LIBRARY
OF
WELLESLEY
COLLEGE
PURCHASED
FROM
in
2011
with
funding
from
Boston
Library
Consortium
Member
Libraries
HARMONIC
MATERIALS
OF
HARMONIC
MATERIALS
OF
MODERN
MUSIC
Resources
of
the
Tempered
Scale
Ilowar3™™lfansoir
DIRECTOR
EASTMAN SCHOOL
OF
MUSICUNIVERSITY
OF
ROCHESTER
New
York
n.
Copyright
©
1960 byAPPLETON-CENTURY-CROFTS,
INC.610-1
All rights reserved. This hook, or parts thereof, must not he reproduced in any
form withoutpermissionofthepublisher.
LibraryofCongressCard Number: 58-8138
PRINTEDINTHEUNITED STATESOF AMERICA
MUSIC LIBRARY.
'v\t:
who
lovesmusic but
does notentirely
approve
of the twelve-tone scale,This
volume
represents the results of over a quarter-century ofstudyofthe
problems
ofthe relationships oftones.The
convictionthat there is a
need
forsuch
a basic text hascome
from
theauthor's experience as a teacher of composition,
an
experiencewhich
hasextended
over a period ofmore
than
thirty-five years.It has
developed
inan
effort to aid giftedyoung
composers
grop-ing in the vast uncharteredmaze
ofharmonic
and
melodic
possibilities,
hunting
for anew
"lost chord,"and
searching foran
expressivevocabulary
which would
reach out intonew
fieldsand
at the
same
time satisfy theirown
esthetic desires.How
can
theyoung
composer be guided
in his search for thefar horizons? Historically, the training of the
composer
hasbeen
largely a matter of apprenticeship
and
imitation; technic passedon from master
to pupil undergoing, for themost
part, gradual change, expansion, liberation, but, at certain points in history,radical
change
and
revolution.During
themore
placid days theapprenticeship
philosophy—
which
is in effect a study ofstyles-was
practicaland
efficient.Today,
although stillenormously
im-portant to the
development
ofmusical understanding, it doesnot,hy
itself, give theyoung
composer
the helphe
needs.He
might, indeed, learn to write in the styles of Palestrina, Purcell, Bach, Beethoven,Wagner,
Debussy,
Schoenberg,and
Stravinskyand
still
have
difficulty incoming
to gripswith
theproblem
of hisown
creative development.He
needs
aguidance
which
ismore
basic,
more
concerned
with a studyof the material of theartand
PREFACE
less
with
themanner
of its use, although thetwo
can
neverbe
separated.This universality of concept
demands,
therefore,an
approach
which
is radicaland even
revolutionary in its implications.The
author has
attempted
to present here such a technic in the fieldof tonal relationship.
Because
of the complexity of the task, thescope of the
work
is limited to the study of the relationship oftones in
melody
orharmony
without reference to the highlyim-portant
element
of rhythm. This is notmeant
to assign a lesserimportance
to therhythmic
element. It rather recognizes thepractical necessity of isolating the
problems
of tonal relationshipand
investigatingthem
with
the greatest thoroughness if thecomposer
is to develop a firm grasp of his tonal vocabulary.I
hope
that thisvolume
may
serve thecomposer
inmuch
thesame
way
that a dictionary or thesaurus serves the author. It isnot possible to bring to the definition of musical
sound
thesame
exactness
which
one
may
expect in the definition of a word. It ispossible to explain the derivation of a sonority, to analyze its
component
parts,and
describe its position in the tonal cosmos. Inthiswav
theyoung
composer
may
be
made
more
aware
ofthewhole
tonal vocabulary;he
mav
be
made
more
sensitive to thesubtleties of tone fusion;
more
conscious of the tonalalchemy
by
which
a mastermay, with
the addition ofone
note, transformand
illuminatean
entire passage.At
thesame
time, it shouldgive tothe
young
composer
a greater confidence, a surer grasp ofhis material
and
a validmeans
of self-criticism of the logicand
consistency of his expression.
It
would
notseem
necessary to explain that this is not a"method"
of composition,and
yet in these days of systems itmay
be
wise toemphasize
it.The
most complete knowledge
oftonal material cannot create a
composer
any
more
than thememorizing
of Webster's dictionarycan
produce
a dramatist orpoet.
Music
is, or should be, ameans
ofcommunication,
a vehiclefortheexpression oftheinspiration ofthecomposer.
Without
thatinspiration, without the
need
tocommunicate,
without—
in otherwords—
the creative spirit itself, the greatestknowledge
will availnothing.
The
creative spirit must,however,
have
amedium
inwhich
to express itself, avocabulary
capable of projectingwith
the
utmost
accuracyand
sensitivity those feelingswhich
seekexpression. It is
my
hope
that thisvolume
may
assist theyoung
composer
in developing hisown
vocabulary so that his creativegift
may
express itselfwith
thatsimplicity, clarity,and
consistencywhich
isthemark
of allgreatmusic.Since this text differs radically
from
conventionaltextson
"har-mony,"
itmay
be
helpful to point out the basic differencestogether
with
thereasonfor those diflFerences.Traditional theory,
based
on
theharmonic
technics of theseventeenth, eighteenth,
and
nineteenth centuries, has distinctlimitations
when
applied to themusic
of thetwentieth—
oreven
the late
nineteenth—
century.Although
traditionalharmonic
theory recognizes the twelve-tone equallytempered
scale asan
underlying basis, itsfundamental
scales are actually the seven-tonemajor
and
minor
scales;and
the onlychordswhich
itadmitsare those consisting of
superimposed
thirds within these scalestogether
with
their "chromatic" alterations.The
many
othercom-binations of tones that occur in traditional
music
areaccounted
for as modifications of these chords
by means
of"non-harmonic"
tones,
and
no
further attempt ismade
to analyze or classifythese combinations.
This
means
that traditionalharmony
systematizes only a verysmall proportion of all the possibilities of the twelve-tones
and
leaves all the rest in a state of chaos. In
contemporary
music,on
the other hand,
many
other scales are used, in addition to themajor
and minor
scales,and
intervals other than thirds areused
in constructingchords.
