Howard.hanson Bw

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

who

wife, Peggie,

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

a need for such a basic text has

is

come from the

author's experience as a teacher of composition, an experience

which has extended over a period It

of

more than

thirty-five years.

has developed in an effort to aid gifted young composers grop-

maze

and melodic and searching for an expressive vocabulary which would reach out into new fields and at the same time satisfy their own esthetic desires.

ing in the vast unchartered possibilities,

How

hunting for a

new

of harmonic

"lost chord,"

can the young composer be guided in

far horizons? Historically, the training of the

his search for the

composer has been

and imitation; technic passed on from master to pupil undergoing, for the most part, gradual

largely a matter of apprenticeship

change, expansion, liberation, but, at certain points in history, radical change

and

revolution.

During the more placid days the

apprenticeship philosophy— which

was

is

in effect a study of styles-

and efficient. Today, although still enormously important to the development of musical understanding, it does not, hy itself, give the young composer the help he needs. He might, practical

indeed, learn to write in the styles of Palestrina, Purcell, Bach,

Beethoven, Wagner, Debussy, Schoenberg, and Stravinsky and still

own

have

difficulty in

coming

creative development.

basic,

more concerned with

He

to grips

with the problem of his

more the art and

needs a guidance which

a study of the material of

is

vn

PREFACE less

with the manner of

its

use,

although the two can never

be separated. This universality of concept demands, therefore, an approach in its implications.

The

author has attempted to present here such a technic in the

field

which

radical

is

and even revolutionary

of tonal relationship.

Because of the complexity of the

scope of the work

limited to the study of the relationship of

tones in

melody

or

portant element of

is

harmony without reference rhythm. This is not meant

importance to the rhythmic element.

It

task, the

to the highly im-

to assign a lesser

rather recognizes the

practical necessity of isolating the problems of tonal relationship

and investigating them with the greatest thoroughness composer

is

the

if

to develop a firm grasp of his tonal vocabulary.

hope that this volume may serve the composer in much the same way that a dictionary or thesaurus serves the author. It is I

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

component

its

and describe its position in the tonal cosmos. 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

In this

wav

parts,

the

consistency of his expression.

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 It

Without that communicate, without— in other

for the expression of the inspiration of the composer.

inspiration, without the viii

need to

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

composer gift

is

is

hope that

in developing his

may express

which

my

the

itself

mark of

this

volume may

own vocabulary

with that simplicity,

all

assist

the young

so that his creative

clarity,

and consistency

great music.

Since this text differs radically from conventional texts on "har-

mony,"

may be

it

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

when

limitations

the

late

applied to the music of the twentieth— or even

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

tones,

and no further attempt

is

by means

made

of "non-harmonic"

to analyze or classify

these combinations.

This means that traditional harmony systematizes only a very small proportion of leaves

all

the possibilities of the twelve-tones and

all

the rest in a state of chaos. In contemporary music, on

the other hand,

major and minor

many

other scales are used, in addition to the

scales,

and

intervals other than thirds are

used

in constructing chords. I

have, therefore, attempted to analyze

all of

of the twelve-tone scale as comprehensively

traditional

chords

it

classified

and

the possibilities as

thoroughly as

harmony has analyzed the much smaller number

covers. This vast

and thus reduced

of

and bewildering mass of material is to comprehensible and logical order IX

PREFACE

by four

chiefly

devices: interval analysis, projection, involution,

and complementary Interval analysis

is

scales.

explained in Chapter 2 and applied through-

out. All interval relationship

perfect

reduced to

is

six basic categories

:

the

the minor second, the major second, the minor third,

fifth,

each— except the tritone— conabove and below the initial tone.

the major third, and the tritone, sidered in both

its

relationship

This implies a radical departure from the classic theories of interterminology, and their use in chord and scale construc-

vals, their tion.

Most

of

Western music has

perfect-fifth category.

for centuries

Important as

been based on the

this relationship

has been,

it

should not be assumed that music based on other relationships

cannot be equally valid, as Projection logical

I

believe the examples will show.

means the construction

and consistent process

of scales or chords

of addition

and

by any

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

order,

it

compared with a is

similar series arranged in descending

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

might

The

result

was

felt,

however, that confusion

because of the traditional use of the term inversion.

)

any sonority and its 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 relation of

Chapter

3,

involution

is

discussed in

PREFACE intensive study, for

sets

it

up

a basis for the logical expansion of

tonal ideas once the germinating concept has been decided

mind

in the

The

of the composer.

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 I

upon

must

my

reiterate

sidered a "method" nor a "system." of harmonic-melodic material. Since

It

is,

rather, a

it is

in his lifetime use

of the material studied.

own

it is

all,

Each composer

those portions which appeal to his

compendium

inclusive of

basic relationships within the twelve-tones,

any composer would

scales.

passionate plea that this text not be con-

all

of the

hardly likely that

or even a large part,

will,

rather, use only

esthetic taste

and which

own creative needs. Complexity is no guarantee 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

contribute to his of excellence,

mode I

and

of procedure, namely, that the student study carefully Parts II,

Chapters

exercises— although will enlighten

During the

to 16, without undertaking the creative

I

if

there

is

and inform the first

sufficient

time the creative exercises

theorist as well as the composer.

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 Professor

his

deep debt of gratitude

many

his

help-

manuEastman School of Music Wayne Barlow, Allen Irvine McHose, Charles Riker,

and

faculty,

and

for his meticulous reading of a difficult

to his colleagues of the

and Robert Sutton, also

acknowledge

Herbert Inch of Hunter College for

ful suggestions script,

to

for valuable criticism.

His appreciation

is

extended to Clarence Hall for the duplication of the chart,

to Carl A.

Rosenthal for his painstaking reproduction of the

examples, and to their

Mary Louise Creegan and

Janice Daggett for

devoted help in the preparation of the manuscript.

His

warm

generous finally

thanks go to the various music publishers for their

permission

and

especially

to

quote to

from

works

copyrighted

Appleton-Century-Crofts

for

and their

co-operation and for their great patience. Finally,

students

my

devoted thanks go to

who have borne with me

my

hundreds of composition

so loyally all these

many

years.

H. H.

Contents

Preface

vu

1.

Equal Temperament

1

2.

The Analysis of Intervals The Theory of Involution

7

3.

Part

I.

17

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 Key Modulation

56

7.

60

Minor Second

8.

Projection of the

9.

Projection of the Major Second

10.

Projection of the Major Second

11.

Projection of the Minor Third

65 77

Beyond the Six-Tone

Series

90 97

12.

Involution of the Six-Tone Minor-Third Projection

13.

Projection of the

14.

Projection of

15.

Projection of the Major Third

16.

Recapitulation of the Triad Forms

17.

Projection of the Tritone

18.

Projection of the Perfect-Fifth-Tritone Series

19.

The pmn-Tritone

20.

Involution of the pmn-Tritone Projection

158

21.

Recapitulation of the Tetrad Forms

161

Six

110

Minor Third Beyond the Six-Tone the Major Third

Series

Beyond the Six-Tone

Series

118 123 132 136 139

Beyond

Tones

148 151

Projection

xiii

CONTENTS

CONSTRUCTION OF HEXADS BY THE SUPERPOSITION OF TRIAD FORMS Part

II.

pmn

22.

Projection of the Triad

23.

Projection of the Triad pns

24.

Projection of the Triad

177

25.

Projection of the

182

26.

Projection of the Triad nsd

167 172

pmd Triad mnd

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

31.

Simultaneous Projection of the Perfect Fifth and Major Third 211

32.

Simultaneous Projection of the Major Third and Minor

Second

207

Second 33.

215

Simultaneous Projection of the Perfect Fifth and Minor

219

Second

Part IV.

PROJECTION BY INVOLUTION AND AT FOREIGN INTERVALS

Projection

225

35.

by Involution Major-Second Hexads with Foreign Tone

36.

Projection of Triads at Foreign Intervals

236

37.

Recapitulation of Pentad Forms

241

34.

Part V.

38. 39.

THE THEORY OF COMPLEMENTARY SONORITIES

The Complementary Hexad The Hexad "Quartets" xiv

232

249 254

CONTENTS

COMPLEMENTARY SCALES

Part VI.

40.

Expansion of the Complementary-Scale Theory

4L 42.

Projection of the Six Basic Series with Their Complementary Sonorities 274 Projection of the Triad Forms with Their Complementary

43.

The pmn-Tritone

44.

Projection of

263

285

Sonorities

with

Projection

Its

Complementary 294

Sonorities

Two

Similar Intervals at a Foreign Interval

with Complementary Sonorities 45.

46. 47.

48.

Simultaneous

Projection

of

298

with

Intervals

Their

303 Complementary Sonorities 314 Projection by Involution with Complementary Sonorities 331 The "Maverick" Sonority Vertical Projection by Involution and Complementary 335

Relationship 49.

Relationship of Tones in Equal

346

50.

Translation of Symbolism into

Temperament Sound

356

Appendix: Symmetrical Twelve-Tone Forms

373

Index

377

The Projection and Equal Temperament Chart:

Interrelation of Sonorities in

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

that a study confined to equal

first is

though complex, a

-finite

retical

within just intonation

A

possibilities

temperament

would be

simple example will illustrate this point.

major

third, E,

G#, above E,

al-

is,

study, whereas a study of the theo-

If

we

infinite.

construct a

above C, and superimpose a second major

we produce

the sonority C-E-G#i

superimpose yet another major third above the tone B#. In equal temperament, however, equivalent of C, and the four-tone sonority

GJj:,

Now we

third,

we

if

reach the

B# is the enharmonic 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,

the equivalent of C.

A

B# would not be

projection of major thirds above

C

would therefore approach infinity. The second reason is a- corollary of the first. Because

in

just intonation

pitches

possible

intonation

ments or

is

for

in

just

intonation

approach

infinity,

the just

not a practical possibility for keyboard instru-

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 and since it is unlikely that keyboard, keyed, and valve instruments will be done away with, o£ these resources simultaneously,

at least within the generation of living composers, the system

of equal

temperament

Another

is

advantage

the logical basis for our study.

of

equal

temperament

simplicity possible in the symbolism

is

the

greater

of the pitches involved.

Because enharmonic equivalents indicate the same pitch, possible to concentrate

upon the sound

than upon the complexity of

its

it

of the sonority rather

spelling.

Referring again to the example already cited,

if

we were

find ourselves involved in endless complexity.

to

we

continue to superimpose major thirds in just intonation

would soon

is

The

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 temperament, 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

major third above

first

three.

Example Pure Temperament

"% !

1-1

Equal Temperament

]

ip"

)

This point of view has the advantage of freeing the composer

from certain inhibiting preoccupations with academic symbolization as such.

For the composer, the important matter

sound of the notes, not

G-B-D-F sounds

their "spelling."

like a

is

the

For example, the sonority

dominant seventh chord whether

it

is

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 spelled

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 fifth,

since

embraces

all

the

may

represent the interval of a perfect

superposition

of

twelve

perfect

also

fifths

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,

the circle at

BJf,

which

in equal

as C. Similarly, the pitch

names

we

complete

temperament has the same pitch of

C# and D^, D# and

Ej^,

and

so forth, are interchangeable.

Example

1-2

GttlAb)

D« (Eb

MK

The term

sonority

of tone relationship,

When we

is

used in

whether

this

book

in terms of

to cover the entire field

melody or of harmony.

speak of G-B-D-F, for example,

we mean

ship of those tones used either as tones of a

harmony. This

may seem

(Bb)

the relation-

melody

to indicate a too easy fusion of

or of a

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 consecutively, as in Example l-3b:

HARMONIC MATERIALS OF MODERN MUSIC

Example

1-3

(fl)

i The of

first

^

problem

component

its

in the analysis of a sonority

A

parts.

sonority sounds as

it

is

the analysis

does primarily

because of the relative degree of consonance and dissonance of elements, the position and order of those elements in relation

its

to the tones of the clarity in tion,

harmonic

series,

the degree of acoustical

terms of the doubling of tones, timbre of the orchestra-

and the

like.

further affected

It is

which the sonority

is

by the environment

in

placed and by the manner in which

experience has conditioned the ears of the listener.

Of these

factors, the first

would seem

to

be

For example,

basic.

the most important aural fact about the familiar sonority of the

dominant seventh thirds than of

that

any other

of the perfect fifth of the

is

it

contains a greater

number

interval. It contains also the

of

minor

consonances

and the major third and the mild dissonances

minor seventh and the

tritone.

This

is,

so to speak, the

chemical analysis of the sonority.

Example

Minor thirds

It

is

of

f

Perfect

fifth

1-4

Mojor third

paramount importance

to the

Minor seventh

Tritone

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

a scholar and a craftsman.

all,

even though he

may be both

EQUAL TEMPERAMENT This does not imply a lack of importance of the secondary analyses already referred

The

to.

various styles and periods,

in

tonality

implied— and the

is

analyses strengthen the

historic position of a sonority

its

function in tonality— where

Such multiple

like are important.

young composer's grasp

of his material,

providing always that they do not obscure the fundamental analysis of the

sound as sound.

we

Referring again to the sonority G-B-D-F,

should note

historic position in the counterpoint of the sixteenth century its

harmonic

position

in

the

tonality

eighteenth, and nineteenth centuries, but

observe

its

construction, the elements of

of these analyses are important

the

we

should

and contribute

and

seventeenth,

of

which

its

it is

first

of all

formed. All

to an understand-

ing of harmonic and melodic vocabulary.

As another example of multiple

analysis, let us take the familiar

contains two perfect

chord C-E-G-B.

It

one minor

and one major seventh.

third,

Example

Perfect

It

may be

fifths

fifths,

two major

thirds,

1-5

Mojor thirds

Minor third

*

Major seventh

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

and the minor

triad

E-G-B

or the triads*

Example

C-G-B and C-E-B:

1-6

ofijiij^ij ii i

*The word

triad

is

used

to

mean any

C-E-G

three-tone chord.

HARMONIC MATERIALS OF MODERN MUSIC Historically, ties

it

represents one of the important dissonant sonori-

of the baroque and

may be

classic periods. Its function in tonality

subdominant or tonic seventh of the major

scale,

the mediant or submediant seventh of the "natural" minor

scale,

and

as the

so forth.

Using the pattern of analysis employed

and

1-6,

Examples

1-4, 1-5,

analyze as completely as possible the following sonorities

Example 4.

i

in

ji8

ijia^

1%

5.

^

1-7 e.

±fit ift

7.

9.

=^Iia^itftt«^

10.

i

«sp

The Analysis of

Intervals

In order again to reduce a problem of theoretically proportions to a finite problem, an additional device

is

infinite

suggested.

Let us take as an example the intervallic analysis of the major triad

C-E-G:

Example

Perfect fifth

This triad

is

Major

commonly described

combination of a perfect

fifth

analysis

is

incomplete, since

as long as the triad

however, the chord

is is

third

Minor third

in conventional analysis as a

and a major third above the lowest

or "generating" tone of the triad. It

the minor third between

2-1

it

obvious, however, that this

is

omits the concomitant interval of

E and

G. This completes the analysis

in the simple

form represented above.

present in a form in which there are

If,

many

doublings in several octaves, such a complete analysis becomes

more complex. If

we examine

Transfiguration

the scoring of the final chord in Death and

by Richard

Strauss

Example

:i

* *

^m

we 2-2

find a sixteen -tone chord:

HARMONIC MATERIALS OF MODERN MUSIC These sixteen tones combine to form one hundred and twenty

The

different intervals.

between

relationship

C and G

is

repre-

sented not only by the intervals

Example

2-3

eta

r

J J

U^J JuJ

Ij

jiiJ

3-10

Ji'^

i

j^jjg^J jj i

Ji'-^t

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

t

^^rrrr

IJJ

i

juJJf

J ^ ^ i;ii.JtJbJ

(j^^jiJ^ri|J^^rr Note:

We

ijJjtJ

U|J

i

J

^

^^^

jjJ^^

i^^^^^UJ^tJ^P

Uj jj^^*^ i>J^^ i

''^^rrri>ji>J^^^

have defined an isometric sonority

as

one which

has the same order of intervals regardless of the direction of projection.

The student should note

character of a sonority

is

that

this

bidirectional

not always immediately evident. For

example, the perfect-fifth pentad in the position C2D2E3G2A3(C)

does not at position

apparent.

24

first

glance seem to be isometric.

D2E3G2A3C2(D),

its

isometric

However

character

is

in the

readily

1^

:

PartJ

THE SIX BASIC TONAL SERIES

4

Projection of the Perfect Fifth

We if

have seen that there are

we

types of interval relationship,

consider such relationship both "up" and "down": the

perfect fifth and

and

six

its

the minor sixth;

inversion,

its

inversion, the perfect fourth; the major third

inversion, the major

the minor third and

major second and

sixth; the

minor seventh; the minor second and seventh; and the tritone,— the fifth— which

and

t,

we

its

its

inversion, the

inversion, the major

augmented fourth by the letters,

are symbolizing

its

or diminished p,

m,

n,

s,

d,

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

most

sonorities fall into

fifth types,

There

is

one of the

intervals. In other words,

six great categories: perfect-

major-third types, minor-third types, and so forth.

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

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

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 is,

TONAL

SIX BASIC

SERIES

therefore, in terms of the projection of each of the six basic

intervals discussed in

Chapter

By

2.

"projection"

we mean

the

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 consistency by superimposing one above the other until all of the tones of the equally tempered scale have been used. These two building of sonorities or scales

consider

first

We

and the minor second.

are, of course, the perfect fifth

shall

the perfect-fifth projection.

we add

Beginning with the tone C,

and then the perfect

fifth,

fifth,

G,

D, to produce the triad C-G-D

or,

first

the perfect

reduced to the compass of an octave, C-D-G- This triad contains,

two fifths, the concomitant may be analyzed as ph.

in addition to the

major second.

It

Example

m

4-1

Perfect Fifth Triad,

^^ 2

The

p^

5

tetrad adds the fifth above D, or A, to produce

contains three perfect

time in

fifths,

this series— a

C-G-D-A,

C-D-G-A. This sonority

or reduced to the compass of the octave,

first

interval of the

two major seconds, and— for the

minor

third,

Example

A

to C,

4-2

Perfect FifthTetrad.p^ns^

m

^^ 2

The analysis is, therefore, p^ns^. The pentad adds the next C-G-D-A-E, or the melodic

5

2

fifth,

E,

forming the sonority

C-D-E-G-A, which

will

be

recognized as the most familiar of the pentatonic scales.

Its

components are four perfect 28

scale

fifths,

three major seconds, two

PROJECTION OF THE PERFECT FIFTH

minor

thirds,

and— for

the

time— a major

first

third.

The

analysis

therefore, p^mnh^.

is,

Example Perfect

4-3

Fifth Pentad,

p^mn^s^

i .

S

^^ o

2

2

The hexad adds

B,

2

3

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

its

components being

2

3

2

2

five perfect fifths, four

major seconds, three

two major thirds, and— for the first time— the dissonant minor second (or major seventh), p^m^n^s'^d. minor

thirds,

The heptad adds F#:

Example Perfect Fifth

i a

Heptod.p^m^n^s^d^t

^^ '2

2

4-5

2

^ '

•I I

2

2

29

)

THE producing the

first

TONAL

SIX BASIC

scale

which

in

its

SERIES

melodic projection contains

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), interval larger than a major

no

C

to

This sonority contains

FJf.

six perfect

seconds, four minor thirds, three major thirds,

and one is

the

tritone: p^m^n'^s^dH. (It will

first

sonority to contain

The octad adds

fifths,

five

major

two minor seconds,

be noted that the heptad

of the six basic intervals.

all

Cfl::

Example

4-6

Perfect Fifth Octod.

p^m^ n ^s^ d^

t^

Am

«5i=

5

12 Its

components are seven perfect

minor

thirds,

12

2

2

fifths, six

major seconds,

five

four major thirds, four minor seconds, and two

tritones: p^m'^n^s^dH^.

The nonad adds G#:

Example

m—

4-7

Nonad, p^m^n^s^d^t^

Perfect Fifth

J^

m

m iff

Its

I

?

components are eight perfect

minor

thirds,

six

major

tritones: p^m^n^s^dH^.

30

thirds,

=

9

fifths,

six

seven major seconds,

six

minor seconds, and three

PROJECTION OF THE PERFECT FIFTH

The decad adds D#:

Example «*!" Perfect

u

-

4-8

Decad, p^m^n^s^d^t^

Fifth

^^

m IT"

Its

I

I

I

I

I

I

components are nine perfect

O I

I

2

eight major seconds, eight

fifths,

and four

minor thirds, eight major thirds, eight minor seconds, tritones: 'p^m^n^s^dH'^.

The undecad adds A#:

Example

4-9

Undecad p'°m'°n'°s'°d'°t^

? s"** Perfect Fifth

,

Isjf

^^

m 1*"^ I

Its

r

components are ten perfect

minor

thirds,

2

I

I

fifths,

II

I

I

ten major seconds, ten

ten major thirds, ten minor seconds, and five

tritones: p^'^m'V^s'Od/'^f^

The duodecad adds the

E#:

last tone,

Example

A^

I s

4-10 I2_l2j2„l2jl2^6

Perfect Fifth Duodecad, p'^m'^n'^s'^d

V^

r

I

r

I

I

I

I

I

I

I

I

31

)

:

THE Its

TONAL

SIX BASIC

components are twelve perfect

SERIES

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 components of the

intervallic

carefully the progression 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

should be noted. The

first

is

on perfect

fifths,

several points

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

whether or not

major second. this

is

(It

interesting to speculate as to

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,

The its

last of all,

third observation

esthetic implications.

related

by the

sonority, there

32

•

is

the tritone.

of the greatest importance because of

From

the

first

sonority of

interval of the perfect fifth, is

up

two

tones,

to the seven-tone

a steady and regular progression.

