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Full text of "Harmonic materials of modern music; resources of the tempered scale"

LIBRARY OF 
WELLESLEY COLLEGE 




PURCHASED FROM 

BUNTING FUND 



Digitized by the Internet Archive 

in 2011 with funding from 

Boston Library Consortium Member Libraries 



http://www.archive.org/details/harmonicmaterialOOhans 



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^ij i ii 



*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 



>. . i i ..i'« t i' 



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'^ i j^jjg^J i jj Ji'-^t 



'fiJ^J^rU^jjtJJUjtJJtf^^i^rr^r'Ti^^ 



t ^^rrrr i juJJf ijJjtJ ^ i^^^^^UJ^tJ^P 



I J J J ^ ^ i;ii.JtJbJ U|J J ^^^ Uj jj^^*^ i>J^^ 



(j^^jiJ^ri|J^^rr i jjJ^^ 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 



^ i jj i i' 



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 i gu ^ 



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" i iJJ-' 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 /TT l ^Nf l jJi I jjr 



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 ^rrrJ r^ 







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 b J 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, ^ I jti 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'JlJ J 



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 ■* ' ^ b J 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. y j ^ ^ 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 jitJ i j^JtJ i |tJtJitJ i itJ<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 |J ii J i [jitJiif'rM | tJiiJY'r'l4ii i ir 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 >i JitJi^tJ<tJ« r'^'i Jii-'J '^T' i ' i i jJi tJrr it>.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 



(| i JuJiiTnY 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' 



^^^=t F 



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|i i iJ|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 J l J l . 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 l U 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 




fp j ^^dripi"^LJ ' ^ - ^'u^Lii 



i^ i^^^ 




i^n^^^^dlifj^alLLS^^^'iLlL 



i 



bm \ ;mr 



bf^ M 



I) 1, iff ^^^ JJ I t-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:tfLtfr i jy..^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, i rWr 



'cmrftc^^rrT c'TrTT yrT_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: ..k J^"^* 



^ l ^rt^< 



^^^^ 



^ 



^ 



i 



-0 — 
@ S 



!?• lj v\ • 



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 fl es 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>o p 



^=*^ 



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^ 



U l Jl iJbJ^J^J^J^r ^ 



^^ 



P^bo^° " tl' 



ibJ^J^J I ^^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 I J J J J 



1-1. I 



>>;j.^^X 



J 



'^m 






mm 



ro - tl - - nem me -am, 

J3 iri J7J 



rp'rrmr i 



^^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> */ */ g g 



-^-»- 



^ 



i 



3_^ pizz. 



orco 
p — J- 



•OS? 



pizz. 



-»-y- 



^m 



^ 



a Jj^»^i 5 



pocosj 



06p ^•^ ^J5? 



^4? 



orco 

I — s ■ 









pizz. 

-1~ t» 



a n^y W 



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- itJi i tfe aS 




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 







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 ' i r 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^ i Vrnmi ^ 



lU ' jJm^-"^'l^ 



^^ 



m 



^ 



Is 



■^' ^^JbJ\J 



^^ 



(|j7;pja^i-^ i jjr, . [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 i f i i^ 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'A scension" 



( ^^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 



ijj i iJ.i i >'^-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 



k i^ 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'\ iJ in ii^LiF 



(|i J JjtJ7^ij^i;3itJ Jl^ i Jpp^cJ 



i'ijj J7^ iiJiiJ)tJltJ«^ijitJ«J|'LL/ 



145 



THE SIX BASIC TONAL SERIES 



j^ JJlt^^iJ^tl^ l JJJ ^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 i i"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 



^^^ 



j jo 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^d f 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'i i ii»""*-° 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 i tJ 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^^'^^^ i j-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 T 



^^ 



^v% ot i ' 



^ 



g > tt *^ 



13 2 13 



Modulotion @ t^ 



@ il 



>>; .. It.. » lt° ' ^ 



-it..it"«-' i ""' 



bo » bo " 



*s^ 



# 






^^^= 



^^^^3 



:«=« 



@ d. 



@ £ 



@m 



@S 



!>^ . , - »tf' 



^^ 



, I ' fj t * 



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 



ibJi l J ^IjJI t J- 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 i tr 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] i j j Jf i Jrrr 



^ 



^ 



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 d 1 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. , .b eg: 



^ 



: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^ 




» tf ogo " 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 y i r^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. 



» ll iii M k f ^ 



kf i wrijitiiiiiig 



#|*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; J I 



I pns @ p 2 2 3 2 i_pns @n 4 2 12 i pns @£ 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 

Shostokovic h, Sym phony 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 Uhj W 



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 



"r i " "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 



P o — 
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 ^^ i Ti if J'^iVrri^J I' l r^ 



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 



i f=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,JbJ I ^ 

^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^ /- l U J P 



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 l l l Hi 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 



^ ^ ^ f g 



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^J tJ ^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 I jo 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 son de" . 



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 



# 



b i' 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^iJ aJ ■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|, iJ i l J 



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 



b o .^- i>o tj o 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^rr s 



=^©: 



=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 



^ 



k i 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 a t i 



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 J i 



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"^ hl M 






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^ 



^ " JJJ I ^ 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 i i '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 t t^ 



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 ^ rrJ i 




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 



j g^ 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 
He xad, 




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.;tlly l f/yl|JH¥^ 



>/ »l l*P 



ii » t tii % ^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 I I 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^JJ i ^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/jt j \i O \i>r^ ' I ..rj j j;^ 



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^r i JbJ 



' \'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^f l T^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^ p bp 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 



f l ^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 



^f i '^Jj J 



? 



m 



3 3 3 (3) 
4 yi Minor Third Pentad pmn^sdt^ 



j bJ ^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 . 