I have, therefore,
attempted
to analyze all of the possibilitiesof the twelve-tone scale as
comprehensively
and
as thoroughly astraditional
harmony
has analyzed themuch
smallernumber
ofchords it covers. This vast
and
bewilderingmass
of material isclassified
and
thusreduced
tocomprehensible
and
logical orderPREFACE
chiefly
by
four devices: interval analysis, projection, involution,and complementary
scales.Interval analysis isexplainedin
Chapter
2and
appliedthrough-out. Allinterval relationshipis
reduced
tosixbasic categories: theperfectfifth, the
minor
second, themajor
second, theminor
third,the
major
third,and
the tritone,each—
except thetritone—con-sidered in
both
its relationshipabove
and
below
the initial tone.This implies a radicaldeparture
from
theclassic theories ofinter-vals, theirterminology,
and
their use inchord
and
scaleconstruc-tion.
Most
ofWestern music
has for centuriesbeen based on
theperfect-fifth category.
Important
as this relationship has been, itshould not
be assumed
thatmusic based on
other relationshipscannot
be
equally valid, as I believe theexamples
will show.Projection
means
the construction of scales or chordsby
any
logical
and
consistent process of additionand
repetition. Severaltypes of projection are
employed
indifferent sections ofthebook.Ifaseries of specifiedintervals, arrangedinadefiniteascending
order, is
compared
with
a similar series arranged indescending
order, it is
found
that there is a clear structural relationshipbetween
them.The
second
series is referred to here as theinvolution ofthefirst.
(The
term
inversionwould seem
tobe
more
accurate, since the process is literally the "turning
upsidedown"
ofthe original
chord
or scale. Itwas
felt,however,
thatconfusionmight
resultbecause
of the traditionaluse oftheterm
inversion.)The
relation ofany
sonorityand
its involution is discussed inChapter
3,and
extensivelyemployed
later on.Complementary
scales refer to the relationshipbetween
any
series oftones selected
from
the twelve-tonesand
the othertoneswhich
are omittedfrom
the series.They
are discussed in PartsV
and
VI. This theory,which
is perhaps themost important—
and
also the
most
radical—contribution of the text, isbased
on
thefact that every
combination
of tones,from
two-tone to six-tone,has its
complementary
scalecomposed
of similar proportions ofthe
same
intervals. Ifconsistency ofharmonic-melodic
expressionintensive study, for it sets
up
a basis for the logical expansion oftonal ideas
once
the germinating concept hasbeen decided
upon
in the
mind
of the composer.The
chartattheend
ofthetextpresents graphically therelation-ship ofall of the combinations possible inthe twelve-tonesystem,
from
two-tone intervals to theircomplementary
ten-tone scales.I
must
reiteratemy
passionate plea that this text notbe
con-sidered a
"method"
nor a "system." It is, rather, acompendium
of
harmonic-melodic
material. Since it is inclusive of all of thebasic relationshipswithin the twelve-tones, it is hardlylikelythat
any composer
would
in his lifetime use all, oreven
a largepart,of the material studied.
Each
composer
will, rather, use onlythose portions
which
appeal to hisown
esthetictasteand
which
contribute to his
own
creative needs.Complexity
isno
guaranteeof excellence,
and
a smallerand
simpler vocabularyused with
sensitivity
and
convictionmay
produce
the greatest music.Although
this textwas
written primarily for the composer,my
colleagues
have
felt that itwould
be
useful as a guide to theanalysis of
contemporary
music. If it isused
by
the student oftheory rather than
by
the composer, Iwould
suggest a differentmode
ofprocedure, namely, thatthestudent studycarefullyPartsI
and
II,Chapters
I to 16, without undertaking the creativeexercises—althoughif thereissufficienttime the creative exercises
will enlighten
and
inform the theorist as well as the composer.During
the first part of this studyhe
should try to find in theworks
ofcontemporary composers examples
of the varioushexad
formations discussed.He
will not findthem
in greatabundance,
since
contemporary
composers
have
not written compositionsprimarily toillustrate the
hexad
formations ofthis text!However,
when
he
masters thetheoryofcomplementary
scales,he
willhave
at his disposal
an
analytical technicwhich
will enablehim
toanalyzefactually
any
passage orphrase writtenin the twelve-tone equallytempered
scale.H.
H. Rochester,New
York
Acknowledgments
The
author wishes toacknowledge
hisdeep debt
of gratitudeto Professor
Herbert Inch
ofHunter
College for hismany
help-fulsuggestions
and
for hismeticulous reading ofa difficultmanu-script,
and
to his colleagues of theEastman
School ofMusic
faculty,
Wayne
Barlow, Allen IrvineMcHose,
Charles Riker,and Robert
Sutton, for valuable criticism. His appreciation isalso
extended
to Clarence Hall for the duplication of the chart,to Carl A. Rosenthal for his painstaking reproduction of the examples,
and
toMary
LouiseCreegan
and
JaniceDaggett
fortheir
devoted
help in the preparation of the manuscript.His
warm
thanksgo
to the variousmusic
publishers for theirgenerous permission to
quote
from
copyrightedworks
and
finally
and
especially to Appleton-Century-Crofts for theirco-operation
and
for their great patience.Finally,
my
devoted
thanksgo
tomy
hundreds
of compositionstudents
who
have borne
withme
so loyallyall thesemany
years.Preface
vu
1. Equal
Temperament
12.
The
Analysis of Intervals 73.
The Theory
of Involution 17Part I.
THE
SIX
BASIC
TONAL
SERIES
4. Projection of the Perfect Fifth 27
5. Harmonic-Melodic Material ofthe Perfect-Fifth
Hexad
406.
Modal
Modulation 567.
Key
Modulation 608. Projection of the
Minor
Second 659. Projection of the Major Second 77
10. Projection of theMajor Second
Beyond
theSix-ToneSeries 9011. Projection of the
Minor
Third 9712. Involution of the Six-Tone Minor-Third Projection 110
13. Projection ofthe
Minor
ThirdBeyond
theSix-Tone Series 11814. Projection of the Major Third 123
15. Projection oftheMajor Third
Beyond
theSix-ToneSeries 13216. Recapitulation of the Triad
Forms
13617. Projection of the Tritone 139
18. Projection of the Perfect-Fifth-Tritone Series
Beyond
Six Tones 148
19.
The
pmn-Tritone Projection 15120. Involution of the pmn-Tritone Projection 158
21. Recapitulation of the Tetrad
Forms
161CONTENTS
Part II.
