:
14
Example 2-17
G<»(Ab)
THE ANALYSIS OF INTERVALS
It has been my experience that although the young composer
who has been thoroughly grounded in academic terminology
may at first be confused by this simplification, he quickly
embraces the new analysis because it conforms directly to his
own aural impression.
In analyzing intervals, the student will find it practical to form
the habit of "measuring" all intervals in terms of the "distance"
in half-steps between the two tones. Seven half-steps (up or
down), for example, will be designated by the symbol p; four
half-steps by the symbol m; three half-steps by the symbol n, and
so forth, regardless of the spelling of the tones which form
the interval:
perfect fifth
7 half
-Steps
V
perfect fourth
5
If
II
major third
4
II
II
m
minor sixth
8
II
II
minor third
3
II
II
n
major sixth
9
II
II
major second
2
II
II
s
minor seventh
10
II
II
minor second
1
II
II
d
major seventh
11
II
II
augmented fourth
6
II
II
t
diminished fifth
6
II
II
Example 2-18
p
m
n
9f=
^4S=
-^c^^ 44«
i."*
— A?—
4.^
#^
Perfect
fifth
Perfect Major
fourth third
Minor
sixth
1
^inor
third
Major
sixth
15
HARMONIC MATERIALS OF MODERN MUSIC
s d t
=f^rt^
^
2T*t°
w
=15=
'^XXt
Minor Major
second seventh
^M
«»i
Major Minor
second seventh
Augmented Diminished
fourth fifth
In speaking of sonorities we shall apparently make little
distinction between tones used successively in a melody and
tones used simultaneously in a harmony. It is true that the
addition of the element of rhythm, the indispensable adjunct of
melody, with its varying degrees of emphasis upon individual
notes by the devices of time length, stress of accent, and the like,
creates both great and subtle variance from the sonority played
as a "block" of sound. Nevertheless, the basic relationship is the
same. A melody may grow out of a sonority or a melody may
itself be a sonority.
Analyze the following sonorities in the same manner employed
In Examples 2-15 and 2-16, pages 13 and 14, giving first the
conventional interval analysis, and second the simplified analysis:
Example 2-19
i
^ jt# I ^i ^^
^1^
w
S3S:
^»S^
#
3
^S^
^S
^
r^
'^BT
Repeat the same process with the chords in Example 1-7, page 6.
16
The Theory of Involution
Reference has already been made to the two-directional
aspect of musical relationship, that is, the relationship "up" and
"down" in terms of pitch, or the relationship in clockwise or
counterclockwise rotation on the circle already referred to. It
will be readily apparent that every sonority in music has a
counterpart obtained by taking the inverse ratio of the original
sonority. The projection dovon from the lowest tone of a given
chord, using the same intervals in the order of their occurrence
in the given chord, we may call the involution of the given
chord. This counterpart is, so to speak, a "mirror" of the
original. For example, the major triad C-E-G is formed by the
projection of a major third and a perfect fifth above C. However,
if this same relationship is projected downward, the interval C
to E has as its counterpart the interval |C to Aj^; and the interval
C to G has as its counterpart [C to F.
Example 3-1
B
17
HARMONIC MATERIALS OF MODERN MUSIC
It will be noted that the involution of a sonority always contains
the same intervals found in the original sonority.
There are three types of involutions: simple, isometric, and
enharmonic.
In simple involution, the involuted chord differs in sound from
the given chord. Let us take, for example, the major triad C-E-G,
which is formed by the projection of a major third and a perfect
fifth above C. Its involution, formed by the projection downward
from C of a major third and a perfect fifth, is the minor triad
'[F-A^-C. The major triad C-E-G and its involution, the minor
triad ^F-A^-C, each contain a perfect fifth, a major third, and a
minor third, and can be represented by the symbols pmn.
Example 3-2
i
^m
^
In the second type of involution, which we may call isometric
involution, the involuted sonority has the same kind of sound as
the original sonority. For example, the tetrad C-E-G-B has as its
involution jDb-F-Ab-C.
Example 3-3
«
^
18
THE THEORY OF INVOLUTION
Each of these is a major seventh chord, containing two perfect
fifths, two major thirds, a minor third, and a major seventh, and
can be characterized by the symbols p^irrnd, the exponents in
this instance representing two perfect fifths and two major thirds.
In the third type, enharmonic involution, the invohited sonor-
ity and the original sonority contain the same tones in different
octaves (except for one common tone). For example, the
augmented triad C-E-G# involutes to produce the augmented
triad ^F^-Ab-C, F^ and A^ being the equal-temperament equiva-
lents of E and G#. Another common example of enharmonic
involution is the diminished seventh chord :
Example 3-4
« 3 .CK.
m
I
All sonorities which are formed by the combination of a
sonority with its involution are isometric sonorities, since they
will have the same order of intervals whether considered "up"
or "down," clockwise or counterclockwise. We have already seen
that the involution of the triad C-E-G is jC-Ab-F. The two
together produce the sonority F3Ab4C4E3G, which has the same
order of intervals upward or downward.*
*The numbers indicate the number of half-steps between the tones of the
sonority.
19
HARMONIC MATERIALS OF MODERN MUSIC
If the tone E of the triad C4E3G is used as the axis of involu-
tion, a diflFerent five-tone sonority will result, since the involution
of E3G5C will be J^EsC^gGJ, forming together the sonority
GJgCfllsEaGgC. If the tone G is used as the axis of involution, the
involution of G5C4E will be J,G5D4Bb, forming together the
sonority Bb4D5G5C4E. These resultant sonorities will all be seen
to be isometric in structure. ( See Note, page 24. )
Example 3-5
) (2) (3)
"^^", 44 ■ '^%j|§ii r 33 . ' bS-i|-2"3 3 2
If two tones are used as the axes of involution, the result will
be a four-tone isometric sonority:
Example 3-6
5=^
313 343 434
In the first of the above examples, C and G constitute the
"double axis"; in the second C and E; and in the third E and G.
The discussion of involution up to this point does not differ
greatly from the "mirror" principle of earlier theorists, whereby
"new" chords were formed by "mirroring" a familiar chord and
combining the "mirrored" or involuted chord with the original.
At this point, however, we shall expand the principle to the
point where it becomes a basic part of our theory. When a major
triad is involuted— as in Example 3-2— deriving the minor triad
as the "mirrored" image of the major triad seems to place the
minor triad in a position of secondary importance, as the
reflected image of the major triad.
In the principle of involution presented here, no such second-
ary importance is intended; for if the minor triad is the reflected
image of the major triad, it is equally true that the major triad is
20
THE THEORY OF INVOLUTION
also the reflected image of the minor triad. For example, the
involution of the major triad C4E3G is the minor triad |C4Ab3F,
and the involution of the minor triad C3E[74G is the major triad
jCaA^F.
In order to avoid any implication that the involution is, so to
speak, a less important sonority, we shall in analyzing the sonori-
ties construct both the first sonority and its involution upward
by the simple process of reversing the intervallic order. For
example, if the first triad is C4E3G the involution of this triad
will be any triad which has the same order of half-steps in
reverse, for example F3Ab4C, the comparison being obviously 4-3
versus 3-4.
In this sense, therefore, the involution of a major triad can be
considered to be any minor triad whether or not there is an axis
of involution present.
In Example 3-7, therefore, the B minor, B^ minor, G$ minor,
F# minor, E^ minor, and D minor triads are all considered as
possible involutions of the C major triad, although there is no
axis of involution. When the C major triad is combined with any
one of them, the resultant formation is a six-tone isometric
sonority.
Example 3-7
■m Ha
m=^
b<i fc^
« o
3 bo "
ffi
Hi^ "^
222 I
OgP
^=
m
^
^=j^
2 13 12
3 13 13
3 2 12 3
Wt
>..ii..i'«ti'
btxhc^feo-
m
=»=si
i^
3 12 13
M\i "°
2 2 3 2 2
21
HARMONIC MATERIALS OF MODERN MUSIC
Note that the combination of any sonority with its involuted
form always produces an isometric sonority, that is, a sonority
which can be arranged in such a manner that its foraiation of
intervals is the same whether thought up or down. For example,
the first combination in Example 3-7, if begun on B, has the
configuration BiC2D2E2FJj:iG, which is the same whether con-
sidered from B to G or from G to B,
The second combination, C major and B^ minor, must be
begun on Bj^ or E to make its isometric character clear:
BbsCiDbsEiF^G or E,F,GsB\),C,Dh-
The isometric character of the third combination, C major and
G# minor, is clear regardless of the tone with which we begin:
C3D#iE3GiG#3B; D^,E,G,GJi^,B,C, etc.
If, however, for the sake of comparison, we combine a major
triad with another major triad, for example, the combination of
C major with D major, the resultant formation is not isometric,
since it is impossible to arrange these tones so that the configura-
tion is the same up or down:
CsD^E^FSiG^A; D^EsFliG^AsC; E^FSiG^AsC^D;
FitiGsAgCsDsE; G2A3C2D2E2F#; A.C^D^E^Fi.G.
There is one more phenomenon which should be noted. There
are a few sonorities which have the same components but which
are not involutions one of the other, although each has its own
involution. Examples are the tetrads C-E-fJ-G and C-F#-G-Bb.
Each contains one perfect fifth, one major third, one minor third,
one major second, one minor second, and one tritone (pmnsdt),
but one is not the involution of the other— although each has
its own involution.
We shall describe such sonorities, illustrated in Example 3-8,
as isomeric sonorities.
22
the theory of involution
Example 3-8
Involution'.
^ IIIVUIUMUIIi
ife
pmnsdt
pmnsdt
^#
Using the lowest tone of each of the following three-tone
sonorities as the axis of involution, write the involution of each
by projecting the sonority downward, as in Example 3-5.
i
Example 3-9
2. 2o. 2b.
3o.
Sn
3b.
^5^=
:x«o=
c^"* g o*> ^ W
5^
i
4o.
4b.
7rt
5a.
5b.
6o.
6b.
tU^
(*^ I o*^
=S^Q=
^
:^^
^^
sT-^n 2og> Q '^'
7o.
7b.
8.
9.
10.
lOo. lOb.
10 ^-=
ytbb-Q^^bt^oo'ro U'1% I ^tbt^^ 'ith^'^
llo.
Mb.
12.
12a.
ft |bo gboM 2b<j^^
i
^nrgr
Solution:
^^f
5=33=
^
12b.
ng»-
Ib.
fl/C.
^^D»=
m
SijO-
Zl-oU
"^^If
^15
-«s^l2
The following scales are all isometric, formed by the combina-
tion of one of the three-tone sonorities in Example 3-9 with its
involution. Match the scale in Example 3-10 with the appropriate
sonority in Example 3-9.
23
harmonic materials of modern music
Example 3-10
^ J J J >r U^J JuJ Ij jiiJ Ji'^ ij^jjg^J ijj Ji'-^t
'fiJ^J^rU^jjtJJUjtJJtf^^i^rr^r'Ti^^
t ^^rrrrijuJJf ijJjtJ ^ i^^^^^UJ^tJ^P
I J J J ^ ^ i;ii.JtJbJ U|J J ^^^ Uj jj^^*^ i>J^^
(j^^jiJ^ri|J^^rrijjJ^^i''^^rrri>ji>J^^^
Note: We have defined an isometric sonority as one which
has the same order of intervals regardless of the direction of
projection. The student should note that this bidirectional
character of a sonority is not always immediately evident. For
example, the perfect-fifth pentad in the position C2D2E3G2A3(C)
does not at first glance seem to be isometric. However in the
position D2E3G2A3C2(D), its isometric character is readily
apparent.
24
1^: PartJ
THE SIX BASIC
TONAL SERIES
4
Projection of the Perfect Fifth
We have seen that there are six types of interval relationship,
if we consider such relationship both "up" and "down": the
perfect fifth and its inversion, the perfect fourth; the major third
and its inversion, the minor sixth; the minor third and its
inversion, the major sixth; the major second and its inversion, the
minor seventh; the minor second and its inversion, the major
seventh; and the tritone,— the augmented fourth or diminished
fifth— which we are symbolizing by the letters, p, m, n, s, d,
and t, respectively.
In a broader sense, the combinations of tones in our system of
equal temperament— whether such sounds consist of two tones
or many— tend to group themselves into sounds which have a
preponderance of one of these basic intervals. In other words,
most sonorities fall into one of the six great categories: perfect-
fifth types, major-third types, minor-third types, and so forth.
There is a smaller number in which two of the basic intervals
predominate, some in which three intervals predominate, and a
few in which four intervals have equal strength. Among the
six-tone sonorities or scales, for example, there are twenty-six
in which one interval predominates, twelve which are dominated
equally by two intervals, six in which three intervals have
equal strength, and six sonorities which are practically neutral
in "color," since four of the six basic intervals are of equal
importance.
The simplest and most direct study of the relationship of tones
27
THE SIX BASIC TONAL SERIES
is, therefore, in terms of the projection of each of the six basic
intervals discussed in Chapter 2. By "projection" we mean the
building of sonorities or scales by superimposing a series of
similar intervals one above the other. Of these six basic intervals,
there are only two which can be projected with complete con-
sistency by superimposing one above the other until all of the
tones of the equally tempered scale have been used. These two
are, of course, the perfect fifth and the minor second. We shall
consider first the perfect-fifth projection.
Beginning with the tone C, we add first the perfect fifth, G,
and then the perfect fifth, D, to produce the triad C-G-D or,
reduced to the compass of an octave, C-D-G- This triad contains,
in addition to the two fifths, the concomitant interval of the
major second. It may be analyzed as ph.
Example 4-1
Perfect Fifth Triad, p^
m
^^
2 5
The tetrad adds the fifth above D, or A, to produce C-G-D-A,
or reduced to the compass of the octave, C-D-G-A. This sonority
contains three perfect fifths, two major seconds, and— for the
first time in this series— a minor third, A to C,
Example 4-2
Perfect FifthTetrad.p^ns^
m
^^
2 5 2
The analysis is, therefore, p^ns^.
The pentad adds the next fifth, E, forming the sonority
C-G-D-A-E, or the melodic scale C-D-E-G-A, which will be
recognized as the most familiar of the pentatonic scales. Its
components are four perfect fifths, three major seconds, two
28
PROJECTION OF THE PERFECT FIFTH
minor thirds, and— for the first time— a major third. The analysis
is, therefore, p^mnh^.
Example 4-3
Perfect Fifth Pentad, p^mn^s^
i
S
. o
^^
2 2 3 2
The hexad adds B, C-G-D-A-E-B, or melodically, producing
C-D-E-G-A-B,
Example 4-4
Perfect Fifth Hexod,p^nn^n^s^d
m
1 4JJ
2 2 3 2 2
its components being five perfect fifths, four major seconds, three
minor thirds, two major thirds, and— for the first time— the
dissonant minor second (or major seventh), p^m^n^s'^d.
The heptad adds F#:
i
Example 4-5
Perfect Fifth Heptod.p^m^n^s^d^t
a
^^
•I ^ '
'2 2 2 I 2 2
29
THE SIX BASIC TONAL SERIES
producing the first scale which in its melodic projection contains
no interval larger than a major second— in other words, a scale
without melodic "gaps." It also employs for the first time the
interval of the tritone (augmented fourth or diminished fifth),
C to FJf. This sonority contains six perfect fifths, five major
seconds, four minor thirds, three major thirds, two minor seconds,
and one tritone: p^m^n'^s^dH. (It will be noted that the heptad
is the first sonority to contain all of the six basic intervals. )
The octad adds Cfl::
Example 4-6
Perfect Fifth Octod. p^m^ n ^s^ d^ t^
Am
♦
«5i=
5
12 2 12 2
Its components are seven perfect fifths, six major seconds, five
minor thirds, four major thirds, four minor seconds, and two
tritones: p^m'^n^s^dH^.
The nonad adds G#:
Example 4-7
Perfect Fifth Nonad, p^m^n^s^d^t^
J^
m — =
m
iff I ? 9
m
Its components are eight perfect fifths, seven major seconds, six
minor thirds, six major thirds, six minor seconds, and three
tritones: p^m^n^s^dH^.
30
PROJECTION OF THE PERFECT FIFTH
The decad adds D#:
Example 4-8
- u «*!" Perfect Fifth Decad, p^m^n^s^d^t^
m
IT" I I I O
^^
I I I
I I 2
Its components are nine perfect fifths, eight major seconds, eight
minor thirds, eight major thirds, eight minor seconds, and four
tritones: 'p^m^n^s^dH'^.
The undecad adds A#:
Isjf
Example 4-9
? s"** Perfect Fifth Undecad, p'°m'°n'°s'°d'°t^
^^
m
1*"^ I r I 2 I I II I
Its components are ten perfect fifths, ten major seconds, ten
minor thirds, ten major thirds, ten minor seconds, and five
tritones: p^'^m'V^s'Od/'^f^
The duodecad adds the last tone, E#:
Example 4-10
I
A^ Perfect Fifth Duodecad, p'^m'^n'^s'^d
I2_l2j2„l2jl2^6
V^
s
r I r I I I I I I I I
31
THE SIX BASIC TONAL SERIES
Its components are twelve perfect fifths, twelve major seconds,
twelve minor thirds, twelve major thirds, twelve minor seconds,
and six tritones: p'^^m^^n^^s^^d^H^.
The student should observe carefully the progression of the
intervallic components of the perfect-fifth projection, since it has
important esthetic as well as theoretical implications:
doad:
P
triad:
p^s
tetrad:
p^ns^
pentad:
p^mn^s^
hexad:
p^m^n^s^d
heptad:
p^m^n^sHH
octad:
p'm^nhHH^
nonad:
p^m^n^s^dH^
decad:
p^m^n's^dH''
undecad:
plO^lO^lO^lO^lO^B
duodecad :
p'^m^^n'^s^^d'H'
In studying the above projection from the two-tone sonority
to the twelve-tone sonority built on perfect fifths, several points
should be noted. The first is the obvious affinity between the
perfect fifth and the major second, since the projection of one
perfect fifth upon another always produces the concomitant
interval of the major second. (It is interesting to speculate as to
whether or not this is a partial explanation of the fact that the
"whole-tone" scale was one of the first of the "exotic" scales to
make a strong impact on occidental music. )
The second thing which should be noted is the relatively
greater importance of the minor third over the major third in
the perfect-fifth projection, the late arrival of the dissonant
minor second and, last of all, the tritone.
The third observation is of the greatest importance because of
its esthetic implications. From the first sonority of two tones,
related by the interval of the perfect fifth, up to the seven-tone
sonority, there is a steady and regular progression. Each new
32 •
PROJECTION OF THE PERFECT FIFTH
tone adds one new interval, in addition to adding one more to
each of the intervals already present. However, when the pro-
jection is carried beyond seven tones, no new intervals can be
added. In addition to this loss of any new material, there is also
a gradual decrease in the difference of the quantitative formation
of the sonority. In the octad there are the same number of major
thirds and minor seconds. In the nonad the number of major
thirds, minor thirds, and minor seconds is the same. The decad
contains an equal number of major thirds, minor thirds, major
seconds, and minor seconds. When the eleven- and twelve-tone
sonorities are reached, there is no differentiation whatsoever, ex-
cept in the number of tritones.*
The sound of a sonority— either as harmony or melody-
depends not only upon what is present, but equally upon what is
absent. The pentatonic scale in the perfect-fifth series sounds as
it does not only because it contains a preponderance of perfect
fifths and because of the presence of major seconds, minor thirds,
and the major third in a regularly decreasing progression, but
also because it does not contain either the dissonant minor
second or the tritone.
On the other hand, as sonorities are projected beyond the
six-tone series they tend to lose their individuality. All seven-tone
series, for example, contain all of the six basic intervals, and the
difference in their proportion decreases as additional tones
are added.
This is probably the greatest argument against the rigorous
use of the atonal theory in which all twelve tones of the chro-
matic scale are used in a single melodic or harmonic pattern,
since such patterns tend to lose their identity, producing a
monochromatic effect with its accompanying lack of the essential
element of contrast.
All of the perfect-fifth scales are isometric in character, since if
any of the projections which we have considered are begun on
* See page 139 and 140.
33
THE SIX BASIC TONAL SERIES
the final tone of that projection and constructed downward, the
resultant scale will be the same as if the projection were upward.
The seven-tone scale C2D2E2F#iG2A2B, for example, begun on
the final tone of the projected fifths— that is, F+f— and projected
downward produces the same tones: J,F#2E2D2CiB2A2G.
Every scale may have as many versions of its basic order as
there are tones in the scale. The seven-tone scale, for example,
has seven versions, beginning on C, on D, on E, and so forth.
i
Example 4-11
Seven "versions" of the Perfect Fifth Heptad
^
f^o^
rtn*
v> o^^ » ^\
^^
o *^
f
2 2
^
2 2 2
2 2 (1) 2 2 I 2 2 I (2)
2 (2)
=^33
*^
;x4^M
:&:xsi
_Ql
i^
:^=KS
3s:«i
=0^5
O*^"*
bcsr^
2 2 1 2 2 (2)
2 2 I 2 2 2 (I)
2 12 2 2
(2)
#
(-C^)
v^g>
2 2 2
2 (2)
The student should distinguish carefully between an involu-
tion and the different versions of the same scale. An involution
is the same order of progression but in the opposite direction and
is significant only if a new chord or scale results.
Referring to page 29, you will see that the perfect-fifth penta-
tonic scale on C, C-D-E-G-A, contains a major triad on C and a
minor triad on A. The six-tone perfect-fifth scale adds the major
triad on G and the minor triad on E. Analyze the seven-, eight-,
nine-, ten-, eleven- and twelve-tone scales of the perfect-fifth
projection and determine where the major, minor, diminished,
and augmented triads occur in each.
Construct the complete perfect-fifth projection beginning on
the tone A. Indicate where the major, minor, diminished, and
augmented triads occur in each.
34
PROJECTION OF THE PERFECT FIFTH
Since the perfect-fifth projection includes the most famihar
scales in occidental music, innumerable examples are available.
The most provocative of these would seem to be those which
produce the greatest impact with the smallest amount of tonal
material. To illustrate the economical use of material, one can
find no better example than the principal theme of Beethoven's
overture, Leonore, No. 3. The first eight measures use only the
first five tones of the perfect-fifth projection: C-D-E-G-A. The
next measure adds F and B, which completes the tonal material
of the theme.
Example 4-12
Beethoven, Overture, Leonore No.3
^^
*
m
^^
wm
^ijjii'
o ^^
In the same way. Ravel uses the first five tones of the perfect-
fifth projection G-D-A-E-B— or, in melodic form, E-G-A-B-D— in
building to the first climax in the opening of Daphnis and Chloe,
Suite No. 2.
Example 4-13
Ravel, Daphnis end Chloe
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
The principal theme of the last movent ent of the Beethoven
Fifth Symphony is only slightly less economical in its use of
material. The first six measures use only the pentatonic scale
C-D-E-F-G, and the seventh measure adds A and B.
Beethoven, Symphony No. 5
Example 4-14
35
THE SIX BASIC TONAL SERIES
However, even Beethoven with his sense of tonal economy
extended his tonal material beyond the seven-tone scale without
implying modulation. The opening theme of the Eighth Sym-
phony, for example, uses only the six tones F-G-A-B^-C-E of the
F major scale in the first four measures but reaches beyond the
seven-tone perfect-fifth scale ^r an additional tone, Bt] (the
perfect fifth above E ) in the fifth measure.
Example 4-15
Beethoven, Symphony No. 8
f T^^r igu ^
Such chromatic tones are commonly analyzed as chromatic
passing tones, non-harmonic tones, transient modulations, and
the like, but the student will find it useful also to observe their
position in an "expanded" scale structure.
Study the thematic material of the Beethoven symphonies and
determine how many of them are constructed in the perfect-fifth
projection.
A useful device of many contemporary composers is to begin
a passage with only a few tones of a particular projection and
then gradually to expand the medium by adding more tones of
the same projection. For example, the composer might begin a
phrase in the perfect-fifth projection by using only the first four
tones of the projection and then gradually expand the scale by
adding the fifth tone, the sixth tone, and so forth.
36
PROJECTION OF THE PERFECT FIFTH
Examine the opening of Stravinsky's Petrouchka. The first five
measures are formed of the pure four-tone perfect-fifth tetrad
G-D-A-E. The sixth measure adds Bt], which forms the perfect-
fifth pentad G-D-A-E-B. The following measure adds a C#,
forming the hexad G-A-B-Cj|-D-E. This hexad departs momen-
tarily from the pure perfect-fifth projection, since it is a combina-
tion of a perfect-fifth and major-second projection— G-D-A-E-B
+ G-A-B-C#.
Measure 11 substitutes a C for the C# and measure 12 substi-
tutes a Bb for the previous B, forming the hexad G2A1BI72C2D2E
which is the involution of the previous hexad G2A2B2C#iD2E.
Measure 13 adds an F, establishing the seven-tone perfect-fifth
scale Bb-F-C-G-D-A-E.
Continue this type of analysis to rehearsal number 7,
determining how much of the section is a part of the perfect-
fifth projection.
Analyze the thematic material of the second movement of the
Shostakovitch Fifth Symphony. How much of this material con-
forms to the perfect-fifth projection?
Excellent examples of the eight-tone perfect-fifth projection
are found in the beginning of all three movements of the
Stravinsky Symphony in C. In the first movement, for example,
the first seven measures are built on the tonal material of the
seven-tone perfect-fifth scale on C: C-G-D-A-E-B-F#. In the
eighth measure, however, the scale is expanded one perfect
fifth downward by the addition of the Fki in the violas, after
which both F and Ffl: are integral parts of the scale. Note the
scale passage in the trumpet:
Stravinsky, Symphony in C
Example 4-16
Copyright 1948 by Schott & Co., Ltd.; used by permission of Associated Music Publishers, Inc., New York.
37
THE SIX BASIC TONAL SERIES
Similarly, the following theme from the first movement of the
ProkofieflF Sixth Symphony may be analyzed as the expansion of
the perfect-fifth projection to nine tones:
Example 4-17
Prokotieff, Symphony No. 6
© 1949 by Leeds Music Corporation, 322 West 48th St., New York 36, N. Y. Reprinted by permission; all
rights reserved.
i
ft-
m
Even when all of the tones of the chromatic scale are used, the
formation of individual sonorities frequently indicates a simpler
basic structure which the composer had in mind. For example,
the first measure of the Lyrische Suite by Alban Berg employs
all of the tones of the chromatic scale. Each sonority in the
measure, however, is unmistakably of perfect-fifth construction:
Albon Berg, Lyrische Suite
Example 4-18
Copyright 1927 by Universal Editions, Vienna; renewed 1954 by Helene Berg; used by permission of Asso-
ciated Music Publishers, Inc., New York.
38
PROJECTION OF THE PERFECT FIFTH
Analyze the first movement of the Stravinsky Symphony in C
and determine how much of it is written in the perfect-fifth
projection.
In any analysis, always try to discover how the work is
constructed, that is, how much should be analyzed as one frag-
ment of the composition. It will be observed, for example, that
some composers will use one scale pattern for long periods of
time without change, whereas others will write in a kind of
mosaic pattern, one passage consisting of many small and
diflPerent patterns.
39
Harmonic-Melodic Material
of the Perfect-Fifth Hexad
Since, as has been previously stated, all seven-tone scales contain
all of the six basic intervals, and since, as additional tones are
added, the resulting scales become increasingly similar in their
component parts, the student's best opportunity for the study of
different types of tone relationship hes in the six-tone combina-
tions, which offer the greatest number of different scale types.
We shall therefore concentrate our attention primarily upon the
various types of hexads, leaving for later discussion those scales
which contain more than six tones.
In order to reduce the large amount of material to a manage-
able quantity, we shall disregard the question of inversions. That
is, we shall consider C-E-G a major triad whether it is in its
fundamental position— C-E-G; in its first inversion— E-G-C; or in
its second inversion— G-C-E. In the same way, we shall consider
the pentad C-D-E-G-A as one type of sonority, that is, as a
sonority built of four perfect fifths, regardless of whether its form
is C-D-E-G-A, D-E-G-A-C, E-G-A-C-D, and so forth. It is also
clear that we shall consider all enharmonic equivalents in equal
temperament to be equally valid. We shall consider C-E-G a
major triad whether it is spelled C-E-G, C-F^-G, B#-E-G, or in
some other manner.
Examining the harmonic-melodic components of the perfect-
fifth hexad, we find that it contains six types of triad formation.
These are in order of their appearance:
1. The basic triad C2D5G, p^s, consisting of two superimposed
40
THE PERFECT-FIFTH HEXAD
perfect fifths with the concomitant major second, which is
dupHcated on G, D, and A:
Example 5-1
Perfect Fifth Hexad Perfect Fifth Triads
2 2 3 2 2
2 5
2 5
2 5
2 5
2. The triad C7G2A, pns, with the involution C2D7A, which
consists of a perfect fifth, a major second, and a major sixth (or
minor third ) . These triads are duphcated on G and on D :
Example 5-2
Triad pns and involutions
7 2 ,27^
(involution)
7 2 2 7
(involution)
2 7
(involution)
3. The triad C4E3G, pmn, with the involution A3C4E, which
consists of a perfect fifth, a major third, and a minor third, form-
ing the familiar major and minor triads. The major triad is
duplicated on G, and the minor triad on E:
Example 5-3
^ Triad
pmn
and involu
fions
h 1 J
-i : 1 1 J
J r r : J
— J r II
\ J- ^
;i> •
' \ —
-■ ■ — m
1 — s — 1 u
4 3
3 4
4 3
3 4
4. The triad C7G4B, pmd, with the involution C4E7B, consist-
ing of the perfect fifth, major seventh (minor second), and
major third:
Example 5-4
Triad pmd and invoiution
*
i-H^^t-h-t-
7 4
4 7
41
THE SIX BASIC TONAL SERIES
5. The triad C2D2E, ms^, which consists of two superimposed
major seconds with the concomitant major third, an isometric
triad, which is reproduced on G:
Example 5-5
Triad ms^
^
-S- • —
2 2 2 2
6. The triad BiCoD, nsd, with the involution A2B1C, which
consists of a minor third, a major second, and a minor second:
Example 5-6
Triad nsd and involution
I
^
^
,2 I. .
(involution)
The tetrads of the perfect-fifth hexad consist of seven types.
The first is the basic tetrad C2DgG2A, p^ns^, aheady discussed
in the previous chapter, duphcated on G and D:
Example 5-7
Perfect Fifth Tetrads p^ns^
i
2 5 2
2 5 2
2 5 2
The second is the tetrad C2D2E3G, also duplicated on G
(G2A2B3D), and the involutions A3C2D2E and E3G2A2B. This
tetrad contains two perfect fifths, two major seconds, one major
third, and one minor third: p^mns^.
Example 5-8
2 2
Tetrads p mns and involutions
223 *3 22 223 ,32 2,
(involution) (involution)
42
THE PERFECT-FIFTH HEXAD
It is one of the most consonant of the tetrads, containing no
strong dissonance and no tritone. Not only does it contain an
equal number of perfect fifths and major seconds, but it is also
the first example of the simultaneous projection of two different
intervals above the same tone, since it consists of the two perfect
fifths above C plus the two major seconds above C, that is,
C-G-D plus C-D-E, or-above G-G-D-A plus G-A-B. (These
formations will be discussed in Part III. )
Example 5-9
m
Tetrad p^mns^ as p^+s^
^
fe
i^^
J J r r
i
?
p
2 2 3
+ s'
+ 32
2 2 3
The involutions may also be considered to be formed by the
simultaneous projection of two perfect fifths and two major
seconds downward, that is J,E-A-D + J,E-D-C: and J,B-E-A
+ jB-A-G:
Example 5-10
Involution
II Jj I IT I IT^^
; p2 + s2 =223 i p2
^^
^g^
^
iTt
2 2 3
The third is the tetrad C4E3G2A, duphcated on G (G4B3D2E),
also a predominantly consonant tetrad, which consists of two
perfect fifths, C to G and A to E; two minor thirds, A to C and
E to G; the major third, C to E; and the major second, G to A:
p^mnrs. This is an isometric tetrad since, if we begin on the tone
E and form the same tetrad downward, J^E4C3A2G, we produce
the identical tones:
Example 5-11
Tetrads p^m n^ s.
m
^
J .11 .1 r r r I
4 3 2
4 3 2
(Isometric involution)
4 3
^
4 3 2
(isometric involution)
43
THE SIX BASIC TONAL SERIES
It may be considered to be formed of the relationship of two
perfect fifths at the interval of the minor third, indicated by the
symbol p @ n; or of two minor thirds at the interval of the
perfect fifth, indicated by the symbol n @ p:
Example 5-12
p @ n
il@_P
It contains the major triad C4E3G and the involution A3C4E;
Example 5-13
m.
43 34
)mn + involution
and the triad C7G2A, pns, with the involution G2A7E :
Example 5-14
J J:j J
m
7 2 2 7
pns + involution
The fourth tetrad, C4E3G4B, is also isometric, since if we begin
on the tone B and form the same tetrad downward, we produce
the identical tones, IB4G3E4C:
Example 5-15
Tetrad p^m^n d
ij •' r^T^jj^
434 434
(isometric involution)
It is a more dissonant chord than those already discussed, for it
contains two perfect fifths, C to G and E to B; two major thirds,
44
THE PERFECT-FIFTH HEXAD
C to E and G to B; one minor third, E to G; and the dissonant
major seventh (or minor second), C to B: p^m^nd. It may be
considered to be formed of two perfect fifths at the interval
relationship of the major third, C to G, plus E to B; or of two
major thirds at the relationship of the perfect fifth, C to E plus
G to B:
Example 5-16
ii J U
p @ m E @ P
It contains the major triad C4E3G and the involution, the minor
triad E3G4B;
Example 5-17
ji J j ij ^ r
•^ ^^4 3 3 4'
pmn + involution
and the triad C7G4B, pmd, and the involution C4E7B :
Example 5-18
j,^r ^i J r
7 4 ^4 7
pmd + involution
The fifth tetrad C2D5G4B, p^mnsd, consists of two perfect
fifths, C to G and G to D, with the dissonance, B. This tetrad
may also be considered as the major triad G-B-D with the added
fourth above, or fifth below, G, that is, C. It is the first of the
tetrads of this projection which contains all of the intervals of
the parent hexad.
Together with this tetrad is found the involution C4E5A2B,
which consists of the minor triad A-C-E with the perfect fifth
above, or the perfect fourth below, E, namely, B :
45
THE SIX BASIC TONAL SERIES
Example 5-19
iTetrad p^mnsd and involution
. <li j^'jiUJ ^ r (i)
*^ 2 5 4 ^ 4 5 2' 9-'
(involution)
The sixth tetrad, G2A2B1C, pmns^d, contains one perfect fifth,
one major third, one minor third, two major seconds, and a
minor second. We also find the involution B1C2D2E :
Example 5-20
Tetrad pmns^d and involution
*
2 2' I
2 2
(involution)
And finally, we have the isometric tetrad A2B1C0D, pnh^d,
which consists of a perfect fifth, two minor thirds, two major
seconds, and a minor second. It may be analyzed as the com-
bination of two minor thirds at the interval of the major second,
or two major seconds at the interval of the minor third. It
contains the triad B1C2D, nsd, and the involution A2B1C; also the
triad D7A2B, pns, and the involution C2D7A:
Example 5-21
Tetrad pn^s^d
L @ 1. S. @I}. Q^ ■*" involution
7 2' -^277
pns + involution
The parent hexad contains three pentad types. The first is the
basic perfect-fifth pentad C2D2E3G2A, p^mn^s^, also duplicated
on G, G2A2B3D2E:
Example 5-22
i
Perfect Fifth Pentads p'^nnn^s^
^
^
^
^
46
THE PERFECT-FIFTH HEXAD
The second pentad, C2D0E3G4B, p^m^n^s^d, predominates in
perfect fifths, hke its parent scale, but has an equal number of
major thirds, minor thirds, and major seconds. It may be identi-
fied more easily as the superposition of one major triad upon the
fifth of another, C-E-G + G-B-D; its involution is C4E3G0A2B
with, of course, the same analysis, and consists of two minor
triads projected downward, J^B-G-E plus J,E-C-A:
Pentad p^m^n^s^d
Example 5-23
and involution
i J J ^ r j Mi ^ ^ ^ r ii J
22 34 .jTr^L ^43 22 ,^*
pmn @ p
pmn@ p
The final pentad consists of the tones G2A2B1C2D, p^mn^s^d.
This pentad will be seen to have an equal number of perfect
fifths and major seconds, two minor thirds, one major third, and
one minor second. The involution is A2B1C2D2E:
i
Example 5-24
Pentad p'^nnn^s^d and involution
S
#=F
J^j^r^irJjJ
m
p
f
m
pns @ s
2 2 I
2 2
\ pns @ s
These pentads may be analyzed further as consisting of two
triads pns at the interval of the major second, projected up
or down.
The scales formed of perfect fifths, which have been discussed
in this and the previous chapter, account for a very large segment
of all occidental music. The five-tone scale in this series is the
most important of all the pentatonic scales and has served as the
basis of countless folk melodies. The seven-tone scale upon ex-
amination proves to be the most familiar of all occidental scales,
the series which embraces the Gregorian modal scales, including
the familiar major scale and the "natural" minor scale.
47
THE SIX BASIC TONAL SERIES
We have found in the previous chapter that the perfect-fifth
hexad contains two isometric triads, p^s and ms^, and four triads
with involutions, pns, pmn, pmd, and nsd. These triads are
among the basic words, or perhaps one should say, syllables, of
our musical vocabulary. They should be studied with the
greatest thoroughness since, unlike words, it is necessary not only
to "understand" them but to hear them.
For this reason the young composer might well begin by play-
ing Example 5-25, which contains all of the triad types of the
perfect-fifth hexad, over and over again, listening carefully until
all of these sounds are a part of his basic tonal vocabulary. I
suggest that the student play the first measure at least three
times, with the sustaining pedal down, so that he is fully con-
scious of the triad's harmonic as well as melodic significance; and
then proceed with measure two, and so forth.
Example 5-25
n ^iiiiM^ I , I P* I I r . r"Ti I I
^ - m -J- f» r F^^^-*'-* p r r r r " ■ — ^
'^^ ^ 9 — ^ r I J — ^ r I J — [_ I -^ --J- V — 9 ^
f^. ^r^r^rr '_rr r~F~ r i ■ _ ^ r » r • — r
p^s
pns
involution
^^^^^
i^^rf"
pnrid involution
pmn
involution
ms^ nsd involution
48
THE SIX BASIC TONAL SERIES
In Example 5-26 play the same triads but as "block" chords,
listening carefully to the sound of each.
Example 5-26
When the student comes to measures 8 and 9, and 10, the triads
may sound too "muddy" and unclear in close position. Experi-
ment with these sounds by "spreading" the triads to give them
harmonic character, as in Example 5-27.
Example 5-27
^
gpw
etc.
^^
etc.
^
etc.
^
The sound of each of these triads will be affected both by its
position and by the doubling of its tones. In the Stravinsky
Symphony of Psalms, familiar sonorities take on new and some-
times startling character merely by imaginative differences in the
doubling of tones. In Example 5-28, go back over the ten triad
forms and experiment with the different character the triad can
assume both in different positions and with different doublings.
Example 5-28
1*^ IT
m
£
m
■»" ^ etc.
49
THE SIX BASIC TONAL SERIES
In Example 5-29a play the tetrads in arpeggiated form, and
in Example 5-29b play them as "block" harmonies.
Example 5-29
(«)
IW^P^an Ld"iiJJ-' JT^ J^^"^
^^^^^^^^
!iy^^iiLlal!\JPJ^al}iiil
,|j]Tl.r7T3^^l^j
»— J * ~ » - - r at — ^ 9
In Example 5-30 experiment with different positions and
different doublings of the tones of the tetrads.
Example 5-30
m
etc.
-^ =
$
^^-t-
etc.
etc.
* 5
^#
/if|-,^l /TTl^NfljJi Ijjr
e/c.
ete.
f
r
50
THE PERFECT-FIFTH HEXAD
^^i
^
^
^
J=J
^
^^^^^^
T
In Example 5-31, repeat the same process with the five pentad
types.
Example 5-31
(a)
ilTT^SV^ P^^iOUi
SlUr^fTT^n-^oiU
^^ jJJ-iJJJ^^"^ ^rr ^rrrJr^
iriirrfirr
(b)
^
e/c.
e/c.
m
m
iriMfjii
51
THE SIX BASIC TONAL SERIES
In Example 5-32 repeat the same procedure with the hexad.
Example 5-32
(a)
^!^nJ^ir^P'^ai:!StimJai^
(b)
etc.
^
The student will find upon experimentation that although the
basic tetrad seems to keep much of the same character regardless
of its position, the remaining tetrads vary considerably in sound
according to the position of the tetrad— particularly with regard
to the bass tone. Play Example 5-30 again, noting the changes
which occur in the sound when different tones of the tetrad are
placed in the lowest part.
Repeat the experiment in relation to the five pentads in
Example 5-31b and the one hexad in Example 5-32b and notice
that as the sonority becomes more complex, the arrangement of
the tones of the sonority becomes increasingly important. ( Note
especially the complete change in the character of the sonority
in the second measure of Example 31b when the G major triad
is shifted from its position above the C major triad to a position
below it. ) *
The melody in Example 5-33 includes all of the triads in the
perfect-fifth hexad in melodic form. Play the example through
several times and then finish the analysis.
*See Note, page 55.
52
#
2 2
PS p's
the perfect-fifth hexad
Example 5-33
pmn
' «^^ Q
O «^
o o
P'S P'3
pmn
Example 5-34 harmonizes each triad by the same tones in the
left hand in block harmony. Play this through several times and
notice how the change of harmony in the left hand gives to the
melodic line a certain pulse which we may call harmonic rhythm.
Experiment with the changing of this harmonic rhythm by shift-
ing the grouping of the tones in the melody, thereby changing
the harmonic accompaniment. (For example, group the eighth,
ninth, and tenth notes in the melody together and harmonize
them with an E minor triad under the melodic tone B, and shift
the following A minor triad one eighth note earher. ) Continue
this type of change throughout the melody.
Example 5-34
'■'f T r V ^r t' }'■ f
JIL^.' P n f^mm
m
f
' i. i. L
53
THE SIX BASIC TONAL SERIES
Example 5-35 contains all of the tetrads, the pentads, and the
hexad of the six-tone perfect-fifth scale. Play this exercise several
times in chorale style and listen to each change of harmony. Now
analyze each sonority on the principle that we have discussed in
the previous chapter.
Example 5-35
h'UTiiiJ i^
^W
.- ^r f f f f
Finally, using as much or as little as you wish of the material
which we have been studying, compose a short work in your
own manner. Do not, however, use even one tone which is not in
the material which we have studied. If you have studied orches-
tration, it would be desirable to score the composition for string
orchestra and if possible have it performed, since only through
actual performance can the composer test the results of his tonal
thinking. Use all of your ingenuity, all of your knowledge of
form and of counterpoint in this exercise.
54
THE PERFECT-FIFTH HEXAD
Note: It is interesting to speculate upon the reason why two sonorities containing
identical tones should sound so differently. The most logical explanation is
perhaps that Nature has a great fondness for the major triad and for those
sonorities that most closely approximate the overtone series which she has ar-
ranged for most sounding bodies — with the exception of bells and the like. The
human ear seems to agree with Nature and prefers the arrangement of any
sonority in the form which most closely approximates the overtone series. In
the case of the combination of the C major and the G major triads, for example,
if C is placed in the bass, the tones D-E-G-B are all found approximated in the
first fifteen partials of the tone C. If G is placed in the bass, however, the
tone C bears no close resemblance to any of the lower partials generated by
the bass tone.
Example 5-36
55
Modal Modulation
Most melodies have some tonal center, one tone about which
the other tones of the melody seem to "revolve." This is true not
only of the classic period with its highly organized key centers,
but also of most melodies from early chants and folk songs to
the music of the present day— with, of course, the exceptions of
those melodies of the "atonal" school, which deliberately avoid
the repetition of any one tone until all twelve have been used.
( Even in some of these melodies it is possible to discern evidence
of a momentary tonal center.)
The advantage of a tonal center would seem to be the greater
clarity which a melody derives from being organized around
some central tone. Such organization avoids the sense of con-
fusion and frustration which frequently arises when a melody
wanders about without any apparent aim or direction. The tonal
center, however, is not something which is immutably fixed. It
may, in fact, be any one tone of a group of tones which the
composer, by melodic and rhythmic emphasis or by the con-
figuration of the melodic line, nominates as the tonal center.
For example, we may use the pentatonic scale C-D-E-G-A
with C as the tonal center, by having the melody begin on C,
depart from it, revolve about it, and return to it. Or we might
in the same manner nominate the tone A as the tonal center,
using the same tones but in the order A-C-D-E-G. Or, again,
we might make either D, E, or G the tonal center of the melody.
One illustration should make this principle clear. If we begin
56
MODAL MODULATION
a melody on C, proceed upward to D, return to C, proceed
downward to A, return to C, proceed upward to D, then upward
to G, down to E, down to A and then back to C, we produce
a melodic line the configuration of which obviously centers about
C. If, using the same tones, we now take the same general con-
figuration of the melodic line beginning with A, we produce a
melody of which A is the tonal center:
#
»i O i^t __ ri O *^
Example 6-1
i^ *^ o
S3I
M VI r VI — %T g. fc:t
-^
Finally, we may move from one tonal center to another, within
the same tonal group, by changing our emphasis from one tone
to another. In other words, we might begin a melody which was
centered about C, as above, and then transfer that emphasis to
the tone A. Such a transition from one tonal center to another is
usually called a modulation. Since, however, the term modula-
tion generally implies the adding— or more properly, the substi-
tution—of new tones, we may borrow an old term and call this
type of modulation modal modulation, since it is the same
principle by which it is possible to modulate from one Gregorian
mode to another without the addition or substitution of new
tones. (For example, the scale C-D-E-F-G-A-B-C begun on the
tone D will be recognized as the Dorian mode; begun on the
tone E, as the Phrygian mode. It is therefore possible to "modu-
late" from the Dorian to the Phrygian mode simply by changing
the melodic line to center about the tone E rather than D.
The six-tone perfect-fifth scale has four consonant triads which
may serve as natural key centers: two major triads and two
minor triads. The perfect-fifth hexad C-D-E-G-A-B, for example,
contains the C major triad, the G major triad, the A minor triad,
and the E minor triad. We may, as we have seen, nominate any
57
THE SIX BASIC TONAL SERIES
one of them to be the key center merely by seeing to it that the
melodic and harmonic progressions revolve about that particular
triad. We may modulate from one of these four key centers to
any of the others simply by transferring the tonal seat of govern-
ment from one to another.
This transferral of attention from one tone as key center to
another in a melody has already been discussed on page 57. We
can assist this transition from one modal tonic to another (har-
monically) by stressing the chord which we wish to make the
key center both by rhythmic and agogic accent, that is, by
having the key center fall on a strong rhythmic pulse and by
having it occupy a longer time value. The simplest of illustra-
tions will make this clear. In the following example, 6-2a, the
first three triads seem to emphasize C major as the tonic, while
in Example 6-2b we make F the key center merely by shifting
the accent and changing the relative time values. In the slightly
more complicated Example 6-2c, the key center will be seen to
be shifted from A minor to E minor merely by shifting the
melodic, harmonic, and rhythmic emphasis.
^
(b[
9
3=
^^
58
MODAL MODULATION
^^
'>'■■ r r r f
r r r r
Compose a short sketch in three-part foiin using the hexad
C-D-E-G-A-B. Begin with the A minor triad as the key center,
modulating after twelve or sixteen measures to the G major
triad as the key center and ending the first part in that key.
Begin the second part with G major as the key center and after
a few measures modulate to the key center of E minor. At the
end of part two, modulate to the key center of C major for a few
measures and back to the key of A minor for the beginning of
the third part. In the third part, pass as rapidly as convenient
from the key center of A minor to the key center of E minor,
then to the key center of G major and back to A minor for the
final cadence.
In writing this sketch, try to use as much of the material
available in the hexad formation as possible. In other words, do
not rely too heavily upon the major and minor triads. Since these
modulations are all modal modulations, it is clear that the only
tones to appear in the sketch will be the tones with which we
started, G-D-E-G-A-B.
At first glance it may seem difficult or impossible to write an
interesting sketch and to make convincing modal modulations
with only six tones. It is difficult, but by no means impossible,
and the discipline of producing multum in parvo will prove
invaluable.
59
7
Key Modulation
In projecting the perfect-fifth relationship, we began with the
tone C for convenience. It is obvious, however, that in equal
temperament the starting point could have been any of the
other tones of the chromatic scale. In other words, the pentatonic
scale C0D2E3G2A may be duplicated on D^, as Db2Eb2F3Ab2Bb;
on D, as DoE2F#3A2B; and so forth. It is therefore possible to use
the familiar device of key modulation to modulate from any
scale to an identical scale formation begun upon a different tone.
The closeness of relationship of such a modulation depends
upon the number of common tones between the scale in the
original key and the scale in the key to which the modulation
is made. The pentatonic scale C-D-E-G-A, as we have already
observed, contains the intervals p*mnV. Therefore the key
modulation to the fifth above or to the fifth below is the closest
in relationship. It will have the greatest number of common
tones, for the scale contains four perfect fifths. Since the scale
contains three major seconds, the modulation to the key a major
second above or below is the next closest relationship; the modu-
lation to the key a minor third above or below is the next order
of key relationship; the modulation to the key a major third
above or below is next in order; and the last relationship is to the
key a minor second above or below, or to the key related to the
original tonic by the interval of the tritone.
A practical working-out of these modulations will illustrate
this principle:
60
KEY MODULATION
C-D-E-G-A modulating to the:
perfect fifth above gives
" below
major second above "
below
minor third above "
below
major third above "
below
minor second above "
below
tritone above
or
below gives
G-A-B-D-E
F-G-A-C-D
D-E-F#-A-B
Bb-C-D-F-G
Eb-F-G-Bb-C
A-B-Cif-E-F#
E-F#-G#-B-C#
Ab-Bb-C-Eb-F
Db-Eb-F-Ab-Bb
one new tone
t> If II
two " tones
II II II
three "
// n II
four "
// // //
all new tones
V%-G%-A%-C%-D% (all new tones)
*
Perfect Fifth Pentad
Example 7-1
Modulation
to Perfect Fifth above
to Major Second above
^
o o
to Major Second below
Modulation
to Perfect Fifth below
- o
to Minor Third above to Major Third above to Minor Second obove
i
^
^^
^
^
|;> 1^'
i
to Minor Third below
to Major Third below
to Minor Second below
f
i* > ff* *'
*
^^%* °"'
b,: 17»-
*
to Augmented Fourth above
It. %- i' ^' *•
i
to Augmented Fourth below
, _ I — L-- !;• ty*
\,-9- ?♦ "^
61
THE SIX BASIC TONAL SERIES
The student should learn to distinguish as clearly as possible—
though there will be debatable instances— between, for example,
(1) a modulation from the pentatonic scale C-D-E-G-A to the
pentatonic scale A-B-CJj:-E-Ffl:, and (2) the eight-tone perfect-
fifth scale, C-C#-D-E-F#-G-A-B, which contains all of the tones
of both pentatonic scales. In the former instance, the two
pentatonic scales preserve their identity and there is a clear point
at which the modulation from one to the other occurs. In the
latter case, all of the eight tones have equal validity in the scale
and all are used within the same melodic-harmonic pattern.
In the first of the two following examples, 7-2, there is a
definite point where the pentatonic scale on C stops and the
pentatonic scale on A begins.
Example 7-2
^i^^^ 4 i hJ-
^
^^
In the second example, 7-3, all of the eight tones are members
of one melodic scale.
Example 7-3
I i ^ti r^^^
Although modal modulation is the most subtle and delicate
form of modulation, of particular importance to the young com-
poser in an age in which it seems to be the fashion to throw the
entire tonal palette at the listener, it does not add new material
to the tonal fabric. This task is accomplished either by the
"expansion" technic referred to on page 36 or by the familiar
device of key modulation.
Key modulation offers the advantages of allowing the com-
poser to remain in the same tonal milieu and at the same time to
62
KEY MODULATION
add new tones to the pattern. A composer of the classic period
might— at least in theory— modulate freely to any of the twelve
major keys and still confine himself to one type of tonal material,
that of the major scale. Such modulations might be performed
deliberately and leisurely— for example, at cadential points in the
formal design— or might be made rapidly and restlessly within
the fabric of the structure. In either case, the general impression
of a "major key" tonal structure could be preserved.
This same device is equally applicable to any form of the
perfect-fifth projection, or to any of the more exotic scale forms.
The principle is the same. The composer may choose the tonal
pattern which he wishes to follow and cling to it, even though
he may in the process modulate to every one of the twelve
possible key relationships.
It is obvious that the richest and fullest use of modulation
would involve both modal modulation and key modulation used
successively or even concurrently.
Write an experimental sketch, using as your basic material
the perfect-fifth-pentatonic scale C-D-E-G-A. Begin in the key
of C, being careful to use only the five tones of the scale and
modulate to the same scale on E (E-F#-G#-B-CJj:). Now modu-
late to the scale on F# (F#-Gif-A#-C#-D#) and from F# to Eb
(Eb-F-G-Bb-C). Now perform a combined modal and key
modulation by going from the pentatonic scale on E^ to the
pentatonic scale on B (B-C#-D#-F#-G#), but with G# as the key
center. Conclude by modulating to the pentatonic scale on F,
with D as the key center ( F-G-A-C-D ) , and back to the original
key center of C.
You will observe that the first modulation— C to E— retains
only one common tone. The second modulation, from E to F#,
retains three common tones. The third, from F# to E^, has two
common tones. The fourth, from E^ to B, like the first modula-
tion, has only one common tone. The fifth, from B to F, has no
common tones, and the sixth, from F to C, has four common tones.
If you play the key centers successively, you will find that
63
THE SIX BASIC TONAL SERIES
only one transition offers any real problem: the modulation from
B, with Gif as the key center, to F, with D as the key center. It
will require some ingenuity on your part to make this
sound convincing.
Work out the modulations of the perfect-fifth hexad at the
intervals of the perfect fifth, major second, minor third, major
third, minor second and tritone, as in Example 7-1.
64
8
Projection of the Minor Second
There is only one interval, in addition to the perfect fifth,
which, projected above itself, gives all of the tones of the
twelve-tone scale. This is, of course, the minor second, or its
inversion, the major seventh.
Proceeding, therefore, as in the case of the perfect-fifth pro-
jection, we may superimpose one minor second upon another,
proceeding from the two-tone to the twelve-tone series.
Examining the minor-second series, we observe that the basic
triad C-C#-D contains two minor seconds and the major second
C-D: s(P.
The basic tetrad, C-C#-D-D#, adds another minor second,
another major second, and the minor third: ns^cP.
The basic pentad, C-CJ-D-Dif-E, adds another minor second,
another major second, another minor third, and a major third:
The basic hexad, C-CJj:-D-D#-E-F, adds another minor second,
another major second, another minor third, another major third,
and a perfect fourth: pm^nh^d^:
Minor Second Triad sd^
Example 8-1
Minor Second Tetrad ns^d^
^
t^
i
Minor Second Pentad mn'^s d
2.3^4
Minor Second Hexad pm^n^s^d^
"X5 yes ^
I I I
I I
65
THE SIX BASIC TONAL SERIES
The seven-, eight-, nine-, ten-, eleven- and twelve-tone minor-
second scales follow, with the interval analysis of each. The
student will notice the same phenomenon which was observed
in the perfect-fifth projection: whereas each successive projection
from the two-tone to the seven-tone scale adds one new interval,
after the seven-tone projection has been reached no new inter-
vals can be added. Furthermore, from the seven-tone to the
eleven-tone projection, the quantitative diff^erence in the propor-
tion of intervals also decreases progressively as each new tone
is added.