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 



r r ^ 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 



rTT i Tl '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 ti l 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.. i i ' .- ' ^ ' 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^r irJ ^^ 






^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'r i Jj 



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 J i J 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 ^« Lt t i r 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 In volution 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 ^ 

^• ( 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^l pmn'-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 ; Jb J l , Jl' J i^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 



^ 



1 1 \ 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 Complement ary 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 i t 



?= 



^ 



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&. b is =^^ 

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 j i 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 Ji l 



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 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 i J 



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^j t i* ^ 



^^ 



^ 



^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 



jr npiem 



r ^r V ^r ^ ^ ^■' 



2 2 4 
2^s^m^@s Hexad m^s^t ^ 



2 2 2 II 2 I 
Complementary Hexad 

k 



j i JttJ^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'-r i '-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) 



^ 



i iJJJ ;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 fr iJ 



^^^^ 



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 nJ i ii, 



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^ . l JJ | 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 . i iJUp 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 jj jJ l ^ 



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 



"rr i '^'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 • " ' ^ g V3 



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^i i J i iJJj 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^':: i tij^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^ :*rr i i^»^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' i Vii^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 



ijjL i 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, mrtf jjij J [riip i J||j] \ \m n 



212. 2 II 2' ~ 342 334 

9.- pSm^nSsSd^tS * «2^2c2 



# 



i p HD rl^ 



i- 



jj ^jJt JJJr 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 



nJ i ^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 p m^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^rr i i^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^ i i^ i^r"^ ^ 



J| ; JbJfcj J 



i 



2 1 4 

p^m^n^sd^t 



3 I 3 



j> jbJuJ ''''^i i ^ '■ 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 i i' 



^ 



§= 



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.! ||llui l »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^ jj i JiiJp i ,ii l i;;^B"i«r " r r tiJ 



^p 



^^ 



12 1 3 112 
9- p5m7n5s*d5t2 



I 3 4 



# 



Jm^ n^ d^^'l 



j^ jtjJ i J i iJf 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 ^ n J 



^^ 



«= 



I 2 6 
3- tn^d'l mn^sdt 



II 12 3 1 2 



t m2 s2 d2 p' t 



f'""aijJrfri-^ijJ-rV i XiiiriiV iij i u^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 f J i J ^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'lrVrii JtJ 



^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^ ri J 



^^ 



^ 



2 2 4 2 



12 2 4 2 



7y) p2m4n4'^d'^t^'^ i ni^ £■ ^ Complementary Pentad 2 2 

/I . I,. I . I IlL^ bfc>9 l ;'l#H :r^rrh.l.J .. "i.. ^ 



i 



ly l J 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^p J !> • 



■ 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'ntrri iJii i ii fj ^ 



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'iri i rrJ/::« 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 



jJ J JJf 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^p l Jj 



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 I f J iJ 



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 J J ^jJ i J i iJ 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 






ibj ii j,^ 



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 



b n''^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 
p3m2n4 s2d2t2 



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)bo l ^"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> -^ 

+ 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-% l ji 



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 ».>jU i ^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 



bo l qo 



>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 ^i i^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 



_ b L - *| ;| 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 jt o 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^ jt k . 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„^ 



^")'"ti l? 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) 



^ 



F F re^^ 



^ 



^ 



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^ 



^ol j obotio^ 



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" i t« ' 



=^33 



2 1 2 (7) 

N^ 004,002 



i 



2 11 II I 2 (3) 
C.N^ 448,444 



>bo("J ; 



^^^ 



. l> «jboln < 



^ob i 



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) 
C n^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^ 



^bo t job * 



^ 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^^ 



*-" ■ '»'' OPO c^ 



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 



'boo l> ^ 



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(");"b ei^ 



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 



, b o t )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 



= — ' ■ O P Ui 

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 



|" i l"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^^"^ :"boo b t 



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(« > ^):^>obo g^ 



>>) :"otyo t;^ 



^=^ 



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 



to rO^ 



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 ^ 



^bo ll |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) \ ll i ?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 



,bo t i^) : 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-m S 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") : " obo i 



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 



l u-< * »! "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 i fj^Ji i Jur^r'^nijj j||jjJ ^JttJr i ir*r"r 



'^ '^■^2' I 2 2^ ^'22322 ^2^2 2: I 2 21222 



# 



-> <- 



nsd 



-> <- 



iijjtfj J ^ tJ ^I'^'i^t^r i UJ^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, i ii i ^J^ i r ' r^r'rV ^ n ijj.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 



> <- 



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^i i ^'irV'^r i ujJ^ir "r^r^^ ^ 



^ 



wf 



3 13 3 13 13 



13 13 3 13 13 



pmn 




I 14 11 

pmn@t 



f|.i,i"jiiJJ i i^ iiJ^tJ^ r Tj^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 i iJnrt.i^riiJ'ir i ^. 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.J ii J | .ii ii ^i ,'i; I I 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 




Ha„so„,lw?002 00339 1492 

Harmonic materials of modern music; reso 





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Hanson^ Howard, l&SG- 

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