CONSTRUCTION OF
HEXADS
BY
THE
SUPERPOSITION
OF TRIAD
FORMS
22. Projection of the Triad
pmn
16723. Projection of the Triad pns 172
24. Projection of theTriad
pmd
17725. Projection of the Triad
mnd
18226. Projection of the Triad nsd 187
Part III.
SIX-TONE
SCALES
FORMED
BY
THE
SIMULTANEOUS PROJECTION OF
TWO
INTERVALS
27. SimultaneousProjection of the
Minor
ThirdandPerfect Fifth 19528. SimultaneousProjection oftheMinor Third
and
Major Third 20029. Simultaneous Projection of the Minor Third
and
MajorSecond 204
30. Simultaneous Projection of the Minor Third
and
MinorSecond 207
31. SimultaneousProjection ofthePerfect Fifth
and
Major Third 21132. Simultaneous Projection of the Major Third
and Minor
Second 215
33. Simultaneous Projection of the Perfect Fifth and
Minor
Second 219
Part IV.
PROJECTION
BY INVOLUTION
AND
AT
FOREIGN INTERVALS
34. Projection
by
Involution 22535. Major-Second
Hexads
with ForeignTone
23236. Projection of Triads at Foreign Intervals 236
37. Recapitulation of Pentad
Forms
241Part V.
THE
THEORY
OF
COMPLEMENTARY
SONORITIES
38.
The Complementary Hexad
249Part VI.
COMPLEMENTARY
SCALES
40. Expansion of the Complementary-Scale Theory 263
4L
Projection of the Six Basic Series with TheirCom-plementary Sonorities 274
42. Projection of the Triad
Forms
with TheirComplementary
Sonorities 285
43.
The
pmn-Tritone Projection with ItsComplementary
Sonorities 294
44. Projection of
Two
Similar Intervals at a Foreign Intervalwith
Complementary
Sonorities 29845. Simultaneous Projection of Intervals with Their
Complementary
Sonorities 30346. Projection
by
Involution withComplementary
Sonorities 31447.
The
"Maverick" Sonority 33148. Vertical Projection
by
Involutionand
Complementary
Relationship 335
49. Relationship ofTones in Equal
Temperament
34650. Translation of Symbolism into
Sound
356Appendix: Symmetrical
Twelve-Tone
Forms
373Index
377Chart:
The
Projectionand
Interrelation of Sonorities inEqual
Temperament
inside back coverHARMONIC
MATERIALS
OF
Equal Temperament
Since
the
subject of our study is the analysisand
relationshipof all of the possible sonorities contained in the twelve tones of
the equally
tempered
chromatic scale, inboth
theirmelodic and
harmonic
implications, our first task is to explain the reasons forbasing our study
upon
that scale.There
aretwo
primary
reasons.The
first is that a study confined to equaltemperament
is,al-though
complex, a -finite study,whereas
a study of thetheo-retical possibilities within just intonation
would
be
infinite.A
simpleexample
will illustrate this point. Ifwe
construct amajor
third, E,above
C,and
superimpose
asecond
major
third,G#,
above
E,we
produce
the sonorityC-E-G#i
Now
ifwe
superimpose
yet anothermajor
thirdabove
the GJj:,we
reach the tone B#. In equaltemperament, however,
B#
is theenharmonic
equivalent ofC,and
the four-tone sonorityC-E-G#-B#
is actuallythe three tones C-E-Gfl:
with
the lower tone, C, duplicated attheoctave. In just intonation,
on
the contrary,B#
would
notbe
the equivalent of C.
A
projection ofmajor
thirdsabove
C
injust intonation
would
thereforeapproach
infinity.The
second
reason is a-corollary of the first.Because
thepitches possible in just intonation
approach
infinity, justintonation is not a practical possibility for
keyboard
instru-ments
or forkeyed
and
valve instruments of thewoodwind
and
brass families. Just intonation
would
be
possible for stringedinstruments, voices,
and
one
brass instrument, the slideHARMONIC
MATERIALS OF
MODERN
MUSIC
o£ these resources simultaneously,
and
since it is unlikely thatkeyboard, keyed,
and
valve instruments willbe done
away
with,at least within the generation of living composers, the system
of equal
temperament
is the logical basis for our study.Another
advantage
of equaltemperament
is the greatersimplicity possible in the
symbolism
of the pitches involved.Because
enharmonic
equivalents indicate thesame
pitch, it ispossible to concentrate
upon
thesound
of the sonority ratherthan
upon
the complexity of its spelling.Referring again to the
example
already cited, ifwe
were
tocontinue to
superimpose
major
thirds in just intonationwe
would
soon find ourselves involved in endless complexity.The
major
thirdabove
BJj:would
become
D
double-sharp; themajor
third
above
D
double-sharpwould
become
F
triple-sharp; thenext
major
third,A
triple-sharp;and
so on. In equal tempera-ment, after the first three toneshave
been notated—
C-E-Gjj:—theG#
is considered the equivalent of Aj^and
the succeedingmajor
thirds
become
C-E-Gfl:-C,merely
octave duplicates of thefirst three.
Example
1-1Pure Temperament Equal Temperament
"%
!
]ip"
)This point of
view
has theadvantage
of freeing thecomposer
from
certain inhibiting preoccupationswith academic
symboliza-tion as such.
For
the composer, the important matter is thesound
of thenotes, not their "spelling."For
example, the sonorityG-B-D-F
sounds
like adominant
seventhchord
whether
it isspelled
G-B-D-F,
G-B-D-E#,
G-B-CX-E#,
G-Cb-C-:^-F, or insome
othermanner.
The
equallytempered
twelve-tone scalemay
be
conveniently thought of as a circle,and any
pointon
the circumferencemay
circumference
may
thenbe
divided into twelve equal parts,each
representing aminor
second, orhalf-step. Or,with
equal validity,each
of the twelve partsmay
represent the interval of a perfectfifth, since the superposition of twelve perfect fifths also
embraces
all of thetwelve tones ofthe chromatic scale—as in thefamiliar "key-circle."
We
shallfindthelatterdiagram
particularlyuseful.
Beginning
on
C
and
superimposing
twelveminor
secondsor twelve perfect fifths clockwise
around
the circle,we
complete
the circle at BJf,
which
in equaltemperament
has thesame
pitchas C. Similarly, the pitch
names
ofC#
and
D^,
D#
and
Ej^,and
so forth, are interchangeable.