Each new

PROJECTION OF THE PERFECT FIFTH

new

tone adds one

interval, in addition to

adding one more to

each of the intervals already present. However, when the projection

beyond seven

carried

is

added. In addition to this

no new

tones,

intervals

any new material, there

loss of

can be is

also

a gradual decrease in the difference of the quantitative formation

same number

of the sonority. In the octad there are the

of major

thirds and minor seconds. In the nonad the number of major thirds,

minor

and minor seconds

thirds,

contains an equal

number

is

the same.

of major thirds,

When

seconds, and minor seconds. sonorities are reached, there

is

no

minor

The decad major

thirds,

the eleven- and twelve-tone

differentiation whatsoever, ex-

number of tritones.* The sound of a sonority— either as harmony or melodydepends not only upon what is present, but equally upon what is absent. The pentatonic scale in the perfect-fifth series sounds as cept in the

it

does not only because

fifths

and because

second or the

On

contains a preponderance of perfect

of the presence of major seconds,

and the major third also because

it

minor

thirds,

in a regularly decreasing progression, but

does not contain either the dissonant minor

it

tritone.

the other hand, as sonorities are projected beyond the

six-tone series they tend to lose their individuality. All seven-tone

example, contain

series, for

difference

in

their

all

of the six basic intervals,

proportion

decreases

as

and the

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

any of the projections which * See

page 139 and

we have

if

considered are begun on

140.

33

THE

SIX BASIC

TONAL

SERIES

the final tone of that projection and constructed

same

resultant scale will be the

The seven-tone

scale

as

C2D2E2F#iG2A2B,

the

same

Every scale may have

F+f— and projected

is,

tones: J,F#2E2D2CiB2A2G.

many

as

versions of

The seven-tone

there are tones in the scale.

begun on

for example,

the final tone of the projected fifths— that

downward produces

downward, the

the projection were upward.

if

basic order as

its

example,

scale, for

has seven versions, beginning on C, on D, on E, and so forth.

Example Seven "versions"

i 2

2

Perfect Fifth Heptad

of the

^^ o 2

rtn*

*^

o^^ »

v>

2

2

2

2

(1)

*^ =0^5 O*

2

2

1

2

2

2 (2)

2

2

I

^f

2

2

2

^ 2

(2)

;x4^M

^"*

I

^\ 2

I

=^33 bcsr^ 3s:«i

4-11

(I)

2

2

12

2

(2)

i^

_Ql

:^=KS :&:xsi 2

f^o^

2

(2)

(-C^)

v^g>

#

2 2

2

2

(2)

The student should tion

and the

distinguish carefully

different versions of the

same

between an involu-

An

scale.

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 pentatonic scale on C,

C-D-E-G-A, contains a major

minor triad on A. The six-tone triad

on

G

and the minor

nine-, ten-, eleven-

triad

triad

perfect-fifth scale

on

C

adds the major

on E. Analyze the seven-,

and twelve-tone

and a eight-,

scales of the perfect-fifth

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.

projection

34

PROJECTION OF THE PERFECT FIFTH Since the perfect-fifth projection includes the most famihar scales in occidental music,

The most provocative

innumerable examples are available.

of these

would seem

to

be those which

produce the greatest impact with the smallest amount of tonal

To illustrate the economical use of material, one can no better example than the principal theme of Beethoven's overture, Leonore, No. 3. The first eight measures use only the

material. find

five tones of the perfect-fifth projection:

first

next measure adds

F and

B,

C-D-E-G-A. The

which completes the tonal material

of the theme.

Example

4-12

wm ^^ ^^

Beethoven, Overture, Leonore No.3

*

m

^

i

o

jj i' i

In the same way. Ravel uses the fifth

projection

building to the Suite No.

G-D-A-E-B— or, first

in

^^

first five

tones of the perfect-

melodic form, E-G-A-B-D— in

climax in the opening of Daphnis and Chloe,

2.

Example

4-13

Ravel, Daphnis end Chloe

Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Philadelphia, Pa., agents.

The

principal

theme

of the last

movent ent of the Beethoven

Symphony is only slightly less economical in its use of material. The first six measures use only the pentatonic scale Fifth

C-D-E-F-G, and the seventh measure adds Beethoven, Symphony No. 5

Example

A

and

B.

4-14

35

8

THE

SIX BASIC

TONAL

However, even Beethoven with

SERIES

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

F major

scale in the

first

four measures but reaches

^r an

seven-tone perfect-fifth scale perfect fifth above

E

)

tones F-G-A-B^-C-E of the

six

beyond the

additional tone,

Bt]

(the

in the fifth measure.

Example

4-15

Beethoven, Symphony No.

f T^^r gu i

^

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

adding the

36

fifth tone,

the sixth tone, and so forth.

by

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

measure adds

sixth

which forms the

Bt],

perfect-

pentad G-D-A-E-B. The following measure adds a C#,

fifth

forming the hexad G-A-B-Cj|-D-E. This hexad departs momen-

from the pure

tarily

perfect-fifth projection, since

a combina-

and major-second projection— G-D-A-E-B

tion of a perfect-fifth

+

it is

G-A-B-C#.

Measure 11 substitutes a tutes a

which

Bb is

for the

C# and measure

12 substi-

forming the hexad G2A1BI72C2D2E

the involution of the previous hexad G2A2B2C#iD2E.

Measure 13 adds an scale

C

for the previous B,

F, establishing the seven-tone perfect-fifth

Bb-F-C-G-D-A-E.

Continue

type

this

determining

how much

of

analysis

to

of the section

is

rehearsal

number

7,

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

Stravinsky the

first

in

the beginning of

Symphony

all

movements of the movement, for example,

three

in C. In 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

downward by the addition of the Fki which both F and Ffl: are integral parts of fifth

in the violas, after

the scale. Note the

scale passage in the trumpet:

Example Stravinsky, Symphony

in

C

Copyright 1948 by Schott

&

Co., Ltd.; used

by permission

4-16

of Associated

Music Publishers,

Inc.,

New

37

York.

THE

SIX BASIC

TONAL

SERIES

theme from the first movement of the Sixth Symphony may be analyzed as the expansion of

Similarly, the following ProkofieflF

the perfect-fifth projection to nine tones:

Example Prokotieff,

©

4-17

Symphony No. 6

1949 by Leeds Music Corporation, 322 West 48th

St.,

New

York 36, N.

Y.

Reprinted by permission;

all

rights reserved.

ft-

i m Even when

all of

the tones of the chromatic scale are used, the

formation of individual sonorities frequently indicates a simpler

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 basic structure

measure, however,

is

unmistakably of perfect-fifth construction:

Example

4-18

Albon Berg, Lyrische Suite

Copyright 1927 by Universal Editions, Vienna; renewed 1954 by Helene Berg; used by permission of Associated Music Publishers, Inc.,

38

New York.

PROJECTION OF THE PERFECT FIFTH

Analyze the

first

movement

and determine how much

of the Stravinsky

of

it

is

Symphony

in

C

written in the perfect-fifth

projection.

In

any

analysis,

constructed, that

is,

always try to discover

how much

how

the

work

is

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

diflPerent patterns.

39

and

Harmonic-Melodic Material

Hexad

of the Perfect-Fifth

Since, as has been previously stated, of the six basic intervals,

all

and

all

seven-tone scales contain

since, as additional tones are

added, the resulting scales become increasingly similar in their

component

different types of tone relationship hes in the six-tone

which

tions,

We

study of

parts, the student's best opportunity for the

offer the greatest

number

combina-

of different scale types.

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 manageable quantity, is,

we

we

shall disregard the question of inversions.

shall consider

C-E-G a major

fundamental position— C-E-G; in its

triad

its first

whether

C-D-E-G-A

we

fifths,

its

shall consider

as one type of sonority, that

sonority built of four perfect

That in

inversion— E-G-C; or in

second inversion— G-C-E. In the same way,

the pentad

is

it

regardless of whether

is,

as a

its

form

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

is

temperament

to

be equally

major triad whether

it is

valid.

We

shall consider

spelled C-E-G, C-F^-G, B#-E-G, or in

some other manner. Examining the harmonic-melodic components fifth

hexad,

These are 1.

The 40

we

find that

C-E-G a

it

of the perfect-

contains six types of triad formation.

in order of their appearance:

basic triad C2D5G, p^s, consisting of

two superimposed

:

1

THE PERFECT-FIFTH HEXAD perfect

with the concomitant major second, which

fifths

is

dupHcated on G, D, and A:

Example Perfect

2

2.

3

2

The

Perfect Fifth Triads

Hexad

Fifth

2

2

2

2

5

.

triads are

duphcated on

Example pns

and

7

The

5

G and on D

5-2

2

7

2

7

2

(involution)

(involution)

3.

2

involutions

,27^

2

5

major second, and a major sixth (or

consists of a perfect fifth, a

minor third ) These

7

2

5

C7G2A, pns, with the involution C2D7A, which

triad

Triad

5-1

(involution)

C4E3G, pmn, with the involution A3C4E, which

triad

consists of a perfect fifth, a

major

third,

and a minor

ing the familiar major and minor triads.

third,

The major

form-

triad

is

duplicated on G, and the minor triad on E:

Example ^

h \

-i J ^

1

J4

4.

The

pmn

Triad

triad

3

5-3

and involu fions

:

1

J

1

• ;i> 4 3

'

—

J

r

r

-

\

4

3

—mJ —Js—r

:

1

3

u

II

4

C7G4B, pmd, with the involution C4E7B,

ing of the perfect

fifth,

consist-

major seventh (minor second), and

major third:

Example Triad

pmd and

5-4 invoiution

* i-H^^t-h-t7

4

4

7

41

— THE 5.

The

triad

SIX BASIC

TONAL

C2D2E, ms^, which

SERIES

which

triad,

is

Triad

2

consists

an isometric

5-5

ms^

^

•

-S-

The

third,

reproduced on G:

Example

6.

two superimposed

consists of

major seconds with the concomitant major

2

2

2

BiCoD, nsd, with the involution A2B1C, which of a minor third, a major second, and a minor second: triad

Example Triad nsd and

I^

5-6 involution

^

,2 (involution) .

I.

The tetrads of the perfect-fifth hexad consist of seven types. The first is the basic tetrad C2DgG2A, p^ns^, aheady discussed duphcated on

in the previous chapter,

Example

i The second

is

2

2

5

2

5

and D:

5-7 p^ns^

Fifth Tetrads

Perfect

G

2

2

5

2

the tetrad C2D2E3G, also duplicated on

G

(G2A2B3D), and the involutions A3C2D2E and E3G2A2B. This

two perfect fifths, two major seconds, one major and one minor third: p^mns^.

tetrad contains third,

Example Tetrads

2 p

223 42

mns

2

*3

5-8

and involutions

22

(involution)

223

,32

2,

(involution)

.

)

THE PERFECT-FIFTH HEXAD one of the most consonant of the tetrads, containing no

It is

strong dissonance and no tritone. Not only does

equal number of perfect the

it

contain an

and major seconds, but

fifths

it is

also

example of the simultaneous projection of two different

first

above the same tone, since

it consists of the two perfect 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

intervals

C

above

fifths

plus the

formations will be discussed in Part

Example

m^ Tetrad

The

p^mns^ as p^+s^

+

fe

2

may

involutions

5-9

i^^

i

s'

III.

2

3

be considered

also

simultaneous projection of two perfect

seconds downward, that

+

is

J J r r

p3

32

to

be formed by the

2

2

and two major J,E-D-C: and J,B-E-A fifths

jB-A-G:

Example

Jj

II ;

The also a

perfect to

I

p2

third

IT =223 I

+

s2

2

p2

i

the tetrad C4E3G2A, duphcated on

is

G

2

3

(G4B3D2E),

predominantly consonant tetrad, which consists of two

C

fifths,

is

G

to

G; the major

p^mnrs. This

E and

5-10

^g^ IT^^ iTt ^ ^^

Involution

E

+

J,E-A-D

?

+

A to E; two minor thirds, A to C and C to E; and the major second, G to A:

and

third,

an isometric tetrad

since,

if

we

begin on the tone

form the same tetrad downward, J^E4C3A2G,

we produce

the identical tones:

Example Tetrads

m

4

3

p^m

2

^

n^

5-11

s.

4

3

J

.11 2

(Isometric involution)

r

r

1

4

3

r

I

^

4

3

2

(isometric involution)

43

:

THE It

may be

TONAL

SIX BASIC

SERIES

considered to be formed of the relationship of two

perfect fifths at the interval of the minor third, indicated

symbol p perfect

@

fifth,

n; or of

indicated

two minor thirds at the by the symbol n @ p:

Example

@

p

It

contains the major triad

n

43

the involution A3C4E;

5-13

m. 34 + involution

C7G2A, pns, with the involution G2A7E

Example

m tetrad,

5-14

J:j

J

7 pns

The fourth

5-12

C4E3G and

)mn

triad

interval of the

il@_P

Example

and the

by the

C4E3G4B,

J

7 + involution

2

2

is

we begin downward, we produce

also isometric, since if

on the tone B and form the same tetrad the identical tones, IB4G3E4C:

Example p^m^n

Tetrad

5-15 d

ij 434 434 r^T^jj^ •'

(isometric involution)

It is

a more dissonant chord than those already discussed, for

contains two perfect

44

fifths,

C

to

G

and E

to B;

two major

it

thirds,

:

:

THE PERFECT-FIFTH HEXAD

C

E and G

to

to B;

one minor

E

third,

C

major seventh (or minor second),

to B: p^m^nd. It

considered to be formed of two perfect relationship of the major third,

C

and the dissonant

to G;

fifths

to G, plus

E

major thirds at the relationship of the perfect

G

to B; or of

C

fifth,

to

E

two plus

to B:

Example

ii @ contains the major triad

5-16

UE @

J

m

p

It

may be

at the interval

P

C4E3G and the

involution, the

minor

E3G4B;

triad

Example

ji^^4 J •^

triad

ij

j

^ 3

3

pmn

and the

5-17

+

r

4'

involution

C7G4B, pmd, and the involution C4E7B

Example

5-18

J j,^r ^i ^4 7 r 7 4 pmd

The fifths,

may

fifth

C

also

to

+

involution

tetrad C2D5G4B, p^mnsd, consists of

G

and

G

to

be considered

fourth above, or

fifth

D, with the dissonance, B. This tetrad

as the

major triad G-B-D with the added

below, G, that

tetrads of this projection

two perfect

is,

C. It

which contains

is

all of

the

first

of the

the intervals of

the parent hexad.

Together with

which

this

consists of the

tetrad

is

found the involution C4E5A2B,

minor triad A-C-E with the perfect

above, or the perfect fourth below, E, namely,

B 45

fifth

:

THE

TONAL

SIX BASIC

Example p^mnsd

iTetrad

5-19

and involution

j^'jiUJ ^



three

above

E-F#-G#-B-C#

"

four //

Ab-Bb-C-Eb-F

below minor second above

Db-Eb-F-Ab-Bb

"

n

//

A-B-Cif-E-F#

" II

" //

//

all

new

tones

(all

new

tones)

below above

tritone

or

below

gives

V%-G%-A%-C%-D%

Example

7-1

Modulation Perfect Fifth Pentad

to

Perfect Fifth above

Major Second above

to

*

o o

^

Modulation to Perfect Fifth

below

to Major

Second below

-

to Minor Third

i

^ to

i to

*

to

Minor Third below

^

Major Third above

"

b,:

17»-

to

Minor Second obove

|;>

to Major Third below

^^%* ° f'

*

to

1^

^^

'

Minor Second below

i*

>

ff*

Augmented Fourth above It.

*• %- i' ^'

Augmented Fourth below

to

i

above

^

,

_? —

\,-9-

I

L--

"^

!;•

ty

o

*

61

*'

THE

The student should though there

will

TONAL

SIX BASIC

SERIES

learn to distinguish as clearly as possible—

be debatable instances— between,

(1) a modulation from the pentatonic scale

for example,

C-D-E-G-A

to the

pentatonic scale A-B-CJj:-E-Ffl:, and (2) the eight-tone perfectfifth scale,

of

C-C#-D-E-F#-G-A-B, which contains

both pentatonic

In

scales.

the

all

of the tones

former instance, the two

pentatonic scales preserve their identity and there

is

a clear point

which the modulation from one to the other occurs. In the 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. at

latter case, all of the eight tones

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

I i

7-3

^ti ^^^ r

Although modal modulation

is

the most subtle and delicate

form of modulation, of particular importance poser in an age in which

it

entire tonal palette at the listener, to

to the

young com-

seems to be the fashion to throw the

the tonal fabric. This task

is

it

does not add

new

material

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 might— at least major keys and

to the pattern.

in

A

composer of the

classic period

theory— modulate freely to any of the twelve

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

made

formal design— or might be

rapidly and restlessly within

the fabric of the structure. In either case, the general impression of a "major key" tonal structure could

This same device

is

equally applicable to any form of the

perfect-fifth projection, or to

The

principle

is

be preserved.

the same.

any of the more exotic scale forms.

The composer may choose the

pattern which he wishes to follow and cling to

he

may

in the process

modulate

it,

tonal

even though

one of the twelve

to every

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

same scale on E (E-F#-G#-B-CJj:). Now moduon 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

modulate

to the

late to the scale

,

key center of C.

You

will observe that the first

modulation— C

to

E— retains

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 modulation, 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. only one

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

require

will

some

ingenuity

on

D

your

as the

part

key center.

to

make

It

this

sound convincing.

Work

out the modulations of the perfect-fifth hexad at the

intervals of the perfect fifth, third,

minor second and

64

major second, minor

tritone, as in

Example

7-1.

third,

major

8

Minor Second

Projection of the

There

is

only one

interval, in addition to the perfect fifth,

which, projected above

itself,

twelve-tone scale. This

is,

gives

of the

all

of course, the

tones of the

minor second, or

its

inversion, the major seventh.

Proceeding, therefore, as in the case of the perfect-fifth projection,

we may

superimpose one minor second upon another,

proceeding from the two-tone to the twelve-tone

Examining the minor-second triad

C-C#-D

C-D:

s(P.

The

series,

we

series.

observe that the basic

contains two minor seconds and the major second

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 tetrad,

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^:

Example Minor Second Triad

8-1

Minor Second Tetrad

sd^

ns^d^

^

t^ 2.3^4 mn'^s d

Minor Second Pentad

i

Minor Second Hexad

I

^

yes

"X5 I

pm^n^s^d^

I

I

I

65

THE

SIX BASIC

TONAL

SERIES

The seven-, eight-, nine-, ten-, eleven- and twelve-tone minorsecond 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 adds one new interval,

from the two-tone

to the seven-tone scale

been reached no new interbe added. Furthermore, from the seven-tone to the

after the seven-tone projection has

vals can

eleven-tone projection, the quantitative diff^erence in the propor-

new

tion of intervals also decreases progressively as each is

tone

added.

Example p^^n'^s^d^t

Minor Second Heptad

I

I

I

I

I

r

I

m

I

I

I

I

"^j^o^o o

III

I

Minor Second Undecad p

Octod p'^m^n^s^d^t^

Minor Second Decad

^ v»jtoO^^^»tt« I

Minor Second

I

MinorSeoond Nonad p^m^n^s^d^^

I

8-2

n s d

t

I

I

p^m^n^s^d^t'*

t.^t^^e^f^

III

Minor Second Duodecod p

m

n

s

d

t

^^ojto°"*"°1t°"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., Philadelphia, Pa., agents.

Whereas the minor-second hexad may not be 84

as

bad

as

it

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

improvisors. Because of the homogeneity of

its

casual

material,

it

is

often used in the most obvious manner, which destroys the subtle nuances of

which

capable and substitutes a "glob" of

it is

"tone color."

The author tone scale in this scale

is

its

not making a plea for the return of the whole-

unadulterated form, but

it

must be

said that

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) etc.

i ^ ^r

'^/^^ (h)

(c) etc.

titijt.

^4

^^r

In Example 9-2 la, experiment with the triad types in various

Repeat the same process for the

positions.

tetrads, as in 21b; for

the pentad, as in 21c; for the hexad, as in 21d.

Example

9-21

(a)

i

i

r

F

m ^^^ 86

K

4

J

^^ f^

^

PROJECTION OF THE MAJOR SECOND (b)

(hi i

i\l

'}

f^f

f

«hi

i \

i

J

/h^^^

^ ^^

"F

^fe

(c)

i

ii ^i

'>t

ile

itJ

^tl:^^b*)b^

w^ 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

A somewhat

SERIES

more complicated illustration Alban Berg song, "Nacht," already referred to the pure whole-tone scale:

Example

found in the

is

as

beginning in

10-12

Albon Berg, "Nacht

Copyright 1928 by Universal Editions, Vienna; renewed 1956 by Helene Berg; used by permission of Associated Music Publishers, Inc.

m

,i'°,^'

The student should now be ready

ii

",b»(it.^

to write a free improvisatory

sketch employing the materials of this scale (Example 10-1). will notice that the scale has

C

major and one on

G

two natural resting

He

points,

one on

G

minor,

minor. Begin the sketch in

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

much

C-D-E-Ff-G-Ab-Bb are employed

in this sketch, but get as

variety as possible from the harmonic-melodic material

of the scale.

96

11

Projection of the

The next

Minor Third

which we

series of projections

shall consider

is

the

C we

projection of the minor third. Beginning with the tone

superimpose the minor third E^, then the minor third G^, forming the diminished triad CgEbsGb, which consists of two minor thirds

and the concomitant

tritone,

from

C

to G^.

