relating musical contours - extensions of a theory for contour

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RELATINGMUSICALCONTOURS:

EXTENSIONSOF A THEORY

FOR CONTOUR

ElizabethWest Marvinand

Paul A. Laprade

Cognitivepsychologists ndmusictheoristshave, ormanyyears,under-stood that humanperceptionof pitchcannotsimplybe modelledalonga

singlecontinuumrom ow to high.'Thusrepresentational odelsforpitchperceptionhavebeendevelopedbypsychologistso reflecta number f re-lateddimensions,2among hem hetendency f listeners amiliarwith West-ern tonalmusic to groupoctave-related itchesinto equivalenceclasses.Nevertheless,n

spiteof this

tendency,istenersare forthe most

partunable

to recognizefamiliarmelodies which havebeen distortedby octavedis-

placement nlessthemelodiccontour emainsnvariant. o importants therole of contour ntheretention ndrecognition f well knownmelodies hateventhesize of the intervalbetweensuccessivepitchesmaybe altered,and

subjectswill usuallyrecognize he tune if the contourremainsunaltered?Further,experimentation as shown that listenersfrequentlyconfuse a

fugue subjectwith its tonalanswer-thatis, theyidentify he twoas identi-cal onthe basisof theirequivalentontours nddiatonic caletypes,despitethe fact thattheir

pitchcontents

differ.By extention o a non-tonal ontext,we may predict hatlistenerswillbe morelikely to assumethatnon-equivalentets belongto the same setclass if theircontoursare the same. In fact,W.J. Dowlingand D. S. Fugi-tani haveofferedexperimentalustification or the premisethat listeners

225

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retainbrief non-tonalmelodiessolely in termsof theircontours.Thus we

maysurmise hatgiventhe sameor similarrhythmicpattern, istenersare

generallyable to perceiveequivalenceor similarityamongmusical con-tours moreeasilythanamongpitch-class ets in melodicsettings. Figure

1, for example, llustrates wo instancesdrawn rom the musicof AlbanBerg in which melodicpatternsshare contouridentitybut not set-class

identity.The melodiesof Figure lA appearaboutsix bars apartin the

secondmovement f theLyricSuite.Surely he listenerwill associate hese

twoon thebasis of their dentical ontoursandrhythmic imilarity,n spiteof the fact that theirintervallic ndpitchcontentsdiffer.The firstmelodyis a member f the set class 10-4,while the secondbelongs o set class 10-3.

The melody of Figure IB, drawnfrom the second movementof Berg'sViolinConcerto,maybe divided nto twopartsas marked.Thesecondunit

is an intervallic xpansion f the first,butmaybe heardas a same-contourimitationof the first. As in the previousexample,each unitbelongsto a

different et class-the first to 4-27 andthe secondto 4-20.Forpurposesof musicalanalysisanddescription,music theoristshave

also found t useful to dividemusicalspaceinto a numberof interrelated

spaces,7mostcommonly ntopitch space(a linearspaceof pitcheswhich

extends romthe lowestaudiblerange o thehighest)andpitch-class pace(a cyclicalspaceof twelve pitchclasses that assumesoctaveequivalenceand, becauseof its closed groupstructureundertranspositionaddition

mod-12) nablesequivalence lassesnotpossiblein pitchspace)?Recentlya numberof theoristshavefocused heir attentionupontheexaminationf

another ype of musicalspace, which has been called contourspace?In

formulatinghis concept,music theoristsrecognize he factthatlisteners

mayperceivesimilarityor equivalence mong he contoursof twophrasesquite apartfromaccurately ecognizingpitchor pitch-classrelationshipsbetweenthem, as notedabove.In order to reflectthis aspectof musical

perceptionn analysis,newtheories orcomparing ontoursarenecessary.Criteria ywhichcontoursmaybejudgedequivalent avealreadyappearedin the literaturen publications y RobertMorrisandMichaelFriedmann.This articletakesMorris's ontour-spacequivalence elationsas its pointof departure, evelopsa prime ormalgorithm ndtableof c-spacesegmentclasses,positssimilaritymeasurementsorc-spacesegmentsandsegment-classes of the same or differingcardinalities,and appliesthese tools in

musicalanalysis.

ContourEquivalence.RobertMorrisdefinescontour pace (c-space)asa

typeof musical

space"consisting f elementsarrangedrom low to high

disregardinghe exact intervalsbetweenthe elements."'•These elementsaretermed"c-pitches""cps") ndare"numberedn order rom ow tohigh,

beginningwith0 upto n-l,"wheren equals hecardinality f the segment,and where the "intervallicdistancebetweenthe cps is ignoredand left

226








a.Berg:LyricSuite, (mvt. II), vln. I,

mm. 66-67 and72-73

b. Berg:ViolinConcerto: mvt. II), bassoon, mm. 35-36

1 2

AJwa

Figure1. Same-ContourMelodies

227








undefined."IISeeGlossary ordefinitions f technicalerms.)The decisionnot to definethe intervallicdistancebetweenc-pitchesreflectsa listener's

ability o determinehatonec-pitch s higher han, ower han,or the sameas another,butnot to quantifyexactlyhow muchhigheror lower. In this

respect,Morris'sheorydiffers romthatof MichaelFriedmann.Manyofthe issues addressedn the latterpartof Friedmann'srticlehinge uponthe

conceptof contour ntervalshatmeasure he distancebetweenc-pitches.'2In ourformulation, owever, s in Morris, he intervallicdistancebetween

cps will remainundefined.

Musicalcontoursarebydefinition rdered; hus,we will definea c-seg-ment (cseg) as an orderedset of c-pitchesin c-space.13Csegs will be

labelledthroughouthis paperby capitalletters;the cps which makeupcsegs will be denotedbylower-caseetters.Further,we defineanyordered

sub-groupingf a given cseg as a c-subsegmentor csubseg).A csubsegmaybe comprisedof eithercontiguousor non-contiguous -pitchesfrom

theoriginalcseg, as shown n Figure2. Thecontourdiagramsusedin this

figureappear hroughoutur discussionas graphicrepresentationsf con-

tourshape.Suchdiagramsmakerelationships mongcontours airlyeasyto spotvisually;thus,we see thatcsubsegsB andC are inversionally e-

lated,whileA andD appear o be equivalentontours.Moreformaldefini-

tions of contour quivalence,heoperation f inversion,andotherrelations

amongcontours ollow.

Weproposea "normalorm" orcsegsandanoperationbywhichcsegsthat arenot in normal ormmaybe reduced o this form.The elementsof

a cseg of n distinctc-pitchesare listedin normal ormwhenthe cseg'sc-

pitchesarenumberedrom0 to (n - 1)and arelistedin temporal rder.A

csubseg'selementsmayretain he samenumbersassignedto these cps in

theoriginalcseg, or maybe renumberedhrough"translation."ranslation

is an operationhroughwhich a csubsegof n distinctc-pitches,not num-

bered n register rom0 to (n - 1), is renumberedrom0 forthe lowestc-

pitchto (n - 1)for the highestc-pitch n the csubseg,as illustrated y the

asterisks n Figure2 ."Morris'scomparisonmatrix(COM-matrix)will be used to compare

contours n c-space, to defineequivalencerelations,and to developour

similaritymeasurementsor musicalcontours.Thecomparisonmatrix s a

two-dimensionalrraywhichdisplays he resultsof the comparison unc-

tion, COM(a,b), oranytwoc-pitches n c-space.If b is higher hana, the

functionreturns"+1"; f b is the same as a, the functionreturns"0";and

if b is lowerthana, COM(a,b)returns"-1."'5 The repeatednstancesof

theinteger

"1" re omitted ntheCOM-matrix, s shown n Figure3 below.

Eachof the matrices hroughouthisarticle,has symmetrical ropertiesn

whichthe diagonalof zeros from the upper eft-hand o lowerright-handcomer (the"main" iagonal) ormsan axisof symmetry.Eachvalue n the

upperright-handriangles mirrored n the othersideof themaindiagonal

228








VVn.

Webern,op. 10/1, mm. 7-10

54

3 =<502314>

0

Selectedcsubsegsof cardinality :

A: 5

A: 3 =<5023>==<3012>*

0

B: 3

"

=<0231>

0

C:54 =<5314>=<3102>*

3

1

D: 5

4

=<50 14>= <30 12>*

0

*Normal rderby translation.

A andB are contiguous;C andD are non-contiguous -subsegments.

Figure2. C-Subsegments229








7

4 4

3 3 5

2 2 4

1 1 3

0 0 0

A=<03241> B=<04231> C=<05473>*

03241 04231 05473010++++ 010++++ 010++++31- 0 -+ - 41- 0 - - - 51- 0 -+-21- +0+ - 21-+ 0+ - 41-+ 0+-41- - -0- 31-+ - 0 - 71-- - 0-1 - + + + 0 1 I- + + + 0 3 1- + + + 0

Csegs A and C areequivalentbecausetheygenerate denticalCOM-matrices.

* Normal formof < 0 5 4 7 3 > = < 0 3 2 4 1 > by translation.

Figure3. ComparisonMatrices

230








by its inverse.This symmetrical tructures a natural onsequenceof thefactthatcontour-pitchCOM-matrices nly comparea cseg with itself.