Example
1-2GttlAb)
D«(Eb
MK(Bb)
The
term
sonority isused
in thisbook
to cover the entire fieldof tone relationship,
whether
in terms ofmelody
or ofharmony.
When
we
speak ofG-B-D-F,
for example,we
mean
therelation-ship of those tones
used
either as tones of amelody
or of aharmony.
Thismay
seem
to indicate a too easy fusion ofmelody
and
harmony,
and
yet theproblems
of tone relationship areessentially the same.
Most
listenerswould
agreethat thesonorityin
Example
l-3a is a dissonant, or "harsh,"combination
of toneswhen
sounded
together.The
same
efl^ect of dissonance,however,
persists in our aural
memory
if the tones aresounded
HARMONIC
MATERIALS OF
MODERN
MUSIC
Example
1-3(fl)
i
^
The
firstproblem
in the analysis of a sonority is the analysisof its
component
parts.A
sonority sounds as it does primarilybecause
of the relative degree ofconsonance
and
dissonance ofits elements, the position
and
order of those elements in relationto the tones of the
harmonic
series, the degree of acousticalclarity in terms of thedoubling of tones, timbre of the
orchestra-tion,
and
the like. It is further affectedby
theenvironment
inwhich
the sonority is placedand
by
themanner
inwhich
experience has conditioned the ears of the listener.
Of
thesefactors, the firstwould seem
tobe
basic.For
example,the
most
important aural fact about the familiar sonority of thedominant
seventh is that it contains a greaternumber
ofminor
thirds than of
any
otherinterval. It contains alsothe consonancesof the perfect fifth
and
themajor
thirdand
themild
dissonancesof the
minor
seventhand
the tritone. This is, so to speak, the chemical analysis ofthe sonority.Example
1-4f
Minor thirds Perfect fifth Mojor third Minor seventh Tritone It is of
paramount
importance
to the composer, since thecomposer
shouldboth
loveand
understand
thebeauty
of sound.He
should "savor"sound
as the poet savorswords
and
thepainter
form
and
color.Lacking
this sensitivity to sound, thecomposer
is not acomposer
at all,even
though he
may
be both
This does not
imply
a lack ofimportance
of the secondaryanalyses already referred to.
The
historic position of a sonorityin various styles
and
periods, its function in tonality—where
tonality is
implied—and
the like are important.Such
multipleanalyses strengthen the
young
composer's grasp of his material,providing always that they
do
not obscure thefundamental
analysis ofthe
sound
assound.Referring again to the sonority
G-B-D-F,
we
should note itshistoric position in the counterpoint of the sixteenth century
and
its
harmonic
position in the tonality of the seventeenth,eighteenth,
and
nineteenth centuries,but
we
should first of allobserve its construction, the elements of
which
it is formed. Allof these analyses are important
and
contributetoan
understand-ing of
harmonic
and
melodic
vocabulary.As
anotherexample
ofmultiple analysis,letus takethefamiliarchord
C-E-G-B.
It containstwo
perfect fifths,two
major
thirds,one minor
third,and
one major
seventh.Example
1-5*
Perfect fifths Mojor thirds Minor third Major seventh
It
may
be
considered as thecombination
oftwo
perfect fifths atthe interval of the
major
third;two
major
thirds at the perfectfifth; or
perhaps
as thecombination
of themajor
triadC-E-G
and
theminor
triadE-G-B
or the triads*C-G-B and
C-E-B:
Example
1-6ofijiij^ij
iii
HARMONIC
MATERIALS
OF
MODERN
MUSIC
Historically, it represents
one
of the important dissonantsonori-ties of the
baroque
and
classic periods. Its function in tonalitymay
be
as thesubdominant
or tonic seventh of themajor
scale,the
mediant
orsubmediant
seventh of the "natural"minor
scale,and
so forth.Using
the pattern of analysisemployed
inExamples
1-4, 1-5,and
1-6,analyze ascompletelyaspossiblethefollowingsonorities:
i
Example
1-7 4. 5. e. 7.±
fiti
9. 10.ft
of
In
order again
toreduce
aproblem
of theoretically infiniteproportions to a finite problem,
an
additionaldeviceis suggested.Let
us take asan example
the intervallic analysis of themajor
triad
C-E-G:
Example
2-1Perfect fifth Majorthird Minor third
This triad is
commonly
described in conventional analysis as acombination
of a perfectfifthand
amajor
thirdabove
the lowestor"generating" tone of thetriad. It is obvious,
however,
thatthisanalysis is incomplete, since it omits the
concomitant
interval ofthe
minor
thirdbetween
E
and
G. This completes the analysisas long as the triad is in the simple
form
represented above. If,however,
thechord
is present in aform
inwhich
there aremany
doublings in several octaves,
such
acomplete
analysisbecomes
more
complex.If
we
examine
the scoring of the finalchord
inDeath
and
Transfiguration
by
Richard
Strausswe
find a sixteen-tone chord:Example
2-2:i
*
*
HARMONIC
MATERIALS OF
MODERN
MUSIC
These
sixteen tonescombine
toform one
hundred
and twenty
different intervals.
The
relationshipbetween
C
and
G
isrepre-sentednot only
by
theintervalsExample
2-3eta
<^ -o-
«
a
o ^
but
alsoby
theintervalsExample
2-4i
^
5
-o—
©
—
©
—
o
—
»^^ f^ »^ *^'*^
in
which
casewe
commonly
call thesecond
relationship the "inversion" of the first.The same
is true of the relation ofC
toE
and
E
to G.However,
the composite of all of the tones still gives theimpression of the
C
major
triad in spite of the complexity ofdoubling. In other words, the interval
C
toG
performs thesame
function in the sonority regardless of the
manner
of the doublingof voices.
The
similarity ofan
intervaland
its inversionmay
be
furtherillustrated if
one
refers again to thearrangement
of theExample
2-5Here
it willbe
seen thatC
hastwo
perfect-fifth relationships,C
toG
and
C
to F; the one,C
to G,proceeding
clockwise (ascending)and
the other,C
to F,proceeding
counterclockwise (descending). In thesame
manner,
C
hastwo
major-secondrelationships,
C
toD
and
C
to B^;two
major-sixth relationships,C
toA
and
C
to E^;two
major-third relationships,C
toE
and
C
to A\);and
two
major-seventh relationships,C
toB
and
C
toDt>. It has only
one
tritone relationship,C
up
to F#, orC
down
to G\). It will
be
helpful in ,our analysis ifwe
use onlyone
symbol
to representboth
the intervalunder
considerationand
its inversion. This is not
meant
toimply
that the intervaland
itsinversion are the same,
but
rather that theyperform
thesame
function in a sonority.