Upon this we we shall call

superimpose the minor third above G^, B^^, which

by

its

enharmonic equivalent, A, forming the familiar tetrad of

the "diminished seventh," consisting of four minor thirds:

Eb,

Eb

to Gb,

Gb and Eb

Gb

to

Bbb (A), and

A

to C;

and two

tritones:

C C

to

to

to A; symbol, nH^\

Example

11-1

Minor Third Tetrad

i

i.

o

^o

u^\^

^^^C^-)

3

As

in the case of the

major-second

projected in pure form beyond

scale,

six tones,

which could not be so the minor third

cannot be projected in pure form beyond four tones, since the next minor third above

wish to extend

A

duplicates the starting tone, C. If

this projection

beyond four tones we must,

introduce an arbitrary foreign tone, such as the perfect

and begin a new

series

of minor-third projections

we

again,

fifth,

G,

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

The minor-third pentad,

TONAL

therefore,

Example

SERIES

becomes C3Eb3GbiGt]2A:

11-2

pmn^sdt^

Minor Third Pentad

>obo "h*

jJt^^tjJ 3

3

It contains, in

^

12

addition to the four minor thirds and two tritones

aheady noted, the perfect the major second,

G

analysis of the scale

fifth,

C

to G; the

therefore, pmn^sdt^.

is,

major

third,

and the minor second, G^

to A;

preponderance of minor thirds and

The

tritones,

E^

to G;

to G.

scale

still

The

has a

but also contains

the remaining intervals as well.

The

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 six-tone scale adds a

perfect

fifth,

from E^ to

Bj^;

a major third, from

major second, from B^ to C, and the minor second,

G^

A

to B^; a

to B^, the

analysis being p^m^n^s^dH^:

Example Minor Third Hexad

>o^» The component

--

11-3

p^m^n^s^d^t^ ^'^'

jbjbJtiJ ^^r

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 Minor Third Triads

98

t

C3Eb4G and Eb3Gb4Bb, pmn, with the one the major triad Eb4G3Bb, which are characteristic of

the minor triads involution,

n

11-4

PROJECTION OF THE MINOR THIRD the perfect-fifth series;

Example pmn

Triads

the triads

6^)2^70;

C7G0A and

found

in

and

4

3

11-5 involution

4

4

3

Ej^yBl^aC, pns,

72

series;

11-6

and

pns

Triads

with the one involution

and minor-second

the perfect-fifth

Example

3

involution

27

72

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 nsd

Triads

and

I the triads

we have

G(;)4B|72C,

involution

2

I

Eb4G2A and

11-7

2

I

mst, with no involution, which

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

tonal

six basic

Example Triads

mnd

series

11-9

and

involution

U J^f ^jJl^J 3 3 l

ibJ^J^f 3

1

I

I

and the triads CeG^iG and E^eAiBb, fdt, without which are new in hexad formations

Example Triads

11-10

pdt

6

The student should study

I

series

introduces.

doubtedly, be thoroughly familiar with the

diminished

triad,

but he will probably be

triad ipdt. Since, as I

first

He

and

will,

less familiar

with the

have tried to emphasize before, sound

"new" sounds, experimenting with

un-

of these, the

all-important aspect of music, the student should play to these

new

carefully the sound of the

which the minor-third

triads

involution,

and

is

the

listen

diflFerent inversions

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

Minor Third Tetrad

^

4 2 n

t

the isometric tetrads C^¥.\)4GzB\), p^mn^s, and GsAiBbsC, pn'^s^d,

both of which fifth

we have

already met as a part of the perfect-

hexad, the latter also in the minor-second hexad;

100

projection of the minor third

Example p^mn

Tetrad

4

3

four

one

new

tetrad types,

11-12 Tetrad pn

s

2

3

s

d

2

I

consisting of a diminished triad plus

all

tone: C3Eb3Gb4Bt) and A3C3Et)4G, pmn^st; and C3Eb3GbxG4 Eb3Gb3AiBb, pmnHt; GbiGt^^AgC and AiBbsCg Eb, pn^sdt; Eb3GbiG^2A and Gb3AiBb2C, mnhdt;

"foreign"

Example pmn

Tetrads

^J

ji,j 3

11-13

pmn

st

t

kfA r

^

I

3

r-[

4

3

3

J

mn J

^J 2

the

tetrads

r

*

I

I

2

'

C6GbiGtl3Bb,

analysis pmnsdt, the

first

3

I

J^p

j^j 3

3

I

sdt

-^

3

^^

!

i^J 3

I

2

and Eb4G2AiBb, both having the appearance in any hexad of the twin

tetrads referred to in Chapter 3,

Example

Example

jbJuJ^r 6

3-8;

11-14

pmnsdt

Tetrads

^

^jfej

hj^J^j

r

r

"r

3

dt

J 3

11^

pn^sdt

2

^

ibJ

^'^r

4

3

I

2

I

and the two isometric tetrads EbsGbiGtisBb, prn^n^d, which will to consist of two major thirds at the interval of the minor third, or two minor thirds at the relationship of the major third;

be seen

Example pm^n^d

Tetrad

(j

I,

J

i'^

3

11-15

t^ I

^r 3

hi^^ ^if^ \

ni

@ —

a.

@j]i

101

:

THE

TONAL

SIX BASIC

SERIES

and GbiGtisAiBb, mn^sd^, which consists of two minor thirds at the interval relationship of the minor second, or two minor seconds

minor third

at the interval of the

Example mn'^sd

Tetrad

2

I

The pentads

11-16

*" n.

1

@d

d_

@ji

consist of the basic pentads C3Et)3GbiGfcj2A,

and

EbaGbsAiBbsC, pmn^sdt^;

Example

11-17

Minor Third Pentads pmn'^sdt^

li^J'J^^^ 3

I

12

3

^J^^^ 12 3

3

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^ 13 mn @ 3

3

p

n^

the pentad C3Et)4G2AiBb, p^mn^s^dt, which as

two

triads

pns

at the interval of the

Example

ti

-J-

''• I

3

102

4

11-19

3^2,

.2

2

1

-il-

pns

may

minor

P*"

@£

also

third;

be analyzed

PROJECTION OF THE MINOR THIRD the pentad E^aGbiGtioAiBb, pm~n^sdH, which

lyzed as the combination of two triads

minor

mnd

may

also

be ana-

at the interval of the

third;

Example

11-20

Pentad pm^n^sd^t ffl [

.JbJ^J

and

the

pentad

Ji^f

12

3

l^jJ^JbJJ^f 3

3

1

mnd

1

@

1

ji

pmrfs^dH,

GbiGl:]2AiBb2C,

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 The

contrast

between the

is

_n_

and the be immediately apparent. Whereas

limited to various combinations of major thirds,

major seconds, and of harmonic

'2

I

@

six-tone major-second scale

six-tone minor-third scale will

the former

2

I'

nsd

tritones, the latter contains a

and melodic

possibilities.

of course, in the interval of the

The

wide variety

scale predominates,

minor third and the

tritone,

but

contains also a rich assortment of related sonorities.

Subtle

examples

of

the

minor-third

Debussy's Pelleas et Melisande, such

Example

hexad are found

in

as:

11-22

Debussy, "Pelleas and h^elisande"

Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co., Inc., Philadelphia, Pa., agents.

103

THE

SIX BASIC

TONAL

SERIES

Play each of the triads in the minor-third hexad in each of three versions, as indicated in

Example

11-23. Play

several times slowly, with the sustaining pedal held. If sufficient pianistic technic,

play

all

hands in octaves, otherwise the one each

its

each measure

you have

of the exercises with both

line will suffice.

Now

analyze

triad.

Example

11-23

rn. ^jjii.mi^irmi^LJ ^^LjLLJ 1;

-

i>

1

^

^ p^r

''

fp ^^dripi"^LJ j

'

^-

^'u^Lii i^

i^^^

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

i I)

1,

iff ^^^ JJ

104

I

t-i^^

[^

;mr

\

^cU 1^^^

bf^

M

k^ ^LL

PROJECTION OF THE MINOR THIRD

Repeat the same process with the tetrads of the

Example

scale:

11-24

|jP^.mc:tfLtfr jy..^clJ^Lffl i

,,f,

jw n^^^a!J\^.P^i^ ^ci^ ^

(liP^i-^crJcdJ

F

Lph

JJ^^ft^^MJ^^^yrJ^cfT bp

^f^F ^f^f-j^

p^^crtfrdT^cttri^ffl^^ciLrigj

^^

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

Example

^

jM

^

M^'^^ r^r

six

pentads and the hexad

11-25

r

r'l

^

^r r'll [/'tT^ 105

THE

#

SIX BASIC

TONAL

SERIES

J^JJ^^^^^^ypJ

bJ^JjJ^

vH

I

jjt-^t>''

(|

jn7i:^,jT3T:^cxUlrciiir

''

r^r

Y

r 't

I,

^

r'TT ^c_r

r'T

b.

1,

-Vlir,

i

rWr

'cmrftc^^rrT c'TrTT yrT_l

^^^^^^

V4

y^kr

^^^^^^^^^^^^ ^H One

of the

degree

of

'te--±^

fe^

most important attributes of any sonority

its

is

consonance or dissonance, because the "tension"

induced by the dissonance of one sonority

may be

reduced, or released by the sonority to which interesting

^rk^'t'''

f-

and important study, therefore,

is

it

increased,

An

progresses.

the analysis of the

relative degrees of dissonance of diiferent sonorities.

At

first

glance, this

may seem

to

be an easy matter. The

vals of the perfect octave; the perfect fifth

perfect fourth; the major third and

and the minor third and generally sonority.

106

its

its

its

inter-

inversion, the

inversion, the

inversion,

considered to perform

The major second and

its

and

minor

sixth;

the major sixth,

are

a consonant

function

inversion, the

minor seventh;

in

a

PROJECTION OF THE MINOR THIRD the minor second and

and the

inversion, the major seventh;

its

(augmented fourth or diminished

fifth)

are generally

considered to perform a dissonant function.

When

these intervals

tritone

are

mixed together, however, the comparative degree

sonance in different sonorities

is

indeed, cannot be answered with

We may

Some

not always clear.

of dis-

questions,

finality.

assume that the dissonance of the major seventh and minor second is greater than the dissonance of the safely

minor seventh, major second, or however, there

listeners,

is

tritone.

much

not

To

the ears of

difference

many

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 since

of tones.

C-E-F#-G

conclude that the

is

the tritone, whereas the

minor second, the might

contains three dissonances— the

first

tritone,

and the major second. However,

it

be argued that whereas the sonority C-E-F#-G con-

also

tains a larger

number

of dissonant intervals, C-FJf-G contains a

greater proportion of dissonance. is

we might

For example,

more dissonant than the sonority C-F#-G, the second contains two dissonances— the minor second and

sonority

The

analysis of the

first

sonority

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 Tetrad

pmnsdt

m' i i Finally,

it

11-26 Triad pdt

»'

would seem

ii^i^U

I

fe°

i

v^-

d

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

more dissonant than the

TONAL

SERIES

tetrad C-E-Bt>-D,

which contains four

mild dissonances:

Example Tetrad

pm 2 n 2 d

11-27

m 2s 3

Tetrad

With the above

t

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 indissonant

creasingly

sonorities.

Play through Example

11-28

carefully, listening for the increasing tension in successive sonorities.

Note where the degree of "tension" seems

approximately the same. Analyze

you agree with the order

all

to

of the sonorities

of dissonance in

them. Have someone play the example

for

which

I

remain

and see

if

have placed

you and take

it

down

from dictation:

Example

'^

r^

J

tl-~"

—

3

r

108

r

3i

1

iittii.-.

1-^

11-28

LJJiW=^

\rh 'i

J

J

^

hN i4

p

J

f^Tw ffi 3

PROJECTION OF THE MINOR THIRD

Reread Chapters 6 and 7 on modal and key modulation.

it is

hexad

minor-third

the

Since

has

the

p^m^n^s^dH^,

analysis

evident that the closest modulatory relationship will be at

the interval of the minor third; the next closest will be at the

and the third order

interval of the tritone;*

be at major

the

interval

minor

or

third,

of the minor third

four

common

will

perfect

the

of

Modulation

second.

have

common

five

Example

f^

^

Minor Third Hexad

@

Modulation i:

^

..k J^"^*

i @—

!?•

-0

lj

7- bo^'

'1'

'

p

m

n

s

^

@m

^

the

interval

at

two common

tones.

d t^

^ n^

^rt^<

M'^'^' @P

@1

^

^

^^^^

rWV4^

v\ •

S

l

second,

11-29

^^^ ^53

pr^^^

major

tones; at the tritone,

tones; at the other intervals

Modulation of

of relationship will

fifth,

\

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

modulate

to the

key a

fifth

C

(that

above (E), and then back

is.

key of C. See Chapter

17,

A);

to the

pages 139 and 140.

109

12

Involution of the Six-Tone

Minor-Third Projection

The

first

three series of

projections, the perfect fifth,

second, and major second, have

For example, the

all

produced isometric

perfect-fifth six-tone scale

minor scales.

C2D2E3G2A2B, begun

on B and constructed downward, produces the identical

B2A2G3E2D2C. This tion. If

The same

we

is

scale,

not true of the six-tone minor-third projec-

projection

downward produces

a different scale.

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

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:. produce the same scale downward,

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 Minor Third Pentad

110

12-2

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.

t-

A

simpler

method would be

7

^^

^

J

^^

to take the configuration of the

minor third hexad, 3 3 121, beginning on C, but in reverse, 1213 3, which produces the same tones, CiCJsEbiEtjs original

GsBb:

Example Minor Third Hexad upward [.o

bo

12-4

Involution

^

t?o

t^ If

to

we examine

be the same

involution.

the components of this scale

We

4

|

;)

we

shall find

them

conceived upward but in

as those of the scale

The

p^m^n^s^dH^.

£

bo

analysis of the scale

is,

of course, the same:

again, the four basic diminished triads

find,

C^gEsG, EgGsBb, G3Bb3Db(C#), and A#(Bb)3C#3E;

Example

12-5

Minor Third Triads

4V

i-jl

\,^{tr)iHf^

the major triads— (where before

and Eb4G3Bb, with the one

we had minor

involution, the

Example Triads

t

n^^t

12-6

pmn and

H

triads)— C4E3G

minor triad C3Eb4G;

involution

''i>i

111

THE the

triads

TONAL

SIX BASIC

SERIES

BbsCyG and Db(Cfl:)2Eb7Bb,

pns,

with the one

involution, Eb7Bt)2C;

Example and

Triads pns

4

12-7

^ 2i 7

involution

pjur

•>

Y-~

'bJ 7

7

2

the triads Bl:)2CiDb(C#) and CJfaDJiE,

T2r

nscZ,

together with the

one involution CiDb2Eb;

Example Triads

and

nsd

iJ'aiU 2

12-8 involution

lijJ^J " 2

1

I

^^ I

2

the triads Bb2C4E and Db2Eb4G, mst;

Example

12-9

Triads mst

ITlTg

^

2

4

the triads CiCJgE and DJiEgG, mnc/, with the one involution

CsDfiE;

Example Triads

mnd

12-10

and

involution

4 ^,t^3 ^^A 13 ^

'^jt

3

and the

triads CiCJfeG

and D^iEgBb, pdt:

Example Triads

i 112

12-11

pdt

WFWf \v

6

I

6

INVOLUTION OF THE MINOR-THIRD PROJECTION

The first

Bb,

tetrads consist of the

same isometric

tetrads found in the

minor- third scale: the diminished-seventh tetrad, CifgEgGa the

nH^,

other

isometric

pmVc/,

CsDJiEsG,

C3Eb4G3Bb,

tetrads,

and

pnVc/,

BbsCiDbsEb,

jrmn^s,

CiDbsEbiEl^,

rmnrsd^;

Example ^Tetrad

12-12

1343

333

212

313

mn^d

Tetrad

Tetrad p^mn^s Tetrad pm^n^d Tetrad pn^s^d

n^^t^

121

four tetrads consisting of a diminished triad and one foreign tone, each of

which

tetrad

similar

Eb4G3Bb3Db,

GgBbsCiDb

be discovered

will

the

in

first

to

minor-third

be the involution of a scale:

C4E3G3Bb and

pmnht; CiCJfsEsG and D^iEgGsBb, pmnHtand Bb3Db2EbiEl^, pnhdt; and Bb2CiC#3E and

C^sDftiEsG, mn^sdt;

Example

12-13

pmn^st

Tetrads

A 4

^ 3

Tetrads

A

y^ 3

y,

and the

'

Tetrads

p 2

y

\?m I

I

"twins",

3

3

t^p

\?m

^

^

yk

Tetrads pn^sdt

3

pmn ^dt

'7 2

3

mn^

I

3

sdt

^

vi-y^

I

^

\-ii^ 2

3

^ I

CgD^iEsBb and CiDb2Eb4G, pmnsdt, the

involu-

tions of similar tetrads discussed in the previous chapter:

Example

12-14

Isomeric Tetrads

J

jt^

3

I

J 't 6

pmnsdt

Ui-J I

1^ 2

3

4

113

THE

The pentads

SIX BASIC

TONAL

SERIES

CJaDJiEgGsBb and

consist of the basic pentads

Bb2CiC|:3E3G, pmn'^sdt^ (the involutions of the basic pentads in the previous chapter);

Example

12-15

pmn

Minor Third Pentads

sdt^

^^^^^^ 13

2

3

13

2

the pentad CgEj^iEtjsGgBb, p^m^n^sdt, a

which may be analyzed

combination of two major triads

minor

3

at

the

Pentod

I

the

12-16

p^m^n^sdt

J ^ V i 3^J 13 3

i0

•'

^

ii

amn

@

n_

may be

the pentad CiDb2Eb4G3Bb, p^mn^s^dt, which

minor

of

third;

Example

as the

interval

as

combination of two

analyzed

pns, at the interval of the

triads,

third;

Example Pentad

12-17

p^mn's^dt

l-iJ^J^YLi^J^Jt 12 4 3 2

^2

7

7

may be

the pentad CiDb2EbiEti3G, pm^n^sdH, which the

combination of two

minor

triads,

rand,

at

the

third;

Example g,

Pentod CIIIUW pm^n^sd^t ^111 3U I

II

12 13 114

12-18 I

I

3

mnd

_

@

3

I

_n_

analyzed as

interval

of the

INVOLUTION OF THE MINOR-THIRD PROJECTION

and the pentad BboCiDboEbiEt], pmn^s^dH, which may be analyzed as the combination of two triads, nsd, at the interval of the minor third:

Example

12-19

Pentad pmn^s^d^t

^^^^^m 12

2

2'

1

All of the above pentads will

2

1

nsd

@

1

_n_

be seen

to

be involutions of

similar pentads discussed in the previous chapter.

many examples of the involution of the minor-third hexad we may choose two, first from page 13 of the vocal score From

the

of Debussy's Pelleas et Melisande;

Example Debussy, "Pelleas and

U]ITi\'^^

^y-

12-20

Melisande"^

____^

m--^-m--r-0-r-

p?

^Jt o

13-2

^^

^*

i J

4 4 8 4

Minor Third Octad

t;o

bo

m

p

"

n

s

d

3

p^

m^

n^ s® d^

b« ^*

2

I

If.

ij =

r

I

S

^^

p' ^m'^n'^

Minor Third Undecod

k-

.

,

11111112

iT^b-*-

=S

the

seven-tone

I

Jg 1

^^^S^^ I

I

I

I

I

I

I

p'^nn'^n'^ s '.^d'^t^

^^b»b»

-K;:b^ b,l;i'

U

iJbJ^J^J^J^r ^^Il^ ibJ^J^J Jl I

these

^r 2

I

^

I

Minor Third Duodecod

of

2

I

I I

J

^J

i^J

^jgiJllJbJ^J

s'^d'^t

*?:

All

I

p^m^n^s^d^ f^

Minor Third Decad

tl'

2

I

t"^

•*-

"

I

|^^^^§^^ ^

iP^bo^°

^

t

2

^*

2

I

4 4

tl>

Minor Third Nonad

^^

^J ^^ 2

I

scales scale,

are

isometric

the

involution

I

I

l I

I

I

I

I

I

^

I

with the exception of

which produces

different scale:

119

of

a

:

THE

TONAL

SIX BASIC

Example Minor Third Heptad

These

scales

SERIES

13-3

Involution

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

Example this scale,

13-4,

movement

where the

of Stravinsky's

first

of Psalms,

seven measures are consistently in

EiF2GiG#2A#iB2C#iD

Example

2 Strovinsky, /1AIT03 AlTos

Symphony

2

1

Symphony

2

I

of

I

13-4

(2)

Psalms

I

J

J

^'

Ex

1-1.

I

>>;j.^^X

J

'^m

J

J

ro -

tl

I

-

J

J -

nem

J3 iri

mm

J J me -am,

J7J

rp'rrmr

^^m ^^ ^m "^m Copyright by Edition Russe de Musique. by permission of Boosey & Hawkes, Inc.

120

Revised version copyright 1948 by Boosey

& Hawkes,

Inc.

Used

i

—

:

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

same composer's Sijmphony

number

rehearsal

in

first

movement

of the

Three Movements, beginning

and continuing without deviation

7,

at

for

twenty-three measures

Example Symphony

Stravinsky,

in

13-5

Three Movements

^^"

,

^•^mr

marcato

V

mf I-

-^-»-

'i^'r^

*""*^°

Dizz

i

^

^^T-Vi

pizz.

m

f

f

^

b

l?Qj>

j}

*/

*/

g

g

•OS?

^

orco

3_^ -»-y-

i

—

pizz.

^•^

pocosj 06p

J-

^m p

pizz.

a Jj^»^i ^4?

^J5?

pizz.

orco I

5

s

-1~ t»

n^y

a s^

W

^

pizz.

vT pocosjz

J

J

I

r

p

J

r

mf

^ ^

^ ^ pizz.

Copyright 1946 by Associated Music Publishers,

^^-i

12

3

Inc.,

12

New

York; used by permission.

I

121

THE

SIX BASIC

TONAL SERIES

Another interesting example of the eight-tone minor-third scale

found

is

siaen's

at the

opening of the third movement of Mes-

VAscension: Example

Messloen

13-6

,"l' Ascension"

Vif

^M\h

>

"/g^^^

-.