The comparisonmatrixprovidesa conciseprofileof a cseg'sstructurein much the samewayas Friedmann's ontourAdjacencySeries(CAS),'6

except hat he COM-matrixurnishesa muchmorecompletepicture inceit is not limitedsimplyto relationships etweenadjacent ontourpitches.Indeed, the CAS appearsas a subset of the COM-matrix,as the first

diagonalaboveandto the rightof the maindiagonal,as shown in Figure4A. Each of the diagonalsto the rightof the main diagonalis termed

INT,,'7 wheren stands or the differencebetweenorderpositionnumbersof the twocps compared;hat s, INT4 ompares ps thatarefourpositionsapart.INT,displays heresultsof thecomparisonunction oreachpairof

adjacent ps as Figure4B shows: < + - + + > for the comparisons

to3, 3 to 1, 1 to2, and2 to 4. INT2 howseachcomparison etweena givenc-pitchand a secondcp twice removed rom the first: < + - + > for0to 1, 3 to 2, and 1 to 4. Likewise,INT3displayseachcomparison etweentwo cps threepositionsapart:< + + > for 0 to 2, and 3 to 4. Finally,INT4shows the comparisonbetweentwo cps fourpositionsapart.In this

case, the predominancef plusesoverminuses in each of the INTsillus-tratesthe generallyupwardmotion of this contour.

The information rovidedby the COM-matrix ives a much more ac-curateprofileof cseg structure hanINT, alone, since c-pitches may be

comparednot only consecutively,butalso non-consecutivelywith respectto relativeheight. By way of example, Figure5 contrastsseveralcsegswhichsharean identical NT, but whichdiffera greatdealwithrespect otheiroverallmusicalcontours,a fact which is reflected n theirrespectivecomparisonmatrices.

Twocontour quivalence lassesbasedupon hecomparisonmatrixhavebeenproposedby Morris.The first of these is madeup of all c-segmentswhichshare he samecomparisonmatrix; hus,thefirstandthirdcontoursof Figure3 precedingwereequivalent segs sincetheyproduceddentical

COM-matrices.Further,equivalentcsegs may be reduced to the samenormalorderby ourtranslation peration,as shown n Figure3. The sec-ondcontourequivalence elation, hec-spacesegmentclass (csegclass),isan equivalence lass madeup of all csegs relatedby identity, ranslation,retrograde,nversion,andretrograde-inversion.he inversionof a cseg P

comprisedof n distinctcps is written P,andmaybe foundby subtractingeachc-pitchfrom(n - 1).18The retrogradef a cseg P (writtenRP)or itsinversion writtenRIP)consists of the c-pitches n cseg P or IP in reverseorder.Twocsegs belonging o the samec-spacesegmentclass maybe re-duced to the sameprimeformaccording o the primeformalgorithmweintroducebelow.Csegclasses,as distinctfromcsegs, will be labelledwithunderscoredapital etters.

Figure6 showsrepresentativesf csegclassP, consistingof its prime

231








A:

4

3

20 1

031240 -- INT4

3 - -- INT31 I- + -- INT22- + - -- INT14 -- maindiagonal

INT1 = < + - + +> ( = CAS) INT2 = <+ -+ >

INT3= <++>

INT4=

<+>

B:

INT1--<O 3 1 2 4> INT2--<O 3 1 2 4>

+-++ + +

+

INT3--<O 3 1 2 4> INT4--<O0 3 1 2 4>

+ +

Figure4. Structure f the

Com-Matrix

232








5 5 5 5

A-<021435>B=<120435> C=<043512>

D-<452301>

021435 120435 043512 452301

21- - +++ 21- -+++ 41-5.

51-

11-+ ++ 01++ ++ 31 21++

41- - u-,+

41- - - + 51 31++ -

1 31 3 31 +++ 4301+++

- 0 -+++ 01 ++ ++1+---21- 21- -+++ 41 -+ - 51-

51 51 21- +++ - III++++ -

Eachcontour has INT1 = < + - + - + >

As shown by contourgraphs,contoursA andB are most similar,

A and D most dissimilar.

Figure 5. Comparisons Among Selected Csegs Where

INT1= < + - + -+ >

233








form,< 0 1 3 2 >, togetherwith itsinversion, etrograde,ndretrograde-inversion,and the COM-matrixor each. The inversion,retrograde nd

retrograde-inversionf a contourP arealso definedby Morris n termsof

specifictransformationsf the COM-matrixorP,9 as illustratedn Figure

6. TheCOM-matrixorIP,forexample,merelyexchanges ach"+" fromthe P matrix for "-" in the IP matrix and likewise exchanges "-" for "+."

The matrixfor RIP is related n a somewhatmore abstractmanner,as

though heP matrixhadbeen"flipped"round hesecondarydiagonal the

diagonalproceedingrom helower eft-handorner otheupperright-handcorner).Finally,the COM-matrixor RP combinesboththe flip and the

exchange eatures.Twocsegsbelonging o thesamec-spacesegmentclassmaybe reduced

to thesameprimeform.Simplyexpressed,ourprimeformalgorithm on-

sists of threesteps:

1) If necessary,ranslate hecseg so itscontent onsistsof integers rom0 to (n - 1),

2) If (n - 1)minus the last c-pitchis less than the firstc-pitch,invertthe cseg,

3) If the lastc-pitch s less thanthe firstc-pitch, retrogradehe cseg?0

If forsteps2 and3 the firstand lastcps are the same,compare he second

and the second-to-last ps, and so on until the "tie" s broken.Figure7illustrates he use of this algorithm or severalcsegs andshows that eachis a memberof the samecsegclass.A listingof all c-space segmentclasses

of cardinalities through6 maybe found n the Appendix o this article.

Weexclude argercsegs because of limitationsof space.

SimilarityRelations.The similarityof twocsegs or csegclassesmaybe

measuredntwoways:eitherbycomparingheirstructuralrofilesas sum-marizedn theCOM-matrix,rbyexaminingheircommoncsubsegstruc-

ture.The firstof thesewe will call the contour imilarity unction CSIM)and the second, the contourembeddingfunction(CEMB)?•Both are

designed o return decimalnumberwhichapproaches 1" scsegsbecome

more similar.A functionwhichreturnshe value"1" ompares woequiv-alentcsegs?2

The contoursimilarity unction,CSIM(A,B),measures he degreeof

similaritybetween wo csegs of the samecardinality.t compares pecific

positions n the upperright-handriangleof the COM-matrixor cseg A

withthecorresponding ositions

n the matrixofcseg

B in orderto total

the numberof similaritiesbetween hem?3For eachcomparedpositionof

identicalcontent, his total s incremented y 1. Sucha similaritymeasure,if it weresimplyto totalthe numberof identicalmatrixpositions,would

not yet yield a uniformmodel of similarity among csegs of various

234








3 3 3 32 2

11 Z2 Z2 1

0 0 0 0

P: <0132> I: <3201> RI: <1023> R: <2310>

0 1 2 3201 1 0 2 3 2310010 + + + 310 - - - 110 - + + 210+ - -

11- 0 + + 21+ 0 - - 01+ 0 + + 31- 0 - -31- - 0 - 01+ + 0 + 21- - 0 + 11++0 -21- - + 0 11+ + - 0 31 - - - 0 01+++O

Inversion= Retrograde=

Exchange only Exchange& flip

Retrograde nversion=

Flip only

Figure6. C-Space SegmentClass < 0132 >

235








Csegs: <0312> <3021> <1204> <3241>1. TRANSLATE, if I I

not consecutive Translated= Translated=integers0 to(n- 1): OK OK <1203> <2130>

2. INVERT, if Inverted= Inverted=(n- 1) minuslast

cp< first

cpOK < 0 3 12 > < 2 1 30 > OK

3. RETROGRADE, if Retrograde= Retrograde=lastcp < firstcp

OK OK <0312> <0312>

PRIMEFORM: <0312> <0312> <0312> <0312>

All four csegs belong to the same c-space segment class.

Operations:

To translate, enumber hecseg with consecutiveintegersfrom0 to (n - 1), where n

equals thecardinalityof thecseg.

To invert,subtracteach cp from(n - 1).

To retrograde,place the cps in reverseorder.

Figure 7. Prime Form Algorithm

236








cardinalities.That s, a similaritymeasurementf 3 between wothree-note

csegswouldsignifya muchhigherdegreeof similarityhanwoulda similar-

ity measurementf 3 between wo seven-note segs?4In orderto createamoreuniformmeasurement,he number f identicalpositionswillbedivid-

edbythe totalnumber fpositionscompared;25husCSIM(A,B)will returna decimalnumberwhichsignifiesgreater imilaritybetweencsegs as thisnumberapproaches . Figure8 illustratesCSIM(A,B) or variouscsegs of

cardinality . As thecontourdiagrams f Figure8 show,contoursA andDhavean inversional elationship.Theyare,in fact,RI-related nd aremem-bersof the samecsegclass,c4-4. OurmeasurementCSIM(A,B)as yet ac-countsonlyforsimilaritybetweencsegs,notcsegclasses; hus,anextensionof the similaritymeasurements needed.

We define he similarity unctionCSIM(A,B) o compare he similarity

between wo csegclasses.CSIM(A,B)returnshe largestdecimalnumber,or 1,obtainedbycomparingheCOM-matrix f one cseg representativef

csegclassA with fourcseg representativesP,I, R andRI)of csegclassB.

