Duke University Press and Yale University Department of Music
are collaborating with JSTOR to digitize, preserve and extend
access to Journal of Music Theory.http://www.jstor.orgYale
University Department of MusicSome Intervallic Aspects of
Pitch-Class Set Relations Author(s): Alan Chapman Source:Journal of
Music Theory, Vol. 25, No. 2 (Autumn, 1981), pp. 275-290Published
by: on behalf of theDuke University Press Yale University
Department of MusicStable
URL:http://www.jstor.org/stable/843652Accessed: 11-07-2015 20:58
UTCYour use of the JSTOR archive indicates your acceptance of the
Terms & Conditions of Use, available at
http://www.jstor.org/page/info/about/policies/terms.jspJSTOR is a
not-for-profit service that helps scholars, researchers, and
students discover, use, and build upon a wide range of content in a
trusted digital archive. We use information technology and tools to
increase productivity and facilitate new forms of scholarship. For
more information about JSTOR, please contact [email protected]
content downloaded from 128.59.222.12 on Sat, 11 Jul 2015 20:58:05
UTCAll use subject to JSTOR Terms and ConditionsSOME
INTERVALLICASPECTS OF PITCH-CLASS SETRELATIONS Alan Chapman
Pitch-class setsarea necessary toolforthe analysis ofnon-tonal
music. Theyprovide an organized meansof classifying harmonic
structures (orany other pitchcombinations), but they also pose a
problem, one which can most easily be exposed through a tonal
analogy. In Example 1 a, the tonal analyst would recognize the
structuralcon- nectionbetweenthefirstandthird chords and describe
thesecond chord as a secondarypitch structurewhich is linearin
origin. This judg- ment is both facilitated and supportedby the
lack of a tonal name (for example, "triad"or "seventh chord") for
the second chord. In Example ib, the third chord, taken out of
context, has a name ("minortriad"), butincontext this is not its
meaning. It toois linear in origin and to place the Roman
numeral"iii" beneath it would be erroneous. In the analysis
ofnon-tonal music, pitch-class sets allow us to assign a name to
every conceivable pitch combination.Caseslike Example l a,
therefore, do notarise. Every non-tonalchord successionis
comparable to Example ib, but becausewe lack the
structuralcriteriaof functional harmony, we run the risk of
consideringonly the name of a pitch-class set at the expense of its
contextual role. How, then, can weevaluate the structural
significance ofdifferent pitch-class sets in a composition? Such
evaluationsare invariably based 275 This content downloaded from
128.59.222.12 on Sat, 11 Jul 2015 20:58:05 UTCAll use subject to
JSTOR Terms and Conditionsupon therecurrenceofcertain pitch-class
sets, an approach which is dependent upon the assumption that a
composer chooses certain pitch combinations as musical materials
and presents them invaried forms throughout a composition. Allen
Forte writes: The determinationofa significant set, as distinct
from a nonset, is not always easy. Some informal guides are: (1)
the set occurs con- sistently throughout-it isnot merely "local;"
(2) the complement ofthe set occurs consistently throughout;(3) if
the set is a member ofa Z-pair, the other member also occurs;(4)
the set is an "atonal" set, not a set that would occur in a tonal
work.1 Forte's first twocriteriaare based upon set
recurrence.Criterion (2) is problematic in that non-Z hexachordsare
their own complements; thus criteria (1) and (2) are simultaneously
satisfied by multiple occurrences of these sets. In his analyses of
significant sets, Forte emphasizes segmentation, "the analytical
procedure by which the significant musical units ofa composition
are determined."2 A primary segment is "determined by conventional
means, such as a melodic configuration[or a simultaneous vertical
combination]."3By criterion (2) above, a significant tetrachord
would require recurrenceofits complement, an eight-note set.
Because occurrences ofsuch large sets as primary segments are
relatively in- frequent, analysis must rely tosome extenton
composite segments which are "formed bysegments or subsegments that
are contiguous or linkedinsome way."4Employment of
compositesegments often obviates the necessity of dealing with
structuralrelations among indi- vidualvertical pitch combinations.