Proceeding
on
this theory,we
shall choose thesymbol p
torepresent the relationship of the perfect fifth
above
orbelow
thefirst tone,
even
though
when
the lower tone ofeach
of thetwo
intervals is raised
an
octave the relationshipbecomes
actuallyharmonic
materials
of
modern
music
Example
2-6#
Perfect fifth p Perfect Perfect- fifth fourth
The
symbolization is arbitrary, the letterp being chosen because
it connotes the designation "perfect,"
which
apphes
to bothintervals.
The
major
thirdabove
orbelow
the given tojie willbe
desig-nated
by
theletterm:
Example
2-7^
^^
Major third,m (or minorsixth)
The
minor
thirdabove
orbelow
the given tone willbe
representedby
thelettern:Example
2-8i
Minor third,B^
n (or majorsixth) themajor second
above
orbelow,by
s:i
Example
2-9(i'ji)
t»to
tib<^
Majorsecond, s (or minorseventh)
the dissonant
minor
secondby
d:Example
2-10Minor second, d
(or mojor seventh)
and
thetritoneby
t:Example
2-11 (bo'iM
*^ Augmented^
fourth,^(ordiminished fifth) (Tritone)
The
letterspmn,
therefore, represent intervalscommonly
considered consonant,
whereas
the letterssdt represent theinter-vals
commonly
considered dissonant.The
symbol
pmn,
sdt'*would
therefore represent a sonoritywhich
containedone
perfectfifth or its inversion, the perfect fourth;
one major
third or itsinversion, the
minor
sixth;one
minor
third or its inversion, themajor
sixth;one
major second
or its inversion, theminor
seventh;one minor second
or its inversion, themajor
seventh;and one
augmented
fourth or its inversion, the diminishedfifth; thethreesymbols
attiie left ofthecomma
representing consonances, thoseat the right representing dissonances.
A
sonority represented,for example,
by
thesymbol
sd^, indicating a triadcomposed
ofone major second
and
two
minor
seconds,would
be
recognizedasahighly dissonant sound, while the
symbol
pmn
would
indicate a consonant sound.The
complexity of the analysis willdepend,
obviously,upon
thenumber
of diflFerent tones present in the sonority.A
three-tone sonority such as
C-E-G would
contain the three intervalsC
to E,C
to G,and
E
to G.A
four-tone sonoritywould
contain3+2+1
or6
intervals; a five-tone sonority,4+3+2+1
or 10in-tervals,
and
so on.Since
we
are considering all tones in equaltemperament,
ourtask is
somewhat
simplified.C
toD#,
for example, representsthe
same
sound
as the intervalC
to E^i;and
since thesound
is "Forthe sake ofuniformity, analysesof sonorities willlisttheconstituent inter-vals inthisorder.HARMONIC
MATERIALS OF
MODERN
MUSIC
the same, they
would
both be
representedby
the singlesymbol
n.
A
table of intervalswith
their classification would, therefore,be
as follows:C-G
(orG-C),B#-G,
C-F^K<,etc.=
p
C-E
(orE-C),
B#-E, C-Fb, BJf-Fb, etc.= m
C-Eb
(orEb-C),C-Dif,
B#-Eb, etc.=
n
C-D
(orD-C),
Bif-D,C-Ebb,
etc.=
sC-Db(orDb-C),C-C#,B#-Db,etc.
=
d
C-F#
(orF#-C), C-Gb, B#-Gb,
etc.=
txo
efc.Example
2-12it\^
^°
tf^g'1 '^» r i^i »j|o^^
SE
t^
i
ife efc. etc£
^^g
bo
bo^^'^'^^
i--*0 fv^»
m
Qgyi
Ed
-XT*
¥^
For
example, theaugmented
triadC-E-G#
contains themajor
third
C
to E; themajor
thirdE
to G#,and
the intervalC
to G^.Since,
however,
C
toG#
sounds likeC
to Ab, the inversion ofwhich
isAb
toC—
also amajor
third—the designation of theaugmented
triadwould
be
threemajor
thirds, or m^.A
diagram
of these three notes in equal
temperament
quickly illustrates thevalidity of this analysis.
The
joining of the three notesC-E-G#
(Ab)
formsan
equilateraltriangle— atrianglehaving
threeequalsides
and
angles:Example
2-13B /-^ ' -\
CJt G<t(Ab)
It is, of course, a figure
which
has thesame
form
regardless ofwhich
side isused
as its base:Example
2-14Similarly the
augmented
triadsounds
thesame
regardlessof
which
ofthethreetonesisthelowest:G#
•B#(C)
E
E
G#
C
C
E
G#(Ab)
One
final illustration will indicate the value of this techniqueof analysis.
Let
us consider the following complex-lookingsonor-ity in the light of conventional
academic
analysis:Example
2-15f
The
chord
contains six notesand
therefore has5+4+3+2
+1,
or 15 intervals, as follows:C-D#
and Ab-B
C-E
and
G-B
C-G
and
E-B
C-Ab and
D#-B
augmented
secondsmajor
thirds perfect fifthsminor
sixthsHARMONIC
MATERIALS OF
MODERN
MUSIC
C-B
=
major
seventhD#-E
and
G-A^
=
minor
seconds Djf-Gand
E-Aj^=
diminished fourthsD#-Ab
=
double-diminished fifthE-G
=
minor
thirdHowever,
in thenew
analysis it converts itself into only fourtypes of intervals, or their inversions, as follows:
3perfectfifths:
C-G,
E-B,and Ab-Eb
(Dif).6
major
thirds: C-E,Eb
(D^f )-G,E-G#
(Ab), G-B,Ab-C,
and B-D#.