Therefore,CSIM(A,B) ndicates he degreeof highestpossiblesimilaritybetween wocsegclasses.If thetwocsegsaremembers f thesamec-spacesegmentclass, CSIM(A,B)will returna value of "1".Figure9 offerstwo

examples: f we compare hecsegs A: < 0 2 3 1 > andB: < 3 1 0 2 >for similarity,CSIM(A,B)accuratelyreflects heir totaldissimilarityandinversional elationshipwithrespect o contour CSIM(A,B)= 0), but not

the fact hat hesecsegsbelong othe samec-spacesegment lass. CSIM(A,B), however, eturnshevalue"1"rincethe twocsegsaremembersof cseg-class c4-4. Inthesecondexampleof Figure9, csegs C andD arenotmem-bers of the samecsegclass;

CSIM(_C,D)

returns he value .80.Oneof the mostintuitively atisfyingwaysofjudgingsimilarityn csegs

of differing ardinalitiess to countthe numberof timesthe smallercsegis embeddedn the larger?6We can do this in one of two ways:eitherbyexamining he two COM-matrices o determine he numberof times thesmallercseg'sCOM-matrixs embeddedn the COM-matrix f the largercseg, orbylookingat allpossiblecsubsegswithin helarger seg anddeter-mining by translationhow manyare equivalent o the smallercseg. We

proposea contour mbeddingunction CEMB(A,B))n whichthe numberof timescseg A is embedded n cseg B is dividedby the totalnumberofcsubsegsof the samecardinality s A possible, in orderto returna valuewhichapproaches forcsegs of greater imilarity.The formula ordeter-

miningthe numberof m-sizedsubsetsof an n-sizedset is:27

n!

m! (n - m)!.

Figure 10 illustrates wo ratherdissimilarcsegs of unequalcardinality:CEMB(A,B)= 2/20 = .10.Csegc3-1 < 0 1 2 > is embedded nlytwiceincsegc6-96 < 4 5 2 3 6 1 >, as thecontiguous subset< 2 3 6 > and

237








3 3 3 3

0 0 0 0

A=<2013> B=<0123> C=<1302> D= <0231>

0 01 3 1302 0 231

1 21+ 21 -o001+ 31-O

3---O0 21-+-1 -1+0-

CSIM(A,B) = 4/6 = .67CSIM(A,C) 3/6= .50CSIM(A,D) = 2/6 = .33

Figure8. CSIM as SimilarityMeasurementor Csegs of the SameCardinality

238








CSIM(A,DL): A=<0231> B=<31'02>

3 3 3 3 32

/ -A2

2 1 1 2 1 1 1

0A PB IB RB RIB

0231 . 3102 0231 2013 1320010+++ 310--- 010+++ 210 - -+ 110++ -

21- 0+ - 11+0 -+ 21- 0+ - 01+0++ 31- 0 - -

31- - 0 - 01++0+ 31- -0- 11+- 0+ 21-+ 0 -

11-++ 0 21+ -- 0 11- ++0 31- - -0 01+++0

CSIM(A, PB)= 0/6 = 0CSIM(A, IB) = 6/6 = 1

CSIM(A, RB) = 2/6 = .33

CSLM(A,RIB)= 4/6 = .67

CSIM(A., 1) =1

CSIM(C,D): C=<10432> D=<12403>

4 4 4 4 43 3 3 3 3

2 2 2 1223

C PD ID RD RID

10432 12403 32041 30421 14023110 -+++ 110++ -+ 310 - -+ - 310 -+ - - 110+ -++01+0+++ 21- 0 +-+ 21+0-+- 01+0+++ 41- 0-- -

41- - 0 - - 41- -0-- 01++0++ 41- - 0 - - 01++ 0++31- -+ 0 - 01+++0+ 41- - - 0 - 21+-+ 0 - 21- + - 0+21- -++ 0 31- -+ - 0 11++-+ 0 11+-++ 0 31- + - - 0

CSIM(C, PD) = 6/10 = .60

CSIM(C, ID) = 4/10 = .40CSIM(C, RD) = 8/10 = .80

CSIM(C, RID) = 2/10 = .20

CSIM(C, f2.) = .80

Figure9. CSIM for C-SpaceSegmentClasses

239








thenoncontiguous 4 5 6 >. InFigure10B, hecompletematrixof cseg< 0 1 2 > is foundas a contiguous ubsetof thelargecseg'smatrix,while

Figure10Cshowsthe matrixof < 0 1 2 > embedded s a noncontiguoussubset.Thec-pitchesassociatedwith eachpositionof the embeddedmatrix

are membersof the csubseg < 0 12 >. Note thatin the noncontiguousinstance, he entire structure f each embeddedrow andcolumn mustre-mainintact n order o reflect hecsubsegrelationaccurately.t is for thisreason that CEMB(A,B)must considerthe structureof the embeddedCOM-matrix s a whole ratherhan heupperright-handrianglealone.In

figure10D,theplusesof theupperright-handriangleof the smallercseg'smatrixhavebeen circledin non-adjacent ositionsof the largercseg'sma-trix. If the rowsandcolumnsare not violated,the correspondingmatrixentriesforthe maindiagonaland lower eft-handriangle indicatedn the

figureby squares)are incorrect or the embedded ubset.Thus,the infor-mationgivenintheupperright-handriangles notalonesufficient o iden-

tify c-subsegments.Since theembeddingunction hecksfornon-contiguousubsetsas well

ascontiguous nes,itaccountsorsuch nstances s a contourwhichweper-ceive asgenerally ising,eventhough talsoincludes omedescents. nFig-ure11A, orexample,he embeddedsubseg< 0 1 2 > appears epeatedly,bothas a non-contiguousnda contiguous ubsetof < 0 2 1 3 4 >, and

its role in our perceivingthis contour as an ascendingline is clearly

audible. As the comparisonmatrix and correspondingontourdiagramsshow,< 0 1 2 > is embedded even imesin thelargercseg. CEMB(A,B)can alsobe foundby extracting ll three-notesubsegs rom helarger seg,

translatingeach to normal form, and counting the numberof times

< 0 1 2 > is found,as shownin Figure11B.

Although heCSIMand CEMBfunctionsprovideanadequatemeasure

of similaritybetweenmostcsegs (of equalor unequal ardinality),heyare

not alone sufficient o describerelationships etweenany two csegs. For

example,our embedding unctiononly describesrelationshipsbetween

csegsof differing ardinalities.Whatof the situationn whichtwocsegsofequalcardinalityhareoneor morecommon segs?FollowingRahn'sener-alization f DavidLewin's mbeddingunction,28e propose woadditional

functionswhich count thecsubsegsmutually mbeddedn csegs A andB.

CsegsA andB maybe of equalor unequal ardinality.CMEMBn(X,A,B)countsthe numberof timesthe csegs, X (of cardinalityn), areembedded

in bothcsegs A andB. The variable"X"maysuccessivelyrepresentmore

thanone cseg-typeduring he courseof the function,as shown in Figure12. Eachcseg X must be embeddedat leastonce in bothA andB; then,

all instances f X arecountednbothA andB. Thetotalnumber fmutually-embedded segs of cardinality is dividedbythe numberof n-cardinality

csubsegspossiblein bothcsegs to returna decimalnumberapproachingasthecsegsA andB aremoresimilar.Generally,CMEMBn(X,A,B)eturns

240








CEMB(A,B)

A = <0 1 2 > = c3-1B = < 4 5 2 3 6 1 > = c6-96

A - MATRIX OF c3-1:

01 0 + +11- 0 + <012>21 - - 0

B - MATRIX OF c3-1 EMBEDDEDAS CONTIGUOUSSUBSET OFc6-96:

4 5 ~6) 1

410 + - - +-51-0 - - +QI+ + - <236>=<012>

U3+-I+ + +++0

C - MATRIXOF c3-1 EMBEDDEDAS NON-CONTIGUOUSSUBSET OFc6-96:

21+ + 0++ - <456>=<012>31+ + - 0+-

@le( - - @-11+ + +++0

D - UPPERRIGHT-HANDTRIANGLE:

012

4 5 2 3 6 452361

51--0- -

21[-] [E 0 + - 21+ + + + -31+ + - 0 + -

3,1• - o+-61E] [ - -

[M- 6 - -

11 + + + + + 0 1 1l1]++

0

Figure10. CEMB(A,B)241








Matrixembedding: A=<012> B=<02134>

012

01 0 + +11- 0 +21 - - 0

4 4 4

30

2 13

-

2 1

02134 02134 021340 o-8+00 +& 0rg)+?-+e2- +

2- 0 + 21-0-++1 I-+ 0++ 1l-+ 0++

104-+03 - -@+ 33--- -0+ 31-

- - 0+4 I- - - - 0 4Q(Qa--( 4) - (-

csubsegs: <0 2 3 > <0 2 4 > <014>

4 42

1

02134 02134 0213404Q+@)+ 0 100 (++ 0I0++++21- 0 -++ 21- 0 -++ 21- 0 -++

0E E l l - + 0 + + 1 l - + 0

41----O 01-~o

csubsegs: <013> <0 3 4 > <134>

4 02134010++++

1 1 I-+ 0++ CEMB(A,B) - 7/10 = .70

41csubseg:2 3

csubseg: < 234 >

Figure 11A. CEMB(A,B): Additional Examples

242








A=<012> B=<02134>

Possiblecsubsegs are: <021 >=<02 1><023>=<012>*<024>=<0 12 >*

<013>=<0 1 2 >*<0 1 4 >= <0 1 2>*<034>=<012>*<213>=<102><214>=<102><234>=<0 12 >*<134>=<012>*

* Embedded< 0 1 2 > identifiedby tlanslation.

CEMB(A, B) = 7/10 = .70.