Example 8-2
Minor Second Heptad p^^n'^s^d^t Minor Second Octod p'^m^n^s^d^t^
I I I I I I r I I I I I I
MinorSeoond Nonad p^m^n^s^d^^ Minor Second Decad p^m^n^s^d^t'*
■^ v»jtoO^^^»tt« "^j^o^o o t.^t^^e^f^
I I I I
III III
Minor Second Undecad p m n s d t Minor Second Duodecod p m n s d t
^^ojto°"*"°1t°"<'"Lj^v>j)»°"jl"°«°"''
I I I
i I I I I I I I I I
I I I I I I I
Proceeding again, as in Chapter 5, we may now examine the
harmonic-melodic material of the minor-second hexad. First, we
have the basic triad C-C#-D, sd-, duplicated on the tones C|:,
D, and D#:
Example 8-3
)Minor Second Hexad Minor Second Triads sd^
I I I I I
I I
I I
I I
I I
The triad CiCJsDJj:, nsd, a form observed in the perfect-fifth
hexad, duplicated on C# and D, with their involutions:
66
projection of the minor second
Example 8-4
Triads nsd and involutions
f^ Triads
J|J ■ i t}J bJ Uj J J ■ J jtJ J I J bJ J ^ J t|J
'2 El^ia'^ZI 12 2
2' I
(involution)
2 I
(involution)
I 2 "^ 2 I 12
(involution)
The triad CiC^gE, mnd, duplicated on C#, with their
involutions :
Example 8-5
Triads mnd and involutions
J-t^J ^i|J,^ Ijti J^^lJ J^
r 3
3 I
(involution)
13 ^31
(involution)
The triad CiDb4F, pmd, with its involution C4E1F; which has
already been found in the perfect-fifth hexad:
Example 8-6
Triad pmd and involution
i>J>U
I 4 4 I
(involution)
The isometric triad C-D-E, ms^, which has already occurred
as a part of the perfect-fifth hexad; duplicated on D^;
i
Example 8-7
Triads ms^
^
^F^
2 2 2 2
67
THE SIX BASIC TONAL SERIES
and the triad C2D3F, pns, with its involution, CsE^oF, which
form also has been encountered in the perfect-fifth series:
i
Example 8-8
Triad pns and involution
J^XJ,^U>
"2 3
3 2
involution
The minor-second hexad contains the basic tetrad CiCflliDiDJ):,
ns'^d^, duplicated on Cfl: and on D:
Example 8-9
2 3
Minor Second Tetrads ns d
i^JJ^j'j^iJjtjJ'JlJJ
I I I
I I I
The tetrad CiCjiDoE, mns-d^, duplicated on C#, with their
respective involutions;
Example 8-10
Tetrads mns^d^and involutions
r' I 2 2
(involution
2 ^ 2' I I
(involution)
which may be analyzed as the simultaneous projection of two
minor seconds and two major seconds above C, or, in its involu-
tion, below E:
Example 8-11
i
Tetrad mns d
2h2
d2+s2
^W
^
-^ — *-
ld2 + -I s2
68
PROJECTION OF THE MINOR SECOND
The isometric tetrad CjC^aDitiE, mn^sd^, duplicated on C#;
Example 8-12
Tetrads mn sd
2.^2
r 2 I I 2^*^ r * I 2 I I 2 r
(isometric involution) (isometric involution)
which may be analyzed as two minor thirds at the relationship
of the minor second, or two minor seconds at the relationship
of the minor third:
Example 8-13
j> bj nU ' b^Ljt^ ^ ^^juM
n (S d
d @ n
I 2 I
or as a combination of the triad nsd and the involution onCj^, or
the triad mnd and its involution:
Example 8-14
l«^2" « 2" I
nsd + involution
3 ^3^ I
mnd -t- involution
The isometric tetrad CiDbgEiF, pm^nd^;
Example 8-15
Tetrad pm^nd^
* leiraa pm-ng-
(Ji jl,J J Jl J J^
I 3 I
I 3 I
(isometric involution)
which may be analyzed as consisting of two major thirds at the
interval of the minor second, or of two minor seconds at the
interval of the major third;
69
THE SIX BASIC TONAL SERIES
Example 8-16
i
j. J hjm^
m @ d
d @ m
or as a combination of the triad mnd, and the involution on D^,
or the triad pmd, and its involution:
Example 8-17
i
J 1;J J J J ^ I i |;J ^ ^
13 3 1 "14
mnd -t- involution pmd
4
involution
The tetrad CiCJiDsF, pmnsd", and its involution:
Example 8-18
Tetrad pmnsd^ and involution
#
iitiJ ^ UitJ
I I 3
3 I I
The tetrad CiDbsEb^F, pmns^d, and its involution, which has
already been found in the perfect-fifth projection;
Example 8-19
[Tetrad pmns^ and involution
^ letraa pmns'-o ana invc
12 2 2 2 1
and the isometric tetrad CoDiE^oF, pn^s^d, which is also a part
of the perfect-fifth hexad, and which may be analyzed as a
combination of two minor thirds at the interval of the major
second, or of two major seconds at the interval of the minor third :
Example 8-20
I
Tetrad pn^s^d
J J bJ ■* ' ^ bJ J i 1^
(isometric involution)— ^ 1
^
3 @ n
70
PROJECTION OF THE MINOR SECOND
The student will observe that the tetrad C-D-E^-F may also
be analyzed as a combination of the triad nsd and the involution
on D, or the triad pns and its involution:
Example 8-21
1^
^
^"2 1 12 ^2 3
risd + involution pns
3 2
involution
Finally, the pentads in the minor-second hexad consist of the
basic pentad CiCJiDiDJiE, mn^s^d^, duplicated on C#;
Example 8-22
Minor Second Pentads mn^s^d^
■ij^i JffJ ^ 'j|i J j^J
III I
I I I I
the pentad CiC^iDsEiF, pm^n~s~d^, with its involution,
C,C#2D#iEiF;
Example 8-23
2 2 2 3
Pentad pm n s d and involution
» renraa pm n s a ana invoi
I 12 1
2 I I
which may be analyzed as the relationship of two triads mnd, at
the interval of the minor second:
i
Example 8-24
2 2 2 3
Pentad pm n s d as mnd @ d
^^
P^W
r 3
I 3 I
i involution
71
THE SIX BASIC TONAL SERIES
and the pentad CiCJiDiEbsF, pmn^s^d^, with its invokition,
CsDiDJiEiF, which may be analyzed as the combination of two
triads nsd, at the interval of the major second:
i
Example 8-25
Pentad pmn^s d and involution
J J J J i J 1^^
^
lt>
J. yj ^^ i|*
12 1
nsd @ s
tr-^itw
2 I
I I
2 I 2
involution
The minor-second hexad is, quite obviously, a highly dissonant
scale. For this reason it has perhaps less harmonic than melodic
value. It may be effectively used in two-line or three-line con-
trapuntal passages where the impact of the thick and heavy
dissonance is somewhat lessened by the rhythmic movement of
the melodic lines.
Example 8-26 constitutes a mild puzzle. It is constructed to
have the same arithmetic, or perhaps I should say geometric
relationships, as the melodic line in Example 5-33. It should take
only a short examination to discover what this relationship is.
Example 8-26
mnd
obo'jjoy o *^oj^o|j^k3'^;_o^"t^l;otlot;v3.^
The six-tone minor-second scale will be found to be too
limited in compass to give the composer much opportunity in
this restricted form. Nevertheless, it is valuable to become
intimately acquainted with the small words and syllables which
72
PROJECTION OF THE MINOR SECOND
go to make up the vocabulary of this series, since these small
words constitute an important part of the material of some
contemporary music. Therefore, I suggest that you play through
Example 8-26 slowly and thoughtfully, since it contains all of
the triads of the minor-second hexad. Since I have kept all of
these triads in close position, the melody is even "wormier" than
such melodies need be.
Complete the analysis of all of the melodic triads under the
connecting lines and then play through the melody at a more
rapid tempo with the phrasing as indicated in Example 8-27.
See if you can sing the melody through without the aid of a
piano and come out on pitch on the final Ej^.
Example 8-27
Example 8-28 is a four-measure theme constructed in the
minor-second hexad. Continue its development in two-part
simple counterpoint, allowing one modulation to the "key" of G—
G-G#-A-A#-B-C— and modulating back again to the original "key"
of C.
Example 8-28
a^^i^iJ
"^^^'CJW^iPrr^^
G=p
^
73
THE SIX BASIC TONAL SERIES
etc.
It is difficult to find many examples of the effective use of the
minor-second hexad in any extended form in musical literature
because of its obvious limitations. A charming example is
found in "From the Diary of a Fly" from the Mikrokosmos of
Bela Bartok. The first nine measures are built on the six-tone
scale F-Gb-Gkj-Ab-Ati-Bb. The tenth measure adds the seventh
tone, C^,
Example 8-29
Bortok, Mikrokosmos
{hi Lb}^ J ^ JjfjL^^^sWmJ \i^\\^i^\J^'i \>^^i\^^
|^^5
m
^pi^r'"^p''t
m
^P
Copyright 1943 by Hawkes & Son (London), Ltd. Used by permission of Boosey & Hawkes, Inc.
On the other hand, examples of the utilization of the entire
chromatic scale within a short passage abound in contemporary
music, one of the most imaginative of which can be found in
the first movement of the Sixth Quartet of the same composer:
Bortok, Sixth Quartet
I^J2^-k
'■>'■ «r ' - a
Example 8-30
Mljl.
Ml
SUA-
g
Copyright 1941 by Hawkes & Son (London), Ltd. Used by permission of Boosey & Hawkes, Inc.
74
PROJECTION OF THE MINOR SECOND
A more obvious example of the use of the minor-second scale
is found at the beginning of the second movement of the Bartok
Fourth String Quartet:
Example 8-31
Bartok, Fourth Quartet, 2 movement
^TT-v ''jT" ^u l ,, , _ ,-T^ >.. . etc.
Copyright 1929 by Universal Editions; renewed 1956. Copyright and renewal assigned to Boosey & Hawkes,
Inc., for the U.S.A. Used by permission.
A more subtle example— and one very characteristic of the
Hungarian master— is found in the twenty-fifth measure of the
first movement of the same quartet. Here the tonal material
consists of the seven-tone minor-second scale B^-Btj-C-CJ-D-DJ-
E, but divided into two major-second segments, the cello and
second violin holding the major-second triad, B-C#-D#, and the
first violin and viola utilizing the major-second tetrad, B^-C-D-E:
Example 8-32
Bartok, Fourth Quortet
i
*
^
P
^
Y TT
m
ifiw
E i ^if-^i^y H-^
Copyright 1929 by Universal Editions; renewed 1956. Copyright and renewal assigned to Boosey & Hawkes,
Inc., for the U.S.A. Used by permission.
75
THE PERFECT-FIFTH HEXAD
Analyze the first movement of the Bartok Sixth Quartet to
determine how much of it is constructed in the minor-second
projection.
Modulation of the rninor-second pentad follows the same
principle as the perfect-fifth pentad. Modulation at the minor
second produces one new tone, at the major second two new
tones, at the minor third three new tones, at the major third four
new tones, and at the perfect fifth and tritone five new tones.
Work out all of the modulations of the minor-second pentad
and hexad.
76 'i
Projection of the Major Second
Since the major second is the concomitant interval resulting
from the projection of either two perfect fifths or of two minor
seconds, it would seem to be the most logical interval to choose
for our next series of projections.*
The basic, triad of the major-second series is C2D2E,
Example 9-1
M Major Second Triad msf
2 " 2
two major seconds with their concomitant interval of the major
third: ms^. We have already observed this triad as a part of both
the perfect-fifth and the minor-second hexads. The third major
second produces the tetrad C2D2E2F#, adding the new interval
of the tritone, C to FJj:. The analysis of this sonority becomes
three major seconds, two major thirds, and one tritone: m^sH.
Example 9-2
Major Second Tetrad m^^
^ >. *3 ^ <=» ^ ^'
2 2
• The major second would also seem to follow the perfect fifth and minor
second, since it can be projected to a pure six-tone scale, whereas the minor
third and the major third can be projected only to four and three tones,
respectively.
77
THE SIX BASIC TONAL SERIES
Superimposing another major second produces the pentad
C2D2E2FJj:2G#, which consists of four major seconds; four major
thirds, C to E, D to Ft, E to G#, G# ( Ab ) to C; and two tritones,
C to F# and D to G#: m^sH\
Example 9-3
Major Second Pentad rrfs^t^
^^
C5 »-
2 2 2
The superposition of one more major second produces the
"whole-tone" scale C2D2E2F#2GJl:2AJj::
Example 9-4
Major Second Hexad m^s^t^
i
2 2 2 2 2
t" <t^ ^*'
This scale will be seen to consist of six major thirds— C to E, D to
F#, E to G#, FJf to A#, G# to BJf (C) and A# (Bb) to D; six
major seconds-C to D, D to E, E to F^, F# to G#, G# to A#, and
AS (Bb ) to C; and three tritones-C to F#, D to G#, and E to A#.
Its analysis is m^sH^. It will be obvious that the scale cannot be
projected beyond the hexad as a pure major-second scale, since
the next major second would be BJ, the enharmonic equivalent
of C.
The major-second hexad is an enharmonic isometric scale; not
only is its form the same whether thought of clockwise or
counterclockwise, up or down, but its involution produces the
identical tones. Analyzing its components, we find that it has
"three different types of triads: the basic triad C2D2E, ms^,
duplicated on D, E, F#, GJf, and A#;
78
projection of the major second
Example 9-5
Major Second Triads ms'^
jij jitJij^JtJi|tJtJitJiitJ<Mir^'i^ri"^r r
~St — w
2 2
2 2
2 2
2 2
2 2
2 2
the augmented triad C4E4G#, m^, duplicated on D (since the
remaining four augmented triads are merely inversions of those
on C and D ) ;
Example 9-6
Major Third Triads m^
^i .1 ftJ I J |JiiJ i[jitJiif'rM|tJiiJY'r'l4iiiir r \^^^^
'^■•'4 4 44 44 44 44 44
and the triad C2D4FJJ:, mst, and its involution, C4E2F#, also
duplicated on the other five notes of the scale:
Example 9-7
Triads mst and involutions
ItJij JitJ Ij Jtl^ ;j||J|^ljjJ<tJ :.li|J|^
2 4
4 2
2 4
4 2
2 4
4 2
j^jijjtJ«rr':nJiiJ<r'f'iiJiiJiYii^r H'Jjj* i^jj i^r^^
The basic triad we have already analyzed as containing two
major seconds and a major third, ms^. The augmented triad
contains three major thirds, C to E, E to GJf, and G# {A\)) to C,
m^. The triad C2D4F# and its involution C4E2F#, contain one
major second, one major third, and one tritone, mst.
The major-second hexad contains three different types of
tetrads: the basic tetrad C2D2E2F#, 7n-sH, duplicated on D, E,
F#, Ab, and Bb;
Example 9-8
Major Second Tetrads mst
THE SIX BASIC TONAL SERIES
the isometric tetrad C2D2E4G#, duplicated on D, E, G^, A^, and
B\), containing three major thirds, two major seconds, and one
tritone, rrfsH;
Example 9-9
Tetrads m^s^t
^ iBiiuu^ III a I
4
■^24 224 224
2 2 4 224 224
which may also be considered to be formed by the simultaneous
projection of two major seconds and two major thirds;
Example 9-10
i
[f' i J ^ i J fr
and the isometric tetrad C4E2F#4AJf, duplicated on D and E,
which contains two major thirds, two major seconds, and two
tritones, m^sH^:
Example 9-11
I Tetrads m^s^t^
f^ letraas m^s t~ ^ ^^^
9>iJitJi^tJ<tJ«r'^'iJii-'J'^T'i'iijJitJrrit>.ii'r'^'ri>('fr'nii
4 2 4 4 2 4
4 2 4
424 424 424
This may also be analyzed as two major thirds at the interval of
the tritone; as two tritones, at the interval of the major third; as
two major seconds at the interval of the tritone, or as two tri-
tones at the interval of the major second.
Example 9-12
m @ t
t @ m
s @ t
t @ s
80
PROJECTION OF THE MAJOR SECOND
This highly isometric sonority was a favorite of Scriabine,
particularly in the Poeme de TExtase.
There is only one type of pentad in the six-tone major-second
scale, since the remaining five pentads are merely transpositions
of the first :
Example 9-13
Major Second Pentads m'^s f
ij|j I J j|jjtJji^ I jttJiiJiJ<ir^''it^''^^^
%
2222 2222 2222 2222
(| iJuJiiTnY r n Y r I* I" *ni
2 222 2222
An examination of this series will show both its strength and
weakness. Its strength lies in the complete consistency of its
material. It is one of the most homogeneous of all scales, since it is
made up exclusively of major thirds, major seconds, and tritones.
It is only mildly dissonant in character, since it contains no pri-
mary dissonances (the minor second or major seventh).
Its very homogeneity is also its weakness, for the absence of
contrasting tonal combinations gives, in prolonged use, a feeling
of monotony. Also, the absence of the perfect fifth deprives the
scale of any consonant "resting-place," or tonic, so that its pro-
gressions sound vague, lacking in contrast, and without direction.
Nevertheless, it is an important part of the tonal vocabulary and,
in the hands of a genius, adds a valuable color to the tonal palette
which should not be lightly discarded by the young composer.
Its effective use is illustrated in Debussy's "Voiles," the first
thirty measures of the first section of which are written entirely
in the whole-tone scale.
The same composer's "La Mer" contains extended use of the
81
THE SIX BASIC TONAL SERIES
same scale in the excerpt below:
Example 9-14
Debussy, "lo Men"
^m.
*=*.:
fVff^rp
ayrt-tjj
'/■hh S'
^^^=tF
P^- 'f^ff:
^
^ ^
iji^g
^E
■■'■e-
#^
■^fi^^ bi
^
^
Z
^
^
rt
^m
^
^^
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
An example of the whole-tone scale where it might not be
expected is found in the opening of an early song, "Nacht," of
82
PROJECTION OF THE MAJOR SECOND
Alban Berg, the first five measures of which are in one of the
two forms of the whole tone scale :
Alban Berg, Nacht
Example 9-15
J0_
Copyright 1928 by Universal Editions, Vienna; renewed 1956 by Helene Berg; used by permission of Asso-
ciated Music Publishers, Inc.
*J o o o o o
It will be observed that whereas the perfect-fifth and minor-
second series may be transposed to eleven different pitches, giv-
ing ample opportunity for modulation, there is only one effective
modulation for the whole-tone scale— the modulation to the
whole-tone scale a half-tone above or below it, that is, from the
scale C-D-E-F#-G#-AJj: to the scale Db-Eb-F-G-A-B. Modal
modulation is' impractical, since the whole-tone scales on C, D,
E, etc., all have the same configuration:
Example 9-16
The two Major Second Hexads
i
^
tt.. ^^ ^" '-"^
^*.^ tjo 1
%T o
^
: — a —
2 2
(2)
(2)
In the introduction to Pelleas et Melisande Debussy begins
with the material of the perfect-fifth pentad for the first four
measures— C-D-E-G-A, changes to the pure whole-tone scale for
83
THE SIX BASIC TONAL SERIES
the fifth, sixth, and seventh measures, and returns to the perfect
fifth-series in measures 8 to 11:
Example 9-17
Debussy, "Pel leas and Melisonde*
bi Jtr
J.f^
i — J^'' ^J*"|r >T
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
From the same opera we find interesting examples of the use
of whole-tone patterns within the twelve-tone scale by alternat-
ing rapidly between the two whole-tone systems:
Example 9-18
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
Whereas the minor-second hexad may not be as bad as it
84
PROJECTION OF THE MAJOR SECOND
sounds, the careless use of the whole-tone scale frequently makes
it sound worse than it is, particularly when used by casual
improvisors. Because of the homogeneity of its material, it is
often used in the most obvious manner, which destroys the
subtle nuances of which it is capable and substitutes a "glob" of
"tone color."
The author is not making a plea for the return of the whole-
tone scale in its unadulterated form, but it must be said that
this scale has qualities that should not be too lightly cast aside.
Example 9- 19a gives the triads; 19b the tetrads, 19c the pentad,
and 19d the hexad, which are found in the six-tone scale. Play
them carefully, analyze each, and note their tonal characteristics
in the di£Ferent positions or inversions.
Example 9-19
(«)
(b)
^3=- = ii=;,^^^% = bEE-^^ = ^^^tb^
liPPjyftjj;^^
(c)
(d)
i>jJWii^iW^W*^^ir¥[lS.
hrrr^
4
85
THE SIX BASIC TONAL SERIES
Play the triad types in block form as in Example 9-20a. Repeat
the same process for the tetrad types in 20b; for the pentad type
in 20c; and for the hexad in 20d.
Example 9-20
(a)
i
^ ^r '^/^^
etc.
(h)
(c)
titijt. ^4 ^^r
etc.
In Example 9-2 la, experiment with the triad types in various
positions. Repeat the same process for the tetrads, as in 21b; for
the pentad, as in 21c; for the hexad, as in 21d.
Example 9-21
(a)
i i r F K 4 J ^^
f^
m
^^^
^
86
PROJECTION OF THE MAJOR SECOND
(b)
(hi i i\l i \«hi i J /h^^^
'}■■ f^f f "F
^
^^
^fe
(c)
i ii ^i itJ u ^
'>t ile tit <lp i
(|iiiij|g|iiiJ|itdiii
'>'.^^tp f#«f»f:
Experiment with different doublings and positions of all of the
above sonorities, as in Example 9-22.
Example 9-22
m
^
i
Have the material of Example 9-21 played for you in different
order and take it down from dictation, trying to reproduce not
only the notes but their exact position.
Analyze in detail the first section of Debussy's "Voiles" and note
87
THE SIX BASIC TONAL SERIES
not only his use of the widest resources of the scale but also his
employment of the devices of change of position and doubling.
In detailed analysis it seems generally wise to analyze every
note in a passage regardless of its relative importance, rather
than dismissing certain notes as "nonharmonic" or "unessential"
tones, for all tones in a passage are important, even though they
may be only appoggiaturas or some other form of ornamentation.
Occasionally, however, the exclusion of such "unessential" tones
seems obvious. The thirty-first measure of Debussy's "Voiles"
oflFers an excellent example of such an occasion. Every note in
every measure preceding and following this measure in the
first section of the composition is in the six-tone major-second
scale, Ab-B^-C-D-E-FJI:, with the exception of the two notes G
and D^ in measure 31, Since both of these notes were quite
obviously conceived as passing tones, it would seem unrealistic
to analyze them as integral parts of the tonal complex.
Debussy, "Voiles"
Example 9-23
4
^0^- — ^i^r —
*
^^
^^
^
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phik-
delphia. Pa., agents.
In using any of the tonal material presented in these chapters,
one all-important principle should be followed: that the com-
poser should train himself to hear the sounds which he uses
before he writes them. There is reason to fear that some young
composers— and some not so young— have been tempted at times
to use tonal relationships which are too complex for their own
aural comprehension. This is comparable to the use by a writer
of words which he does not himself understand— an extremely
hazardous practice!
88
PROJECTION OF THE MAJOR SECOND
When you feel confident of your understanding of the material,
write a short sketch which begins with the use of the major-
second hexad on C, modulates to the major-second hexad on G,
and returns at the end to the original hexad on C. See to it that
you do not mix the two scales, so that the sketch consists entirely
of major-second material.
89
10
Projection of the Major Second
Beyond the Six-Tone Series
We have already observed that the major-second scale in its
pure form cannot be extended beyond six tones, since the sixth
major second duphcates the starting tone. We can, however,
produce a seven-tone scale which consists of the six-tone major-
second scale with a foreign tone added, and then proceed to
superimpose major seconds above this foreign tone. We may
select this foreign tone arbitrarily from any of the tones which
are not in the original whole-tone scale. If we take, for example,
the perfect fifth above C as the foreign tone to be added, we
produce the seven-tone scale CoDoEoF#iGi*G#2A#(Co). (The
foreign tone is indicated by an asterisk to the right of the letter
name.) This again proves to be an isometric scale having the same
configuration of half-steps downward, 2221122; since if we begin
on the tone D and form the scale downward with the same order
of whole- and half-steps, we shall produce the same scale,
jD,aBb2AbiGiF#,E,2,(D):
Example 10-1
Major Second Heptad p4n n s d^
I
Jti» ' ^^ ^'
2 2 2 112
' It should be noted that the choice of G as the added foreign tone is arbitrary. The
addition of any other foreign tone would produce only a different version of the same
scale; for example, CiC#iD2E2F#2G#2A#,2)(C).
90
FURTHER PROJECTION OF THE MAJOR SECOND
We may now form the eight-tone scale by adding a major
second above G, that is, A: CJD^EM.C'GtjA.^'A^^^AC):
$
Example 10-2
Major Second Octod p'^m^n'^s^d'^t'^
^^ « tfo ' tf'
2 2 2
I I
The nine-tone scale becomes, then, the above scale with the
major second above A added, that is, B :
C2D2E2FiG,*G#,A,*A#iB,„*(C):
I
Example 10-3
Major Second Nonod pmnsdt
^^^^^^^
2 2 2
The ten-tone scale adds the major second above B, namely, C#,
CiC#,*D,E,F#,Gi*G#iA,*A#,B(,, * ( C ) :
*
Example 10-4
8 8 8 9 8 4
Major Second Decod p m n sdt
J^.. #^o 'i^^
r 1 2 2
I I I
The eleven-tone scale adds the major second above C#,
namely, Dfl:, C,C#i*DiDJfi*E2F#iGi*G#iA/A#iBa,*(C):
Example 10-5
». • c ^ n ^ , 10 10 10 10,10.5
Major Second Undecad p m n s d t
^1^ ojj. O It" 'i" 'J'" •
III 211
91
THE SIX BASIC TONAL SERIES
The twelve-tone scale adds the major second above DJ;, that
is, E#, and merges with the chromatic scale,
Example 10-6
.. • ,- . r^.. J 12 12 12 12 .12,6
Major Second Duodecad p m n s d t
■^ v> ij, o tf, ^> ■ ^^ ' fi-*
^^
I I
If we diagram this projection in terms of the twelve-tone
perfect-fifth series, we find that we have produced two hexagons,
the first consisting of the tones C-D-E-F#-G#-Ajj:, and the second
consisting of the tones G-A-B-Cij:-D#-Efl:. We employ first all of
the tones of the first hexagon, and then move to the second
hexagon a perfect fifth above the first and again proceed to add
the six tones found in that hexagon.
Example 10-7
A-'
The following table gives the complete projection of the
major-second scale with the intervallic analysis of each:
92
FURTHER PROJECTION OF THE MAJOR SECOND
C D
s
C D E
ms^
C D E F#
rn^sH
CD E F# G#
mHH^
CD E F# G# A#
m^sH^
CD E F# G G# A#
fm^nhHH^
CD E F# G Gt A
A#
fm^n's'dH^
CD E F# G G# A
A#
B
p^ni^n^s^dH^
C C# D E F# G Gif
A
A#
B
p'm'n's'dH^
C C# D D# E F# G
G#
A
A#
B
^10^10^10^10^10^5
C C# D D# E E# F#
G
G#
A
H B
^12^12^12^12^12^6
We have already observed that the six-tone major-second scale
contains only the intervals of the major third, the major second,
and the tritone. The addition of the tone G to the six-tone scale
preserves the preponderance of these intervals but adds the new
intervals of the perfect fifth, C to G and G to D; the minor
thirds, E to G and G to B^; and the minor seconds, F# to G and
GtoAb.
It adds the isometric triad ph, C2D5G; the triad pns, G7D2E,
and the involution Bb2C7G; the triad pmn, C4E3G, and the
involution GsB^^D; the triad pmd, G7D4F#, and the involution
Ab4C7G; the triad mnd, EsGiAt), and the involution F^iGsBb;
the triad nsd, GiAb2Bb, and the involution E2FJt:iG; the two iso-
metric triads, sd^, FJiGiAb, and nH, EgGsBb; and the triad pdt,
CeF^iG, with the involution GiAfjeD.
The addition of these triad forms to the three which are a
part of the major-second hexad, ms^, rrf, and mst, gives this
seven-tone scale all of the triad types which are possible in the
twelve-tone scale.
£!i
Example 10-8
pns and involution pmn and involution
THE SIX BASIC TONAL SERIES
pmd and involution mnd and involution nsd and involution
*J 7 4 47 ' 3 ^ ^'l ? I 2^ ' f I
Sd
7 4
2
n^t
pdt ond involution
^iiiJ JlJl .1 J^r I J i|J J;Ji-J f=
^ ^ I I 3 3 ■'■e I 16
The seven-tone impure major-second scale therefore has cer-
tain advantages over the pure six-tone form, since it preserves
the general characteristic of the preponderance of major seconds,
major thirds, and tritones but adds a wide variety of new
tonal material.
For the reasons given earlier, we shall spend most of our time
experimenting with various types of six-tone projections, since
we find in the six-tone scales the maximum of individuality and
variety. We shall make an exception in the case of the major-
second projection, however, and write one sketch in the seven-
tone major-second scale, since the addition of the foreign tone
to the major-second hexad adds variety to this too homogeneous
scale without at the same time entirely destroying its character.
It is a fascinating scale, having some of the characteristics of a
"major" scale, some of the characteristics of a "minor" scale, and
all of the characteristics of a whole-tone scale.
Begin by playing Example 10-9, which contains all of the
triads of the scale. Listen carefully to each triad and then com-
plete the analysis.
Example 10-9
^? iiiwiwi^i j^j :NiiJiJ,it^it^j ^'H»^".
fi<^\
^
s
u^
g
m
^^
^^
Example 10-10 contains all of the tetrad types, but in no
regular order. Play the example tRrough several times as sensi-
94
FURTHER PROJECTION OF THE MAJOR SECOND
tively as possible, perhaps with a crescendo in the third and
fourth measures to the first beat of the fifth measure, and then
a diminuendo to the end. Note the strong harmonic accent
between the last chord of the fifth measure and the first chord
of the sixth measure, even though the tones of the two chords
are identical.
Have another student play the example for you and write it
accurately from dictation. Now analyze all of the chords as to
formation including the sonorities formed by passing tones.
Tfc^* — \ — \ — r-
EXAIV
— \ h
[PLE 10-10
i-j J t*y
rthh
=f=r^
-f r ^
r r r
bp f p
#r^
\/ h f r £^
\
1 r 1 ^
— 1— ■
. 1
^^
^
m
^
TIT
3
C///77.| I I
^^
J ^J i
^.Lt^Ul
bJ^ J
^E^
r
F
g'T [j*
^
r^
fi^
T r r ' y.-
r
The following measure from Debussy's Pelleas et Melisande
offers a simple illustration of the seven-tone major-second scale,
the foreign tone, E^, merely serving as a passing tone:
Example 10-11
Debussy ," Pelleas ond Melisande"
±^
jij^.i^rribj'^ fofv t
w^
>^tl:^^b*)b^
^^fr^
2 2 2 I
Permission for reprint granied by Diirand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
95
THE SIX BASIC TONAL SERIES
A somewhat more complicated illustration is found in the
Alban Berg song, "Nacht," already referred to as beginning in
the pure whole-tone scale:
Example 10-12
Albon Berg, "Nacht
Copyright 1928 by Universal Editions, Vienna; renewed 1956 by Helene Berg; used by permission of Asso-
ciated Music Publishers, Inc.
m
,i'°,^'ii",b»(it.^
The student should now be ready to write a free improvisatory
sketch employing the materials of this scale (Example 10-1). He
will notice that the scale has two natural resting points, one on
C major and one on G minor. Begin the sketch in G minor,
modulate modally to C, establish C as the key center, and then
modulate back to the original key center of G. See that only the
tones C-D-E-Ff-G-Ab-Bb are employed in this sketch, but get as
much variety as possible from the harmonic-melodic material
of the scale.
96
11
Projection of the Minor Third
The next series of projections which we shall consider is the
projection of the minor third. Beginning with the tone C we
superimpose the minor third E^, then the minor third G^, form-
ing the diminished triad CgEbsGb, which consists of two minor
thirds and the concomitant tritone, from C to G^. Upon this we
superimpose the minor third above G^, B^^, which we shall call
by its enharmonic equivalent, A, forming the familiar tetrad of
the "diminished seventh," consisting of four minor thirds: C to
Eb, Eb to Gb, Gb to Bbb (A), and A to C; and two tritones: C to
Gb and Eb to A; symbol, nH^\
Example 11-1
Minor Third Tetrad u^\^
i
3
i.o ^o ^^^C^-)
As in the case of the major-second scale, which could not be
projected in pure form beyond six tones, so the minor third
cannot be projected in pure form beyond four tones, since the
next minor third above A duplicates the starting tone, C. If we
wish to extend this projection beyond four tones we must, again,
introduce an arbitrary foreign tone, such as the perfect fifth, G,
and begin a new series of minor-third projections upon the
foreign tone.*
** The choice of the foreign tone is not important, since the addition of any
foreign tone would produce either a different version, or the involution, of the
same scale.
97
THE SIX BASIC TONAL SERIES
The minor-third pentad, therefore, becomes C3Eb3GbiGt]2A:
Example 11-2
Minor Third Pentad pmn^sdt^
>obo "h* jJt^^tjJ ^
3 3 12
It contains, in addition to the four minor thirds and two tritones
aheady noted, the perfect fifth, C to G; the major third, E^ to G;
the major second, G to A; and the minor second, G^ to G. The
analysis of the scale is, therefore, pmn^sdt^. The scale still has a
preponderance of minor thirds and tritones, but also contains
the remaining intervals as well.
The six-tone scale adds a minor third above the foreign tone
G, that is, Bb, the melodic scale now becoming C3Eb3GbiG2AiBb.
The new tone, Bj^, adds another minor third, from G to Bj^; a
perfect fifth, from E^ to Bj^; a major third, from G^ to B^; a
major second, from B^ to C, and the minor second, A to B^, the
analysis being p^m^n^s^dH^:
Example 11-3
Minor Third Hexad p^m^n^s^d^t^
>o^» -- ^'^' jbjbJtiJ ^^r
The component triads of the six-tone minor-third scale are the
basic diminished triad CgE^gGb, nH, which is also duplicated on
Eb, Gb, and A;
Example 11-4
Minor Third Triads n t
the minor triads C3Eb4G and Eb3Gb4Bb, pmn, with the one
involution, the major triad Eb4G3Bb, which are characteristic of
98
PROJECTION OF THE MINOR THIRD
the perfect-fifth series;
Example 11-5
Triads pmn and involution
3 4 3 4 4 3
the triads C7G0A and Ej^yBl^aC, pns, with the one involution
6^)2^70; found in the perfect-fifth and minor-second series;
Example 11-6
Triads pns and involution
72 72 27
the triads Gt)iGk]2A and AiB^aC, nsd, with the one involution
GsAiBb, which we have also met as parts of the perfect-fifth and
minor-second projection;
Example 11-7
Triads nsd and involution
I
I 2
2 I
the triads Eb4G2A and G(;)4B|72C, mst, with no involution, which
we have encountered as part of the major-second hexad;
Example 11-8
Triads mst
i
4 2
4 2
the triads E^aGbiGt] and Gb3AiBb, mnd, with the one involution
GbiGtjsBb; which is a part of the minor-second hexad;
99
the six basic tonal series
Example 11-9
Triads mnd and involution
^
jJl^J lU J^f ibJ^J^f
3 I
3 I
1 3
and the triads CeG^iG and E^eAiBb, fdt, without involution,
which are new in hexad formations :
Example 11-10
Triads pdt
6 I
The student should study carefully the sound of the new
triads which the minor-third series introduces. He will, un-
doubtedly, be thoroughly familiar with the first of these, the
diminished triad, but he will probably be less familiar with the
triad ipdt. Since, as I have tried to emphasize before, sound is the
all-important aspect of music, the student should play and listen
to these "new" sounds, experimenting with diflFerent inversions
and different doublings of tones until these sounds have become
a part of his tonal vocabulary.
The tetrads of the six-tone minor-third scale consist of the
basic tetrad CgE^gGbgA, the familiar diminished seventh chord,
consisting of four minor thirds and two tritones, nH^, already
discussed;
Example 11-11
4 2
Minor Third Tetrad n t
^
the isometric tetrads C^¥.\)4GzB\), p^mn^s, and GsAiBbsC, pn'^s^d,
both of which we have already met as a part of the perfect-
fifth hexad, the latter also in the minor-second hexad;
100
projection of the minor third
Example 11-12
Tetrad p^mn s Tetrad pn s d
3 4 3
2 I 2
four new tetrad types, all consisting of a diminished triad plus
one "foreign" tone: C3Eb3Gb4Bt) and A3C3Et)4G, pmn^st;
C3Eb3GbxG4 and Eb3Gb3AiBb, pmnHt; GbiGt^^AgC and AiBbsCg
Eb, pn^sdt; Eb3GbiG^2A and Gb3AiBb2C, mnhdt;
Example 11-13
Tetrads pmn st
kfA
2
pmn dt
ji,j ^J t I ^ r-[ r 11^ J ^jfej I j^j J^p
3 3
pn^sdt
3 3 4
3 3 I
3 3
mn sdt
J ^J J r I ■* "r r r hj^J^j -^ i^J ■! ^^
2 3
I ' 2 3
3 I 2
the tetrads C6GbiGtl3Bb, and Eb4G2AiBb, both having the
analysis pmnsdt, the first appearance in any hexad of the twin
tetrads referred to in Chapter 3, Example 3-8;
Example 11-14
Tetrads pmnsdt
^ jbJuJ^r ibJ ^ ^'^r
6 I 3
4 2 I
and the two isometric tetrads EbsGbiGtisBb, prn^n^d, which will
be seen to consist of two major thirds at the interval of the minor
third, or two minor thirds at the relationship of the major third;
Example 11-15
Tetrad pm^n^d
(j I, J i'^ t^ ^r hi^^ \^if^
3 I 3
ni @ — a. @j]i
101
THE SIX BASIC TONAL SERIES
and GbiGtisAiBb, mn^sd^, which consists of two minor thirds at
the interval relationship of the minor second, or two minor sec-
onds at the interval of the minor third :
Example 11-16
Tetrad mn'^sd
I 2 1 *" n. @d d_ @ji
The pentads consist of the basic pentads C3Et)3GbiGfcj2A, and
EbaGbsAiBbsC, pmn^sdt^;
Example 11-17
Minor Third Pentads pmn'^sdt^
li^J'■J^^^I^J^^^
3 3 12
3 3 12
the pentad CgE^gGbiGtisBb, p^m^nhdt, which may also be ana-
lyzed as a combination of two minor triads at the interval of the
minor third;
Example 11-18
Pentad p^m^n^sdt
liU^V^I^i^
3 3 13 p mn @ n^
the pentad C3Et)4G2AiBb, p^mn^s^dt, which may also be analyzed
as two triads pns at the interval of the minor third;
Example 11-19
.2 3^2,
ti -J- ''• I -il- P*"
3 4 2 1 pns @ £
102
PROJECTION OF THE MINOR THIRD
the pentad E^aGbiGtioAiBb, pm~n^sdH, which may also be ana-
lyzed as the combination of two triads mnd at the interval of the
minor third;
Example 11-20
Pentad pm^n^sd^t
ffl [.JbJ^J Ji^f l^jJ^JbJJ^f
3 12 1
3 1 3 1
mnd @ ji
and the pentad GbiGl:]2AiBb2C, pmrfs^dH, which may be
analyzed as the combination of two triads nsd at the interval
of the minor third;
Example 11-21
Pentad pmn^s^d^t
12 12
I' 2 I '2
nsd @ _n_
The contrast between the six-tone major-second scale and the
six-tone minor-third scale will be immediately apparent. Whereas
the former is limited to various combinations of major thirds,
major seconds, and tritones, the latter contains a wide variety
of harmonic and melodic possibilities. The scale predominates,
of course, in the interval of the minor third and the tritone, but
contains also a rich assortment of related sonorities.
Subtle examples of the minor-third hexad are found in
Debussy's Pelleas et Melisande, such as:
Example 11-22
Debussy, "Pelleas and h^elisande"
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
103
THE SIX BASIC TONAL SERIES
Play each of the triads in the minor-third hexad in each of its
three versions, as indicated in Example 11-23. Play each measure
several times slowly, with the sustaining pedal held. If you have
sufficient pianistic technic, play all of the exercises with both
hands in octaves, otherwise the one line will suffice. Now analyze
each triad.
Example 11-23
1;- 1 rn. i>^ ^p^r
^jjii.mi^irmi^LJ''^^LjLLJ
fpj^^dripi"^LJ'^-^'u^Lii
i^ i^^^
i^n^^^^dlifj^alLLS^^^'iLlL
i
bm \;mr
bf^ M
I) 1, iff ^^^ JJ It-i^^ [^ ^cU 1^^^ k^ ^LL
104
PROJECTION OF THE MINOR THIRD
Repeat the same process with the tetrads of the scale:
Example 11-24
|jP^.mc:tfLtfrijy..^clJ^Lffl
,,f, jw n^^^a!J\^.P^i^ ^ci^
(liP^i-^crJcdJ
^F Lph
JJ^^ft^^MJ^^^yrJ^cfT
p^^crtfrdT^cttri^ffl^^ciLrigj
bp ^f^F ^f^f-j^
^^
k-.^ b>f- b*r^i
^
b[B bet?
^a^^'LlU^^.W^W
(| i'i?^^r£jc!lin''cll
Repeat the same process with the six pentads and the hexad
Example 11-25
^ jM ^ M^'^^r^r r r'l ^^r r'll [/'tT^
105
#
J^JJ^^^^^^ypJ
THE SIX BASIC TONAL SERIES
I jjt-^t>'' bJ^JjJ^ ''r^r Y r 't ^r'T r'TT^c_r
vH
(| jn7i:^,jT3T:^cxUlrciiir
I, b. 1,-Vlir, irWr
'cmrftc^^rrTc'TrTTyrT_l
^^^^^^
V4
y^kr f- ^rk^'t'''
^^^^^^^^^^^^
'te--±^
^H
fe^
One of the most important attributes of any sonority is its
degree of consonance or dissonance, because the "tension"
induced by the dissonance of one sonority may be increased,
reduced, or released by the sonority to which it progresses. An
interesting and important study, therefore, is the analysis of the
relative degrees of dissonance of diiferent sonorities.
At first glance, this may seem to be an easy matter. The inter-
vals of the perfect octave; the perfect fifth and its inversion, the
perfect fourth; the major third and its inversion, the minor sixth;
and the minor third and its inversion, the major sixth, are
generally considered to perform a consonant function in a
sonority. The major second and its inversion, the minor seventh;
106
PROJECTION OF THE MINOR THIRD
the minor second and its inversion, the major seventh; and the
tritone (augmented fourth or diminished fifth) are generally
considered to perform a dissonant function. When these intervals
are mixed together, however, the comparative degree of dis-
sonance in different sonorities is not always clear. Some questions,
indeed, cannot be answered with finality.
We may safely assume that the dissonance of the major
seventh and minor second is greater than the dissonance of the
minor seventh, major second, or tritone. To the ears of many
listeners, however, there is not much difference between the
dissonance of the minor seventh and the tritone.
Another problem arises when we compare the relative con-
sonance or dissonance of two sonorities containing a different
number of tones. For example, we might conclude that the
sonority C-E-F#-G is more dissonant than the sonority C-F#-G,
since the second contains two dissonances— the minor second and
the tritone, whereas the first contains three dissonances— the
minor second, the tritone, and the major second. However, it
might also be argued that whereas the sonority C-E-F#-G con-
tains a larger number of dissonant intervals, C-FJf-G contains a
greater proportion of dissonance. The analysis of the first sonority
is pmnsdt—one-hali of the intervals being dissonant; whereas the
analysis of the second sonority is pcff— two-thirds of the intervals
being dissonant:
Example 11-26
Tetrad pmnsdt
Triad pdt
m' i i »' ii^i^U I fe° i v^- d
Finally, it would seem that the presence of one primary dis-
sonance, such as the minor second, renders the sonority more
dissonant than the presence of several mild dissonances such as
the tritone or minor seventh. For example, the sonority C-D#-E-
G, with only one dissonant interval, the minor second, sounds
107
THE SIX BASIC TONAL SERIES
more dissonant than the tetrad C-E-Bt>-D, which contains four
mild dissonances:
2 2
Tetrad pm n d
Example 11-27
2 3
Tetrad m s t
With the above theories in mind, I have tried to arrange all
of the sonorities of the minor-third hexad in order of their
relative dissonance, beginning with the three most consonant
triads— major and minor— and moving progressively to the in-
creasingly dissonant sonorities. Play through Example 11-28
carefully, listening for the increasing tension in successive sonori-
ties. Note where the degree of "tension" seems to remain
approximately the same. Analyze all of the sonorities and see if
you agree with the order of dissonance in which I have placed
them. Have someone play the example for you and take it down
from dictation:
Example 11-28
'^ r^ J tl-~"
— 1 —
■3i
3
\rh 'i J J
^
iittii.-.
hN
■
i4 J f^Tw
1-^ r r LJJi
W=^
p
3
ffi
108
PROJECTION OF THE MINOR THIRD
Reread Chapters 6 and 7 on modal and key modulation.
Since the minor-third hexad has the analysis p^m^n^s^dH^,
it is evident that the closest modulatory relationship will be at
the interval of the minor third; the next closest will be at the
interval of the tritone;* and the third order of relationship will
be at the interval of the perfect fifth, major second,
major third, or minor second. Modulation at the interval
of the minor third will have five common tones; at the tritone,
four common tones; at the other intervals two common tones.
Example 11-29
Modulation of Minor Third Hexad p m n s d t^
^
M'^'^'
^53
^
pr^^^
^^^
f^
Modulation @ n^
@1
@P
i: ..kJ^"^*
^l^rt^<
^^^^
^
^
i
-0 —
@ S
!?• ljv\ •
7- bo^' '1' '
^
rWV4^
@ m
^
^
P^
\}m bot|<
, k^b* ^
Write a sketch using the material of the minor-third hexad.
Begin with C as the key center and modulate modally to E^ as
the key center, and back to C. Now perform a key modulation to
the minor-third hexad a minor third below C (that is. A);
modulate to the key a fifth above (E), and then back to the
key of C.
See Chapter 17, pages 139 and 140.
109
Involution of the Six-Tone
Minor-Third Projection
12
The first three series of projections, the perfect fifth, minor
second, and major second, have all produced isometric scales.
For example, the perfect-fifth six-tone scale C2D2E3G2A2B, begun
on B and constructed downward, produces the identical scale,
B2A2G3E2D2C. This is not true of the six-tone minor-third projec-
tion. The same projection downward produces a different scale.
If we take the six-tone minor-third scale discussed in the
previous chapter, C3Eb3GbiGti2AiBb, and begin it on the final
note reached in the minor-third projection, namely, B^, and
produce the same scale downward, we add first the minor third
below B\), or G; the minor third below G, or E; and the minor
third below E, or Cjj:.
Example 12-1
Mi nor Third Tetrad
downward
^
at^
We then introduce, as in the previous chapter, the foreign tone
a perfect fifth below B\), or E\), producing the five-tone scale
BbsGsEkiiEbsCJ:
Example 12-2
Minor Third Pentad
110
INVOLUTION OF THE MINOR-THIRD PROJECTION
By adding another minor third below E^, or C, we produce the
six-tone involution BbgGsEtiiEboCjfiCfc]:
Example 12-3
Minor Third Hexad
*
b. ^^ 7 ^ J ^^
t-
A simpler method would be to take the configuration of the
original minor third hexad, 3 3 121, beginning on C, but in
reverse, 1213 3, which produces the same tones, CiCJsEbiEtjs
GsBb:
Example 12-4
Minor Third Hexad upward
Involution
^
£
[.o bo t?o
bo |;)4
t^
If we examine the components of this scale we shall find them
to be the same as those of the scale conceived upward but in
involution. The analysis of the scale is, of course, the same:
p^m^n^s^dH^. We find, again, the four basic diminished triads
C^gEsG, EgGsBb, G3Bb3Db(C#), and A#(Bb)3C#3E;
Example 12-5
Minor Third Triads n^^t
4
V
i-jl \,^{tr)iHf^
the major triads— (where before we had minor triads)— C4E3G
and Eb4G3Bb, with the one involution, the minor triad C3Eb4G;
t
Example 12-6
Triads pmn and involution
H ''■■i>i
111
THE SIX BASIC TONAL SERIES
the triads BbsCyG and Db(Cfl:)2Eb7Bb, pns, with the one
involution, Eb7Bt)2C;
Example 12-7
4
Triads pns
and
involution
Y-~
^ i •> pjur 'bJ T r
2 7
2 7
7 2
the triads Bl:)2CiDb(C#) and CJfaDJiE, nscZ, together with the
one involution CiDb2Eb;
i
Example 12-8
Triads nsd and involution
J'aiU lijJ^J^^
2 1 " 2 I
the triads Bb2C4E and Db2Eb4G, mst;
Example 12-9
Triads mst
I 2
ITlTg ^ 2 4
the triads CiCJgE and DJiEgG, mnc/, with the one involution
CsDfiE;
Example 12-10
4
Triads mnd
and involution
^,t^3^^^A '^jt
3 13
and the triads CiCJfeG and D^iEgBb, pdt:
Example 12-11
Triads pdt
i
WFWf
\v 6
I 6
112
INVOLUTION OF THE MINOR-THIRD PROJECTION
The tetrads consist of the same isometric tetrads found in the
first minor- third scale: the diminished-seventh tetrad, CifgEgGa
Bb, nH^, the other isometric tetrads, C3Eb4G3Bb, jrmn^s,
CsDJiEsG, pmVc/, BbsCiDbsEb, pnVc/, and CiDbsEbiEl^,
rmnrsd^;
Example 12-12
^Tetrad n^^t^ Tetrad p^mn^s Tetrad pm^n^d Tetrad pn^s^d Tetrad mn^d
333 1343 313 212 121
four tetrads consisting of a diminished triad and one foreign
tone, each of which will be discovered to be the involution of a
similar tetrad in the first minor-third scale: C4E3G3Bb and
Eb4G3Bb3Db, pmnht; CiCJfsEsG and D^iEgGsBb, pmnHt-
GgBbsCiDb and Bb3Db2EbiEl^, pnhdt; and Bb2CiC#3E and
C^sDftiEsG, mn^sdt;
Tetrads pmn^st
Example 12-13
Tetrads pmn ^dt
A ^ y^ A y, yk ^ ^
4 3 3
Tetrads pn^sdt
3 3
Tetrads mn^ sdt
I 3 3
\?m p t^p I \?m y '7 ^
^
vi-y^ ^ \-ii^
3 ' 2 I
3 2 I
2 I 3
and the "twins", CgD^iEsBb and CiDb2Eb4G, pmnsdt, the involu-
tions of similar tetrads discussed in the previous chapter:
Example 12-14
Isomeric Tetrads pmnsdt
J jt^ J 't Ui-J 1^
3 I 6
I 2 4
113
THE SIX BASIC TONAL SERIES
The pentads consist of the basic pentads CJaDJiEgGsBb and
Bb2CiC|:3E3G, pmn'^sdt^ (the involutions of the basic pentads in
the previous chapter);
Example 12-15
Minor Third Pentads pmn sdt^
^^^^^^
2 13 3
2 13 3
the pentad CgEj^iEtjsGgBb, p^m^n^sdt, which may be analyzed as
a combination of two major triads at the interval of the
minor third;
Example 12-16
Pentod p^m^n^sdt
I i ^J ii J ^ V •' i0^
3 13 3
amn @ n_
the pentad CiDb2Eb4G3Bb, p^mn^s^dt, which may be analyzed
as the combination of two triads, pns, at the interval of the
minor third;
Example 12-17
Pentad p^mn's^dt
l-iJ^J^YLi^J^Jt
12 4 3
2 7 ^2 7
the pentad CiDb2EbiEti3G, pm^n^sdH, which may be analyzed as
the combination of two triads, rand, at the interval of the
minor third;
Example 12-18
Pentod pm^n^sd^t
g, I CIIIUW ^111 II 3U I
12 13 I 3 _ I 3
mnd @ _n_
114
INVOLUTION OF THE MINOR-THIRD PROJECTION
and the pentad BboCiDboEbiEt], pmn^s^dH, which may be an-
alyzed as the combination of two triads, nsd, at the interval of
the minor third:
Example 12-19
Pentad pmn^s^d^t
^^^^^m
2 12 1
2' 1 2 1
nsd @ _n_
All of the above pentads will be seen to be involutions of
similar pentads discussed in the previous chapter.
From the many examples of the involution of the minor-third
hexad we may choose two, first from page 13 of the vocal score
of Debussy's Pelleas et Melisande;
Example 12-20
Debussy, "Pelleas and Melisande"^ ____^
■m--^-m--r-0-r-
^y- U]ITi\'^^
p? ^Jt<al;„tg
p
m
"3:
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
and from the second movement of Benjamin Britten's Illumina-
tions for voice and string orchestra:
Example 12-21
Benjomin Britten, "les Illummations"
772./ espress.
espress. e sost.
Copyright 1944 by Hawkes & Son (London), Ltd. Used by permission of Boosey & Hawkes, Inc.
115
THE SIX BASIC TONAL SERIES
Analyze the following two measures which come at the end of
a section of Debussy's "Les fees sont d'exquises danseuses." If all
of the notes of the two measures are considered as integral parts
of one scale, we have the rather complex scale iC-Cb-B^-A-Ab-
G-Gb-F-E^-D composed of the two minor-third tetrads, jC-A-Gb-
E^ and iF-D-C^-Ab, plus the minor third, Bt)-G (forming the
ten-tone minor-third projection).
A closer— and also simpler— analysis, however, shows that the
first measure contains the notes of the minor-third hexad
|F-D-Cb-Ab-Bb-G, and the second measure is the identical scale
pattern transposed a perfect fifth, to begin on C, I C-A-Gb-E^-
F-D.
This simpler analysis is much to be preferred, for most com-
posers, whose desire is to communicate to their listeners rather
than to befuddle them, tend to think in the simplest vocabulary
commensurate with their needs.
Example 12-22
Debussy, "Les fles sont d'exauises danseuses"
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
*
'a\^>-^
i^U^be»[;t,k.^=^
_:H-^^»t>op
^=*^
A detailed comparison of the material of the minor-third
hexad discussed in Chapter 11 with that of the material in
Chapter 12 will indicate that the isometric material of the two
116
INVOLUTION OF THE MINOR-THIRD PROJECTION
hexads is identical, but that where the sonorities have involu-
tions, each sonority of one scale is the involution o£ a similar
sonority in the other. For example, the minor-third hexad dis-
cussed in Chapter 11 contains two minor triads and one major
triad, whereas the involution of the hexad contains two major
triads and one minor triad. The involution does not, therefore,
strictly speaking, add any new types of sonorities, but merely
substitutes involutions of those sonorities.
117
13
Projection of the Minor Third
Beyond the Six-Tone Series
We produced the six-tone minor-third scale in Chapter 11 by be-
ginning on any given tone, superimposing three minor thirds
above that tone, adding the foreign tone of the perfect fifth, and
superimposing another minor third above that tone.