3
minor
thirds:C-Eb
(D#), E-G,
and
G#
(Ab)-B. 3minor
seconds: DJf-E,G-Ab,
and
B-C.The
descriptionis,therefore, p^m^n^d^.Example
2-16i
Perfect fifths ii Mojorthirdso
\fv:w^
vObB
l^»^
^
b8(tfo)tlit^^
i
Minor thirds Minor seconds
b^||o)t8
tfS^^^
Ijl^
^'^'=^a=
A
diagram
willindicatetheessentialsimplicity ofthe structure:14
Example
2-17It has
been
my
experience that although theyoung
composer
who
hasbeen
thoroughlygrounded
inacademic
terminologymay
at firstbe
confusedby
this simplification,he
quicklyembraces
thenew
analysisbecause
itconforms
directly to hisown
aural impression.In analyzing intervals, the studentwill find itpractical to
form
the habit of"measuring"
all intervals in terms of the "distance"in half-steps
between
thetwo
tones.Seven
half-steps(up
ordown),
for example, willbe
designatedby
thesymbol
p; fourhalf-steps
by
thesymbol
m;
three half-stepsby
thesymbol
n,and
so forth, regardless of the spelling of the tones
which
form
the interval:
perfect fifth
7
half-StepsV
perfect fourth5
If IImajor
third4
II IIm
minor
sixth 8 II IIminor
third 3 II IIn
major
sixth9
II IImajor second
2
II II sminor
seventh 10 II IIminor second
1 II IId
major
seventh 11 II IIaugmented
fourth 6 II II t diminished fifth 6 II IIExample
2-18 pm
n9f=
^4S=
-^c^^44«
i."*—
A?—
4.^#^
Perfect fifth Perfect Major fourth third Minor sixth 1^inor third Major sixth 15HARMONIC
MATERIALS OF
MODERN
MUSIC
s d t =f^rt^
^
2T*t°w
=15= '^XXt Minor Major second seventh^M
«»i Major Minor second seventh Augmented Diminished fourth fifthIn speaking of sonorities
we
shall apparentlymake
littledistinction
between
tonesused
successively in amelody and
tones
used
simultaneously in aharmony.
It is true that theaddition of the
element
ofrhythm,
the indispensable adjunct ofmelody,
with
its varying degrees ofemphasis
upon
individual notesby
the devices of time length, stressof accent,and
thelike,creates
both
greatand
subtle variancefrom
the sonorityplayed
as a "block" of sound. Nevertheless, the basic relationship is the same.
A
melody
may
grow
out of a sonority or amelody
may
itself
be
a sonority.Analyze
the following sonorities in thesame manner employed
In
Examples
2-15and
2-16, pages 13and
14, giving first the conventional interval analysis,and second
the simplified analysis:Example
2-19i
^
jt#
I^i
^^
^1^
w
S3S:^»S^
#
^S^
^
r^
3
^S
'^BTof Involution
Reference
has
already
been
made
to the two-directionalaspect of musical relationship, that is, the relationship "up"
and
"down"
in terms of pitch, or the relationship in clockwise orcounterclockwise rotation
on
the circle already referred to. Itwill
be
readily apparent that every sonority inmusic
has acounterpart obtained
by
taking the inverse ratio of the originalsonority.
The
projection dovonfrom
the lowest tone of a givenchord, using the
same
intervals in the order of their occurrencein the given chord,
we may
call the involution of the givenchord. This counterpart is, so to speak, a "mirror" of the
original.
For
example, themajor
triadC-E-G
isformed
by
theprojection of a
major
thirdand
a perfect fifthabove
C.However,
if this
same
relationship is projecteddownward,
the intervalC
to
E
has as its counterpart the interval|C
to Aj^;and
the intervalC
toG
has as its counterpart[C
to F.Example
3-1HARMONIC
MATERIALS
OF
MODERN
MUSIC
Itwill
be noted
that the involution of a sonority always containsthe
same
intervalsfound
in the original sonority.There
are three types of involutions: simple, isometric,and
enharmonic.
In simple involution, the involuted
chord
differsinsound from
the given chord.Let
us take, for example, themajor
triadC-E-G,
which
isformed
by
the projection of amajor
thirdand
a perfectfifth
above
C. Its involution,formed
by
theprojectiondownward
from
C
of amajor
thirdand
a perfect fifth, is theminor
triad'[F-A^-C.
The
major
triadC-E-G
and
its involution, theminor
triad
^F-A^-C, each
contain aperfect fifth, amajor
third,and
aminor
third,and
can
be
representedby
thesymbols
pmn.
Example
3-2i
^m
^
In the
second
type of involution,which
we
may
call isometricinvolution, the involuted sonority has the
same
kind ofsound
asthe original sonority.
For
example, the tetradC-E-G-B
has as itsinvolution
jDb-F-Ab-C.
Example
3-3THE THEORY
OF INVOLUTION
Each
of these is amajor
seventh chord, containingtwo
perfectfifths,
two
major
thirds, aminor
third,and
amajor
seventh,and
can
be
characterizedby
thesymbols
p^irrnd, the exponents inthis instancerepresenting
two
perfect fifthsand
two
major
thirds.In the third type,
enharmonic
involution, the invohitedsonor-ity
and
the original sonority contain thesame
tones in differentoctaves (except for
one
common
tone).For
example, theaugmented
triadC-E-G#
involutes toproduce
theaugmented
triad
^F^-Ab-C,
F^ and
A^
being
theequal-temperament
equiva-lents of
E
and
G#.Another
common
example
ofenharmonic
involutionisthediminished seventh
chord
Example
3-4« 3 .CK.
m
I
All sonorities
which
areformed
by
thecombination
of asonority
with
its involution are isometric sonorities, since theywill
have
thesame
order of intervalswhether
considered "up"or
"down,"
clockwise or counterclockwise.We
have
already seenthat the involution of the triad
C-E-G
isjC-Ab-F.
The
two
together
produce
the sonorityF3Ab4C4E3G,
which
has thesame
order of intervals
upward
ordownward.*
*The numbers indicate the number of half-steps between the tones of the sonority.
HARMONIC
MATERIALS
OF
MODERN
MUSIC
If the tone
E
of the triadC4E3G
isused
as the axis ofinvolu-tion, a diflFerent five-tone sonoritywill result, since theinvolution
of
E3G5C
willbe
J^EsC^gGJ,forming
together the sonorityGJgCfllsEaGgC. Ifthe tone
G
isused
as the axis of involution, theinvolution of
G5C4E
willbe
J,G5D4Bb,forming
together thesonority
Bb4D5G5C4E. These
resultant sonorities will allbe
seento
be
isometric in structure. (See Note,page
24.)