A pitch-class setname is usually thought ofas a designation for a
specific collection of pitches. The present paper shows that itis
often appropriate tothinkofthesetname as a designation fora
specific collection of intervallic properties. The association
ofset names with pitch classes is reinforced by the use, inset
identification, of prime forms, inwhich only a superficial
representation oftheset's totalinterval content is readily
apparent. There is, of course, the interval vector, but no
efficient analyst would identify a set bycalculating the total
intervalcontent of a collection of pitches and looking up
theinterval vector. (This method would not discriminatebetween
Z-relatedsets in any case.) Iftotalinterval contentis relegated toa
position of secondary importance, then the intervallic properties
of pitch-class sets and their intervallic relations
withothersetsbecome tertiary. Theintervallic similarityrelations,
Ro, R1 and R2 (based upon comparison ofinterval vectors), are
abstract reflections ofintervallic relations;they play only 276
This content downloaded from 128.59.222.12 on Sat, 11 Jul 2015
20:58:05 UTCAll use subject to JSTOR Terms and Conditions(a)(b)
Example 1.A Tonal Analogy (a)(b) Example 2.Two Tetrachords 4-Z15
4-Z15 (soprano-bass): 16 (alto-bass): 48 (tenor-bass): 611 Example
3.The AB (above bass) intervalset is the set of intervals (in
semitones) relativeto the bass. 277 This content downloaded from
128.59.222.12 on Sat, 11 Jul 2015 20:58:05 UTCAll use subject to
JSTOR Terms and Conditionsa minor role in most atonal
analysis,usually appearing as supplemental observations upon
analyses based on pitch-class set recurrence. Interval sets:anew
system. Investigation ofintervallic properties and relations calls
for consideration of pitch-class set intervalcontent as itis
expressed inmusical contexts.This study utilizes tetrachords and,
where appropriate, trichords as model harmonic structures.It is
evident that the properties and relationsexhibited are adaptable to
sets of larger cardinalities. Because ofthe present focus on
tetrachords, this study will employ four-part voicings
todemonstrate theoretical principles. (The four voices
willbeidentified as soprano, alto, tenor and bass.) Let us con-
sider the measurementof the intervals represented in the two
tetrachords of Example 2. We can first reckon intervals (in
semitones) relativeto the bass.5 These intervalsare displayed in
Example3, in descending vertical orderbelow the staff. The
intervalset thus obtainedwill be calledtheAB (for above bass)
intervalset and will be written in the form AB:X-Y-Z, where X, Yand
Zare theintervals in ascending numerical order; the intervalsets
represented in Example 3 are AB:1-4-6and AB:6-8-11. An AB set which
consists ofthree different intervals may appear in sixdistinct
permutations. Example 4shows the permutations ofAB: 1-4-6.Each AB
set which consists of three different intervalsis unique toa
specific tetrachord. It can beseen in Example 4 that permutation
oftheAB intervalsdoes not change the pitch-class set (4-19). In
this example, withthebass held constant, these permutations are
brought about by different registralplacements of the upper voices.
A given tetrachord may have up to twenty-four distinctforms based
on pitch content, but it may have a maximum of eight AB sets
associated with it.The eight AB intervalsets of the pitch-class set
4-19 are shown in Example 5. The numberof different AB intervalsis
equal to (n-1), where n is the cardinality ofthe pitch-class set.
Therefore, a trichord will have twodifferent intervals, a five-note
set will have four, and soforth.For the purposes ofthis study, AB
sets oftrichordsare expressed as sets ofthree intervals, with
the"extra"interval reflecting theoctave duplication ofoneofthe
upper voices, as in Example 6. This practice willallow usto explore
certain intervallic relationships between trichordsand tetrachords.
Let us now return tothe two tetrachordswe examinedin Examples
2and3andmeasure theintervals (insemitones) between adjacent voices.
These intervalsare displayed(Ex. 7) in descending verticalorder
below the staff. This interval set will be called theVP (for voice
pairs) intervalset and will be written in the form VP:X/Y/Z, where
X, Y and Zaretheintervals in ascending numerical order; theinterval
sets represented in Example 7 are VP:6/9/10 and VP:9/10/11.6 278
This content downloaded from 128.59.222.12 on Sat, 11 Jul 2015
20:58:05 UTCAll use subject to JSTOR Terms and Conditions4-Z 15 II
i A14641 6 [AB:] 461164 614641 Example 4.An AB set which consists
ofthree differentintervals may appear in six distinct permutations.