We may now complete the series by superimposing two more
minor thirds, thereby completing a second diminished-seventh
chord, then adding a second foreign tone a perfect fifth above
the first foreign tone, and superimposing three more minor thirds,
thereby completing the third diminished-seventh chord. For the
student who is "eye-minded" as well as "ear-minded," the
following diagram may be helpful:
Example 13-1
118
FURTHER PROJECTION OF THE MINOR THIRD
Here it will be seen that the minor-third projection divides the
twelve points in the circle into three squares, the first beginning
on C, the second on G, and the third on D. We begin by super-
imposing E\), G[}, and A above C, then adding G and super-
imposing B\), D\), and F^ (E), and then adding D and super-
imposing F, A\), and Cj^ (B):
The scales thus produced, with their respective analyses,
become:
Example 13-2
i
Minor Third Heptad p^m^n^s'^d^t^
l>o ^g* " ^*
^^
i J ^J ^^ ^^ ^
I 2 3 I 2 I
4 4 8 4 4 4
Minor Third Octad p m n s d t
|^^^^§^^
t;o bo " tl>
2 I 2 I 2 I
Minor Third Nonad p^ m^ n^ s® d^ t"^
^
b« ^* If.
ij i^J ^J J ^r
•*- = I r I I
r I I 2 I 2 I
Minor Third Decad p^m^n^s^d^ f^
^
S
^jgiJllJbJ^J Jg
^
^
*
11111112 1
Minor Third Undecod p'^m'^n'^ s'^d'^t^
. k- , iT^b-*-
^^^S^^
*?:
=S
I I I I I I I I
Minor Third Duodecod p'^nn'^n'^ s '.^d'^t^
i
b»b» b,l;i'
-K;:b^
UlJliJbJ^J^J^J^r^
^^
P^bo^° " tl'
ibJ^J^JI^^Il^
I I I I I I I I I
All of these scales are isometric with the exception of
the seven-tone scale, the involution of which produces a
different scale:
119
THE SIX BASIC TONAL SERIES
Minor Third Heptad
Example 13-3
Involution
These scales with their rich variety of tonal material and their
generally "exotic" quality have made them the favorites of many
contemporary composers.
A beautiful example of the eight-tone minor-third scale will be
found in the first movement of Stravinsky's Symphony of Psalms,
Example 13-4, where the first seven measures are consistently in
this scale, EiF2GiG#2A#iB2C#iD :
Example 13-4
2 1 2 I 2 I (2)
Strovinsky, Symphony of Psalms
I AlTos
/1AIT03
Ex
J J ^'
J J IJ J J J
1-1. I
>>;j.^^X
J
'^m
mm
ro - tl - 0 - nem me -am,
J3 iri J7J
rp'rrmri
^^m
^^
^m
"^m
Copyright by Edition Russe de Musique. Revised version copyright 1948 by Boosey & Hawkes, Inc. Used
by permission of Boosey & Hawkes, Inc.
120
FURTHER PROJECTION OF THE MINOR THIRD
A completely consistent use of the involution of the seven-tone
minor-third scale will be found in the first movement of the
same composer's Sijmphony in Three Movements, beginning at
rehearsal number 7, and continuing without deviation for
twenty-three measures :
Example 13-5
Stravinsky, Symphony in Three Movements ,
^
^"
■^•^mr
mf
marcato
I- ^^T-Vi '■i^'r^^
V
i
Dizz *""*^°
pizz.
m
f f b
j} l?Qj> */ */ gg
-^-»-
^
i
3_^ pizz.
orco
p — J-
•OS?
pizz.
-»-y-
^m
^
a Jj^»^i5
pocosj
06p ^•^ ^J5?
^4?
orco
I — s ■
pizz.
-1~ t»
a n^yW
pizz.
s^
^
vT J J r
I J r
pocosjz p
■mf
pizz.
^
^
^
^
Copyright 1946 by Associated Music Publishers, Inc., New York; used by permission.
^^-i 12 3 12 I
121
THE SIX BASIC TONAL SERIES
Another interesting example of the eight-tone minor-third
scale is found at the opening of the third movement of Mes-
siaen's V Ascension:
Example 13-6
Messloen ,"l' Ascension"
Vif
>^M\h "/g^^^ -.
% i- itJiitfeaS
Reproduced with the permission of Alphonse Leduc, music publisher, 175 rue Saint-Honore, Paris. Copy-
right by Alphonse Leduc.
I
^ 4- *■ fl"0 = I 2 I 2 I 2 I (21
Analyze further the Stravinsky Symphony of Psalms and try to
find additional examples of the minor-third projection.
122
14
Projection of the Major Third
We have observed that there are only two intervals which can
be projected consistently through the twelve tones, the perfect
fifth and the minor second. The major second may be projected
through a six-tone series and then must resort to the interjection
of a "foreign" tone to continue the projection, while the minor
third can be projected in pure form through only four tones.
We come now to the major third, which can be projected only
to three tones. Beginning again with the tone C, we superimpose
the major third, E, and the second major third, E to G#, produc-
ing the augmented triad C-E-G# consisting of the three major
thirds, C to E, E to G#, and G# to B# (C), m^:
Example 14-1
Major Third Triad mj
I ^ ° tf° ^°'"-'
To project the major third beyond these three tones, we again
add the foreign tone Gtj*, a perfect fifth above G, producing the
basic major-third tetrad G4E,oGiGJj: having, in addition to the
three major thirds already enumerated, a perfect fifth, from C
to G; a minor third, from E to G; and a minor second from
GtoG# {k\));pnv'nd:
• Here the choice of the foreign tone is more important, since the addition of D, F|, or
A# with their superimposed major thirds would duplicate the major-second hexad. The
addition of any other foreign tone to the augmented triad produces the same tetrad in a
different version, or in involution.
123
the six basic tonal series
Example 14-2
Major Third Tetrad pm^nd
t^ ^. J J^g
4 3 I
To produce the pentad, we superimpose a major third above
G, or B, forming the scale C4E3GiG#3B, and producing, in addi-
tion to the major third, G to B, the perfect fifth, E to B; the
minor third, G# to B; and the minor second, B to C; p^m^n^d^:
Example 14-3
.Major Third Pentad p^m^n^d^
To produce the six-tone major-third scale, we add the major
third above B, or D^, giving the scale CgDJiE.sGiGJfsB. The new
tone, D^, in addition to forming the major third, B to DJj:, adds
an additional major third, from Dfl: (El^) to G. It also adds
another perfect fifth, G# to DJf; a minor third, C to DJj: (E^);
and a minor second, DJj: to E; p^m'^n^(P.
Example 14-4
Major Third Hexod p^m^n'^d^
i
iitJ ^ ^ «^ r
*
If we proceed to analyze the melodic-harmonic components of
this six-tone major-third scale, we find that it contains the
augmented triad, which is the basic triad of the major-third
scale, m^, on C and on G. It contains also the major triads
C4E3G, E^GifsB and G#4B#3(C)D#, pmn, with their involutions,
the minor triads C3Eb4(D#)G, E3G4B, and Gjj^,B,Djj^;
124
projection of the major third
Example 14-5
Triads pmn and involutions
and the triads C,G4B, E,B4D#, and Ab(G#),Eb(D#),G, pmd,
together with their invohitions C^E^B, E4G#7D# and Ah(G#)4
C^G:
Example 14-6
Triads pmd
and involutions
74 74 74 47 47 47
Finally, it contains the triads CJD^-JE, EgGiGJ, and GJyBiC,
mnd, with the involutions BiCsDfl:, DJiEgG, and GiG^sB, which
have already been seen as parts of the minor-second and minor-
third scales but which would seem to be characteristic of the
major-third projection:
Example 14-7
Triads mnd
I id J J ^^^ i^r r
and involutions
ji^j tfjj^^ ^m
13 13 13
^
3 I
3 I
The tetrads consist of the basic tetrads, new to the hexad
series, C4E4G#3B, E4G#4B#3(C)D#, and Ab(G#)4C4E3G, which
are a combination of the augmented triad and the major triad,
pm^nd, together with their involutions C3Eb4G4B, Etj3G4B4D]|:,
and G#3B4Dfl:4F-)<(-(Gt:]), which consist of the combination of the
augmented triad and a minor triad;
Example 14-8
Major Third Tetrads pm^nd and involutions
THE SIX BASIC TONAL SERIES
the isometric tetrads C4E3G4B, E^G];^^Bj:>g and Ab4(G#)C3Eb4
(DJj:)G, p~m~nd, which we first observed in the perfect-fifth
projection;
Example 14-9
2 2
•Tetrads p m nd
^"%34 434 4 -54
the isometric tetrads CgDSiEaG, EsGiGJsB, and GI^BiCsDJ,
pm^n-d, which we have encountered as parts of the minor-
third series;
Example 14-10
.Tetrads pm n^d
3 13 3 13 3 13
and the isometric tetrads B^C^Dj^^E, DJiEgGiGJ, and GiGJgBiC,
pmrnd^, which can be analyzed as two major thirds at the inter-
val of the minor second, or two minor seconds at the interval of
the major third, previously observed in the minor-second series:
^Tetrads pm^nd^
7' 1
Example 14-11
1 \-rr-\ ! — \-r, — 1 — = — w—
r-^ \ — 1
-^r:iitJ ^ i^ •
J J(t^ ^JjtJ r ^
Ni— ^
^ ' ^ ^J- "
I 3 I
I 3 I
I 3 I
m @ d
d (g m
The pentads consist only of the basic pentads C4E3GiG#3B,
E4G#3BiC3D#, and Ab4(Gt)C3D#iE3G^ p-m^n'd^ together with
their involutions C3DtfiE3G4B, E3GiG#3B4D#, and Ab3(G#)Bi
C3Eb4(D#)Gti.
Example 14-12
Major Third Pentads p^m^n^d^
and involutions
PROJECTION OF THE MAJOR THIRD
From this analysis it will be seen that the six-tone major-third
scale has something of the same homogeneity of material that
is characteristic of the six-tone major-second scale. The scale
includes only the intervals of the perfect fifth, the major third,
the minor third, and the minor second, or their inversions. It
does not contain either the major second or the tritone. It is,
however, a more striking scale than the whole-tone scale, for it
contains a greater variety of material and varies in consonance
from the consonant perfect fifth to the dissonant minor second.
The six-tone major-third scale is an isometric scale, because if
we begin the scale CgDSiEgGiGJgB on B, and project it in
reverse, the order of the intervals remains the same. There is,
therefore, no involution as was the case in the minor-third scale.
A clear example of the major-third hexad may be found in the
sixth Bartok string quartet:
Bartok, Sixth Quartet
Vivacissimo
Example 14-13
Copyright 1941 by Hawkes & Son (London), Ltd. Used by permission of Boosey & Hawkes, Inc.
P^
(b«^
3 13 13
127
THE SIX BASIC TONAL SERIES
An harmonic example of the same scale is illustrated by the
following example from Stravinsky's Petrouchka:
Stravinsky, "Petrouchko"
Example 14-14
^^
P
p
m
cresc.
i
0
g VIos. ^ _
j'^^bS ^% l^s ^
b*-! # V-l ^^
j! [b^§ ^^ J jiJ ^r t r ]
3 13 13
Copyright by Edition Russe de Musique. Revised version copyright 1958 by Boosey & Hawkes, Inc. Used
by permission of Boosey & Hawkes, Inc.
A purely consonant use of this hexad may be found in the
opening of the author's Fifth Symphony, Sinfonia Sacra:
Example 14-15
Honson, Symphony No. 5
Bossesby- b^S- . tt ! — ^. M^ y-
H.
W
Copyright © 1957 by Eastman School of Music, Rochester, N. Y.
A charming use of this scale is the flute-violin passage from
Prokofieff's Peter and the Wolf:
Example 14-16
Prokofieff, "Peter and the Wolf"
Fl.
Copyright by Edition Russe de Musique; used by permission.
128
PROJECTION OF THE MAJOR THIRD
P
\^A
SE
b^N
'r 'ir r TT
3 13 13
Play the triads, tetrads, pentads, and the hexad in Example
14-17 which constitute the material of the major-third hexad.
Play each measure slowly and listen carefully to the fusion of
tones in each sonority:
Example 14-17
rPi^iVrnmi^
lU'jJm^-"^'l^
^^
m
^
Is
■^' ^^JbJ\J
^^
(|j7;pja^i-^ijjr,.[Trpi^
129
THE SIX BASIC TONAL SERIES
Experiment with different positions and doublings of the
characteristic sonorities of this scale, as in Example 14-18:
Example 14-18
^
etc.
%
etc.
P
etc.
^=H
W
(j d n J
d i^ ^i
^H i^ ^«»
etc .
etc.
etc.
/ u, ■
i ^ 'T H
Hi i ii
The following exercise contains all of the sonorities of the
major-third hexad. Play it through several times and analyze
each sonority. Have someone play through the exercise for you
and take it down from dictation :
Example 14-19
^^
^^^ff
^«
^^
^^
#^^
^^
^m
^
m
130
PROJECTION OF THE MAJOR THIRD
(|^ "^ n'-JlJ Ji.^ Lnj 4 d liti w ^
tfc^
^
^
¥
*
S
Write a short sketch Hmited to the material of the major-third
hexad on C.
Example 14-20 illustrates the modulatory possibilities of this
scale. Modulations at the interval of the major third, up or down,
produce no new tones; modulations at the interval of the perfect
fifth, minor third, and minor second, up or down, produce three
new tones; modulations at the interval of the major second and
the tritone produce all new tones.
Example 14-20
p^m^n^d^
S
3 13 13
og»
oflo
Modulation @ m
Modulation @ p
^S
^3
n-e-
^
#
^
^
,j|. olt'""'
>^°'«°'
^^
^^
@ n
@d
7- .»^."'«"
@1
^^
^
Write a short sketch which modulates from the majors-third
hexad on C to the major-third hexad on D, but do not "mix"
the two keys.
131
Projection of the Major Third
Beyond the Six-Tone Series
15
If we refer to the diagram below we see that the twelve
points in the circle may be connected to form four triangles: the
first consisting of the tones C-E-Gif; the second of the tones
Gt]-B-D#; the third of the tones Dt^-F#-A#; and the fourth of the
tones Ati-C#-E#:
Example 15-1
We may, therefore, project the major third beyond the six
tones by continuing the process by which we formed the six-tone
scale. Beginning on C we form the augmented triad C-E-G#;
132
FURTHER PROJECTION OF THE MAJOR-THIRD
add the foreign tone, Gt|, and superimpose the augmented triad
G-B-DJj:; add the fifth above the foreign tone G, that is, Dt], and
superimpose the augmented triad D-F#-AJ|:; and, finally, add the
fifth above the foreign tone D, or At], and superimpose the
augmented triad A-Cj-E^f. Rearranged melodically, we find the
following projections :
Seven tone: C-E-G# + G-B-D# +• D^ = CsDiDJiEaGiG^gB,
p^m^n^s^dH, with its involution CaDJiEgGiGJiAaB:
Example 15-2
Major Third Heptad p'''m®n'*s^d'*t
and involution
■^ = 2T13I3 ^1^119
3 13 1 12
Eight tone-. C-E-G# + Gt^-B-DJ + Dtj-FJ = CaDiDJiEaFJiGi
G#3B, fm'nhHH^, with its involution CgD^iEiFaGiGJiAaB:
Example 15-3
Major Third Octad p^m^ n ^ s'^d ^ t^ and involution
^^
3^
iJiJjit^^«-'r ■ j|j JJ ^11^^ I
2 113
3 I I 2 I 12
ISline tone: C-E-G# + Gti-B-D# + Dt^-FJf-AJ, =■ CaDiDJiEs
F#iGiG#2A#iB, p^m^n^s^dH^:
Example 15-4
Major Third Nonod p^m^n^s^d^t'
li ^ M ^H , 1 1 1 1
J H J itJ , 1
«^ -»- -H = ^2 r 1 2 1
1 2 1
(This is an isometric scale, for if we begin the scale on A# and
proceed downward, we have the same order of whole and half
steps, 21121121.)
133
THE SIX BASIC TONAL SERIES
Ten tone: C-E-Gif + G\\-B-Dj^ + Dt^-Ff-AJ -f Al^
E2F#iGiG#iAiA#iB, fm^nhHH'-.
Example 15-5
Major Third Decad p^m^n^s^d^t"*
QDiDfti
^
^
j J^J Jj[J-'tfJ^1^
2 112 1 II II
(This scale is also isometric, for if we begin the scale on F# and
progress downward, we have the same order of whole and
half-steps. )
Eleven tone: C-E-G# + Gti-B-D# + Dt;-F#-Afl + Al^-Cft =
CiC#iDiD Jf lE^FSiG.GJi AiAliB, f'w}'n''s''fH' :
Example 15-6
Major Third Undecod p'^m'^n'^s'^d'S^
^^^^^^
^
-*-
11 I 12 1 I I I I
Twelve tone: C-E-Gif + Gtj-B-Dft + Dt^F#-A# + Al^-C#-Et
C,C#,D,D#,E,E#,F#,G,G#,AiAif,B,p^WW^c/^-T:
Major Third Duodecac
^^
Example 15-7
12 I2„I2,I2 .12.6
p m n s d t
rff^i r I I I I I III
( The eleven- and twelve-tone scales are, of course, also isometric
formations. )
The student will observe that the seven-tone scale adds the
formerly missing intervals of the major second and the tritone,
while still maintaining a preponderance of major thirds and a
proportionately greater number of perfect fifths, minor thirds,
and minor seconds. The scale gradually loses its basic character-
istic as additional tones are added but retains the preponderance
of major thirds through the ten-tone projection.
134
FURTHER PROJECTION OF THE MAJOR-THIRD
The following measure from La Nativite du Seigneur by Mes-
siaen, fourth movement, page 2, illustrates a use of the nine-tone
major-third scale:
Example 15-8
Messiaen^La Nativite du Seigneur"
^
f if ii^ p ^'f f
Reproduced with the permission of Alphonse Leduc, music publisher, 175 rue Saint-Honore, Paris. Copy-
right by Alphonse Lediic.
i/ii il 8
Sr
m
iJ^J^JitJJi'^^r^^
2 I I 2 I I 2 I (I)
The long melodic line from the second movement of the same
composer's V Ascension is a striking example of the melodic use
of the same scale:
Example 15-9
Mes3ioen,"L'Ascension"
(^^i^\r[^\>^-^^^ \iIiJ?\^-} a
Reproduced with the permission of Alphonse Leduc, music publisher, 175 rue Saint-Honore, Paris. Copy-
right by Alphonse Leduc.
2 I I 2 11 2 I (I)
Analyze further the second movement of Messiaen's V Ascen-
sion and try to find other examples of the major-third projection.
135
16
Recapitulation of the Triad Forms
Inasmuch as the projections that we have discussed contain all
of the triads possible in twelve-tone equal temperament, it may
be helpful to summarize them here. There are only twelve types
in all if we include both the triad and its involution as one form,
and if we consider inversions to be merely a different arrange-
ment of the same triad.
There are five triads which contam the perfect fifth in their
composition: (1) the basic perfect-fifth triad p^s, consisting of
two perfect fifths and the concomitant major second; (2) the
triad pns, consisting of a perfect fifth, a minor third, and a major
second, with its involution; (3) the major triad pmn, consisting
of a perfect fifth, major third, and minor third, with its involu-
tion, the minor triad; (4) the triad fmd, consisting of a perfect
fifth, a major third, and a major seventh with its involution; and
(5) the triad pc?f, in which the tritone is the characteristic
interval, consisting of the perfect fifth, minor second, and tritone
with its involution. Here they are with their involutions:
i
2
I. p s
2. psn
Example 16-1
and involution 3. pmn
and involution
=f
1/ •#-
2 5
7 2
2 7
4 3
3 4
m
4. pmd and involution 5. pdt ond involution
r ^ J J r I J ^[J m
I 6
136
RECAPITULATION OF THE TRIAD FORMS
The first, p^s, has appeared in the perfect-fifth hexad. The second,
pns, has appeared in the perfect-fifth, minor-second, and minor-
third hexads. The third, pmn, is found in the perfect-fifth, minor-
third, and major-third hexads. The fourth, pind, has been
encountered in the perfect-fifth, minor-second, and major-third
hexads. The fifth, pdt, has appeared only in the minor-third
hexad, but will be found as the characteristic triad in the projec-
tion to be considered in the next chapter.
There are, in addition to the perfect-fifth triad p^s, four other
triads, each characteristic of a basic series: ms^, nH, m^, and sd~:
2 2
Example 16-2
3 3
4 4
It* I
The triad ms^ is the basic triad of the major-second scale, but is
also found in the perfect-fifth and minor-second hexads. The
triad nH, has occurred only in the minor-third hexad. The triad
m^ has been found only in the major-second and major-third
hexads. The triad sd^ is the basic triad of the minor-second pro-
jection and is found in none of the other hexads which have
been examined.
There remain three other triad types: mnd, nsd, and mst:
Example 16-3
10. mnd and involution II. nsd and involution 12 .mst and involution
31 r3l2 21 24 42
The triad mnd is found in the major-third, minor-third, and
minor-second hexads. The triad nsd is a part of the minor-second
hexad and is also found in the perfect-fifth and minor-third
hexads. The twelfth, mst, has occurred in the major-second and
minor-third hexads.
137
THE SIX BASIC TONAL SERIES
Since these twelve triad types are the basic vocabulary of
musical expression, the young composer should study them
carefully, listen to them in various inversions and with various
doublings, and absorb them as a part of his tonal vocabulary.
If we "spell" all of these triads and their involutions above
and below C, instead of relating them to any of the particular
series which we have discussed, we have the triads and their
involutions as shown in the next example. Notice again that the
first five triads— basic triads of the perfect-fifth, minor-second,
major-second, minor-third, and major-third series— are all iso-
metric, the involution having the same "shape" as the original
triad. The remaining seven triads all have involutions.
p^s
Example 16-4
s^d ms^ n^t m' pdt and involution
ijjiiJ.ii>'^-MitJ«^i?i'r:r
ifll 05 -^X A. A Rl
^^
25 ri 22 33 44 61 61
mst and involution pmn and involution pns and involution
m^
fe^^
^
r r -r : r V ^J I r
^
^ 1 1 ' '^^
7 2 7 2
24 24 43 43
#
pmd and involution mnd and involution nsd and involution
^
74 74 31 31 12 12
138
17
Projection of the Tritone
The student will have observed, in examining the five series
which we have discussed, the strategic importance of the tritone.
Three of the six-tone series have contained no tritones— the
perfect-fifth, minor-second, and major-third series— while in the
other two series, the major-second and minor-third series, the
tritone is a highly important part of the complex.
It will be observed, further, that the tritone in itself is not use-
ful as a unit of projection, because when one is superimposed
upon another, the result is the enharmonic octave of the first
tone. For example, if we place an augmented fourth above C we
have the tone F#, and superimposing another augmented fourth
above F# we have BJf, the enharmonic equivalent of C:
Example 17-1
^^
t^^^
For this very reason, however, the tritone may be said to have
twice the valency of the other intervals. An example will illus-
trate this. The complete chromatic scale contains, as we have
seen, twelve perfect fifths, twelve minor seconds, twelve major
seconds, twelve minor thirds, and twelve major thirds. It con-
tains, however, only six tritones: C to F#, D^ to G, Dt] to G#, E^
to A, Ft] to Aij:, and F to B, since the tritones above F|:, G, A^,
At], Bb, and Bti are duplications of the first six. It is necessary,
139
THE SIX BASIC TONAL SERIES
therefore, in judging the relative importance of the tritone in any
scale to multiply the number of tritones by two.
In the whole-tone scale, for example, we found six major
thirds, six major seconds, and three tritones. Since three tritones
is the maximum number of tritones which can exist in any six-
tone sonority, and since six is the maximum of major seconds or
major thirds which can exist in any six-tone sonority, we may say
that this scale is saturated with major seconds, major thirds, and
tritones; and that the three tritones have the same valency as
the six major seconds and six major thirds.
Since the tritone cannot be projected upon itself to produce a
scale, the tritone projection must be formed by superimposing
the tritone upon those scales or sonorities which do not them-
selves contain tritones. We may begin, therefore, by super-
imposing tritones on the tones of the perfect-fifth series.
Starting with the tone C, we add the tritone Fif; we then add
the perfect fifth above C, or G, and superimpose the tritone C#;
and, finally, we add the fifth above G, or D, and superimpose the
tritone G#, forming the projection C-F#-G-C#-D-G#, which
arranged melodically produces the six-tone scale CiC^iD^Fj^i
GxG#:.
Example 17-2
Tritone- Perfect Fifth Hexad p'*m^s^d'*t'
tf^"" i^iU^
I I 4 I I
This scale will be seen to consist of four perfect fifths, four minor
seconds, two major thirds, two major seconds, and three tritones:
p'^m^s^dH^. Multiplying the number of tritones by two, we find
that this scale predominates in tritones, with the intervals of the
perfect fifth and the minor second next in importance, and with
no minor thirds. This is an isometric scale, since the same order
of intervals reversed, 11411, produces the identical scale.
If we superimpose the tritones above the minor-second projec-
140
PROJECTION OF THE TRITONE
tion we produce the same scale: C to Ffl:, D^ to Gt], Dk] to G#, or
arranged melodically, CiDbiDl:]4F#iGiGJj::
Example 17-3
Tritone - Minor Second Hexad p^m^s^d^t^
I I
I I
The components of this perfect-fifth— tritone projection are the
characteristic triads CeF^iG, CJeGiGJ, FJeCiCS, and GeC^iD,
pdt, and their involutions CiCJsG, CJiDgGiJ:, FJiGgCJ, and
GiGifeD, which, though they have been encountered in the
minor-third scale, are more characteristic of this projection;
Triads pdt
Example 17-4
end involutions
^6 I «^6 I 6 1 6 1 ^% * . c , c . c
16 16 16 16
the triads C2D5G and FJaGfgCjj:, p^s, the characteristic triads of
the perfect-fifth projection;
Example 17-5
Triads p^s
M iiiuu:> p :>
2 5 2 5
the triads CiC#iD and FJfiGiGJj:, 5<i^, the characteristic triads
of the minor-second projection;
Example 17-6
Triads sd^
r I II
141
THE SIX BASIC TONAL SERIES
the triads C#,G#4B#(C) and G^D^Fj^, pmd, with the involutions
Ab4C7G and D4F#7C#, which have been found in the six-tone
perfect-fifth, minor-second, and major-third projections;
Example 17-7
Triads pmd
and involutions
^ -? A. -t A. A -I A -I
and the triads C2D4F# and Y%.Q%^%(^C), mst, with the involu-
tions D4F#2G# and Ab(G#)4eoD, which have been met in the
major-second and minor-third hexads:
Triads mst
Example 17-8
and involutions
J j^j ^J^J^t^'r'Nlt^^t^ ''^(^^
2 4
2 4
4 2
4 2
The series contains five new forms of tetrads which have not
appeared in any of the other hexads so far discussed:
1. The characteristic isometric tetrads of the series, CiC^gF^iG
and GjfiDgGiGJ, p^cPf, which contain the maximum number of
tritones possible in a tetrad, and which also contain two perfect
fifths and two minor seconds. These tetrads may also be con-
sidered to be formed of two perfect fifths at the interval of the
tritone, of two tritones at the interval of the perfect fifth, of two
minor seconds at the interval of the tritone, or of two tritones
at the interval of the minor second :
Example 17-9
Tetrads p^d^t^
151 151 P@t t@p d@t t@d
2. The isometric tetrads CiC^iDsG and F#iGiG#5C#, p^sdH,
142
PROJECTION OF THE TRITONE
which also contain two perfect fifths and two minor seconds, but
which contain only one tritone and one major second. These
tetrads may be considered to be formed by the simultaneous
projection of two perfect fifths and two minor seconds :
Example 17-10
Tetrads p^sd t
.tf , c lie _2 . MZ*f
r I 5 115
p2 + d'
3. The isometric tetrads CiCflieGiGJ and F^iGeCjfiD, p^mdH,
which contain two perfect fifths, two minor seconds, one major
third and one tritone; and which will be seen to embrace two
relationships : the relationship of two perfect fifths at the interval
of the minor second, and the relationship of two minor seconds
at the interval of the perfect fifth :
Example 17-11
Tetrads p'^md'^t
2_^2.
@ d
d @ p
4. The tetrads CiCtiDeGit and FJiGiGSgD, pmsdH, with
their involutions CgF^iGiGiJ: and FJeCiCJfiD:
Example 17-12
Tetrads pmsd t
and involutions
ijijjit^ tfJ^ii^r ijiiJ JitJ ^Jfg^
116 116 6 11
6 I I
5. The tetrads CsD^GiGJ, and FtsG^gCifiD, p^msdt, with
their involutions CiC#5F#oG# and F^iGgCoD:
143
the six basic tonal series
Example 17-13
Tetrads p'^msdt
and involutions
oRi o*;! i+i*io I*;
2 5 I
2 5 I
r 5 2
I 5 2
The remaining tetrad is the isometric tetrad C2D4F#2G#,
m^sH^, which we have aheady discussed as an important part of
the major-second projection:
Example 17-14
Tetrad m^s^t^
^^P
2 4 2
The series contains two new pentad forms and their involu-
tions: the characteristic pentads CiC#iD4F#iG, p^msdH^, and
F}fiGiGif4CiC#, with the involutions C#iD4F#iGiG# and
GiG#4CiC#iD;
Example 17-15
Pentads p'msd^t^ and involutions
iijj«J^ iiJ^»^r"ri^tiJ)iJJiJ k^^m
I I 4 I
114 1
14 11
14 11
and ClCil:lD4F#2G#,.p2mVc^2f^ and its involution C2D4F#iGiG#,
which also predominate in tritones:
Example 17-16
Pentad p^m^s^d^t^and involution
ii^j.ji^i^ 'ijjt^ ^0^
r I 4 2
2 4 11
The characteristics of the hexad will be seen to be a pre-
dominance of tritones, with the perfect fifths and minor seconds
144
PROJECTION OF THE TRITONE
of secondary importance, and with the major third and the
major second of tertiary importance. It will be noted, further-
more, that the six-tone scale contains no minor thirds.
Listening to this scale as a whole, and to its component parts,
the student will find that it contains highly dissonant but
tonally interesting material. The unison theme near the beginning
of the Bartok sixth quartet dramatically outlines this scale:
Bartok , Sixth Quartet
Example 17-17
i
^W
T^^afBl
?
^
F
^
ri
^
^
i
^
^^
r- 'T^r-^r
Copyright 1941 by Hawkes & Son (London), Ltd. Used by permission of Boosey & Hawkes, Inc.
i
tssz
ki^ ti"
See also the beginning of the fifth movement of the Bartok
fourth quartet for the use of the same scale in its five-tone form.
Play several times the triad, tetrad, pentad, and hexad material
of this scale as outlined in Example 17-18.
Example 17-18
hii^^^i^ii^^ ij,j'\iJinii^LiF
(|i J JjtJ7^ij^i;3itJ Jl^iJpp^cJ
i'ijj J7^ iiJiiJ)tJltJ«^ijitJ«J|'LL/
145
THE SIX BASIC TONAL SERIES
j^ JJlt^^iJ^tl^lJJJ ^fJ^I u-^W
^ ^i^
I isometric \
I involution/
This scale adds five new tetrad forms, two new pentad forms,
and, of course, one new hexad form. Experiment with these new
sonorities as in Example 17-19, changing the spacing, position,
and doublings of the tones of each sonority.
Example 17-19
%
^
^
i^
^
etc.
P^^
^
etc.
etc.
§
m
*
^i
^
fct.
p i.j H
$
^
^^
'>■■ F ) i l|ii J ii"F "{^
^^
p
146
PROJECTION OF THE TRITONE
I if^ 4i
I
SeH
I^MPt
^
^
i
3— i*-|%
f
Now write a short sketch based on the material of the perfect-
fifth— tritone hexad.
Example 17-20 indicates the modulatory possibilities of the
perfect-fifth— tritone hexad. Write a short sketch employing any
one of the five possible modulations, up or down.
i
Example 17-20
^^^
jjo o i*
^^
f^^
^-S^> ftr^ OflO
114 1
Modulation @t
@ P
'/' .. iU. o <t» *^
• ^"^t
|o o^«
' ^^ ^
iM
3^^
b»|^» ' "^''
^^
«=?=
o *
f^
1^
@ d
@_m
»3
@_n
!...[,, b> ^^"
b>b---"*^
^^
147
18
Projection of the Perfect-Fifth-
Tritone Series Beyond Six Tones
Beginning with the six-tone perfect-fifth— tritone scale CiC#iD4
Ffl:iGiG#, we may now form the remaining scales by continuing
the process of superimposing tritones above the remaining tones
of the perfect-fifth scale. The order of the projection will, there-
fore, be C to FJ, G to C#, D to G#, A to D#, E to A#, B to E#:
Example 18-1
ayp
^^
O go 331
W
i|i JjiJ JiJn
£
Seven tone: CiC|iD4F#iGiG#iA, fm^nhHH^, with its involu-
tion CiC#xDiD#4GiG#iA:
Example 18-2
Perfect Fifth - Tritone Heptad p^n
1
d^t
3
and involuti
on
, J.J J
^h^ ^'° iU J
bt-^
td
•■ ia J
w
^i^ ^
14 11
11)411
Eight tone: CiC#iDiD#3F#iGiG#iA (isometric), fm^n^s^dH^:
Example 18-3
Octad p^m'^n'^s'^ d^l^*
*
^^
Ties
•ffc*
IT" I 11
^
r I I 3 I I I
148
FURTHER PROJECTION OF THE TRITONE
Nine tone: C.CiJD.DJl^.E^FJl^.G.Gj^.A, p'm^nhHH\ with its
involution CiC#iDiD#2FtiiFitiGiG#iA:
Example 18-4
., . 7 6 6 6 .7.4
Nonad p m n s d t
tlH
4- '-
and involution
Ten tone: CiC#iDiD#iE2F#iGiG#iAiA# (isometric),
p^m^n^s^dH^'.
Example 18-5
Decad p^m^n^s^df t^
. I. iS^Js
i
iU JmJ J|J -^^
r I I 12 1
I I
Eleven tone: CiC#iDiD#iE2F#iGiG#iAiA#iB (isometric),
Example 18-6
P ,,. »9 ^^- ^ ^
1 1 1 hn (ti s —
g^fe '°
ii^
1 jjjj Jtt^ ^ti'' "^^^^ r i
Twelve tone: CiC#iDiD#iEiE#iF#iGiG#iAiA#iB, pi2^i2^i25i2
Example 18-7
Duodecod p m n. s d t
m
*
fl'°' ° j^ jjiJJitJ^-'^it^^i'^r
^
I I I I I I I i I I
The melodic line in the violins in measures 60 to 62 of the first
of the Schonberg Five Orchestral Pieces, is an excellent example
of the eight-tone perfect-fifth— tritone projection:
149
the six basic tonal series
Example 18-8
SchOnberg, Five Orchestral Pieces, No. 1
113 1 r I ' -■
By permission of C. F. Peters Corporation, music publishers.
Measures 3 and 4 of the Stravinsky Concertino for string
r quartet are a striking example of the seven-tone perfect-fifth—
tritone projection in involution:
Stravinsky, Concertino
sfz p
^rt
Example 18-9
^ lA "^ ,^ ^
m
m
^i
^F^
:ot|A3.
I I 4 I I I
^J-i
- »F
Afp
f ^P '\ Iz
pizz. t 1< '
*
g> «^l
Copyright 1923, 1951, 1953 by Wilhelm Hansen, Copenhagen. By permission of the publishers.
The following diagram is a graphic representation of the
perfect-fifth— tritone projection.
150
Example 18-10
B^""-"^^
r^^\c#
f
<^^
Xg»
/
M^^~^~~~
\
k
"y
V
yA«
G^^~^__
_^.^E«
D«
19
The pmn-Tritone Projection
There are nine triads which contain no tritones, the triads
already described by the symbols p^s, sd^, ms^, m^, pmn, pns,
pmd, mnd, and nsd.
Example 19-1
i
P^s
sd'
ms
pmn and involution
tt^ I J J J i
i
^
2 5 *^l?f^l ^2 2 ^4
i
4 4 3 3 4
pns and involution pmd and involution mnd and involution
^
m
^
w
7 4 4 7 3^ I \V
i
72 27 74 47 3'1
nsd and involution
i bJ fc'J ^ i ^
2 I
It would seem, therefore, logical to assume that we might pro-
duce a six- tone tritone projection using each of these triads.
However, if we use each of the above triads as a basis for the
projection of the tritone, we find that only one new scale is
produced. The projection of tritones upon the triads p^s and
sd^, as we have already seen, produces the same scale,
CiCjj^iD^F^iGiGjf^. The projection of tritones on the triad pmd
151
THE SIX BASIC TONAL SERIES
also produces the same scale, C-G-B + F#-CJj;-EJj: = B^CiCjl^J^Jl^i
F#iG:
Example 19-2
pmd + tritones
The projection of tritones above the triads ms^ and m^ pro-
duces the major-second scale, C-D-E + F#-G#-A# = C2D2E2Ffl:2
G#2A#; and C-E-G# + F#-A#-C>^-(D) = C2D2E2F#2G#2A#:
ms^ + tritones
Example 19-3
m^ + tritones
ItU <IS fl» ' fe fo' <t^°'-' ^ ^ o o ^" tl° fl'
2 2
The projection of tritones above the major triad, however,
produces a new six-tone scale (Example 19-4a). The projection
of tritones above the triads pns and nsd produces the involution
of the same scale, that is, two minor triads, C-E^-G and F-jf-A-C^j:,
at the interval of the tritone ( Example 19-4??, c ) . The projection
of the tritone above the triad mnd also produces the involution
of the first scale: two minor triads, A-C-E and DJj:-F#-A||:, at the
interval of the tritone (Example 19-4<i).
o) pmn + tritones
Example 19-4
b) pns + tritones
it» <i« '° i^ JjtJ JB^r'iiii»""*-° j^itJ^J J ^'ri|[^4 i}«M J
3 2 I 3 (2) 12 3 1 2(3)
c)^nsd + tritones d) mnd + tritones
i
M
flJJiiJir'H"
'Jl^ig-tf'J J|JitJ»^ ' "f'ita-jtU't; itf
I 2 3 I 2 (3)
3 I 2 3 I (2)
152
THE pmn-TRITONE PROJECTION
Beginning with the major triad C-E-G, we project a tritone
above each of the tones of the triad: C to F#; E to A#, and G to
C#, producing the six-tone scale CiCJaE.FifiGaAJ. This scale
has two perfect fifths, two major thirds, four minor thirds, two
major seconds, two minor seconds, and three tritones:
p^m^n^sWf. It predominates, therefore, in tritones, but also
contains a large number of minor thirds and only two each of the
remaining intervals. Its sound, is, therefore, somewhat similar to
that of the six-tone minor-third scale which predominates in
minor thirds but also has two of the possible three tritones.
The components of this scale are the two major triads C4E3G
and F#4A#3C#, pmn; the diminished triads CJaEgG, E3G3Bb(A#),
G3Bb(AiJ:)3Db(C#), and A#3C#3E, nH; the triads (A#)Bb2C,G
and EsF^^Cj, pns; the triads CiC#3E and F#iG3A#, mnd; the
triads EsFJ^G and A#oCiC#, nsd; the triads EoFJ^AJ and
Bb2(A#)G4E, mst, with the involutions FJ^AJfoC and C4E2F#;
and triads CgFJiG and FJfeCiGJf, pdt, with their involutions
CiC^gG and FJiGeCfl:; all of which we have already met:
Example 19-5
prnn - tritone p^m^n^s^d^t^ pmn Triads n^t
Triads pns
Triads mnd
Triads nsd
Ljj ^ J w^ 'r I i^j J iiJ J itJ I J jiJ J 1^
2727 13 13 21 21
Triads mst and involutions Triads pdt and involutions
jt I I lUU 9 mo I UIIU IIIVUIUII\.'IIO IIIUUO ^J\J l Ul lU II IV\Jt U III^IIO
§ jttJit^^r<i'^>rr;itJji>iriJiJUi[JJit^ri'riu^ i^^'t
24 24 4242 61 61 1^+6 16
It contains the isometric tetrads CjfsEsGaAfl:, nH^, CiC#5F#aG,
p^dH^ (which will be recalled as the characteristic tetrad of the
previous projection), and C^EzFJl^iAjf;, m^sH^; the tetrads
C4E3G3Bb(A#) and F#4A#3Cif3E, pmnht; CxC#3E3G and FJ^Gs
153
THE SIX BASIC TONAL SERIES
A#3C#, pmnHt; C^sEsFJiG and GsAJsCiCt, pnhdt; and EoF^i
GgAfl: and AfaCiCJfgE, mn^sdt (which will be recalled as forming
important parts of the six-tone minor-third scale); and the two
pairs of "twins," pmnsdt, C4E2Fij:iG and F|:4AiJ:2CiCfl;, and
CiCJgEsFJ and FJfiGsAJfaC, both of which have the same
analysis, but neither of which is the involution of the other. None
of these tetrads is a new form, as all have been encountered in
previous chapters.
Example 19-6
Tetrads n'^t^ fJ^d^t^
i
m2s2t2
imn'-st
Ki^^'^^^ij-Jti^^iiJif^^^iiJ^t^^'^UJ^^^^^rr
^r-is
3 3 3 15 1 4 2 4
Tetrads pmnsdt pn^sdt
4 3 3 4 3 3
mn^sdt
33 133^321 321 213 213
^
3 3 13 3
Tetrads pmnsdt
4 2 1 4 2 1 I* 3 2 13 2
Finally, we find the characteristic pentads CiCJfgEoFfiG and
F#iG3A#2CiC#, fmnhdH\ and C4E2FtiG3A# and FJf^Ait^CiCJg
E, pm-nh^dt^; and the characteristic pentads of the minor-third
scale, CiC#3E3G3A# and F#iG3A#3C}f3E, pmn'sdt^:
Pentads p^mn^sd^t^
Example 19-7
pm^n^s^dt^
r 3 2 I
iPentads pmn'''sdt^
13 2 1
4 2 13
4 2 13
13 3 3
Of these pentads, only the first two are new forms, the third
154
THE pmn-TRITONE PROJECTION
having already appeared as part of the minor-third projection.
This projection has been a favorite of contemporary composers
since early Stravinsky, particularly observable in Petrouchka.
Strovinsky, Petrouchko
^ Rs.,Obs., EH.
Example 19-8
i
z
tH
CIS.
P*^ ^l^^b
"t^
Bsns.
^
S
^^^i
<lf*
Horns
i
Tpts., Comets
0t
3
Piano, Strings
Copyright by Edition Russe de Musique. Revised version copyright 1958 by Boosey & Hawkes, Inc. Used
by permission of Boosey & Hawkes, Inc.
A striking earlier use is found in the coronation scene from
Boris Goudonov by Moussorgsky:
Example 19-9
Moussorgsky, "Boris Godounov", Act I, Scene 2
155
THE SIX BASIC TONAL SERIES
A more recent example may be found in Benjamin Britten's
Les Illuminations, the entire first movement of which is written
in this scale:
Example 19-10
Benjamin Britten, Les Illuminations, Fanfare
1Vlns.i
^^botlQ°jlfc(g=^
^
m
o "
1¥
i^L
VIOS. Pr
. 3. ^
'Cellos /
Bosses
^
13 2 13
^
i^
p ^
^
Copyright 1944 by Hawkes & Son (London), Ltd. Used by permission of Boosey & Hawkes, Inc.
Play over several times Examples 19-5, 6, and 7; then play
the entire six-tone scale until you have the sound of the scale
firmly established.
Play the two characteristic pentads and their involutions, and
the six-tone scale, in block harmony, experimenting with spacing,
position, and doubling as in Example 19-11.
Example 19-11
^ etc.
0* lii tt
tt|l^^ f i^^-^ tt'if
^i^i
etc.
4i>;J n^
w^
H^»"^ ii^
Write a short sketch using the material of the six-tone pmn-
tritone projection.
Example 19-12 indicates the possible modulations of this
scale. It will be noted that the modulation at the tritone changes
no tones; modulation at the minor third, up or down, changes
two tones; modulation at the perfect fifth, major third, major
second, and minor second changes four of the six tones.
156
THE pmn-TRITONE PROJECTION
Example 19-12
I
p rm n s 0 T
^^
^v% oti'
^
g> tt*^
13 2 13
Modulotion @ t^
@ il
>>; .. It.. » lt° '^
-it..it"«-'i""'
bo » bo "
*s^
#
^^^=
^^^^3
:«=«
@ d.
@ £
@m
@S
!>^ . , - »tf'
^^
, I'fjt*
5^
N^
1
Write a short sketch employing any one of the possible
modulations.
Analyze the third movement of Messiaen's V Ascension for the
projection of the major triad at the interval of the tritone.
157
20
Involution of the
pmn-Tritone Projection
If, instead of taking the major triad C-E-G, we take its involu-
tion, the minor triad |G E^i C, and project a tritone below each
tone of the triad— G to C#, E^ to A, C to F|;— we will produce
the six-tone scale J,GiF#3E[^2C#iCt]3A(2)(G) having the same
intervallic analysis, p^m^n'^s^dH^.
This scale will be seen to be the involution of the major triad-
tritone scale of the previous chapter.
Example 20-1
^
13 2 I
Minor Triad pmn + tritones
.,Q
17-0-
r^ ...
:h." r «r t i^r^
m
(2)
The components of this scale are the involutions of the
components of the major triad-tritone projection. They consist of
the two minor triads C3Eb4G and F^^A^Cj^, pmn; the diminished
triads C3Eb3Gb(F#), D#3(Eb)F#3A, FJt3A3C and AaCsEb, nH;
the triads C7G2A and F#7C#2D}f(Eb), pns; the triads EbsFJiG
and A3CiC#, mnd; the triads CiDb2Eb and FJiGgA, nsd; the
triads Eb4G2A and A4CJj:2DJj:(Eb), mst, with the involutions
G2A4C# and Db(C#)2Eb4G; and the triads CiC#eG and FJiGeCif,
pdt, with their involutions CgFJfiG and FifsGiCJ.
158
involution of the pmn-tritone projection
Example 20-2
.Triads pmn Triads n t
Triads pns
Triads mnd
Triads nsd
7 2 7 2
Triads mst
3 I 3
and involutions
Triads pdt
12 4 2 4 2
and involutions
24 24 l-ffe 16 61 61
It contains the isometric tetrads CgEbsFjIgA, nH^, CiCifgFJiG,
p^d'f, and Eb4G2A4C#, mVf^; the tetrads D#3(Eb)F#3A4C}f and
A3C3Eb4G, pmn^st; CsEbsFJfiG and FJsAsCiCJ, pmnHt;
CiCjj:2DlJ:(Eb)3Fi|: and FJiGoAsC, pnhdt; EbsFftiGsA and
AaCiC#2D#(Eb), mn^sdt (all of which will be seen to be
involutions of the tetrads in the major triad-tritone projection);
and the involutions of the two pairs of the "twins," CiCfl:2Eb4G
and F#iG2A4Cif, and C#2D#(Eb)3F#iG and GaAsCiCif, pmnsdt.
Example 20-3
Tetrad n^ Tetrad _^dftf Tetrad m^£t5 Tetrads pmn^st
ibJilJ ^IjJItJ-J
^Ti^JibijitJ^^r ^1
5
t^
3 3 3 r' 5 I 4 2 4
' ''* Tetrads pn sdt
3 3 4 3 3 4
Tetrads mn^dt
M letrods pmn dt letrads pn sdt lerraas mn^sai
m
3 3 1 3 3 1 1^* 2 3
.Tetrads pmnsdt
12 3 3 12 3 12
F24 124 **23l 231
Finally, we have the characteristic pentads CiCJoEbsFSiG and
FitiG2A3CiC#, p^mnhdH^; and Eb3FJfiG2A4C# and A3CiC#2Eb4G,
159
THE SIX BASIC TONAL SERIES
pm^nh^dt^; and the characteristic pentads of the minor- third
scale, EbsFifgAsCiCij; and AsCgEbsFJiG, pmn^sdf, all of which
are involutions of the pentads of the major triad-tritone
projection:
Example 20-4
,2 .2*2
Pentads p'^mrrsd^r
pm^n^s^dt^
jj^JttJJ <iJ.iJr"rUJjtJJ^^ir ^rY'^
w
2 3 I
I 2 3
3 12 4
3 12 4
pmn sdt
ff, t,i „j J r itr Jiif''^
3 3 I
3 3 3 1
Since the triad has only three tones, it is clear that the resultant
scale formed by adding tritones above the original triad cannot
be projected beyond six tones. The complementary scales beyond
the six-tone projection will be discussed in a later chapter.
Write a short exercise, without modulation, employing the in-
volution of the pmn- tritone hexad.
160
21
Recapitulation of the Tetrad Forms
We have now encountered all of the tetrad forms possible in
the twelve-tone scale, twenty-nine in all, with their respective
involutions. The young composer should review them carefully,
listen to them in various inversions, experiment with different
types of doubling and spacing of tones, until they gradually
become a part of his tonal material.
The six-tone perfect-fifth projection introduces the following
tetrad types with their involutions (where the tetrad is not
isometric ) :
Example 21-1
i
p^ns^
p^mn^s
3
.1 J j[Jrrr]ij j JfiJrrr
^
^
252 432 34 3 434 212
■■ p^mns^ and involution p^mnsd and involution pmns d and involution
223 322 254 452 221 122
The six-tone minor-second projection adds five new tetrad types:
Example 21-2
ns^d'^ m
n sd pm ncr fpn^s d1 mns^d and involutic
III 121 131 212 Tl22
I I
161
THE SIX BASIC TONAL SERIES
pmnsd^ and involution [pmns^d and involution!
J ^ '^ 'il^Jt^
r I 3 3" I I 2 2 1 12 2
The six-tone major-second scale adds three new tetrad types:
i
m2^3.
m s t
Example 21-3
3^2. r«2c2t2
m s t m s t
iJ Ji-^Uj J«^ii^«^*
222 2. 24 424
The six-tone minor-third scale adds eight new tetrad types;
4*2
n'^t
Example 21-4
pmn^st and involution pmn^dt and Involution
333 334 433 331 r33
pn^sdt and involution mn^sdt and involution pmnsdt and Involution
2„2,
pmnsdt and involution pm'^n'^d
4 2 1 12 4 3 13
The six-tone major- third scale adds one new tetrad;
Example 21-5
pm'^nd and involution
#
J jit^ r : i^J^
4 4 3 3 4 4
The tritone-perfect-fifth scale adds five new tetrads:
162
recapitulation of the tetrad forms
Example 21-6
V 5 1 r I 5
and involution
p^msdt
iJ^t^ guit^it
2 5
1^
5 2
6 I I
The pmn-tritone projection adds no new tetrads.
If we build all of the tetrads on the tone C and construct their
involutions— where the tetrads are not isometric— below C, we
have the sonorities as in Example 21-7. The sonorities are
arranged in the following order: first, those in which the perfect
fifth predominates, then those in which the minor second pre-
dominates, then the major second, minor third,* major third, and
finally, those in which the tritone predominates. These are
followed by the tetrads which are the result of the simultaneous
projection of two intervals: the perfect-fifth and major second;
the major second and minor second; two perfect fifths plus the
tritone; two minor seconds plus the tritone; and finally the
simultaneous projection of two perfect fifths and two minor
seconds. These are followed by the tetrads which consist of two
similar intervals related at a foreign interval.
i
p^ ns''
p^mnsd
-O-
EXAMPLE 21-7
ns^d^
pm
nsd^
^^^^^g
EC»I
331
^^^-
^^
-KSI
4 *
2 5 2
m2^t
2 5 4
2 5 4
pmns d
I I
,4*2
I I 3
I r 3
n^t
^35
^
pmn'^st ,
l7o-o-,f I., .beg:
^
:xs
^^
^tet^
=c=^
^^
2 2 2
2 2 I
■2~"2 r
3 3 3
3 3 4
3 3 4
* In the case of the minor-third tetrads it would be more accurate to say that
they are dominated equally by the minor third and the tritone because of the
latter 's double valency.
163
pmn^dt
THE SIX BASIC TONAL SERIES
mn^sdt
pn^sdt
^^
'SSl.
4'^"0° i.
^^
^
^^M
tec
3 3 I
pm^nd
3 3 I
I 2 3
^
2 3 ♦ 3 I 2 3 I 2
£Vt2
xx:
^
fc^ e
IXS
4 3 ♦
4 4
pmnsdt
fe°*^
2 2 4
p^mns^
4 2 4
33l
^^^S
rro-
'^^ii
^^
4 2 I
mns^d 2
4 2 I
fe^
bo(>c^,|'>^
6 13 6
p^msdt
I 3
2 2 3
pmsd^ t
4
2 2 3
* i i fi i i R ♦
eeO:?^=ec
P^
^'^^
I I 2
I I 2
251 251* 116 116
p^m^nd pm^n^d pm^nd^
»tfogo " I iij^w
Play the tetrads of Example 21-7 as indicated in previous
chapters, listening to each carefully and experimenting with
different positions and doublings.
164
Part 11
CONSTRUCTION OF HEXADS
BY THE SUPERPOSITION
OF TRIAD FORMS
22
Projection of the Triad pmn
Having exhausted the possibilities of projection in terms of
single intervals we may now turn to the formation of sonorities
—or scales— by the superposition of triad forms. For reasons
which will later become apparent, we shall not project these
triads beyond six-tone chords or scales, leaving the discussion
of the scales involving more than six tones to a later section.
We have found that there are five triads which consist of three
different intervals and which exclude the tritone : pmn, pns, pmd,
mnd, and nsd. Each of these triads projected upon its own tones
will produce a distinctive six-tone scale in which the three
intervals of the original triad predominate.
Beginning with the projection of the major triad, we form the
major triad upon C— C-E-G— and superimpose another major
triad upon its fifth, producing the second major triad, G-B-D.
This gives the pentad C2D2E3G4B, p^m^n^s^d, which has already
appeared in Chapter 5, page 47, as a part of the perfect-fifth
projection:
Example 22-1
.Pentad p^m^n^s^d
i i f i J J -'g
« pmn @ p = 2 2 3 4
* The symbol pmn @ p should be translated as "the triad pmn projected at
the interval of the perfect fifth."
167
SUPERPOSITION OF TRIAD FORMS
We then superimpose a major triad on the major third of the
original triad, that is, E-G#-B, producing in combination with
the first triad, the pentad C4E3GiG}t:3B, p^m^n^(P (which we have
aheady observed as a part of the major-third projection ) :
Example 22-2
Pentad p^m'^n^d^
fc
Hi J J jj|j^
pmn @ m =
I 3
The triad on E and the triad on G together form the pentad
EgGiGJfsBaD, p^m^n^sdt (which we have observed as a part of
the minor-third projection ) :
Example 22-3
Pentad p^m^n^sdt
4
pmn @ n =
I 3 3
The combined triads on C, E, and G form the six-tone major-
triad projection CsDoEsGiGJsB, p^m^n^s^dH:
Example 22-4
pmn Hexod p^m'^n^s^d^f
— a —
2 2
^
The chief characteristic of this scale is that it contains the
maximum number of major triads. Since these triads are related
at the intervals of the perfect fifth, the major third, and the
minor third, the scale as a whole is a mixture of the materials
from the perfect-fifth, major-third, and minor-third projections
and has a preponderance of intervals of the perfect fifth, major
third, and minor third.
168
PROJECTION OF THE TRIAD pmU
The major-triad projection adds no new triads or tetrads. It
contains, in addition to the pentads aheady mentioned (com-
binations of two major triads at the intervals of the perfect fifth,
major third, and minor third, respectively), three new pentads:
the pentad C2D2E3GiG#, p^m^ns^dt, which may be analyzed
as the simultaneous projection of two perfect fifths and two
major thirds;
Example 22-5
Pentad p^m^ns^dt
^^ J J J ^ «^ j ^i
2 2 3 1 p2 ^ ^2
the pentad C2D2E4GiJ:3B, pm^n^s^dt, which may be analyzed as
the simultaneous projection of two major thirds and two minor
thirds above G# (Ab);
Example 22-6
Pentad pm^n^s^dt
i J J ti^ r 'f W
I
2 2 4 3
and the pentad CsDgGiGJsB, p^m^n^sdH, which may be ana-
lyzed as the simultaneous projection of two perfect fifths and two
minor thirds, downward:
i
Example 22-7
Pentad p^m^n^sd^t
«^ r \i i}^
tj ■•L
2 5 1 3 I p2 + n2i
The involution of the projection of the major triad
C2D2E3GiGiJ:3B will be the same order of half-steps in reverse,
that is, 31322, producing the scale C3EbiEt]3G2A2B:
169
pmn Hexod
SUPERPOSITION OF TRIAD FORMS
Example 22-8
Involution
- o o —
2 2 3
i
bo ^i
o *:^
This will seem to be the same formation as that of the previous
chapter, if begun on the tone B and constructed downward:
Example 22-9
i
* n i4
If we think the scale upward rather than downward, it becomes
the projection of three minor triads: A-C-E, C-E^-G, and Etj-G-B.