Example
3-5) (2) (3)
"^^",
44
'^%j|§ii
r33
. ' bS-i|-2"3 3 2If
two
tones areused
as the axes of involution, the result willbe
afour-tone isometric sonority:Example
3-65=^
313
343
434
In the first of the
above
examples,C
and
G
constitute the "double axis"; in thesecond
C
and
E;and
in the thirdE
and
G.The
discussion of involutionup
to this point does not differgreatly
from
the "mirror" principle of earlier theorists,whereby
"new"
chordswere formed
by
"mirroring" a familiarchord and
combining
the "mirrored" or involutedchord
with
the original.At
this point,however,
we
shallexpand
the principle to thepoint
where
itbecomes
a basic part of ourtheory.When
amajor
triad is involuted—as in
Example
3-2—
deriving theminor
triadas the "mirrored"
image
of themajor
triadseems
to place theminor
triad in a position of secondary importance, as thereflected
image
of themajor
triad.In the principle of involution presented here,
no such
second-ary
importance
is intended; for iftheminor
triad is the reflectedimage
of themajor
triad, it is equally true that themajor
triad isTHE THEORY
OF INVOLUTION
also the reflected
image
of theminor
triad.For
example, theinvolution of the
major
triadC4E3G
is theminor
triad|C4Ab3F,
and
the involution of theminor
triad C3E[74G is themajor
triadjCaA^F.
In order to avoid
any
implication that the involution is, so tospeak, a less important sonority,
we
shall in analyzing thesonori-ties construct
both
the first sonorityand
its involutionupward
by
the simple process of reversing the intervallic order.For
example, if the first triad isC4E3G
the involution of this triadwill
be
any
triadwhich
has thesame
order of half-steps inreverse, for
example F3Ab4C,
thecomparison
being obviously4-3versus 3-4.
In this sense, therefore, the involution of a
major
triadcan
be
considered to
be
any minor
triadwhether
or not there isan
axisof involution present.
In
Example
3-7, therefore, theB
minor,B^
minor,G$
minor,F#
minor,E^
minor,and
D
minor
triads are all considered aspossible involutions of the
C
major
triad, although there isno
axis of involution.
When
theC
major
triad iscombined
with
any
one
ofthem,
the resultant formation is a six-tone isometricsonority.
Example
3-7m
Ha
« o
m=^
b<ifc^ 3bo
ffi
Hi^
"^
222
IOgP
^=
m
^
^=j^
213
12
313
13
3 212
3Wt
>..i i ..i'«ti'btxhc^feo-m
=»=sii^
312
13
M\i
"°
2 2 3 2 2HARMONIC
MATERIALS
OF
MODERN
MUSIC
Note
that thecombination
ofany
sonoritywith
its involutedform always
producesan
isometric sonority, that is, a sonoritywhich
can
be
arranged in such amanner
that its foraiation ofintervals is the
same
whether
thoughtup
ordown. For
example,the first
combination
inExample
3-7, ifbegun
on
B, has theconfiguration BiC2D2E2FJj:iG,
which
is thesame
whether
con-sidered
from
B
toG
orfrom
G
to B,The
second
combination,C
major
and B^
minor,must
be
begun
on
Bj^ orE
tomake
its isometric character clear:BbsCiDbsEiF^G
orE,F,GsB\),C,Dh-The
isometric character of the third combination,C
major
and
G#
minor, is clear regardless of the tonewith
which
we
begin:C3D#iE3GiG#3B;
D^,E,G,GJi^,B,C, etc.If,
however,
for the sake of comparison,we
combine
amajor
triad
with
anothermajor
triad, for example, thecombination
ofC
major with
D
major, the resultant formation is not isometric, since it is impossible to arrangethese tones so that the configura-tion is thesame
up
ordown:
CsD^E^FSiG^A;
D^EsFliG^AsC;
E^FSiG^AsC^D;
FitiGsAgCsDsE;G2A3C2D2E2F#;
A.C^D^E^Fi.G.
There
isone
more
phenomenon which
shouldbe
noted.There
are a
few
sonoritieswhich
have
thesame
components
butwhich
are not involutions
one
of the other, althougheach
has itsown
involution.
Examples
are the tetradsC-E-fJ-G and
C-F#-G-Bb.
Each
containsone
perfectfifth,one
major
third,one
minor
third,one major
second,one
minor
second,and one
tritone(pmnsdt),
but
one
is not the involution of theother—
althougheach
hasits
own
involution.We
shall describe such sonorities, illustrated inExample
3-8,Example
3-8Involution'.
^ IIIVUIUMUIIi
ife
pmnsdt pmnsdt
^#
Using
the lowest tone ofeach
of the following three-tonesonorities as the axis of involution, write the involution of
each
by
projecting the sonoritydownward,
as inExample
3-5.i
Example
3-9 2. 2o. 2b. 3o.Sn
3b. ^5^= :x«o= c^"* go*> ^W
5^
i
4o. 4b. 7rt 5a. 5b. 6o. 6b.tU^
(*^ I o*^ =S^Q=^
:^^
^^
sT-^n
2og
>Q
'^'7o. 7b. 8. 9. 10. lOo. lOb.
10
^
-= ytbb-Q^^bt^oo'roU'1%
I ^tbt^^'ith^'^
llo. Mb. 12. 12a. ft|bo
gboM
2b<j^^i
^nrgr
Solution:^^f
5=33=^
12b. ng»-Ib. fl/C. ^^D»=m
SijO-Zl-oU"^^If
^15
-«s^l2The
following scales are all isometric,formed
by
thecombina-tion of
one
of the three-tone sonorities inExample
3-9with
itsinvolution.
Match
the scale inExample
3-10with
the appropriatesonority in
Example
3-9.harmonic
materials of
modern
music
Example
3-10^
J
J J>r
U^J
JuJ Ij
jiiJ Ji'^ ij^jjg^J
ijj
Ji'-^t'fiJ^J^rU^jjtJJUjtJJtf^^i^rr^r'Ti^^
t
^^rrrr
ijuJJf
ijJjtJ
^i^^^^^UJ^tJ^P
I
J
J J ^ ^i;ii.JtJbJ
U|J
J^^^
Uj
jj^^*^i>J^^
(j^^jiJ^ri|J^^rr
ijjJ^^
i''^^rrri>ji>J^^^
Note:
We
have
definedan
isometric sonority asone
which
has the
same
order of intervals regardless of the direction ofprojection.