Each AB set which consists of three differentintervalsis unique to
a specific tetrachord. 4-19 11334444 [AB:] 45475788 8981188911
Example 5.A given tetrachord may have up to 24 distinct forms based
on pitch content, but it may have a maximum of8AB sets associated
with it. 3-9 2 [AB:] 7 2 Example 6.ABsetsoftrichords may be
expressed as setsofthree intervals. The extra interval reflects
theoctave duplication ofone of the upper voices. (n ) I (b) j 4-Z15
4-Z15 910 [VP:] 109 6 11 Example 7.TheVP (voice pairs) intervalset
is the set ofintervals (in semitones) between adjacent voices. This
content downloaded from 128.59.222.12 on Sat, 11 Jul 2015 20:58:05
UTCAll use subject to JSTOR Terms and ConditionsAVPinterval
setwhich consists ofthree different intervals may appear in six
distinct permutations.Example 8 shows the permutations of
VP:6/9/10. Unlike AB sets, a given VP intervalset may be expressed
by specific voicings of more than one pitch-class set. Because
weare constructing four-part voicings of trichords, itis possible
fortrichords to express some ofthe same VP interval sets as
tetrachords, as illustrated byExample 9. Hitherto, sets ofdifferent
cardinalities have beenconnected solely by thesubset relation. The
relation shown in Example 9is an important extension.
Pitch-classsets which have VP interval sets in common will be
called VP-relatedsets. Thus, in example 8,4-12, 4-13and4-Z15are
VP-related sets; in example 9, 3-3,3-11, and 4-17are VP-related.
Linearconsiderations:VP-relatedsets. Example 10a shows the entire
two-measure refrain, afour-chord progression from4-Z15to 4-18,
fromthethirdmovementof Stravinsky's Three Piecesfor String Quartet.
Consideration oftheinterval setsinvolved reveals thatthe
structureofthe phrase is an embellished VPinterval exchange, as
illus- trated in Example 10b. (The formal statements ofthe
voice-leading procedures which permute VP sets are beyond the scope
of this paper.) Within 3-5and 4-14, the outer voices provide upper
and lower neigh- bors to D and GM. In addition, the anticipation of
4-18'sA and F as the inner voices of4-14 emphasizes the new
position offour semitones as the middle VP interval. Another
exchange involving thesame VP interval set occurs in the same
movement between successivetetrachords (Ex.11). Here a sym-
metrical structurecan be observed; the VP exchange is framed by two
identically-voicedtranspositions of4-18.In Example 11 it can be
seen that a VP intervalset can be transposed in the same sense as a
pitch-class set.Here VP:4/5/9 is transposed down onesemitone as
theinterval exchange occurs. Another symmetricalstructure,occupying
a longer span, is shown in Example 12.Framed by two
rhythmically-stressed occurrencesofthe same form of3-5are two
appearances of VP:2/3/6, one within 4-12 andonewithin4-13.The
viola's motionfrom F$ to A, which will effectthe exchange, ismade
immediately, onthedownbeat ofthe second measure.The exchange would
occur at this point were it not for the displacement ofDand DW(in
violin Iand 'cello) by their upper neighbors, Eb and E. The viola's
A is emphasizedby its lower neighbor G?. Finally, on the last chord
ofthe second measure, D and D$ return in the outer voices and the
exchange is completed. It is noteworthy that violinII retains C
throughout the symmetrical structure, making a prolonged
common-toneconnection between 4-12and 4-13. In Example 13, 4-25and
4-24, which have VP:2/4/4 in common, 280 This content downloaded
from 128.59.222.12 on Sat, 11 Jul 2015 20:58:05 UTCAll use subject
to JSTOR Terms and Conditions4-134-134-Z154-Z154-124-12 61096109
[VP:] 99101066 10669910 Example 8.AVP setwhich consists ofthree
different intervals may appear in six distinct permutations. Unlike
AB sets, a given VP set may be expressedby specific voicings of