Figure 11B. Embedded Csubsegs by Translation

243








CsegA = c5-26: < 10 4 3 2

>Csubsegs of A:

<10432>=<10432>

<1043>=<1032><1432>=<0321><1032>=<1032><1042>=<1032><0432>=<0321>

CsegB = c5-24: < 2 0 1 4 3

>Csubsegs of B:

<20143>=<20143>

<2014>=<2013><2013>=<2013><2043>=<1032><0143>=<0132><2143>=<1032>

CMEMB4(X,A, B) = 5/10= .50

<104>=<102><103>=<102><102>=<102><143>=<021><142>=<021><132>=<021><043>=<021><042>=<021><432>=<210><032>=<021>

<201>=<201><204>=<102><203>=<102><214>=<102><213>=<102><243>=<021><014>=<012><013>=<012><043>=<021><143>=<021>

CMEMB3(X, A, B) = 16/20 = .80

Commoncsubsegs are underlined.

Figure 12. CMEMBN (X, A, B)

244








a higherdecimalnumber or embedded segs of smallercardinality incethere are fewercseg types, andtherefore higherprobability f inclusionin bothcsegs A and B. Thus a refinement f the function s necessary.

ACMEMB(A,B)ounts hetotalnumber f significantmutually-embed-

dedcsegs of cardinality through he cardinality f the smallercseg, andadjusts his to a decimalvalueby dividingby thetotalnumberof possiblesubsetsof A andB (excludinghe null sets foreachand the one-notecsub-

segs).9 Figure13Ashowstheadjustedmutualembeddingunction ortwo

csegs of the samecardinality, nd 13Bforcsegs of differingcardinalities.

Finally,we generalizeourembeddingunctions orcsegclasses n muchthe same manneras the CSIM function.Thatis, CEMB(A,B),CMEMBn(X,A,B) andACMEMB(A,B)will compare hecsubsegcontentof cseg Awitheach of thefourtransforms f cseg B (PB,IB,RBandRIB)and return

thehighestof these values.Thus,if A andB aremembers f thesamecseg-class, each of these functionswill returna value of "1."

Extensionsof the Theoryor Context-Dependentnalysis.Up to this

point, we have considered relationsamong contours without extensivereferenceo themusicalcontextsn whichthesecontoursappear.Theappli-cationof contour heoryto context-dependentnalysisposes a numberof

problems,not theleastof which s thesegmentationf the music ntomean-

ingful units. Friedmannhas discussedsegmentationn some detail;his

examplesprovideconsiderablensight nto this difficultproblem.30 sec-ond context-dependentssue withconsiderableheoretical amificationssthecommonoccurrence f repeatednotes withina musicalcontour?'Con-secutiverepeated otesposenoproblem, incetheymaybetreated s singlecontourpitches,as shown n Figure14A.Wepropose hatcsegs containingnonconsecutive epeated -pitchesbe numberedn orderfromlow to highwith0 representinghelowestpitchand(n - 1 - r)thehighest;repetitionsof ac-pitcharerepresentedythe same nteger.Here he variable"n" tandsfor the cardinality f the cseg, while "r"equalsthe numberof times anyc-pitch is repeated.Thus, the contour of the melody in Figure 14B is< 1 2 3 0 3 1 >. Thecardinality f thecseg is 6, cp 1 is repeated nce and

cp 3 is repeatedonce; thus the cps are numbered rom 0 to 3, since(n - 1 - r) equals (6 - 1 - 2) or 3. Translation of a cseg including re-

peatednotes s definedas therenumberingf thecseg withintegers angingfrom 0 to (n - 1 - r). The inversionof a repeated-noteseg is calculated

by subtractingachcp from(n - 1 - r). Previously tateddefinitions f RandRIstillhold. Ourprime ormalgorithm lsoholds,although"ties"mayoccur more

frequentlyif for

steps2 and 3 the firstand last

cpsare the

same,the secondand the second-to-last ps arecompared,andso on untilthe"tie" s broken).TheCOM-matrices f repeated-notesegs differ rom

previousCOM-matrices nly in the fact that the repeatednotesgeneratezeros in positionsother thanalongthe maindiagonal.

245








A: CSEGS OF EQUALCARDINALITY

A=<0123> B=<0213>

Csubsegs of A: < 0 1 > = < 0 1 > Csubsegs of B: < 0 2 > = < 0 1><02>=<01> <01>=<01><03>=<01> <03>=<01><12>=<01> <23>=<01><23>=<01> <13>=<01><13>=<01> <21>=<10><012>=<012> <021>=<021><013>=<012> <023>=<012><023>=<012> <013>=<012>

<123>=<012> <213>=<102><0123>=<0123> <0213>=<0213>

17 csegs mutually embedded in both csegs; ACMEMB(A, B) = 17/22 = .77

B: CSEGSOF UNEQUALCARDINALITY

C=<02134>

Csubsegs of C: <0214>=<0213> <021>=<021> <02>=<01><0234>=<0123> <023>=<012> <01>=<01><0134>=<0123> <024>=<012> <03>=<01><0213>=<0213> <013>=<012> <04>=<01><2134>=<1023> <014>=<012> <23>=<01>

<213>=<102> <24>=<01><214>=<102> <13>=<01>

<02134>=<02134><234>=<012> <14>=<01><134>=<012> <34>=<01><034>=<012> <21>=<10>

29csegs mutually

embedded incsegs

A and C; ACMEMB(A, C) = 29/37 = .7833 csegs mutuallyembedded n csegs B and C; ACMEMB(B, C) = 33/37 = .89

Figure 13. ACMEMB(A,B) for Sets of Equal Cardinality

246








A. Repeatea-Note Segments Consecutive Repeated Notes:

Webern,op. 10/1,mm. 10-11

r-

<0 53241> NOT <0533241>

B.Repeated-Note egments,Non-ConsecutiveRepeatedNotes:Webern, op. 10/1, mm. 3-6

1230313 3

II ++-+0Z2 21-0+-+-

1 31--0-0-01 +++0++

P=<123031> 31--0-0-

110++-+0

3 3 3 32 2 222

1 2 1 1> RIP< 2 0 3 0 1 2

IP=<210302>* RP=<130321> RIP=<203012>

*To invert, each cp is subtracted from (n-l-r), where n represents the cardinality of the csegand r is the number of times a particular cp is repeated. In this instance, r=2, since cp I isrepeated once and cp 3 is repeated once.

Figure 14 247








Thecsegclassnameof a repeated-noteseg is a hyphenated ompositelabel, basedon the cseg's similarity o nonrepeated-notesegclasses.The

cardinality f thecseg appearso the leftof thehyphen.To the rightof the

hyphen,separated y slashes,are the ordinalnumbers f two related seg-classes. The firstordinalnumber epresentshecsegclass abelof somecsegwhose COM-matrixs identical o thatof therepeated-toneseg except hatit containsa plus in the placeof each0 in the upperright-handriangle.The second ordinalnumber epresentshecseg whichcontainsa minus n

each of those positions.In Figure15A, csegclassesc5-2 and c5-4 differ

fromtherepeatednotecseg in onlyone positioneach;thecomposite abel

is rc5-2/4("rc" tands or"repeated-notesegclass").32Tworepeatednotes

will result n two zerosin theupperright-handriangle,as shown n Figure15B,and so on. TheCSIMfunctionwill return he samevaluewhenmea-

suredbetweena repeated-note seg and the csegclasses representedn its

composite abel (or betweenthose two csegclasses), since each of these

csegclassesdifferspreciselyin the positionsof the COM-mtarixwherea

"O" ppearsor therepeated-noteet. Therefore,he nameof therepeated-notecseg allowsus to generate he COM-matrix f the repeated-noteseg

(andthereforehenormal ormof thecseg itself)merelyby comparinghe

csegclasses n its name.Finally,our similarityandembeddingunctions33still hold for repeated-notesegs, as for nonrepeated-notesegs.

AnalyticalApplications.We have chosen to illustrate ome analytical

applications f thepreceding ontour heories n thefirstof AntonWebern's

FanfStackear Orchester,Opus10. The movementdividesinto fourtwo-

andfour-barphrases-A (mm. 1-2), B (mm. 3-6), C (mm. 7-10), andD

(mm. 10-11)-plus a concluding1-bar"codetta"f a singlereiterated itch.The two centralphrasesarejoined in an antecedent-consequentelation-

ship.Bothconsistof a broadsolo line played n the upperregisterovera

sustained elestatrill. Bothmelodieshave substantial ccompaniments:

seriesof chordsbeneath heantecedent

hrase,anda

thicker,morecontra-

puntalaccompanimento theconsequent.Flanking his centralportionon

eitherside areopeningandclosingsectionsof sparser exture,consistingof solo lines withoutaccompaniment. he firstandlast barsof the move-

mentfeaturestriking nstancesof Klangfarbenmelodie,hile the second

andpenultimate arsconsistof unaccompaniedolo lines on distinctive,coloristicinstruments.Thus the openingand closing sectionsframethe

centralportion n a symmetrical rrangement,s shown n Figure16.

Eachof thefourprincipalmelodies ormsa melodiccontourof cardinal-

ity six. Yetin eachcase the six cps arepartitioned ifferentlyn termsofrhythm,register,and/or imbre: hefirstas 3 13, the secondas 4 12, and

thethirdas5 11. Thefinalmelody s interruptedyrestsandhasnochangein instrumentation;husit formsa 6 10 partition.Comparison f set-class

membershipeveals hatno pairof melodiesbelongsto the sameset class.

248








A: CSEGWITH ONE REPETITION

A=<01232>01232

01 211- ++

31- - -21- - 0+

RelatedMatrices:

B = c5-2: <0 12 4 3 > C = c5-4: <0 13 4 2 >

01243 0134201 +++ 0 ++II- ++ +- +21 - - 31--41- - - 41 - - -31- 21

Therefore: A = rc5-2/4.

CSIM(A,B) = CSIM(A,C) = CSIM(B,C).