The scale contains six pentads, the first three of which are
formed of two minor triads at the interval of the perfect fifth,
major third, and minor third, respectively :
Example 22-10
|j ^ r^^JjN bi r^J^jjibi^ ^JJjJ
i pmn @p = 2 2 3 4 I pmn @m^ = 4 3 13 pmn @n = 3 133
The remaining pentads are:
Example 22-11
|r^^j^j|4 M^^^i/'f yir^j^i j I
2 2 3 1 i £2 + rn2l 2 2 4 3 i n^ + m^ 2 5 13 t p2 + nf f
All of these will be seen to be involutions of the pentads
discussed in the first part of this chapter.
A short but clear exposition of the mixture of two triads pmn
at the interval of the perfect fifth may be found in Stravinsky's
Symphony of Psalms:
170
*
PROJECTION OF THE TRIAD pmn
Example 22-12
I
m
V^ O
mn @ p
StravinsKy, "Symphony of Psalms'
Sop.
;i J I n 'r
Lou - do - te
Boss
^
i^
Lou
do
te
Si
' ■©'■
Copyright by Edition Russe de Musique. Revised version copyright 1948 by Boosey & Hawkes, Inc. Used
by permission of Boosey & Hawkes, Inc.
The short trumpet fanfare from Respighi's Pines of Rome, first
movement, constitutes another very clear example of the projec-
tion of the triad pmn:
Example 22-13
Respighi "Pines of Rome" , 3
Tpts.
»lliii Mkf^
kf iwrijitiiiiiig
#|*H t^iH
w
ff
By permission of G. Ricordi & Co., Inc.
An exposition of the complete projection of the triad pmn in
involution is found in the opening of the seventh movement,
Neptune, from Gustav Hoist's suite. The Planets:
Example 22-14
Gustov Hoist, "Neptune" from "The Planets"
Flute
i
*
^^
Bossflute
By permission of
J. Curwen & Sons, Ltd.
171
23
Projection of the Triad pns
To PROJECT THE TRIAD pus, wc may begin with the triad on C—
C-G-A— and superimpose similar triads on G and A. We produce
first the pentad C7G2A + G7D2E, or C2D2E3G2A, p^mn^s^, which
we recognize as the perfect-fifth pentad:
Example 23-1
Pentad p'^'mn^s^
I i T- i J J^
pns @ p
2 2 3 2
Next we superimpose upon C7G2A the triad A7E2F#, producing
the pentad C4E2F#iG2A, p^mn^s^dt;
Example 23-2
Pentad p^mn^^dt
<| j- 7 j JjJ ^ ^
pns @ n = 4 2 12
and, finally, the pentad formed by the combination of G7D2E
and A7E2F#, or G2A5D2E2F#, p^mnhH:
Pentad p'mn^s^
=1
Example 23-3
pns @ _3
2 5 2 2
2 2 12
172
PROJECTION OF THE TRIAD pUS
Together with the original triad C-G-A, they produce the six-
tone scale C2D2E2F#iG2A, p'^m^n^s'^dt. This scale has two other
equally logical analyses. It may be considered to consist of two
major triads at the interval o£ the major second, that is, C-E-G
+ D-FJf-A; and it may also be formed by the simultaneous pro-
jection of three perfect fifths and three major seconds above the
first tone, that is, C-G-D-A-(E) + C-D-E-F# = C-D-E-FS-G-A:
Example 23-4
pns Hexad p^m^n^s^dt
^
*
— a —
2 2
pmn @ 3
It is a graceful scale in which to write, deriving a certain
pastoral quality from its equal combination of perfect fifths and
major seconds and having among its intervals one strong dis-
sonance of the minor second, and one tritone.
This scale contains, in addition to the pentads already dis-
cussed, three more pentads, none of which has appeared before.
1. The isometric pentad C2D2E2F#3A, p^m^n^sH, formed by
the projection of two major seconds above and two minor thirds
below C, which we shall consider in a later chapter:
Example 23-5
p^m^n^s^t
^m
fjjj(»^)
w
2 2 2 3 32
2. The pentad C2D2E2F#iG, p^m^ns^dt, which may be analyzed
as the simultaneous projection of two perfect fifths and three
major seconds:
173
SUPERPOSITION OF TRIAD FORMS
Example 23-6
p^m^ns^dt
^liJ J|J.^lj ,^P
2 2 2 1 2 3
3. The pentad C2D4FJj:iG2A, p^mn^s^dt, which may be analyzed
as the projection of two (or three) perfect fifths above and two
minor thirds below C :
Example 23-7
p3 mn^s^dt
f ?*= + • n^i
The involution of the projection C2D2E2FiJ:iG2A, pns, will have
the same order of half-steps in reverse, 21222, forming the
scale C2DiEb2F2G2A:
Example 23-8
i
pns Hexad
Involution
^-o — » —
2 2 2
o *■»
: — a — "
2 I
>^ o »-^
This scale will be seen to be the same formation as the original
pns hexad if begun on the tone A and constructed downward:
Example 23-9
2 2 2 12
The scale contains six pentads, all of which are involutions of
those found in the original hexad, except, the first and fourth
pentad, which are isometric. The first pentad contains the involu-
174
PROJECTION OF THE TRIAD pUS
tion of two triads pns at the interval of the perfect fifth; the
second at the interval of the major sixth; and the third at the
interval of the major second:
Example 23-10
J»l ^''■'JJ.' I b^^-t " ■'bJ J J 1^"'* bJ"''!'! l*'"'t;JI
I pns @ p 2 2 3 2 i_pns @n 4 2 12 ipns @£ 2522 2212
This scale contains, in addition to the pentads already dis-
cussed, three more pentad forms, all of which will be found to be
involutions of the pentads discussed in the first part of this
chapter:
1. The isometric pentad AoGoFsE^gC, p^mrn^sH, which may
be analyzed as the projection of two major seconds below, and
two minor thirds above, A:
Example 23-11
^ 2 2 2 3^ *s2 '7 L^t
2. The pentad A2G2F2EbiD, p^m^ns^dt, which may be
analyzed as the simultaneous projection of two perfect fifths and
three major seconds below A:
Example 23-12
2 2 2 1 *^-
1 p" + s^4
3. The pentad A2G4EbiD2C, p^mn^s^dt, which may be ana-
lyzed as the projection of two perfect fifths below A and two
minor thirds above A:
175
SUPERPOSITION OF TRIAD FORMS
Example 23-13
m
W
M
i p2 + n^t
The smooth, pastoral quahties of this scale are beautifully
illustrated by the following excerpt from Vaughn-Williams' The
Shepherds of the Delectable Mountains:
Example 23-14
^^
^■=?
m
b.o '■^
o fc^
i
pns or pmn @ ^
Voughn -Williams "The Shepherds of the Delectable Mountoins"
|b Jijuij J J l^}^i.^
see ev'-ry day flowers op - peer in the (and
zzf
Copyright 1925 Oxford University Press; quoted by permission.
The involution of this scale is clearly projected in the theme
from the Shostakovich Fifth Symphony, first movement:
Example 23-15
Shostokovich, Symphony No. 5
h^
m
hi ^^JJJj
m
If
ricaca
-T^^s^-j^r)'
i pmn @ 3 2 2 2 12
b^-b^-h>^-h
Copyright MCMVL by Leeds Music Corporation, 322 West 48th Street, New York 36, N. Y. Reprinted
by permission. All rights reserved.
176
24
Projection of the Triad pmd
The projection of the triad C-G-B, fmd, produces the pentad,
frnd @ p, C,G4B + G^D^Fjj^, or C2D4FJfiG4B, fm^nsdH;
Example 24-1
, Pentad p^m^nsd^t
^^^
pmd @ £ = 2 4 14
the pentad, pmd @ d, C7G4B + B^Fj^^Ag or CeFJfiGsAftiB,
p^m^nsdH;
Example 24-2
Pentad p^m^nsd't
^
Jf
r ii^ ^"^ r
m
pmd @ d = 6 I 3 1
and the pentad, pmd @ m, G7D4F# + B7F#4A#, or G3A#iB3D4FiJ:,
p^m^n^d^;
Example 24-3
Pentad p^m^n^d^
pmd @£L = 5 ' 3 4
177
SUPERPOSITION OF TRIAD FORMS
which we have aheady observed as the involution of the
characteristic pentad of the major-third series. The triad pmd
and the two projections together form the six-tone scale
Example 24-4
pmd Hexad p^m'^n^s^d^t
^^3
^^ o
: — CT
2 4 I
In addition to the three pentads already described, the prnd
projection contains three other pentads :
1. The pentad CoDiFJiGgAJ, p^m^ns^dt, the projection of two
perfect fifths and two major thirds below D, already found in
the involution of the projection of the triad pmn:
i
Example 24-5
Pentad p^m'ns^dt
HJflJ UhjW
i
2 4 13
^p^ + m**
2. The pentad C2D4F#4A#iB, pm^ns^dH, which, if begun on
Afl:, may be analyzed as the simultaneous projection of two major
thirds and two minor seconds above AJf ( or B^ ) :
Example 24-6
Pentad pm-^ ns^d^t
2 4 4 1 m2 + d^
3. The pentad CoDgGsAJfiB, p^rn^n^s-d^, which may be ana-
lyzed as the projection of two perfect fifths above C and two
minor seconds below C:
178
PROJECTION OF THE TRIAD pmd
Example 24-7
Pentad p^m^n^s^d^
fc
s
"ri" "i ^ji^
Wi
5 3 I
tp2 + dS
This scale has one major and two minor triads which may
serve as key centers if the scale is begun on G or on B. It bears
the closest affinity to the major-third scale but contains both
major seconds and a tritone, which the major-third scale lacks.
The involution of the projection pmd will have the same order
of half-steps in reverse. Since the order of the original pmd
projection was 24131, the order of the involution will be 13142,
or CiDb3EiF4A2B:
Example 24-8
pmd Hexad
Involution
S
2 4
Po —
I 3
If we begin on B and project the original triad pmd downward,
we produce the same scale :
Example 24-9
If p J ^^rr^-
2 4 I 3 I
The scale contains six pentads, the first three of which are
formed by the relationship of the involution of pmd at the inter-
vals of the perfect fifth, major seventh, and major third,
respectively;
Example 24-10
Jf f ^^iTi if J'^iVrri^J I'lr^
I pmd @^ 2 4 14 pmd @ d 6 13 1 I pmd @m 3 13 4
179
SUPERPOSITION OF TRIAD FORMS
the pentad B2A4FiE3Db, p^m^ns^dt, the projection of two perfect
fifths and two major thirds above A, aheady found in the
major-triad projection;
Example 24-11
*
3
r^^jj
2 4 13
^
p' + m'
the pentad B2A4F4DbiC, pm^ns^dH, which, if begun on D^, may
be analyzed as the simultaneous projection of two major thirds
and two minor seconds downward;
Example 24-12
if=r='
to
?
w
2 4 4 1
and the pentad BsAgEsD^iC, p^m^n^s^d^, which may be analyzed
as the projection of two perfect fifths below B and two minor
seconds above B:
Example 24-13
r^J^Ji'li rrt'
2 5 3 1
i p2 + d2 t
All of the above pentads will be observed to be involutions
of the pentads in the first part of this chapter.
An illustration of the use of the triad pmd at the interval of
the perfect fifth, used as harmonic background, in the Danse
Sacrale from Stravinsky's Le Sacre du Printemps, follows :
Example 24-14
pmd @ p
180
PROJECTION OF THE TRIAD pmd
Stravinsky, "Danse Sacrale"
fh^
w
f»-
^ — i —
fH
^
g y 1.
TlHu
p
r^r— 1
• a
y
P—
•7 r
'-y — i
Vm
• •
p. .
» —
p.
» V J
• 1
p.
• —
* 7 1
ft
k
f- 1
0—
|» T I-
•^^v — *
-^ —
r
F
■
w
r ■
r
^^#
r
r ■
r
r
■ ^ ^
Copyright by Associated Music Publishers, Inc., New York; used by permission.
All of the above pentads will be observed to be involutions
of the pentads in the first part of this chapter.
An illustration of the use of the triad pmd at the interval of
the perfect fifth, used as harmonic background, in the Danse
Sacrale from Stravinsky's Le Sacre du Printemps, follows :
181
25
Projection of the Triad mnd
The projection of the triad CgD^iE, mnd, forms the pentad
mnd @ m, CsDj^jE + EgGiGS, or CgDJiEgGiGJ, fm^nH^,
which, if begun on G#, or Aj^, will be seen to be the characteristic
pentad of the major-third series;
Example 25-1
Pentad p^m'^n^d^
W
J IHi jJJ
^ I „ 3 1
mnd (S rp
the pentad mnd @ n, C-Dif-E + D#-F#-G, or CgDJiEsFJiG,
pm^n^sdH;
Example 25-2
Pentod pm^n^sd^t
3". I "^ 3 I ^ \ 2
3 I 3
mnd @ n
the pentad mnd @ d, DJsFSiG + EgGiGJf, or Dj^.E^F^.G.Gl
pm^n^s^d^:
Example 25-3
.Pentad pm^ n^ s^ d^
^ ^ 3 U 3 1 12 11
mnd (g d
182
PROJECTION OF THE TRIAD mtld
Together they form the six-tone scale CgDJiEsFJiGiGJj:,
Example 25-4
mnd Hexad p^m^n^s^d^t
i
:f^
^% ^ #Q
I I
The remaining pentads are the pentad CgDJiEsFJaGJ,
pm^n^s^dt, which may be analyzed as the simultaneous projection
of two major thirds and two minor thirds, and which has already
appeared as a part of the pmn projection;
Example 25-5
3 12 2
the pentad C4E2F#iGiGfl:, pm^ns^dH; which has already been
observed as a part of the pmd projection, and which may be
analyzed as the combination of two major thirds and two minor
seconds below G|f;
Example 25-6
pm 3ns2d2t
ilH^ Ji|J|||j t)JiiJ|tJ II
4 2 11
4 m^ + d^ I
and the new pentad CaDJfgFJiGiGfl:, p^m^rfsdH, which may be
analyzed as a combination of two minor seconds above, and
two minor thirds below F# :
Example 25-7
-Pentad p^m^n^sd^t
3 3 11
t d'
+ n';
183
SUPERPOSITION OF TRIAD FORMS
This hexad has a close affinity to the six-tone major-third scale
C-D#-E-G-G#-B. The presence of the tritone and two major
seconds destroys the homogeneity of the major-third hexad but
produces a greater variety of material.
Since the projection of the triad mnd has the order 31211,
the involution of the projection will have the same order in
reverse, 11213, or CiCJiDsEiFgAb:
i
Hexad mnd
Example 25-8
Involution
^
Fo o — ^t'^—
3 12 1
1^
If we begin with the tone A^ and project the triad mnd down-
ward, we obtain the same results :
Example 25-9
'3-,H -|^H ,-rm
3 I
This scale has six pentads, the first three of which are formed by
combinations of the involution of the triad mnd at the intervals
of the major third, the minor third, and the minor second:
Example 25-10
^
W '{'{Mh.^ ^^
W
5 I
♦ mnd @ m
3 1 3 1
I mnd @ Ji
3 I 2
^^^^^^
mnd @ d
The Others are the pentad A^gFiEsDaC, pm^nh^dt, which may
be analyzed as the simultaneous projection of two major thirds
184
PROJECTION OF THE TRIAD mnd
and two minor thirds below A\) (or G#);
Example 25-11
Pentad pm^^n^s^dt
4j reniuu pm~n~5 ar
2 2 i m2 t n2|
the pentad Ab4EoDiC#iCt|, pm^ns^dH, which may be analyzed
as the simultaneous projection of two minor seconds and two
major thirds above C;
Example 25-12
f
and the pentad AbaFsDiCjfiC, p^m^n^sdH, which may be ana-
lyzed as being composed of two minor seconds below and two
minor thirds above D :
Example 25-13
Pentad p^m^n^sd^t
3 3 I
^
A nineteenth-century example of the involution of this scale
may be found in the following phrase from Wagner's King des
Nibelungen:
Example 25-14
Wagner,
3^
I
w
o po
^^
3 I 2^ I I
■or
185
SUPERPOSITION OF TRIAD FORMS
Another simple but effective example of the involution of this
projection from Debussy's Pelleas et Melisande follows :
Example 25-15
Debussy, ' Pelleas and Melisande"
i=i^
i
kit :~gT
s^
m
P.p. p\P
pp
^
i^i
it^^'PvCs p i^
^^h^ h
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
3 12(1
186
26
Projection of the Triad nsd
Finally, we come to the last of the triad projections, the projec-
tion of the triad nsd. Beginning with the triad C-Db-E^, we form
the three pentads:
1. CiDbsEb + DbiD^sE^ == CiDbiDfc^iEbiEti, mnhH\ which
is the basic pentad of the minor second series :
Example 26-1
I Pentad mn^s^d'^
nsd @ d.
2, The pentad nsd @ n,
CiDb2Eb + EbiFb2Gb = CiDb2EbiFb2Gb, pmn'sWt:
Example 26-2
Pentad pmn^s^d^t
s
s
ibJbJ bJ^'^t'^ Jl,JbJI^
^12+ I 2 =^l 2 I
2 + 12 =
nsd @ n
12 12
3, The pentad nsd @ s,
DbiDt^^E + EbxFb^Gb = DbiDtiiEbiFb2Gb, pmnhH':
187
SUPERPOSITION OF TRIAD FORMS
Example 26-3
Pentad pmn^s^d^
M. remaa pmn s a
^ V Z + I 2 ; f I I 2
nsd @ 2
The three together produce the scale CiD^iDtiiEbiFbsGb'
pm^n^s^dH, which may also be analyzed as the simultaneous
projection of three minor seconds and three major seconds above
C; or as two triads mnd at the interval of the major second:
Example 26-4
nsd Hexod pm^n^s^d^t
b^ Ho bo k^ ^^ ljj^iJjtJt,J[)J Jjt^ I \)'^^
mnd @ _s
This scale contains three other pentads :
1. CiCJiDtjsEsFfl;, pm^ns^dH, which may be analyzed as the
projection of two major seconds above D and two minor seconds
below D; or as the simultaneous projection of three major
seconds and two minor seconds above C :
Example 26-5
Pentad pm^ns'^d^t
2. The pentad CJD\)-J)[\iE\)sG[), pmn^s^dH, which may be
analyzed as the simultaneous projection of two minor thirds and
two minor seconds above C :
Example 26-6
Pentad pmn^s^d^t
I r I 3 n2 + d2
188
PROJECTION OF THE TRIAD nsd
3. The isometric pentad CsDiEbiFboGb, m^n~s^dH, which may
be analyzed as the simultaneous projection of two minor thirds
and two major seconds above C :
Example 26-7
Pentad m^n^s^d^t
2 112 n2 + s^
^^
This hexad will be seen to have a strong affinity to the minor
second six-tone scale. It does, however, have somewhat more
variety with the addition of the tritone.
Since the projection of the triad nsd has the order 11112,
the involution of the projection will have the same order in
reverse: 21111, or CaDiE^iEtiiFiGb. This hexad may be ana-
lyzed as the simultaneous projection of three minor seconds and
three major seconds below G^ (FJj:), or as two triads mnd at the
interval of the major second :
Example 26-8
nsd Hexad pm^n^s^d^t Involution
If we begin with the tone G^ and project the triad nsd down-
ward, we obtain the same result:
Example 26-9
1 O I " O 1 O^ II I I o^
4
1 2
I ' 2
I 2"'
III 12^
This scale has six pentads, three of which are formed by
combinations of the involution of the triad 7^sd at the interval
of the minor second, minor third, and major second:
189
superposition of triad forms
Example 26-10
•^ 12 r2 = llll ^ 12 12'*= 1212
12 I ' 2
nsd @ d
I 2
nsd @ n
J,J,J UIJ^ /-lU JP
nsd @ _3
1112
It contains also the pentad GbiFiEoDoC, pmrns^dH, which may
be analyzed as the projection of two major seconds below
E and two minor seconds above E; or as the projection of three
major seconds and two minor seconds below G^ (F# ) ;
Example 26-11
i ^^ ^ J Jiijj Ji J J^^Ti|j(U)jj J jtj^j
112 2 I s2 + d2 t
is-
+ d2 \
the pentad GbiFiEiEbgC, pmn^s^dH, which may be analyzed as
the simultaneous projection of two minor thirds and two minor
seconds below G^;
Example 26-12
fe
13 |n2 d^i
and the isometric pentad GbsFbiEbiDoC, m^n^s^dH, which may
be analyzed as two minor thirds and two major seconds below
Gb(Fif):
Example 26-13
i
))»JJ|,JJ lllHi 1^
2 112
I nZ s2*
190
PROJECTION OF THE TRIAD Usd
All of these pentads are, again, involutions of the pentads dis-
cussed in the first part of this chapter.
The remaining triads add no further possibilities. The super-
position of the triads p~s, ms^, and 5<i^ form the perfect-fifth,
major-second, and minor-second scales, already discussed.
The superposition of the augmented triad, nv\ upon its own
tones duplicates itself:
Example 26-14
(| ij i^i ij«-ti
The superposition of the diminished triad, nH, produces only
one new tone:
Example 26-15
^^^^^P
The projection of the triad mst merges with the five-tone
major-second scale:
Example 26-16
ij.j^^ J J»^ m-ii^^r iJ JttS
2 2 2 2
The projection of the triad pdt merges with the five-tone tri-
tone— perfect-fifth projection :
Example 26-17
jij J |jir<'r J^ri' j^^jw
I" I 4 I
191
SUPERPOSITION OF TRIAD FORMS
An excellent example of the projection of the triad nsd, with
its characteristic combination of four half-steps plus a whole
step, is found in the first movement of the fourth Bartok string
quartet where the first and second violins project the scale with
a stretto imitation at the major ninth below in the viola and cello :
Example 26-18
Bortok, Fourth Quartet
^
#
^
1^
i^^'^^$^h^'»'^ ' i
j* ^' 4 h^
Copyright 1929 by Universal Editions; renewed 1956. Copyright and renewal assigned to Boosey & Hawkes,
Inc., for the U.S.A. Used by permission.
iuj^j i^jj|jjJi^^jjjsiJ J I'^'^iJ J^^W
I I I I 2
I I I I 2
Review the material of the projections of the triads pmn, pns,
pmd, mnd, and nsd. Choose the one which seems best suited to
your taste and write a short sketch based exclusively on the six
tones of the scale which you select.
192
|.^.. '.. x 3,rL JJ^X-^
SIX-TONE SCALES FORMED
BY THE SIMULTANEOUS
PROJECTION OF
TWO INTERVALS
27
Simultaneous Projection of the
Minor Third and Perfect Fifth
We have already seen that some of the six-tone scales formed
by the projection of triads (see Example 23-4) may also be
explained as the result of the simultaneous projection of two
difiFerent intervals. We may now explore further this method
of scale structure.
We shall begin with the consideration of the simultaneous pro-
jection of the minor third with each of the other basic intervals,
since these combinations offer the greatest variety of possibilities.
Let us consider first the combination of the minor third and
perfect fifth.
If we project three perfect fifths above C, we form the tetrad
C-G-D-A. Three minor thirds above C produce the tetrad
C-E^-Gb-A. Combining the two, we form the isometric hexad,
CsDiEbaGbiGtiaA, fm~n^s-dH^:
Example 27-1
Hexad p^m^n'^s^d^ t^
bo bo ^1 t^^ bo bo tv
^& «i: — "^^ «i: cr
_p' + n3 =21312
This scale, with its predominance of minor thirds and perfect
fifths, is closely related to the minor-third hexad (see Example
11-3) except for the relatively greater importance of the
perfect fifth.
195
SIMULTANEOUS PROJECTION OF TWO INTERVALS
It contains three pentads, each with its own involution :
Example 27-2
Minor Third Pentad and involution
i U ^^ ^^ '^ ^ i J bJ ^^^
3 3 1
2 13 3
which are the characteristic pentads of the minor third scale; and
Example 27-3
Pentad p^rr? n^sd^t and involution
J J bJ ^^ ^^ 4 b^i N bJ ^^ li^ ^ \i^H
^
2 13 1
p' + n'
13 12
V + n^
which we have already encountered as a part of the pmn projec-
tion (Chapter 22); and which is formed by the simultaneous
projection of two perfect fifths and two minor thirds; and
Example 27-4
Pentad p^mn^s^dt
i^±
and involution
m
J J ttJU'^H'
^m
4 I
r ttti
tp2+ n^ *
4 2
ip'
n2t
which v/e have met as a part of the pns projection ( Chapter 23 ) ,
and which is formed by the projection of two perfect fifths above
and two minor thirds below C,
One interesting fact that should be pointed out here is that
every isometric six-tone scale formed by the simultaneous pro-
jection of two intervals has an isomeric "twin" having the
identical intervallic analysis. For example, if, instead of super-
imposing three perfect fifths and three minor thirds above C,
we form the relationship of two minor thirds at the interval of
the perfect fifth we derive the scale C-E^-Gb + Gl:]-Bb-Db,
or CiDbsEbsGbiGl^aBb, p'm~n'sWt^:
196
minor third and perfect fifth
Example 27-5
Hexod p3m2n4s2d2t2
^^ ^fg
1,^ bo t'g^^^
n2 @ p I 2 3 1 3
Analyzing this scale we find it to contain three perfect fifths, C
to G, Eb to B\), and Gb to D^i; two major thirds, E^ to G, and G^
to B\); four minor thirds, C to E^, E^ to G^, G[\ to B^, and Bb to
Db; two major seconds, D^ to E^, and B^ to C; two minor
seconds, C to D^ and G\) to Gt|; and two tritones, C to G^, and
D\) to Gt^; p^m^n^s^dH^, the same interval combinations that
existed in the scale formed by simultaneous projection of three
perfect fifths and three minor thirds. It will be observed that
neither scale is the involution of the other.
This scale also contains three pentads and their involutions :
Pentad p^m^n^sdt
Example 27-6
and involution
^neiiiuu yj III II 3UI uiiu iiiYuiu 1 luii
pmn @ n
?mn @ ji^
which were found in the projection of the triad pmn as the com-
bination of two major or two minor triads at the interval of the
minor third; and
Example 27-7
Rented p^mn^s^dt and involution
pns @ n
pns @ n_
197
SIMULTANEOUS PROJECTION OF TWO INTERVALS
which were found in the projection of the triad pns at the minor
third; and
Example 27-8
Pentad p^ mn^sd^t^ and Involution
U^JtJ ^i ^Ji ■■■\>h^^r rt 1'^^
^*=^
u
J^iWsi
f*f
12 3 1
r
pmn + \_
2(1-3)
13 2 1
pmn + t'
2(1-5)
which was found in the pmn tritone projection (Chapter 19), as
a major or minor triad with added tritones above the root and
the fifth.
An example of the six-tone scale formed by the simultaneous
projection of three perfect fifths and three minor thirds is found
in the following excerpt from Stravinsky's Petrouchka, which
can, of course, also be analyzed as a dominant ninth in C# minor
followed by the tonic:
Example 27-9
Stravinsky, Petrouchka
Bsn.
I ■ »i^ ItJJJj
VIn.pizz.
^
^
^
Copyright by Edition Russe de Musique. Revised version copyright 1958 by Boosey & Hawkes, Inc. Used
by pennission of Boosey & Hawkes, Inc.
Its "twin" sonority, formed of two minor thirds at the interval
of the perfect fifth, is illustrated by the excerpt from Gustav
Hoist's Hymn of Jesus, where the sonority is divided into two
triads pmn, one major and one minor, at the interval of the
tritone:
198
minor third and perfect fifth
Example 27-10
Hoist, Hymn of Jesus
m
r ir r r
Oi - vine Grace is done
ing
(| 4 ij a
^
t»
^t^tff f '
T^
Wff
m
^
^
W^
m^
By permission of Galaxy Music Corporation, publishers.
,JUJ Jfi^y l,JtJUJ^«^
_n2 @ p
12 3 1 3
199
28
Simultaneous Projection of the
Minor Third and Major Third
Projecting three minor thirds above C and two major thirds
above C, we form the isometric six-tone scale C-E^-Gb-A +
C-Et^-GJf, or CgEbiEt^oGboGftiA, having the analysis p^m^n's^dH\
This scale bears a close relationship to the minor-third series but
with a greater number of major thirds:
Example 28-1
Hexad p^m^n'^s^d^t^
^
^
^
bo tjo t'Q ^
+ m"
3 12 2 1
This scale contains two new isometric pentads:
Example 28-2
Pentad p^m^n^d^t
i J bJ tiJ ii
3 1 4
t m'
which is formed of a major third and a minor third above and
below C, tm~n^; and ^ ^^ ^
Example 28-3
Pentad p^mn^s^d^t
|^J(a^JjW,^ j^n^U i
^
200
MINOR THIRD AND MAJOR THIRD
which is formed of a minor third and a major second above and
below Fjl; and two pentads with their involutions,
Example 28-4
4 2
Minor Third Pen tod pmn sdt involution
fj! J r 'T "r *r : 11.^^
3 3 12
I 3 3
which are the basic pentads of the minor-third series; and
Pentad pm^n^s^dt
Example 28-5
and involution
^ A o n I i_2 . _2i T« I
^J ^ t^^ I'^^nyit
*
4 2 2 1
4m^ + A
I 2 24 tm2+n2t
which is a part of the ipvfin and the mnd projection, and which
may be analyzed as the simultaneous projection of two major
thirds and two minor thirds.
If we now project two minor thirds at the interval of the major
third, we form the isomeric twin having the same intervallic
analysis, p^m^n^s^dH^:
Example 28-6
p^m^n^s^d^t^
to ^fg
|,o Ijo I*" ^
_n_2 @ _m.
This scale contains three pentads, each with its involution:
i
Pentad p^m^n\dt
Example 28-7
and involution
iP jbJiiJ ^'T ki ^' ^jbJ^^^^V hi d
3 1 3 3 pmn @ n
3 3 13
pmn @ n
201
SIMULTANEOUS PROJECTION OF TWO INTERVALS
which has already appeared in the pmn projection as two triads,
pmn, at the interval of the minor third; and
Pentad pm^n'sd^t
Example 28-8
and involution
which has already appeared in the projection mnd as two triads
mnd at the interval of the minor third, and
Example 28-9
Pentad pm^n^s^dt^
and involution
^
^4213 r I 3124 f^ '
which has already been found in the tritone-pinn projection.
Two quotations from Debussy's Pelleas et Melisande illustrate
the use of the two hexads. The first uses the scale formed by the
simultaneous projection of minor thirds and major thirds:
Example 28-10
Debussy, "Pelleos and Mel i sonde" .
V- J- rrrr
HP
^^
b^
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
Jlo-
bo *'
bo "
<^
_n^ + m^ 3 12 2 1
The second employs the hexad formed of two minor thirds at
the interval of the major third :
202
minor third and major third
Example 28-11
ibid.
*>•• ij^ ■ p i i i 1 i i p p I ,,j{o^«^
ugQ
^^
«.^BO
J2J0
n' @ m^
3 12 13
•^•- muuiuu
The following interesting example of the second hexad is
found in the second of Schonberg's Five Orchestral Pieces:
Example 28-12
Schonberg , Five Orchestral Pieces, No. 2
n.,E.H. ^
Cello,
'ueiio,^
aa.,BsWP
By permission ot C. F. Peters Corporation, music publishers.
t;vi tio i^
@ JL
3 12 13
203
29
Simultaneous Projection of the
Minor Third and Major Second
Projecting three minor thirds and three major seconds above C,
we form the six-tone scale C-E^-Gb-A + C-D-El^-Ffl, or
C2DiEbiEt|2F#3A, with the analysis phn^n^s^dH^:
Example 29-1
i
Hexad p^m^n'^s^d^t ^
bo ^° '^ ^ V5 \^ tf" ^ .^ bo t}o :^
+ 33 2 112 3
which will be seen to be similar to the minor-third series, but
with a greater number of major seconds.
This scale contains two isometric pentads:
Example 29-2
Pentad p^m^n^s^t
^^
2 2 2 3 , „2 „2
\ P £L
which has appeared in the projection pns (see Example 23-5),
and may also be considered as the projection of a perfect fifth
and a minor third above and below A; and
204
minor third and major second
Example 29-3
Pentad m^n^s^d^t
i
y j JbJtiJjiJ bo t^g.i^o
2 112
J n} d^
which has been found in the projection nsd and may also be con-
sidered as the projection of a minor third and minor second
above and below E^). There are also two pentads, each with its
involution :
Example 29-4
IVlinorThird Pentad involution
pmn^sdj^ . a
^ 3 3 12 ^2133
which are basic pentads of the minor third series; and
Example 29-5
.Pentad p^mn^s^d^ Involution
#
^^
I r 2
3 2 11
tt» J|.JI|
^
g
P' + 3'
+ d'
P' -^ 3*
^ d^
which appears here for the first time and may be analyzed as
the simultaneous projection of two perfect fifths, two major
seconds, and two minor seconds above D or below E.
If we now project two minor thirds at the interval of the major
second, we produce the isomeric twin C-E^-Gb + D-F-Aj^, or
CsDiEbaFiGbsAb, with the same analysis, p^m^n^s^dH^:
Example 29-6
#
bi'a "0
n^ @ 3
CF
2 I
bo '■'*
be ^^
205
SIMULTANEOUS PROJECTION OF TWO INTERVALS
This scale contains three pentads, each with its involution:
Example 29-7
Pentad p^mn^^dt Involution
3 2 12 pns @ ji
which has already appeared in the pns projection as two triads
pns at the interval of the minor third; and
Pentad pmn^s^d^t
Example 29-8
Involution
■212 n-M fn\ n O I O I ...^ /^
nsd @ n^
2 12 1 nsd @ n
which has appeared in the projection nsd as a combination of
two triads nsd at the interval of the minor third; and
Pentad pm^n^s^dt^
Example 29-9
Involution
i4±
ti.
2 13 2
2 3 12
which has appeared in the p^nn-tritone projection.
The climactic section of the author's Cherubic Hymn begins
with the projection of two minor thirds at the interval of the
major second and gradually expands to the eight-tone minor-
third scale:
Example 29-10
Hanson/TVie Cherubic Hymn"
^i ^j\ rr r r ri^
■>-z % rr r r r^
rg bhj
n^ @ s
Copyright © 1950 by Carl Fischer, Inc., New York, N. Y.
206
30
Simultaneous Projection of the
Minor Third and Minor Second
Projecting three minor thirds and three minor seconds above C,
we form the six-tone scale C-Eb-Gb-A + C-Db-Dt|-Eb, or
CiDbiD^iEb3Gb3A, with the analysis fm^n'^s^dH^:
Example 30-1
Hexad p^nnVS^d^t?
I r I 3 3
This scale is, again, similar to the minor-third series, but with
greater emphasis on the minor second.
This scale contains three pentads, each with its involution:
Example 30-2
Min^yji^d Pentad involution
i
3 3 12 - 2 I 3 3
which is the basic pentad o£ the minor third series; and
Pentad p^m^n^sd^t
Example 30-3
Involution
3 4 I I Td^' * +^n2 t
207
SIMULTANEOUS PROJECTION OF TWO INTERVALS
which has occurred in the projection mnd and appears here as
the projection of two minor seconds above and two minor thirds
below C; or, in involution, as two minor seconds below and two
minor thirds above D#; and
Example 30-4
Pentad pmn^s^d^t Involution
\>m tip y \\'F • i?|» I^P . J
jj^juu^ ij^jjj^^^rTrriTrT''^
1113 d2 +• n2
3 111 4d2
n 2 ^
which has occurred in the projection nsd. This may be analyzed
as the simultaneous projection o£ two minor seconds and two
minor thirds above C or below E^.
If we project two minor thirds at the interval of the minor
second, we produce the isomeric twin C-E^-Gb + CJ-Eti-Gti, or
CiCJfsEbiEtisGbiGtl, with the same analysis, fm^n^sHH^:
Example 30-5
i
g^ b^^i- i^a ^ j^ t^^ tio t^" ^»
n} @ ±
I 2 I
This scale contains three pentads, each with its involution:
Example 30-6
Pentad pm^n^sd^t Involution
^ reniau pm ri su i mvumiiuii
r^
3 1 3 1 ,■ r, i ■>.
mnd @ n 12 13
.3 13
mnd @ n
which has appeared in the projection mnd as a combination of
two triads mnd at the interval of the minor third; and
208
minor third and minor second
Example 30-7
Pentad pmn ^s^d^t Involution
Wyi^'^' W ^^H^'ȴ:^^^f M^ W
nsd @ n^
I 2 1
nsd @ J2
which has appeared in the projection nsd as a combination of
two triads nsd at the interval of the minor third; and
Pentad p^mn^sd^ t^
Example 30-8
Involution
pmn @ j;^
which has aheady occurred in the pmn-tritone projection.
A review of Chapters 27 to 30, which have presented the
simultaneous projection of the minor third with the intervals of
the perfect fifth, major third, major second, and minor second
respectively, will show that all of the hexads so formed fall
naturally into the minor-third series, since all of them contain a
preponderance of minor thirds with their concomitant tritones.
The short recitative from Debussy's Pelleas et Melisande ade-
quately illustrates the hexad formed by the simultaneous pro-
jection of minor thirds and minor seconds:
Example 30-9
Debussy, Pelleas and Melisande
j)i ^' ^' '/ g'j^^jT I'/pipp^^' J^ )iM)i\^
^
#
n-
^
"^
bo ^i
Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Phila-
delphia, Pa., agents.
The quotation from Stravinsky's Petrouchka is an excellent
example of the projection of two minor thirds of the interval of
the minor second:
209
simultaneous projection of two intervals
Example 30-10
Stravinsky, Petrouchko
Harp
Copyright by Edition Russe de Musique. Revised version copyright 1958 by Boosey & Hawkes, Inc. Used
by permission of Boosey & Havifkes, Inc.
Review the projections of Chapters 27 to 30, inclusive. Select
the hexad which most appeals to you and write a short sketch
based exclusively on the material of the scale which you select.
210
31
Simultaneous Projection of the
Perfect Fifth and Major Third
If we project three perfect fifths above C, C-G-D-A, and two
major thirds above C, C-E-Gfl:, we produce the six-tone isometric
scale CsDsEsGiGifiA, fm^nhHH:
Example 31-1
1
p3 + m^
«s
2- 2
I I
It bears a close relationship to the perfect-fifth series because
it is the perfect-fifth pentad above C with the addition of the
chromatic tone G#.
It contains two isometric pentads :
Example 31-2
Perfect Fifth Pentad
P^mn2s3
I J .1 J J
2 2 3 2
already described as the basic perfect-fifth pentad; and
Example 31-3
Pentad p^m^n^d^
*J -0- J. 1. \ I M. ~ 9 .9
4 3 1 I
t m^ 1'
2 aZ
211
SIMULTANEOUS PROJECTION OF TWO INTERVALS
which is a new isometric pentad, and which may be analyzed as
the formation of a major third and a minor second above and
below G#, Wd\
It also contains two pentads, each with its involution:
Example 31-4
p^mns ^d^t
W
f-f^
Involution
^m
^m
-ZgL W-
2 5 11
p3+ d2
115 2 -^ »^it'
t p3 + d2
which may be analyzed as the simultaneous projection of three
perfect fifths and two minor seconds, and which has not before
been encountered; and
p^m^ns^dt
Example 31-5
Involution
i^\h^ nj. «i : iiJ ^ r r
2 2 3 1
p' + m'
13 2 2
^m
s
i p2 + m^ I
which we have met before as a part of the projection of both the
triads 'pmn (Chapter 22) and 'pmd (Chapter 24) and is formed
by the simultaneous projection of two perfect fifths and two
major thirds.
If we now project two perfect fifths at the interval of the
major third, we form another isomeric twin having the same
intervallic analysis as the previous scale, but not constituting an
involution of the first scale. The scale thus formed is C-G-D +
E-B-F#, or C2D2E2F#iG4B, which also has the intervallic forma-
tion p^m^nrs^dH:
Example 31-6
i
p? @ m
^€i^ a
2 2
212
PERFECT FIFTH AND MAJOR THIRD
This scale will be seen also to have a close resemblance to the
perfect-fifth series, for it consists of the tones of the seven-tone
perfect-fifth scale with the tone A omitted.
It contains three pentads, each with its involution:
p-^m2 n^s^d
Example 31-7
Involution
J.J^r i h^n'r i
M
2 2 3 4 pmn @ p
4 3 2 2 pmn @ p
which has already occurred in the pmn projection as the relation-
ship of two triads pmn at the interval of the perfect fifth; and
p3m2nsd2 t
^
^^
Example 31-8
Involution
ii
m
s^
pmd @ p
4 14 2
pmd @ p
which has already occurred as the projection of two triads pmd
at the interval of the perfect fifth; and
Example 31-9
p^m^ns^dt
Involution
fe=*
^=m
^^
2 2 2 1
p« + s^
2 2 2 I ^2 + s3 i
which we have met' in the projection of the triad pns as the
simultaneous projection of two perfect fifths and three
major seconds. ,
A striking example of the projection of two perfect fifths at
the interval of a major third is found in the opening of the
Stravinsky Symphony in C:
213
simultaneous projection of two intervals
Example 31-10
Strovinsky, Symphony in C
#
e
iSj,
n
J
i
^
^
^
^
f
¥
m
Strgs., Hns.,
*'"^- him
m
Winds
^--.^v.^
p2@nn-
rimp.
Copyright 1948 by Schott & Co., Ltd.; used by permission of Associated Music Publishers, Inc., New York.
An excellent example of the simultaneous projection of two
perfect fifths and two major thirds, giving the pentatonic scale
C D E G Ab, may be found in Copland's A Lincoln Portrait:
Example 31-11
Copiond,"A Lincoln Portrait"
Hns.
nnti
^m
■^Sr
4u
bo ,Trb.,' Cellos, Basses
iuDa,Tro., cellos,
r
^
r
r
r
Copyright 1943 by Hawkes & Son (London), Ltd. Used by permission of Boosey & Hawkes, Inc.
214
32
Simultaneous Projection of the
Major Third and Minor Second
Projecting major thirds and minor seconds simultaneously, we
form the six-tone scale C-E-G# + C-Ci|:-D-D#, or CiCJfiDiDJi
E4G#, with the analysis p^m^n^s^dH. This scale is very similar to
the six-tone minor-second series with the exception of the addi-
tion of the tritone and greater emphasis on the major third:
Example 32-1
Hexad p^m^n^s^d^ t
^' %.T3ft^ "tt" ^J-
Ss
I I
This scale contains two isometric pentads :
Example 32-2
Pentad p^m^n ^sd^
P 2^ I 4 t '^ ^2 ™2
which is formed of a perfect fifth and a major third above and
below G#; and
Example 32-3
#
Minor Second Pentad mn^s^d^
^ ,»^ I ^ I ^'
215
SIMULTANEOUS PROJECTION OF TWO INTERVALS
which is the basic minor-second pentad. There are two additional
pentads, each with its involution:
.Pentad pm^ns^d^t
Example 32-4
Involution
1^ I 2 4 tm2 + d
-* 2 I I 4m2 ,+ d2
which has been found as a part of the projection pmd and mnd,
and is analyzed as the simultaneous projection of two major
thirds and two minor seconds; and
Pentad p^mns^d^t
Example 32-5
Involution
J^iJaJ■ltiJ|<j,iJ,J■li^l^||^lVKi"m
•^ tt^l !+♦ I 4 1^p2+^7i ^ 4 111 %*fp2+d3
which consists of the simultaneous projection of two perfect
fifths and three minor seconds, and which appears here for the
first time.
If we project two minor seconds at the interval of the major
third, we form the isomeric twin C-C#-D + E-F-Ffl:, or
CiCifiDaEiFiFfl:, having the same analysis, p^m^n^s^dH:
Example 32-6
Hexod p^m^n^s^d'^t
o " ^^
=°=#^
^#^."
@
r I
I I
This scale contains three pentads, each with its involution:
Pentad pm^n^s^d^
Example 32-7
Involution
216
MAJOR THIRD AND MINOR SECOND
which is a part of the projection mnd, being formed of two triads
mnd at the interval of the minor second; and
Example 32-8
Pentad p^m^nsd^t Involution
(|j,,iJ/iri r'ff :J3J,||J|, iJilJ
pmd @ d
pmd @ d[
which is a part of the projection pmd, being formed of two
triads pmd at the interval of the minor second; and
Example 32-9
Pentad pm^ns^d^t Involution
Iff I 2 2
s3 + d^ 2
2 11 I s3 + d2
which is a part of the nsd projection and may be considered as
the simultaneous projection of three major seconds and two
minor seconds.
Copland's A Lincoln Portrait contains the following example
of the projection of two minor seconds and two major thirds,
producing the pentad J,At)-G-F}f-E-C:
Example 32-10
Copland, "Lincoln Portrait"
Copyright 1943 by Hawlces & Son (London), Ltd. Used by permission of Boosey & Hawkes, Inc.
217
SIMULTANEOUS PROJECTION OF TWO INTERVALS
An example of the hexad formed by the simultaneous
projection of three minor seconds and major thirds will be found
at the beginning of Le Tour de Passe-Passe from Stravinsky's
Petrouchka:
Example 32-11
Stravinsky,"Petrouchkgl_^
^
S
^"^r-[jir
53l^
m
Bsns^p-r-[Jr
^
^
^
BasSj
C.Bsn,
Bab
Copyright by Edition Russe de Musique. Revised version copyright 1958 by Boosey & Havv-kes, Inc. Used
by permission of Boosey & Haw'xes, Inc.
An unusual example of the projection of two minor seconds
at the interval of the major third is found in the cadence at the
end of the first of the Five Orchestral Pieces of Schonberg:
Example 32-12
Schonberg, "Five Orchestral Pieces'
By permission of C. F. Peters Corporation, music publishers.
218
33
Simultaneous Projection of the
Perfect Fifth and Minor Second
The simultaneous projection of three perfect fifths and three
minor seconds produces the scale C-D-G-A + C-C#-D-D#, or
CiCJiDiDJfiGaA, p^m^n^s^dH^, which may also be analyzed as
the triad pdt at the interval of the major second:
Example 33-1
Hexad p^m^n^s^d^ t^
j'+d'*" 11142 pdt &_ 3
This does not form an isometric six-tone scale but a more
complex pattern, a scale which has its own involution and also
has its isomeric "twin" which in turn has its own involution. This
type of formation will be discussed in detail in Chapter 39.
If we project two perfect fifths at the interval of the minor
second, we form the six-tone scale C-G-D + D^-Ab-Eb, or
CiDbiDtiiEb4GiAb, with the analysis p^m^ns^dH^:
Example 33-2
Hexad p^mSnsSd^fS
i
bo .^- i>o tjo P'
p^ @ 6_ I I I
This scale is most closely related to the projection of the tritone
discussed in Chapter 17.
219
SIMULTANEOUS PROJECTION OF TWO INTERVALS
It contains three pentads, each with its involution:
Example 33-3
^Pentad
p
^ msd
3t2
Involution
4=1
1 \r2
f—
L ^i'^^ ! J
-t-
^
f-^P \k-^
#d
kJ
^
«l 17 ■
■-IT*
W ^v W .-.'■--J
t'l
-^=^^^'—
114 1 2dt @ p
14 11
^t @ p
which is a part of the tritone-perfect-fifth projection and may be
analyzed as the triad fdt at the interval of the perfect fifth; and
Example 33-4
Pentad p^m^nsd^t Involution
i U i,J ^ 1'^ r H tp ■ ^j J ^-i r y if ^
2 4 14
pmd@ Q^
4 14 2
pmd @ p
which has appeared previously as the triad pmd at the interval
of the perfect fifth; and
Pentod p^mns^d^t
Example 33-5
Involution
which may be analyzed as the simultaneous projection of two
perfect fifths and three minor seconds.
If we now reverse the projection and form two minor seconds
at the interval of the perfect fifth, we form the scale C-C#-D +
G-G#-A, or CiC^iDgGiGJfiA, having the same analysis,
Example 33-6
Hexed p'^m^ns^d^tg
&^I^
ft
dz
@
I*' I 5
jto "
I I
220
PERFECT FIFTH AND MINOR SECOND
This scale contains three pentads, each with its involution:
Example 33-7
Pentad p^ msd^t^
Involution
tt^l 5 11 pdt ® p I" I 5 I ^ '
pdt @ p
which is a part of the tritone-perfect-fifth projection, being a
combination of two triads pdt at the interval of the perfect
fifth; and
Example 33-8
^ Pentod p'^m'^ nsd'^t # Involution
^ ^ifi^ie I nmd ra d *^*6II T* '
It* I 6 I pmd
pmd @ d
which has occurred in the projection 'pmd as the combination
of two triads fmd at the interval of the major seventh; and
Example 33-9
Pentad p^mns^ d'^t Involution
,3„„^2 ^2*
)3 + d?^ 2 5 11 i|p3 + d2
r I 5 2
which may be analyzed as the simultaneous projection of three
perfect fifths and two minor seconds.
The first of the hexads discussed in this chapter has a pre-
dominance of tritones, while the second and third have an equal
strength of tritones, perfect fifths, and minor seconds. This means
that all three scales have a close resemblance to the tritone-
perfect-fifth projection. The following measure from the Stravin-
sky Concertino illustrates the simultaneous projection of three
minor seconds and three perfect fifths. It will be seen to be a
variant of the illustration of the tritone projection of Ex-
ample 18-9.
221
simultaneous projection of two intervals
Example 33-10
Stravinsky. Concertino .
pizz.
Copyright 1923, 1951, 1953 by Wilhelm Hansen, Copenhagen. By permission of the publishers.
This concludes the discussion of the simultaneous projection
of two intervals, since the only pair remaining is the combination
of the major second and the major third, the projection of which
forms the major-second pentad.
Review the hexads of Chapters 31 to 33, inclusive. Select one
and write a short sketch confined entirely to the material of the
scale you select.
222
Part lY
PROJECTION BY
INVOLUTION AND
AT FOREIGN INTERVALS
34
Projection by Involution
If we examine again the perfect-fifth pentad C-D-E-G-A,
formed of the four superimposed fifths, C-G-D-A-E, we shall
observe that this combination may be formed with equal logic
by beginning with the tone D and projecting two perfect fifths
above and below the starting tone:
Example 34-1
i
-^
o *>
All such sonorities will obviously be isometric.
Using this principle, we can form a number of characteristic
pentads by superimposing two intervals above the first tone and
also projecting the same two intervals below the starting tone.
Referring again to the twelve-tone circle of fifths, we note that
we have six tones clockwise from C: G-D-A-E-B-Fjj:, and six
tones counterclockwise from C: F-B^-Eb-Ab-Db-Gb, the G^
duplicating the F}. The following visual arrangement may be
of aid:
12 3 4 5
G D A E B
C F#(Gb)
F Bb Eb Ab Db
G and F form the perfect fifth above and below C; D and Bb
225
INVOLUTION AND FOREIGN INTERVALS
form the major second above and below C; A and E^ form the
major sixth above and below C; E and A\) form the major third
above and below C; and B and D^ form the major seventh above
and below C.
Taking the combination of 1 and 2, $p^s^, we duplicate the
perfect-fifth pentad:
^ ^ Example 34-2
$ p2s2 2 3 2 3 C 2 2 3 2
The combination of 1 and 3 forms the pentad |5^n^| ( Example
23-5):
G A
F Eb
tp^n%
or, arranged melodically C3Eb2F2G2A, p^m^nhH:
Example 34-3
$
p m n ST
i^^ b<
^ p''n
2«2
^
iS
3 2 2 2
The combination of 1 and 4 forms the pentad
G E
C , tp^rn^ or
F Ab
C4EiF2GiAb, p''m^nh(P:
Example 34-4
4=^
-r*^
p m
n^sd^
-r. —
'g> u"
\jvs
=^
— ■ •-
vu
% p2m2
4 12 1
226
PROJECTION BY INVOLUTION
The combination of 1 and 5 forms the pentad
G B
C , tp'd',
F Db
or CiDb4F2G4B, p^mhHH^:
Example 34-5
*
t
^" bo j-bJ ^ ^ r
♦ p2d2
14 2 4
The combination of 2 and 3 forms the pentad
D A
C , ts^n^
Bb Eb
or CsDiEbeAiBb, p^mnhHH:
Example 34-6
p^ mn^s^d^t
X s^n'
bo iJj^T
2 16 1
The combination of 2 and 4 duphcates the major-second pentad
D E
C , Xs^rn\
Bb Ab
or C2D2E4Ab2Bb, m^sH^:
#
Example 34-7
^'t^e \M
% «2m2
i J J 1'^ ^r (''^ ^1
2 2 4 2
2 2 2 2
227
INVOLUTION AND FOREIGN INTERVALS
The combination of 2 and 5 duplicates the minor-second pentad
D Bti
C , ts^d%
Bb Db
or CiDbiDtisBbiBti, mnh'd^:
#
Example 34-8
^
iJW^r^r iv^rrs
=^©:
=F^
t s2d2
116 1
I I I I
The combination of 3 and 4 forms the pentad
A E
C , tn^m\
Eb Ab
or C3EbiEl^4AbiAl^, p'^m^nHH:
Example 34-9
Ml y I II I I \J I
I n£m2
3 14 1
The combination of 3 and 5 forms the pentad
A B
C , XnH\
Eb Db
or CiDb2Eb6A2B, m^n^s^dH, which has also been analyzed in
Example 26-7 as the projection of two major seconds and two
minor thirds, A-B-Cjj: + A-Ct^-Eb:
228
projection by involution
Example 34-10
m n s d 1
fej^
\,j^r [^r rV
5
12 6 2
2 112
And finally, the combination of 4 and 5 forms the pentad
E B
C , tm^d^
Ab Db
or CiDb3E4Ab3B, p^m^n^sd^:
Example 34-11
*
m
t m^d^
bo J.^
g
1^
13 4 3
The only way in which an isometric six-tone scale can be
formed from the above pentads is by the addition of the tritone
F# (or Gb). For example, if we take the first of these pentads
and add the tritone above and below C, we produce the
six-tone scale C2D3FiF#(Gb)iGli3Bb, p^m^nhHH-.
Example 34-12
#
n \}Q
*
t p2s2t
i J J tt^ ^ T
m
2 3 1 13
The remaining pentads with the tritone added become
C3Eb2FiF#iG2A, fm^n^sHH^:
229
INVOLUTION AND FOREIGN INTERVALS
Example 34-13
p2m2n^s3d2t2
li^J ^»^ ^
Jp2n2t 3 2 112
C4EiFiF#iGiAb, p^rrfnhHH:
Example 34-14
p2m3n2s3d^t
I J J h\^ ^ '-^
$p2ri5
4 I I I I
CiDb4FiF#iG4B, p'mhH'f:
Example 34-15
P^m2s2d^t3
1 p^d^t I 4 I I 4
CaDiEbsFJsAiBb, p^m^n's^dH^:
Example 34-16
p2m^n^s2d2 t2
Jl |/ 111 I I 9 u I
S ,2n2T ? I f 3 I
C2D2E2F#2Ab2Bb, m«s«^3.
Example 34-17
;1f 2 "^ 2 ^' 2^ 2 2
TCT €»^
} s2m
230
PROJECTION BY INVOLUTION
CiDbiD^4F#4BbiB^, fm^nhHH; t sHH (duplicating 34-14)
CgEbiEoFJ.AbiAl^, fm^n's-dH^- % n^mH (duplicating 34-16)
CiDbsEbsFSsA.B, p^^Vs^cZ^^^; t nHH (duplicating 34-13)
CiDbsEsFSsAbsB, fm^nhHH; mHH ( duplicating 34-12 )
Since all of the six-tone scales produced by the addition of
the tritone have already been discussed in previous chapters,
we need not analyze them further.