The
student should note that this bidirectionalcharacter of a sonority is not always
immediately
evident.For
example, the perfect-fifth
pentad
in the positionC2D2E3G2A3(C)
does not at first glanceseem
tobe
isometric.However
in theposition
D2E3G2A3C2(D),
its isometric character is readilyapparent.
1^
PartJ
THE
SIX
BASIC
Projection
of
the Perfect Fifth
We
have
seen
that there are six types of interval relationship,if
we
considersuch
relationshipboth
"up"and
"down":
theperfectfifth
and
its inversion, the perfect fourth; themajor
thirdand
its inversion, theminor
sixth; theminor
thirdand
itsinversion, the
major
sixth; themajor second
and
itsinversion, theminor
seventh; theminor second
and
its inversion, themajor
seventh;
and
the tritone,—theaugmented
fourth or diminishedfifth—
which
we
are symbolizingby
the letters, p,m,
n, s, d,and
t, respectively.In a
broader
sense, the combinations of tones in oursystem
ofequal
temperament—
whether such sounds
consist oftwo
tones ormany—
tend
togroup
themselves into soundswhich
have
apreponderance
ofone
of these basic intervals. In other words,most
sonorities fall intoone
of the six great categories:perfect-fifth types, major-third types, minor-third types,
and
so forth.There
is a smallernumber
inwhich two
of the basic intervalspredominate,
some
inwhich
three intervals predominate,and
afew
inwhich
four intervalshave
equal strength.Among
thesix-tone sonorities or scales, for example, there are twenty-six
in
which
one
interval predominates, twelvewhich
aredominated
equallyby
two
intervals, six inwhich
three intervalshave
equal strength,and
six sonoritieswhich
are practically neutralin "color," since four of the six basic intervals are of equal
importance.
THE
SIX BASICTONAL
SERIESis, therefore, in terms of the projection of
each
of the six basicintervals discussed in
Chapter
2.By
"projection"we
mean
thebuilding of sonorities or scales
by
superimposing
a series of similar intervalsone above
the other.Of
thesesixbasic intervals,there are only
two which
can be
projectedwith complete
con-sistency
by
superimposing
one
above
the other until all of thetones of the equally
tempered
scalehave
been
used.These
two
are, of course, the perfect fifth
and
theminor
second.We
shallconsider first the perfect-fifth projection.
Beginning with
the tone C,we
add
first the perfect fifth, G,and
then the perfect fifth,D,
toproduce
the triadC-G-D
or,reduced
to thecompass
ofan
octave,C-D-G-
This triad contains,in addition to the
two
fifths, theconcomitant
interval of themajor
second. Itmay
be
analyzedasph.
Example
4-1Perfect FifthTriad,
p^
m
^^
2 5
The
tetradadds
the fifthabove D,
or A, toproduce
C-G-D-A,
or
reduced
to thecompass
of the octave,C-D-G-A.
This sonoritycontains three perfect fifths,
two
major
seconds,and—
for thefirst time in this series— a
minor
third,A
to C,Example
4-2Perfect FifthTetrad.p^ns^
m
^^
2 5 2
The
analysis is, therefore, p^ns^.The
pentad
adds
the next fifth, E,forming
the sonorityC-G-D-A-E,
or themelodic
scaleC-D-E-G-A,
which
willbe
recognized as themost
familiar of the pentatonic scales. Itsminor
thirds,and—
for the firsttime—
amajor
third.The
analysisis, therefore,
p^mnh^.
Example
4-3Perfect Fifth Pentad,p^mn^s^
i
S
.^^
o
2 2 3 2
The
hexad adds
B,C-G-D-A-E-B,
or melodically,producing
C-D-E-G-A-B,
Example
4-4PerfectFifthHexod,p^nn^n^s^d
m
1
4JJ
2 2 3 2 2its
components being
five perfectfifths,fourmajor
seconds, threeminor
thirds,two
major
thirds,and—
for the firsttime—
thedissonant
minor second
(ormajor
seventh), p^m^n^s'^d.The
heptad adds
F#:i
Example
4-5Perfect FifthHeptod.p^m^n^s^d^t
a
^^
•I^
'THE
SIX BASICTONAL
SERIESproducing
the first scalewhich
in itsmelodic
projection containsno
interval larger than amajor
second—
in other words, a scalewithout
melodic
"gaps." It alsoemploys
for the first time theinterval of the tritone
(augmented
fourth or diminished fifth),C
to FJf. This sonority contains six perfect fifths, fivemajor
seconds, four
minor
thirds,threemajor
thirds,two
minor
seconds,and one
tritone: p^m^n'^s^dH. (It willbe noted
that theheptad
is the first sonority to contain all of the six basic intervals.
)
The
octadadds
Cfl::Example
4-6Perfect Fifth Octod.
p^m^
n^s^ d^
t^Am
«5i=
5
12
212
2Its
components
are seven perfect fifths, sixmajor
seconds, fiveminor
thirds, fourmajor
thirds, fourminor
seconds,and two
tritones: p^m'^n^s^dH^.
The nonad
adds
G#:
Example
4-7Perfect Fifth Nonad,p^m^n^s^d^t^
J^
m
—
=
m
iff I ? 9
m
Its
components
are eight perfect fifths, sevenmajor
seconds, sixminor
thirds, sixmajor
thirds, sixminor
seconds,and
threeThe
decad adds
D#:
Example
4-8- u «*!" Perfect FifthDecad,p^m^n^s^d^t^
m
IT" I I I O
^^
I I I I I 2
Its
components
are nine perfect fifths, eightmajor
seconds, eightminor
thirds, eightmajor
thirds, eightminor
seconds,and
fourtritones: 'p^m^n^s^dH'^.
The
undecad
adds A#:
Isjf
Example
4-9?s"** Perfect FifthUndecad,p'°m'°n'°s'°d'°t^
^^
m
1*"^
I
r
I 2 I III
IIts
components
are ten perfect fifths, tenmajor
seconds, tenminor
thirds, tenmajor
thirds, tenminor
seconds,and
fivetritones: p^'^m'V^s'Od/'^f^