B: CSEGWITHTWO REPETITIONS

D=<123031>123031

+ + - +

21- + -+-31- -- -0 I++ + + Csegclass label= 6-?31- -0--110+ + -+

Relatedmatrices:

E=c6-145: <134052> F=c6-154: <235041>

134052 2350411I +-F+] 21 +-+[31- -+- 31- -+-41-- J - 51---\ -

01+++•+

01+++ +

21-++ -+ 1 +++-+

Therefore: D = rc6-145/154.

CSIM(D,E) = CSIM(D,F) = CSIM(E,F).

Figure 15. Csegclass Labels for Repeated-Note Csegs249








A B C D "codetta"

3 + 3 4 + 2 5 + 1 6 (+ 0) 1solo solo clarinet solo violin

KlangfarbenGlock. "antecedent"".consequent" solo harp Klangfarben

sot

re•chordal contrapuntal solo texture

Glock.

AHp.et al.

Contour A: mm. 1-2

A = < 0 10 4 3 2 > rc6-29/133

Contour B: mm. 3-6

B = <1 2 3 0 3 1 > rc6-145/154

Vl.+ Glock

A Vln.

Contour C: mm. 7-10 /

C= <5 0 2 3 1 4 > c6-104

' 3--"--

ContourD: mm. 10-11

D = < 0 5 3 2 4 1 > c6-104

Figure16. PrimaryMelodic Contours n Webern,op. 10/1

250








Infact,sincetwo arerepeated-notesegs, the cardinalities f thepitch-classsets differ;the firstis a pentachord,he'seconda tetrachord, nd the lasttwo, hexachords.Although he two hexachords o not belongto the sameset class (C = 6-Z44, D = 6-Z6), theyaremembersof the samec-space

segmentclass, c6-104. ContourD immediatelyollows C musically,andisits contour nversion.This, of course,is a muchmorepreciserelationshipthansimplyreversing he patternof ups and downs betweenadjacentc-

pitches (a reversalof signs in the INT1);thatis, changing < + - - + - >to < - + + - + >. In this case, such a reversal of signs is instead re-

flectedthroughouthe entireCOM-matrix.Further,he orderingof cps inc6-104producesa successivepattern f preserved djacenciesbetween he

inversionally-relatedontours:34

C =<5023 14 >

D= <05 32 41 >.

Therelationship etweensuccessivecontours s, forthe mostpart,oneof highdissimilarity:CSIM(A,B)andCSIM(B,C) qual .27 andCSIM(C,D) equals0. Ontheotherhand,connectionsbetween heopeningmelodiesandthe concludingone are muchstronger CSIM(A,D)= .53 andCSIM(B,D) = .60). Thusthethirdmelody,atthehighpoint f themovement,hasthecontourmost dissimilar romthose whichprecedeandfollowit, a con-tourwhichsetsit apart romthe others(CSIM(A,C)= .40, CSIM(B,C)=

.27 and CSIM(C,D) = 0).All four of theprimarymelodiesarerelatedby theircsubsegstructure.

Each hasc4-6 embeddedat least once as foursuccessivecps, oftenpromi-nentlypositioned.Yet n no case do thesesuccessivepitchesbelongto thesame set class, despitetheirmembershipn the samecsegclass.Forexam-

ple, contourA endswith < 0 4 3 2 > (or,bytranslation,< 0 3 2 1 >),and s immediatelyollowedbyitsretrogradenthefirstfourcpsof contour

B, < 1 2 3 0 >. This segmentationntofours is aurallysuggestedby theisolationof thesetetrachords y restson eitherside. LikecontourB, con-

toursC and D beginwithc4-6 as thefirstfourcps. ContourC beginswith< 5 0 2 3 >, whichis the inversionof the originalcsubsegas stated nA (by translation < 5 0 2 3 > becomes < 3 0 1 2 >, and by inversion,< 0 3 2 1 >). ContourD's initialtetrachords a return o < 0 3 2 1 >as initiallyappeared.Finally,csegclassc4-6 appears mbeddedas noncon-tiguouscsubsegs n contoursA, C, and D as well. It occursa totalof threetimes in A andfive times in D, andis in factthe only four-note subsegthese two contours hare(CMEMB4(X,A, D) = 8/30 = .27). ContourCalso contains ive embedded tatements f c4-6, but in the inverted orm.

Secondarymelodicmaterial of cardinalityour or greater)s shown n

Figure 17 as contoursE throughH. In contourF, c4-6 appearsagaininexactlythe same formas in contourC, the melodywhichit accompanies.Thus the contoursof the violin and cello lines (mm. 7-8) forma heter-

251








Fl.Vc.

Contour E: mm. 4-6

3i-- 3-• • 3- '

E = < 2 0 1 3 > c4-4

VA

Contour F: mm. 7-87 A .. - , I

F = <3 0 1 2 > c4-6

ContourG: mm.6-7 1.t;| t !

G = < 0 2 1 3 > c4-3

Fl.

Contour H: mm. 8-9

H = < 2 0 1 0 3 > c5-14/20

Figure17. SecondaryMelodicMaterial:Webern,op. 10/1

252








V i n .

( a , .

B o t h c o n t o u r s c 4 - 6 ,

Figure18. ContourHeterophony:Webern,op. 10/1,mm. 7-8

253








ophonictextureof overlappingtatements f c4-6 in close temporalprox-imity,as shown n Figure18.Contourheterophonyccursonlyatthishigh-pointof thepiece,where hecontrapuntalextures mostcomplex.Ineveryothercase, the csegclass of the accompanyingine is not an embedded

csubsegof themelody t accompanies;husthedistinctionbetweenmelodyandaccompaniments clear.

Finally,onlytwopossiblecsegclassesexist forc-segmentsof cardinalitythree.Therefore, ccasional nstancesof recurringhree-note subsegsmaybe of relatively rivialanalyticalmportance.The distinctive epeated-note

csubsegrc3-2/2, < 0 1 0 >, occurswithenough requencyhroughouthe

movement o warrantdiscussion,however.This "neighbornote"motive

opensthe movementwith its vivid Klangfarben coring.Its inverted ormis embedded epeatedlyn contourB whichfollows,as thecontiguous ps

< 3 0 3 > and as the noncontiguous csubsegs < 1 2 1 >, < 1 3 1 >(twice),and < 10 1 >. Further,t occursas the central hreeconsecutive

cps of contourH. Most striking,however, s its prolonged tatement ver

the courseof measures3 through10-first in the extended rill (whichin

itself containsrepeatednstancesof < 0 10 >) and thenin the continua-tionof this line in thetrumpet/harpf m. 9 andcelesta/celloof m. 10.This

extended< 0 10 > clearlyrefersbackto theopeninggesture,even with

respectto its instrumentation.

If musictheoristsmodelanalyticalheories o reflectauralperceptions,thena theorywhichdescribesrelationshipsmongmusicalcontourss cer-

tainlyoverdue.Thetheorydetailedabovedefinesequivalence ndsimilarityrelationsfor contours n contourspace. The analysisthatfollowsbrieflyillustrates owspecificcontour elationshipsmaybe used to shapea formal

scheme,to differentiatemelodyfromaccompaniment,o associatemusicalideasthatbelongto different etclasses,andto createunitythrough aried

repetition.

254








GLOSSARY

COM-matrix (comparison matrix) - a two-dimensionalarray hatdisplays he

resultsof the comparisonfunction, COM(a,b) for anytwo c-pitches in c-space.

If b is higherthana, the functionreturns"+"; if b is the same as a, the functionreturns"0";and if b is lower than a, COM(a,b) returns"-."

C-pitches (cps) - elements in c-space, numbered in order from low to high,

beginning with 0 up to (n - 1), where n equals the number of elements.

C-segment (cseg) - an orderedset of c-pitches in c-space.

C-space (contour space) - a type of musical space consisting of elements ar-

rangedfrom low to high disregarding he exact intervals between elements.

C-space segment class (csegclass) - an equivalenceclass made up of all csegsrelatedby identity,translation,retrograde, nversion,and retrograde-inversion.

C-subsegment (csubseg) - any orderedsubgroupingof a given cseg. May be

comprisedof either contiguous or non-contiguousc-pitches from the original

cseg.

INTn - any of the diagonalsto the rightof the main diagonal (upperleft-hand

to lowerright-hand orner)of the COM-matrix, n which n standsfor the differ-

ence between order position numbers of the two cps compared;that is, INT3

compares cps which are 3 positions apart.

Inversion - the inversionof a cseg S comprisedof n distinctcps is writtenIS,andmaybe foundby subtractingachc-pitch rom(n - 1).

Normal form - an orderedarrayin which elements in a cseg of n distinct c-

pitches are numberedfrom 0 to (n - 1) and listed in temporalorder.

Prime form - a representative orm for identificationof cseg classes, derived

by the following algorithm: 1)if necessary,translate he cseg so its contentcon-

sists of integersfrom 0 to (n - 1); (2) if (n - 1) minus the last c-pitch is lessthan the firstc-pitch invertthe cseg; (3) if the last c-pitch is less than the first

c-pitch, retrograde he cseg. AppendixOne lists the csegclasses and their cor-

responding abelsas used in this paper.The first numberof the label representsthe cardinalityof the csegclass andthe second numberrepresents ts ordinalpo-sition on the list: thusc5-12represents he twelfth contouron the list of five-note

csegclasses.

Translation - an operation hroughwhich a csubseg is renumbered rom 0 for

the lowest c-pitch to (n - 1) for the highest.