231
35
Major-Second Hexads
with Foreign Tone
Examining the seven-tone major-second scale C-D-E-Fjf-G-Ab-
Bb, we find that it contains the whole-tone scale C-D-E-F#-Gfl:-
A#: and three other six-tone scales, each with its involution:
Example 35-1
p m n s d t
*
ff" "
t;cH bo
o ©-
2 2 2 11
1. CaDsEsFifiCsBb with the involution EgGiAbaBbaCaD,
Example 35-2
p2m4n2s4dt2
Involution
M \) III MS U I ~ IIIVUIUIIUII
2 2 2
3 12 2 2
which may also be considered to be formed of four major
seconds above, and two minor thirds below B\) or, in involution,
four major seconds below and two minor thirds above E;
Example 35-3
m
i=F
^m
2 2 2 2 1
^^m
ts''
HI* is.'*
^g
i
+ n^ t
232
MAJOR-SECOND HEXADS WITH FOREIGN TONE
2. CoD.EoFJfiGiAb with the involution F^iGiAbsBbsCsD,
Example 35-4
p'^m^ns^d^t'
Involution
'2 2 2
12 2 2
which may also be considered as the projection of four major
seconds and two perfect fifths above C, or below D;
Example 35-5
+ p2 I s*
+ p2 I
3. C4E2F#iGiAb2Bb with the involution E2F#iGiAb2Bb4D,
Example 35-6
nvolution
*^ ■*-4 2 I 12 2 112 4
which may also be considered as the projection of four major
seconds and two minor thirds above E, or below B^:
Example 35-7
#
12 2 s"
+ n2 \s'
The theory of involution provides an even simpler analysis.
Example 35-2 becomes the projection of two major thirds and two
major seconds above and below D, and one perfect fifth below D;
and the involution becomes two major thirds and two major
233
INVOLUTION AND FOREIGN INTERVALS
seconds above and below C, and one perfect fifth above C—
that is X'^^s^pi or |mVp|. Similarly, Example 35-4 becomes
Xm^s^n^ or :|)mVn|. Example 35-6 becomes t:mV<i| or
Example 35-8
^m 1 l* ^m 1 £ ^ *J!? 5 11^ *i!? i il^ ^^ i 1* *2! 1 £
All of these impure major-second scales will be seen to have
the characteristic predominance of the major second, major
third, and tritone.
A striking use of the impure major-second scale of Example
35-6, where one might not expect to find it, will be seen in the
following excerpt from Stravinsky's Symphony of Psalms:
Example 35-9
StrovinsKy," Symphony ot Psalms"
Copyright by Edition Russe de Musique. Revised version copyright 1948 by Boosey & Hawkes, Inc. Used
by permission of Boosey & Havifkes, Inc.
i
^
ki b«
4 2 11
An earlier use of the scale illustrated in Example 35-2 will be
234
MAJOR-SECOND HEXADS WITH FOREIGN TONE
found in the excerpt from Scriabine's Prometheus:
Scriobine, "Prometheus"
7' <tr if
Hns . '
Example 35-10
4«^
m
'^ i^ ^^
m
fep
ttf-
^
it'
m
2 2 2
A more familiar example is found at the beginning of the
same composer's Le Poeme de TExtase :
Example 35-11
Scriobine, "Le Poeme de TExtose"
5
i
^
m
Q \i^y b«
o ati
2 2 2 13
i
Write a short sketch using the material of the hexads of Ex-
amples 35-2, 35-4, or 35-6.
235
36
Projection of Triads
at Foreign Intervals
In Part II we discussed the projection of triads upon the inter-
vals which were a part of their own composition, for example,
fmn @ p, fmn @ ra, pmn @ n, each of which forms a pentad,
and the three together forming the six-tone pmn projection. It is
obvious that we may form a six-tone scale directly from a triad
by projecting it at a foreign interval, that is, at an interval which
is not in the original triad. For example, pmn at the interval of
the major second produces the six-tone scale which we have
already discussed in Chapter 23, C-E-G + D-F#-A =
C2D2E2F#iG2A, which has been analyzed both as the projection
of the triad pns and as the simultaneous projection of three
perfect fifths and three major seconds :
Example 36-1
i tj iJ J»J Ji
pmn @ ^ 2 2 2 12
We have noticed, also, that the six-tone scale formed by the
projection of the triad nsd may be analyzed as the relationship
of two triads mnd at the major second ( see Example 26-4 ) .
Certain of these projections, however, form new hexads which
have not so far appeared.
The triad pmd at the interval of the major second produces
the scale C-G-B + D-A-C#, or CiCJiDsGsAsB, p^m^nh^dH, with
236
PROJECTION OF TRIADS AT FOREIGN INTERVALS
its involution CsDsEgAiBbiBtl:
Example 36-2
p3m2n2s^d3t Involution
I i "J itiJ ^r ^ J J J ^^r^r j ^i
pmd @ s
r I 5 2 2
2 2 5 11
pmd @ s
The same triad pmd at the interval of the minor third forms
the scale C-G-B + Eb-Bb-D, or C2DiEb4G3BbiB^, with its
involution CiC#3E4G#iA2B, fm^nhH^:
Example 36-3
p^m^n^s^d' Involution
^
m
i ju ^^r
pmd @ n_
2 14 3 1
r 3 4 12
pmd @ in
The triad ins^ at the interval of the minor third forms the new
isometric six- tone scale, C-D-E + Eb-F-G, or CsDiEbiEtjiFaG,
p^m^n^s'^d^, which predominates in major seconds, but which also
may be analyzed as a projection of three perfect fifths above, and
three minor seconds below F (F-C-G-D + F-E-Eb-D):
Example 36-4
p3nn2n3s4(j3
s^ @ n 2 I I I 2 tp^ d'i
The triad mst at the interval of the perfect fifth forms the
scale C2D4F# + G2A4G#, or CiC#iD4F#iG2A with its involution
C2DiEb4GiAbiAI::], p'^m^n^s^dH^, which is most closely related to
the tritone-perfect-fifth series:
Example 36-5
p4m2n2 s2d3t2 Involution
mst @ p
I 2
2 I 4 I I
mst @ p
237
INVOLUTION AND FOREIGN INTERVALS
The same triad, mst, at the interval of the minor second forms
the scale C^D^Fif + Db2Eb4G = CiDbiD^iEbsFJiG, with its
involution CiDbsEiFiFJiG, p^m^n^s^dH^, which also resembles
the tritone-perfect-fifth projection:
Example 36-6
p3m2n2s2d4^2
Involution
mst @ d
mst @ d^
There are, finally, eight projections of triads at foreign tones,
in which the scales and their involutions follow a pattern some-
what similar to the projections discussed in Chapters 27 to 33.
They should, for the sake of completeness be mentioned here,
but will be discussed in detail in a later chapter. They are:
The projection of the triad pvm at the interval of the minor
second, which forms the scale CiDbsEiFaGiAb, with its involu-
tion CiDb2EbiFb3GiAb, p^m'^n^sdH; the triad pns at the major
third, CiCJfgEgGaAaB, with its involution CaDoEaGsBbiBt^,
p^m^n^s^dH; the triad pns at the minor second, CiDbeGiAbiAtji
Bb, with its involution CiDbiDtjiEbeAiBb, p^m^n^s^dH; the triad
pdt at the major second, C2D4F}t:iGiGJ|:iA, with its involution
CiDbiDt|iEb4G2A, p^m^rrs^dH^, which may also be analyzed
as the simultaneous projection of three perfect fifths and three
minor seconds; the triad pdt at the major third, C4E2F#iG3A#iB,
with its involution CiDb3EiF2G4B, p^m^n^s^dH^; the triad nsd at
the perfect fifth, CiDb2Eb4GiAb2Bb, with its involution
C2DiEb4G2AiBb, p^m^n^s^dH; the triad nsd at the major
third, CiDb2EbiE^iF2G, with its involution C2DiEbiEti2F#iG,
p^m^n^s^dH; and the triad mnd at the perfect fifth, CgD^iEaGg
A#iB, with its involution CiDbsEgGjAbsB, p^m^n^sdH.
238
^^^
projection of triads at foreign intervals
Example 36-7
p^m'*n^sd'^t Involution
it J J J ^''^^JJUtJ -l"^ hlM
I 3 I 2 I 12 13 1
^3m3n3c:3r|2
p-^m-^n'^S'^d'^t Involution
pmn @ d^
pns @m_ P3322 2233
p^na^n^s^d^t Involution
pns @ m
I 6 I I I
p3rTi2n2s3d3j2
pns @ d.
Involution
_pdt@£ 2 4 I I I ilp3+<^| I r I 42
p3m3n2s2(j3t2 Involution
)3+d5
pdt @ m
p^m^n'^s^d^t Involution
pdt @ n_
12 4 12 2 14 2
p^m^n^s^d^t Involution
nsd @ £
nsd @ _m^
I 2 I I 2 2 I I 2 I
nsd @ Tji
i
p^m^n^sd^t
Involution
S
J bJ r iJ I'J^
S
r UJ J ^''^r ^
^
mnd @2 313 3
3 3 13 innd@p
Of the thirteen new hexads discussed in this chapter, all but
four may also be explained as projection by involution, as illus-
trated in Example 36-8.
239
involution and foreign intervals
Example 36-8
pmd @n
pmn@ d
pn8@ m
pns@d
pdt@s
i h jt» ^"^ I T: ^*> ^^ I ^ «" ^" I t^^e i^» ^^^ I g tt^» ^
»2 ^2 „U ♦ »2 ~.2 ^1
t m' d; n't tp' m' d" t ^p'= H iH * t n T E* *£ H 1 ♦
dt@m
pdT
nsd@p
nsd@m
bl-g i,„ ^_ I it^^
nnnd@_g_
^
ra
t£2 d2 m'i tn2 s} p^l tji2 d2 mU jm^ j^:
2 Ti
2' *
The four new hexads which cannot be arranged in similar
manner are: pmd @ s, s^ @ n, mst @ p, and mst @ d.
240
37
Recapitulation of Pentad Forms
We have now encountered all the pentad forms which are
found in the twelve-tone equally tempered scale. It is wise,
therefore, to summarize them here. The student should review
them carefully, play them and listen to them in all of their
inversions and experiment with them, both melodically and
harmonically. All of the pentads are projected above C for
comparison and, where the pentad is not isometric, the involution
is projected downward from C.
Pentads numbered 1 to 5 predominate in perfect fifths, while
number 6 contains an equal number of perfect fifths and major
seconds. Pentads numbered 7 to 11 predominate in minor
seconds, with number 12 containing an equal number of minor
seconds and major seconds. Pentad number 13 has major thirds,
major seconds, and tritones in equal strength. Pentads numbered
14 to 17 predominate in major seconds. Pentad number 18 pre-
dominates in minor thirds and tritones; numbers 19 to 22
predominate in minor thirds. Pentads 23 to 29 predominate in
major thirds. The tritone, considering its double valency,
dominates pentads 30 to 33, and the remaining pentads, numbers
34 to 38, are neutral in character.
Example 37-1
i
p^mn^s^ 2. p^m^^s^d
Involution
mn (S) n 2. 2. 3 4
=E
=§
^— *
2 2 3 2
2 2 3 4
pmn @ p
j pmn @ p
241
INVOLUTION AND FOREIGN INTERVALS
i
3 p^m^nsd^t
Involution
^2414 nX, ra n 2414 ^^W
2 4 14 ^ @ ^
. p^mn^s^dt /^^ Involution
5^
2 4 14 b^
j pmd @ p
^i i r'Tt'J
te
i
^!^
?
2 4 I 2 t p2 + n^'^*.
p3mns2(j2t
2 4 12
Involution
i pf + _n2 t
2 5 2 2 pns @ s_
■7 mn^s^d"* 8 P^rn^nsd^ t
2 5 2 2
I pns @ _s
nvolution
M J mil 5 U g p III IISU I Ll lilVUIUIIUII ^^
1^ I r' I 6 13 1 pmd @ d 6 13 1 j, pmd @ d^
pm^n^s^d^ Involution
I 2^ I I ^mnd @ d^ 12 11 j mnd @ ^
10. pmn^^d^t Involution
itiJU^^ JJi^J-^^^j ^rr^r^jjJ lejtjmi^
I** I I 3 d2*^ + n2 1113 ,h2 ^ „2
F I I 2 nsd @ 3
I3.J
lVt2
14
p^m^n^s^t
J.nsd @ _s^
ig=i.
iJJltJil^ UbJj-l^ Jr. ■■ :^j.i^jJ^^
2 2 2 2
3 2 2 2 J p2„2 or 2 2 2 3 f^Z + ^c^
242
RECAPITULATION OF THE PENTAD FORMS
15 p^m^ns^dt
^
Involution
jjJtJ^
^ " JJJI^ i
J
2 2 2 1 £3 + p2 2 2 2 1 13^^ 4*^'
16. pm ^ns^'d^t Involution
r I 2 2 3^ + d2^ II
f pl^plJj-^'^4j[;J ptlpb|
r I
1 17 m^n^s^d^t
112 2 is
' . d?
^17 m-n-5-o-i
12 6 2 t IL^ 1^ or ^^
Involution
IQ pmn^sdt^
bJ ijj J ' r ^
^
^^^
^33! 2
ig p^m^n^sdt
3 3 12
Involution
I
^
iH,^^s-'/i- i
J ii'i ^^
pmn @ n
3 13 3
{ pmn @ n
i
20 p mn'^s^dt
Involution
S
r .:^ i,''j' ^ r ^ J J J ly^
F^f
3 4^ 2 I pns @ n
2i.pm^n3sd^t
3 4 2 I ; pns @ n
Involution
3* I 2 I
pmn^s^d^t
mnd @ n
3 12 1
Involution
I mnd @ n
12 12
nsd @ r^
12 12
4 nsd @ n
23. p^m^n^d^
Involution
i J J itJ r ^ r ''^ ^
■•- /I 1 I 1 A. Tf.
4 3 13
24. P^rn^ns^dt
4 3 13
nvolution
2 2 3 1
p"^ + m''
2 2 3 1
I p^ + m'
243
INVOLUTION AND FOREIGN INTERVALS
3n2c2,
25 prtT^n^s^dt
nvolution
ft /''\ 1- IIIVUIUIIUII
3 12 2 m! +J1^
1 26. pm^ns^d^t
3 12 2
Involution
i m-^ + n'
27 p^m~n^sd^
d^ + m2
\ CT
28. P^^n2d2t
id'' + m^
i^J^Jl-^'l^ I'-^^ i
S
^
4 I 2 I J p^2 ^2
29 p^m^n^sd^
S
3 14 1 t m^2 n^
l.» "8
i . bJ . J
13 4 3 % n^ ^
30 p^msd^t^ Involution
J It- Jif. ° ; r T T i'^ -1^
I't' I 4
31 p2mn2sd^t2
^^^^
II 4 1
Involution
ijJiJ ^ ^^ ^-i^ir'r'^^^j^^
-^
3 2 I
32 pm^n^s^dt^
13 2 1
nvolution
e
^ J|J Jil-' Ig^ lis ° :rl'^^i' Jb^J tw.-tf
» ^>
i
"4213
33.p!m!sVt2
4 2 13
^^
i . I).
r *
34
4 2 4 J p_'
p^m^n^sd^t involution
d2
iJi,j^^^^ ii b^i ■' rV ^1
V 1 J tt^
2 13 1
p^m^n^s^d^
p2 + n2
2 13 1 i^p2 ^- n2
Involution
+ p2 + d2 I
4 p2+ d2t
244
RECAPITULATION OF THE PENTAD FORMS
36
p^m^rr^sd^t
Involution
J tri^rrJi
p2+32 4 d2
I p2+ s^ + d<
:^fl p^mn^s^d^t
¥^
J t'* ^"it t.a
2 «2
2 I 6 I Ins'
245
|^^Part,.,_Y,.
THE THEORY OF
COMPLEMENTARY
SONORITIES
38
The Complementary Hexad
We come now, logically, to the rather complicated but highly
important theory of complementary sonorities. We have seen
that the projection of five perfect fifths above the tone C
produces the hexad C-G-D-A-E-B.
Example 38-1
Referring to our twelve-tone circle, we note that these six tones
form a figure having five equal sides and the baseline from C
to B. We note, also, that the remaining tones form a com-
plementary pattern beginning with F and proceeding counter-
249
THE THEORY OF COMPLEMENTARY SONORITIES
clockwise to G^. This complementary hexad has the same
formation as its counterpart and, of course, the same intervallic
analysis.
Example 38-2
#
p^m^n's^d
pSm^n^s^d
''" ^" I'o u> i^
m Q-
f^
■gy. tJ « ==
2 2 3 2 2
m
k^z
bo
Since the hexad |F-Eb-Db-Bb-At)-Gb is the isometric involution
of the original, it will be clear that the formation is the same
whether we proceed clockwise or counterclockwise. That is, if
instead of beginning at F and proceeding counterclockwise, we
begin at G^ and proceed clockwise, the result is the same. We
note, also, that the complementary hexad on G^ is merely the
transposition of the original hexad on C:
Example 38-3
A more complicated example of complementary hexads occurs
where the original hexad is not isometric. If we consider, for
example, the hexad composed of major triads we find an
important difference. Taking the major triad C-E-G, we form a
second major triad on G— G-B-D, and a third major triad on E—
E-GJf-B. Rearranging these tones melodically, we produce the
hexad CsD^EsGiGifsB:
Example 38-4
^
^ m i J .1 •' ti^ r
2 2
250
THE COMPLEMENTARY HEXAD
If we now diagram this hexad, we produce the pattern indicated
in the following example, the major triad hexad being indicated
by solid lines and the complementary hexad by dotted lines:
Example 38-5
(Bbb)A
G^(Ab)
;^Eb
i
jg^ a ^g il^^ ^g ^k
Here it will be observed that the complementary hexad
F-Bb-Eb-Db-Gb-A (Bbb) is not the transposition but the
involution of the original, and that the pattern of the first can
be duplicated only in reverse, that is, by beginning at F and
proceeding counterclockwise. The validity of this statement may
be tested by rotating the pattern of the complementary hexad
within the circle and attempting to find a position in which the
second form exactly duplicates the original. It will then be
discovered that the two patterns cannot be made to conform
in this manner. They will conform only if the point F is placed
upon C and the second pattern is turned ouer— similar to the
turning over of a page. In this "mirrored" position, the two
patterns will conform.
Transferring the above to musical notation, we observe again
251
THE THEORY OF COMPLEMENTARY SONORITIES
that the complementary hexad to the hexad CoDoEgGiGJoB is its
involution, |F2EboDb3BbiBbb3Gb. It will be noted further that as
the first hexad was produced by the imposition of major triads
upon the tones of a major triad, so the second hexad is a
combination of three minor triads, the minor triad being the
involution of the major triad:
Example 38-6
p3m''n3s2d2t
p^m'^n^s^d^t
2 2 3 13
2 2 3 1 3
As might be expected, the intervallic analysis of the two
sonorities is identical: three perfect fifths, four major thirds,
three minor thirds, two major seconds, two minor seconds, and
one tritone, p^m'^n^s^dH.
The third, and most complicated, type of complementary
hexad occurs when the remaining six tones form neither a trans-
position nor an involution of the original hexad but an entirely
new hexad, yet having the same intervallic analysis. For example,
the triad C-E-G at the interval of the minor second forms the
hexad C-E-G + Db-F-A^, or CiDbsEiFoGiAb. Its complementary
hexad consists of the remaining tones, DiEbsFJsAiBt^iBti. Both
hexads have the same intervallic analysis, p^m'^n^sdH but, as will
be observed in Example 38-7, the two scales bear no other
similarity one to the other.
Example 38-7
j^-j J J "^ : j^JjjJ ^^rp
pmn @ d
I 3 I 2 I
I 3 3 I I
252
THE COMPLEMENTARY HEXAD
Ftt
THE COMPLEMENTARY HEXAD
A fourth type includes the "isomeric twins" discussed in Part
III, Chapters 27 to 32, If, for example, we superimpose three
perfect fifths and three minor thirds above C we produce the
hexad C-G-D-A plus C-EbGbA, or C2-Di-Eb3-Gbi-Gti2-A. The
remaining tones, C#3EtiiE#3G#2AJfiB, will be seen to consist of
two minor thirds at the interval of the perfect fifth, A#-C#-Et^
plus E#-G#-B.
Example 38-8
i
, p3m2n4s2d2t2
p3m2n4s2(j2 t2
bo^°'^ -ijg^^t^^ :^tJ|j«Jj^
^
W
pi +
2 13 12 3 13 2 1 n^ @ p
253
39
The Hexad 'Quartets
??
We are now ready to consider the more complex formations
resulting from the projection of triads at intervals which are
foreign to their own construction. We have already noted in the
previous chapter that every six-tone scale has a complementary
scale consisting in each case of the remaining six tones of the
twelve-tone scale.
We have also noted that these complementary scales vary in
their formation. In certain cases, as in the example of the six
tone— perfect-fifth projection cited in Example 38-3, the com-
plementary scale is simply a transposition of the original scale.
In other cases, as in the major-triad projection referred to in
Example 38-5, the complementary scale is the involution of the
original scale. However, in fifteen cases the complementary
scale has an entirely diflFerent order, although the same inter-
vallic analysis.
We have already observed in Part III, Chapters 27 to 33, the
formation of what we have called the isomeric twins— seven pairs
of isometric hexads with identical intervallic analysis. A still
more complex formation occurs where the original hexad is not
isometric, for here the original scale and the complementary
"twin" will each have its own involution. In other words, these
formations result in eight quartets of hexads: the original scale,
the involution of the original scale, the complementary scale,
and the involution of the complementary scale.
The first of these is the scale formed by two major triads pmn
254
THE HEXAD QUARTETS
at the interval of the minor second, already referred to. Its
involution will have the order, 12131, or CiD^oEbiFbaGiAb,
having the same analysis and consisting of two minor triads at
the interval of the minor second. The complementary scale of the
original will consist of the tones DiEbsF^aAiBbiBti, also with the
analysis p^m^n^sdH. Begun on B, it may be analyzed as B3D1EI9
+ FJsAiBb, or two triads mnd at the interval of the perfect
fifth. This scale will in turn have its involution, having again the
same analysis:
Example 39-1
p^m^n^sd^t Involution
Complementary Involution
Hexad,
mnd @
The triad pns at the interval of the minor second forms the
six-tone scale C-G-A + Db-Ab-B^, or CiDbeGiAbiA^iBb,
p^m^n^s^dH. Its involution becomes CiDbiD^iEbeAiBbi- The
complementary scale of the original is DiDifiEiFiFJfgB, with its
involution :
Example 39-2
p^m^n^s^d^t Involution Comp. Hexed Involution
*^ PM ©" d I 6 I I I ' I !• I 6 I
r I I I 5 5 II I I
The triad pns at the interval of the major third forms the
six-tone scale C-G-A + E-B-C#, or CiCJgEgGaAsB, p^m^nh^dH.
255
THE THEORY OF COMPLEMENTARY SONORITIES
Its involution becomes CaDaEgGaBbiBti. The complementary
scale is DiE|:)2FiGb2Ab2Bb, with its involution:
Example 39-3
p'm^n^s^d^t Involution Comp. Hexad Involution
12 12 2 2 2 12 1
The triad pdt at the interval of the major second forms the
six-tone scale C-F#-G + D-G|-A, or CoD4F#iGiG#iA,
p^m^n^s^dH^. Its involution becomes CiC#iDl:iiEb4G2A. The com-
plementary scale is DboEbiEl^iFgBbiBt], with its involution:
Example 39-4
p^m^n^s^d^t^ Involution Comp. Hexad Involution
pdt @ _s_
I 5 I I 2
The triad pdt at the interval of the major third forms the
six-tone scale C-F#-G + E-A#-B, or C4E2FJfiG3A#iB, p^m^nh^dH^.
Its involution is CiDt)3EiFoG4B. The complementary scale of
C4E2F#iG3A#iB is DbiDl^iEboFgAbiAti, with its involution:
Example 39-5
p^m^s^d^t^ Involution Comp. Hexod Involution
pdt @ m
3 124
112 3 1 I 3 2 I I
The triad nsd at the interval of the perfect fifth forms the
six-tone scale C-Db-Eb + G-Ab-Bb, or CiDb2Eb4GiAb2Bb,
p^m^nh^dH. Its involution becomes C2DiEb4G2AiBb. The com-
plementary hexad of C-Db-Eb-G-Ab-Bb is D2EaFiFjj:3A2B, with
256
THE HEXAD QUARTETS
its involution. These hexads, with their preponderance of perfect
fifths and secondary strength in major seconds and minor thirds,
are most closely related to the perfect-fifth series:
Example 39-6
p*m^n^s^d^t Involution Comp.Hexod Involution
The triad nsd at the interval of the major third forms the
six-tone scale C-Db-Eb + Et|-F-G, or CiDbsEbiEt^iFoG,
p^m^nh^dH. Its involution becomes C2DiEbiEtl2FJt:iG. The com-
plementary hexad of C-Db-Eb-Et^-F-G is D4FiJ:2G#iAiAifiB, with
its involution. This quartet of hexads is neutral in character,
with an equal strength of major thirds, minor thirds, major sec-
onds, and minor seconds:
Example 39-7
p m n s d t Involution Comp.Hexod Involution
nsd @ _nn
The last of these quartets of six-tone isomeric scales is some-
what of a maverick, formed from the combination of the inter-
vals of the perfect fifth, the major second, and the minor Second.
If we begin with the tone C and project simultaneously two
perfect fifths, two major seconds, and two minor seconds, we
form the pentad C-G-D + C-D-E + C-C#-D, or GiC#iD2E3G,
with its involution C3Eb2FiFJ|:iG, p^^mnrs^dH:
Example 39-8
Pentad p^mn^sTl t Involution
p2 f S2 + d^
If we now form a six-tone scale by adding first a fifth below C,
257
THE THEORY OF COMPLEMENTARY SONORITIES
we form the scale CiCifiD2EiF2G, with its involution
C^DiEbsFiFJiG:
Example 39-9
p^m^n^s^d^t Involution
iff I ? I 2 2 12 1
2 I 2 I I
If we add the minor second below C, we form the six-tone scale
CiCfliDsEgG^B, with its involution C4E3G2AiA#iB:
Example 39-10
p^m^n^s^d^t Involution
['"1234 43211
Upon examining these four scales, Examples 9 and 10, we find
that they all have the same intervallic analysis, p^m^n^s^dH. We
also discover in Example 11 that the complementary hexad of
Example 9 is the same scale as the involution of the scale in
Example 10:
Example 39-11
I Original Hexod Comp. Hexed Transposition above C
12 12
4 3 2 11
4 3 2 11
(If we take the third possibility and add a major second
below C, we form the six-tone scale CiC^iDaEsGsBb, which is
an isometric scale with the analysis p^m^n^s^dH^, already dis-
cussed in Chapter 29. It will be noted that this scale contains
both the pentad CiCjfiDsEsG and its involution jDiCJfiCl^sBbsG.
Example 39-12
p2m2n4s3d2t2
Involution
258
THE HEXAD QUARTETS
The complementary hexads of Examples 39-1 to 39-7, inclusive,
may all be analyzed as projection by involution, as illustrated in
Example 39-13:
Example 39-13
m
^
JB fs
<t°tj {t^O
W^
i
3. _ ^ ^ 4
d^ p' t J s^ d^ p' i
^^^^^^
h u^° ^g ^-
^
t p^ m2 s' * tp2 m^ s' t $ n^
7.
S'^ TTl' I
:|,l^fi^8^«
tm^ n^ s't Jm^ n^ s' t tp^ s^ d' t Jp^ s^ d'i J m^ d^ s'Um^d^s't
259
^Part^
COMPLEMENTARY SCALES
40
Expansion of the
Complementary- Scale Theory
We have noted that every six-tone scale has a complementary
six-tone scale consisting of all of the notes which are not present
in the original scale, and that these scales have the same inter-
vallic analysis. An analysis of all of the sonorities of the twelve-
tone scale will reveal the fact that every sonority has a
complementary sonority composed of the remaining tones of the
twelve-tone scale and that the complementary scale will always
have the same type of intervallic analysis, that is, the
predominance of the same interval or intervals. In other words,
every two-tone interval has a complementary ten-tone scale,
every triad has a complementary nine-tone scale, every tetrad
has a complementary eight-tone scale, every pentad has a
complementary seven-tone scale, and every six-tone scale has
another complementary six-tone scale.
For example, the major triad will be found to have a nine-tone
scale as its counterpart, a scale which is saturated with major
triads and whose intervallic analysis has a predominance of the
intervals of the perfect fifth, major third, and minor third which
make up the major triad. This nine-tone scale we shall call the
projection of the major triad, since it is in fact the expansion or
projection of the triad to the nine-tone order. The importance of
this principle to the composer can hardly be overestimated, since
it allows the composer to expand any tonal relation with
complete consistency.
263
COMPLEMENTARY SCALES
The process of arriving at such an expansion of tonal resources
is not an entirely simple one, and we shall therefore examine it
carefully, step by step, until the general principle is clear. The
major triad C-E-G has a complementary nine-tone scale consist-
ing of the remaining nine tones of the chromatic scale, the tones
C#-D-D#-F-F#-G#-A-A# and B. We shall observe in analyzing
this scale that it has seven perfect htths, seven major thirds, and
seven minor thirds, but only six major seconds, six minor seconds,
and three tritones— that it predominates in the same three
intervals which form the major triad.
If we again revert to our circle and plot the major triad C-E-G,
we find, proceeding counterclockwise, the complementary figure
E#-A#-D#-G#-C#-F#-B-A and D:
Since, as has already been noticed, clockwise rotation implies
proceeding "upward" in perfect fifths and counterclockwise
rotation implies proceeding "downward" in perfect fifths, we
may transfer the above diagram to musical notation as follows:
264
i
expansion of the complementary-scale theory
Example 40-2
Triad pmn Complementary Nonad p m n^s^d^t^
*r V ^'r r tt^ 1'^ <t^ j^^j
I 1 O 1 I I O /IV"
^
I I
1112 (ir
If we analyze the complementary nine-tone scale, we find that
it consists of a nine-tone projection downward from E#, or
upward from GJf, not of the major triad, but of its involution,
the minor triad :
Example 40-3
I
''I'V^^j^il ii|j.,ji;^ ^
If we now form the involution of the nine-tone sonority by
constructing a scale which has the same order of half and whole
tones proceeding in the opposite direction, we construct the
following scale:
Example 40-4
Involution of ttie Complementary Nonad
i J jj .1 ii^ J t^ ^ r
2 II
II I 2 (I)
Analyzing this scale, we find it to consist of the nine-tone
projection of the major triad :
Example 40-5
We may therefore state the general principle that the nine-tone
projection of a triad is the involution of its complementary scale.
We shall find, later, that this same principle applies also to the
projection of tetrads and pentads.
The tone which is used as the initial tone of the descending
265
COMPLEMENTARY SCALES
complementary scale— in this case E# (or Ft])— we shall call the
converting tone. Its choice in the case of the superposition of
perfect fifths or minor seconds is simple. For example, if we
superimpose twelve perfect fifths above C, the final tone
reached is F, which becomes the initial tone of the descending
complementary scale. The complementary heptad of the perfect-
fifth pentad C-D-E-G-A becomes the scale FsE^oDbaCbiBboAba
Gb(i)(F). The seven-tone projection of C-D-E-G-A becomes
therefore the complementary heptad projected upward from C,
or C2DoE2F#iG2A2B(i)(C) (See Ex. 41-1, lines 4 and 6.)
The converting tone of any triad is almost equally simple to
determine being the final tone arrived at in the upward projection
of the original triad. For example, if we superimpose major
triads upon the tones of the major triad C-E-G and continue
superimposing major triads upon each resultant new tone until
all twelve tones have been employed, the final tone arrived at
will be the converting tone for the complementary scales of that
formation. Beginning with the major triad C-E-G, we form the
triads (E)-GJ|:-B and (G)-B-D, giving the new tones G#, B, and
D. Superimposing major triads above G#, B, and D, we form the
triads (G#)-(B#)-D#, (B)-D#-F|t and (D) F# A giving the new
tones D#, Fjj:, and A. Again superimposing major triads on DJ,
F#, and A, we form the triads (DJf )-(F-^ )-A#, (F|;)-A#-C#, and
(A)-CJj:-(E), giving the new tones A# and C}f.
Finally, superimposing major triads above A# and Cf, we
form the triads (A#)-(C>x< )-E# and (C#)-E#-(G#), giving the
final twelfth tone E#. This tone becomes the converting tone,
that is, the initial tone of the descending complementary scale.
Example 40-6
j> ^jiy.;tllylf/yl|JH¥^
>/ »l l*P
ii» ttii% ^1^
266
EXPANSION OF THE COMPLEMENTARY-SCALE THEORY
The complementary heptad of the pentad composed of two
major triads at the perfect fifth, C-E-G + G-B-D, or C2D2E3G4B,
becomes, therefore, iFaEbsD^gBbiAiAbsGb. The projection of
C-D-E-G-B is therefore CsDsEsGiG^iAaB. (See Ex. 42-1, lines
2 and 5. )
In many other cases, however, the choice of the convert-
ing tone must be quite arbitrary. For example, in the case
of any sonority composed entirely of major seconds, the
choice is entirely arbitrary. The whole-tone hexad above C,
for example, is C2D2E2F#2G#2AJ1:. Since this scale form super-
imposed on the original tones produces no new tones but merely
octave duplications, it is obvious that the converting tone of
the scale C2D2E2F#2G#2A# will be B-A-G-F-Eb or D^, giving
the complementary scales J,B2A2G2F2Et)2Db' iA2G2F2Eb2
DbsB, iG2F2Eb2Db2B2A, jF2Eb2Db2B2A2G, jEb2Db2B2A2G2F, or
iDb2B2A2G2F2Eb. The choice of F as the converting tone in
Example 40-7 is therefore entirely arbitrary.
Example 40-7
Major Second Hexod Complementary Hexads
I I ^ I II r r r r 't b^ i r r
J J jf-' ff"" ti" " 'I I I I — L
22222 22^22
22222
^|rr'rhvJir"rh-|.Jji'rVr^JJi^rp^^T
22222 22222 22222 22222
Take, again, the major-third hexad in Example 40-8,
CsDJiEaGiGfligB. If we superimpose this intervallic order, 31313,
upon each of the tones of the hexad, we form the hexads
CsDJtiEeGiGSsB; (D'i^)sFUG)sAUB),D^; iE),(G),{G^),
(B)i(C)3(D#); (G)3A#,(B)3D,(D#)3F#; (G#)3(B),(C)3
(D#)i(E)3(Gti); and (B),D,(Dj)^)sFUG)sAl giving the new
tones F#, A#, and D and producing the nine-tone scale
267
COMPLEMENTARY SCALES
C2DiD#iE2F#iGiG#2A#iB(i)(C.) The remaining tones, F, Db,
and A, are all equally the result of further superposition and are
therefore all possible converting tones, giving the descending
complementary scales iFgDiDbsBbiAsGb, iDbsBbiAsGbiFsDl^,
and lAaGbiFgDiDbsBb- Our choice of F is therefore an arbitrary
choice from among three possibilities.
Example 40-8
^m
o "
^
« -^fr
^"*'^" av^^^' '^'
3 13 13
3 13 13
Complementary Hexads
3 13 13
3 13 13
3 13 13
One final example may suffice. The tritone hexad of Example
40-9 contains the tones CiC#iDjFJfiGiG#. This scale form super-
imposed upon the original tones gives the hexads CiCj^J^^Fjj^iGi
G#; (C#),(D),D#4(G),(G#)iA; (D),DS,E,(G#)AAJf; (F#),
(G),(G#),(C)i(C#),(D); (G),(GJf)iA4(C#),(D),D#; and
(G#)iAiA#4(D)iD#iE, with the new tones D#, E, A and A#,
producing the ten-tone scale CiC#iDiD#iE2F#iGiG#iAiA#,2) (C).
The remaining tones, F and B, are therefore both possible con-
verting tones giving the descending complementary scale of the
hexad CiCjf.D^FJfiGiGJf as iF,EiEb4BiBbiA or jBiBbiA^FiEiEb-
Our choice of F is therefore an arbitrary choice between two
possibilities. ( See the Appendix. )
268
expansion of the complementary-scale theory
Example 40-9
V- ..|t» o<l^ "^" it- '<^ ^
^
i^-^=^
I 4 I
I I 4 I I
I I 4 I I
^
-^4*^
1l--«t°
ft. °fr
i"^:
I I 4 I I
I I 4 I I
i
Complementary Hexads
I I 4 11
114 11
In certain cases where sonorities are built-up from tetrads or
pentads through connecting hexads to the projection of the
complementary octads or heptads respectively, the converting
tone of the connecting hexad is used. *
An understanding of the theory of complementary scales is
especially helpful in analyzing contemporary music, since it
shows that complex passages may be analyzed accurately and
ejffectively by an examination of the tones which are not used
in the passage. Let us take, for example, the moderately
complex tonal material of the opening of the Shostakovitch
Fifth Symphony:
Example 40-10
Shostakovitch, Symphony No. 5
Moderoto S^
/
Copyright MCMVL by Leeds Music Corporation, 322 West 48th Street, New York 36, N. Y Reprinted
by permission. All rights reserved. '
* A "connecting hexad" is defined as any hexad which contains a specific
pentad and is also a part of that pentad's seven-tone projection.
269
COMPLEMENTARY SCALES
r_yJP
i
o tfv> o ^
o> -P=
^
12 12
2 I
i iiJ j^ ^r I jij ^^f ^
omitted tones
An examination of the opening theme shows not only the
presence of the tones D-D#-E-F#-G-A-B[;,-C-CJf, but the absence
of the tones F, G#, and B. Since F-G#-B is the basic minor third
triad, it becomes immediately apparent that the complementary
nine-tone theme must be the basic nine-tone minor third scale.
A re-examination of the scale confirms the fact that it is composed
of two diminished tetrads at the interval of the perfect fifth plus
a second foreign tone a fifth above the first foreign tone— the
formation of the minor-third nonad as described in Chapter 13.
This type of "analysis by omission" must, however, be used
with caution, lest a degree of complexity be imputed which was
never in the mind of the composer. The opening of the Third
Symphony of Roy Harris offers a fascinating example of music
which, at first glance, might seem much more complex than
it actually is.
Example 40-11
Vios.i K
Harris, Symphony No. 3 _ ^^ ^J^-J).
270
EXPANSION OF THE COMPLEMENTARY-SCALE THEORY
1 =.
^
«*-«»
V^ -
^>,, |Trr [rrrri^r^rr^ir «r^
By permission of G. Schirmer, Inc., copyright owner.
If we examine the first twenty-seven measures of this
symphony, we shall find that the composer in one long
melodic line makes use of the tones G-Ab-Ati-Bb-Bfc|-C-C#-D-D#-
E-F-F#; in other words, all of the tones of the chromatic scale.
Upon closer examination, however, we find that this long line is
organized into a number of expertly contrived sections, all linked
together to form a homogeneous whole. The first seven measures
consist of the perfect-fifth projection C-G-D-A-E-B, or
melodically, G-A-B-C-D-E, a perfect-fifth hexad with the tonality
apparently centering about G.
The next phrase, measures 8 to 12, drops the tone C and adds
the tone B^. This proves to be another essentially perfect-fifth
projection: the perfect-fifth pentad G-D-A-E-B (G-A-B-D-E)
with an added B^i, producing a hexad with both a major and
minor third. (See Example 39-6, Chapter 39, complementary
hexad.) Measure 15 adds a momentary A[) which may be ana-
lyzed as a lowered passing tone or as a part of the minor-second
tetrad G-Ab-Atj-B^. Measures 16 to 18 establish a cadence consist-
ing of two major triads at the relationship of the major third—
Bb-D-F plus D-F#-A (D-F-F#-A-Bb-Example 22-2).
271
COMPLEMENTARY SCALES
Measures 19 to 22 establish a new perfect-fifth hexad on D—
D-A-E-B-F#-C# (D-E-F#-A-B-C#), which will be seen to be a
transposition of the original hexad of the first seven measures. In
measure 23 the modulation to a B tonality is accomplished by
the involution of the process used in measures 16 to 18, that is,
two minor triads at the relationship of the major third:
AJf F#
DJf B
Measures 24 to 27 return to the pure-fifth hexad projection
G-D-A-E-B-F#, in the melodic form B-D-E-F#-G-A, a transposi-
tion of the hexad which introduced the theme.
The student may well ask whether any such detailed analysis
went on in the mind of the composer as he was writing the
passage. The answer is probably, "consciously— no, subconscious-
ly—yes." Even the composer himself could not answer the
question with finality, for even he is not conscious of the
workings of the subconscious during creation. What actually
happens is that the composer uses both his intuition and his
conscious knowledge in selecting material which is homogeneous
in character and which accurately expresses his desires.
A somewhat more complicated example may be cited from
the opening of the Walter Piston First Symphony:
Example 40-12
Piston, Symphony No
'Cellos, Bosses pizz
'-n : j]^ ^'v/jtj \iO \i>r^ ' I ..rjjj;^
By permission of G. Schirmer, Inc., copyright owner.
Here the first three measures, over a pedal tone, G, in the
tympani, employ the tones G-G#-A-Bb-Bti-C-C#(Db)-D-E, all
of the tones except F, Ffl:, and D#, in which case the nine-tone
scale might be considered to be a projection of the triad nsd.
272
EXPANSION OF THE COMPLEMENTARY-SCALE THEORY
Such an analysis might, indeed, be justified. However, a simpler
analysis would be that the first five beats are composed of two
similar tetrads, CiDboEgG and GiG^gBgD, at the interval of the
perfect fifth; and that the remainder of the passage consists of
two similar tetrads, B[)iBl:|iC4E and GiGj^iA^Cjl^, at the interval
of the minor third. Both analyses are factually correct and
supplement one another.
273
41
Projection of the
Six Basic Series with Their
Complementary Sonorities
We may now begin the study of the projection of all sonorities
with the simplest and most easily understood of the projections,
that of the perfect-fifth series. Here the relationship of the
involution of complementary seven-, eight-, nine-, and ten-tone
scales to their five-, four-, three-, and two-tone counterparts will
be easily seen, since all perfect-fifth scales are isometric.
Referring to Chapter 5, we find that the ten-tone perfect-fifth
scale contains the tones C-G-D-A-E-B-F||-C#-Gif-Dif or, ar-
ranged melodically, C-C#-D-Dit-E-FJf-G-Git-A-B. We will observe
that the remaining tones of the twelve-tone scale are the tones
F and B^. If we now examine the nine-tone-perfect-fifth scale,
we find that it contains the tones C-G-D-A-E-B-F#-C#-G# or,
arranged melodically, C-C#-D-E-F#-G-G#-A-B. We observe that
the remaining tones are the tones F, B^, and E^.
If we now build up the entire perfect-fifth projection above C,
we find that the complementary interval to the ten-tone scale
is the perfect fifth beginning on F and constructed downward;
the complementary three-tone chord to the nine-tone scale con-
sists of two perfect fifths beginning on F and formed downward,
F-B^-E^; the complementary four-tone chord to the eight-tone
scale consists of three perfect fifths below F, F-Bb-E^-Ab; and
the complementary five-tone scale to the seven-tone scale consists
of four perfect fifths below F, F-Bb-Eb-Ab-Db-
The first line of Example 41-1 gives the perfect fifth with its
complementary decad. The projection of the doad of line 1 is
274
PROJECTION OF THE SIX BASIC SERIES
therefore the decad of Hne 9, which is the involution of the
complementary decad of line 1.
Line 2 gives the perfect-fifth triad v^ith its complementary
nonad. The projection of the triad becomes the nonad, line 8,
vi^hich is the involution of the complementary nonad of line 2.
Compare, therefore, line la with line 9, 2a with 8, 3a with 7,
4a with 6, and 5<2 with 5. Note also that 9a is the involution of 1,
8a the involution of 2, 7a the involution of 3, 6a the involution of
4, and 5a the involution of 5.
Example 41-1
Perfect Fifth Doad p
lo.
Complementary Decad
^r'r^r Ji^Ji^
i
7 (5)
Perfect Fifth Triad p^s
II I I ^ I I I 2 (I)
^°- Complementary Nonad
r r'T^r'f r ^'^^^J
^
2 5(5)
Perfect Fifth Tetrad p^ns^
I I 2 2 r I I 2 (I)
3a.
S
Complementary Octad
Z
r r'T^rir^riJbJ
' \'2 2 {0
^
2 5 2 (3)
Perfect Fiftti Pentod p^mn^s^
I I 2 2 12 2 (I
4a.
^m
omplementqry Heptad
Lompiementq
2 1' 2 2 (1^
2 2 3 2 (3)
$
2 2 2 I 2 2 {I
Perfect Fifth Hexad p m n^s d " Complementary Hexad
^P
fi
^^^
"^^ ^
i
2 2 3 2 2 (I)
2 2 3 2 2 (I)
60.
Perfect Fifth Heptad p m n s d^ Complementary Pentad
^
^^
m
-^ 5t
i
^
2 2 2 I 2 2 (I)
2 2 3 2(3)
7a.
il^i Perfect Fifth Octad p^m^n^s^d^t^ Complementary Tetrad
r I 2 2 I 2 2 (I)
2 5 2 (3)
275
COMPLEMENTARY SCALES
8. 8a.
j^ Perfect Fifth Nonad pfim^n^s^d^t^ Complementary Triad
4
i^
*
m
^
12 2 1 I I 2 (I)
2 5 (5)
9a.
jl-e- Perfect Fifth Decod p^m^nQs^d^t^ Complementary Doad
^
1<'^I r I 2 I I I 2 (1) 7 (5)
lOo.
10.
fej;
Perfect Fifth Undecod pjVWW*
^^^
Q »" Jul*'}
r^i I I 2 I I I I I (I)
4t^ #^ Perfect Fifth Duodecad p'^m'^n'^s'^d'^t^
^m
r I T I I I r I I I I (I)
The minor-second series shows the same relationship between
the two-tone interval and the ten-tone scale; between the triad
and the nine-tone scale; the tetrad and the eight-tone scale, and
the five-tone and the seven-tone scale. Line 9 is the involution
of la; line 8 of 2a; line 7 of 3a, line 6 of 4a, and hne 5 of 5a.
Conversely, line 9a is the involution of 1, line 8a the involution
of 2, line 7a of 3, line 6a of 4, and line 5a of 5.
Example 41-2
Minor Second Dead d
Minor Second Triad sd^
IQ Complementary Decad
2a.
Complementary Nonad
^^^^
i
^
^
f
(10)
Minor Second Tetrad ns^d^
I I I
I I I
Complementary Octad
^^
S
r I I (9)
276
I I I
I I I (5)
PROJECTION OF THE SIX BASIC SERIES
4 Minor Second Pentad mn^s^d^ 4o Co,mplenr|entary Heptod
•P-f- f»-bf- ,,&,
i
m
£^
^^^
f^
(8)
I I I I I (6)'
5. Minor Second Hexed pm^n^s^d^ 5q. Complementary Hexod
^ *^^ ^^ -km
i
^^£#^
'^ itr I 1^ I I (7)
m
i
'l+*'l 1^ I I (7) '^
Minor Second Heptad p^m^^s^d^t Complementary Pentad
Lpmpiementa
60. ^W ^bi^
^
|J J ^IHJ
s^
^
^
*^ "^ r I I I- (6)
I I I I (8)'
s4^4„ 5,6^7*2
7 Minor Second Octod p^m^n^s^dM*^ 7^ Complementary Tetrad
^
m
r I r I I I I (5)
p
I I I (9)
Minor Second Nonod p^m®n^s^d®t' ^ Complementary Triad
8./I ■'^ 8q. ^ ^
^
^
iTi I r' I II
^
r I r I III I (4)
9 Minor Second Decad £^nri^n^s^d2t'*9Q Cqmpiementary Doad
I I (loy
I I I I I I I I (3)
... o J II J . 10 10 10 10 .10.5
iO/» Minor Second Undecod pmnsdt loa.
II ii Minor Second Duodecod p'^m'^n'^s'^d'^t^
I I I I I I I I I I I (I)
The major-second projection follows the same pattern, even
though it is not a "pure" scale form. Note again that the decad
in line 9 is the involution of the complementary decad, la; the
nonad 8 is the involution of the complementary nonad 2a; and
so forth. Note also that 9a is the involution of 1, 8a the involu-
tion of 2, and so forth.
277
complementary scales
Example 41-3
I. ^ Major Second Dood £ la. Complementary Decad
2 (10)
2. /I Major Second Triad ms^
2o.
I i 2 2 1 I II I (i)
Complementary Nonad
2 2 (8)
3.^ Major Second Tetrad m^s^t
3a.
2 2 2 11 I I I (I)
Complementary Octad
r'Tt^flT^r Ji^
njJi[Jr
^
^
2 2 2 (6)
4yi Major Second Pentad m^s^t^
4a.
2 2 2 1 I I I (2)
Complementary Heptad
2 2 2 2 (4)
5y, Major Second Hexad m^s^t^
5a.
1^
2 2 2 r I 2 (2)
Complementary Hexad
r'T^'r ^^
i J J^rt^
55
2 2 2 2 2 (2)
Q CO c o ^
6.^ Major Second Heptad p m n s d t 6o.
i
2 2 2 2 2 (2)
Complementary Pentad
^^
'^pt^pbp J
«
^^
^
2 2 2 I I 2 (2)
7y, Major Second Octad p\n^nV^d^^ 7o.
jjjjj Jti^^it^r'
2 2 2 2 (4)
Complementary Tetrad
fl^bphp
:ti^
2 2 2 11 I I (2)
8.^ Major Second Nonad p^m^n^s^d^t^ Ba.
2 2 2 I I I I I (1)
2 2 2 (6)
Complementary Triad
^
(I)
9Jj Major Second Decad p^m^n^s^d^t^ 9a.
flJ ^1^ ^^1^ r 'f
2 2 (8)
Complementary Doad
m
m
I 2 2 I I I I I (I)
10/. Major Second Undecad p'Om'Qn'Qs'Qd'QtS |q^
2 (10)
10/}
Is
jj[j ^jt^ ^ii^ r 'f^^
r I r I 2 I I I I I (I)
lly, Major Second Duodecad p'^m'^n'^s'^d'^t^
PROJECTION OF THE SIX BASIC SERIES
The minor-third projection follows the same pattern, with the
exception that the minor-third scale forms are not all isometric.
It should be noted that while the three-, four-, eight-, and
nine-tone formations are isometric, the five-, six-, and seven-tone
scale each has its involution. (See Chapters 11 through 13.)
The student should examine with particular care the eight-
tone minor-third scale, noting the characteristic alternation of a
half-step and whole step associated with so much of con-
temporary music.
Example 41-4
I- ^ Minor Third Doad n
i
la Complementary Decod
II I I I I 19 1 m/
^
3 (9)
2.M Minor Third Triad n^t
I I I
2a. Complementary Nonad
^^
I I I I 2 I. 2 T (?r
3 3 (6)
4 2
3.^ Minor Third Tetrad n t
3o. Complementary Octad
P^p
^fi'^Jj J
?
m
3 3 3 (3)
4 yi Minor Third Pentad pmn^sdt^
jbJ^Jft^^r'
2 12 12 -tt-o\-^)
4q Complementary Heptad
^
^
m
3 3 1 2 (3)
5.^ Minor Third Hexod p^m^n^s^d^t^
I 2 3 I 2 T (2)
5a. Complementary Hexad
i^M^^Yr'
s
^W
3 31 2 I (2)
6- MinorThird Heptad p'm^ n^s^d^t^
3 3 I 2 T (2)
6o. Complementary Pentad
jtj J^rr>
^
^p
?
m
I 2 3 1 2 1 (2)
7.^ Minor Third Octad p%i^ n^s'^d'^t'^
3 3 12 (^f")
7a. Complementary Tetrad
1 2 1 2 1 2 1 (2)
279
COMPLEMENTARY SCALES
,6w,6„8e6H6t4
QM Minor Third Nonod p°m°n°s°d°t^ 8a. Complementary Triad
3 3 (6)^
9.^ Minor Third Decad p^m^n^s^d^t'^ 9o. Complementary Doad
I I I I I I I 2 I (2)
,„_ Minor Third Undecod p'°m'°n'°s'°d'° t^ ,^
10^ — i : 10 0 .
11.^ MinorThird Duodecod p'^m'^n'^s'^d'^t®
I I I I I I I I I I I (I)
The major-third projection forms isometric types at the three-,
six-, and nine-tone projections; the four-, five-, seven-, and
eight-tone projections all having involutions, (See Chapters 14
and 15.) The student should examine especially the nine-tone
major-third scale with its characteristic progression of a whole
step followed by two half-steps, or vice-versa.
I. ^ Major Third Doad m^
i
Example 41-5
io. Complementary Decad
^T^r^fii'^l.
m
w
A (8)
2 ii Major Third Triad m^
i^A Major ihird
2 I I 2 I I I I I (I)
2a. Complementary Nonad
■$
r^V^fJ JJ^j
4 4 (4)
3/5 Major Third Tetrad pm^nd
2 I I 2 I I 2 I (I)
3o. Complementary Octad
jJ^tt^T
g
S
rrr'ri^r^a
2 I I 2 11 3 (I)
4a. Complementary Heptad
4 3 I (4)
4.^ Major Third Pentad p^ m^n^d^
jj^ti^rT
iF=i=^
rr^ry-^^JJ
4 3 1 3 (I)
280
2 1 I 3' I 3 (I)
PROJECTION OF THE SIX BASIC SERIES
5y. MajorThird Hexad p^m^n^d^ 5a. Complementary Hexad
^m
r^r^^hi
te
^
^
it-*-
3 I 3 1 3 (I)
3 I 3 I 3 (1)
6y, MajorThird Heptad p'^m^n^s^d^t 6a. Complementary Pentod
j-jjiJ J ^i^ r f
fe
^¥^
2 r I 3 I 3(1) 4 3 I 3 (I)
7- Major Ttiird Octad p^nn^nSs^d^t^ 7a. Complementary Tetrad
8^ Major Ttiird Nonod p°m n°s d°t^ 8a. Complementary Triad
»
jj.jJiiJJttJjt^rt'
s
F^*r
w
2 (■ I 2 I I 2 I (I) 4 4 (41
9. A Mojor Third Decad pOm^nQs^d^H 9a , Complementare Doad
jiJjtfJ^ii^r'r^
^
w
2 r I 2 I I I I I (I)
4 (8)
MajorThird Undecod p'^m'^n'^s'^d'^t^ IOq
I I I I 2 I I .1 I I (I)
u K, ■ T^- A r^ ^ A '2 12 12 12 .12,6
"y) MajorThird Duodecad p m n s d t
■ yj major iniru uuuuBt;uu p rti ii a u i
r I I I I I I I I I I (I)
The projection of the tritone upon the perfect-fifth series
produces a series of scales which predominate in tritones—
remembering the double valency of the tritone discussed in
previous chapters. All of the scales follow the general pattern of
the triad pdt, with a preponderance of tritones and secondary
importance of the perfect fifth and minor second. The four-,
six-, and eight-tone forms are isometric, whereas the three-,
five-, seven-, and nine-tone forms have involutions.
281
I.^Tritone t
complementary scales
Example 41-6
lo . Complementary Decad
^
^^^^^
n I I (i
6 (6)
2. - Perfect Fifth - Tritone Triad pdt 2a.
i I I 2 M I I I (^)
Complementary Nonad
^m
i i I I o' i~i i r~
I I I 2 r I I (3)
Complementary Octod
6 I (5)
3.A Tetrad p^d^t^
3.^ leiruu p u
1^ IJiiJ J^
3o.
I i~^ 1 ' I I i T
g
^
I I 3 1'! I (3)
Complementary Heptad
r 5 I (5)
4./I Pentad p^msd^t^
4a.
M
rr^r^r Ji'^
#^
^3*^5
r I 4 I (5)
5^, Hexad p'^m^s^d'^t'
ifl I 4. I \ t<X)
5o.