%

i-

itJiitfe

Reproduced with the permission of Alphonse Leduc, music publisher, 175 rue Saint-Honore, right by Alphonse Leduc.

I

4-

^

*

fl"0

=

I

2

I

2

I

2

I

aS

Paris.

Copy-

(21

Analyze further the Stravinsky Symphony of Psalms and try to find additional examples of the minor-third projection.

122

14

Major Third

Projection of the

We

have observed

that there are only

two

intervals

which can

be projected consistently through the twelve tones, the perfect

and the minor second. The major second may be projected

fifth

through a six-tone

and then must

series

resort to the interjection

of a "foreign" tone to continue the projection, while the

form through only four

third can be projected in pure

We

come now

to the

major

third,

which can be projected only

the major third, E, and the second major third,

C-E-G# consisting G#, and G# to B# (C), m^:

ing the augmented triad

C

to E,

E

to

Example

I ^ project the major third

add the foreign tone

of the three major

14-1

Major Third Triad

To

we superimpose E to G#, produc-

Beginning again with the tone C,

to three tones.

thirds,

minor

tones.

mj

^°'"-'

°

tf°

beyond these three

Gtj*, a perfect fifth

tones,

we

again

above G, producing the

basic major-third tetrad G4E,oGiGJj: having, in addition to the

three major thirds already enumerated, a perfect to G; a

GtoG# •

minor

from

E

to

from

C

G; and a minor second from

{k\));pnv'nd:

Here the choice

A# with

third,

fifth,

their

of the foreign tone

is

more important,

since the addition of D, F|, or

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

tonal

six basic

Example

To produce

14-2

pm^nd

Major Third Tetrad

^.

t^

series

^g

J4J

3

we superimpose

the pentad,

I

a major third above

G, or B, forming the scale C4E3GiG#3B, and producing, in addi-

G

tion to the major third,

minor

G#

third,

to B;

and the minor second, B

Example

to B; the

to C; p^m^n^d^:

14-3

p^m^n^d^

.Major Third Pentad

To produce

E

to B, the perfect fifth,

we add

the six-tone major-third scale,

the major

third above B, or D^, giving the scale CgDJiE.sGiGJfsB.

D^,

tone,

in addition to

an additional major another perfect

fifth,

and a minor second,

forming the major

third,

G#

DJj:

from

Hexod

i If

minor

we proceed six-tone

augmented scale,

to G.

It

C

to

third,

The new DJj:,

adds

also DJj:

adds

(E^);

p^m^n'^d^

iitJ to analyze the

which

C and on

is

G.

^

^

«^

r

melodic-harmonic components of

major-third scale,

triad,

m^, on

to

14-4

*

this

B

to E; p^m'^n^(P.

Example Major Third

(El^)

Dfl:

to DJf; a

third,

we

find

that

it

contains

the

the basic triad of the major-third 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

124

Gjj^,B,Djj^;

projection of the major third

Example Triads

14-5

pmn

and

involutions

C,G4B, E,B4D#, and Ab(G#),Eb(D#),G, pmd, together with their invohitions C^E^B, E4G#7D# and Ah(G#)4

and the

triads

C^G:

Example

14-6

pmd

Triads

and

Finally,

74

74

74 it

involutions

47

47

47

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 minorthird scales but which would seem to be characteristic of the major-third projection:

Example Triads

mnd

id ^

I

The

J

14-7 and

^^^

J 3

i^r 3

I

involutions

ji^j 13

r I

tetrads consist of the basic tetrads,

tfjj^^

13

new

^m 13

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

and

their involutions

G#3B4Dfl:4F-)g and (DJj:)G, p~m~nd, which

we

Ab4(G#)C3Eb4

observed in the perfect-fifth

first

projection;

Example •Tetrads p

2

14-9

m 2 nd

^"%34

434

-54

4

the isometric tetrads CgDSiEaG, EsGiGJsB,

we have encountered

pm^n-d, which

as

and GI^BiCsDJ,

parts of the minor-

third series;

Example .Tetrads

pm n^d

13

3

14-10

13

3

3

13

and the isometric tetrads B^C^Dj^^E, DJiEgGiGJ, and GiGJgBiC, pmrnd^, which can be analyzed as two major thirds at the interval of the

minor second, or two minor seconds

at the interval of

the major third, previously observed in the minor-second series:

Example ^Tetrads

7'

1

\-rr-\

•J

J(t^

1

-^r:iitJ I

14-11

pm^nd^

3

^

i^ I

I

The pentads

3

I

— = ^JjtJ r

!

—

w—

1

\-r,

I

3

r-^

^ Ni— m @

I

^

d

consist only of the basic pentads

'

^ d

—

\

^Jm (g

"

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 Major Third Pentads

p^m^n^d^

14-12 and

involutions

PROJECTION OF THE MAJOR THIRD

From

this analysis

scale has

it

will

be seen that the six-tone major-third

something of the same homogeneity of material that

The

characteristic of the six-tone major-second scale.

is

includes only the intervals of the perfect

fifth,

scale

the major third,

the minor third, and the minor second, or their inversions.

does not contain either the major second or the tritone.

however, a more striking scale than the whole-tone

It

scale, for

It is,

it

contains a greater variety of material and varies in consonance

from the consonant perfect

The

we

fifth to

six-tone major-third scale

is

the dissonant minor second.

an isometric

scale,

because

begin the scale CgDSiEgGiGJgB on B, and project

reverse, the order of the intervals remains the same.

therefore,

A

clear

sixth

no involution

as

was the case

it

in

There

is,

in the minor-third scale.

example of the major-third hexad

may be found

in the

Bartok string quartet:

Bartok, Sixth Quartet

Example

14-13

Vivacissimo

Copyright 1941 by Hawkes

& Son (London),

Ltd.

P^

Used by permission

of Boosey

& Hawkes,

Inc.

(b«^ 3

if

13 13 127

— THE

TONAL

SIX BASIC

An harmonic example

of the

same

SERIES

scale

is

illustrated

by the

following example from Stravinsky's Petrouchka:

Example

Stravinsky, "Petrouchko"

i

g

VIos.

j'^^bS

j!

[b^§

^

P

m

^^

p

cresc.

_

^% ^s l

^^

14-14

J

jiJ 3

^

b*-!

^r t r 13 13

Copyright by Edition Russe de Musique. by permission of Boosey & Hawkes, Inc.

#

V-l

^^

]

Revised version copyright 1958 by Boosey

& Hawkes,

Inc.

Used

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

!

W

yM^ H.

^.

Copyright

A

charming use of

Prokofieff's Peter

this scale

©

is

1957 by Eastman School of Music, Rochester, N.

Y.

the flute-violin passage from

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 Play the

b^N

\^A

SE

'r

i

r

r

TT

13 13

3

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

^

m

^^ rPi^ Vrnmi lU jJm^-"^'l^ ^ i

'

Is ^'

^^JbJ\J

^^

(|j7;pja^i-^ jjr, [Trpi^ i

.

129

:

.

THE

SIX BASIC

TONAL

SERIES

Experiment with different positions and doublings of the characteristic sonorities of this scale, as in

Example

W (j

^ d

d

u,

i

Hi

P

i^

^=H

^H

^i

i^

^«» etc.

etc.

i

ii

The following

etc.

etc.

%

n J

14-18:

14-18

etc.

etc

/

Example

^

'

T

H

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

^^

^« ^^ ^^ ^^^ff #^^ ^^

m

130

14-19

^m

^

PROJECTION OF THE MAJOR THIRD

"^

(|^

n'-JlJ

Ji.^ Lnj

4

d

^

S

liti

^

tfc^

w

^

¥

*

Write a short sketch Hmited to the material of the major-third

hexad on C.

Example 14-20 scale.

illustrates the

modulatory

possibilities of this

Modulations at the interval of the major

up

third,

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^

og»

oflo S 3

13 13

Modulation

n-e-

#@ 7-

,j|.

olt'""'

n

>^°'«°' @d

@m

^S

Modulation

^

@

p

^3

^^ ^ ^^^ @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

15

Projection of the

Beyond

If

we

the Six-Tone Series

we

refer to the diagram below

points in the circle first

Major Third

may be connected

to

see that the twelve

form four

triangles: the

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

We tones scale.

15-1

may, therefore, project the major third beyond the six by continuing the process by which we formed the six-tone 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# p^m^n^s^dH, with

+

CsDiDJiEaGiG^gB, involution CaDJiEgGiGJiAaB:

its

Example Major Third Heptad

^

Eight

G#3B,

tone-.

15-2 and

p'''m®n'*s^d'*t

+

C-E-G#

+

Gt^-B-DJ

fm'nhHH^, with

ISline

tone:

p^m^ n

3^

^1^119 13 12 3

Dtj-FJ

^ s'^d ^ t^

and

+

Gti-B-D#

involution

j|j JJ ^11^^ I

113

2

+

CaDiDJiEaFJiGi

15-3

iJiJjit^^«-'r

C-E-G#

=

1

involution CgD^iEiFaGiGJiAaB:

its

Example

^^

involution

2T13I3

=

Major Third Octad

=

G-B-D# +• D^

3

I

I

2

=

Dt^-FJf-AJ,

12

I

CaDiDJiEs

F#iGiG#2A#iB, p^m^n^s^dH^:

Example Major Third Nonod

li «^

(This

is

^

M

steps,

-H

an isometric

proceed downward,

p^m^n^s^d^t'

^H

-»-

15-4

1

1

1

1

J

,

=

^2

scale, for if

we have

the

r

we

1

2

1

H

1

J

,

itJ

2

1

1

begin the scale on

same order

of

A# and

whole and half

21121121.)

133

))

THE

Ten

tone: C-E-Gif

TONAL

SIX BASIC

+

+

G\\-B-Dj^

SERIES

Dt^-Ff-AJ -f

QDiDfti

Al^

E2F#iGiG#iAiA#iB, fm^nhHH'-.

Example Major Third

Decad

p^m^n^s^d^t"*

^^ (This scale

^

j J^J Jj[J-'tfJ^1 II II 2 112 1

also isometric, for

is

15-5

we have

progress downward,

if

we

begin the scale on F# and

the same order of whole and

half-steps.

Eleven tone: C-E-G#

+

+

Gti-B-D#

CiC#iDiD lE^FSiG.GJi AiAliB, f'w}'n''s''fH'

Major Third Undecod

^

=

15-6

p'^ m'^n'^s'^d'S ^

^^^^^^

-*-

Twelve tone: C-E-Gif

Al^-Cft

:

Jf

Example

+

Dt;-F#-Afl

11

+

12

I

Gtj-B-Dft

+

1

I

I

+

Dt^F#-A#

I

I

Al^-C#-Et

C,C#,D,D#,E,E#,F#,G,G#,AiAif,B,p^WW^c/^-T:

Example Major Third Duodecac

^^ (

The

eleven-

12

p

m

15-7

I2„I2,I2 .12.6 n s d t

rff^i

and twelve-tone

r

I

I

I

I

I

III

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

and minor seconds. The

scale gradually loses

istic as

of

additional tones are

of perfect fifths,

added but

its

134

thirds,

basic character-

retains the

major thirds through the ten-tone projection.

minor

preponderance

FURTHER PROJECTION OF THE MAJOR-THIRD

The following measure from La siaen, fourth

movement, page

Nativite

du Seigneur by Mes-

2, illustrates a

use of the nine-tone

major-third scale:

Example Messiaen^La Nativite

^

i

f

15-8

du Seigneur"

f

i

i^ p

f ^'f

Reproduced with the permission of Alphonse Leduc, music publisher, 175 rue Saint-Honore, right by Alphonse Lediic.

Sr m i/ii

il

8

The long melodic of the

same

line

VAscension

Copy-

iJ^J^JitJJi'^^r^^ 2

composer's

Paris.

I

I

2

I

I

2

I

(I)

from the second movement of the same is

a striking

example of the melodic use

scale:

Example

15-9

Mes3ioen,"L'A scension"

^^i^\r[^ >^-^^^

(

\iIiJ?\^-} a

\

Reproduced with the permission right by Alphonse Leduc.

of

Alphonse Leduc, music publisher, 175 rue Saint-Honore,

2

I

I

2

11

2

I

and

try to find other

Copy-

(I)

Analyze further the second movement of Messiaen's sion

Paris.

VAscen-

examples of the major-third projection. 135

16

Recapitulation of the Triad

Inasmuch as the projections

that

we have

Forms

discussed contain

of the triads possible in twelve-tone equal temperament,

and

if

if

ment

all

may

summarize them here. There are only twelve types we include both the triad and its involution as one form,

be helpful in all

it

we

to

consider inversions to be merely a different arrange-

of the

same

There are composition:

two perfect

triad.

five triads

which contam the perfect

fifth in their

(1) the basic perfect-fifth triad p^s, consisting of

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 involution, 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: fifths

Example 2

I.

i 1/

p s

•#-

2.

2

m 4.

pmd

136

and involution

psn

=f 7

5

2

2

and involution

r

^

J

J

16-1

r

5.

I

pdt

J

^[J

3.

pmn 4

7

ond involution

m

I

6

and

3

involution

3

4

RECAPITULATION OF THE TRIAD FORMS

The

has appeared in the perfect-fifth hexad.

p^s,

first,

The second,

pns, has appeared in the perfect-fifth, minor-second,

The

third hexads.

third,

pmn,

is

found

and major-third hexads. The

third,

encountered in the

The

hexads.

fifth,

perfect-fifth,

and minor-

in the perfect-fifth,

pind,

fourth,

minor-

has

been

minor-second, and major-third

pdt, has appeared only in the minor-third

hexad, but will be found as the characteristic triad in the projection to

be considered

There triads,

in the next chapter.

are, in addition to the perfect-fifth triad p^s, four other

each characteristic of a basic

Example

2

2

The

triad

also

found in the

ms^

is

nH, m^, and sd~:

16-2

4

4

3

3

series: ms^,

It*

I

the basic triad of the major-second scale, but perfect-fifth

is

and minor-second hexads. The

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 projection and is found in none of the other hexads which have triad nH, has occurred only in the minor-third hexad.

been examined. There remain three other

triad types:

Example 10.

mnd

and involution

triad

nsd

and

r3l2

31 The

II.

mnd

is

found

hexads.

The

is

and mst:

nsd,

16-3 involution

21

12 .mst

24

and involution

42

in the major-third, minor-third,

minor-second hexads. The triad nsd

hexad and

mnd,

is

and

a part of the minor-second

also found in the perfect-fifth and minor-third

twelfth, mst, has occurred in the major-second

minor-third hexads.

137

and

THE

TONAL

SIX BASIC

SERIES

Since these twelve triad types are the basic vocabulary of

young composer should study them 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

musical expression,

the

carefully, listen to

first

five triads— basic triads of the perfect-fifth,

minor-second,

major-second, minor-third, and major-third series— are

same "shape" as the all have involutions.

metric, the involution having the

The remaining seven

triad.

triads

Example p^s

ijj ri

25 r -r

24

:

r

16-4

n^t

m'

pdt and

i

05 22

^J

V 24

and involution

I

involution

i

44A

-^X 33

and involution

r

fe^^ ^

mnd

Rl 61

A.

pmn

43

iso-

original

iJ.i >'^-MitJ«^ ?i'r:r

and involution

pmd

#

i

ifll

m^ mst

r

ms^

s^d

all

43

and involution

pns

1

7 nsd

61

^ ^^^ and

involution

'

'

1

^^

2

7

2

and involution

^

74

138

74

31

31

12

12

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

It will

a highly important part of the complex.

be observed,

further, that the tritone in itself

when one

is

not use-

superimposed

ful as a unit of projection,

because

upon another, the

the enharmonic octave of the

tone.

For example,

result if

we

is

is

place an augmented fourth above

first

C we

have the tone F#, and superimposing another augmented fourth

we have

above F#

BJf,

the enharmonic equivalent of C:

Example

^^ For

this

17-1

t^^^ may be said to have An example will illusscale contains, as we have

very reason, however, the tritone

twice the valency of the other intervals. trate this.

The complete chromatic

seen, twelve perfect fifths, twelve

minor seconds, twelve major

seconds, twelve minor thirds, and twelve major thirds. It contains,

however, only

to A, Ft] to At],

Bb, and

Aij:,

six tritones:

and F

C

to F#,

D^

to G, Dt] to G#,

to B, since the tritones

Bti are duplications of the

above

first six. It is

F|:,

E^

G, A^,

necessary,

139

THE

TONAL

SIX BASIC

SERIES

therefore, in judging the relative importance of the tritone in scale to multiply the

number

of tritones

by two.

we found

In the whole-tone scale, for example, thirds, six is

the

major seconds, and three

maximum number

tone sonority, and since

major thirds which can that this scale

is

exist in

major

which can

maximum

any six-tone

any

exist in

six-

of major seconds or

sonority,

we may

say

saturated with major seconds, major thirds, and

and that the three

tritones;

the

six

tritones. Since three tritones

of tritones

six is

any

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 must be formed by superimposing the tritone upon those scales or sonorities which do not themselves contain tritones. We may begin, therefore, by super-

scale, the tritone projection

imposing tritones on the tones of the perfect-fifth Starting with the tone C,

the perfect and,

finally,

tritone

G#,

fifth

we add

the tritone

series.

Fif;

we

then add

above C, or G, and superimpose the tritone C#; the fifth above G, or D, and superimpose the

we add

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'

i^iU

tf^""

I

I

4

I

I

This scale will be seen to consist of four perfect seconds, two major thirds,

fifths,

four minor

two major seconds, and three

p'^m^s^dH^. Multiplying the

that this scale predominates

number

of tritones

in tritones,

tritones:

by two, we

find

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 -

Tritone

17-3

Minor Second Hexad p^m^s^d^t^

I

The components and

I

of this perfect-fifth— tritone projection are the

characteristic triads

pdt,

I

I

CeF^iG, CJeGiGJ, FJeCiCS, and GeC^iD, CiCJsG, CJiDgGiJ:, FJiGgCJ, and

involutions

their

GiGifeD, which, though they have been encountered in the minor-third scale, are more characteristic of this projection;

Example

^6

end

pdt

Triads

I

the triads

«^6

17-4

I

6

C2D5G and

6

1

1

involutions

^% 16

*

16c .

16c

16c

,

.

FJaGfgCjj:, p^s, the characteristic triads of

the perfect-fifth projection;

Example iiiuu:> M Triads

2

the triads

CiC#iD and

17-5

p^s p :>

5

FJfiGiGJj:,

2

5rr;itJji>iriJiJUi[JJit^ri'riu^ § jttJit^^r< 24 24 4242 61 61 1^+6 i'

It contains

i^^'t

16

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

pmnsdt, C4E2Fij:iG and F|:4AiJ:2CiCfl;, and and CiCJgEsFJ 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 pairs

"twins,"

of

previous chapters.

Example Tetrads

n'^t^

3

^

4 2

1

4

4 3

4

3

3

3

mn^sdt

pn^sdt

133^321 13 3

33

3 3

213

321

213

pmnsdt

Tetrads

4

15

3

pmnsdt

Tetrads

imn'-st

j-Jti^^iiJif^^^iiJ^t^^'^UJ^^^^^rr

i

^r-is

3

m2s2t2

fJ^d^t^

iKi^^'^^^

19-6

2

4

1

Finally,

we

2

13

3 2

I*

1

2

find the characteristic pentads CiCJfgEoFfiG

fmnhdH\

F#iG3A#2CiC#,

and and C4E2FtiG3A# and FJf^Ait^CiCJg

and the characteristic pentads of the minor-third CiC#3E3G3A# and F#iG3A#3C}f3E, pmn'sdt^:

E, pm-nh^dt^; scale,

Example p^mn^sd^t^

Pentads

r

3

2

I

19-7 pm^n^s^dt^

13

2

1

13

3

3

4 2

13

4

2

13

iPentads pmn'''sdt^

Of these pentads, only the 154

first

two are new forms, the

third

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

z

i tH CIS.

"t^ Bsns.

P*^

^

^l^^b

S

^^^i

;J

i^^-^

tt'if

n^

Write a short sketch using the material of the six-tone pmntritone projection.

Example 19-12 scale. It will

indicates

possible

be noted that the modulation

no tones; modulation

two

the

at the

minor

third,

modulations of at the tritone

up

or

this

changes

down, changes

tones; modulation at the perfect fifth, major third, major

second, and minor second changes four of the

156

six tones.

THE pmn-TRITONE PROJECTION

Example rm

p

I >>;

n

s

g

>

13 ..

It..

T

tt

^v%

*^

2

»

13

lt°

#

'

^ :«=«

@ £ !>^

.

1

,

19-12

- »tf'

ot

^

' i

Modulotion

@

@

t^

bo " *s^ bo »

^^

il

""' -it..it"«-'

i

^^^= ^^^^3

@m

@S

5^

^^ N^

@ ,

I

'

fj

d.

*

t

Write a short sketch employing any one of the possible modulations.

Analyze the third movement of Messiaen's

VAscension 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,

tion, the

minor triad

|G

E^i C,

tone of the triad— G to C#,

E^

and project a to A,

C

we

take

tritone

its

involu-

below each produce

to F|;— we will

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 triadtritone scale of the previous chapter.

Example

13 pmn

Minor Triad .

,Q

r^

2

20-1

I

+ tritones

17-0...

:h."

r «r

^ ^m

t

i^r

(2)

The components

of

components of the major

this

scale

are

the

involutions

triad-tritone projection.