255








SIMILARITYMEASUREMENTS:

ACMEMB(A,B) - counts the total number of mutually-embedded segs of

cardinality2 throughthe cardinalityof the smaller cseg and adjuststhis to a

decimal value by dividingby the total numberof possible subsegs of A and B

(excludingthe null set for each and the one-note csubsegs).

CEMB(A,B) - countsthe numberof times cseg A is embedded n cseg B, then

divides this sum by the total numberof csubsegs of the same cardinalityas A

possible, to returna value that approaches 1 for csegs of greatersimilarity.

CMEMBn(X,A,B) - counts the numberof times the csegs, X (of cardinality

n), are mutuallyembedded n both csegs A andB. (The variable"X"may suc-

cessively representmore thanone cseg-type duringthe courseof the function.)

Eachcseg X mustbe embeddedat leastonce in bothA and

B; then,all instances

of X are counted n bothA andB. The totalnumberof mutually-embeddedsegsof cardinalityn is then dividedby the numberof n-cardinality subsegs possiblein orderto returna decimal numberapproaching1 as csegs A and B are more

similar.

CSIM(A,B) - measures he degreeof similaritybetweentwo csegs of the same

cardinalityby comparing specific positions in the upper right-hand riangleof

the COM-matrix or cseg A with the correspondingpositions in the matrixof

cseg B in order to total the numberof similaritiesbetween them. This sum is

divided by the totalnumberof positions comparedto returna decimalnumberthat signifies greatersimilaritybetween csegs as the value approaches1.

In addition, ACMEMB(A,B), CEMBA(A,B), CMEMBn(X,A,B), and

CSIM(A,B), generalizeeach of the functionsabove to measuresimilaritybe-

tween csegclasses.

256








APPENDIX

C-SPACE SEGMENT-CLASSES

OF CARDINALITIES 2 THROUGH 6

The followingtable of csegclasses, cardinalities2 through6, is a portionof

the outputof a computerprogramwrittenin March 1986. The program,written

in standardPascal language,was implementedon a Digital PRO-350using the

Xenix Pascal compiler and editor.

The csegclasses are listed in prime form, groupedby cardinality,and num-

bered in ascending order by prime form considered as an integer value. An

asterisk (*) following the csegclass name indicates identity under retrogradeinversion.For referentialpurposes, the INTi of a csegclass is listed at the right

of its csegclass representative.

C-space segment classes for cseg cardinality2

Csegclass/RIinv. Prime form INT(1)c 2-1* <01 > < + >


Csegclass/RIinv. Primeform

INT(1)c 3-1* < 0 12 > < + + >

c 3-2 < 02 1 > < + - >


Csegclass/RIinv. Prime form INT(1)c4-1* <0123 > < + + + >c 4-2 < 0 132 > < + + - >

c 4-3* < 02 13 > < + - + >

c4-4 <0231 > < + + - >c4-5 <0312 > < + - + >

c 4-6 < 032 1 > < + - - >

c 4-7* < 1032 > < - + - >c 4-8* < 1302 > < + - + >


Csegclass/RIinv. Prime form INT(1)

c5-1* < 01234 > < + + + + >c5-2 <01243 > < + + + - >

c 5-3 < 0 1 3 2 4 > < + + - + >

c5-4 <01342> < ++ + - >

c 5-5 < 0 1 4 2 3 > < + + - + >

257








c 5-6 < 0 14 3 2 > < + + >

c5-7 < 02143 > < + - + - >

c5-8 < 02314 > < + + - + >

c5-9 < 02341 > < + + + - >

c 5-10 < 024 13 > < + + - + >c 5-11 < 0243 1 > < + + >

c 5-12 < 03 142 > < + - + - >

c 5-13* < 032 14 > < + - + >

c 5-14 < 0324 1 > < + - + - >

c 5-15 < 034 12 > < + + - + >

c 5-16 < 0342 1 > < + + >

c5-17 < 04 123 > < + - + + >

c 5-18 < 04 132 > < + - + - >

c5-19 < 04213 > < + -+ >c5-20 < 04231 > < + - + - >

c 5-21 < 043 12 > < + -+ >

c 5-22 < 0432 1 > < + - - - >c 5-23* < 10243 > < - + + - >

c 5-24 < 10342 > < - + + - >

c 5-25 < 1 0423 > < - + - + >

c 5-26 < 10432 > < - + >

c 5-27 < 1 2403 > < + + - + >

c 5-28 < 1 3042 > < + - + - >

c 5-29 < 1 3402 > < + + - + >

c5-30 < 14032 > < + - + - >

c 5-31 < 14203 > < + - + >

c 5-32 < 14302 < < + -+ >


Csegclass/RIinv. Prime form INT(1)c6-1* <012345 > < + + + + + >

c6-2 <012354 > < + + + + - >

c6-3 <012435 > < + + + - + >

c 6-4 < 012453 > < + + + + - >

c6-5 < 012534 > < + + + - + >

c6-6 <012543 > < + + + >

c6-7* < 013245 > < ++ - ++ >

c6-8 < 0 13254 > < + + - + - >

c6-9 < 013425 > < + + + - + >

c6-10 < 013452 > < + + + + -

c 6-11 < 0 1 3524 > < + + + - + >

c6-12 < 013542 > < + + +-- >

c6-13 < 014235 > < + + - + + >c6-14 < 014253 > < + + - + -

258








c6-15 <014325 > < + + - + >

c 6-16 < 0 1 4352 > < + + - + - >

c6-17 <014523 > < + + + - + >

c6-18 <014532 > < + + + - >

c6-19 <015234 > < + + - + + >c6-20 <015243 > < + + - + - >c 6-21 < 0 1 5324 > < + + - + >

c 6-22 < 0 1 5342 > < + + - + - >

c6-23 < 015423 > < + + + >

c6-24 <015432 > < + + - - - >

c 6-25 < 02 1354 > < + - + + -c6-26* < 02 1435 > < + - + - + >c6-27 <021453 > < + - + + - >

c 6-28 < 02 1 534 > < +- + - + >c 6-29 < 02 1 543 > < + - + >

c6-30 < 023 154 > < + + - + - >c6-31 < 0234 15 > < + + + - + >c6-32 < 02345 1 > < + + + + - >c6-33 < 0235 14 > < + + + - + >c 6-34 < 02-354 1 > < + + + - >

c 6-35* < 024 135 > < + + - + + >

c6-36 < 024153 > < + + - +->

c6-37 < 0243 15 > < + + - + >c6-38 < 02435 1 > < + + - + - >c 6-39 < 0245 13 > < + + + - + >

c6-40 < 02453 1 > < + + + - >c6-41 < 025 134 > < + + - + + >

c6-42 < 025 143> < + + - + - >

c6-43 <025314> < + + -- + >c6-44 < 02534 1 > < + + --+ -c6-45 < 0254 13 > < + + - - + >

c6-46 < 02543 1 > < + + - --

c6-47 < 03 1254 > < + - + + - >c6-48* < 03 1425 > < + - + - + >

c6-49 <031452> < + - + + - >c6-50 <031524 > < + -+ -+ >c6-51 < 03 1542 > < + - + >c 6-52 < 032 154 > < + - - + - >c6-53 <032415> < +- + - + >c 6-54 < 03245 1 > < + - + + - >c 6-55 < 0325 14 > < + - + - + >c6-56 < 032541 > < + + - >c 6-57* < 034 125 > < + + - + + >c6-58 < 034 152 > < + + - + - >