more than one pitch-class set. 3-113-11 4-17 4-173-3 3-3 6 Id IiI
498 498 [VP:] 889 9 44 944889 Example 9.Hitherto sets of
differentcardinalitieshave been connected solelyby thesubset
relation. However, sets ofdifferent cardinalities may be
VP-related. 4-Z15 4-18 VIo. Example 10.AVPInterval Exchange:
Stravinsky, Three Pieces for StringQuartet,III, mm. 8-9,15-16,
22-23 281 This content downloaded from 128.59.222.12 on Sat, 11 Jul
2015 20:58:05 UTCAll use subject to JSTOR Terms and
Conditions4-184-18 4-Z154-18 3-5 5555 0 [VP:] 64966 99 4. 95
Example 11.AVPInterval Exchange: Stravinsky, Three Piecesfor
StringQuartet,III, mm. 17-18 yiVi Vc . I I 3-54-124-133-5 22 [VP:]
63 Example 12. AVPInterval Exchange: Stravinsky, Three Pieces for
StringQuartet,III, mm. 19-21 282 This content downloaded from
128.59.222.12 on Sat, 11 Jul 2015 20:58:05 UTCAll use subject to
JSTOR Terms and Conditionsare connected by a"chromatic passing
tone,"DM, which generates 4-27.The sense ofan end tothis
prolongational unit comes with the motion to4-5:the upper voices
all move, bass II leaps down an octave and a distinctly different
intervalset is introduced. Structuralconsiderations: VP
representation. As shown in Example 4,any three-intervalAB set,
regardlessof ordering, can be expressed in theAB domain by one and
only one pitch-class set. This pitch-class set is called the AB
source set ofthat interval set. Example 14 shows that 4-19is the AB
source set of AB:4-8-9. A given three-intervalVP set may be
expressed in the VP domain by one, two or threedifferent
pitch-class sets (depending on the intervalcon- tentofthe VP set).
Different vertical orderingsare, in most cases, ex- pressedby
different pitch-class sets. Pitch-classsets which reproduce, as VP
sets, the intervalcontent of a given AB set, will be called
VPrepresen- tatives of the AB sourceset. In Example9,3-3,3-11
and4-17 areseen to be VP representatives of 4-19 (Ex. 14). The
following musical excerpts demonstratethe usefulness of inter- val
sets in determining structuralrelations. NotethatVP sets appear
above the score and AB sets below. Example 15 shows multiple
instances ofVP representation in a Bar- tok string quartet. The
structureofthe first measureis based not only upon the repetition
of4-18and 4-12, but also upon the function of thesecond form
of4-12as aVP representative ofthefirst form of 4-18.In thesecond
measure, 4-Z29and 4-16introduce AB:1-6-10 and AB:5-6-10.
4-16returnsat the beginning of the third measurewith adifferent
ordering of AB:5-6-10. Then 4-6and 4-Z29,functioning as VP
representatives of4-Z29and 4-16,respectively, recreate (in the VP
domain) thesuccession ofinterval setswhichoccurred inthe previous
measure. Example 16, from Schoenberg'sGeorgeLieder, contains two
pitch- class sets which are each VP-representedby twoother
pitch-class sets. In the first measure, after a VP interval
exchange which connects 3-11 and 3-4, 4-20 appears with the
setAB:3-7-8.In the third and fourth measures, 4-18and 4-Zl15
function as VP representatives of4-20.In the second and third
measures, 4-5and 4-6function as VP representa- tives oftheform of
4-Z15 (AB:2-5-6) which follows. As in the Bartok example, a
recurrent pitch-class set (in this case 4-Z15) is also involved in
a system of intervallicrelations. Example 17, from Pierrot Lunaire,
may be viewed as a variationon the succession ofinterval sets which
ends example 16. In Example 16, 4-19, in the form AB:4-5-8, is
immediatelypreceded and immediately followed by two orderings of
VP:3/7/8, expressedby 4-18and 4-Z15. Example 17 contains the same
AB form of 4-19, with the same vertical 283 This content downloaded
from 128.59.222.12 on Sat, 11 Jul 2015 20:58:05 UTCAll use subject