They

of

the

consist of

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, the two minor triads

pdt, with their involutions CgFJfiG

158

and FifsGiCJ.

involution of the pmn-tritone projection

Example pmn

.Triads

Triads n

20-2 Triads pns

t

mnd

Triads

3

12

24

24

4 2

4

2

and involutions

pdt

Triads

2

mst

Triads

3

I

and involutions

It

nsd

Triads

7

2

7

61

61

16

l-ffe

contains the isometric tetrads CgEbsFjIgA, nH^, CiCifgFJiG,

D#3(Eb)F#3A4C}f and and FJsAsCiCJ, pmnHt; A3C3Eb4G, pmn^st; CsEbsFJfiG and FJiGoAsC, pnhdt; EbsFftiGsA and CiCjj:2DlJ:(Eb)3Fi|: AaCiC#2D#(Eb), mn^sdt (all of which will be seen to be p^d'f, and Eb4G2A4C#,

mVf^; the

tetrads

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

n^

Tetrad

Tetrad _^dftf Tetrad m^£t5

JibJi J ^IjJI t^ l

M letrods

3

3

pmn

3

.Tetrads

'

5

r'

5

3 3

1

m 1^*

2

pmn^st

^Ti^JibijitJ^^r 4

I

Tetrads

2

4

3

^1

3 4

3

3

3

4

Tetrads lerraas mn^sai mn^dt

Tetrads letrads pn sdt

''* dt

1

12

3

3

12

3

12

pmnsdt

F24 Finally,

J

t

3

3

20-3

124

we have

**23l

231

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

and AsCgEbsFJiG, pmn^sdf,

scale, EbsFifgAsCiCij;

are

involutions

the

of

pentads

of

the

all

major

of

which

triad-tritone

projection:

Example Pentads

,2 .2*2 p'^mrrsd^r

w

20-4

pm^n^s^dt^

jj^JttJJ

c^

2

I

,

P^

|'>^

4

2

I

3

iij^

2

2

3

t

4 ^'^^

*

p^m^nd

tf

'^^ii

2 2

251* 116

251

» ogo "

rro-

pmsd^

p^msdt

fe^eeO:?^=ec I

^^M

3

p^mns^

fe°*^ mns^d

i.

2

I

^

e

fc^

pmnsdt

4

3

3

£Vt2

^

4

SERIES

i

pm^n^d

i

fi

116R i

i

pm^nd^

w

Play the tetrads of Example 21-7 as indicated in previous chapters, listening to each carefully different positions

164

and doublings.

and experimenting with

Part

11

CONSTRUCTION OF HEXADS BY THE SUPERPOSITION OF TRIAD FORMS

22

pmn

Projection of the Triad

Having exhausted the single intervals

of projection in terms of

possibilities

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

We have

found that there are

will

nsd.

six

tones to a later section.

five triads

and which exclude the

different intervals

mnd, and

more than

Each

which

tritone

of these triads projected

produce a distinctive six-tone scale

:

consist of three

pmn,

upon

in

its

pns,

pmd,

own

tones

which the three

intervals of the original triad predominate.

Beginning with the projection of the major

C— C-E-G— and

major triad upon triad

upon

its

fifth,

triad,

we form

the

superimpose another major

producing the second major

triad,

G-B-D.

This gives the pentad C2D2E3G4B, p^m^n^s^d, which has already

appeared in Chapter

page

5,

47, as a part of the perfect-fifth

projection:

Example .Pentad

i i pmn «

*

The symbol pmn

@

the interval of the perfect

p^m^n^s^d

i

f

@

22-1

p

=

p should be

2

-'

J

J 2

3

g 4

translated as "the triad

pmn

projected at

fifth."

167

— :

:

SUPERPOSITION OF TRIAD FORMS

We

then superimpose a major triad on the major third of the

original triad, that

the

first triad,

E-G#-B, producing

is,

aheady observed

Pentad

Hi pmn @ m

on

E and

we have

)

22-2

p^m'^n^d^

fc triad

combination with

as a part of the major-third projection

Example

The

in

the pentad C4E3GiG}t:3B, p^m^n^(P (which

jj|j

J

J =

the triad on

EgGiGJfsBaD, p^m^n^sdt (which

G

^ 3

I

together form the pentad

we have

observed as a part of

the minor-third projection )

Example

22-3

Pentad p^m^n^sdt

4 The combined triad projection

@

pmn

triads

=

n

G

on C, E, and

3

form the six-tone major-

CsDoEsGiGJsB, p^m^n^s^dH:

Example pmn Hexod

22-4

p^m'^n^s^d^f

^

—a 2

2

The

3

I

chief characteristic of this scale

maximum number

is

third, the scale as a

from the

whole

perfect-fifth, major-third,

and has a preponderance third, and minor third. 168

it

contains the

of major triads. Since these triads are related

at the intervals of the perfect fifth, the

minor

that

is

major

third,

and the

a mixture of the materials

and minor-third projections

of intervals of the perfect

fifth,

major

pmU

PROJECTION OF THE TRIAD

The

new

major-triad projection adds no

triads or tetrads. It

aheady mentioned (comtwo 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 contains, in addition to the pentads

binations of

major

thirds;

Example

p^m^ns^dt

Pentad

^^

22-5

j

^ «^

J2 J 2 J

3

p2

1

the pentad C2D2E4GiJ:3B, pm^n^s^dt,

^i ^2

^

which may be analyzed

as

the simultaneous projection of two major thirds and two minor thirds

above

G# (Ab); Example Pentad

I

i

pm^n^s^dt J

J 2

22-6

2

ti^

4

r

W

'f

3

and the pentad CsDgGiGJsB, p^m^n^sdH, which may be anafifths and two

lyzed as the simultaneous projection of two perfect

minor

thirds,

downward:

Example Pentad

i tj

The

involution

p^m^n^sd^t «^

•L

2

of

22-7

5

the

1

r 3

i}^

\i I

p2 +

projection

of

n2i

the

major

triad

C2D2E3GiGiJ:3B will be the same order of half-steps in reverse, that

is,

31322, producing the scale C3EbiEt]3G2A2B:

169

:

SUPERPOSITION OF TRIAD FORMS

Example pmn Hexod -

Involution

o o

—

2

3

2

22-8

i

bo

o

^i

*:^

This will seem to be the same formation as that of the previous chapter,

if

begun on the tone B and constructed downward:

Example

i we

*

upward

22-9

n

i4

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 If

think the scale

formed of two minor major

third,

and minor

rather than

triads at the interval of the perfect fifth, third, respectively

Example 22-10

|j ^ r^^JjN pmn @p

i

2 3 4

2

=

I

The remaining pentads

r^J^jjibi^ ^JJjJ

bi

pmn @m^

4 3

=

3

1

i

£2

+

rn2l

=

3

133

22-11

|r^^j^j|4 M^^^i/'f 2

pmn @n

are:

Example

2

13

2 2 4 3

i

ym^ r^j^i j I 13 i

n^ +

2

t p2 +

5

nf

f

All of these will be seen to be involutions of the pentads

discussed in the

A

first

part of this chapter.

short but clear exposition of the mixture of

at the interval of the perfect fifth

Symphony 170

of Psalms:

may be found

two

triads

pmn

in Stravinsky's

PROJECTION OF THE TRIAD

Example

*

Imn

StravinsKy, "Symphony of

Lou

^

do

-

V^

O

p

n

I

J

@

22-12

Psalms'

Sop.

;i

m

pmn

-

'r

te

Boss

i^ do

Lou

te

Si ©'

'

Copyright by Edition Russe de Musique. by permission of Boosey & Hawkes, Inc.

The

Revised version copyright 1948 by Boosey

short trumpet fanfare from Respighi's Pines of

movement,

constitutes another very clear

tion of the triad

pmn:

Respighi "Pines of

Rome"

Example Tpts.

»

ll

iii

M

k kf f

Rome,

example of the projec-

wrijitiiiiiig

w

#|*H

By permission

t

is

found

in the

^iH

of G. Ricordi

exposition of the complete projection of the triad

involution

first

3

^ i

Used

Inc.

22-13

,

ff

An

& Hawkes,

&

Co., Inc.

pmn

in

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

J.

By permission of Curwen & Sons, Ltd.

171

23

pns

Projection of the Triad

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@ Next

we

i

T-

pns

J

J 2

p

2

^ 2

3

superimpose upon C7G2A the triad A7E2F#, producing

the pentad C4E2F#iG2A, p^mn^s^dt;

Example Pentad

h^^r

^

pmn

Involution

^*=^

Ji f*f

^i

1

27-8

and

13

2(1-3)

2

1'^

1

^u

pmn

\_

+

2(1-5)

t'

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

formed by the simultaneous

of the six-tone scale

projection of three perfect fifths and three minor thirds in the following excerpt

can, of course, also

is

found

from Stravinsky's Petrouchka, which

be analyzed

as a

dominant ninth

in

C# minor

followed by the tonic:

Example Stravinsky,

Petrouchka

Bsn.

I

27-9

»i^

ItJJJj VIn.pizz.

Copyright by Edition Russe de Musique. by pennission of Boosey & Hawkes, Inc.

^ ^

^

Revised version copyright 1958 by Boosey

& Hawkes,

Inc.

Used

"twin" sonority, formed of two minor thirds at the interval

Its

by the excerpt from Gustav where the sonority is divided into two pmn, one major and one minor, at the interval of the

of the perfect Hoist's triads

Hymn

tritone:

198

fifth,

is

of Jesus,

illustrated

minor third and perfect fifth

Example 27-10 Hoist,

Hymn

m (|

4

of Jesus

r Oi

ij

ir -

vine

a t»

r

r

Grace

is

done

^

ing

Wff ^t^tff W^ ^ m^ '

T^

^

m

By permission

,JUJ _n2

@

Jfi^y p

f

of

Galaxy Music Corporation, publishers.

l,JtJUJ^«^

12

3

1

3

199

28

Simultaneous Projection of the

Minor Third and Major Third C and two

Projecting three minor thirds above

above C,

we form

C-Et^-GJf, or CgEbiEt^oGboGftiA,

major thirds

C-E^-Gb-A

the isometric six-tone scale

+

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 Hexad

p^m^n'^s^d^t^

^ ^ +

m"

This scale contains two

^

new

J 3 bJ

t'Q

12

^ 2

1

28-2

ii

tiJ 1

4

which is formed of a major below C, tm~n^; and ^

t

third

Example

m'

and a minor third above and ^^ ^ 28-3

p^mn^s^d^t

|^J(a^JjW,^ 200

3

tjo

p^m^n^d^t

Pentad

Pentad

bo

isometric pentads:

Example

i

28-1

j^n^U

i

^

:

MINOR THIRD AND MAJOR THIRD which below

formed of a minor third and a major second above and Fjl; and two pentads with their involutions, is

Example 4

Minor Third Pen tod pmn sdt J

fj!

'T

r 3

3

"r 12

*r

which are the basic pentads

which

is

may be

n 2

involution

^^ 3

I

3

and

of the minor-third series;

Pentad pm^n^s^dt

o 2

2

11.

Example

^ 4A

28-4

28-5

and involution

i_2 4m^

I 1

a part of the

.

+

A

T«

_2i

ipvfin

^J ^ * 24 2

I I

mnd

and the

t^^

I'^^nyit

tm2+n2t

projection,

analyzed as the simultaneous projection of

and which two major

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 thirds

analysis, p^m^n^s^dH^:

Example

28-6

p^m^n^s^d^t^

to _n_2

|,o

^fg @ _m.

jo

I*"

I

^

This scale contains three pentads, each with

Example Pentad

i iP

p^m^n\dt

jbJiiJ ^'T 3

1

3

3

ki

^' @ n

involution:

28-7 and

pmn

its

involution

^jbJ^^^^V 3

3

13

hi

pmn

d

@

201

n

: i

SIMULTANEOUS PROJECTION OF TWO INTERVALS

which has already appeared in the pmn projection pmn, at the interval of the minor third; and

Example

involution

which has already appeared in the projection mnd at the interval of the minor third, and

Example Pentad

pm^n^s^dt^

^4213

as

two

triads

28-9

3124

I

which has already been found

Two

mnd

^

and involution

r

triads,

28-8

and

pm^n'sd^t

Pentad

two

as

f^ '

in the tritone-pinn projection.

quotations from Debussy's Pelleas et Melisande illustrate

the use of the two hexads.

The

first

by the

uses the scale formed

simultaneous projection of minor thirds and major thirds:

Example

V-

Mel son de"

and

Debussy, "Pelleos

28-10 .

rrrr

J-

HP

^^b^ Permission for reprint granted by Durand et Cie, Paris, France, copyright owners; Elkan-Vogel Co.,

Inc., Phila-

delphia, Pa., agents.

Jlo-

bo

.

.3 r^

mnd

@

13 n

which has appeared in the projection mnd as a combination two triads mnd at the interval of the minor third; and 208

of

minor third and minor second

Example Pentad pmn ^s^d^t

Wyi^'^' which has appeared two triads nsd at the

W @

nsd

Involution

interval of the

nsd

minor

@

1

J2

combination of

as a third;

and

30-8

Involution

t^

pmn

which has aheady occurred

A

2

I

nsd

n^

in the projection

p^mn^sd^

M^ W

^^H^'ȴ:^^^f

Example Pentad

30-7

@

j;^

in the pmn-tritone projection.

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,

show

that

respectively, will

major second, and minor second all

of the hexads so

naturally into the minor-third series, since

all

of

formed

preponderance of minor thirds with their concomitant

The

fall

them contain a tritones.

short recitative from Debussy's Pelleas et Melisande ade-

quately illustrates the hexad formed by the simultaneous projection of

minor thirds and minor seconds:

Example

30-9

Debussy, Pelleas and Melisande

j)i

#

^'

^'

'/

g'j^^jT I'/pipp^^'

^

n-

Permission for reprint granted by

"^ bo Durand

^

J^

)iM)i\^

^

i

et Cie, Paris, France, copyright owners;

Elkan-Vogel Co.,

Inc., Phila-

delphia, Pa., agents.

The quotation from example

Stravinsky's Petrouchka

of the projection of

is

an excellent

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. by permission of Boosey & Havifkes, Inc.

Review the projections

Revised version copyright 1958 by Boosey

& Hawkes,

Inc.

Used

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

210

select.

:

31

Simultaneous Projection of the

and Major Third

Perfect Fifth

If

we

project three perfect

above C, C-G-D-A, and two

fifths

we produce

major thirds above C,

C-E-Gfl:,

scale CsDsEsGiGifiA,

fm^nhHH: Example

1 p3

+

the six-tone isometric

31-1

«s

m^

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

2

J

J

J

.1

2

2

3

already described as the basic perfect-fifth pentad; and

Example Pentad

*J

-0-

4 J.

31-3

p^m^n^d^

1.

3

\ 1

I

M.

I

t

~

9 2

m^

.9

aZ 1'

211

SIMULTANEOUS PROJECTION OF TWO INTERVALS

which

a

is

new

which may be analyzed as and a minor second above and

isometric pentad, and

the formation of a major third

below G#,

Wd\

contains two pentads, each with

It also

Example p^mns ^d^t W-

-ZgL

involution:

31-4

^m 115 ^m Involution

W

f-f^

11

5

2

its

-^

2

p3+ d2

»^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

Example p^m^ns^dt

Involution

i^\h^ 2

2

3

nj.

«i

p' +

1

which we have met before

^ r

iiJ

:

r

13

m'

2

2

^ms i

p2

+ m^

by the simultaneous projection major

two perfect

of

fifths

both the

is

formed

and two

thirds.

we now

major

I

as a part of the projection of

'pmn (Chapter 22) and 'pmd (Chapter 24) and

triads

If

31-5

third,

two perfect

project

we form

fifths at

intervallic analysis as the previous scale,

involution of the

the interval of the

another isomeric twin having the same

first scale.

The

but not constituting an

scale thus

formed

is

C-G-D

+

E-B-F#, or C2D2E2F#iG4B, which also has the intervallic formation p^m^nrs^dH:

Example

i p?

212

@

m

31-6

a

^€i^

2

2

PERFECT FIFTH AND MAJOR THIRD This scale will be seen also to have a close resemblance to the perfect-fifth series, for perfect-fifth scale It

it

consists of the tones of the seven-tone

with the tone

A

omitted.

contains three pentads, each with

Example n^s^d

p-^m2

2

i

pmn

@

which has already occurred

^

p3m2nsd2

pmn

31-7

h^n'r 4

p

Example

31-8

m

Involution

pmd

4

@p

14

2

M @

p

as the relation-

pmd

as the projection of

and

two

@

p

triads

pmd

and

Example

31-9

p^m^ns^dt

Involution

^^

fe=*

2

simultaneous

striking

2

I

^2 +

in the projection of the triad

projection

major seconds.

2 2

p« + s^

1

which we have met'

of

two

perfect

fifths

^=m s3

i

pns

as the

and

three

,

example of the projection of two perfect

the interval of a major third Stravinsky

i

pmn

2

s^

ii

at the interval of the perfect fifth;

A

2

pmn projection

in the

which has already occurred

2

3

at the interval of the perfect fifth;

t

^^

2

involution:

Involution

J.J^r 2 3 4

ship of two triads

its

Symphony

is

fifths at

found in the opening of the

in C:

213

simultaneous projection of two intervals

Example Symphony

Strovinsky,

#^ m

in

31-10

C

e ^ ^ n Jf iSj,

Strgs., Hns.,

*'"^-

him

m

¥

i^

Winds

p2@nn-

^--.^v.^

rimp. Copyright 1948 by Schott

An

&

Co., Ltd.; used

by permission

of Associated

Music Publishers,

Inc.,

New

York.

perfect fifths

example of the simultaneous projection of two and two major thirds, giving the pentatonic scale

CDEG

may be found

excellent

Ab,

in Copland's

Example

A

Lincoln Portrait:

31-11

Copiond,"A Lincoln Portrait"

^m

Hns.

nnti

^

^Sr

4u iuDa,Tro., bo ,Trb.,' cellos, Cellos, Basses

r

r

Copyright 1943 by Hawkes

214

r

& Son (London),

r Ltd.

Used by permission

of

Boosey & Hawkes,

Inc.

:

32

Simultaneous Projection of the

Major Third and Minor Second Projecting major thirds and minor seconds simultaneously,

form the six-tone scale C-E-G#

+

we

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 addition of the tritone

and greater emphasis on the major

Example Hexad

^'

third:

32-1

p^m^n^s^d^ t

%.T3ft^ " "

Ss

^J-

tt

I

I

This scale contains two isometric pentads

Example Pentad

P which is formed below G#; and

32-2

p^m^n ^sd^

2^

4

I

t

of a perfect fifth

Example

,»^

™2

32-3

mn^s^d^

^'

^ I

^2

and a major third above and

Minor Second Pentad

#^

'^

I

215

SIMULTANEOUS PROJECTION OF TWO INTERVALS

which

the basic minor-second pentad. There are two additional

is

pentads, each with

its

involution:

Example pm^ns^d^t

.Pentad

1^

I

2

4

32-4

Involution

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

Example Pentad p^mns^d^t

32-5 Involution

aJ ltiJ|

%

226

duplicate the

34-2

2 3 2 3 C

$ p2s2

The combination

and

p2m2

m

34-4 n^sd^

=^ 12 —

4

—

-r.

•- vu 1

(

Example

PROJECTION BY INVOLUTION

The combination

and 5 forms the pentad

of 1

G B C

tp'd',

,

F Db or

CiDb4F2G4B, p^mhHH^:

Example

34-5

t

* ^"

bo

j-bJ

14

p2d2

The combination

^

^ 2

r 4

and 3 forms the pentad

of 2

D A C

ts^n^

,

Bb Eb

p^mnhHH:

or CsDiEbeAiBb,

Example

34-6

p^ mn^s^d^t

iJj^T 16

bo X

The combination

2

s^n'

1

and 4 duphcates the major-second pentad

of 2

D E C

Xs^rn\

,

Bb Ab or C2D2E4Ab2Bb, m^sH^:

Example

#

^'t^e %

«2m2

\

M

i2J 2 J

1'^

4

34-7

^r 2

^

(''^

2

1

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

# =^©:

iv^rr iJW^r^r 116

=F^

t

34-8

s2d2

I

1

The combination

of 3

I

^ s I

I

and 4 forms the pentad

A E C

tn^m\

,

Eb Ab or C3EbiEl^4AbiAl^, p'^m^nHH:

Example y

Ml

I

The combination

n£m2

of 3

I

II

3

34-9 I

I

\J

14

I

1

and 5 forms the pentad

A B C

,

XnH\

Eb Db or CiDb2Eb6A2B, m^n^s^dH,

Example 26-7 minor

as the projection of

thirds, A-B-Cjj:

228

which has

+ A-Ct^-Eb:

also

been analyzed

in

two major seconds and two

projection by involution

Example m

n

s

d

34-10

1

fej^

5 12\,j^r 6

And

finally,

[^r

2

2

rV 112

the combination of 4 and 5 forms the pentad

E

B

C

tm^d^

,

Ab Db or CiDb3E4Ab3B, p^m^n^sd^:

Example

* m bo t

The only way

in

J.^ 13

m^d^

(or Gb). For example,

and add the six-tone scale

g 1^ 4 3

which an isometric

formed from the above pentads

F#

34-11

tritone

if

is

we

six-tone scale can

by the addition

take the

first

above and below C,

of the tritone

of these pentads

we produce

C2D3FiF#(Gb)iGli3Bb, p^m^nhHH-.

Example

#

n

\}Q

34-12

*

t p2s2t

i 2J

The remaining pentads with the

J 3

tt^ 1

tritone

be

mT ^

13

added become

C3Eb2FiF#iG2A, fm^n^sHH^: 229

the

INVOLUTION AND FOREIGN INTERVALS

Example

34-13

p2m2n^s3d2t2

^»^

li^J Jp2n2t

^

112

2

3

C4EiFiF#iGiAb, p^rrfnhHH:

Example 34-14 p2m3n2s3d^t

I$p2ri5J 4 J

'-^

^

h\^ I

I

I

I

CiDb4FiF#iG4B, p'mhH'f:

Example

34-15

P^m2s2d^t3

1 p^d^t

I

4

I

4

I

CaDiEbsFJsAiBb, p^m^n's^dH^:

Example 34-16 p2m^n^s2d2 9 u |/

Jl

S C2D2E2F#2Ab2Bb,

111

,2n2T ?