259








c 6-59 < 0 3 4 2 1 5 > < + + + >

c6-60 < 03425 1 > < + + - + - >

c6-61 < 0345 12 > < + + + - + >

c 6-62 < 03452 1 > < + + + >

c6-63 < 035 124 > < + +- + +c 6-64 < 035 142 > < + + - +- >

c 6-65 < 0352 14 > < + + + >

c 6-66 < 03524 1 > < + + - + - >

c6-67 < 0354 12 > < + + + >

c6-68 < 03542 1 > < + + - - - >

c6-69 <041253 > < + - + + - >

c6-70 < 04 1352 > < + - + + - >c6-71 <041523> < + - +- + >

c 6-72 < 04 1532 > < + - + >

c6-73 < 042153 > < + -+ ->

c6-74* < 0423 15 > < + - + - + >

c6-75 < 04235 1 > < + - + + - >

c6-76 < 0425 13 > < + - + - + >

c6-77 < 04253 1 > < + - + >c6-78 < 043 152 > < + - -+ - >

c 6-79* < 0432 15 > < + - - - + >

c 6-80 < 04325 1 > < + - - + - >

c6-81 < 0435 12 > < + - + - + >

c6-82 < 04352 1 > < + -+ >

c6-83 < 045 123 > < + + - + + >

c6-84 < 045132 > < + + - + - >

c 6-85 < 0452 13 > < + + -- + >

c 6-86 < 04523 1 > < + + - + - >

c6-87 < 0453 12 > < + + - - + >

c6-88 < 04532 1 > < + + - - - >

c6-89 < 05 1234 > < + - + + + >

c6-90 < 05 1243 > < + - + + - >

c6-91 < 05 1 324 > < + - + - +

c6-92 < 05 1 342 > < + - + + - >

c6-93 < 05 1423 > < + - + - +>

c6-94 < 05 1432 > < + - + >

c6-95 < 052 134 > < + -+ + >

c6-96 < 052143 > < + -+ - >

c6-97 < 0523 14 > < + - + - + >

c6-98< 05234 1 > < + - +

+ ->

c 6-99 < 0524 13 > < + - + - + >

c 6-100 < 05243 1 > < + - + >

c 6-101 < 053 124 > < + - + + >

c 6-102 < 053 142 > < + -+ - >

260








c 6-103 < 053214 > < + - - - + >

c 6-104 < 05324 1 > < + + - >

c 6-105 < 0534 12 > < + - + - + >

c 6-106 < 05342 1 > < + - + - >

c 6-107 < 054 123 > < + + + >

c 6-108 < 054 132 > < + + - >

c 6-109 < 0542 13 > < + - -- + >

c 6-110 < 05423 1 > < + - - - >

c 6-111 < 0543 12 > < +- - - >

c 6-112 < 05432 1 > < + ->

c 6-113* < 102354 > < - + + + - >

c 6-114 < 1 02453 > < - + + + - >c 6-115 < 1 02534 > < - + + - + >

c 6-116 < 1 02543 > < - + + >

c 6-117* < 1 03254 > < - + - + - >

c 6-118 < 1 03452 > < - + + + - >

c 6-119 < 103524 > < - + + - + >

c 6-120 < 103542 > < - + + >c 6-121 < 104253 > < - + - + - >

c 6-122 < 1 04352 > < - + - + - >

c 6-123 < 104523 > < - + + - + >

c 6-124 < 104532 > <-

++

-

>c 6-125 < 105234 > < - + - + + >

c 6-126 < 105243 > < - + - + - >c 6-127 < 105324 > < - + + >

c 6-128 < 105342 > < - + - + - >

c6-129 < 105423 > < - + + >

c6-130 < 105432 > < - + - - - >

c 6-131 < 120453 > < + - + + - >c 6-132* < 120534 > < + - + - + >

c 6-133 < 1 20543 > < + - +-

>c 6-134 < 123504 > < + + + - + >c 6-135 < 1 24053 > < + + - + - >c 6-136 < 1 24503 > < + + + - + >

c 6-137* < 1 25034 > < + + - + + >

c 6-138 < 125043 > < + + - + - >c 6-139 < 125304 > < + +-- + >c 6-140 < 125403 > < + + - - + >c 6-141 < 130452 > < +- + + - >

c 6-142* < 130524 > < + - + - + >c 6-143 < 130542 > < + - + >c 6-144 < 132504 > < + - + - + >c 6-145 < 1 34052 > < + + - + - >c 6-146 < 134502 > < + + + - + >

261








c 6-147 < 1 35024 > < + + - + + >

c 6-148 < 135042 > < + + - + - >

c 6-149 < 135204 > < + +- - + >

c 6-150 < 1 35402 > < + + + >

c 6-151 < 140253 > < +- + + ->

c 6-152 < 140352 > < + - + + - >c6-153 < 140523 > < + - + - + >

c 6-154 < 140532 > < + - + >c6-155 < 142053 > < + - - + - >

c6-156 < 142503 > < + - + - + >

c 6-157 < 143052 > < + - - + - >

c6-158 < 143502 > < + - + - + >

c 6-159 < 145023 > < + + - + + >

c 6-160 < 145032 > < + + - + - >

c 6-161 < 145203 > < + + -+ >c 6-162 < 145302 > < + + -+ >

c 6-163 < 1 50243 > < + - + + - >

c 6-164 < 1 50342 > < + - + + - >

c 6-165 < 1 50423 > < + - + - + >

c 6-166 < 1 50432 > < + - + >

c 6-167 < 1 52043 > < + - - + - >

c 6-168* < 152304 > < + - + - + >

c 6-169 < 152403 > < + - - + >

c 6-170 < 153042 > < + -+ - >

c 6-171 < 1 53204 > < + - - - + >

c 6-172 < 153402 > < + - + - + >

c 6-173 < 1 54023 > < + - + + >

c 6-174 < 1 54032 > < + - + - >

c6-175 < 154203 > < + - - - + >

c 6-176 < 154302 > < + - + >

c 6-177* < 20 1453 > < - + + + - >

c 6-178 < 20 1543 > < - + + - - >

c6-179* < 204 153 > < - +- + -

c 6-180 < 2045 13 > < - + + - +

c 6-181 < 205 143 > < - + - + - >c 6-182 < 2054 13 > < - + + >

c 6-183* < 2 10543 > < -+ - - >

c 6-184 < 2 14503 > < - + + - + >c 6-185* < 2 15043 > < - + - + - >

c 6-186 < 2 15403 > < - + - - + >

c 6-187*< 2405 13 > < +- + - +

c 6-188 < 241503 > < +- + - +

c 6-189* < 245013 > < + + - + +

c6-190 < 245 103 > < + + -- +

c6-191 < 25 1403 > < +- + - +

c 6-192* < 254 103 > < + - -> >

262








NOTES

1. A bidimensional model for pitch, distinguishing pitch (or pitch height) from pitchclass (calledpitch qualityor chroma)has existed in the psychological iterature ince

the mid-nineteenthcentury. Christian Ruckmick ('A New Classificationof TonalQualities,"PsychologicalReview 36 [1929]: 172), for example,cites an M. W. Dro-

bisch article from 1846 ("Uberdie mathematischeBestimmungder musikalischen")

as the earliestattempt o depict pitch perceptionas a helical model. This model shows

the close perceptualproximityof octaves as distinct from rising pitch heightby the

verticalalignmentof octave-relatedpitches within each turnof the helix.

2. In recentyears, severalpsychologistshaveposited representationalmodels for pitch

perceptionon thebasis of experimentation, mongthemDianaDeutsch, Carol Krum-

hanslandRogerN. Shepard.Shepard'smulti-dimensionalmodel for pitchis a double

helix wrappedarounda helical cylinder,where ascent representspitch height with

octave-related hromaaligned verticallywhile a downwardprojectionof each pitch

producesa circle of fifthsmodel. Further,a verticalplanepassing through he double

helix model divides those tones which are diatonic to a given key from those which

are not. See Shepard's"StructuralRepresentationsof Musical Pitch," in Diana

Deutsch, ed., ThePsychologyof Music (NY: AcademicPress, 1982), pp. 343-390,for an overview of representationalmodels for pitch perception.Shepardnotes else-

where, however, that certain aspects of pitch perception differ markedly amonglistenersdependingupontheirmusicalbackgrounds. nexperimentsundertakenoint-

ly with Krumhanslin 1979, Sheparddiscovered that musical listeners perceivedoctave-relatedpitches as functionally equivalent,whereassubjectswith less musical

experiencedid not perceive such an equivalence.See his "IndividualDifferencesin

the Perceptionof Musical Pitch," in DocumentaryReportof the Ann ArborSympo-sium (Reston, VA: Music Educators National Conference, 1981),pp. 152-174, for

furtherdetails of this phenomenon.For purposesof this article, we will therefore

assume experiencedmusical listeners in discussions relatingto perceptualissues.

3. See Diana Deutsch, "The Processing of Pitch Combinations,"The Psychology ofMusic, pp. 277-289, for an overview of experimentson recognitionof melodies dis-

torted by octave displacementor by alteration of interval size. W. J. Dowling and

A. W. Hollombe's tudy,"ThePerceptionof Melodies DistortedBy SplittingInto Sev-

eral Octaves: EffectsofIncreasingProximity

and MelodicContour,"Perception

and

Psychophysics21 (1977):60-64, generalizesDeutsch'sfindingsas publishedin "Oc-

tave Generalization nd TuneRecognition,"Perceptionand Psychophysics11(1972):411-412,over a numberof familiarmelodies. See also W. L. Idson and D. W. Mas-

saro,"A BidimensionalModel of Pitch in the Recognitionof Melodies," Perceptionand Psychophysics24 (1978):551-565 and W. J. Dowling and D. S. Fujitani,"Con-

tour, Interval,and Pitch Recognitionin Memory for Melodies," TheJournalof the

AcousticalSociety of America 49 (1971):524-531.

4. W. J. Dowling, "Scaleand Contour: Two Componentsof a Theory of Memory for

Melodies,"PsychologicalReview85 (1978):341-354, and "MentalStructuresThrough

Which Music is Perceived,"DocumentaryReportof the AnnArborSymposium Res-ton, VA: Music Educator'sNationalConference, 1981),pp. 144-151.

5. W. J.DowlingandD. S. Fugitani n the firstof twoexperimentsdescribed n "Contour,

Interval,and PitchRecognitionin Memory for Melodies"(Journalof the Acoustical

Society ofAmerica49 [1971]:524-431) discovered hat istenerswere likely to confuse

263








the exact transpositionof a novel non-tonalmelody with a second non-tonalmelodyif the latterretained he same contour.Thus, theyconcludedthat listeners retainnon-

tonal melodies in memory solely in terms of contour.The authorsadmitted,however,that their subjects'confusion of same-contour melodies with transpositionsof the

original melody mayhaveresulted rom the severe constraintsplacedon the intervallic

constructionof the melodies used in thisexperiment.Onlyminorseconds, majorsec-

onds, and minor thirds were used (pp. 527-528). See also Dowling, "Mental Struc-

tures,"p. 146.

6. James C. Bartlett and W. Jay Dowling in "Recognitionof TransposedMelodies: A

Key-DistanceEffect in Developmental Perspective"(Journalof ExperimentalPsy-

chology:HumanPerceptionand Performance6 [1980]: 501) give a brief overview of

severalexperiments, concludingthat"in all of these taskswith unfamiliarmelodies,

subjectsseem to have little troublereproducingor recognizingthe melodic contour,

but they havea greatdeal of troublewith the exact-pitch ntervalsamongthe notes."