to JSTOR Terms and Conditions4-254-27 4-244-5 TenorI IiI"r T A j 4
2 4 [VP:]246 4 4 1 Example 13.A VP Interval
Exchange:Stravinsky,Zvezdoliki, m. 11 4-19 4 [AB:] 8 9 Example
14.The AB source set ofAB: 4-8-9is 4-19.In Example 9, 3-3, 3-11and
4-17are seen to beVP representatives of 4-19. 284 This content
downloaded from 128.59.222.12 on Sat, 11 Jul 2015 20:58:05 UTCAll
use subject to JSTOR Terms and Conditions610 11 16 [VP:] 3 105
4-184-124-184-124-Z294-164-164-64-Z29 11665 AB:] 6 10106 31510
Example 15. Bartok, Second StringQuartet,op. 17, (third movement)
572638 [VP:] 756583 4-64-2154-184-194-215 [AB:] 3 28 8 65 Example
16. Schoenberg,GeorgeLieder, op. 15/4 (mm. 5-8) [VP:] 7 3 756 5 83
4-274-19 4-2158 4-94-Z 7 54 [AB:] 8 8655 Example 17. Schoenberg,Pi
GeorgeLieder, op. 21/17 (mm.8-9) Example
17.Schoenberg,tP'errotLunaire,op. 21/]17 (mm. 8-9) This content
downloaded from 128.59.222.12 on Sat, 11 Jul 2015 20:58:05 UTCAll
use subject to JSTOR Terms and Conditionsordering of intervals.
Here 4-19is also preceded and followed by two orderings of
VP:3/7/8. Thesecond ordering, asin Example 16, is expressedby
4-Z15. The first ordering,however, is expressedby 4-27 rather than
4-18.It is the interval sets, not the pitch-classsets, which
confirm the structural similarity. Interval setsina larger context.
Example 18isa primarily tetra- chordal excerpt from the second
ofCarl Ruggles's Evocations (1943). Thetetrachords are identified
by ordinal number (forexample, "4" means "4-4") between thestaves.
The table which accompanies the excerpt shows that eighteen
tetrachords appear atleast once. Twelve ofthese tetrachords might
be considered structurallysignificantsolely on the basis of
recurrence, but these tetrachordsare also involved in a numberof
intervallicrelations. Intervallicrelationsalso provide a means of
integrating the single occurrences ofsixother tetrachords intoa
structural analysis of the excerpt. Example 19isa summary
ofrecurrentinterval sets in the excerpt. Examples 19a-ddeal
withinterval sets which first appear in measure 7;Examples 19e-g
show other significant interval sets. Pitch-classsets whose names
are outlined are those which occur only once. In example 19a,
4-4and 4-Z15are seen tobe VP representatives(VP:1/8/10) of 4-11
(AB:1-8-10). Example 19b shows that twoof these voicings of 4-11are
VP-relatedto4-2 (VP:1/9/10); the AB source set, 4-3, also
appears.Example 19c groups 4-10and 4-12with their VP representa-
tives. 4-10occurs only onceinthe excerpt, while 4-12 appears four
times, but they are of comparable intervallic status. In example
19d, 4-5is a VP representative(VP:1/7/10) of four subsequent
occurrences of4-13 (AB:1-7-10). These voicings of4-13in turn
express the set VP: 1/9/9. Becausethis VP set contains two, not
three, different intervals, its AB source set is a trichord; this
trichord,3-3, is includedin example 19d. In example 19e, 4-9, which
appearsonly once, is associatedwith its VP representative, 4-5. The
tetrachordsof example 19f are presented in the orderin which they
occur in the composition, demonstrating an interesting succession
of relationships: 1.4-12 expresses its recurrentAB set (AB:1-3-9)
and introduces VP: 1/2/6. 2.4-12isaVP representative of4-5
(AB:1-2-6), which intro- duces VP:1/5/8. 3.4-5isaVP representative
of4-20 (AB:1-5-8), which intro- duces VP:1/7/9. 4.4-20isaVP
representative of 4-Z15 (AB:1-7-9), which in thesetwo
appearancesexpresses VP:1/8/10, itselfarecurrentVP set (see Example
19a). 286 This content downloaded from 128.59.222.12 on Sat, 11 Jul
2015 20:58:05 UTCAll use subject to JSTOR Terms and Conditionsa
tempo 9poco oco 7ITea F.reqeni 42 2 1 4 55 3 1 1012 Poio .r
4-814-19 4113 Z293917135138 5141920115 .44-ab.44- 2odeato- 4-9 1
4-200 11 ..... 7 1213 i , 16417 4-1014-Z292 Example 18.