I I

I

f

t2 I

3

I

m«s«^3.

Example 34-17

}

230

s2m;1f 2

TCT "^

€»^

2

^' 2^

2

2

)

PROJECTION BY INVOLUTION

CiDbiD^4F#4BbiB^,

fm^nhHH;

sHH

(duplicating 34-14)

n^mH

(duplicating 34-16)

t

CgEbiEoFJ.AbiAl^, fm^n's-dH^- %

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 Bb,

we

find that

it

contains the whole-tone scale C-D-E-F#-Gfl:-

A#: and three other six-tone

scales,

Example m

p

* 1.

n

s

d

o 2

CaDsEsFifiCsBb

©- ff" 2

2

with

the

MS U ~ p2m4n2s4dt2 III

\)

2

which may

I

2

each with

its

involution:

35-1

t

Example M

scale C-D-E-Fjf-G-Ab-

2

bo

;cH

"

t

11 involution

EgGiAbaBbaCaD,

35-2 Involution IIIVUIUIIUII

12

3

2

2

be considered to be formed of four major

also

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 ^m^^m ^g i i=F

2

232

2

2

2

1

ts''

HI*

is.'*

+

n^ t

MAJOR-SECOND HEXADS WITH FOREIGN TONE 2.

with

CoD.EoFJfiGiAb

the

involution

Example p'^m^ns^d^t'

'2

2

F^iGiAbsBbsCsD,

35-4

Involution

12

2

2

2

which may also be considered as the projection of four major seconds and two perfect fifths above C, or below D;

Example

+

3.

C4E2F#iGiAb2Bb

p2

with

35-5

the

Example

+

s*

I

involution

p2 I

E2F#iGiAb2Bb4D,

35-6 nvolution

*^

*-4

2

I

12

112

2

4

which may also be considered as the projection of four major seconds and two minor thirds above E, or below B^:

Example

#

12

2

s"

35-7

+

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 that

is

|mVp|.

X'^^s^pi or

Xm^s^n^

Example

:|)mVn|.

or

Example

^m

1 l* ^m 1 £

^

5

*J!?

fifth

C—

above

Example 35-4 becomes 35-6 becomes t:mV

* J

d2

p_'

involution

iJi,j^^^^

ii p2

1

b^i +

n2

p^m^n^s^d^

'

^1 rV 2 13 V 1

t^

t

1

i^p2

^-

n2

Involution

+

244

^

^-i^ir'r'^^^j^^ -^ 13 2

34 p^m^n^sd^t

13

-1 1

nvolution

33.p!m!sVt2

2

i'^

1

I

J|J Jil-' ^"4213

.

T T 4

II

Involution

32 pm^n^s^dt^

i

r

4

I

^

n^

%

Involution

J I't'

"8

l.»

30 p^msd^t^

i

n'

+

m-^

i

p^m^n^sd^

i

31

2

Involution

^ 27 p^m~n^sd

4

12

3

+J1^

p2 + d2

I

4

p2+ d2t

RECAPITULATION OF THE PENTAD FORMS

36

p^m^rr^sd^t

Involution

J

tri^ rrJ

p2+32 :^fl

i

4 d2

I

p2+ s^

+ d<

p^mn^s^d^t J 2

I

6

¥^Ins'

t'*

^"it

2

I

t.a

«2

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. that

the projection of five perfect

fifths

We

have seen

above the tone

C

produces the hexad C-G-D-A-E-B.

Example

Referring to our twelve-tone

form a figure having to B.

We

note,

also,

five

38-1

circle,

we

note that these

six

tones

equal sides and the baseline from

C

that the remaining tones form a com-

plementary pattern beginning with

F and proceeding

counter-

249

THE THEORY OF COMPLEMENTARY SONORITIES clockwise

formation as

complementary hexad has the same counterpart and, of course, the same intervallic This

G^.

to

its

analysis.

Example

38-2

pSm^n^s^d

p^m^n's^d

#

f^

m

k^z bo

2

m Q

«

gy. tJ

2

''"

-

^"

I'o

==

2

3

u> i^

2

Since the hexad |F-Eb-Db-Bb-At)-Gb

the isometric involution

is

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

note, also, that the

clockwise, the result

is

complementary hexad on G^

the same.

We

merely the

is

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,

second major triad on

G— G-B-D,

on

E— the

triad

hexad CsD^EsGiGifsB:

Example

^ 250

^

m

38-4

i2

.

J 2

•' 1

a

we produce

and a third major

E-GJf-B. Rearranging these tones melodically,

we form

ti^

r

THE COMPLEMENTARY HEXAD

we now diagram

If

this

hexad,

in the following example, the

by

solid lines

we produce

the pattern indicated

major triad hexad being indicated

and the complementary hexad by dotted

Example

lines:

38-5

G^(Ab)

(Bbb)A

i Here

it

j

will

;^Eb

a

g^

^g

il^^ ^ g

^k

be observed that the complementary hexad

F-Bb-Eb-Db-Gb-A

(Bbb)

is

not

the

transposition

involution of the original, and that the pattern of the

be duplicated only

in reverse, that

is,

this

the

first

can

F and statement may

by beginning

proceeding counterclockwise. The validity of

but

at

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.

discovered that the two patterns cannot be in this

manner. They will conform only

upon C and the second pattern

is

if

It

will then

made

the point

to

F

be

conform is

placed

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

|F2EboDb3BbiBbb3Gb.

involution,

the

It will

triads,

as

triads

the tones of a major triad, so the second hexad

combination of three minor

is its

be noted further that

hexad was produced by the imposition of major

first

upon

hexad CoDoEgGiGJoB

to the

is

a

the minor triad being the

involution of the major triad:

Example

38-6 p^m'^n^s^d^t

p3m''n3s2d2t

2 2 3

13

As might be expected, the sonorities

identical:

is

2

2

intervallic

three perfect

analysis

fifths,

3

of

3

1

the

two

four major thirds,

three minor thirds, two major seconds, two minor seconds, and

one

tritone, p^m'^n^s^dH.

The

and most complicated, type of complementary

third,

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

+

hexad C-E-G hexad

minor second forms the

at the interval of the

Db-F-A^, or CiDbsEiFoGiAb.

Its

complementary

consists of the remaining tones, DiEbsFJsAiBt^iBti.

Both

hexads have the same intervallic analysis, p^m'^n^sdH but, as will

be observed similarity

in

one

Example

38-7,

the two scales bear no other

to the other.

Example

j^-j pmn

252

@

d

I

3

J

J I

2

38-7

"^ I

:

^^rp

j^JjjJ I

3

3

I

I

THE COMPLEMENTARY HEXAD Ftt

THE COMPLEMENTARY HEXAD

A III,

fourth type includes the "isomeric twins" discussed in Part

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 plus

thirds at the interval of the perfect

A#-C#-Et^

E#-G#-B.

Example ,

fifth,

p3m2n4s2d2t2

i

bo^°'^ pi +

38-8

p3m2n4s2(j2

t2

-ijg^^t^^ :^tJ|j«Jj^ 2

13 12

3

13

2

1

^

W n^ @

253

p

39

??

The Hexad

We

are

'Quartets

now ready

to consider the

more complex formations which are

resulting from the projection of triads at intervals

own

foreign to their

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

tone— perfect-fifth projection cited plementary scale

is

example of the

Example

six

38-3, the

com-

simply a transposition of the original

scale.

in

In other cases, as in the major-triad projection referred to in

Example

38-5, the

original

scale.

complementary scale is the involution of the However, in fifteen cases the complementary

scale has an entirely diflFerent order, although the

same

inter-

vallic analysis.

We

have already observed in Part

formation of what

we have

III,

Chapters 27 to 33, the

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 254

triads

pmn

:

THE HEXAD at

QUARTETS

the interval of the minor second, already referred

involution will have the order,

Its

to.

12131, or CiD^oEbiFbaGiAb,

having the same analysis and consisting of two minor triads

The complementary

the interval of the minor second.

at

scale of the

original will consist of the tones DiEbsF^aAiBbiBti, also with the

analysis p^m^n^sdH.

+

fifth.

same

Begun on

two

FJsAiBb, or

B,

may be

it

mnd

triads

analyzed as B3D1EI9

at the interval of the perfect

This scale will in turn have

its

involution, having again the

analysis:

Example p^m^n^sd^t

39-1 Complementary

Involution

Involution

He xad,

@

mnd

The

triad

six-tone

pns

C-G-A

scale

p^m^n^s^dH.

at the interval of the

or CiDbeGiAbiA^iBb, becomes involution CiDbiD^iEbeAiBbi- The

Its

complementary

+

minor second forms the

Db-Ab-B^,

scale of the original

is

DiDifiEiFiFJfgB, with

its

involution

Example p^m^n^s^d^t

*^

PM

The

©" d

triad

six-tone scale

I

6

pns

I

I

39-2 Comp. Hexed

Involution

'

I

I

!•

I

6

I

at the interval of the

C-G-A

+

r

I

I

I

5

Involution

5 II

I

I

major third forms the

E-B-C#, or CiCJgEgGaAsB, p^m^nh^dH.

255

THE THEORY OF COMPLEMENTARY SONORITIES Its

becomes CaDaEgGaBbiBti. The complementary

involution

scale

DiE|:)2FiGb2Ab2Bb, with

is

its

Example

involution:

39-3 Comp. Hexad

p'm^n^s^d^t Involution

12 12

The

+

C-F#-G

scale

p^m^n^s^dH^.

involution

Its

plementary scale

is

D-G|-A,

The

DboEbiEl^iFgBbiBt], with

39-4 Comp. Hexad

triad pdt at the interval of the

C-F#-G is

C4E2F#iG3A#iB

is

+ E-A#-B,

3

nsd

scale

p^m^nh^dH.

Its

2

its

involution:

39-5 Comp. Hexod

112

124

3

1

at the interval of the perfect fifth

C-Db-Eb

+

G-Ab-Bb,

or

is

Involution

I

3 2

I

I

forms the

CiDb2Eb4GiAb2Bb,

involution becomes C2DiEb4G2AiBb.

plementary hexad of C-Db-Eb-G-Ab-Bb

256

I

I

or C4E2FJfiG3A#iB, p^m^nh^dH^.

Involution

@ m

triad

5

major third forms the

DbiDl^iEboFgAbiAti, with

p^m^s^d^t^

six-tone

Involution

CiDt)3EiFoG4B. The complementary scale of

Example

The

involution:

I

involution

pdt

its

_s_

six-tone scale Its

CoD4F#iGiG#iA,

or

p^m^n^s^d^t^ Involution

@

1

becomes CiC#iDl:iiEb4G2A. The com-

Example

pdt

12

2 2

pdt at the interval of the major second forms the

triad

six-tone

2

Involution

The com-

D2EaFiFjj:3A2B, with

THE HEXAD its

These hexads, with

involution.

and secondary strength

fifths

QUARTETS

in

preponderance of perfect

their

major seconds and minor

thirds,

are most closely related to the perfect-fifth series:

Example

nsd

triad

six-tone

scale

p^m^nh^dH.

Its

Comp.Hexod

Involution

p*m^n^s^d^t

The

39-6

+

Et|-F-G,

or

CiDbsEbiEt^iFoG,

involution becomes C2DiEbiEtl2FJt:iG.

plementary hexad of C-Db-Eb-Et^-F-G its

major third forms the

at the interval of the

C-Db-Eb

is

involution. This quartet of hexads

with an equal strength of major

thirds,

Involution

The com-

D4FiJ:2G#iAiAifiB, with

is

neutral in character,

minor

thirds,

major

sec-

onds, and minor seconds:

Example p

@

nsd

m

n s d

t

39-7 Comp.Hexod

Involution

Involution

_nn

The last of these quartets of six-tone isomeric scales is somewhat of a maverick, formed from the combination of the intervals of the perfect fifth, the

major second, and the minor Second.

C and project simultaneously two two major seconds, and two minor seconds, we form the pentad C-G-D + C-D-E + C-C#-D, or GiC#iD2E3G, If

we

begin with the tone

perfect

fifths,

with

involution C3Eb2FiFJ|:iG, p^^mnrs^dH:

its

Example

39-8

Pentad p^mn^sTl

p2 f S2

If

we now form

+

t

Involution

d^

a six-tone scale

by adding

first

a fifth below C,

257

THE THEORY OF COMPLEMENTARY SONORITIES

we

form

the

with

CiCifiD2EiF2G,

scale

involution

its

C^DiEbsFiFJiG:

Example p^m^n^s^d^t

iff

If

we add

I

?

I

39-9

Involution

2

2

122 I

the minor second below C,

CiCfliDsEgG^B, with

1 I

I

we form

the six-tone scale

involution C4E3G2AiA#iB:

its

Example 39-10 p^m^n^s^d^t

Involution

43211

['"1234 Upon examining that they

all

have the same

also discover in

we

find

intervallic analysis, p^m^n^s^dH.

We

these four scales, Examples 9 and 10,

Example 11 that the complementary hexad of

Example 9 is the same Example 10:

scale as the involution of the scale in

Example Hexod

Original

39-11 Transposition above

Comp. Hexed

C

I

12 12 (If

we

4

3

11

2

4

3

2

11

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 discussed in Chapter 29.

It will

be noted that

both the pentad CiCjfiDsEsG and

Example p2m2n4s3d2t2

258

its

this scale contains

involution jDiCJfiCl^sBbsG.

39-12 Involution

THE HEXAD

The complementary hexads

may

all

be analyzed

of

QUARTETS

Examples 39-1

as projection

by

to 39-7, inclusive,

involution, as illustrated in

Example 39-13:

Example

^

mW^ ih

^

_

3.

^

u ° t

^g

^-

m2

p^

s'

*

x< )-E#

A# and

Cf,

we

and (C#)-E#-(G#), giving the

twelfth tone E#. This tone becomes the converting tone, is,

the initial tone of the descending complementary scale.

Example j>

>/

A# and

superimposing major triads above

40-6

^jiy.;tlly

»l

266

l*P

»

ii

t

% ^1^

tii

l

f/yl|JH¥^

)

EXPANSION OF THE COMPLEMENTARY-SCALE THEORY

The complementary heptad major triads

pentad composed of two

of the

at the perfect fifth,

C-E-G

+

G-B-D, or C2D2E3G4B,

becomes, therefore, iFaEbsD^gBbiAiAbsGb. The projection of

C-D-E-G-B 2 and In

therefore CsDsEsGiG^iAaB.

is

(See Ex. 42-1, lines

5.

many

other cases, however, the choice of the convert-

ing tone must be quite arbitrary. For example, in the case

any

of

choice

composed

sonority entirely

is

for example,

arbitrary.

entirely

major

of

seconds,

the

The whole-tone hexad above C,

C2D2E2F#2G#2AJ1:. Since this scale form super-

is

imposed on the original tones produces no new tones but merely octave duplications,

is

it

obvious that the converting tone of

the scale C2D2E2F#2G#2A# will be B-A-G-F-Eb or D^, giving

complementary

the

scales

J,B2A2G2F2Et)2Db'

iA2G2F2Eb2

DbsB, iG2F2Eb2Db2B2A, jF2Eb2Db2B2A2G, jEb2Db2B2A2G2F, or

iDb2B2A2G2F2Eb. The choice of

Example 40-7

is

I

I

^ ti"

I

22222 jf-'

ff""

40-7

Complementary Hexads

Major Second Hexod

J

as the converting tone in

therefore entirely arbitrary.

Example

J

F

I I

"

r r 'I r r

't b^

22^22 I

I

i

I

—22222

|rr'rhvJir"rh- .Jji'rVr^JJ

^

22222

Take,

again,

22222 |

the

CsDJiEaGiGfligB. If

upon each

of the

CsDJtiEeGiGSsB;

22222

major-third

we superimpose tones

hexad

in

i

^rp^^ 22222 Example

of the hexad,

we form

F#,

A#,

(G)3A#,(B)3D,(D#)3F#;

and

D

40-8,

the hexads

iE),(G),{G^), (G#)3(B),(C)3

(D#)i(E)3(Gti); and (B),D,(Dj)^)sFUG)sAl giving the tones

T

this intervallic order, 31313,

(D'i^)sFUG)sAUB),D^;

(B)i(C)3(D#);

r L

r

and producing the nine-tone 267

new scale

)

COMPLEMENTARY SCALES C2DiD#iE2F#iGiG#2A#iB(i)(C.) The remaining tones, F, Db,

and A, are therefore

equally the result of further superposition and are

all

possible converting tones, giving the descending

all

complementary

iFgDiDbsBbiAsGb, iDbsBbiAsGbiFsDl^, and lAaGbiFgDiDbsBb- Our choice of F is therefore an arbitrary

choice from

scales

among

three possibilities.

Example

^mo

40-8

^

"

«

^"*'^" av^^ ^ '

' ^'

13 13

3

-^fr

13 13

3

Complementary Hexads

13 13

3

One

final

13 13

3

example may

suffice.

The

tritone

3

13 13

hexad of Example

40-9 contains the tones CiC#iDjFJfiGiG#. This scale form superimposed upon the original tones gives the hexads CiCj^J^^Fjj^iGi

G#;

(C#),(D),D#4(G),(G#)iA;

(G),(G#),(C)i(C#),(D);

(D),DS,E,(G#)AAJf;

(F#),

(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

Our choice possibilities.

268

of (

F

is

as

iF,EiEb4BiBbiA or jBiBbiA^FiEiEb-

therefore an arbitrary choice between

See the Appendix.

two

5

expansion of the complementary-scale theory

Example

"^"

V- ..|t» o

m

\;

m

Jj

(I)

^ ^ j|

12

3

1

Complementary Pentad (2)

? 13

3

tff^

J

itrl^r

comp. Npnad p^m^n^s^d^t^

Involution of

I

Complementary Hep Heptad

3

^p^mSn^ S'=^d^

3

I

12

1

I

comp. Heptad

Involution of

Jg

J

i

^^m

3

^I'^ii^ 9^ j^j J;iJ p3m4n5s3d4t2

^A

I

I

Complementary Heptad

1

Hexad _pVnJs^dft

3 o.A In 5./ Involution volution of

pm^n^sd^t

12

Pentad

''y5mnd@n +

J

I'i'p

2

I

I

Jit.

m

mnd @ mnd(g

tif

J

Pentad

3 3-^ ^f^

42-4

1

Complementary Triad

(

I

°"^

mnd

2

I

I

I

II Complementary Nonad

Triad

III IQ5 ncmd

@d

+ ji

i

Hexad

p^m^nVd^t

p^^n^s'^d^t comp.Heptad ept

14

Complementary Hexad

I

"/5 Involution of

I

2

I

I

4

Complernentary

Pentad ,

i^

I

r

I

14

1211 291

COMPLEMENTARY SCALES 12.^

Involution of comp. Nonaid

p^m^n^s^^t^

Example nsd Triad

2

I

nsd

@

d

Pentad (I) mn^s^d^

^A risd 7^ nsd

@

n n_

Pentad

^

@

nsd

+

I

pmn3^d^t

comp.Heptad

iiHV^^?V.' B.A Involution of

i

r

I

I

8y( nsd

Triad

^ " nsd

@

I

I

I

Ji Jb J ;

s^

l

,

Jl' J

-

^

r^r^;N 2

I

(2)

-"-"^-^^J

2

Complementary Triad

2

Pentad pmn^s^d ^

dl i^jJi|J

I

I

2

3

Complementary Heptad

1112

12

Hexad pm^n^s^d^t

Complementary Hexad

^W 12 I

292

•QJiJiJJ

I

Complementary Nonad

_s

@

I

(I)

Complementary Pentad

(2)

III

'°-^nsd

(2)

3

JiJ J

r:r I

I

12

I

ri>r ^

comp Nonad p^m^n^s^d^t^

I

I

Complementary Pentad

(I)

JHH^f.^Vp3.3...,e Involution of

I

1112

'pe.3n.s5.s,

comp.Heptad

I

(I)

Complementary Hexac

III 5/) Involution of

I

I

II

12

Hexad_gnA^|s^d^

ji

1112

I

I

Lomplemen Complementary Heptad

(2) pmn'-y-d^t l^l

12

I

Complementary Heptad

I

I

^/5

42-5 Complementary Nonad

I

2fl

Complementary Triad

I'

I

2 6

5

1112^

1

r

PROJECTION OF THE TRIAD FORMS Complementary Pentad

"yj Involution of comp.Heptad fAn^n'^s^dSf

112

I

5

1

comp.Nonad p^m

Involution of

12-5

m ww^

J^pi^r

¥W=*

I

I

I

I

'

^

Complementary Triad

^ '^^^ '

2

I

3

2

I

VsVf

I

Since the triad mst cannot be projected to the hexad by the

superposition,

counterpart

is

and proceed

simplest

to consider

it

Example

as in

method

Example

4

Nonad

da

r

(6)

Involution of

nine-tone

42-6 Complementary

^^ 2

its

major-second hexad,

42-6:

mst Triad

I

forming

of

as a part of the

comp.Nonad

pmnsdt

r

T

^r ^r ^r

112

2

m

J li I

I

(2)

Complementary Triad

:^ IT

I

The p^s,

9

?

I

I

I

I

f?)

4

2

(6)

projection of the triad forms of the six basic series—

sd^,

nH, m^, and pdt— were shown in Chapter 41.

ms^,

The opening

of the author's Elegy in

Memory

of Serge Kous-

sevitzky illustrates the projection of the minor triad first six

third,

pmn. The

notes outline the minor triad at the interval of the major

C-Eb-G

+

Et]-G-B.

The addition

of

D

and fourth measures forms the seven-tone A-B, the projection of the pentad

pmn

@

and

A

in the

second

scale C-D-E^-Eti-Gp.