I I 4 r I I (3)
Complementary Hexad
rTTiTl'r J
^fH^
I"" I 4 1 I (4)
6.^ Heptad p^m^n^s^d^t^
i
I I 4 I I (4)
6a. Complementary Pentad
rT'Tl'r^r^
^^iJitJ^tiJ^if
r I 4 I I I (3)
7.i5 Octad pgm^^nMd^t^
7a.
^^
II 4 I (5)
mplementary Tetrad
3
s
I I I 3
m
I I 3 I I I (3)
^g Nonad p^m^nQsQd^t'*
I 5 I (5)
Complementary Triad
III 1211 I (3)
9yi Decad p^m^n^s^d^t^
9o.
6 I (5)
Complementary Doad
1^ I I I 2 I I I I (2) 6 (6)
r I I 12 1 I I I (2)
'Oj^ Undecad p'Om'On'Os'Od'QtS
lOa.
JJJ^J JtlJ J«^ ^^^ P
r I I I 2 I I I I I (I)
r I I 12 11
11./, Duodecad p'^m'^n'^s'^d'^t^
1.^ uuoaecaa p m n s a t
I II I I I I I I I I (I)
282
PROJECTION OF THE SIX BASIC SERIES
An excellent example of the gradual expansion of the projec-
tion of perfect fifths will be found in Bernard Rogers' "Portrait"
for Violin and Orchestra (Theodore Presser Company). The first
two and a half measures consist of the tones D-E-F (triad nsd).
The third, fourth, and fifth measures add, successively, the tones
G, A, and C, forming the perfect-fifth hexad, D-E-F-G-A-C
(F-C-G-D-A-E).
This material suffices until the fifteenth measure which adds
the next perfect fifth, B. The seventeenth measure adds Cfl:, the
nineteenth measure adds F#, and the twenty-first measure adds
G#, forming the perfect-fifth decad, Ft|-C-G-D-A-E-B-F#-C#-G#.
In the twenty-third measure this material is exchanged in favor
of a completely consistent modulation to another perfect-fifth
projection, the nonad composed of the tones Ab-Eb-B^-F-C-G-D-
At^-Et^. This material is then used consistently for the next
twenty-four measures.
In the forty-seventh measure, however, the perfect-fifth pro-
jection is suddenly abandoned for the harmonic basis F#-G-A-
Cp, the sombre, mysterious pmnsdt tetrad, rapidly expanding to
a similar pmnsdt tetrad on A (A-B^-C-E), and again to a
similar tetrad on C# (CJf-Dti-E-Gf), as harmonic background.
The opening of the following Allegro di molto makes a similar-
ly logical projection, beginning again with the triad nsd
(F-G^-A^) and expanding to the nine-tone projection of the
triad nsd, E^-Etj-F-Gb-Gti-AI^-Akj-Bb-C, in the first four measures.
The projection of the most complex of the basic series, the
tritone, is beautifully illustrated by a passage which has been the
subject of countless analyses by theorists, the phrase at the
beginning of Wagner's Tristan and Isolde. If we analyze the
opening passage as one unified harmonic-melodic conception, it
proves to be an eight-tone projection of the tritone-perfect-fifth
relationship, that is, AiA#iB3DiD}t:iEiF3G#(i)(A). Sensitive
listening to this passage, even without analysis, should convince
the student of the complete dominance of this music by the
tritone relationship. ( See Example 41-6, line 7. )
283
COMPLEMENTARY SCALES
This consistency of expression is, I believe, generally charac-
teristic of master craftsmen, and an examination of the works of
Stravinsky, Bartok, Debussy, Sibelius, and Vaughn-Williams— to
name but a few— will reveal countless examples of a similar ex-
pansion of melodic-harmonic material.
The keenly analytical student will also find that although no
composer confines himself to only one type of material, many
composers show a strong predilection for certain kinds of tonal
material— a predilection which may change during his lifetime.
It might in many cases be more analytically descriptive to refer
to a composer as essentially a "perfect-fifth composer," a "major-
third composer," a "minor-second-tritone composer," and the like
—although no composer limits himself exclusively to one vocabu-
lary—rather than as an "impressionist," "neoclassicist," or other
similar classifications.
284
42
Projection of the
Triad Forms with Their
Complementary Sonorities
Before beginning the study of the complementary sonorities or
scales of the triad projections, the student should review Part II,
Chapters 22 to 26 inclusive. We have seen that any of the triads
fmn, pns, pmd, mnd, and nsd, projected upon one of its own
tones or intervals, produces a pentad. The triad projected upon
all three of its tones produces a hexad which is "saturated" with
the original triad form. The seven-tone scales have the same
characteristics as their five-tone counterparts, and the nine-tone
scale follows the pattern of the original triad.
Let us now examine Example 42-1, which presents the projec-
tion of the major triad pmn. Since the projection of the triads
pns, pmd, mnd, and nsd follow the same principle, the careful
study of one should serve them all.
I , pmn Triad
i
Example 42-1
Complementary Nonad
^^ A 1
r'Tr^r'fr^iJj
F*
4 3
Z.A pmn @ p
Pentad p^m^n^s^d
2 112 1'! I 2
Complementary Heptod U)
^
r'T^r^f Ji-J
^
*
2 2 3 4
3-/1 pmn @ m
Pentad p^m'^n^^
2 2 31 12
Complementary Heptad (2)
f til jj.ifl.ir : I'Tr^rh-AJ
4 3 13
2 113
285
COMPLEMENTARY SCALES
4-ji pmn@ p - m Hexad p^rri^n^s^d^t Complementary Hexad
kf.^l^lMIl^ \J III IIC^VJVJ \J Ml M o VJ I \^\./iii(^n^iii\*iiiv«ijr > i\*^*.jvi
2 2 3 1 3 (I)
5./I Involution of comp.Heptad(l)
^^
2 2 3 13
Complementary Pentad II)
I'T^rtT^J
^^
1
n
m
2 2 3 I I p'p5m4n4s4d3t
6/1 Involution of com p. Heptad (2)
2 2 3 4
Complementary Pentad'(2)
iJitJ J^tf^r
f^^i'^Y^ \^^
2 f I 3 13 P"""^"^'-^"' 4 3 13
7.A Involution of comp.Nonod p^m^n^sQd^^ Complementary Triad
i
iJMJ..ii'.-'^'Ar:f "^
*
P
2 r I 2 I I I 2 (I)
S/i pmn Triad
3^ pnn
4 3
Complementary Nonad
flf^
^P
9/1 P'T'" @ n
4 3
Pentad p^m^n^sdt
I 2 12 I I 2M
Complementary Heptad
3 1-33 I 2 I 3 I 21
10^ pmn@n + ml Hexad p^m^n^s^d^t Complementary Hexad
*^ * b* ^3 r 3 I 2' 3 13 15
*=*
^
^
b* ^3 r 3 I 2' 3 13 12
I'yj Involutionof comp.Heptad p'^m'^n^s^d^t^ Complementary Pentad
m
m
«M
i^
^
^
^'2^ P"3 I 2' 3 13 3
I2y) Involution of comp.Nonod p^mVs^d^t^ Complementary Triad
'^y^ mvoiuTion or comp.iNonaa p^mrrs^^ElL ^.Ajinpu
I ? I 9 I I O I Ml 4 3
i
I 2 I 2 I I 2 I (I)
The first line of Example 42-1 shows the major triad C-E-G
and, separated by a dotted line, its complementary nonad— the
remaining tones of the chromatic scale begun on F and projected
downward. The second line shows the pentad formed by the
superposition of a second major triad, on G, again with its
complementary scale. The third line shows the second pentad
formed by the superposition of a major triad upon the tone E
with its complementary scale.
286
PROJECTION OF THE TRIAD FORMS
The fourth hne shows the hexad formed by the combination
of the three major triads, on C, on G, and on E, with its
complementary hexad. It will be noted that the complementary
scale has the same relationship in involution— in other words,
the similar projection of three minor triads.
The fifth line shows the projection of the first pentad, line 2,
by taking the order of intervals in the complementary heptad
(second part of line 2) and projecting them upward. Its com-
plementary pentad (second part of line 5) in turn becomes the
involution of the pentad of line 2, having the same order of
half-steps— 2234— but projected downward and therefore repre-
senting the relationship of two minor triads at the perfect fifth.
The sixth line shows the projection of the second heptad
(line 3) by taking the order of half-steps in the com.plementary
heptad in the second part of line 3 and projecting it upward.
Its complementary pentad (second part of line 6) becomes in
turn the involution of the pentad of line 3 and presents, therefore,
the relationship of two minor triads at the interval of the
major third.
Line seven is formed by the projection upward of the order of
half-steps in the complementary scale of the original triad
(second part of line 1). Its complementary triad in turn is the
involution of the original triad of line 1, that is, the minor triad.
Note the consistency of interval analysis as the projection
progresses from the three-tone to the six-tone to the nine-tone
formation: three tone— pmn, six-tone— p^m^n^s^dH; nine-tone—
p'^m'^n's^dH^. In all of them we see the characteristic domination
of the intervals p, m, and n.
In examining the hexad we discover the presence of one
additional relationship, that of two major triads at the con-
comitant interval of the minor third— E-GJf-B and Gt|-B-D. Lines
8 to 12 explore this relationship by transposing it down a major
third so that the basic triad is again C major. Line 8 gives the
major triad C-E-G with its complementary nonad begun on A
287
COMPLEMENTARY SCALES
and projected downward (A being the converting tone of the
connecting hexad of hne 10 ) .
Line 9 gives the pentad formed by the relationship of two
major triads at the interval of the minor third, with its com-
plementary heptad. Line 10 is the transposition of line 4,
beginning the original hexad of line 4 on E and transposing it
down a major third to C, the order of half-steps becoming 313
(1)22; with its accompanying complementary hexad which is al-
so its involution.
Line 11 is the projection of the order of half-steps of the
complementary heptad (second part of line 9) upward. Its
complementary pentad will be seen to be the involution of
line 9, or the relationship of two minor triads at the interval of
the minor third.
Line 12 gives the projection upward of the order of half-steps
of the complementary nonad (second part of line 8), its
complementary sonority being the minor triad D-F-A, which is
the involution of the major triad of line 8. It should be observed
that the nonads of lines 7 and 12 are the same scale, line 12
having the same order of half-steps as line 7, if we begin the
nonad of line 12 on E, a major third above C.
Study the relationships within the pmn projection carefully
and then proceed to the study of the projection of the triad pns
(Example 42-2), the triad pmd (Example 42-3), the triad mnd
(Example 42-4), and the triad nsd (Example 42-5).
Example 42-2
1^ pns Triad Complementary Nonad
?A pns@p^
'7 2 I I I I 2 I 2 2
Pentod(i) p^mn^s^ Complementary Heptad (l)
19^ iT WJJ
rr'fbpbJu
\^A pns| n
"2" 232 222122
Pentod (2)p^mn^s^dt Complementary Heptad (2)
m
-hnfi
rrVr^r'^^
4 2 12
2 112 1
288
PROJECTION OF THE TRIAD FORMS
2 2 2 12
5-/5 Involution of comp.Heptod(l)
i
2 2 2 12
Complementary Pentad(l)
S
~~9 9~V9 I"
6^^==!'
iJilH^T
17b J*
2 2 2 1 2 2 P^m^n^^s^d^t 2 2 3' 2
^1^ Involution of comp.Heptad (2)
i
Complementary Pentad (2)
r^r'r^rirJ ^^
^3
¥
p^i^^/ri p p^m^n^s^d^t^ 4 2 12
7/5 Involution of compNonodp^rn^nV^d^B Complementary Triad
I
^
P
^^
^
^
f
*ff*
itt^ I r I 2122
8-/} pns Triad
7 2
Complementary Nonad
'Tl'f'riJj
7 2 I I 2 I 2 2 I I
9.^ Bn§@s Pentad p^n^s^d Complementary Heptad
i j- ij^
^^
> t?(»
^
2 5 2 2
*■'[;»
2 3 2 2 1 I
10.
.4„2„3r4
pns@s + pt Hexad p^m^n^s^dt Complementary Hexad
"y* Involution of comp.Heptad p^m^'S^d^ Complementary Pentad
'2 32211 2522
'2i« Involution of comp. Nonad p^mVVd^^ Complementary Triad
r 12 I 2 2 I I
I- ^ pmd Triad
Example 42-3
Complementary Nonod
7 4
I I 3 I I 2 I I
289
COMPLEMENTARY SCALES
2>. pmd(a p Pentad (I) p^m^nsd^t Complementary Heptad (I)
m
s^^g
iwr
2 pmd (3 d_
2 4 14
Pentad (2) p^m^nsd^t
2 3 1 13 1
Complementary Heptad (2)
6 13 1
I I 4 I 3 I
Complementary Hexac
^ prnd@p+^ Hexad p3m4n2s2d3t Complementary Hexad i
jj'r'PiJii'i'" 1 1 .III' J
n
2 4 13 1
5/5 Involution of comp. Heptad (I)
m
2 4 13 1
Complementary Pentad id)
^g
miiiury retiiuu ivi /
^_ JttJ jg^ r
♦„•',■'. " ' pSm^n
3s3d4t2
2 3 1 13 1
6-/5 Involution of comp. Heptad (2)
o-ij Involution ot comp. He
I"" I 4 I 3 I
2 4 14
Complementary Pentad ,(2)
complementary Kentod ^\d]
6 13 1 ^
p4m4n^s^d^t^
7/) Involution of comp.Nonad p^m^n^^d^t'
7.^ involution ot comp i\onad p'
i^' I "^ I I 9 I I
6 13 1
Complementary Triad
i
^
r I 3 I I 2 I I
7 4
8,/, pmd Triad
m
Complementary Nonad
^^^^P
t^
7 4
3 11 2 1111
9.pmd@rn Pentad p^m^n^d^ Complementary Heptad
«
jjr '^r^r''^<t^iJ^J J
3 13 4
10^ pmd @ m + pi Hexad p^mVs^d^t
3 I I 2 I 3
Complementary Hexad
"■^wnvu^^^T yjj, Hexad p-in^n'^s'^t Complementary Hexad ji
if'l4i4jjJ-ir-^rt'MJjt"jj"P
*3II24 31124 P^
3 112 4
'75 Involution of comp. Heptad p'^m^n^s^d^t
l^lnvolut
3 1 I 2 4
Comolementary Pentad
^ I ■* A VW
^
I I 2 I 3
*^^ Involution of comp. No[;iad p^m^n^s^d^t^
g^ iiJ J J ^fJ ^<H r i ^1
3 13 4
Complementary Triad
3 I I 2 I I II
7 4
i
290
projection of the triad forms
Example 42-4
••/I mnd Triad Complementary Nonad
i
tif I'i'p JiJ Jg
t>j. M J
2/1 mnd@ n
3 I
Pentad (I j pm^n^sd^t
3 I 2 I I I I I
Complementary Heptad (I)
^^m
j^«Lttir Jit.^
3-^ mnd @ m
3 12 1
Pentad 12) p^m'^n^d^
3 12 1 II
Complementary Heptad (2)
^■f^ mnd(g m Pentad Ui) p m n d Complementary Hep
3 13 1
''y5mnd@n + m Hexad _pVnJs^dft
3 12 1 13
Complementary Hexad
3 I 2 I I
5./ Involution of comp. Heptad II)
o.A Involution of comp. hff.
9^ j^j J;iJ ^I'^ii^
3 I 2 I I
Complementary Pentad (I)
> m \;m Jj^ j|^
p3m4n5s3d4t2
3 I 2 I I I
^A Involution of comp. Heptad (2)
3 12 1
Complementary Pentad (2)
itrl^r J tff^
3
^^
^
?
p^mSn^S'=^d^
3 I 2 I I 3 ^
^•(0 Involution of comp. Npnad p^m^n^s^d^t^
3 13 1
Complementary Triad
I 2 I I I II
°"^ mnd Triad
Complementary Nonad
IQ5 ncmd @ d + ji i Hexad p^m^nVd^t
III I 14
Complementary Hexad
"■/5 Involution of comp.Heptad p^^n^s'^d^t
i
^ I r I 14
ept
I 2 I I 4
Complernentary Pentad ,
1211
291
COMPLEMENTARY SCALES
12.^
Involution of comp. Nonaid p^m^n^s^^t^ Complementary Triad
nsd Triad
Example 42-5
Complementary Nonad
2fl nsd @ d
I 2
Pentad (I) mn^s^d^
I I I I 1112
Complementary Heptad (I)
^A nsd @ n
I I
Pentad (2)pmn3^d^t
7^ risd @ n_ Pentad l^lpmn'-y-d^t Lomplemen
I I I I I I
Complementary Heptad (2)
12 12
^/5 nsd @ ^ + ji Hexad_gnA^|s^d^
II I 12 3
Complementary Hexac
III
5/) Involution of comp.Heptad (I)
i
1112
Complementary Pentad (I)
ri>r JiJ J •QJiJiJJ
iHV^^?V.''pe.3n.s5.s, ^ r:r
B.A Involution of comp.Heptad (2)
I I I I
Complementary Pentad (2)
^
JHH^f .^Vp3.3...,e - r^r^;N -"-"^-^^J
i
Involution of comp Nonad p^m^n^s^d^t^
2 I 2
Complementary Triad
I r I I I I I 2
8y( nsd Triad
Complementary Nonad
^ " nsd @ _s Pentad pmn^s^d^
III I I 2 3 r
Complementary Heptad
1112
'°-^nsd @ s^ dl Hexad pm^n^s^d^t
12 5 1
Complementary Hexad
12 6 1112^
Ji;JbJl,Jl'Ji^jJi|J^W
I I' I 2 6
292
PROJECTION OF THE TRIAD FORMS
"yj Involution of comp.Heptad fAn^n'^s^dSf Complementary Pentad
I 112 5 1
m
12-5 Involution of comp.Nonad p^m VsVf Complementary Triad
J^pi^r '■ ^''^^^ ^
ww^
¥W=*
I I I I I 2 3 I
I 2
Since the triad mst cannot be projected to the hexad by
superposition, the simplest method of forming its nine-tone
counterpart is to consider it as a part of the major-second hexad,
and proceed as in Example 42-6:
I
mst Triad
Example 42-6
Complementary Nonad
da
r r T ^r ^r ^r J li
^^
m
2 4 (6)
112 2
I I (2)
Involution of comp.Nonad pmnsdt Complementary Triad
IT I 9 ? I I I I f?) 2 4
:^
2 4 (6)
The projection of the triad forms of the six basic series—
p^s, sd^, ms^, nH, m^, and pdt— were shown in Chapter 41.
The opening of the author's Elegy in Memory of Serge Kous-
sevitzky illustrates the projection of the minor triad pmn. The
first six notes outline the minor triad at the interval of the major
third, C-Eb-G + Et]-G-B. The addition of D and A in the second
and fourth measures forms the seven-tone scale C-D-E^-Eti-G-
A-B, the projection of the pentad pmn @ p. The later addition
of Ab and F# produces the scale C-D-Eb-E^-FJf-G-Ab-A^-B,
whioh proves to be the projection of the major triad pmn.
(See Ex. 42-1, line 7.)
293
43
The pmn-Tritone Projection with
Its Complementary Sonorities
We may combine the study of the projection of the triad mst
with the study of the pmn-tritone projection, since the triad mst
is the most characteristic triad of this projection. Line 1 in
Example 43-1 gives the pmn-tritone hexad with its complement-
ary hexad. Line 2 gives the triad mst with its complementary
nonad, begun on A and projected downward.
Lines 3 and 4 give the two characteristic tetrads pmnsdt,
with their respective complementary octads. Lines 5 and 6 give
the two characteristic pentads with their complementary heptads,
and line 7 gives the hexad with its complementary involution,
two minor triads at the interval of the tritone.
Line 8 forms the heptad which is the projection of the pentad
in line 5 by the usual process of taking the order of half-steps
of the complementary heptad (second part of line 5) and
projecting that order upward. Its complementary pentad ( second
part of line 8) will be seen to be the involution of the pentad
in line 5.
Line 9 forms the second heptad by taking the complementary
heptad of line 6 and projecting the same order of half-steps
upward. Its complementary pentad becomes the involution of
the pentad in line 6.
Line 10 forms the first eight-tone projection by taking the
first complementary octad ( second part of line 3 ) and projecting
the same order of half-steps upward. Its complementary octad
is the involution of the tetrad of line 3.
Line 11 forms the second eight-tone projection in the same
manner, by taking the complementary octad of line 4 and
294
THE pmn-TRITONE PROJECTION
projecting the same order of half-steps upward. Its complement-
ary tetrad becomes the involution of the tetrad of line 4.
Finally, line 12 is derived from the complementary nonad of
line 2 projected upward, its complementary triad being the
involution of the triad mst of line 2.
'•/I P'^'^ @ ^
Example 43-1
Hexad Complementary Hexad
-^P^
13 2 13
2^ Triad mst
13 2 I 3 or I 3 2 1 3
Complementary Nonad
Lpmpie
^^
^%"^H,^,^^M^
'4 2
3y) Tetrads pmnsdt
i
I I 2 2 I I
Complementary Octads
^
11 \ J id ->
I 3 2 112 r
4 2 I
pmnsdt
^m
^^
^
1^ 3 2
^75 Pentads p^mn^sd^t^
i
^
I I 2 2 r y r
Complementary Heptads
3 2 I
pm^n^'s
2„2.2d^2
I 3 2 I 3 I
^
7-^ Hexad p^m^nVd^t^
^
^^i^
13 2 1 12'
Complementary Hexad
'l^ Hexad pmnsdt Complementary Hexad ^
1^' 3 2 I 3 ' pT
a 4 3 4 3 i4 3
°y) Involution of comp.Heptads p m n s d^t
i
Si
3 2 I 3' " I T f-j^
Complementary Pentads
^3
i
Iff :^ ?
^
^^
*i
*rt
3 2 13 1
3 2 I
p^mVs'^d^tS
r
r 3 2 112
i ¥^;^
4 2
295
COMPLEMENTARY SCALES
10^ Involution of comp.Octads p^m^n^s^d^t^ Complementary Tetrads
pi
^
^^
^
3 2 11
p^m^n^s^d^t^
^
J Ji^ ^"^ r
m
^^
|v I 2 2 ■ I 3 I 13 2
'2« Involution of comp.Nonad p^m^n^s^d^t^ Complementary Triad
I 2 2 I I 2
This projection offers possibilities of great tonal beauty to
composers who are intrigued with the sound of the tritone. It is
clearly allied to the minor-third projection but is actually
saturated with tritones, the minor thirds being, in this case,
incidental to the tritone formation. Notice the consistency of
the projection, particularly the fact that the triad and the nonad,
the two tetrads and the two octads, and the two pentads and the
two heptads keep the same pattern of interval dominance.
The opening of the Sibelius Fourth Symphony— after the first
sixteen measures (discussed in Chapter 45)— shows many aspects
of the pmn-tritone relationship. The twentieth measure contains
a clear juxtaposition of the C major and G^ major triads, and
the climax comes in the twenty-fifth measure in the tetrad
C-E-FJf-G, pmnsdt, which with the addition of C# in measures
twenty-seven and twenty-eight becomes C-Cfli-E-Fif-G, the C
major triad with a tritone added below the root and fifth.
The student will profit from a detailed analysis of this entire
symphony, since it exhibits a fascinating variation between
earlier nineteenth-century melodic-harmonic relationships and
contemporary material.
The opening of the author's Symphony No. 2, Romantic, illus-
trates many aspects of this projection. The opening chord is a D^
major triad with a tritone below the root and third, alternating
with a G major triad with a tritone below its third and fifth. Later
the principal theme employs the complete material of the projec-
tion of the pentad Db-F-G-A^-B, that is, Db-Dt^-F-G-Ab-Al^-B.
296
THE pmn-TRITONE PROJECTION
However, it is not necessary to examine only contemporary
music or music of the late nineteenth century for examples of
exotic scale forms. The strange and beautiful transition from the
scherzo to the finale of the Beethoven Fifth Symphony is a mag-
nificent example of the same projection. Beginning with the
tones h.\) and C, the melody first outlines the configuration
Ab-C-Eb-D-FJj:, a major triad, A^-C-Eb, with tritones above the
root and third— D and Ffl:. It then rapidly expands, by the addi-
tion of G, A, and then E, to the scale Ab-A^-C-D-Eb-E^-F#-G
which is the eight-tone counterpart of A^-C-D-Eb, pmnsdt, a
characteristic tetrad of the pmn-tritone projection.
This projection is essentially melodic rather than harmonic,
but the relationship is as readily apparent as if the tones were
sounded simultaneously.
297
44
Projection of Two Similar Intervals
at a Foreign Interval
with Complementary Sonorities
The next projection to be considered is the projection of those
tetrads which are composed of two similar intervals at the
relationship of a foreign interval. We shall begin with the
examination of the tetrad C-E-G-B, formed of two perfect fifths
at the interval of the major third, or of two major thirds at the
interval of the perfect fifth. ( See Examples 5-15 and 16. )
Line 1, Example 44-1, gives the tetrad p @ m with its com-
plementary octad. Line 2 gives the hexad formed by the
projection of this tetrad at the major third— (p @ m) @ m, with
its complementary hexad. Line 3 forms the eight-tone projection
of the original tetrad by the now familiar process of projecting
upward the order of the complementary octad (second part of
line 1).
Since all of these sonorities are isometric in character, there
are no involutions to be considered.
Example 44-1
1 75 p @m Tetrad p ^ m^ nd Complementary Octad
^m
r ^ri^ JiJ^
4 3 4
2j^p^@rn@m Hexod p^m^n^d^
2 11 3 1 12
Complementary Hexad
H id J Jtf^r M' r^ry J^
3 I 3
3 I 3
298
PROJECTION OF TWO SIMILAR INTERVALS
3yi Involution of comp.Octad p m n s d t Complementary Tetrad
i
^m
i JiiJ J '^' 'W
2 113 1 12
4 3 4
The remaining tetrads are projected in similar manner:
Example 44-2 presents the interval of the minor third at the
relationship of the perfect fifth:
Example 44-2
£ @ £ Tetrad p^mn^s Complementary Octad
i
jj ^T
r M^ i|J l;J J
s=
^
3 4 3
2ji«_n@£@p Hexad p^m^n^s^d
^S
2 12 1 12 1
Complementary Hexad
^D^itJ J
^^
^^
"Tpfs
2 12 2 3 Z I 2
^ f\ Involution of comp.Octad p^mfn^s^d^^ Complementary Tetrad
^2121121 34 3ft^
There follows the major third at the tritone;
Example 44-3
' Vj m @ t Tetrad m s t Complementary Octad
^
r 'r r ^r 'r J ^
■It"" i.Jjt
4 2! 4 2 I I 2 2 I I
2- m@t_@ m or § Hexad m^s^t^ Complementary Hexad
fer^^'^t^^
i^j ; r 'r ^r i^r^N
i J Jtf^*>
2 2222 22222
3y) Involution of compOctod p m n s d'^t Complementary Tetrad
i
2 1 12 2 1 I
^
4 2 ' 4
299
COMPLEMENTARY SCALES
the minor third at the interval of the major third;
Example 44-4
2 2
I. n @ m Tetrad pm n d
4
Complementary Octad
r r ^r 't ^r ^^^
b^ ^8 ib^
3 I 3
2'/5n@m@m Hexad p^m^n^d^
3 I 2 I I I 2
Complementary Hexad
gti°ibJ^jT^r ^ ^ r ^r^r ^li
t®<
3 13 13 3 13 13
3y, Involution of comp.Octad p^m^n^s^d^t^ Complementary Tetrad
^
^^
J^bJ |(J it
?=
^
3 I 2 I I I 2 3 13
the major third at the interval of the minor second;
Example 44-5
2 2
''/5 m @ d Tetrad pm nd Complementary Octad
^
S
^
^
:#^
J&. bis =^^
I 3 I
^- m@d.@iTi Hexad p^m^n^d^
II II I 3 I
Complementary Hexad
J ' r ^r J
^^^
1;.€U ^g^ i^
W-g^
I 3 I 3 I I 3 I 3 I
3.^ Involution of comp.Octad p^m^n^s'^d^t^ Complementary Tetrad
i
ft^p jU
iP ibJ IjJ bJ t|J ^ i
I I I I I 3 I 13 1
the minor third at the interval of the major second;
Example 44-6
'•^Jl @ _s Tetrad pn^s^d Complementary Octad
2 I 2
2 1 II 112
Complementary Hexad
2 I I I 2
300
PROJECTION OF TWO SIMILAR INTERVALS
^yjlnvolution of compOctod p^m n^s^d t Complementary Tetrad
9f f ji jtJ
W.
2 I I ■ I I I 2 2 12
the minor third at the interval of the minor second;
Example 44-7
' /I n @ d Tetrad mn2sd2 Complementary Octad
fl !+♦ ? I I I I I I
1^* 2 I
I I I I I I 3
Q 2 3 4 5
'^ji@d@d^ Hexad pm n s d Complementary Hexad
.ft , , . , I I I I I
^S^ ^&S
[III I
I I I I I
Involution of comp. Octad p m n s d t Complementary Tetrad
1^1 III I
^^
r I III 13
I 2 I
and the perfect fifth at the interval of the minor second.
Example 44-8
p @ d Tetrad p^md^t Complementary Octad
izf^ iJ -"'^ ^ '" r'T r'T^^
I 6 I
1113 113
4 2 2 4 3
l^@jd^@j^ Hexad p m s d t C^mplerr^entary Hexad
14 11 I I 4 I I
Involution of comp.Octod p^m^n^s^d^t^ Complementary Tetrad
iu ^J U itJ ^~^ r : I' r It i
^^1 l" I •» I I -X I C I
«Ce:
I r I 3 113
I 6 I
The reverse relationship of (p @ m) @ p; (n @ p) @. n; ( n @
m) @ n; (m @ d) @ d; (n @ s) @ n; and (n@ d) @ n are not
used as connecting hexads in Examples 44-1, 2, 4, 5, 6, and 7
respectively because they all belong to the family of "twins" or
301
COMPLEMENTARY SCALES
"quartets" discussed in Chapters 27-33, 39. The relationships of
{p @ d) @ p; and (p @ d) @ d; are not used as connecting
hexads for the same reason. The reverse relationship of Example
44-3, (m @ t) @ t, is not used because it reproduces itself en-
harmonically.
In the second movement of the Sibelius Fourth Symphony,
the first nineteen measures are a straightforward presentation of
the perfect-fifth heptad on F, expanded to an eight-tone perfect-
fifth scale by the addition of a B^ in measure twenty. (Compare
the Beethoven example. Chapter 4, Example 15).
Measures twenty-five to twenty-eight present the heptad
counterpart of the pmn @ n pentad. Measures twenty-nine to
thirty-six, however, depart from the more conservative material
of the opening being built on the expansion of the tetrad
C-E-Gb-Bb to its eight-tone counterpart C-D-E-F-Gb-Ab-Bb-Bt^.
(See Example 44-3.)
302
45
Simultaneous Projection of
Intervals with Their
Complementary Sonorities
We come now to the projection of those sonorities formed by
the simultaneous projection of different intervals. As we shall
see, some of these projections result in tetrads which may be
projected to their eight-tone counterparts, whereas others form
pentads which may be projected to their seven- tone counterparts.
In Example 45-1 we begin with the simultaneous projection of
the perfect fifth and the major second. Line 1 gives the projection
of two perfect fifths and two major seconds above C, resulting
in the tetrad C-D-E-G with its complementary octad. Line 2
increases the projection to three perfect fifths and two major
seconds, producing the familiar perfect-fifth pentad, with its
complementary heptad; while line 3 gives the pentad formed
by the projection of two perfect fifths and three major seconds,
with its complementary heptad.
Line 6 gives the heptad formed by projecting upward the
order of the complementary heptad in line 2, with its own
complementary pentad— which will be seen to be the isometric
involution of the pentad of line 2. Line seven, in similar manner,
gives the heptad which is the upward projection of the com-
plementary heptad of line 3. Line 8 becomes the octad projection
of the original tetrad.
Lines 4 and 5 are the hexads which connect the pentads of
lines 3 and 4 with the heptads of lines 6 and 7 respectively.
There is a third connecting hexad, C-D-E-G-A-B, which is not
included because it duplicates the perfect-fifth hexad projection.
303
COMPLEMENTARY SCALES
Example 45-1
^ Tetrod p^mns^
Complementary Octad
2 2 3
2./) -^ - PentadlDp^mn^s^
2 2 2 11 I 2
Complementary Heptad (I)
S
7 J i r'T V'T^r''^^^
-0- -^ ^2232
3./) p2 + s^ Pentad (2) p2m2ns3dt
2 2 2 12 2
Complementary Heptad (2)
^ J J jiJ .1 i r 'T V Y ^r ^ ''^
^
2 2 2 I
4.ij Connecting Hexad (l)p'*m^n^s^dt
^^
2 2 2 11 I
Complementary Hexad (1)
r "r ^r I'f ^f iJ
^^
^
& 2 2 I 2
5/^ Connecting Hexad (2)p2m^s^d^t2
I
(?)p2rT
2 2 2 12
Complementary Hexad (2)
'T It 'y ^r J
J J Jil
2 2 2 11
6.^ Inv.of comp.Heptad (I) p nrr n s d^t
^ J J jj|J J 1^
2 2 2
Complementory Pentad (I)
fpplementqry Pen
^
2 2 2 12 2
^ f^ Inv.of comp.Heptad (2)p^m^n^s^d^t^
'■^ Inv.of comp.Heptad ^2)^1"
2 2 3 2
Complementary Pentad (2)
^r ^r i^F k
^
2 2 2 11 I
8./5 Inv.of comp. Octad p^i^n^sV\^
2 2 2 1
Complementary Tetrad
B.^ inv. ot comp.uctad p i n s d t Lompiementi
^
2 2 2 11 I 2
2 2 3
Example 45-2 gives the projection of the minor second and
the major second which parallels in every respect the projection
just discussed:
Example 45-2
'■/) d^ + A Tetrad mnsd Complementary Octad
I*' I 2
II III 12
304
SIMULTANEOUS PROJECTION OF INTERVALS
^fi ^ + s^ Pentad (i)mn^s^d^ Complementary Heptad (1)
r I I I
^fi d^ + s^ Pentad (2) pm^ns^d^t
II I I I I
Complementary Heptad (2)
* 1*1 I 9 9 I I I I 9 9
F I 2 2
4/« Connecting Hexad U) pm^n^s^d^t
I
I III 22
Complementary Hexad (I)
lenrary
m
|1f I 1^ I o
^
^ — d
r I r I 2
5-^ Connecting Hexad (2) p^m^ns'^d^t^
I I I I 2
Complementary Hexad (2)
5-^ Connecting Hexad t2) p'^m^ns^d'^t'^ Complementary Hexad
12 2 2
6y5 Inv.of comp. Heptad (I) pfnrrTs^d^
2™,3„4^5^6.
II 2 2 2
Complementary Pentad (!)
p bp J [J J
iiJ. V«J
*
I*' I I" I II
7. A Inv.of comp. Heptad (2)p^m^n^s^<l^t^
I I I I
Complementary Pentad (2)
7.1^ Inv.of comp. Heptad t2)£^rTrirV2d_t7 Complementary Pe
r I I I 2 2 112 2
r I 112 2
8.<5 Inv.of comp.Octad pVn^s^d^t^
B.ij I nv. of comp.Octad p m^n^s 0 t^ u
j^ jii Jiij J JjtJfl^ ^ r
Iff I i~ I I IP
Complenrientary Tetrad
br J J =
I I
The third illustration is arranged somewhat differently, as it
concerns a phenomenon which we encounter for the first time.
In referring back to the simultaneous projection of the perfect
fifth and the major second, we shall see that if we combine the
two pentads of Example 45-1, line 2, formed of three perfect
fifths plus two major seconds, and line 3, formed of two perfect
fifths and three major seconds, we produce the hexad of line 4
which is a part of both of the heptads of lines 6 and 7.
Line 1 of Example 45-3 gives the tetrad formed by the
simultaneous projection of two perfect fifths and two minor
seconds, together with its complementary octad. Line 2 gives
the pentad formed by the addition of a third perfect fifth— three
305
COMPLEMENTARY SCALES
perfect fifths and two minor seconds— with its complementary
heptad. Line 5 forms the heptad by projecting upward the
complementary heptad of line 2. Its complementary pentad is
the involution of the pentad of line 2. Line 6 forms the octad
by projecting upward the complementary octad of line 1. The
complementary tetrad of line 6 will be seen to be the involution
of the original tetrad of line 1.
P^d2
i
Example 45-3
Tetrad p^sd^t Complementary Octad
m
r j>f P b|> j^^
m
^
F-^ -^.H , ^
2/1 -^
r I 5 I I 4 I I I 2
Pentad p^mns^d^t Complementary Heptad
5 2
3.^ Connecting Hexad p'^m^n^s^d^t^
^
I 4 ' I ' 2 2
Complementary Hexad
r r »r r ^r iJ
iff I 4
^
l« I 4 I 2
4.^ Connecting Hexad p^m^n^^d^t
i
I I 4 ' I ■ 2
Complementary Hexad
i R~1 o 5
*s^r^
r I 5 2 2
^■* Inv.of comp. Heptad p^nn^ n^s^d^ t^
>*/ Inv.ot comp. Heptad p^m^n^
Iff I 4 I ? ?
I I 5 ' 2 2
Complementary Pentad
S
r T h. i,J
^
6/1 Inv.of comp.Octad p^m'^n'^s^d^t^
I I 5 ' 2
Complementary Tetrad
J j j|j jtfJ ^i
Iff 14 11 12
ipieme
r r T ^
p
Example 45-4 is the same as 45-3, except that the pentad of
line 2 is formed by the addition of a minor second— that is, two
perfect fifths and three minor seconds— with its projected heptad
in line 5, and the two connecting hexads of lines 3 and 4.
306
i/l P +d
*
simultaneous projection of intervals
Example 45-4
Tetrad p^sd^t Complementary Octad
S
^r ^ '-^ ^J J }^
^
|tr - I 5 I I I 2 I
Pentad p^mns^d^t Complementary Heptod
I^T I I 4
^•/^ Connecting Hexad p^m^n^s^d^t
^
I I I 2 I I
Complementary Hexad
p ^^ J i7J ^^
w
I 2 2
i
Connecting Hexad p-^m^n^s^d^t^
I I I 2 2
Complementary Hexad
h- ^ 1'^ J I..
I* I I :^ 1
5^5 Inv. of comp. Heptod p^m^n^s^d^t
4^3„3.,4j5*2
I I I 3 I
Complementary Pentad
bJ i^jti* ^
^^
^
^s^
12 11
6,/j Inv. of comp. Octad p^m'^n^s^d^t^
6,|^ mv.
^
1114
Complementary Tetrad
^^
i J ^J ^ it
*^ I I 2
I I
If we compare Examples 45-3 and 4 with Example 45-1, we
shall observe an interesting difference. If we combine the two
pentads in 45-1 formed by the projection of p^ + 5^ and p^ -f s^,
we form the connecting hexad of line 4, C-D-E-F#-G-A, which
consists of three perfect fifths, C-D-G-A, plus three major
seconds, C-D-E-Ffl:. However, if we combine the pentads of
Examples 45-3 and 45-4, formed by the projection of p^ + d^
and f + d^, we form the hexad C-C#-D-G-A + C-Cft-D-Eb-G,
or C-C#-D-Eb-G-A, which is not a connecting hexad for either
projection.
The reason for this is that the hexad C-CJf-D-Eb-G-A is one of
the isomeric "quartets" discussed in Chapter 39. It is the
curious propertv both of the "twins" and the "quartets" of
hexads, as we have already observed, that their complementary
hexads are not their own involutions as is the case with all other
307
COMPLEMENTARY SCALES
hexad forms. This type of hexad, therefore, does not serve as a
connecting scale between a pentad and its heptad projection.
Example 45-5 gives the pentad formed by the projection of
two perfect fifths upward and two minor seconds downward,
with its projected heptad and connecting hexads:
Example 45-5
\.jt ta^ + d^l Pentad p^m^n^s^d^ Complementary Heptad
2 5 3 1
2y, Connecting Hexad U)p^mVs^d^
i
m
2 4 12 11
Complementary Hexad (I)
^^
i
^
2 4
3 1 2 4 13 1
5/5 Connecting Hexa,d (2) p^m^n^ s^d^ Complementary Hexad(2)
I
^
*
r tJ u J i„
i J ^ ^ T "r
2 5 2 1
"^ij Inv.ot CO mp. Heptad p'^m'^n'^s'^d^t
2 5 2 11
Complementary Pentad
2 4 12
j p^ + d^ t
Example 45-6 gives the projection of two major seconds and
two major thirds from the tetrad to the octad which is its
counterpart, using the whole-tone scale as the connecting hexad :
Example 45-6
' ^ -S + nr Tetrad m s^ Conjipleqfientary Octod
iJJtii J J jM
jrnpiem
r ^r V ^r ^ ^ ^■'
2 2 4
2^s^m^@s Hexad m^s^t^
2 2 2 II 2 I
Complementary Hexad
k
jiJttJ^J i r "ry^^
«iw i J J^
2 2 2 2 2
'i* Inv. of comp. Octad p'^m^n^s^d^t^
2 2 2 2 2
Comp Tetrad ' «^
SIMULTANEOUS PROJECTION OF INTERVALS
Example 45-7 gives the projection of the perfect fifth and
major third:
Example 45-7
[.M P^+iH^ Pentad p^m ns dt Complementary Heptad
m
4 i i J -I ^^ I' T ^r 'T h- ^ ^.i
2 2 3 1
2./J Connecting Hexad (I) p^m^ns^d^t^
2 2 2 I ' I 3
Complementary Hexad (I)
3/5 Connecting Hexad (2) p^m'^n^s^d^t
2 2 2
Conriplementary Hexad (2)
2 2 3
4./( Inv.of comp. Heptad p^m^n^s^d^t^
2 2 3 13
Complementary Pentad
2 2 2 1 13
Example 45-8 gives the projection of the minor second and
major third:
Example 45-8
I. /) d5 + m^ Pentad pm^ns^d^t
Complementary Heptad
(■r J J l>J
^^
*
^
^
5*it
^ *
1^ I 2 4
.2_4„3.2^3
I I
I 2
2A Connecting Hexad lOpfnvjTfsfd^t Complementary Hexad ll)
^
m
^
^
^
I*
^ — *
Iff- I 2 I 3 I I 2 I 3
3/5 Connecting Hexad (2) p^m^ns'^d^t^ Complementary Hexad (2)
I
^^
^
^
^
*
ff*^ I ?
iTT- I 2 2 2 112 2 2
4^ Inv.of comp. Heptad p^m5n^^d^l2 Complementary Pentad
III II 9 u I v^v.riiipidiici iiui jr I CI iiuu MHHHM
?
«^ . s '
112 4
M'
+ m'
Example 45-9 gives the projection of the perfect fifth and
minor third; with the second interval in both its upward and
downward projection:
309
COMPLEMENTARY SCALES
Example 45-9
P^lg plm^nZsdZt Comp.Heptod pf-if*g|"r^°n2S2dt, Comp. Heptad
I 12 2 12
^^'^S&.^fz" Conr,p.He«odsU) ffa'miL&^fe"^ ^ CompHexodsU)
^
r r r'T'-ri'-i
1^ I I 3 I I I I 3 I
3.^(2)p3m4n3s2d2t (2)
r I 4 I 2 I I 4 I 2
(2)p4m2n3s4dt (2)
^
iiJJJ ;i"frir^r^J
^
*fi
2 13 13
2 13 13
2 2 2 12
2 2 2 12
a/^^X^"^ CompPentod 'g'5°n!5'??"s4'gi':i'' C°"'l''^"'°''
II3I3 2131 Iff 12212 2412
5.A Combination of Heptads = Nonod p^m^nQs^d^t^ Comp.Triod n^
i
toto|t» ° "^ ^y <I^JtJ
^r^
lit I
I 2 I 2 I
3 3
Example 45-10 gives the projection of the minor second and
minor third:
Example 45-10
_ -Pentad ^ „ ^ Pentad
' /) d'^ + n^pmn^s^d^t Comp. Heptad td^^ ■>• j^i p^m^n^sd^t Comp. Heptad
^
r I I 3 III
2 I
I I 4' I 2 I
„ . Connecting Hexads r«n«r, u«v«He m
^l^(l) pm2n3s4d4t Comp.Hexads il)
J i^^^iJJ
r I 4 3
Connecting Hexads r«r«r, uovnHcn)
(I) p4m2n2s2d3t2 Comp. Hexads U)
r^pb[>iJ
1112 I I I I 2
3.^l2)p3m2n2s2d4t2 (2)
114 12
D.jt Kci p-'m'- n*- s'Ki^T'^ [.£.)
^1 UjJbV :r^^§
(2)p2m4n3s2d5t (2)
m
^m
|ff 14 3 1
Inv.of Comp.Heptad ^ o * j
p4m4n4s3d4t2 Comp. Pentod
<^\ 13 1 I I I 3 I
Inv.ofComp. Heptad ^ □ * ^
4/1 p3m3n4s4d5t2 Comp. Pentad
I I 4 3 I
r I I I 2 I 1113 I* I 4 I 2 I 114 3
310
SIMULTANEOUS PROJECTION OF INTERVALS
^^5 Combination of Heptads = Nonod p^m^nQs^d^t"* Comp. Triad n^t
Example 45-11 gives the projection of the majoi second and
minor third:
Example 45-11
I./1 s2+I?m^l^s5d2t Comp. Heptad 5.2+ n2£lf^n2s5f Comp. Heptod
2./, ( Wn3s4?4r'' Comp.Hexads(l) ^rfJM^T'' Comp.Hexads (!)
m
^
^
^
I*" I I I 9 II
^5
-0~
'2 2212 22212
(2) p2mVs^2 (2)
I I I I 2
3./) (2)pmVs^d^2 (2)
3|^ v^'i
^
S
^
^
JtJ^^r 'r
s^
2 1124 21124 22231 2223
I nv. of comp. Heptad ^ o. ^ ^ Inv.of comp. Heptad ^ d * ^
4.^ p2m4n4s5c|4tf Comp. Pentad p4m4n4s5d2t2 Comp. Pentad
rff I I I 9 4 ? I I
^
^^
1"^ 11124 2112 222121' 2 2' 23
5 ,5 Combination of Heptads = Nonad p^m^n^s^d^t^ Comp.Triad n2t
^
^
t>o [jo f
CI o
f
:i
13 fO DO
r I
1212
3 3
Example 45-12 gives the projection of the major third and
minor third:
Example 45-12
m^-*-!? pm3n2s2dt Comp. Heptad m2^j^2| pm5n2s2dt Comp. Heptad
i
3 122 3l2'l 12 ■*"jf J "^4 2 2 1 13 2 2 1
jf jt ^ d d \ \ :> iL d. \ \
, Connecting Hexads „ ,. ... Connecting Hexads _ ,, . ,,<
^6 (I)p2m4n3s2d.3t Comp.Hexads (I) (i) p3m4n3s2d2t , Comp.Hexads (!)
i
1^
^fl^-3 o 9 I 13 2 2
JtJnJ^''H
3 I 2 I I 3 I 2 I I
3 (2)p2n1»n2s4dt2 (2)
3 2 2 1 13 2 2 1
(2)pm4n2s4d2t2 (2)
^^ I ? 2 ? 3 12 2 2 "^^ 2 2 I I '
4 2 2 1
311
COMPLEMENTARY SCALES
Inv.of comp.Heptad ^ r^ . ^
4./. p3m5n4s4d3t2 Comp. Pentad
Inv.of connp.Heptad _ „ . ^
p3m5n4s4d3t2 Comp. Pentad
T,o.,9 3122 !*♦ 32211 422
3 12 1
5./I Connbination of Heptads = Nonad p°m°n°s°d°t'
4 2 2 1
Comp. Triad n'^t
^
^
)H*" °l" °*^
o ^^
I I
3 3
It will be noted that in Examples 9, 10, and 11, the minor
third is projected both up and down, since in each case a new
pentad results. It will also be observed that in all of these
examples the combination of the heptads produces a minor-
third nonad. In Example 12, however, only the involution of the
first heptad results since the augmented triad is the same whether
constructed up or down.
Finally, Example 45-13 shows the pentad formed by the
simultaneous projection of two perfect fifths, two major seconds,
and two minor seconds, with its seven-tone projection and
connecting hexads.
Example 45-13
p + ^ -t- ^ Pentod p^mn^s^d^t Complementary Heptad
1^- I 2 3 ' ' 2 I 2 4'
2/) Connecting Hexad (I) p^m^n^s^d^t Complementary Hexod tl)
i
J J ^^r '^ ''^ t-^ >J g
^
f*
12 12 112 3 4
3/} Connecting Hexad (2) p3m2n3s3d3t Complementary Hexad (2)
I
i J J J r :^r ^ '^^ ^^ ^ t'
r I 2 3 4
4y^ Inv.of comp.Heptad p'^m^n'^s'^d'^t^
I I 2 I 2
Complementary Pentad
'*fj Inv.of comp.Heptad p m^^n^s d t*^ Complementory Penta
9'iii J J J ^ r <^r ^ ''^ '\M
I* I 2 I 2 4 112 3
The hexads of Example 45-13 have already been discussed in
Chapter 39, Examples 39-8, 9, 10, and 11. It will be noted again
312
SIMULTANEOUS PROJECTION OF INTERVALS
that the complementary hexad of hexad ( 1 ) is the involution
of hexad (2), and vice-versa.
Note: The projections p^ + s~l and d^ + 5^| are not used since
the former is the involution of p^ + s^ (Ex. 45-1, line 3), and the
latter is the involution of d^-\-s^ (Ex. 45-2, line 3). Projections at
m^ are obviously the same whether projected up or down.
The opening of the first movement of the Sibelius Fourth
Symphony, already referred to, furnishes a fine example of the
projection illustrated in Example 45-1. The first six measures
utilize the major-second pentad C-D-E-F#-G|j:. The seventh to
the eleventh measures add the tones A, G, and B, forming the
scale C-D-E-F#-G-G#-A-B, the projection of the tetrad C-D-E-G.
313
46
Projection by
Involution with
Complementary Sonorities
In chapter 34 we observed how isometric triads and pentads
could be formed by simultaneous projection of intervals above
and below a given axis. From this observation it becomes equally
apparent that an isometric series, such as the projection of the
perfect fifth, can be analyzed as a bidirectional projection as well
as a superposition of intervals.
Example 46-1 illustrates this observation graphically. In order
to make the illustration as clear as possible we have "stretched
out" the circle to make an ellipse, placing C at one extreme and
F|; at the other. Now if we form a triad of perfect fifths by
proceeding one perfect fifth above C and one perfect fifth below
C, its complementary scale will be the nine-tone scale formed by
the projection of the remaining tones above and below FJf at
the other extreme of the ellipse.
Example 46-1
314
PROJECTION BY INVOLUTION
Example 46-2 proceeds to illustrate the principle further by
forming the entire perfect-fifth series above and below the axis
C, the complementary scale in each case being the remaining
tones above and below the axis of Fj|.
Example 46-2
i
ESS3
i
^^
=m:
:=^^
^
*J p2
flu *-
a
*s=
p^s
in bQ
^
p3
£_^
*
p3
S^
^
:g=m
^°=^^
¥
''-^V^ b«
£^
if
*
^ «►
p5
It will be obvious that this principle may also be illustrated
equally well by the projection of the minor-second scale above
and below the starting tone.
The projection of the basic series of the perfect fifth or the
minor second by involution rather than by superposition does
not, of course, add any new tonal material, but merely gives
another explanation of the same material. However, if the
projection is based upon the simultaneous involution of two
different intervals, new and interesting sonorities and scales
315
COMPLEMENTARY SCALES
result. Example 46-3a shows the simultaneous projection by
involution of the intervals of the perfect fifth and the major
third above and below C.
The first line gives the perfect-fifth triad formed of a perfect
fifth above and below C, with its complementary nine-tone scale
arranged in the form of two perfect fifths, two major thirds, two
minor thirds, and two major seconds above and below FJf. The
second line adds the major third above C, with its complementary
octad arranged in a similar manner, and the third line shows a
perfect fifth above and below C, with a major third below C—
the two tetrads being, of course, involutions of each other.
The fourth line gives the pentad formed of two perfect fifths
and two major thirds above and below C, with its complementary
heptad. Line 7 forms the projection of line 4 by the usual process
of projecting upward the order of the complementary heptad of
line 4, the tones of this scale being arranged as two perfect
fifths, two major thirds, and two minor thirds above and below
C. The right half of line 7 presents its complementary pentad
arranged as two perfect fifths and two major thirds above and
below F#. Lines 5 and 6 give the connecting hexads between
lines 4 and 7. Lines 8 and 9 form the octad projection by pro-
jecting upward the order of the complementary octads of lines 2
and 3, their complementary tetrads being the involutions of the
original tetrads of lines 2 and 3. Line 10 forms the nonad which
is the prototype of the original triad by projecting upward the
complementary nonad of line 1. The complementary triad of this
nonad is, of course, the same formation as the original triad of
line 1.
Example 46-3a
316
PROJECTION BY INVOLUTION
2.^ } p"" m'tTetrods p'^mnsd Complementary Octads } p^ m^ n^ s' T
m
^^^^f^
4 I 2
3 I I 2 I I I
3.^ J p^ m I
o
i
2
p mnsd
t p2m2 f^s[i
I J friJ
^^^^
jjji'^ [r
5 2 1 4 12
2 11 12 1 I
i
\ P^ n? Pentad p^m^n^sd^
Complementary Heptad | p2 m2 ri2
4 12 1
3 I I 2 I I
,3„4„3„2^3
5y5 Connecting Hexods p m n s d Complementary Hexads
>JtJJ>^T-'
j,Lompiementary
*r riir r nJiii,
12 11 (3)
3 II 2 I (4)
p^m^n^s^d^
jjjMJf
^w^p
^"4 I 2 I I (3) 3 I I 2' I (4)
Iny.of Comp.Heptad
mv.oTUDmp.i-iepToa o o o o •
7.^ p^m^n^s^d^t J^^n? It Complementary Pentad J^*^ m
i
J^.lJJ|J>J i:i."ii'" iTf^rr^^ 1^
m
3 1 12 11
Inv.of Comp.Octads
4 12 1
8-/, P^m^
Lomp.uct
nSs^d^t'
,2_2„2„l
2 I
t P il» J] 1 4 Complementary Tetrads \ p mi
j,.^.i J .iiJUp na;re.:Vr«rr W.
3 I I 2 I II
4 I 2
2m2 n2 eh
9/1 p6m5n5s5d5t2 J p'^m'^n^s't
J p2 m' j
10.
2 11 12 1 I
Inv.of Comp.Nonad
5 2 1 4 12
'fi p8m6n6s7(j6|3 ^ p mr np sf Complementary Triad J p2
2 11 12 1 II
5 2
317
COMPLEMENTARY SCALES
Example 46-3b forms the projection of the same two intervals
of the previous example in reverse, that is, two major thirds plus
the perfect fifth rather than two perfect fifths plus the major
third. The pentad, heptad and connecting hexads are, of course,
the same, but the tetrads and octads are diflFerent.
i
Jm^ Triad m'^
Example 46-3b
Complementary Nonad ^^ m^ n^ d^
S
m
'r<r*r>ftirriiJiiJJ,|i^MH
^
1 2 I I 2 I I 2 XJ- « '
Complementory Octads f p2 r7i2 n2 (jl ^
4 4
2.tm £T Tetrads pm^nd
1.1! J jJ''^
"rV'Trii^^^ ii"M»:g
I 2 I I 2 I I "^ ~j|-C5f«-
tp2 m2 n2 d' t
3. . Jm p I
4 3 I
^"' ' pm^nd
^m
^^^^^s
s
^
3 I I 2 I I 2 *^
Complementary Heptad ^ 2 ^^2 ^2
4 13 4 3 1
4 tm^ ^ Pentad p^m^n^sd^
|l^» 1 jjjJl^
V^r^iirritJiiJ JbMJ
^^'T^
4 12 1
5.^ Connecting Hexads p m n s d
3 112 1 I
Complementary Hexads
m
^fTr^
^
3 112 1
6./J p^m'^n^^d^
4 I 2 I I
P^
^ruf^r^r r"^
f
4 I 2 I I
Inv.of Comp. Heptad ? ? ?