Ruggles,Evocations, II 287 This content downloaded from
128.59.222.12 on Sat, 11 Jul 2015 20:58:05 UTCAll use subject to
JSTOR Terms and Conditions8108 [VP:I1810 1011 -," I VI ,I _
(a)4-4(7)4-Z15(14,15)4-4(16) 4-11(8)4-11(11)4-11(18) 8108 [AB:I
10810 111 91010 [VP:1 199 1011 (b) 4-2(7) 4-11(18)4-11(8)4-3(8) 9
[AB:1 10 1 33 [VP:]110 101 (C) 4-2(7)4-1(16) 3 [AB:1 10 1 4 4 [VP:]
110 101 4-5(7)4-2(16) 4-12(8) 1 [AB:] 10 Example 19.Recurrent
Interval Sets in Evocations, II (measure num- bers in parentheses)
288 This content downloaded from 128.59.222.12 on Sat, 11 Jul 2015
20:58:05 UTCAll use subject to JSTOR Terms and Conditions7999 [VP:]
199 9 10111 (d) 4-5(7)4-13(10) 4-13(14,15) 4-13(17)3-3(14) 7779
[AB:] 1010109 1 111 66 [VP:] 77 11 (e)4-9(10)4-5(11)4-5(11) k-- 7
[AB:] 6 1 68910 [VP:] 2578 (f)4-12(9)4-5(10) 4-Z 15(14,15) 9 257
[AB:J 3 689 1 111 4 78 [VP:] 9 87 1 1 1 (g) 4-2(15) [AB:] 98 11
Example 19 (continued) 289 This content downloaded from
128.59.222.12 on Sat, 11 Jul 2015 20:58:05 UTCAll use subject to
JSTOR Terms and ConditionsExample 19g illustrates that three sets
which occur once each (4-17, 4-19and 4-8) are intervallically
related. 4-17and 4-19are VP repre- sentatives of4-8. (Inaddition,
4-17is VP-representedby 4-2.) Each pitch-class set appearsonly
once; the intervalset appears three times. It is not to be
inferredthat intervalset considerationsare intended to constitute a
complete analysis. It is my hope that this paper has demon-
stratedthe potential for their incorporation into a
largermethodology. NOTES 1.The complement ofa pitch-class
setXisthesetwhichcontainsallthe elementsnotinX.Thesetsofa Z-pair
havethesame totalinterval content, butare notreducibletothesame
prime form.For atonalsets see Allen Forte, "Sets andNonsetsin
Schoenberg's Atonal Music," Perspectives ofNew Music, 11:1
(1972),p. 45. 2.Allen Forte, TheStructure of AtonalMusic (New
Haven: YaleUniv. Press, 1973),p. 210. 3. Forte, Atonal Music,p.
210. 4. Forte, Atonal Music,p. 209. 5.Weshall notreduceintervals
greater than a tritoneand less than an octaveto intervalclasses.
Rather, interval sizes will range from0to11 semitones inclu- sive,
with larger intervals reduced mod12. 6. Forte, inthe"Basic Interval
Patterns," Journal of Music Theory 17 (1973),
referstotheseintervalsetsas"basicinterval patterns"(bips)
andreduces theintervals thereintointerval classes. Thus the bips
for Examples 7a and 7b wouldbe236and 123,respectively. Once again
weshallavoidabstract measurement and maintain absoluteinterval
sizes. 290 This content downloaded from 128.59.222.12 on Sat, 11
Jul 2015 20:58:05 UTCAll use subject to JSTOR Terms and
Conditions