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 Complementary Sonorities

Its

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

ary hexad. Line 2 gives the triad mst with

nonad, begun on

A

its

its

complement-

complementary

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

upward.

line

Its

6 and projecting the same order of half-steps

complementary pentad becomes the involution of

the pentad in line

6.

Line 10 forms the

eight-tone projection

first

by taking the

second part of line 3 ) and projecting the same order of half-steps upward. Its complementary octad first

is

complementary octad

(

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 294

line

4 and

THE pmn-TRITONE PROJECTION

same order

projecting the

of half-steps

upward.

complement-

Its

ary tetrad becomes the involution of the tetrad of line Finally, line 12

derived from the complementary nonad of

is

2 projected upward,

line

its

complementary

involution of the triad mst of line

@

P'^'^

-^P^ 13 2 13 2^

13

^^ '4

2

i 1

1

J id

\

4

I

2

I

I

p^mn^sd^t^

2

2

^

I

I

3

1^' 3

2

I

3

'

pT

°y) Involution of comp.Heptads p

i

Iff

3

:^

? 2

Si ^ 13

m

r

I

^^i^ 12'

1

Complementary Complement ary Hexad

3

4 3 4 3

a

y

r

3

I

^

p^m^nVd^t^ pmnsdt

Hexad

r

^^

^ 2

2

13 2^

'l^

3

Complementary Heptads

pm^n^'s 2„2.2d^2

7-^

1

I

112

2

^m

i 3

2

3

I

I

3

I

^75 Pentads

or

^

->

2

3

3

Complementary Octads

pmnsdt

1^

I

2 2 ^%"^H,^,^^M^

I

I

pmnsdt

3y) Tetrads

2

Complementary Lpmpie Nonad

mst

Triad

43-1 Complementary Hexad

Hexad

^

being the

triad

2.

Example '•/I

4.

i4

3

n s d^t

2

3'

I

^

"

I

T

f-j^

Complementary Pentads

^3 3

1

*i 2

^^ i *rt r

I

p^mVs'^d^tS

r

3

2

112

4

2

i

¥^;^ 295

COMPLEMENTARY SCALES p^m^n^s^d^t ^

10^ Involution of comp.Octads

pi ^

^^ 3

11

2

p^m^n^s^d^t^

^^

Ji^ ^"^

J

|v

2

2

I

r

^

13

I

comp.Nonad p^m^n^s^d^t^

'2« Involution of

2

I

3

I

^

Complementary Tetrads

2

I

m

2

Complementary Triad

2

I

This projection offers possibilities of great tonal beauty to

composers clearly

who

allied

are intrigued with the sound of the tritone. It to

the

minor-third

but

projection

is

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

The opening

of interval

of the Sibelius Fourth

sixteen measures (discussed in Chapter

C

Symphony— after the first 45)— shows many aspects

The twentieth measure

of the pmn-tritone relationship.

a clear juxtaposition of the

dominance.

major and G^ major

contains

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 major triad with a tritone added below the root and fifth.

The student

will profit

symphony, since earlier

it

from a detailed analysis of

exhibits

a

fascinating

C

this entire

variation

between

nineteenth-century melodic-harmonic relationships and

contemporary material.

The opening

many

trates

Symphony No. 2, Romantic, projection. The opening chord is

of the author's

aspects of this

major triad with a tritone below the root and with a

illus-

a

D^

third, alternating

G major triad with a tritone below its third

and

fifth.

Later

the principal theme employs the complete material of the projection of the pentad

296

Db-F-G-A^-B, that

is,

Db-Dt^-F-G-Ab-Al^-B.

THE pmn-TRITONE PROJECTION However,

it

is

not necessary to examine only contemporary

music or music of the exotic scale forms.

late nineteenth

The strange and

century for examples of

beautiful transition from the

scherzo to the finale of the Beethoven Fifth nificent

tones

Symphony

is

a

mag-

example of the same projection. Beginning with the

h.\)

and C, the melody

first

outlines the

configuration

Ab-C-Eb-D-FJj:, a major triad, A^-C-Eb, with tritones above the

and third— D and Ffl:. It then rapidly expands, by the addiand 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 root

tion of G, A,

characteristic tetrad of the pmn-tritone projection.

This projection

is

but the relationship

essentially is

melodic rather than harmonic,

as readily

apparent as

if

the tones were

sounded simultaneously.

297

)

44

Two

Projection of

Similar Intervals

at a Foreign Interval

with Complementary Sonorities

The next projection

be considered

to

which are composed

tetrads

relationship

of

two

similar intervals

We

a foreign interval.

of

the projection of those

is

at

begin with the

shall

examination of the tetrad C-E-G-B, formed of two perfect at the interval of the

interval of the perfect

Line

1,

major fifth.

(

third, or of

two major

See Examples 5-15 and

fifths

thirds at the 16.

@

m with its comp Line 2 gives the hexad formed by the

Example

44-1, gives the tetrad

plementary octad.

@

@

m) m, with complementary hexad. Line 3 forms the eight-tone projection

projection of this tetrad at the major third— (p its

the

of the original tetrad

upward the order

by the now

familiar process of projecting

complementary octad (second part

of the

of

line 1).

Since

all

of these sonorities are isometric in character, there

are no involutions to

be considered.

Example 1

75

p

@m

^m Tetrad

4

2j^p^@rn@m

Hexod

H

id 3

298

p^

m^

nd

Complementary Octad

p^m^n^d ^

3

11

3

1

Complementary Hexad

r^ry

Jtf^r M'

J

JiJ^

^ri^

r 2

4

3

I

44-1

3

I

3

J^

12

TWO

PROJECTION OF 3yi

i

Involution of

comp.Octad p

m

d

n s

SIMILAR INTERVALS Complementary Tetrad

t

W

'^' ' J i 2 JiiJ 113 12

^m

4

4

3

1

The remaining tetrads Example 44-2 presents the

projected

are

similar

in

manner:

minor third

interval of the

at the

relationship of the perfect fifth:

Example

i

£ @ £ s=

p^mn^s

Tetrad

4

^S

f\

Involution of

2

3

p^mfn^s^d^^

comp.Octad

t

It""

i.Jjt 4 2!

2- m@t_@ m

^ m

Tetrad

or

§

s

2

i^j

m

n

^

J "Tpfs

3ft^

44-3

r 'r

Hexad m^s^t^

p

1

Complementary Tetrad

4

compOctod

2

I

Complementary Octad

t

fer^^'^t^^ i J Jtf^*> 2 2222 3y) Involution of

J 12

at the tritone;

Example m @

;J

^D^itJ Z

34

There follows the major third

Vj

l

1

Complementary Hexad

^2121121

'

2

^^12 ^^ 2

M^ 12

i|J

r

3

Hexad p^m^n^s^d

2ji«_n@£@p

^

Complementary Octad

jj ^T 3

44-2

s

d'^t

2

I

I

J

^r 'r

r

2

^ I

I

Complementary Hexad ;

r

22222N 'r ^r

i^r^

Complementary Tetrad

^

i 2

1

12

2

4 1

I

2

'

4

299

COMPLEMENTARY SCALES the minor third at the interval of the major third;

Example @ m

n

I.

4 b^

pm

Tetrad

ib^

^8

3

2'/5n@m@m

2

2

n

Complementary Octad

d

r

3

I

3

13 13

J^bJ

|(J

3

it

2

I

I

^

^

I

I

I

2

^li

13 13

Complementary Tetrad

^^ 13 ^

?= 2

I

I

^r^r

r

^

3

p^m^n^s^d^t^

3y, Involution of comp.Octad

2

I

Complementary Hexad

gti°ibJ^jT^r

t®<

^^^

^r 't ^r

r 3

p^m^n^d^

Hexad

44-4

3

the major third at the interval of the minor second;

Example ''/5

m @

d

J&.

b is

pm

Tetrad

2

=^^ 3S

1;.€U

:#^

i^ 3

3.^ Involution of comp.Octad

iP

ibJ I

bJ

IjJ I

I

3

I

I

I

3

'

^^^ J

^r

r

I

3

3

I

I

p^m^n^s'^d^t^ Complementary Tetrad

ft^p

^ i

t|J I

II

Complementary Hexad

J

W-g^ I

I

i

II

p^m^n^d^

Hexad

^g^

^^ ^ Complementary Octad

I

I

^- m@d.@iTi

nd

44-5

2

3

I

13

jU 1

the minor third at the interval of the major second;

Example '•^Jl

@

_s

pn^s^d

Tetrad

2

I

2

44-6 Complementary Octad

2

II

1

112

Complementary Hexad

2

300

I

I

I

2

I

TWO

PROJECTION OF ^yjlnvolution of

SIMILAR INTERVALS

compOctod p^m n^s^d

Complementary Tetrad

t

9f

W.2

I

I

I

jtJ

12

2

2

I

I

ji

f

the minor third at the interval of the minor second;

Example '

n

/I

@

Tetrad

d

!+ 1^*

fl

Q

? 2

^&S

I I

2 3 4 5 n

,

.

,

,

comp. Octad p

I I

m

n s d

and the perfect

@

izf^ iJ I

'" ^

m 2 s 2 d4

iu r^J U ^^1

l"

I

The

m)

I I

3

•» 3

I I

I

r'T r'T^^ 1113 113

4

I

I

-X

:

I I

i

It

r

I'

r

Hexad

I

p^m^n^s^d^t^ Complementary

113 I

2

I

^~^

itJ

I

C^mplerr^entary

t

11

14 Involution of comp.Octod

I

44-8

I

4

I I

Complementary Octad

-"'^

Hexad p

l^@jd^@j^

3

I

^^

p^md^t

6

I I

the interval of the minor second.

fifth at

Tetrad

d

I I

I

Example p

I I

Complementary Tetrad

t

I

I

I I

I

III 13

1^1 r

I I

Complementary Hexad

d

s

[III .ft

Involution of

Complementary Octad

I I

Hexad pm

'^ji@d@d^

^S^

mn2sd2

44-7

C 6

I

Tetrad

«Ce:

I I

@ m) @ p; (n @ p) @. n; n @ @ s) @ n; and (n@ d) @ n are not

reverse relationship of (p

@ n;

(m

@ d) @ d;

(n

used as connecting hexads in Examples 44-1, respectively because they

all

(

2, 4, 5, 6,

and 7

belong to the family of "twins" or 301

COMPLEMENTARY SCALES The

"quartets" discussed in Chapters 27-33, 39.

relationships of

@ p; and (p @ d) @ d; are not used as connecting {p @ hexads for the same reason. The reverse relationship of Example 44-3, (m @ t) @ t, is not used because it reproduces itself end)

harmonically.

In the second the

first

movement

of the Sibelius

Fourth Symphony,

nineteen measures are a straightforward presentation of

the perfect-fifth heptad on F, expanded to an eight-tone perfectfifth scale

by the addition

of a

B^

the Beethoven example. Chapter

Measures

twenty-five

of

the

twenty-eight

present

the

heptad

@

opening being built on the expansion of the tetrad

C-E-Gb-Bb to (See Example

302

to

measure twenty. (Compare Example 15).

in

n pentad. Measures twenty-nine to however, depart from the more conservative material

counterpart of the thirty-six,

pmn

4,

its

eight-tone counterpart C-D-E-F-Gb-Ab-Bb-Bt^.

44-3.)

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

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 see,

some

of these projections result in tetrads

seconds, producing the familiar perfect-fifth pentad, with

its

complementary heptad; while

line 3 gives the pentad formed 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

by the

projection of

involution of the pentad of line gives the heptad

which

plementary heptad of

is

line 3.

the

2.

Line seven, in similar manner,

upward

projection of the com-

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

which is not hexad projection.

a third connecting hexad, C-D-E-G-A-B,

included because

it

duplicates the perfect-fifth

303

COMPLEMENTARY SCALES

Example

^

p^mns^

Tetrod

2

2

-

-^

2./)

p2

3./)

+ s^

Pentad

5/^

I f^ ^'^

^2 J 2

i

J

2

.1

jiJ

2

S r'T

i

^^

12

2

2

V Y ^r 11 2

r 'T 2 2

r "r ^r

2

2

2

I

2

Connecting Hexad (?)p2rT (2)p2m^s^d^t2

2

^f

'f

(1)

iJ

2

Complementary Hexad

It 'y ^r

'T

J 2 J 2 Ji2 11

I

''^ I

^

^^12

(2)

^

Complementary Hexad

&

(I)

V'T^r''^^^ 2

Complementary Heptad

I

2

I

Complementary Heptad

p2m2ns3dt

J

11

2

Connecting Hexad (l)p'*m^n^s^dt

(2)

J

l

comp.Heptad

6.^ Inv.of

^

(2)

2

2

3

7 ^2232

^ 4.ij

Complementary Octad

PentadlDp^mn^s^

-^

-0-

45-1

J2 J2

(I)

2 p

jj|J

J

2

12

nrr

n s d^t

fpplementqry Pen Pentad Complementory

2

2

2

^r

B.^ inv. Inv.of ot 8./5

2

11

2

2

I

comp. Octad p comp.uctad p^i^n^sV\^ i n s d t

2

2

11

2

Example 45-2

I

2

2

(I)

^2

3

Complementary Pentad ^r

2

2

1^

^1" comp.Heptad ^2) (2)p^m^n^s^d^t^

Inv.of

2

2

(2)

k

i^F

^

1

Complementary Tetrad Lompiementi

2

2

3

^

gives the projection of the minor second

and

the major second which parallels in every respect the projection just discussed:

Example '/)

d^

+

A

I*'

304

mnsd

Tetrad

I

2

45-2 Complementary Octad

II

III 12

SIMULTANEOUS PROJECTION OF INTERVALS

^

^fi

Pentad

s^

+

(i)

r d^

^fi

*

F 1*1

I

r

|1f

1^

r

I

I

I

12 6y5

(I)

A

7. 7.1^

Inv.of

B.ij 8.J

2

^^ ^ ^^ ^112 I

I

I

Complementary Hexad

I

2

ll)

3

I

Complementary Hexad

(2)

*

2

2

II

45-8

I

comp. Heptad p^m5n^^d^l2 III 9 u

? «^ minor

2

2

(r

4 .2_4„3.2^3 2A Connecting Hexad lOpfnvjTfsfd^t 1^

4^

(I)

gives the projection of the minor second

^^

5*it

I^

^.i 3

I

third:

Example I.

'

13

1

Example 45-8

2

^

hI

Conriplementary Hexad

3

comp. Heptad p^m^n^s^d^t^

2

major

p^m'^n^s^d^t

(2)

'T 2

Complementary Hexad

p^m^ns^d^t^

2

3/5

^r 2

2

1

I

2

Complementary v^v.riiipidiici iiui Pentad CI iiuu MHHHM jr

112

I

4

M'

+

gives the projection of the perfect fifth

with the second interval in both

its

m'

and

upward and

projection:

309

COMPLEMENTARY SCALES

Example P^lg plm^nZsdZt Comp.Heptod

45-9 Comp. Heptad

pf-if*g|"r^°n2S2dt,

12

I

^^'^S&.^fz" 1^

I

I

3

13 13

2

^

I

I

I

r r r T'-r '

3

^^

3.^(2)p3m4n3s2d2t

ffa'miL&^fe"^ ^ CompHexodsU)

Conr,p.He«odsU)

I

r

I

a/^^X^"^

i Example 45-10 minor

I

'

/)

d'^

.

^ + n^pmn^s^d^t Comp. Heptad

III

3

I

I

pm2n3s4d4t

I

D.jt Kci p-'m'- n*- s'Ki^T'^ 3.^l2)p3m2n2s2d4t2

UjJbV 13

•

^ r

I

m il)

I

4 3

I

I

I

I

and

I

3

I

2

I

r^pb[>iJ

I

^ * ^ Comp. Pentad

1113

4'

Connecting Hexads r«r«r, uovnHcn) Comp. Hexads U) p4m2n2s2d3t2

114 12

2

^§ I

I

(I)

i^^^iJJ

J

4/1

3

^ Pentad j^i p^m^n^sd^t Comp. Heptad

„ td^^

2

r«n«r, u«v«He Comp.Hexads

1112 ^1

JtJJ>^T-' 3

p^m^n^s^d^

jjjMJf ^"4

2

I

I

^w^p 3

(3)

I

Iny.of Comp.Heptad mv.oTUDmp.i-iepToa

7.^

JJ J>J 12 11

3

J

i

I

i

iJUp

il»

J]

I

(4)

1

4

12

o J^*^

•

m

^

1

1

Complementary Tetrads

2 \

W

na;re.:Vr«rr

p

mi I

.

I

2

11

12

4

II

I

p6m5n5s5d5t2

2

P

2'

I

iTf^rr^^

,2_2„2„l t

I

Complementary Pentad

4

P^m^ nSs^d^t'

j ,.^.

o

i:i."ii'"

Comp.Octads Lomp.uct

Inv.of

9/1

l

1

o

J^^n? It

|

J^ .

3

8-/,

o

p^m^n^s^d^t

im

J

I

2

2m2 n2 eh p'^m'^n^s't

5 1

2

1

J

p2 m'

J

p2

j

4 12

I

Comp.Nonad p8m6n6s7(j6|3 Inv.of

10. 'fi

2

11

12

1

s' T

I

t

fr iJ

I

5 2

p^

}

^^^^f^

2

I

o

i \

Complementary Octads

m

4

3.^ J

p'^mnsd

II

^

p

mr np sf

Complementary Triad

5

2

317

COMPLEMENTARY SCALES Example 46-3b forms the projection example

of the previous

the perfect

fifth

rather than

same two intervals two major thirds plus

of the

in reverse, that

is,

two perfect

fifths

plus the major

The pentad, heptad and connecting hexads

third.

the same, but the tetrads and octads are

are, of course,

diflFerent.

Example 46-3b Jm^

Triad

iS ^

m

2.tm £T 1

.1!

J jJ'' 4 3

3. .

Jm ^"'p

I

|l^»

1

^

13

Pentad

1

p^m^n^sd^

Hexads p

^ 112m I

2

I

m

n s d

2

I

2

I

I

I

2

I

112

1

318

12 11

f

p2

n2

r7i2

*^

2

^

2 ^^2 ^2

JbMJ ^^'T^

I

Complementary Hexads

I

2

I

I

^ruf^r^r 3 ?

? ? m'^IL

I

I

r"^ 2

I

Complementary Pentad

4

12

1

(jl

^

ii"M»:g "^ ~j|-C5f«-

p^m'^n^^d^

t£

'

I

I

1

tes rrr^^ 3

I

«

XJ-

2

Complementary Heptad

4

I

Z^g^m^n'^s^d^t

I

^fTr^

Comp. Heptad

Inv.of

3

3

1

P^

f4

I

V^r^iirritJiiJ

1

6./J

2

I

^^^^^s

4 3

l

3

n^ d^

tp2 m2 n2

jJ ^ jj 4 12

5.^ Connecting

I

"rV'Trii^^^ I

^ ^m

^

2

Complementory Octads

pm^nd

4

tm^

pm^nd

I

'

s 4

1

Tetrads

m^

'rftirriiJiiJJ,|i^MH

4

4

^^

Complementary Nonad

m'^

^^2 ^2

^fy

1

d' t

PROJECTION BY INVOLUTION Comp.Octads

Inv.of

8.^

2

I

2

I

I

J

I

2

I

I

p^m^r^

* t» ^ Complementary Tetrad /-

,

d't

$

1^

1

tm2

13

4

Comp.Nonod p6m9n6s6(j6t3

.2

^^Z^ d'*

2

I

,

4 3

I

I

9^ £^rn[n^s^d£j2

3

o o

o

p5m7n5s4d5t2

4 3

pi t

1

Inv.of

12

1

2

I

I

{p^rr^ji^d^

Complementory Triad ^

^z

2

I

Example 46-4a continues the same process and the minor third;

for the relationship

of the perfect fifth

Example I

^

^£

p^s

Triad

5

i

iS 3

p^mns^

i

Complementary Octads

Jp

2

2

522

mm w 322

£ H^ Pentad p^m^/?

^

p^ m^ n^ s^

i

I

J J

'^'i'^Ji^iiJ I

I

2

,2

s^

r

r^^«^iJ 21222

p2

$2

rr?

^^^B& O

r

"

•

Complermentary Heptad

J

£^ U?

"rrWftjjj 2 2

2

2

5-^ Connecting Hexods p m^n^'^dt

2

12

2 2

I

s^

it^'.'M^-fe

2222

Complementary Hexads

2

12

2

n' i

^ 8^ g V3

'

I

Is:

3

t

|iiM»:;iiB'f'

J

"r

„2 £^ ^2„l n rrr

2 2

)^mns^

I .X3L

J

"rr

2

3.fl^^n'i

4.^ t

Complementary Nonad

2

2 ^ le n.^ Tetrads

±1

46-4a

2

p^m'^nVdt^

319

I

COMPLEMENTARY SCALES Com p.

Inv. of

Heptad

p^m^nVd^t^

7-

2

2

1

2 2 2

3

Jp^m^s^nU

p6m5n5s6d*t2

8.^

2

12

I

1

J2^

2

t

2

I

2

2

p^m^s^

lbg"^°'"

t

p2 n't

n'

ii|iAlJlf 5

Comp.Nonad pSpn^nSs^dSfS

p^

2

n' t

I

I

^

2

2

3

2 2

of

Inv.

10

2

2

3

i jj^jjj^^^r 2

2

4^^^

Complementary Tetrads

2 2

p^m^n^s^dV

a,

Complementary Pentad

Comp.Octads

of

Inv.