Z^g^m^n'^s^d^t t£ m'^IL
3 I I 2 I
Complementary Pentad ^^2 ^2
tes
rrr^^ ^fy
1
3 1 12 11
4 12 1
318
PROJECTION BY INVOLUTION
Inv.of Comp.Octads o o o , /- , * t» ^
8.^ p5m7n5s4d5t2 J p^m^r^ d't Complementary Tetrad .2 1^
I 2 I I 2 I I 4 3 1
9^ £^rn[n^s^d£j2 $ ^^Z^ d'*
tm2 pi t
3 I I 2 I I 2
Inv.of Comp.Nonod
4 13 4 3 1
p6m9n6s6(j6t3 {p^rr^ji^d^ Complementory Triad ^ ^z
12 1 I 2 I I 2
Example 46-4a continues the same process for the relationship
of the perfect fifth and the minor third;
I ^ ^£ Triad p^s
Example 46-4a
Complementary Nonad J p^ m^ n^ s^
5 2
2 ^ le n.^ Tetrads p^mns^
i
,2 „2 ^2„l
±1
Complementary Octads Jp rrr £^ n t
"rri'^'i'^Ji^iiJ J J |iiM»:;iiB'f'
iS
3 2 2
2 I I I 2 2 2
3.fl^^n'i
I
)^mns^
J p2 rr? $2 n' i
mm
"r r^^«^iJ
^B
w
^^
&
8^
O •"'^gV3
522 322 21222 r I
4.^ t £ H^ Pentad p^m^/? s^ r Complermentary Heptad J £^ U? s^
i
.X3L
Is:
"rrWftjjj it^'.'M^-fe
^
2 I 2222
3 2 2 2
5-^ Connecting Hexods p m^n^'^dt Complementary Hexads
2 12 2 2
p^m'^nVdt^
2 12 2 2
319
COMPLEMENTARY SCALES
Inv. of Com p. Heptad
7- p^m^nVd^t^
}p2m2_s2 Complementary Pentad 4^^^ J2^
2 1 2 2 2 2
Inv. of Comp.Octads
8.^ p6m5n5s6d*t2
3 2 2 2
Jp^m^s^nU Complementary Tetrads ^ p^ n' I
2 1 I 12 2 2
a, p^m^n^s^dV
3 2 2
i
t p^m^s^ n' t
jj^jjj^^^r lbg"^°'"
ii|iAlJlf
t p2 n't
2 I 2 2 2 I I
Inv. of Comp.Nonad
10 pSpn^nSs^dSfS
5 2 2 3 2 2
t p nn s nr Complementary Triad t_2
2 I I I 2 2 I I
5 2
and Example 46-4?? gives the reverse relationship— the minor
third plus the perfect fifth:
i
$ n^ Triad n^t
Example 46-4Z7
Complementary Nonad t p^m^s^d^
S
§^
rrii^A^iiJiiJJj i"'' Jii] tfB)|t
s
^
S:
36 33 2I2IIII2
2 y n^ p't Tetrads pmn^t Compiementory Octads j p^m^s^d' ;
i
f'''"'>j^^[^rVn-^'rrit^^WJJ t^'::itij^8 ^
342 334 2122
3.. t n^ p' * pmn^st
I
I p2 m^s^d' t
i
2 12 112 2 "^^
S
m
i
324 334 212
Jn^F? Pentad p^m^ n^s^t Complementary Heptad J p^nn^s^
4./(*- -ti
-ft
""l>jjJ^ :*rrii^»^t^Jj i*"h*
3 2 2 2
2 1 2 2 2 2
320
PROJECTION BY INVOLUTION
4 2 3 4
5^ Connecting Hexads pmrrsdt Complennentary Hexads
rr'iVii^J
^s
i
21222 21222
§ i^jJ^'^r ^^^nMi^j
3 2 2 2 2
3 2 2 2 2
Inv.of Comp. Heptad
7^ p'^m^n'^s^d^t^ ^ p^m^s^ Complementary Pentad ^n^ p2
i
ijjLi N^f 1.811 'fWiU ii^
2 12 2 2 2
Iny.of Comp.Octads
8.>, pSmSn^sSd^fS
i
3 2 2 2
Jp^m^s^d't Complementary Tetrads ^ n^ p' i
J ji,j J J iJi|J f |,g»iM, mrtfjjij J [riipiJ||j] \\m n
212. 2 II 2' ~ 342 334
9.- pSm^nSsSd^tS * «2^2c2
#
i p HD rl^
i-
jj^jJtJJJr l^gn^i^'rii^tJJ[ii^"rrT]rtig
i n2 pi t
-jt»
2 12 1 I 2 2
Inv.of Comp.Nonad
'0/, p6m6n8s6d6t^
3 2 4 3 3 4
$ p2m2s 2^2 Complementary Triad ^ ^2
2 I 2
3 6 3 3
Example 46-5a gives the vertical projection of the perfect fifth
and the minor second, and Example 46-5b the reverse relation-
ship:
Example 46-5a
I ^ ^ P. Tried p^s Complementary Nonod tm^n^ s^ p^
_ci _ _ _ j_
2.^$£ d^ t Tetrads p ^ msdt
2„2 „2 „i
Complementary Octads t HI J] 1 P ♦
COMPLEMENTARY SCALES
4 J£^d2 Pentad p^m^s^d^t^ Complementary Heptad ^ ^ ^^ g^
i
&
^
3
'i¥ JjiJiiJ J ~ Pui'l.'«B
TiiKt
^
^
14 2 4
I 2 2 I I
5.- Connecting Hexads p^m'^ns'^d^t^ Complementary Hexads
f
i
2 2 2 4
p4m2s2d^t3
12 2 2
j> iJ ^llJ ^.^
^
^
I 4 I I 4
Inv.of Comp.Heptod
I 4
7/1 p'^m'^
uomp.i-iep
nSs^d'^t^
(. M |j III II a u I
*f^2 f,2 ^ Complementary Pentad Ap2 ^2
^
S
P
l>J-|g.W
2 2 114
Inv.of Comp.Octods
8. . pSm^n^s^d^t^
4 2
^JT^il^ l^_p'^
Complementary Tetrads ^-2 ^jl i
I 2 2 I I 13
Inv.of Comp. Nonad
'<^/) p8|n6n6s7(j6t3 j ^i^ n2 52 p2 Complementary Triads j p2
nJi^Jt^r'r
^^
rT,^&,^^^u- V'
^^m
2211 121
5 2
J d^ Triad sd^
Example 46-5Z?
Complementary Nonad | ^2 ^2 ^Z ^2
10 I I
2.^ t d^ p'tTetrads pmsd^t
Complementary Octads Im^n^ s^ d' 4
^
=©«
f^i'^l^liViiJiiJ <''UPsi
^s^
6 4
i
J d^ pi I pmsd^t
2 I I I I
I
3./1 '.ii ±
A_2 „2 ^2 jK
|m n s d t
14 6 16 4 I I I 2 r I »*
i^CE
322
PROJECTION BY INVOLUTION
4. I d p Pentad p^m s d t Complementary Heptad J m n s
^
W J|ja_[ ir^^
^
14 2 4 I 2 2 l"
5 - Connecting Hexads p^m^ns d^l^ Complementary Hexods
r i ¥i
ffm
5^?^
2 2 2 4
p4m2s2d^t3
2 2 2
m
s^
^
^
I 4 I I 4
14 1 I
Inv.of Comp. Heptad
7 p'^m'^n^s'^d^t^
$ m^ n^ s2 Complementary Pentad ^ (j2 p2
Inv of j;^omD,Octads
8.^ p5m5n4s5do t3
2 _2 .2 J
2 J,
J m n s d t Complementary Tetrads t d p *
I 2 I I I ! 4 -^^'•^ I g ^
12 1 II ! 4
9. p5m5n4s5d6t3
*
t m2 n^s2 d'l
i^tJjjl^vr^^
J d2 p't
i^j^j >.J ^ ^^^^^rl^^^^^^o^^ ^«^ J J i^X^
Sfeo=
I I I 2 I I 4 ' ' 14 6'
Inv.of Comp. Nonad
10 'p'6;;'6';;6's?d'8t3'"" ^ m^n^s^ d^ Complementary Triad ^ ^2
I I I I I I 14
Example 46-6a presents the relationship of the major third and
minor third, and 46-61? presents the reverse relationship:
Example 46-6a
' ., ^ m2 Triad m^
Complementary Nonod t £^m2d2n2
I 2 I I 2 I I
323
COMPLEMENTARY SCALES
2^,jm^n't Tetrads pnrrnd Complementary Octads { p^m^d^n't
i
^
^^rA
i
^
p^^^^^^
3 I 4
^- Im^nU pm^nd
I 2 I I 2 I I
* p2 m^d^ nU
i,un. J jiJiiJ [r JiJj] i r^r r^^ ^^^^JJ jt*>j
7T
1^^
4 4 1 3 14 I 2 I 3 I
^A^^[^ Pentad p^m^n^d^t Complementary Heptad t p^ rr\^ ^
i4;^j^jiji|j " ;r"rr"^^it^tJ it''""i;jt
3 14 1 I 2 I 3 I I
^ ..Connecting Hexads p^m^n^sd^t Complementary Hexads
jju^j"^'!^ 'U^rhh
I 2 I 4 I
3 I 3 I I
p^m^n^sd^t
j^jjiJiiJ :r«rr»^iiJ»
^
3 13 11
7 Inv. of Comp. Heptad
i, p^m^n^s^d'^t^
12 14
J ETH! ^ Complementary Pentad j rn^ n2
12 1 3 11
Inv. of Comp. Octads
8-p5rT^7n5s4a5t2
3 14 1
I
J p 2 m^cj 2 nU Complementary Tetrads j m^ n' 4
*^r 2 I I 2 I I 3' I 4
2 I I 2 I I
9u£5m^nV^d^
J p2m2d2n't
J m2 n'T
jjjb^'r i[3?:^^s'-":r^ri^^'i^[j^^
te
^w
2 I 3 I I 2
Inv. of Comp.Nonod
'0>, p6m9n6s6d6t5
4 4 1 3 14
m
J 2 ^2^,2^2 Complementary Triad j ni2
jjiJiiJj^JY^rfei-::'^^
^
2 1 12 1 12
4 4
324
projection by involution
Example 46-6??
'• I n^ Triad n^t Complementary Nonad J p2 m^ d^ s^
i
m
^
rffrr'^«^>JiiJtJ^j]i"nt^tBtfe
m
3 6
I 2 I 2 I II
2. ♦ n2 ^1
2 ^2^2 J,
I
♦ n'^m't Tetrads pmn dt Complementary Octads J P. HI ^^ 2 ^
feEdfi
g
r^rrii^it^
^P
iiJiiJ)jj fgi'^^m
o-
3 I 5
^> J n^m'^ pmn^dt
I 2 I 2 I I I
#
J p2 m^d^s' ,1
ti I,. J J iJ ^Jlr^J^j] i TT r it^ J itJiJ^j #f:ttu)iB fc,
351 315 1213111
^- $ n^ m^ Pentad p^m^n^d^t Complementary Heptad ^ p2 ^2 jj2
(|,''iii'); ji,j^jiJiiJ ir"rr«^^iiJ«J if^'tiB
2 I 3
3 4 3 3
5. Connecting Hexods p m n sd t Complementary Hexads
i
t^ii^ i^r"^^
J|;JbJfcjJ
i
2 1 4
p^m^n^sd^t
3 I 3
j> jbJuJ ''''^ii^ '■ r'rr'i'^
3 I 3 I I
Inv.of Comp. Heptad
7- p^mSn^s^d^t^
12 14 r
2 m2
Jl p III 11 O U I
$ p2 m^ d^ Complementary Pentad t n^ m
"I?
e
^
r r f |J ii'
^
§=
2 I 3 I I
Inv.of Comp. Octads
®- - p^mSnSs^dSt 3
3 I 4
,2 _2 m2
2 „l
J pfm d. s* Complementary Tetrads t n*- m' ^
jj^jjbJjiJ^iir (ui.!:teu: r^^r J ^ "^
I 2 I 2 I I I
9- pSmSn^s^d^tS
3 1 5
* P^m^d^ s't
jt *" " ~ L * ii IL' il 2. ' ♦ —
^n2 m' t
I 2 I 3 I
3 5 1 3 15
325
COMPLEMENTARY SCALES
Inv. of Comp. Nonad
'0- p^m^nSs^d^t^ t £2 ^2 ^2 s2 Complementory Triad j n2
-A p-m'"n'"s^a"'T^ j £^ nn- a;- s;-
£
^
12 12 1111
3 6
Examples 46-7a and 46-7Z? show the vertical projection of the
maior third and minor second:
'■ ^ m^ Triad m^
Example 46-7a
Complementary Nonad j m^ ^ ^ p2
4 4
12 1 I 2' I I 2
2. ♦ ^2^1
Jm dt Tetrads pm^nd Complementary Octads % ^ ^ ^ ^^
I 3 4
3- J m^d'l pm^nd
I 2 I 3' I I 2
% m^ rf d^ p' t
^- \ m^d^ Pentad p^nrrrTsd^ Complementary Heptad ^ ^2 ^2 5^2
f ''^' >J J"^
fet
t^fJiy.! ||lluil»i|.
f
13 4 3
I 2 I 4 I 2
5- Connecting Hexads p^m'^n^^d^ Complementary Hexads
'I jjg.ji^^r =^^
12 14 3
13 4 12
p'm^n^s^d^
13 4 12
inv. of Comp. Heptad
12 14 3
"^i. p^m^n^^s^d^t Jm^n^d^ Complementary Pentad | ^ ^
I 2 I 4 I 2
326
13 4 3
PROJECTION BY INVOLUTION
8.
Inv.of Comp.Octads
p5m2nV*d5t2
tm^n^d^p't Complementary Tetrads ^^Z^jl^
j^jjiJiiJp i,iili;;^B"i«r"rrtiJ
^p
^^
12 1 3 112
9- p5m7n5s*d5t2
I 3 4
#
Jm^ n^ d^^'l
j^jtjJiJiiJf i.T#aj*rrtJ_J[||jJ^
J m^ d' t
iS
^
I 2 I I 3 I 2
Inv.of Comp.Nonad
,- inv.oT Lomp.Non
'°^ p6m9n6s^d6t3
4 4 5 13 4
jm^n^d^p^ Complementary Triad .^2
I 2 I I 2 I I 2
4 4
Example 46-7b
I ^ I ^^ Triad sd^ Complementary Nonad | m^ n^ d^ s^
i
^^^^^m
mP€»-
iJ r ir r
I 10 II
I I I 4 I I I
2 t d^ m't Tetrads pmnsd^ Complementary Octods f ^2 ^2 ^z gi |
i
^
rr]:^r*rtVitJiiJ«JtJ iite^Btt'
=o=
»
137 113 I2I4II
^- J d^ m'l pmnsd^
I
""^°'" j j''^ r ^ r M ' ^^'^^^ ^^^^ ^^^
t n? ^ d^ s' *
^
173 113 IIII4I2
4.^ df jT^ Pentad p^^n^sd^ Complementary Heptad j m^ n^ d^
13 4 3
I 2 I 4 I 2
Connecting Hexods p m^n^s^d Complementary Hexods
g
^
Sg^
iiJ jgi'^ r
12 14 3
p^m^n^s^d^
I 3 4
iJ^J^ i*r"r^^r^
J. I; J ^
3 4 12
12 14 3
327
COMPLEMENTARY SCALES
Inv.of Comp. Heptad
{ m2 jl^ d^2 Complementary Pentad | d^ m^
^ 1^ 111 II a u I J rnc nc qc uuiii^iBiiieinui y reiiiuu t Q'- f
2 14 12
Inv.of Comp.Octads
8- p5TT,5n5s5d6t2
3 4 3
tn?Q? d^l'* Complementary Tetrads ^ ^jZ ^plj
^^^^^^^^^
|0«-='» V5
2 14 1 I
9. p5nn5n5s5d6t2
13 7 113
*
im^ n^ d^s't
t d^ m't
jM^i^A^^r i'Hil;;^e^^;tf|Mlrjj|hM„j]^^
7*^
I II 14 12
7 3^11 3
,» Inv.of Comp Nonod r«^rvi«™«^r,+„..„ t^:«^
'°. p6m6n6s7d8t3 | rn2 n2 d2 s^ Complementary Triad ^ ^2
Ji>Jl|J^JljJ
I I I I 4 I I I
I 10
Examples 46-8a and 46-8Z? give the vertical projection of the
minor third and the minor second:
I. ^ n_2 Triad n^t
Example 46-Sa
Complementary Nonad ^m^s^ d^ p^
S
^^^^i^p
m
3 6
^- $ n^d't Tetrads mn^sdt
fc^
s
s
Complementary Octads t m^ s^ d^ p' I
^"r'Trnt^nJ
^^
«=
I 2 6
3- tn^d'l mn^sdt
II 12 3 1 2
t m2 s2 d2 p' t
f'""aijJrfri-^ijJ-rViXiiiriiV iijiu^b,,^'!:
3 6 2 12 6
^ t n2 d^ Pentad m^n^s^d^t
II 12 13 2
i
^^
Complementary Heptad J m2 s2 d^
^
^
r^"rVrii^<J ii^p
I 2 6 2
I 112 4 2
328
PROJECTION BY INVOLUTION
^Connecting Hexods pm^n'sV't Complementary Hexads
^F^
i^
5W.
I 116 2
pm4n2s4d2t2
I I I 6 2
6.
jj J J ^ r - ^^1'^^
I 2 2 4 2
Inv. of Comp.Heptad
7v,p2m^n^s5d4t2
m
^^
12 2 4 2
$ m^ s^ d^ Complementary Pentadj n^ d^
^^g
^^
P
fc^
tj,i>.^^bt!a
I I I 2 4 2
g Inv. of Comp. Octads
p4^m5ri6s5d5t 3
12 6 2
^^^^^^
4 2 2 j2 U Complementary Tetrads o i
^
^
1^
I I 12 3 12
9> p^mSnSsSdSfS
I 2 6
^Hi^^H/.^^rJ ^u^^
^ m^/ d^p'4
A 2 .k
Jn d t
^[J fJiJ^r^^fr^
^^_: I I ^p|.^[il.,[^fq_pg.
II 12 13 2
nv.of Comp.Nonad
'°>)p6m6nSs6d6t4
bo 3 6 2 12 6
A f^2 g2 jj2 p2 Complementary Tried j ^,2
I I I 2 I 2 I 2
Example 46-8Z?
1.^ t d*^ Triad sd
Complementary Nonod J m2 s^ d2 n2
I 10
I I I I ' I 4 I I
2 J d2n't Tetrads mns^d^ Complementary Octads $ H)^!^ ^2 n't
82 128 IIIII42
329
COMPLEMENTARY SCALES
-|2n2 PontnH m2n2c3H2f
4.A ^ ^ - Pentad m^n^s^d'^t Complementary Heptod J m^ s^ d
2 c2 h2
i
^pia
^
'r'lrVriiJtJ
^ci
^^
ii^jt'^ r
©=
2 6 2 I I I 2 4 2
5.^ Connecting Hexads pm^n^s'^d^t Complementary Hexads
^i'^ : r^r^r'Ttpg
^UbbJb
116 2 I I I 6 2
pm''n2s^d2t2
r^rrr^riJ
^^
^
2 2 4 2
12 2 4 2
7y) p2m4n4'^d'^t^'^ i ni^ £■ ^ Complementary Pentad 2 2
/I . I,. I. I IlL^ bfc>9l;'l#H :r^rrh.l.J .. "i..^
i
lylJ ii..o^S
^M^ J"r
?3^
112 4 2
Inv. of Comp. Octads
Q-* p^^m^n^s^d^t^
12 6 2
^^
* 1^2 ^ d^ nU Complementary Tetrads g 1
tfe
«fR^
4
^
^Jl^pJ !>•
■ D'/*.^
I I 2 4 I I
9- p4m5n5s6d6t2
I 2 8
t m 2 ^2 d 2 £ ' t
s
bo : p ^
td'^ n't
^
i^ji^j^j^j'iJ^^r
=&^
1 1 1 4 2
I 8 2
K). p^r|,|^3°' J m^ s2 d2_n2 Com^plementary Triad ^ ^^
111 I I 4 I I
I 10
The vertical projection of the perfect fifth and major second
duphcates the perfect-fifth series; the combination of the major
second and the major third duphcates the major-second series;
and the vertical projection of the minor second and major second
duplicates the minor-second series.
The vertical projection of the minor third and major second
results in a curious phenomenon which will be discussed in the
following chapter.
330
47
The 'Maverick' Sonority
The vertical projection of the minor third and major second
forms a sonority which can be described only as a "maverick,"
because it is the only sonority in all of the tonal material of the
twelve-tone scale which is not itself a part of its own com-
plementary scale. It is, instead, a part of the "twin" of its own
complementary scale. Because of its unique formation, we should
examine it carefully.
In Example 47-1, line 1 gives the tone C with the minor third
and major second above and below it. The second half of line 1
forms the descending complementary scale, beginning on G# and
containing the remaining seven tones which are not a part of
the original pentad, arranged both as a melodic scale and as
two perfect fifths, two major seconds, and two minor seconds-
one above and one below the tone F#.
In line la we follow the usual process of projecting upward
from C the order of the complementary heptad, producing the
scale CiC#iDiEbiEI::|3G2A— also arranged as two perfect fifths,
two major seconds, and two minor seconds, one above and one
below the tone D. We find, however, that the original pentad of
line 1 is not a part of its corresponding heptad (line la). There
can therefore be no connecting hexads.
Line 2 gives the tetrad CsDiE^eA with its complementary
octad, while line 2a forms the octad projection. Lines 3 and 3a
give the tetrad CgEbeAiBb with its octad projection. Lines 4 and
4a form the projected octad of the tetrad CsDiE^^Bb, and lines
331
COMPLEMENTARY SCALES
5 and 5a form the projected octad of the tetrad CoDjAiBb.
The tetrads in Hnes 2 and 3 will be seen to be involutions, one
of the other. In the same way, the tetrads of lines 4 and 5 form
involutions of each other.
Example 47-1
Pentad
^ n5 i p2nir,2s2d2| Complementary Heptad j p2 _s2 d2
rirB^j^jj^r^V'MriirViirr .ii-:^,;^^;
2 16 1 I I I I 3 2
la. Inv. of Comp. Heptad p'^m^n^s^d'^t^
$ p2 s2 d2
^^^
If
^f^
I I I 3 2
.9 1. Tetrad
2/5 *,-^^ pn^sdt
Complementary Octad
I
$ p2 $2 d^ m' t
S
"rV^nir'ntrriiJiiiiifj^
t^
"^2"^ re I I I I 3 2 ■ I
2o. Inv.of Comp.Octod p^m'^n^s^d^t^
#
^
$ p2 j2 ^2 ppl ^
iJbJ^J ^^T
«jt^-8-h
^*^F7^
XT
13 2 1
3 ln2s'| Tetrad Complementary Octad
— ~ pn'^sdt
t p2 s2 d2 mU
^C ~ P" sdt * h i* ii ♦ il. ^ ii
(f "lib, ^j jbp i^rT^fi^r'rriirr ii"'^
Ty
3 6 1 I I I I 2 I 2
3o^ Inv.of Comp. Octad p^m'^n^s^d^t^
t p2 s2 d2 m' t
.^JbJ^JtfJ ^ ■'
^^
I I I I 2 I 2
4.^ t $2 n't Tetrad-^
■* - j— pmnsHJ
$ p2 ^ d 2 £> t
i
Complementary Octad
^^
bri^rV«r'ir'iriirrJ/::«e>ftB""
2 17 I I I I 3 2 2
^5,- Inv. of Comp Octad p5m5n5s6d5t2
t p2 s2 d2 nU
jftjj^j^J ^^ r
II 3 2 2
-nr
^^■8-
332
THE MAVERICK SONORITY
c » ? I . Tetrad
5- t _s^ jiU pmns2d
i
Complementary Octad
$ _p2 s2 d_2 n' j
111112 2"^ ^ ^
bo-^'^Q
^^
2 7 I
or
d^ n't
5g Inv. of Comp. Octad p^m^n^s^d^t^
tP'
Example 47-2 shows the relationship of the pentad of the
previous illustration to its tw^in, the pentad C-Cfl:-D-E-G, which
has the same intervallic analysis, p^mnh^dH. The first line gives
the two pentads, each with its complementary heptad. Line 4
gives the involution of the two complementary heptads but
with the order interchanged, the first heptad of line 4 being the
involution of the second complementary heptad of line 1, and
vice versa. The "maverick" pentad C-D-E-F-B will be seen to be
a part of the complementary scale of its "twin"— second part of
line 4. The first pentad, C-CJ-D-E-G, will be seen to be a part
both of its owTi related heptad and the related heptad of its
maverick twin.
The connecting hexads also show an interesting relationship,
the first connecting hexad of line 2 being the "twin" of the
second connecting hexad of line 2; and the first connecting hexad
of line 3 being the twin of the second connecting hexad of line 3.
Pentad
I- A p2mn?s2d^
Example 47-2
Comp. Heptad (I) pTmr^s2d2t In^f Comp. Heptad (2)
ujjJ :^rMJjjjijjJr'^"'ih-^''^Mi
Iff I 23 II2I2 42 2I6 IIII32
I 2 3
2./) P^m^n3s3d4i
I
Hexad
i2r
I 12 12
Comp. Hexad
2 2 16
Hexad twin
p2m2n3s3d4t
I I I I 3
Comp. Hexad
l~ I i' I 1 I I O I c
^m
iTI I <j I c I
r I I 13
Hexad
3./^ p'^m2n3s3d2t
I I 2 I 6
Comp. Hexad
^
i^4JJj
2 I 6
Hexad twin
p4m2n3s3d2t
I I I I 3
Comp. Hexad
Ifl 12 3 2
2 2 12
w
jJJJJf iV-^^
2 2 12 4
12 3 2
333
COMPLEMENTARY SCALES
X^^ST""^ Comp Pentad 'S°'3n^%^r'""c°"X>.ftntod
I* III 32 2216 1*12124 1123
334
48
Vertical Projection by Involution
and Complementary Relationship
There is a type of relationship which occurs when intervals are
projected by involution, as described in the previous two
chapters, which explains the formation of the hexad "quartets"
described in Chapter 39. If we compare in Example 48-1 the
projection of two perfect fifths and two major thirds, one below
and one above the tone C, together with its complementary
heptad, with a similar projection of perfect fifths and minor
thirds, together with its complementary heptad, we shall notice
a very interesting difference.
Example 48-1
Complementary Heptad . _ Complementary Heptad
J^ m2 p2 n2 m2 ^ V^ It p2 $2 m2
The complementary heptad of
f G E
C
iF Ab
that is, a perfect fifth and major third above and below C, is
335
COMPLEMENTARY SCALES
TCt A A#
n
which forms a perfect fifth, a minor third, and a major third
above and below FJf. The complementary heptad of
TG Eb .
C ,
iF A
a perfect fifth and a minor third above and below C, is
jB E D
which forms a perfect fifth, major second, and major third above
and below F#.
In other words, the projection of ij^p^m^ is X'p'^'m^n^, whereas
the projection of Xp^n^ is X'p'^m-s^. In the first pentad, the vertical
projection of p and m is a part both of its own complementary
heptad and of the complementary heptad of the vertical projec-
tion of p and n. In the case of the second pentad, however, the
vertical projection of p and n is not a part of the vertical
projection of its own complementary heptad, hut is a part of the
vertical projection of the complementary heptad of the pentad
Xp^rrt^, that is, Xp^m^n^.
This phenomenon makes possible a fascinating "diagonal"
relation between pentads and heptads formed by vertical projec-
tion, resulting in quartets of connecting hexads all of the
members of which have the same intervallic analysis. In each
case the "quartet" consists of two hexads having differing
formations but with the same intervallic analysis, each with its
own involution. ( See Chapter 39. )
If the student will re-examine the material contained in
Chapter 46, he will observe that the same phenomenon which
336
VERTICAL PROJECTION
we have just observed in the vertical projection of the projection
p^n^ also occurs in the vertical projections of p^cP, mrn^, mh^, and
n^(P. We have already discussed in detail in Chapter 47 the
peculiarities of the vertical projection of n^s^.
The reason for this phenomenon becomes clear if we examine
Example 48-2. Here again we have the circle of perfect fifths
"stretched out" with C at one extreme of the ellipse and F# at
the other. The letters p, s, n, m, and d at the top of the figure
represent the intervals which the tones G, D, A, E, and B, and
the tones F, B^, E^, A^ and D\), form above and below the tone
C; while the letters d, m, n, s, and p below the figure represent
the relationshhip of the tones E#, A#, D#, G# and C#, and the
tones G, D, A, E, and B, below and above the tone, Ffl:.
Example 48-2
Now if we project the intervals of the perfect fifth and the
major third above and below the tone C, the remaining tones,
which constitute the complementary heptad, become the perfect
fifth, major third, and minor third above and below Ffl:. How-
ever, if we project the perfect fifth and the minor third above
and below C, the complementary projection above and below
F# becomes the perfect fifth, major second, and major third.
Hence it becomes obvious that the projection of the minor third
above and below the axis, C, cannot be found in the com-
plementary scale above and below the axis, F#, since the minor
337
COMPLEMENTARY SCALES
third above and below C are the same tones as the minor third
below and above FJf.
There follows the list of pentads formed by the projection of
two intervals above and below the axis C, with their
complementary heptads arranged above and below the axis F#:
t fs^ X p Vn^
p-n p-^s''m''
22 2 2 9
p^m^ p'^n-m-
p^(P s^n^m^
s^n^ ph^d^
s^m^ p"n^(P
sH^ s^nH^
n^m^ p^m^d^
n^d^ s^m^d^
m^d^ n^m^d^
It will be noted that in four of the ten possible projections,
the pentad contains the same vertical projection as its com-
plementary heptad. In six of the projections, the heptad does not
contain the vertical projection of the same intervals as its
pentad prototype.
Example 48-3 works out all of the relationships based on this
principle which result in the formation of the hexad "quar-
tests." Lines 1 and 2 give the two pentads formed by the
vertical projections p^m~ and p^n^. The heptad of line 1 is the
projection of the pentad of line 2, while the heptad of line 2
is the projection of the pentad of line 1, as indicated by the
dotted lines. The four connecting hexads, upon examination,
prove to have the same intervallic analysis, the second hexad
of line 1 being the involution of the first hexad of line 1; and
the second hexad of line 2 being the involution of the first hexad
of line two; the four together constituting a quartet having the
same intervallic analysis.
All of the other hexads in this illustration are formed on the
same principle and each quartet of scales has the same analysis.
338
VERTICAL PROJECTION
i
— Q -^
Example 48-3
p2 m^ s^ ^p^mSgi I
.09, ,p3m3n3s3d2t
,^. ITTW
I I J J JlJ I Pl^plJj
S
$p2 r^-
p2 m2 n2
a 2 I 2 I
J p2 n^ m' ^
2' 2 I 2 I
p3pn3n3s3(j2f
lp2n2mU'
J J If JiJ
1/^3 ijl
^
^
I 2 2
i
,2 rr,2
2 m2 d2
^^ -^ ^ ^ .-^ fp2m2dlt
^
|p2m2dU'
p3m^n3s d3t
g
^
s^aa
r r ''^ K^ J
^
*=«
i
Jm2 ._h2
^^
n2 m2 n2
3 12 1
^::i^ii JJ^jJiJiiJ I 0^\
13 12 1
„ „ , p^m^n^s d^t
} m^2plf
P^
*
t P'
_5
2 e2
p2 S^ d^
3 I 3 I I
^ £2s2d' t
3 I 3 I I
p4m2h3s3d2t
t p2s2d'i
^^r I rn'O.
^m^
° bS b'^8
il/JtlJ ^
i
XH^ ,5.2
£? s^ _n2
I P 3 2 3
t n2s2^' t
I I ' 3 2 3
p'*m2n3s3d2t
t n2s2£' i
i"lJ J ^^r I rtr J JlJ I
3s:
i
"21 4 2 1
£2 d.2 m.2 ♦ p2d2ml t
ti^^m'*
2' I 4 z r
p3m3n2s2d3f2
;;'""'- ^J7T7]^^
^
^
i
J m2 ^ n2
~ ft"
m2 n2 $2
I 3 12 4
t m2n2sit
I ' 3 I 2
l?R't|u b'H fv^ ^Q J Jb^
^
p3m3n2s2d3t2
^^
#
O L
p2 d2 s2
2 I r 4 I
£2d2slT
3m2n2s3d3t2
ibjiij,^
I' I ' 3 2^
p3m2n2s3d3t2
i
(n2 ,j2
n2 ^
'''11 ^B l^^;
I 13 2 4
$ n2s2m' t
t n2s2mT
^
■*•? r 1^ R 1 2' I I ^
2 1 15 1
339
COMPLEMENTARY SCALES
p3m4n3sd3t
ta f^ — ' - '
^k
S
&^
^
i
»
#w*
?
2i«
12 1 4 1
i
t m2 £■ ' - . m^ d2 _^ | m^dSg ' t
bn''^B \,n ^
I ' 2 I 4 1'
t nrr^i
^
i J J ^ "" r
I 3
I '3 3 17
I I 5 3 I
I 15 3 1
There remains only one other group of hexads to be
considered, the isometric twins discussed in Part III. Example
48-4 indicates that these sonorities may be considered as part
of a projection from a tetrad to its related octad. Line la gives
the tetrad formed by the projection of two minor thirds and a
perfect fifth above C. Line lb gives the isometric twins, the
first formed by the simultaneous projection of three minor thirds
and three perfect fifths, and the second formed by the relation
of two minor thirds at the interval of the perfect fifth. The
combination of these two hexads forms the octad of line Ic,
which is the projection of the tetrad of line la.
Line 2a is similar in construction to line la except that the
perfect fifth is projected below C. Line 2b is similar to line lb
except that in the first isometric twin the perfect fifths are
340
VERTICAL PROJECTION
projected below C, and the second twin is formed of two minor
thirds at the interval of the perfect fifth below C. (It will be
observed that the twins of line 2b are merely different versions
of those of line lb since, if the order of the first twin in line lb
is begun on A, it will be seen to contain the same intervals as the
first twin of line 2b: AgCsDiEbgCbiGt]. In the same way, if the
order of the second twin of Hne lb is begun on G, it will dupli-
cate the intervals of the second twin of line 2b: G3Bb2CiDb2Eb3
Gb.)
Line 2c is the octad formed by the combination of the hexads
of line 2b and proves to be the projection of the tetrad of
line 2a.
In similar manner, lines 3a, 3b, and 3c show the projection of
the tetrad formed of two minor thirds and a major third above
C, while 4a, 4b, and 4c show the projection of the tetrad formed
of two minor thirds above C and a major third below C.
Lines 5a, 5b, and 5c explore the projection of two minor thirds
and a major second above C, while lines 6a, 6b, and 6c show the
projection of two minor thirds above and a major second below C.
Lines 7a, lb, and 7c and lines 8a, Sb, and 8c are concerned
with the projection of two minor thirds and a minor second.
Lines 9a, 9fc, and 9c and lines 10a, IQb, and 10c concern the
projection of two perfect fifths and a major third.
Lines 11a, lib, and lie and lines 12a, 12b, and 12c show the
projection of two minor seconds and a major third.
The relation of two perfect fifths and a minor second, or of
two minor seconds and a perfect fifth, does not follow the same
pattern. It is interesting, however, to observe in lines 13a and 13b
that the combination of the hexads p^ @ d and d^ @ p form a
seven-tone scale which is the involution of the basic perfect-
fifth-tritone heptad.
341
complementary scales
Example 48-4
i
n^+ p = Tetrad pmn dt
WSr^^ JbJ^
3 3 I
Hexad
+ P^ = p3m2nVdgt2 n2 @
^
te
Hexad
p3m2n4s2d2t2
lb.
Ic.
^s
^
^p
>o"
^^0=^
:^
^?^
^^ 2 1312 12313
Combinotion of Hexods = |^5rj6s4(j5f 3 Comp.Tetrod 4 n^ -f £* I
i
yi^ftJfaJ ]|tf§ ^^
^a
^
I I I 3 I 2 I
£^+ p I = Tetrad pn^sdt
3 3 1
2a.
b^g. r^l jt^B
3 2 I
+ p^*
Hexad
= ^m2n^s£d2t2 "."^ |@ ^ *
rc^ -
Hexad
p3m2n4s2d2t2
£
^^m
^t^°""^»bo>jJ^J^^r
xx:
s
2b
2c
3o.
3 b.
3c.
4o.
4 b
4c
3 2 13 1
Combination of Hexads =^^6s^3
3 2 12 3
Comp.Tetrad 4 n2+ p'f
bobi.hi^bol'"
^^
^W
i
3 2 I 2 I I I
n2+rn' = Tetrad mn^sdt
3 2
b^g-fte- ^3^^^,^'^
Hexod
i
+ rn 2 = p2m3n4s2d2t2 _n2 @ m
Hexad
p2m3n4s2d2t2
^°" ^oil^ >JilJb>^^'^ l^bot^^ tletl^'^'' jbJljJt^^^
^
3 1221 ~ '31213
Combination of Hexads = 5^^6s5H5t3^®^P-^®*'"°** * n^+nn' 4
^^
^^
=^^^
3 12 1 I II
n2 + m'l = Tetrod pmn^st
w^^
3 I 2
m
^
g
^
3 3 2
^ + m2^ = p" m^n4s2d2t2 n^ @ m * = pll^gn4s2d2t2
*
3 12 2 1 ^21323
Combination of Hexads -§ai§in6fe5crt5 Comp.Tetrad i n5+ p^lf
^
^^^
^^
>(bbo)bol^"t^°'^"'^^^
2 I 12 2 12
342
3 3 2
VERTICAL PROJECTION
n + s' = Tetrad mrrsdt
S
5a
5b
5c
6a.
6 b.
6c.
7a
7b.
7c.
m
.^k> -^ 0
+ s3
Hexad o
p2rAAVt2 r @ i
iE
Hexad
p2m2nVd2t2
a
■ltv> iJbJtlJfr
^
^^^
^
33C
2 112 3
2 12 12
Octad
I
Combination of Hexad s= n4m5n6s5d5t3 CompTetrod i il^ ■*" l' ^
^
r^ J .Ka'^^^-'t
t7ok-%lji
2 11 112 1
n2 + s ' I = Tetrad pnr^n2st
2 I 3
m
^
i^^
Hexad Hexad
n3 + _s3l = _p2rn2n4s3d2t^2 n2 @ ^ 4 = p2m2n4s3d2t2
3 3 4
^o» "bou,t,„ j^jt,jiJiiJbp i^b^fe", i,»i».>jUi^J^'''r
'^ ■*3'3 2 I 1' "^""^ ^^ " ^12 12 4
Combination of Hexads = D§m5n6c5ri4t3 CompTetrod i_n2 + s' t
^^^^^
<i.^ I^-"'
i
I 2 I 2 2 I I
n2 + d' = Tetrod pn^dt
3 3 4
m
t>^a 1^ ii^J^
I 2 3
Hexad
Hexad
p2m2n4s2d3t2
I 3 3
I 2 I 2 I
Combinotion of Hexads = ^^^^ 6s5d5t3 ^'^^''®^'^°*^ * 11^+^ ^
F^^lJ .li|tf§*^
,ljv,l^e>k^botlotli
1 I' I 12 12
I 2 3
n2 t dU = Tetrad pmrftit
m
8a.
8b.
8c.
!% ^ i,^^r,r
335
Hexad
Hexad
i
n^ + d3 = p2m2n4s2d3t2 ^2 @ _d I = p2m2n4s2d3t 2
^^
>^ijbJJ^^r
^
^
^
^
^
•^- vs^vs^
3 3 311
■XT
*
Octad
2 12 15
Combination of Hexads = -5j^5p6s4ci5t3 Comp.Tetrad i n^-k-6^ t
bolqo
>bo"t^°"
^^P
^g^
2 I 2 I 3 I I
7^
343
COMPLEMENTARY SCALES
,2 4. ,r.l ♦_-r ^ ^2^„^2
f p'^ + m' t sTetrod p^mns
I _2 - neXOa ,^2 ^ r«
+ ill -p4m3n2s3d2t P2 @ m
m
- Hexad
a
9b.
^^
tt""UJjji^^r
2 2 3 1
Octad
=olP=t
2 2 2 14
I
Combination of H9)iods=p6m5n5s6d4t2 Comp. Tetrad I p2 + m U
r'r^r^rk^
^
9c.
10a.
lOb.
lOc.
Ila
lib.
ott"Qft^"
^
2 2 2 II 12
i Q^-^- m' I = Tetrod p^msdt
2 2 3
S
^2^5 I
b1
, j^ o , HexQd , , Hexad
£3 + _r]n2i = p4m3n2s3d2t £2 @ ^ i =p4m3n2s3d2t
^g
S
o "bvn
^
k> ^^
■«- b-O-*^ *^2 I 4 I 2
Combination of Hexods =p^jJ|5°^4g5jj5^3Comp.Tetrad i _p^-»-m't
2 2 3 11
_ Octad
i
^
bo be
L^obii^ii^bo:
r'r^jj j^i
^
2 I I 3 I I I
^
2 5 I
*
d^ + m' = Tetrad mns2d2
^^
«#<=►" 8- -J*
l«- I 2
Hexad
i
+ IIl^=p2m3n2s3d4t
Hexad
i @ ID =p2m3n2s3d4t
«» ^°^° jj^M^i ' ^->" ° """ #^
If 1^ 1 I I 4
Combination of Hexads =p4^°5n5s6d6t2 CompTetrad i d2 + m' j
: [>b|» J J ^obon ^
lie
' I r I I I 2
I I 2
344
VERTICAL PROJECTION
d2 + mU = Tetrad pmsd^t
m
I2a.
^
i«^i
I
:xx
g
a
s
I2b
12 c.
:3o.
13b
Combination of Hexods = p^^T4s5d6t3 Comp.Tetrod | d^ + m' t
i
_bL-*|;|c-ib<
S
^
=&
l~ 1 I
II 14 11
_p2 @ d Hexad p^m^ns^d^t^
fe
d 2 @ p Hexad pWns^d^2
S
at
^
i
$
iff I K I
OBO
r
■^F'
15 1 I
*
Combinotion of Hexods = i^^3j5t3 CompPentod ^^f^
s*
e^
is*
^
I I c^if^
w^
xsr
114 1 I
I I 5^1'
Note: The tetrads of Example 48-4 have all been discussed in
Chapter 46 as projection by involution. For example, tetrad la
of Example 48-4, (n^ + p^), is the same chord as the tetrad of
Example 46-6fo, lirie 2, {%n^m}^), and is itself the involution of
tetrad 8a of Example 48-4, ( n^ + d\^), which appears in Chapter
46-6Z?, line 3, as Xrem}\^.
345
49
Relationship of Tones
in Equal Temperament
We come finally to the formidable but fascinating task of
attempting to show the relationship of these galaxies of tones
within the system of equal temperament. The most complete
presentation, and in many ways the most satisfactory, would
seem to be that involving the abstract symbolism which I have
employed in the large diagram accompanying this text.
Although this symbolism may at first glance seem foreign to
the musician's habit of thinking tones only through the symbol-
ism of written notes, and may, therefore, seem "mathematical"
rather than musical, it has the great advantage of presenting a
graphic, all-embracing picture of tone relationship divorced
from the artificial and awkward complexity of musical notation.
For example, the symbol p^5^ indicates the simultaneous pro-
jection of two perfect fifths and two major seconds on any tone,
up or down, and in any position. This one symbol therefore
represents the sonority C-D-E-G in any of its four positions:
C-D-E-G, D-E-G-C, E-G-C-D, and G-C-D-E, together with its
involution iC-B^-Ab-F, in its four positions: C-Bb-A^-F,
Bb-Ab-F-C, Ab-F-C-Bb, and F-C-Bb-Ab, plus the transposition of
these sonorities to the other eleven tones of the chromatic scale.
The one symbol therefore represents ninety-six sonorities. The
presentation of such a chart using musical notation would
assume a size beyond the realm of the practical. It should be
noted that the order of half-steps of this sonority, represented in
the chart as 223(5)-C-D-E-G-(C)— may also appear in the ver-
346
RELATIONSHIP OF TONES IN EQUAL TEMPERAMENT
sions 235 ( 2 ) , 352 ( 2 ) or 522 ( 3 ) ; and in involution as 322 ( 5 ) ,
225(3), 253(2), or 532(2).
I cannot overemphasize the statement which has reappeared
in different forms throughout this text that my own concern is
not with symbohsm but with sound. The symbols are a means
to an end, a device to aid in clarity of thinking. They have value
to the composer only if they are associated with sound. To me
the symbol p~s^ represents a very beautiful sound having many
diflFerent connotations according to its position, doubling, and
relationship with other sounds which precede and follow it.
One other word of caution should be added before we take off
into the vast realm of tonal space which the chart explores. The
student who has worked his way slowly and perhaps painfully
through the preceding chapters cannot fail to be impressed, not
only with the vast number of possibilities within the chromatic
scale, but also with the subtleties involved in the change or the
addition of one tone. He may feel overwhelmed both by the
amount and the complexity of the material available to him in
the apparently simple chromatic scale, and wonder how any one
person can possibly arrive at a complete assimilation of this
material in one lifetime.
The answer, of course, is that he cannot. For if a composer
were to have a complete aural comprehension of all of the
tonal relationships here presented, he would know more than
all of the composers of occidental music from Bach to Bartok
combined. This would be a formidable assignment for any young
composer and should not be attempted in a one-year course!
The young composer should use this study rather as a means
of broadening his tonal understanding and gradually and slowly
increasing his tonal vocabulary. He may find one series of rela-
tionships which appeals to his esthetic tastes and set about
absorbing this material until it becomes a part of himself. He
will then speak in this "new" language as confidently, as
naturally, and as communicatively, as Palestrina wrote in his
idiom, providing, of course, that he has Palestrina's talent.
347
COMPLEMENTARY SCALES
One of the greatest curses of much contemporary music is
that it uses a wide and comphcated mass of undigested and
unassimilated tonal material. The end result becomes tonal
chaos not only to the listener but, I fear, often to the composer
himself. The complete assimilation of a small tonal vocabulary
used with mastery is infinitely to be preferred to a large
vocabulary incompletely understood by the composer himself.
Let us now turn to an examination of the large chart in the
pocket of this text. Beginning at the extreme right-hand lower
corner we find the letters p, d, s, n, m, and t, symbolizing the
six basic intervals: the perfect fifth or perfect fourth, the minor
second or major seventh, the major second or minor seventh,
the minor third or major sixth, the major third or minor sixth,
and the augmented fourth or diminished fifth.
Below each of the letters you will find a number of crosses,
5 under p, 5 under d, 6 under 5, 5 under n, 6 under m, and 3
under t. These crosses serve as abbreviations of the interval
symbol, that is, every cross under the letter p represents that
interval. A cross indicates that the interval, of which the symbol
appears at the top of the vertical column, is included in the triad,
of which the symbol appears to the left of the horizontal line in
which the cross is located.
Proceeding laterally to the left we find the section of the chart
devoted to triad formations, III. Here, again, the crosses repre-
sent abbreviations of the triad symbols. In other words, each
cross laterally on the line with the triad symbol p^s repre-
sents the triad p^s. The same thing is true of the crosses
marking the positions of triads pns, pmn, pmd, and so forth.
These triads are divided by dotted lines into groups— the first
four all contain the perfect fifth; the next three all contain the
minor second; ms^ is the basic major-second triad; nH is the
basic minor-third triad; m^ is the basic major-third triad; and the
last two triads are those in which the interval of the tritone
predominates. The numbers to the right of the triad symbols
indicate the order of half-steps which form this triad in its basic
348
RELATIONSHIP OF TONES IN EQUAL TEMPERAMENT
position— p^s above the tone C becomes C2D5G5(C), having
the order of half-steps 25(5). Each cross in this section of the
chart indicates that the triad, whose symbol appears at the left
of the horizontal line, is included in the tetrad identified by the
symbol at the top of the vertical column in which the cross occurs.
Proceeding upward from the triads, we find immediately
above them the section of the chart devoted to tetrads, IV. Here
again the crosses represent the tetrad symbol proceeding ver-
tically downward. The tetrad P^, for example, will be found
below the symbol on the first, second, fourth, and fifth spaces of
the chart.
For the sake of space the interval analysis of the tetrad is
given as six numbers, without the interval letters p, m, n, s, d,
and t. The numbers to the right of the interval analysis represent
again the order of the sonority in half -steps. The tetrad P^
should therefore be read: three perfect fifths, having the analysis
301,200, three perfect fifths, no major thirds, one minor third,
two major seconds, no minor seconds, and no tritones; the
order of half-steps being 252(3), that is, above C;
C2DgG2A(3) (C). Each cross in this section of the chart indicates
that the tetrad, whose symbol appears at the top of the vertical
column, is included in the pentad identified by the symbol at the
extreme left of the horizontal column in which the cross occurs.
Proceeding laterally and to the left we come to the section of
pentads, V, which occupies the large lower left-hand section of
the chart. Here, again, the crosses indicate the pentad on the
same lateral line. The pentad P*, for example, is found on the
first, second, fourth, and sixth spaces of the lateral line following
the symbol P* This pentad has the analysis 412,300, and the
order of half-steps 2232(3), which might be represented by the
tones C-D-E-G-A-(C). Each cross in this section of the chart
indicates that the pentad, whose symbol appears at the left of the
horizontal line, is included in the hexad identified by the symbol
at the top of the vertical column in which the cross occurs.
The six-tone scales, or hexads, VI, will be found above the
349
COMPLEMENTARY SCALES
pentads and forming a connection between the pentads below
and the heptads above. The crosses, again, indicate of which
hexads the individual pentads below are a part. The pentad P^
will be seen to be a part of the hexads F^, pns, ^pV^\ and p^m^.
P° has the analysis 523,410, indicating the presence of five per-
fect fifths, two major thirds, three minor thirds, four major
seconds, one minor second, and no tritones. It has the indicated
order of half-steps 22322 ( 1 ) , which would give the scale, above
C, of the tones CsDsEaGaAsBd, ( C ) .
The portion of the chart above the hexads gives the heptads,
VII. These scales are the involutions of the complementary
scales of the pentads below and are so indicated by the letter
"C." The heptad VII p^ is, therefore, the corresponding scale of
the pentad V P^. The scale C, pns/s, corresponds to the pentad
pns/s, the heptad C, p7?in/p, corresponds to the pentad pmn/p,
and so forth. (Pns/s is used as an abbreviated form of the sym-
bol, pns @ s.) Here each cross in this section of the chart indi-
cates that the heptad, whose symbol appears at the left of the
horizontal column, contains the hexad identified by the symbol
below the vertical column in which the cross occurs.
Proceeding now laterally to the right we find the octads,
VIII, above the tetrads. These scales are all the corresponding
scales of the tetrads below, so that it is not necessary to repeat
the symbol, but only to give the intervallic analysis and the order
of half-steps. For example, the corresponding scale to the tetrad,
P^, is the octad opposite, with the analysis 745,642 and the order
1122122(1), giving the scale, above C, of CiCSiDoEoFJfiGsAs
Ba,(C).
Proceeding vertically upward to the top of the chart are the
nonads, IX, which are the counterparts of the triads at the
bottom of the chart.
Proceeding horizontally to the right, we find the relationship
between the nine-and ten-tone scales. It will be observed that
the six ten-tone scales which are on the upper right hand of the
chart are the counterparts of the six intervals which are repre-
350
RELATIONSHIP OF TONES IN EQUAL TEMPERAMENT
sented at the lower right hand portion of the chart.
At first glance, this chart may seem to be merely an interesting
curiosity, but careful study will indicate that it contains a tre-
mendous amount of factual information regarding tone relation-
ship. For example, the relation of two-tone, three-tone, four-tone,
and five-tone sonorities to their corresponding ten-, nine-, eight-,
and seven-tone scales will be discovered to be exact. If we begin
with the pentads on the left of the chart and, reading down, we
add 2 to the number of intervals present in each sonority— except
in the case of the last figure, the tritone, where we add one-half
of two, or one— we automatically produce the intervallic compo-
sition of the sonority's corresponding heptad. For example, the
first pentad has the intervallic analysis 412,300. If we add to
this the number 222,221, we produce 634,521, which will be
found to be the analysis of the corresponding heptad. The
second pentad has the analysis 312,310. Adding to this the
intervals 222,221, we produce the analysis 534,531, which is the
analysis of the heptad C. pns/s. In like manner, the analyses of
all of the heptads may be produced directly from that of their
corresponding pentads.
Proceeding further, we have already pointed out that the
tetrads and octads have a corresponding relationship. This may
be expressed arithmetically by adding to the intervallic analysis
of the tetrad four of each interval, except the tritone, where we
again add half of four, or two. The analysis of the four-tone
perfect-fifth chord we observe to be 301,200. Adding to this
444,442, we produce 745,642, which proves to be the analysis
of the corresponding octad. The second tetrad, p^s^, has the
analysis 211,200. Adding the intervals 444,442, we produce
655,642, which proves to be the analysis of the corresponding
octad. This is true, again, of all tetrad-octad relationships.
The triad-nonad relationship is expressed by the addition to
the triad analysis of six of each interval except the tritone, where
the addition is one-half of six, or three. The first triad at the
bottom of the chart is p^s or, expressed arithmetically, 200,100.
351
COMPLEMENTARY SCALES
Adding to this 666,663, we produce 866,763, which will be found
to be the analysis of the corresponding nine-tone scale at the
top of the chart. The triad pns, 101,100, becomes in its nine-tone
relationship 101,100 plus 666,663, or 767,763, and so forth.
The single interval may be projected to its ten-tone counter-
part by the addition of eight of each interval, p, m, n, s, and d,
and four tritones. The decad projection of the perfect fifth
therefore becomes 100,000 plus 888,884, or 988,884. The projec-
tion of the major third becomes 898,884; of the minor third,
889,884, and so forth.
Since this chart is of necessity biaxial, it may take some prac-
tice to read it accurately. If we begin with the interval of the
fifth, p, at the lower right hand of the chart we find by proceed-
ing laterally to the left that it is contained in five triads p^s, pns,
pmn, pmd, and pdt. Conversely, we find that the perfect-fifth
triad, p^s, contains the intervals p and s. Proceeding now upward
from the triads to the tetrads we find that the triad p^s is con-
tained in the tetrads p^, ph^, p^m^|, p^d^'l, and p^d^. Conversely
the perfect-fifth tetrad p^ will be seen to contain the triads p^s
and pns.
Proceeding laterally to the left, from the tetrads to the pentads,
we observe that the tetrad P^ is found in the pentads P^, pns/s,
^p^n^l, and p^d^. Conversely, the pentad P* contains the tetrads
P^, pV, and p/n.
Proceeding upwards, from the pentads to the hexads, we find
that the pentad P^ is contained in the hexads P^, pns, p^s^d^l,
and p^m^. Conversely, the hexad, P^, contains the pentads P^,
pns/s, and pmn/ p.
Proceeding again upwards, from the hexads to the heptads, we
find that the hexad P^ is a part of the three heptads P®, C.
pns/s, and C. pmn/p. Conversely, the heptad P^ contains the
hexads P^, pns, n^s^p'^X, and p^/m.
Proceeding now laterally and to the right, from the heptads
to the octads, we find that the heptad P^ is a part of the octads
352
RELATIONSHIP OF TONES IN EQUAL TEMPERAMENT
P\ C. pV, and C. p/n. Conversely, the octad P^ contains the
heptads P^ C. pns/s C. jp^n^j, and C. p^d^.