}p2m2_s2

2

I

I

I

2

and Example

2

t

p

Complementary Triad

s nr

nn

2

5

I

I

t_2

gives the reverse relationship— the minor

46-4??

third plus the perfect fifth:

Example $

n^

Triad

y

n^

p't

Tetrads

t

rrii^A^i J iJJj

i"''

i

i

36

33

pmn^t

2I2IIII2

Compiementory Octads

if'''"'>j^^[^rVn-^'rrit^^WJJ 342

3.. t

n^

p'

IS

334

m324

Jn^F? -ti 4./(*-

2122

pmn^st

*

p^m^s^d^

Complementary Nonad

§^

^S

is 2

n^t

46-4Z7

j

tfB)|t S:

p^m^s^d

t^'::

I

Jii]

i

Pentad p^m^ n^s^t

2 12 112 212

p2 m^s^d'

"^^

2

Complementary Heptad

J

p^nn^s^

-ft

i ""l>jjJ^ 3 2

320

2

2

:*rr i^»^t^Jj i

2

1

2

2 2 2

;

tij^8

i 334

'

i*"h*

^

t

i

PROJECTION BY INVOLUTION Connecting Hexads

5^

4 2 3 4

pmrrsdt

^s 21222 i§

rr' Vii^J i

21222

i^jJ^'^r 3

2

2

^^^nMi^j

2

2

3

2 2 2 2

Comp. Heptad

Inv.of

p'^m^n'^s^d^t^

7^

Complennentary Hexads

^

iijjL N^f 12

1.811

i

2

2

p^m^s^

2 2

ii^

'fWiU2 3

Comp.Octads pSmSn^sSd^fS

^n^ p2

Complementary Pentad

2

2

Iny.of 8.>,

Jp^m^s^d't

J J iJi J

i J212. ji,j

|

2

II

pSm^nSsSd^tS

9.-

g |, ~

f

2'

*

i

1

I

342

HD

rl^

Inv.of '0/,

2

I

\

n

334

«2^2c2 p

p'

\m

[riip J||j] i

n^

i

n2

pi

t

-jt»

l^ gn^i^'rii^tJJ[ii^"rrT]rtig 3 3 4 3 2 4

2 2

Comp.Nonad p6m6n8s6d6t^

»iM, mrtf jjij J

^

i-

JJJr # jj ^jJt 12 2

Complementary Tetrads

$

p2m2s 2^2

Complementary Triad

3

2

6

^

^2

3 3

Example 46-5a gives the vertical projection of the perfect fifth and the minor second, and Example 46-5b the reverse relationship:

Example I

^^

P.

Tried p^s

46-5a

Complementary Nonod

tm^n^ _ _ _s^

j_

Complementary Octads

2„2 „2

„i

t HI

_ci

2.^$£

d^

t Tetrads

p ^ msdt

J]

p^

1 P

4

COMPLEMENTARY SCALES Complementary Heptad

J£^d2 Pentad p^m^s^d^t^

4

i

&

TiiKt

3'i¥

^ 14 ^4^

2

I

5.- Connecting

12

2 2 4

p4m2s2d^t3

i

iJ 4^llJ

j>

I

I

^.^ 4

I

Pui'l.'«B

f

I

I

nSs^d'^t^ II III a u M p'^m'^ |j 7/1

*f^2

I

(.

S

f,2

2 2

^^ ^P 4

I

uomp.i-iep Inv.of Comp.Heptod

^

Complementary Pentad

l>J-|g.W

114

2 2 Inv.of Comp.Octods .

2

J~

Hexads p^m'^ns'^d^t^ Complementary Hexads

2

8.

JjiJiiJ

2

^ ^^ g^

^

Ap2 ^2

4 2

Complementary Tetrads

pSm^n^s^d^t^

^-2

^jl

i

^JT^il^ l^_p'^

13 2 2 Comp. Nonad p8|n6n6s7(j6t3 I

I

I

Inv.of 'j1^^

r^r r^^ ^^^^JJ

i

^ ..Connecting Hexads p^m^n^sd^t

I

p^m^d^n't

p^^^^^^

^^rA

Im^nU i,un.

{

3

I

I

3

I

I

p^m^n^sd^t

^ 13

j^jjiJiiJ

Inv. of

7 i,

2

I

I

2

I

J

I

3

Complementary Pentad

I

p 2 m^cj 2

nU

J

I

14

p2m2d2n't

4

4

m

J

1

2

324

1

12

1

12

4 4

m^

J

m2

3

2 ^2^,2^2 Complementary Triad

jjiJiiJj^JY^rfei-::'^^

j

te j^ ^ 14

Inv. of

'0>,

rn^

n2

n'

4

i[3?:^^s'-":r^ri^^'i^[

2

Comp.Nonod p6m9n6s6d6t5

I

j

1

Complementary Tetrads

3'

I

jjjb^'r

2

^

3

9u£5m^nV^d^

^w

ETH!

11

3

1

J

Inv. of Comp. Octads 8-p5rT^7n5s4a5t2

*^r

12 14

Comp. Heptad

p^m^n^s^d'^t^

12

I

:r«rr»^iiJ»

11

3

^ j

ni2

n'T

3

projection by involution

Example '•

m^

n^

I

Triad

i m

^1 n2 n'^m't Tetrads

2.

^-

$

dt

i^it^ iiJiiJ)jj

2

I

I

TT r

3

;

JbJfcj J 4 2

J, 2 ^

m4 n 3 sd 3 t

^^

m^d^s'

#f:ttu)iB

itJiJ^j 1213111

^

ir"rr«^^iiJ«J

p2 ^2

,1

fc,

jj2

if^'tiB

3

I

Complementary Hexads

t^ i^

J|

J

it^

HI

I

I

Complementary Heptad

ji,j^jiJiiJ

Connecting Hexods p

P.

J p2

i

^2^2

fgi'^^mo-

i

2

pmn ^dt

n^ m^ Pentad p^m^n^d^t

i

2

r^rr

2

5.

I

J

5

iJ

(|,''iii');

p2 m^ d^ s^

Complementary Octads

I

J ^Jlr^J^j] J 351 315

ti I,.

I

I

pmn

n^m'^

J

#

3

I

J

rffrr'^«^>JiiJtJ^j]i"nt^tBtfe 2 2 II

^P

Ig

feEdfi

^>

Complementary Nonad

n^t

6

3

46-6??

i^r"^

i

3

1

I

^

3

p^m^n^sd^t

i

3

Jl

i

3

I

^

'

r'rr'i'^ 12 14

I

I

p^mSn^s^d^t^ O U p III 11

$

I

Complementary Pentad

p2 m^ d^

^^

§=

2 Inv.of ®-

-

3

I

I

I

,2

J

^

I

jt

I

I

I

pSmSn^s^d^tS " ~ L *"

I

r r 3

I

f4|J

i

m2 m "I? 2 n^

e

t

i'

Comp. Octads

p^mSnSs^dSt

jj 2 jjbJjiJ^iir 2 9-

r

Comp. Heptad

Inv.of 7-

''''^i

jbJuJ

j>

2

I

3

I

_2 d. m2 s* Complementary

pfm

Tetrads

(ui.!:teu: r^^r J 3

I

P^m^d^ il

* ii

IL'

1

2

t n*-

^ ^n2 —

5

"^

5

s't' 2.

3

„l m' ^

1

3

15 325

m'

t

COMPLEMENTARY SCALES of Comp. Nonad p^m^nSs^d^t^ p-m'"n'"s^a"'T^

Inv.

'0-A

j t

£2 ^2 £^ nn- ^2 a;-

Complementory Triad

s2 s;-

^

1111

12 12

Examples 46-7a and

j

6

3

46-7Z?

show the

n2

£

vertical projection of the

maior third and minor second:

Example '

^

m^

Complementary Nonad

m^

Triad

12

4 4 ^2^1 dt Jm

2.

3-

^-

pm^nd

Tetrads

I

3

I

I

I

3'

I

m^d^ Pentad p^nrrrTsd^

>J

13

fet

4 3

I

t^fJiy.! 2

4

I

I

p2

m^

%

^ ^ ^ ^^

%

m^

rf d^ p' t

^

^2

^2

2

I

Complementary Heptad

f

J"^

^^

j

2

pm^nd

\

5-

2

I

m^d'l

''^'

2'

I

1

Complementary Octads

4

J

f

46-7a

2

5^2

||llui »i|. l

Connecting Hexads p^m'^n^^d^ Complementary Hexads

jjg.ji^^r

'I

12 14

=

3

^^ 13

4

12

p'm^n^s^d^

13

12

Jm^n^d^

p^m^n^^s^d^t

I

12 14

3

Comp. Heptad

inv. of "^i.

4

2

326

I

4

I

2

Complementary Pentad |

13

4 3

^^

PROJECTION BY INVOLUTION Inv.of

8.

Comp.Octads

tm^n^d^p't Complementary Tetrads

p5m2nV*d5t2

^^ 12

j^

jj

i

i

i

2

2

I

I

2

I

jm^n^d^p^

I

d' t

jJ^ iS 13

4

Complementary Triad

.^2

4 4

2

I

[||

4 4 5

Comp.Nonad Lomp.Non p6m9n6s^d6t3

'°^

m^

J

2

I

Inv.of inv.oT

,-

tiJ

4

3

i.T#aj*rrtJ_J

i

3

I

I

r r

Jm^ n^ d^^'l

# ^j^ jtjJ J iJf I

^p

"

I

p5m7n5s*d5t2

9-

i;;^B"i«r

,ii l

112

3

1

JiiJp

^^Z^jl^

Example 46-7b I

^

I

^^

Complementary Nonad

Triad

sd^

iJ

r ir r

2

t

i

=o=

d^

^

II

10

I

^-

J

d^

»

4.^

df

I

4

I

M

13

p^^n^sd^

4

3

Connecting Hexods p m^n^s^d

g

jgi'^ r 12 14 3

iiJ

n^ d^ s^

I

f

^2

^2 ^z

I

I2I4II

'

^^'^^^

^^^^

^

^^

t

n?

2

I

I

4

I

d^

^

IIII4I2

Complementary Heptad

^

j

m^

n^ d^

2

Complementary Hexods

^ Sg^ I

3 4

p^m^n^s^d^

J.

I;

J ^ 3

iJ^J^

i*r"r^^r^

12

12 14

4

gi

rr]:^r*rtVitJiiJ«JtJ iite^Btt'

113

j''^ r ^ r j 173 113

jT^ Pentad

I

Complementary Octods

pmnsd^

m'l

""^°'"

I

I

m't Tetrads pmnsd^

137

m^

^^^^^m

mP€»-

i

|

3

327

s' *

|

COMPLEMENTARY SCALES Comp. Heptad

Inv.of

^

111

1^

II

a

I

{ J

jl^

Complementaryy reiiiuu uuiii^iBiiieinui Pentad

3

Comp.Octads

Inv.of

tn?Q? d^l'*

14

2

1

im^

p5nn5n5s5d6t2

t

3^11

7

II

Comp Nonod p6m6n6s7d8t3

'°.

|0«

n^ d^s't

Inv.of

,»

113

7

^

^plj

^jZ

-=

'»

V5

d^ m't

i'Hil;;^e^^;tf|Mlrjj|hM„j]^^

7*^ * jM^i^A^^r 14 12 I

f

4 3

Complementary Tetrads

13

I

m^

Q'd^

t |

^^^^^^^^^

p5TT,5n5s5d6t2

9.

qc d^2

rnc m2 nc

14 12

2 8-

u

| rn2 n2

d2 s^

3

r«^rvi«™«^r,+„..„ Triad Complementary t^:«^

^

^2

Ji>Jl|J^JljJ I

I

4

I

I

I I

I

10

I

Examples 46-8a and

46-8Z? give the vertical projection of the

minor third and the minor second:

Example I.

^

n_2

S

m 3

^-

$

s

^^ 2

I

Complementary Nonad

^m^s^ d^

Complementary Octads

t

mn^sdt

s

^"r'Trnt

6

II

12

3

^

1

m^

J n 2

m2

t

XiiiriiV f'""aijJrfri-^ijJ-rV II 12 13 2 12 6 i

3

^

i

t

^

^ I

328

2 6

2

Complementary Heptad

r^"rVrii^

Complementary Pentadj n^ d^

d^

s^

^^g P

.^^bt!a

12

Inv.

2

4

2

12

I

^

12

3

9> p^mSnSsSdSfS

^

m^/

I

I

I

2

A

2

I

I

f^2 g2

d^p'4

t d*^

2

J

d2n't

^J

I

mns^d^

82

2

.k d t

f J J ^r^ ^fr^ ^p|.^[il.,[^fq_pg. i

12

6 2

6

j ^,2

46-8Z?

Complementary Nonod

10

Tetrads

A Jn

[

I

3

i

2

Triad sd

I

I

o

1^

^

6

p2 Complementary Tried

jj2

Example 1.^

2

I

^u^^ ^^_: ^Hi^^H/.^^rJ II 12 13 2 bo nv.of Comp.Nonad '°>)p6m6nSs6d6t4

^^ fc^

2

U Complementary Tetrads

j2

^^^^^^ I

6

128

I

I

' I

I

4

I

J

m2

s^

d2 n2

I

Complementary Octads

$

H)^!^

IIIII42 329

^2

n't

COMPLEMENTARY SCALES

i 5.^

m2n2c3H2f m^n^s^d'^t

PontnH Pentad

-|2n2

^ ^ci^pia

4.A ^

ii^jt'^ 2

Complementary Heptod

^

J

2

4

2

I

©=

2

Complementary Hexads

Connecting Hexads pm^n^s'^d^t

^i'^

:

^UbbJb

116

I

I

h2 d

c2 s^

^^

'r'lrVrii JtJ r

6

m^2

r^r^r'Ttpg

2

I

I

6 2

I

pm''n2s^d2t2

^^ 2

2

7y)

/I

4 2

p2m4n4'^d'^t^'^

i

IlL^

J"r i ^M^ 112 I,.

.

I

I

.

r^rrr^ riJ ^ 12 2 4 2 bfc>9

l

^

£

ni^

Complementary Pentad

;'l#H

:r^rrh.l.J ly J

Inv.

Q-*

l

Comp. Octads

of

*

1^2

p^^m^n^s^d^t^

^Jl^p J I

!>

•

^^ s

2

I

4

I

^

2

I

m2

^2

d2

£t bo p ^ '

:

i^ji^j^j^j'iJ^^r

K).

1

4 2

1

I

p^r|,|^3°'

111 The

I

I

4

J

m^

g

D

'

/*

.

^

td'^

n't

^

=&

2

^

^^

10

I

vertical projection of the perfect fifth

duphcates the perfect-fifth

and major second

the combination of the major

series;

second and the major third duphcates the major-second

and the

1

8

s2 d2_n2 Com^plementary Triad

I I

ii..o

^ 8

2

2

^

4

«fR^ t

6

Complementary Tetrads

nU

d^ tfe

I

9- p4m5n5s6d6t2

1

?3^

12

4 2

^ ^S

2

"i..

..

vertical projection of the

series;

minor second and major second

duplicates the minor-second series.

The

vertical projection of the

results in a curious

following chapter.

330

minor third and major second

phenomenon which

will

be discussed

in the

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

twelve-tone scale which

plementary

scale. It

complementary examine

it

is,

scale.

is

not

of the tonal material of the

all

itself

a part of

its

own comits own we should

instead, a part of the "twin" of

Because of

its

unique formation,

carefully.

In Example 47-1, line

1 gives

the tone

and major second above and below

it.

forms the descending complementary

C

with the minor third

The second

scale,

half of line 1

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 secondsone 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

octad, while line 2a forms the octad projection.

give the tetrad CgEbeAiBb with

its

complementary Lines 3 and 3a

octad projection. Lines 4 and

4a form the projected octad of the tetrad CsDiE^^Bb, and lines 331

t

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

i

n5

^

Pentad

47-1

Complementary Heptad

p2nir,2s2d2|

rirB^j^jj^r^V'MriirViirr 16 2 3 2 1

Comp. Heptad

Inv. of

la.

^^^ If

I

.9 2/5

I

#

_s2

.ii-:^,;^^;

p2 s2

$

Complementary Octad

"rV^nir'ntrri iJii

"^2"^ t^

re

I

I

I

I

3

2

i

d^

m'

p2 j2

^2

ppl

ii

fj

I

p^m'^n^s^d^t^

$

^

iJbJ^J ^*^F7^ 13 ^^T 2

^ $2

p2

$

Comp.Octod

d2

^f^

2

3

d2

I

pn^sdt

I S

Inv.of

I

I

p'^m^n^s^d'^t^

Tetrad

1.

*,-^^

2o.

I

I

p2

j

^

XT «jt^-8-h

1

Tetrad

ln2s'| —

3

~

^C

"lib,

3 Inv.of

jbp

^j

(f 3o^

Complementary Octad * h i* ii

pn'^sdt P" sdt

6

I

4.^ t

*

I

$2 -

i ^^

I

I

2

Inv.

of

I

2

Comp Octad

II

I

d2

ii

mU

ii"'^ Ty

2

p2

s2

d2 m' t

^^ Complementary Octad $

17

I

p5m5n5s6d5t2

2

^s2

2

^^ 3

I

t

bri^rV«r'ir'iri

jftjj^j^J 332

I

p2 il.

'

^

n't Tetrad-^ pmnsHJ j—

2

^5,-

I

p^m'^n^s^d^t^

.^JbJ^JtfJ I

i^rT^fi^r'rriirr

1

Comp. Octad

t

r 2

I

I

I

i

3

p2

^

d 2 £> t

rrJ/::« e>ftB"" 2

2

t

p2

-nr

s2

d2

^^8-

nU

THE c

5-

i 5g

»

t

? _s^

Tetrad

.

jiU I

2

of

Comp. Octad

7

2"^

111112

I

p^m^n^s^d^t^

Example 47-2 shows the previous illustration to

s2

$ _p2

^^

bo- ^Q

SONORITY

Complementary Octad

pmns2d

^'

Inv.

MAVERICK

d_2 n' j

d^ n't

tP'

relationship of the pentad of the

its tw^in,

the pentad C-Cfl:-D-E-G, which

has the same intervallic analysis, p^mnh^dH. The the two pentads, each with

line gives

first

complementary heptad. Line 4

its

two complementary heptads but

gives the involution of the

with the order interchanged, the

first

heptad of

4 being the

line

involution of the second complementary heptad of line

vice versa.

The "maverick" pentad C-D-E-F-B

a part of the complementary scale of line 4.

The

both of

first

^or

^

will

1,

and

be seen to be

"twin"— second part of

its

pentad, C-CJ-D-E-G, will be seen to be a part

owTi related heptad and the related heptad of

its

its

maverick twin.

The connecting hexads the

first

also

show an

interesting relationship,

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

Example I-

Pentad A p2mn?s2d^

Comp. Heptad

ujjJ Iff

I

23

2 3

(I)

I 3./^

r

I I

i' I

131 I

Hexad p'^m2n3s3d2t

^ Ifl

In^f

42

Comp. Heptad (2)

:^rMJjjjijjJr'^"'ih-^''^Mi II2I2 2I6 16 IIII32 12 12 2 2 3 I

I

I

I

I

Hexad twin

i2r P^m^n3s3d4i

l~

line 3.

47-2 pTmr^s2d2t

Hexad 2./)

hexad of

12

Comp. Hexad

I I

I I

O 2

p2m2n3s3d4t

c 6

I

I

Comp. Hexad

2 2

12

I

Hexad

jU ^J^'''r i

^12 12

i_n2 + s'

4

t

>

-^^I^v^bo^^t,,^^^

C.pnnd 776,673

r

7 2

pmn

ODOO' 2

7 2(3)

(I

C.pmn 777,663

i

5(5)

pns

767,763

C.pns

*

I

I

i^nt

FO

2

I

(2)

^^^» 3 3

(6)

i

3 1(8)

TRANSLATION OF SYMBOLISM INTO SOUND

Cm'

696,663

m^

^^^ 2

I

2

I

C.pdt

I

2

I

iW=

^ 4 4

(1)

I

766,674

^ (4)

pdt (,o)

I

I

I

2

I

I

I

(3)

I

I

I

2

I

I

I

I

I

6

(3)

^ C.mst 676,764

i

2 2

I

I

111

1«T

6

(5)

15)

1

mst

I

I

I

(2)

I

2 2

I

I

I

I

I

Example 50-3 gives the octads with same order as that o£ the chart.

4

2

(2)

4

2

(6)

(6)

their corresponding tetrads

in the

Example Eight-tone Scales ^"'

50-3

Involutions

'^p3 301,200

745,642

C.p"^

^^

^^ 12

2 2

5dS

ots^i :o:'=»

2 5 2(3)

2(1)

p^^ 211,200

Cp^s^ 655,642 tf..o^ov>^^-); 2 2 2

1

I

2

I

"^^-^^*^ 2 2 2

(I)

1

I

2

I

.boM

12

1

3

l

12 11

1

(2)

I

C.p/m 665,452

t

^ ^ 2 113

4

3

p^m'l ^ i

2 2

3

^^^R^ 3

°^obo

112

1

I

^

3(2)

211,110

,^"^i

"u |

4

1(2)

I

p/nn

re^^

FF

4 3

2(1)

IV

2(5)

4

I"

1

r

I I

II M

2(5)

221,010

^

4(1)

T,

^^

!D

II(5) I

w

=®=EE

Df 001,230

M=

r

(S)

joC")

L(2)

i ^o^ n O ^ ;uboN

(5)

nbo[^

212,100

:

C.p^m't 655,552 b

2 3

p/n

^^Qobetjo

12

2

(I

Cp/n 656,542

2

(i>):

(9)

359

COMPLEMENTARY SCALES C.d^s^ 455,662

I

I

I

I

I

d^s^ 011,220

2

I

Cd/norn/d

(4)

I

I

I

I

I

2

I

(4)

456,562

d/n

P^

#

r

I

I

I

C.d^m'

I

r 2

3(3)

I

I

d£ml

555,562

$

:tnti: (8)

111,120

<

^S

ii^

14

2

I

«^?« I

1

(I)

I

I

2

4

I

I

I

I

^^^«=

13

(f)'

7

'l^

121,020

M:

=