Proceeding upward, from the octads to the nonads, we find
that the octad P^ is found in the nonads P^ (C. p^s) and C. pns.
Conversely, the nonad (P^) contains the octads P^, C. pV, C.
p^m% C. p^d't, and C. pH^.
Finally, proceeding laterally, from the nonads to the decads,
we find that the nonad P^ is contained in the decads C. p, and
C. s. Conversely, the decad C p (or P^) contains the nonads P^
(C.p^s), C. pns, C. pmn, C. pmd, and C; pdt.
The arrows on the chart which indicate the progression from
the intervals to the triads, from the triads to the tetrads, the
tetrads to the pentads, and so forth, may be helpful in tracing
various "paths" of tonal relationship.
As the student examines the analyses of the various sonorities
or scales, he will find that they differ in complexity. The
analysis of the triads is simple. The analysis of the tetrads is
comparatively simple, but there are several forms that have at
least two possible analyses. The second tetrad, p^ s^, for example,
may be analyzed as the simultaneous projection of two perfect
fifths and two major seconds (pV); or as the projection of a
perfect fifth above and below an axis tone, together with the
projection of a minor third above or below the same axis
( p^n^X ) ; or, again, as the projection of a major second above and
below an axis tone, together with the projection of a perfect fifth
above or below the same axis (s^p^). The tetrad p @ n may also
be analyzed as n @ p, since the result is the same. The basic
tetrad of the tritone-perfect-fifth projection may also be analyzed
SLS p @t, and so forth.
The pentads have several members which have a double
analysis, as indicated on the chart. The hexads are more com-
plicated, some of them having three or more valid analyses.
There are still other possible analyses which have not been
specifically indicated, since their inclusion would add nothing
of vital importance.
353
COMPLEMENTARY SCALES
One curiosity might be noted. In Chapter 48 the subject of a
"diagonal" relationship was discussed in the case of the isomeric
"twins" and "quartets" among the hexads. The chart makes this
relationship visually clear. The twins and quartets are indicated
by brackets. Now if we examine the position of the crosses indi-
cating the doads, triads, tetrads, octads, nonads, and decads we
find that the upper half of the chart is an exact mirror of the
lower part of the chart. In the case of the pentads and heptads,
the upper half of the chart is a mirror of the lower except where
the connecting hexad is a member of the "twin" or "quartet" re-
lationship, where the order is exactly reversed. In the vertical
column at the extreme left of the chart, the three crosses indicat-
ing pentads one, two, and three are mirrored above by the hep-
tads one, two, and three, in ascending order. In the second
column from the left the crosses marking pentads, one, two, four,
fourteen, fifteen, and twenty are mirrored by heptads in the same
ascending order. The third and fourth columns, however, are
connected with their corresponding heptads by the isomeric hex-
ad "quartets." Here it will be seen that the third column of pen-
tads is "mirrored" in the fourth column of heptads, and, con-
versely, the fourth column of pentads is mirrored in the third
column of heptads. This same diagonal relationship will be ob-
served wherever the twins and quartets occur, although there are
four cases where there is a "double diagonal," that is, where one
pentad and one heptad are related to both members of a
quartet family.
As far as the order of presentation of the sonorities is con-
cerned, I have tried to make the presentation as logical as
possible. The hexads, for example, are arranged in seven groups.
In the first of these, the perfect fifth predominates or, as in the
case of the second hexad, has equal strength with its concomitant
major second. In the second group the minor second pre-
dominates, except in the case of the second of the series where
the minor second has equal strength with its concomitant major
second. In group three the major second predominates, or has
354
RELATIONSHIP OF TONES IN EQUAL TEMPERAMENT
equal strength with the major third and tritone. In group four
the minor third predominates with its concomitant tritone. In
group five the major third predominates throughout. In group
six the tritone predominates, or has equal valency with the
perfect fifth and/or the minor second. In the last group no
interval dominates the sonority, since in all of them four of the
six intervals have equal representation.
This grouping is indicated by the dotted vertical lines and
the solid "stair-steps" which should make the chart more
easily readable.
355
50
Translation of
Symbolism into Sound
For those composers wHq have difficulty in grasping com-
pletely the symbolism of the preceding chapter, I am attempting
here to translate the chart of the relationship of sonorities and
scales in equal temperament back again to the symbolism of
musical notation. It should be stated again that this translation
cannot possibly be completely satisfactory. A nine-tone scale, for
example, will have nine different versions. If the scale has an
involution, that involution will also have nine positions. Each
of these eighteen scales may be formed on .any of the twelve
tones of the chromatic scale. Therefore, in the cases of such
nine-tone scales, one symbol represents 216 different scales in
musical notation, although only one scale form.
The musical translation of the chart can therefore give only
one translation of the many translations possible and must be
so interpreted.
Example 50-1 begins with the twelve-tone and the eleven-tone
scale, each of which is actually only one scale form, and then
proceeds to the six ten-tone scales. Each of these scales, as we
have seen, corresponds to a two-tone interval. The ten-tone scale
C, p is presented with the interval p of which it is the projection.
The ten-tone scale C, d is presented with its corresponding inter-
val d, and so forth. The order of presentation will be seen \p
conform with the order of presentation in the chart. Since all
of the scales are isometric, no involutions are given.
356
translation of symbolism into sound
Example 50-1
I I (I)
^'/i Eleven -tone Scale
I I (2)
i
Ten -tone Scales
C.p ££m£n£s^dfl^
Conresponding Intervals
P
4n^
^ ^^ o jto o fl*^ ^ ff
I" I I I 2 I I I 2 (I)
C.d p8m8n8sQd9t^ A d
■i^^
Iff 1 r I I I
33:
o k^ jtk.1 o !iL^
Iff I r I I I I I I (3)
C.s. p8m8n8s9d8t4
-o- 5-0-
ff 1 ? 9 I I II
o !>")
#
I 2 2
C.n p8m8n9sQd6r4
I I I I (I)
bo t;o *>> bo^
i
bvs t|o bo 1^*-^ ^
I I
Cm p^m^n^sSd^t'*
I 2 I (2)
o to v^=#g=
ivTr
^^
-o-
*F7^
12 11 II
C.t p^m^nQs^d^t^
(II
ff I l" I ? I I I I (?^ V -«»-
I l" I 2 I 1 1 I (2)
357
COMPLEMENTARY SCALES
Example 50-2 gives the nine-tone scales with their involutions,
where they exist, and with the corresponding triads, of which
they are projections, and the triad involution, if any.
The order of presentation is, again, the same as that of the
chart for ready comparison,
IX
Nine-tone Scales
C. p^s 866,763
Example 50-2
Involutions
Cor^^ponding involutions
p2s
ts^
i^nt
^^^
^EC^
I'n 2 2 I I I 2 (1)
C.pns 767,763
FO
2 5(5)
*
^o(o)i olxibo,,b^(^^
pns
-tnfcisx
^
:tjO
safe
=»^
\^ " I 2 I 2 2 (I)
C.pmn 777,663
I I I I 2 I 2 2 (I
i
7 2(3)
pmn
7 2 B)^
ii?^
-^^I^v^bo^^t,,^^^
/>")'^*^l>>
f^
f
ODOO'
I 2 I i I 2 (I) 4 3 (5) 4 3 (5)
pnnd
2 f 12 11 I 2(1)
C.pnnd 776,673
I I 3 I I 2 I I (0 -» >« ni •, /I /■'?=-'
r I 3 I 12 I I (I)
C.sd^ 666,783
sd
7 4(1) 7 4 (H
2
rtEst
:X«3fc:
r I I I I I I I (4)
C.nsd 667,773
r I (10)
nsd
I I I I I I I 2(3) I I I I I I I 2 (3) I 2 (
(")i *'o
pr
i
II I III I 2(3)
C.mnd 677,673
(9) I 2(91
nnnd
^^oit»°''-'^-^^°'^'^'""'^"^"oi>o„^
^")'"til?i
ni
3 I 2 I I I I I (I) 3 I (8) 3 1(8)
3 I 2 I I I I I (I)
C.rns^ 676,863
«3l^O t^
ms'
2 2 2 I I I I 1(1)
C.n£l 668,664
2 2 8
n2t
»
,i4'o(")
■bol^^il'otl*
^
^^
I r I I 2 i 2 I (2)
3 3 (6)
358
TRANSLATION OF SYMBOLISM INTO SOUND
Cm' 696,663 m^
iW=
^
^^^
^
2 I I 2 I I 2 I (1)
C.pdt 766,674
4 4 (4)
pdt
I I I I 2 I I I (3) I I I I 2 I I I (3) 6
(,o) ■ 111
1«T
I (5) 6 1 15)
^ C.mst 676,764 mst
i 2 2 I I I I (2) I I 2 2 I I I I (2)
2 4 (6)
2 4 (6)
Example 50-3 gives the octads with their corresponding tetrads
in the same order as that o£ the chart.
Eight-tone Scales
^"' C.p"^ 745,642
Example 50-3
Involutions
'^p3 301,200
5dS
ots^i
^^
^^
2 2 12 2(1)
:o:'=»
2 5 2(3)
p^^ 211,200
Cp^s^ 655,642
(i>): nbo[^
tf..o^ov>^^-); "^^-^^
2 2 2 1 I I 2 (I)
Cp/n 656,542
*^
2 2 2 1 I I 2 (I
2 2 3 (5) 2 2 3 (S)
p/n 212,100
r
.boM :
joC")
^^Qobetjo
3 4 3(2)
p^m'l 211,110
i
2 12 1 12 L(2)
C.p^m't 655,552
b^l;uboN i ^^^^R^
,^"^i "|u
^o^
n O
°<>^obo
=®=EE
^
w
t
3 1 12 11 I (2)
C.p/m 665,452
3 112 1 I 1(2)
4 I 2(5) 4
p/nn 221,010
2(5)
^
FFre^^
^
^
2 113
2(1)
4 3 4(1)
IV T,
Df 001,230
M=
!D
(9)
I" 1 r I II II
I M I (5)
^^
359
COMPLEMENTARY SCALES
C.d^s^ 455,662 d^s^ 011,220
#
I I I I I I 2 (4) I I I I I I 2 (4) I I 2 (8)
Cd/norn/d 456,562 d/n 012,120
2 (8)
P^
:tnti:
jM^**#*
^
r I I I I I 3(3)
C.d^m' $ 555,562
r 2 I (8)
d£ml < 111,120
ii^
^S
«^?«
^^^«=
'l^
I 2 14 1 I I (I)
C.d/m or m/d 565,462
I 2 I 4 I I I (f)' 13 7(1) 13 7
d/m 121,020
M:
=<?3^
^oljobotio^
S=
I 3 I (7)
S3 020,301
VIII
i
Ml I I 3 1(3)
S7 464,743
IV
^
4.,o^o"ttv^^'-^i
2 2 2 (6)
2 2 2 11 I I (2)
C.s£n't 555,652
4
s£n"| 111,210
^^^^^^\> '^■■'°'"':--'°''° o
bo("); "t?^
II I 113 2(2) II I I I 3 2 12) 2 I 7(2) 2 1
7 (2)
i
C s/n or n/s 546,652
s/n or n/s 102,210
t54:
i^niJ
,bc»[|o"it«'
=^33
2 1 2 (7)
N^ 004,002
i
2 11 II I 2 (3)
C.N^ 448,444
>bo("J ;
^^^
.l>«jboln<
^obi
3 3 3 (3)
n^p'l 112,101
I 2 12 1 2 I (21
C.n^p't 556,543
3 4 2 (3'
F
2 12 2 1 I 2 (I)
Cn^d' t 456,553
2 I 2 2 I I 2 (if ' 3 4 2 (3)
n^d't 012,111
b.■^l^vto^^^) : ^'obo^^
J.") : ^'o
I 2 eBT^
^botjob*
^oboJ;o/^)' .^ob^
I r I 2 3 I 2(1)
II 12 3 1 2 (T)
I 2 6(3)
360
i
TRANSLATION OF SYMBOLISM INTO SOUND
C.n^s' J 546,553 n^s' t 102,111
^bc»(>'') : ^''oboi.^k
^%M ; ^^\?i
^^
^
^^
II I I 3 2 1(2)
Cn^m' t 556,453
■ ■"■ 2 I 6(3) 2 16 (3)
I I I I 3 2 I (2)
i
nfm'j 112,011
^
I 2 I 2 I I I (3) I 2 I 2 I I I (3)
C.n/m or m/n 566,452
3~r 5(3) 3 1 s'T^'P
n/m or m/n 122,010
^^^^^^
^^
^
3 12 1
I 2(1)
3 (5)
C. M^ 575,452
M^lm^p' t) 131,010
T I 2 I I 3 (I) 2 I I 2 I I 3 (I) ^ 4 3 I (4) 4 3 I (4
"4 3 I (4) 4 3 1 ^p
m2s2 030,201
C.m2s2 474,643
^^^
^g=
2 2 2 11 2 I (I)
Cp/td'^) 644,464
.e.og>^ ^
2 2 4(4)
p/t 200,022
^=^
i^
^^
^3CS
^^
^^f
i
3 I I I (3)
C.m/t 464,644
r 5 I (5)
m/t 020,202
H^"^'>^'^^i
i^c^
^
^^IPS
2 1 12 2 1 I (2)
C pmnsdt 555,553
4 2 4(2)
pmnsdt 111,11!
^4"°«°""°^"';" °H^^^„hJ ^.ii»^^
I 3 2 I I 2 I (I) I 3 2 I I 2 I (I)
C. pmnsdt 555,553
4 2 I (5) 4 2 I (5)
pmnsdt 111,111
^^^^^^^^
^^^^
r I 2 2 I 3 I (I) I I 2 2 I 3 I (h
Cp^d' t 654,553
^^^
(6)
r 3 2 (6) 13 2 (6)
p2d' ^ 210,11!
^„k.>.»>.°^°>"'"'""°"°^°
A"): *-^o
^35=
I 2 2 I I 3 I (I) I 2 2 I I 3 I (I) I 4 2 (5) I 4 2 (B)
361
i
COMPLEMENTARY SCALES
C.£^ or d/p 654,463 p/d or d/p 210,021
!^^Mv>^l^"°'"^''^
^^
^ 6 I (4)
p2d2 200,121
i
I r I 3 I I 3(1)
C.p2d2 644,563
^
^°^
1^^
33l^
r I 4 I I I 2 (1)
C. d^p' J 554,563
r I 5 (5)
d^p'^ 110,121
i
^
'Oj^^bfiob
o(o) ; i-t^
^^^M^^
*-"■'»'' OPOc^
S
f
o
I 6 4(1)
— IKJ
I 6 4 (f
r
12 1 III 4(1)
12 1 I I 14
Example 50-4, in like manner, shows the relation of the hep-
tads to their corresponding pentads and involutions.
VII
Seven -tone Scales
Involutions
Example 50-4
Conresponding
Pentads
Involutions
p6
634.521
412,300
^^
T^m
oo{t^>°
Q l-l O
2 2 3 2(3)
pns/s 312,310
2 2 2 I 2 2 (I)
C, pns/s 534,531
botlolvy : »»^o=
>U>):Ob<
=^=si
'bool>^
o *'
•^^toi
^
r
4
2 3 2 2 1 I (I)
C.pmn/p 544,431
2 3 2 2 1 I (I
2 5 2 2 (I)
pmn/p 322,210
2 5 2 2 (
a^o(");"bei^
o(,o)!"bo|^
m
H^
2 2 3 1 I 2(1)
C t p^rf i 534,432
2 2 3 4(1)
tp^n^l 312,211
1^
2 2 3 1 12
(1)
2 2 3 4
^P
r I 2 2 I 2 (3)
C.p^d^ 533,442
'(■©■) «-»'
3) ^ 2 4 I 2 (3)
IT
I I 2 2 I 2 (3)
2 4
2 (3)
i
p'd^ 311,221
^
C") ; *^ obo
^
TT^
^^
:^^
^^,
=^
r I 4 1 2 2(1)
^5~
I 5 2(3)
r
I 14 12 2
I 5 2(3
362
i
TRANSLATION OF SYMBOLISM INTO SOUND
C.pmd/p 543,342 pmd/p 321,121
J„o^^"^'"H.
2 3 1 I 3 I (I)
f^
2 4 I 4 (I) 2 4 1 4 Ifp
2 3 1 13 1
VII
D^ 234,561
D."* 012,340
^
:^
^^P
"^^f
^^
cr«-
I I I (6)
C.nsd/s 334,551
I (8)
nsd/s 112,330
i
(,ki);not>«
bot|o'»'J
,bot)ot>o'
^^
I (f) III
*'o
r
I I I 2 5 I (I)
Cmnd/d 344,451
I I I 2 5
2 (7) I I I 2{
mnd/d 122,230
>'>*J : tipbc^
>')i"o^.|r
I I 4(3) ' r 2
is^:
^^^
I I (7) I 2 I I (7)
^
I I I I I 4 (3)
C.d^n^ 334,452
III 114
d^n^ Il2,23i
l"l 7 I 2 I (5) I I I I 2 I (5) ^ ri I 3 (6) I I I 3 (6)
(5) ' i*n I 3 (6)
dV 211,231
4
C.d5p2 433,452
(fc.t) : tl olyc^
^
II 12 1 I (5)
C.pmd/d 443,352
IT^
r
I I I 2 I I (5)
I I 4 (5)
I I I 4 (I
i
pmd/d 221,131
i^^^^
V):"ot>OL
m
oC^) : m|
ii
TStP
W.
^Rf=
^33-
l^'^l 4 13 I (I)
VII c
-6 262,623
= — ' ■ OPUi
I I 4 I 3 I (
^
6 I 3 I (I)
S^ 040,402
6 13 I (H
r
2 2 2 1 I 2 (2)
C.t£n2| (or p^n^) 444,522
2 2 2 2 (4)
tsfr^lor ££n2<) 222,30!
2 2 2 I 2 I (2)
C.s3p2 443,532
2 2 2 3(3)
s3p2 221,31!
2 2 2 I I I (3) 2 2 2 I I I
2 2 2 1(5) 2 2 2 I (^)
363
i
COMPLEMENTARY SCALES
C.s^d^ 343,542 s^d^ 121,321
r I I I 2 2 (4) I I I I 2 2 14) r I 2 2 (6) I I 2 2 C«
C.s^n^ior n£d5t) 244,542 s^n^ lor n£d ^ J ) 022,321
i
^^
^
^
jbo^
^^^
r I II 2 4(2)
^" _n6 336,333
2
Vm4
I 2(6)
114,112
boC"):
i^
^obo^^'
=^33
|"il"k>,
k>^^<
^^^
^«s
12 3 12 i^^^ 3 3 1 2 (3)
pmn/n 223,111
^
12 3 1 2 I (2)
C. pmn/n 445,332
3 3 1 2 (3)
>,l7o("):^^i^^
W*-*):*.V,bc
^^l?olyoll<
^^^
I 2 I 3 I 2^^ "3 I 3 3(2)
pns/n 213,211
3 I 3 3 (2)
I 2 I 3 I 2 (2)
C. pns/n 435,432
2 I I 2 I 2(3)
C.mnd/n 345,342
2 I I 2 I 2 13)
4 2 I 2(3)
mnd/n 123,12!
4 2 12
(3)
^»oit"°^"'l"''^^^'^"«be^^;U"°""°'^^
r
3 12 1 I I (3)
C.nsd/2 335,442
3 12 II 1(3)
3 1 2 I (5)
nsd/n 113,22!
3 1 2 I (g)
III 12 3(31 I I I I 2 3,1^' I 2 I 2(61
I 2 1 2 (6)
V!!
Ill I 2 3 (3)
m6 464,24!
Vm4
M'^ 242.020
o^o^^"^ :"boobt
ok"): "k
^^
f
=«a:o^
M»?
^^^
ofloo
I 3 I 3 (I)
2 113
(I)
C.p^m^ 453,432
4 3 1 3 (I)
p2m2 23!,2!!
4 3 I i^
2 I I 3 (I) 2 2 2 I I 3 (I) 2 2 3 I (4) 2 2 3 I
2 2 2 I I 3 (I)
C.m2d2 353,442
[4)
n£d2 !3!,22l
^^^^^^^^
^^^^^
^^^
1*^ I 2 I I 2 (4)
;^x»
W
I I 2 I I 2 (4) r I 2 4 (4) I I 2 4 (4)
364
TRANSLATION OF SYMBOLISM INTO SOUND
Cm^n^ 354,432
mV 132,211
3 I 2 I I 2 (2) 3 I 2 I I 2 {^) 3 I 2 2 (4) 3 I 2 2
Cp^m^t 454,341 p£m2 1 232,120
(«0
(4)
^^
S^^
^otlo *-'»
^
i
I 2 1 I (3)
C.m^d^l 454,341
4 12 1 (4)
m£d_^ $ 232,120
^^^^
booC^')
^^
V «^2 r 4 I 2 (I)
C.m^n^ $ 454,242
■eJ^oO
3 4 3 (I)
,2„2
I
mfrf J 232,021
»iasi}&i
3CSt
^^
^uMo'
j?o|:io
Vtl ,-3
I 2 I 3 I I (3)
532,353
3 14 1 (3)
VjS 310,132
I" I /I I I I /'Z\ 11/1111
j)»o^s")|"c>bot,=
(5
I I 4 I I I {31 r I 4 I (5)
p2d2j 220,222
I" I 4 I I 1 (3)
C.p^d^l 442,443
I 4 I (5)
^ 4 2 4 (I)
imn/t(l-5) 212,122
5^n)=
^^^^^^
12 2 11 4(1)
C pmn/t(l-5) 434,343
[f\ I 3 2 I 3 I (I)
C.pmn/t(l-3) 344,433
r 3 2 I (5) 13 2 1 (5)
pmn/t(l-3) 122,212
3 2 I I 2 (2)
i
C.t^2| 444^441
4 2 1 3 (2)
tp2<j2| 222,220
botjo'^Vl *^te=
2 4 I 2 I I (rT 2 5 3 I (I) 2 5 3 I (f) ■'
p2n2 222,121
i
2 4 12 1 I (I)
444,342
C.p2n2
^^^^^^^
Ul);tl|yo^,
^s
p^
^
^^^
r I I 3 I 3 (2) I I I 3 I 3 (^)
21
2 I 3
(5)
2 I 3 I (5)
365
COMPLEMENTARY SCALES
C.tdfn^l 444,342 td^n^l 222,121
j^.,o»t^^^">;"'=>t>ot>o.4^
^^lys) ^iob<^
^
RX
I 2 I (2) I I 4 I 2 I (2) r I 4 3(3) 1 I 4 3 (3)
r I 4
C. p^s^d^ 434,442
«w
p2s2d2 212,221
■ o(«>^):^>obog^
>>) :"otyot;^
^=^
l<^ 2 1 2 4(1)
I I 2 I 2 4 (^'' i^f^i 2 3 (5) I I 2 3 (5)
C.nVj 434,442 n^^j 212,221
I I I I 3 2(3)
Finally Example 50-5 presents the six-tone scales with their
involutions. In most cases, as we have already seen, the involu-
tion of the hexad is also its complementary scale. In the cases
of the isomeric "twins," the complementary scale is given in the
third part of the line. Where the original scale is a part of a
"quartet," the scale is given with its involution, followed by the
complementary scale, followed in turn by its involution.
VI
i
Example 50-5
Six-tone Scales Involutions Six-tone "Twin"
p5 523,410 (also X pVn' )
Involutions
T^JF^
tl ^"Q
2 2 3 2 2 (I)
PNS (p^s^) 423,411 (also pmn@s; Jpfn^s"; t p^s^m')
j|..o»^"'^i"'^l^^
22212 (3) 22212
Jr£s££l 423,321
(3)
tp^s^d' 423,321
.>>c^^) ;^^bo
.bo'»^) : "ote=
i^^
^^
(2) ^ I I :
2 I 4 2 I (2)
iaiso nsd/p )
I I 3 2 3 (2)
2 14 2 1
3 2 3 (2)
m
366
pym 432,321
TRANSLATION OF SYMBOLISM INTO SOUND
2.2* .A _2^2.
p^m3 432,321 (t pVt ;| mVt )
oM=
^?fe
^
^^
sx
torO^
2 2 2 1 411)
^!- D^ 123,450 (also J s^d^n' )
S-C3 O
2 2 3 1 I (3)
NSD (s^d^) 123,441 (olso mnd@ s; t n^dS ; t s'^d^m' )
r I I 2 (6) I 1 I I 2 (6)
tn^s^fl' 223,341 ^ s^d^p' 223,341
,Jto(^i) :»o|yo^
^boll|o(") : "ot?o
^o^ot>o
^«
I I I 6 I (2) I I I 6 I {^) ' I I 5 3 I (1) 1 I 5 3 I [I
(also pns/d)
W^
^&ar
dVm 232,341
d^m^ 232, 341 it p Vt ; $ s^d^t )
:tg3=i:
^=^
^^«^
?cr«v
-otta **'
r I 2 I I (6)
r IT I 4 (4)
VI
_Sf 060,603 (also t m^s^ )
^^
^
?cy^
I
2 2 2 2 2 (2)
ts^2| 242,412 (also I m^s^p'; t p^n^d')
;^^) 1 "
'^^"t^^l^.^^l
^
^D
2 2 2 2 I (3) 2 2 2 2 I (3)
sV 241,422 ($ m^s^n'; t P^d^n'
i
2 2 2 I I (4) 2 2 2 I I (4) ^
sln2 142,422 (J m^s^d' ; J n^dV )
2 I I 2 2 (4) 2 I I 2 2 (4)
367
COMPLEMENTARY SCALES
i
pmd/s 322,431
(ct) \ lli?i\^
-crg»
55^
T I 5 2 2 U) I I 5 2 2 (I)
s^/n 323,430 (also tp^d^i)
fi s-/r
5B^
P^
*^ ^2 I I I 2 (5)
225,222
VI
i
,boti^) : i't,
P boHo'
5R^
'3 3 I 2 I (2) 3 3 12 1*^^
n^/p 324,222 n^+p^ 324,222
W»^) ;
^oN«"^'^^:
,^\>^^[
I 2 3 I 3(2)
n^/d 224,232
2 I 3 I 2 (3)
n^ + d^ 224,232
^
^
>oljo^°fa'
^^^^
r 2 I 2 I (5)
nVs
224,322
1113 3(3)
n^ + s^ 224, 322 (^ pVt ; t n^d^t )
.b^obo^-'-"-'i
^(?^
jb^W^
2 1 I 2 3(3)
2 I 2 I 2 (4)
n2/fn 234,222
n^-i-mS 234,222(4 mVti^n^s^t)
bo(^^) i
^
^otlot'^H
j?ot|ot^i»
3 1 2 I 3 (2)
^'iM6 363.030
2 2 1(3)
it°^
3 I 3 1 3 (I)
PMN 343,221
,||^o^*>):"bokv.^
2 2 3 1 3 (I)
PMD 342,231
^
2 2 3 13
,.„>.»> ''i'H...^
i
2 4 I 3 1(1)
MNP 243,231
2 4 I 3 I (I
r
^^^
t^^^^°^
^izsx
3 12 I I (4)
^fe°^
3 12 1
(4)
368
TRANSLATION OF SYMBOLISM INTO SOUND
pmd/n 343,230 (also t m^d^n'; t p^m^n')
boljo^^*) : *-^^
^^^
2 1 4 3 I [I)
2 14 3 l7f) ^
i p^m^d' 343,131 JmVp' 343,131
i
b..(.")i^^ok.
M"^"^:""!^
*^ *^ 3 I 2 I (4)
lalso pnnn/d)(alsoi
I 3 1 2 I (4) '
J m^n^d' )
^^
" obo/
r
3 I 3 I I (3) . 3 13 I I (3)
(also mnd@p)(al:so t m^d^' )
^
^P
m
)il^{p^) 420,243 (also i p^d^t )
4
3tnt
^^
r I 4 I I (4)
pmn/t 224,223
||..oll'^'":"°l'"||..l|..„:
"3 2 I 3 (2)
mst/p 422,232
13 2 I 3(2)
■ (la) ■ K\oa.
^^
^
fe
ri 4 I 2 (3)
mst/d 322,242
r
I I 4 I 2 (3)
C") : " oboi
I r I 3 I (5) I I I 3 I (5)
p2/d 421,242 d^/p 421,242
^ta^
>j|ovi^"):
^
r I I 4 I (4)
t p^d^s' 322,332
r I 5 I 1(3)
t n^s^m' 322,332
i
;tnt
>W-
tboCfct) : ^bo^'^bc
*^ "^1 r 3 2 4 [I)
(also pdt /s)
M«
113 2 4"(\r ^-"
(also p^d^)
"^r
2 1 I 5 1(2)
i^d3
2 I I 5 I (2)
I I 1 4 2 (3)
369
COMPLEMENTARY SCALES
,2w2„l -^^o '?^'> X m^n^s' 332,232
t £ldfm' 332,232
o(")i"o(,.
^^^^^^S
k»o"
^
^^e^
^
t/ -«■
I 3 1 2 4 (IJ
(also pdt/m)
13 1 2 4 (^)
IFp 2 I I 4 I t3)
2 I I 4 I
IT
* p2n2m' 333,321
X p2rT?s' 333,321
^")!*VJ^V
lu-<*»! "l?o|;i
^0=
^
■OflOJ
f«^
po-o^
^^^^
3 I I 2 2 (3) ' 2 2 I 2 I (4) 2 2 I 2 I (^)
3 1 I 2 2(3)
(also pns/m)
% n^d^m' 233,331
t m2d2s' 233,331
to*
^^5
k^o(»>) :"ot>ok^
-rr^
,i>obot]<
^
^oi]«
^F
"T
21 5 2 1
(also nsd/m)
12 15 2
112 4 3(1)
I I 2 4 3 (f)
p^s^d^H- pi 323,331
,2 e2 A^
p*^ s^ d*^ + d ; 323,531
(.") ! "'^"bo^
^{yO : iibiit^i,.
"^^
mO
r I 2 I 2 (5)
t«^) -o^e^*^
rr
I I 2 I 2 (5) I I 2 3 4 (1) 112 3 4 0)
These relationships of tone will repay endless study and
absorption, for within them lies all of the tonal material of
occidental music, classic and modern, serious and popular. With-
in them lie infinite and subtle variations, from the most
sensuously luxuriant sounds to those which are grimly ascetic;
from the mildest of gentle sounds to the most savagely dissonant.
Each scale or sonority encloses and enfolds its own character.
In parting, let us look at one combination of sounds which we
have used before as an example, the tetrad pV and its octad
projection. It is a sweet and gentle sound used thousands of
times by thousands of composers. It has, for me, a strong per-
370
TRANSLATION OF SYMBOLISM INTO SOUND
sonal association as the opening sonority of the "Interlochen
theme" from my "Romantic" symphony. You will find it and its
octad projection on the second line of Example 50-3. Note that
the tetrad has the sound of C-D-E-G. Notice that its octad is
saturated with this pleasant sound, for the octad contains not
only the tetrad C-D-E-G but also similar tetrads on D, D-E-Fjf-A;
on E, E-FJ-GJf-B, and on G, G-A-B-D. In the hands of an insensi-
tive composer, it could become completely sentimental. In the
hands of a genius, it could be transformed into a scale of
surpassing beauty and tenderness.
In conclusion, play for yourself gently and sensitively the
opening four measures of Grieg's exquisitely beautiful song,
"En Svane." Note the dissonance of the second chord as con-
trasted with the first. Then note again the return of the consonant
triad followed by the increasinglv dissonant sound, where the
Dt> is substituted for the D. Listen to it carefully, for this is the
mark of genius. It took only the change of one tone to transform
the sound from its gentle pastoral quality to one of vague fore-
boding. But it had to be the right note! If this text is of any help
in assisting the young composer to find the right note, the labor
of writing it will not have been in vain.
371
tl
Appendix
Symmetrical Twelve -Tone Forms
For the composer who is interested in the type of "tone row"
which uses all of the twelve tones of the chromatic scale without
repetition, nineteen of the six-tone scales with their comple-
mentary involutions offer interesting possibilities for symmetrical
arrangement. If we present these scales, as in Example 1, each
followed by its complementary involution, we produce the
following symmetrical twelve-tone scales:
Example 1
pns
i
-> <-
-^ <-
J Jp ifj^JiiJur^r'^nijj j||jjJ ^JttJriir*r"r
'^ '^■^2' I 2 2^ ^'22322 ^2^2 2: I 2 21222
#
-> <-
nsd
-> <-
iijjtfj J ^ tJ ^I'^'i^t^riUJ^Jt'-'i'^^-' iiJt^''^"rt''r
I I I I
6
I I I I I I I I I 2 2 I I I I
-etc.
1222 22222 2 2?22 22222
s4p2
^>' «
»■ <r
^lij,iiii^J^ir'r^r'rV^nijj.i<i m.
-0-
2 2 2 2
12 2 2 2 2 2 2 11
2 2 2
373
4 2
s n*^
APPENDIX
pmd@ s
> <-
0 115 9 P 9 I I 9 "^tt I «S 9 P P P «S I I
2 2 2 2 112
s^@Il
iJt.JhJJJ-iiJtJ^t^rTiibJ^^fa^^^r^
m
2 I I I 2
,6
3 3 12
2 13 3
^ <
<|>iJjJ«Jr itJ^ii^'irV'^riujJ^ir "r^r^^^
^
wf
3 13 3 13 13
13 13 3 13 13
pmn
I 14 11
pmn@t
f|.i,i"jiiJJii^ iiJ^tJ^rTj^ji I yTt^
13213 31231 13213 31231
nnst@ p
mst@d
< - T
-^. '
t=
<
— a 1 jiJ * ir
g
-J — 1 — 1 1 — 1
Ff^
r^P
^^
55^£fJ
NH
b!
f^^'^r^
-*
^
W
t^^
1--J —
— L_
In any of the above scales, any series of consecutive tones
from two to five will be found to be projected to its correspond-
ing ten, nine, eight, or seven-tone scale. For example, in the
first scale, p^, not only are the twelve tones the logical projection
of the original hexad but the first ten tones are the projection
of the first two tones; the first nine tones will be seen to be the
projection of the first three; the first eight tones are the
374
SYMMETRICAL TWELVE-TONE FORMS
projection of the first four, and the first seven tones are the
projection of the first five.
In other words, the seven-tone scale C-D-E-FJj:G-A-B is the
projection of C-D-E-G-A, the eight-tone scale C-D-E-F#-G-
G#-A-B is the projection of C-D-E-G, and so forth, as illustrated
in Example 2:
Example 2
h j'jNN
g
iiJ hip |*r "
r «^
It should be clear that the above relationship remains true
regardless of the order of tones in the original hexad as long as
the series is in the form of a six-tone scale— or sonority— with its
complementary involution. For example, the scale of Example 2
might be rearranged as in Example 3:
Example 3
efc. etc.
-^ <r-
j_i-j JjJ iiJnrt.i^riiJ'iri^.i.irJj«riJ||^
The method of determining the "converting tone"— that is, the
tone on which we begin the descending complementary scale-
was discussed in Chapter 40, pages 266 to 269. A quicker, al-
though less systematic, method is by the "trial and error" process,
that is, by testing all of the possibilities until the tone is found
which, used as a starting point, will reproduce the same order
of intervals downward without duplicating any of the original
tones. Referring, again, to Example 1, p^, it will be clear that
E#, or F, is the only tone from which we can project downward
the intervals 22322 without duplicating any of the tones of the
original hexad.
The hexad "twins" and "quartets" cannot be arranged in this
manner for reasons previously explained. This is also true of the
hexad pmd @ n which follows the general design of the
375
APPENDIX
"quartets" although, unhke them, its complementary scale proves
to be its own transposition at the interval of the tritone.
The nineteen hexads of Example 1 contain in their formation
all of the triads, tetrads and pentads of the twelve-tone scale
except the five pentads, p^m^t, m^d^t, m^n^^, p^s^d^, and nV^,
the last of which will be recognized as the "maverick" sonority
of Chapter 47. The first four may be projected to a symmetrical
ten-tone row as in Example 4:
Example 4
i
p^m^ t
missing rn^d^ t
Torres ~zzz >. <-
|,jjji.JiiJ|.iiii^i ,'i; II uj.J':^r ^W^'
missing
tones
4 12 1
m2n2 J
15 4 3 3 4 3
_ 111 — 1 * missing missing
^ ^ >^ < tones > < tones
3 14 1 14 13
p2s2d2
3 14 1 14 13
missing
tones
112 3 3 2 11
376
Index
A
Accent,
agogic,
58
rhythmic,
58
Analysis of intervals,
7
by omission,
270
Axis of involution,
20
- 21
B
Bartok,
From the Diary of a
Fly,
74
Sixth Quartet,
74, 127,
145
Fourth Quartet,
75, 145,
192
Beethoven,
Leonore No. 3,
35
Symphony No. 5,
35,
297
Symphony No. 8,
36
Berg, Alban,
Lyrische Suite,
38
Nacht,
83,
96
Britten, Les Illuminations,
115,
156
Decads,
Clockwise and counterclockwise
progression, 9
Common tones, 60
Complementary hexad, 249
Complementary sonorities,
of the perfect fifth series, 275
of the minor-second series, 276-277
of the major-second series, 278
of the minor-third series, 279
of the major-third series, 280-281
of the perfect-fifth— tritone series, 282
Consonant symbols, pmn, 11
Converting tone, 266-269
Copland, A Lincoln Portrait, 214, 217
D
Debussy, Voiles, 81, 88
La Mer, 82
Pelleas and Melisande, 84, 95, 103, 115
186, 202-203, 209
Les fees sent d'exquises danseuses, 116
perfect-fifth, p^m^n^sHH*,
31, 276,
315
minor-second, p^m^n^s^dH*,
66,
277
major-second, p^m^n^s^dH"^,
91,
278
minor-third, p^m^n^s^dH'^,
119,
280
major-third, p^m^n^s^dH'^,
134,
281
perfect-fifth-tritone.
p^m^n^s^dH^,
149,
282
"Diagonal" relationship
of hexad quartets,
336
Dissonant symbols, sdt.
11
Dissonant triad, sd^.
11
Dominant seventh,
4
Dorian mode.
57
Double valency of the tritone.
139-140
Doubling,
49
Duodecads,
perfect-fifth,
pl2ml2nl2sl2(fl2i6^
31, 276,
315
minor-second,
pl2^12„12jl2cil2i6^
66,
277
major-second.
pl2ml2„125l2dl2t6^
92,
278
minor-third.
pl2ml2„12sl2cil2i6^
119,
280
major-third.
pl2OTl2„125l2dl2f6^
134,
281
perfect-fifth-tritone.
pl2^12„12sl2cil2^6_
149,
282
E
Enharmonic equivalent.
1
Enharmonic isometric hexad,
78
Enharmonic table.
12
Equal temperament,
1
Expansion of
complementary-scale theory.
263
Exponents,
19
Fusion of harmony and melody, 3, 16
Gregorian modes, 47
Grieg, En Schwan, 371
377
INDEX
Harmonic rhythm.
53
Hanson,
Sinfonia Sacra,
128
Cherubic Hymn,
206
Elegy,
293
"Romantic" Symphony,
296, 371
Harmonic-melodic material,
perfect-fifth hexad,
40- 47
minor-second hexad,
67- 72
major-second hexad.
79- 81
minor-third hexad,
98-103
major-third hexad,
125-126
perfect-fifth-tritone hexad.
141-144
pmn-tritone hexad,
153-154
Harris, Symphony No. 3,
270-271
Heptads,
perfect-fifth, p^m^n'^s^d^,
29,
275, 315
minor-second, p^m^n*s^d^t,
66, 277
major-second, p-m^n~s^d^t^,
90,
232, 278
minor-third, p^m^n^s^dH^,
119, 279
major-third, p*m^n*s^dH,
133, 281
perfect-fifth-tritone,
p^m^n-s^dH^,
148, 282
Heptads, complementary.
of pmn projection.
286
of pus projection,
288
of pmd projection,
290
of mnd projection,
291
of nsd projection,
292
of prnn-tritone projection
295
of pentads p^ + s^, p- + s^
304
oi pentads d3 + s2,d2 + s3.
305
o{ pentad p3 + d2,p2 + d3
306-307
of pentad tp2-|-d24,_
308
of pentad p^ + m^,
309
of pentad d^ + m^,
309
of pentad p^ + n^.
310
of pentad d^ + n^.
310
of pentad s^ + n^.
311
of pentad m~ + rfi.
311-312
of pentad p^ + s^ + d^,
312
1, 333-334
of pentad Ip-m^,
317-318, 335
of pentad \p^n^.
320-321, 335
of pentad ^p^d^.
322-323
of pentad fm'^n^,
of pentad ;; m^d"^.
324-325
326-327
of pentad '^n'^d^.
328-330
Hexads, perfect-fifth.
p5m'^nHid,lp^s^n^,
29, 315
pns, pmn@s, p^ + s^,'lp^n^s'^
■,tP'
is2mi.
ptm^nh'^dt.
173, 236
(nsd@p,l n^s^p^ p^m^n^s^d-t,
\
239,
240, 257
\Xp2s2d\
259
p^@m, p'^m^n'^s^d^t,
■ p3 + m2,lp2sH,lm2d%
212
211,
229, 231
Hexads, minor-second.
pm2n^s*d,ls2d2n^,
65
nsd, s3 -1- d3, mnd@s, I n^d^s^
,Js2d2mi,
pm^n^s^dH,
188
215,
204,
200,
( pn5@d,Jn2s2<ii, p^m^n^sHH,
{ 239,
Kls^d^p^,
{d~@m, p^m^n^s^dH,
d^ + m2,lp2m%ls2d%
Hexads, major-second,
m^s^t^,Xm^s2t,
ts%24,, jm2s2pl, Ip2n2d^,
p2m*n2s*dt2,
s4 4- p2, 1 m2s2ni, l p^d^n^,
p2m*ns*d2t2,
Si + n2,lm2s2d'^,ln2d2pi,
pm^n^s^d^t^,
pmd@s, p^m^n^s'^dH,
s2@n,fp^d^ ],,p^m2n^s*d^,
Hexads, minor- third,
p^m^n^s^d^t^,
(n2@p, p^m^n'^s^d^t^,
\n^ + p^,
)n2@d, p^m^n'^s^dH^,
n^ + d^,
)n2@s, p^nfin^s^d^t^,
n3+s3,i;p2n2f,:I;"2d2i,
in2@m, p2m^n*s2d2t2,
n^ +m2, J n2s2f , J m^n^t,
Hexads, major-third,
p^m^n^d^,
pmn, p^m'^n^s^d^t,
pmd, p^m^n^s^dH,
mnd, p2m*n^s2dH,
pmd@n, X rrfid^n^, % p^m^n^,
p^m^n^s^d^,
/ 1 p^m^d^, pmn@d, I rrfin^d^,
I p^m^nHdH, 239,
Ilm2n2pi, mnd@,p,1vrfid2p'^,
239,
Hexads, tritone,
t^, p2@t,lp2d2t, p^m^s^dH^,
pmn@t, p^m^n'^s^d^fi,
mst@p, p^m^n^s^dH^^
mst@d, p^rrfin^s^dH^,
{p2@d, p'^m^ns^dH^,
d2@p,
/|p2d2ji, pdt@s, p^ + d^,
< p^m2n2sUH2^ 219, 239,
v X n^s^m^,
np^d^m'^, pdt@m,
J p^m?rfis2dH2^
(Jm2n2si,
Hexads, neutral,
/ pns@m,Jp2n2mi,
) p^m^n^s^d^t,
\lp2m2s\
ifn^d^m''^, nsd@m,
< p^m^n^s^dH,
Ilm2d2si,
j p2-\-s2-\-d2-\-p\^,p^m2n^s^dH,
\ p2+s2+d2+d|
Hexad quartets, 254,
Hexad "twins,"
Hoist,
The Planets,
Hymn of Jesus,
239,
239,
239,
240, 255
259
216
230, 231
78, 230
232, 234
233, 234
233, 234
237
237
98
197
195
208
207
205
230, 231
201
230, 231
13, 124
168
178
183
237, 240
240, 255
240, 255
140, 230
152
237
238
219
220
240, 256
259
240, 256
259
240, 256
259
240, 257
259
258
258
339- 40
340-345
171
199
378
ESTOEX
Influence of overtones,
55
Intervals,
symbol p,
9-10
m.
10
^,
10
s.
10
d,
10
t,
11
number present in a sonority,
11
table of,
14-15
Inversion,
8, 40
Involution, theory of,
17
simple.
18
isometric.
18
enharmonic.
19
of the six-tone minor-third projection, 110
of the pmn-tritone projection,
158
of the pmn hexad
170
of the pns hexad.
174
of the pmd hexad.
179
of the mnd hexad,
184
of the nsd hexad,
189-190
Isomeric pentad, pmnsdt,
23
Isomeric sonorities.
22-23
Isomeric twins.
196
J
Just intonation,
1
major-third, p^m^n^s'^dH^,
133, 281, 324, 327
perfect-fifth— tritone p'^m^n^s^d''t*,
149, 282
Nonads, complementary,
M
Major-second hexads with foreign tone, 232
"Maverick" sonority, 331
"Maverick" twins, 333
Messiaen,
L'Ascension, 122, 135
La Nativite du Seigneur, 135
"Mirror," 17
Modulation,
key, 60
modal, 56
concurrent modal and key 63
of the perfect-fifth pentad, 61
of the minor-second pentad, 76
of the minor-third hexad, 109
of the major-third hexad, 131
of the perfect-fifth-tritone hexad, 147
of the pmn-tritone hexad, 157
Moussorgsky, Boris Godounov, 155
Multiple analysis, 5, 6
N
Nonads,
perfect-fifth, p^m^n^s'^dH'^,
30, 276, 315, 320, 322
minor-second, p^m^n^s'^d^t^,
66, 277, 323, 328, 330
major-second, p^m'^n^s^d^t^, 91, 278
minor-third, p^m'^n^s^d^t*,
119, 280, 310, 311, 312, 321, 326, 329
of pmn projection.
286
of pns projection.
289
of pmd projection,
290
of mnd projection,
291-292
of nsd projection.
292-293
of mst projection.
293
of tp2.
317
of lm2.
319
O
Octads,
perfect-fifth, p'^m'^n^sHH'^, 30,
275, 315
minor-second, p'^m'^n^s^d'^t'^.
66, 277
major-second, p'^m^n'^s'^dH^,
91, 278
minor-third, p^m'^n^s^dH'^,
119, 279
major-third, p^ni^n^s^dH-,
133, 281
perfect-fifth— tritone.
p6m*n4s4d6^4^
148, 282
Octads, complementary,
of pmn-tritone projection,
296
of tetrad p@m,
299
of tetrad n@p,
299
of tetrad m@t,
299
of tetrad n@m,
300
of tetrad m@d.
300
of tetrad n@s.
301
of tetrad n@d,
301
of tetrad p@d.
301
of tetrad p'^+s^.
304
of tetrad d^+s^.
305
of tetrad p'^+d^.
306-307
of tetrad s'^+nfi.
308
of tetrad Ip^m^,
317
of tetrad Jm^pi,
318
of tetrad Xp^'n-^i
320
of tetrad jn^pi,
321
of tetrad tp^d^,
322
of tetrad Id^p^,
323
of tetrad Im^n^,
324
of tetrad | n^m'^,
325
of tetrad I m'^d^,
327
of tetrad Jd^^i,
328
of tetrad % n^d^,
329
of tetrad Id^n^,
330
of tetrad In-s'^,
332
of tetrad ts^n^.
332-333
of tetrad n2+pi.
342
of tetrad n^-\-m'^.
342
of tetrad n^-\-s'^,
343
of tetrad n^+d^,
343
of tetrad p^+m^,
344
of tetrad d^+m^,
344-345
p
Pentads,
perfect-fifth, pns@p, "Ip-s^, p*mri^s^,
29, 172, 226, 315
379
INDEX
pns@,s, p^mn-s^d.
47,
172
pmn@p, p^m^rfis^d,
47,
167
"[p+^n^i, p^mn^s^dt.
174,
196
p^+d^, p^mns^d^t,
212,
221
pmd®p, p^m-nsd^t,
177
Pentads, minor-second,
mn^s^di, nsd®d,ts-d2, 65, 187,
228,
277
nsd@s, pmn~s^d^.
72,
188
mnd@d, pm^n^s^d^,
71,
182
d2+n2, pmrfisUH,
188,
208
d^-\-p~, p'^mns^dH,
216,
220
pmd@d, p'^m^nsdH,
177
Pentads, major-second,
|m2s2, m'^sH^, 7fi
1, 81,
227
i*2„2 or Ip2„2^ p^m^n^sH,
173,
226
s3-)-p2^ p^m^ns^dt.
174,
213
s3+d2, ts2+d2 4,^ pnfins^d%
188,
217
s2+n2 or InH^, rrfirfisH%
189, 205
., 228-229
Pentads, minor-third,
pmn'^sdfi.
98
pmn@n, p^m^n^sdt.
102,
168
pns@n, p^mn^s^dt.
102,
172
mnd@n, pm^n^sd^t.
103,
182
nsd@n, pmn^s^d^t,
103,
187
Pentads, major-third,
pmn@m, pmd@m, mnd@m,
p2min2d2, 124, 168,
177,
182
p2-\-m2, p^rrfins-dt.
169
m2+d2, pmHs2d%
178,
216
fji2-\-n~, pm^n^s^dt,
169,
201
^p^m^, p^m^n^sd^.
215,
226
'•m^d^, p2m3n2sd2.
211,
229
;:m2n2, p2m3n2d2t.
200,
228
Pentads, tritone.
p^TnsdH2, pdt@p,
144,
220
Ip2d2, p2m2s2d2t2.
144,
227
pmn@tC^ 5), p2mn2sd2t2.
154
P7nn@f(i3), pm2n2s2dt2.
154
\p2+d2i, p2m2n2s2d2,
179
p2-\-n2, p2m2n2sd2t,
169,
196
'td2+n2l, p2m2n2sd2t,
183,
207
p2+s2+d2, p2mn2s2d2t,
205,
257
In2s2, p2mn2s2d2t,
200,
227
Pentad projection by involution,
338
Perfect-fifth— tritone projection.
140
Phrygian mode,
57
Piston, Walter, Symphony No. 1,
272-273
pmn-tritone projection with its
complementary sonorities,
294-296
Projection
of the perfect fifth.
27
of the minor second.
65
of the major second,
77
of the major second beyond the
six-tone series,
90
of the minor third.
97
of the minor third beyond the
six-tone series.
118
of the major third.
123
of the major third beyond the
six-tone series,
132
of the tritone.
139
of the perfect-fifth-tritone
beyond the six-tone series,
148
of the pmn-tritone series.
151
Projection by involution,
225
Projection at foreign intervals.
236
Projection by involution with
complementary sonorities.
314
Perfect-fifth series.
315
[]p2m2,
[ ", p2n2,
316-319
319-321
lp^d2.
321-323
im2n2,
323-326
lTn2d2,
326-328
tn2d2.
328-330
Projection of the triad pmn.
167
pmn®p,
167
pmn@m.
168
pmn@n.
168
pmn hexad,
168
Projection of the triad, pns,
172
Projection of the triad pmd,
177
Projection of the triad mnd,
182
Projection of the triad nsd,
187
Projection of the triad forms with
their complementary sonorities.
pmn,
285-288
pns,
288-289
pmd.
289-290
mnd,
291-292
nsd.
292-293
mst.
293
Projection of two similar intervals at
a
foreign interval.
298
p@m.
298-299
p®n,
299
m@t.
299
n@m.
300
m@d.
300
n@s.
300
n@d.
301
p@d,
301
Prokofieif,
Symphony No. 6,
38
Peter and the Wolf,
128
R
Ravel, Daphnis and Chloe,
35
Recapitulation of the triad forms,
136
Recapitulation of the tetrad forms.
161
Recapitulation of the pentad forms,
241
Relationship of tones in equal
temperament.
346-355
Relative consonance and dissonance,
, 106-108
Respighi, Pines of Rome,
171
Rogers, Bernard, Portrait,
283
S
"Saturation" of intervals.
140
Scale "versions,"
34
380
INDEX
Schonberg, Five Orchestral Pieces,
No. 1, 150, 203, 218
Scriabine,
Poeme de I'Extase,
Prometheus,
Sibelius, Fourth Symphony, 296
Simultaneous projection,
of the minor third and perfect fifth,
of the minor third and major third,
of the minor third and major second,
of the minor third and minor second,
of the perfect fifth and major third,
of the major third and minor second,
of the perfect fifth and minor second.
Simultaneous projection of intervals
with their complementary sonorities,
p2+s2, 303-304
d^+s^, 304-305
p^-\-d^, 306-307
p3+d3, 307-308
s2+m2, 308
p^-\-m^, d^+m^
81, 235
235
302, 313
195
200
204
207
211
215
219
p^+n^,
309
310
310
311
311
312
25
274
193
176, 269
3
210, 218
37, 214
d2-|-n2,
s2-)-n2,
m2-fn2,
p2^s2+d2,
Six basic tonal series,
with their complementary sonorities
Six -tone scales formed by the
simultaneous projection of
two intervals,
Shostakovitch, Symphony No. 5,
Sonority,
Strauss, Richard,
Death and Transfiguration,
Stravinsky, Petrouchka,
37, 128, 155, 198
Symphony in C,
Symphony of Psalms, 49, 120, 171, 234
Symphony in Three Movements, 121
Concertino, 150, 222
Sacre du Print emps, 181
"Tension," 106
Tetrads, perfect-fifth,
p3ns2, 28, 315
p2-j-s2^ p2mns^, 42-43
p@n-n@p, p^mn^s, 43-44
Ip^m''-, p^mnsd, 46
p@m-m@p, p^m^nd, 18, 44-45
Tetrads, minor-second,
ns^d^, 65
d2+s2, mns^d^, 68
d@n-n®d, mn^sd^, 69, 102
td2mi, pmnsd^, 70
d®m-m@d, pm^nd^, 69-70
Tetrads, major-second,
m^sH, 77
ts2ni, pmns^d, 46
s®n-n@s, pn^s^d, 46
Tetrads, minor-third,
nH2, 97
I n^p^, pmn^st, 101
I n2<ii, mn^sdt, 101
Xn'^s^, pn^sdt, 101
tn^mi, pmn^dt, 101
n@m-m@n, pm^n^d, 101
Tetrads, major-third,
pm^nd, 123-124
m^-\-s^, m^s^t, 80
p@t-t@p; d®t-t®d; p^d^t^, 142
m®t-t@m; s®t-t®s; m^sH^, 80
Tetrads, tritone,
pmnsdt, 101
IpH^, p^msdt, 144
p@d-d®p, p^md^t, 143
p2^d2, p2sd% 143
td^p'^, pmsd^t, 143
Theory of complementary scales, 261
Theory of complementary sonorities, 247
Tonal center, 56
Translation of symbolism into sound, 356
Triads,
perfect-fifth, p^s, 28, 315
minor-second, sd^, 11, 65
major-second, ms^, 42, 77
minor-third, n^t (diminished), 98
major-third, m^ (augmented),
12-13, 79, 123
pns, 41
pmn (major-minor), 11, 41
pmd, 41
pdt, 100
mst, 79
mnd, 67
nsd, 42
Twelve-tone circle, 3
Twelve-tone "ellipse," 337
U
Undecads,
perfect-fifth,
piOmWniOsWdiots^ 31, 276, 315
minor-second, piOmiOniOsio^io^s^ 66, 277
major-second, piOmiOnio^io^^io^o^ 91^ 278
minor-third, piOmiOnio^iOc^io^o^ ng^ 280
major-third, piOmiOniOsiOt^iOfS^ 134^ 281
perfect-fifth— tritone,
piOmWnWsiodwt5^ 149, 282
Vaughn-Williams, The Shepherds of
the Delectable Mountains,
Vertical projection by involution and
complementary relationship,
W
Wagner,
Ring des Nibelungen,
Tristan and Isolde,
176
335
185
283
381
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Hanson^ Howard, l&SG-
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