Judy Edworthy, n "Melodic Contourand MusicalStructure,"Musical StructureandCognition(London:AcademicPress, Inc., 1985), confirms these findings. Her ex-

perimentsinvolve transpositionof novel, tonal melodies to variouskeys. She con-

cludes that "intervalnformation s well-definedandprecise only when the listeneris

able to establisha key. . ... Contour information s immediately precise but decays

rapidlyas a melody progressesand its lengthincreases.However,accurateencodingof contourdoes notdependon the listener'sabilityto establisha key" (p. 186).In non-

tonalcontexts,subjectsshould thereforebe able to recognize relationshipsamongcon-

tours more quickly and easily than among pitch-class sets, since only the latter

requires subjectsto perceive intervallic nformation.

7. RobertMorris, in his CompositionwithPitch Classes: A Theory of CompositionalDesign (New Haven: Yale University Press, in press), develops five such spaces.DavidLewin'sGeneralizedMusical Intervalsand TransformationsNew Haven: Yale

UniversityPress, 1987)posits six temporaland six pitch- and/orpc-relatedmusical

spaces (pp. 16-25).

8. John Rahn, in Basic Atonal Theory(New York:Longman, 1980) clearly and con-

sistently distinguishes between pitch relationships and pitch-class relationships,

effectively separating heoreticalconceptswhichapplyonly to pitchspace fromthose

which operatein pitch-classspace.9. In additionto RobertMorris'sCompositionwith Pitch Classes, another mportant e-

source is Michael Friedmann's"A Methodologyfor the Discussion of Contour:Its

Applicationto Schoenberg'sMusic,"Journalof Music Theory29 (1985): 223-248.

Friedmann'swork raises important ssues regardingmusical structure,analysis, and

perception.His article posits a numberof theoretical constructsfor comparingand

relatingmusicalcontours,includingthe contouradjacencyseries and relatedvector,

the contour class with its associatedvector,and the contourintervalsuccession and

array.Althoughthese formulationsdiffer from ours in a numberof crucial aspects,his work has greatly influenced our thinking.

Discussionof musicalcontour s not without earlierprecedents,however,particu-

larlyin the writingsof musictheorist-composers, uchas ArnoldSchoenberg Funda-

mentalsof Musical Composition New York:St. Martin'sPress, 1967], pp. 113-115),Ernst Toch(TheShapingForcesin Music [New York:CriterionMusic Corp., 1948],

Chapter5), and RobertCoganand Pozzi Escot, whose Sonic Design: The NatureofSoundand Music(EnglewoodCliffs,NJ:Prentice-Hall,Inc., 1976)makes extensiveuse

264








of contourgraphs n musicalanalysis.See also Cogan'sNewImages of MusicalSound

(Cambridge:HarvardUniversityPress, 1984).10. Morris, Glossary,under the word "c-space."11. Morris, Definition 1.1.

12. Friedmanndefinescontourintervals(CIs) as "thedistancebetween one element in aCC (ContourClass) and a later element as signified by the signs + or - and anumber.For example, in CC < 0-1-3-2 >, the CI of 0 to 3 is +3, and the CI of 3to 2 is -1" (p. 246). He readilyacknowledgesthat the contourinterval s "infinitelyexpandableor contractable n pitch space," and that "a largerCI contains a greatnumberof interveningpitches in the registralorder of the musical unit ... [and] is

by no meansnecessarilya largerinterval n pitch space" (p. 230). Althoughwe findsuch a conceptinteresting, t seems counterintuitive romtheperspectiveof a listener's

perceptions,since a contour ntervalof +3 maybe considerablysmaller in pitchspacethan a CI of +1. For

example,the

cseg< 0 1 3 2 4 >

maybe realized as

follows:4

21

CI+32

CI+3cI+1

CI+1

0

In this case, CI + 3 (measuredfrom contourpitches 1 to 4) is only a major third,while CI + 1 is a minor tenth. Othermusical realizationsof this cseg may produceeven largerdifferences n CI size. Further,Friedmannuses the contourinterval,con-tour intervalarray,and associated vectors as an equivalencecriterion(pp. 231 and

234), and to comparesimilaritiesamongcontours in his analyses (pp. 240

if).

Since

we choose not to define intervalsin c-space, our equivalencecriteriaand similarityrelationsdiffermarkedlyfrom Friedmann'sn concept.

13. We use a slightly differentdefinitionthanMorris, since we refer to all contoursas

c-segments, not as c-sets.

14. Note that our definitionsdo not account as yet for repeatedtones within a musicalcontour.This is a separate ssue which will be addressedat a laterpoint in the article.

15. Morris, Definition 1.2.16. Friedmann,pp. 226-227.

17. The term INT is used to be consistent with Morris's erminologyfor matricesin p-and

pc-space,where the

integersappearingn each

diagonalgive informationabouta set's intervallicstructure, ncludingpropertiesof invariance.Thus the term INT isretainedhere, even though we do not define intervals in c-space.

18. WerephraseMorris'sDefinition1.4 slightlyto conform with ourterminology:the in-version of a cseg P, of cardinalityn, is the cseg IP. Each IPmequals (n - 1) -

Pmwhere the subscriptm denotes orderpositions with the cseg P.

265








19. Morris, Chapter2.

20. More formally:

Let [cp(1). .. cp(n)] be a cseg with cps numbered n time from 1 to n.

Let "n"equal the cardinalityof the cseg;

Let "x"equal an ordinalposition within the cseg, rangingfrom 1 to n

(thus, "cp(x)" s a particularc-pitch, located "xth"from the left).

1) If necessary,translate he cseg to normalform,

2) If (n - 1) - cp(n) < cp(1), then invert the cseg,

3) If cp(n) < cp(1), then retrogradehe cseg.

21. The design of these functions is modelled, in part, uponthe similaritymeasuresfor

pitch-classsetspreviouslyformulatedbyDavidLewin, RobertMorrisandJohnRahn.

See, inparticular,Lewin's"Forte's ntervalVector,myIntervalFunction,andRegener'sCommon-NoteFunction,"Journalof Music Theory21 (1977): 194-237; Morris's"A

SimilarityIndexfor Pitch-ClassSets,"Perspectivesof NewMusic 18(1979/80):445-

460; and Rahn's"RelatingSets," in the same volume, pp. 438-498.

22. We arefollowingJohnRahn's xample n designingfunctions o returna decimalvalue

approaching"1"as similarityincreases. See his "RelatingSets."

23. As previouslymentioned,the entries in the lowerleft-handtriangleof the COM-ma-

trices used here simply mirror(with inversevalues) those in the upper right-hand

triangle.We thereforebase our similaritymeasurementupon comparedpositions in

the uppertrianglesalone.

24. Rahn, "RelatingSets," p. 490.

25. Thistotalnumberofcomparisons

betweenrighttriangles

ssigma(n);

which we define

as:

n-1

E

(S)

S=1

(in other words, the summationof an arithmetic series from 1 to (n - 1), where n

equals the cardinalityof the cseg).26. We choose this methodof comparingcsegs of unequalcardinalityover an expansion

and generalizationof the CSIM measurement or two reasons. First,the

embeddingrelation s easier to hear and therefore s intuitivelymoresatisfying.Second, any gen-eralizationof CSIM to csegs of unequalcardinalitywould, in effect, createanother

type of embeddingfunction, since it would involve comparingmatricesof unequal

size (thusembeddingone matrix within anotherand systematically hiftingthe posi-

tion of the embeddedsmallermatrix to makecomparisonswith each positionof the

largermatrix).27. Rahn, Basic Atonal Theory,p. 122.

28. Rahn,"RelatingSets," p. 492. RahngeneralizesDavid Lewin'sembeddingfunction

as formulated n Lewin, "Forte's ntervalVector,"pp. 194-237.

29. More formally:c

CMEMBn(X,A,B)

ACMEMB(A,B)= n = 2

2#A + 2#B - (#A + #B + 2)

266








where c = cardinalityof the largerof the 2 csegs,n = cardinalityof x,x = mutuallyembeddedcseg, and

# stands for "cardinality f."The numerator f this fractionloops through he CMEMBn(X,A,B)functionsucces-

sively for cardinalities2 throughthe cardinalityof the largercseg. The denominator

divides this figure by the totalnumberof csegs possible (2#A + 2#B) minus the one-note csubsegs (#A + #B) and minus the null set for each (2).

30. Friedmann,pp. 234-236.

31. The introductionof repeatednotes into contourtheory,as formulated o this point,strikes at the heartof the distinctionbetweenpitch space and contourspace. Because

our definitionof c-space, following Morris,disregards he exact intervalsbetweenc-

pitchesandchooses to leave this distanceundefined, he perceptionof a repeatednote

must be seen as a pitch-space rather than a c-space phenomenon. In considering

analyticalapplicationsof contourtheory,we must thereforedepartslightly fromour

previousc-spacedefinition n orderto accommodate hose segmentsin whichpitchesare repeated.

32. In symmetrically-structuredsegs of odd cardinality (i.e., < c b r x r b c > or< 1 3 2 0 2 3 1 >), the compositelabel will reflect the cseg's symmetry.Forexam-

ple, the COM-matrix or the repeated-note seg < 1 0 2 0 1 > is shown below with

the two matrices which determine its composite label:

1 0 2 0 1 2 0 4 1 3 3 1 4 0 2

1 0 -+ -0 2 0- + - 3 00+0 o0+ 0 +0+8+ 1 +02 - -

.

4

--4•

4-- -+0+0 1 +-+ + 0 +++0

1 0 -+-0 3 --+- 2 +- + - 0

rc5-28/28 c5-28 also c5-28

In cases such as these, the relatedcsegs thatdetermine he compositelabel belong tothe same c-space segmentclass. The compositelabel reflects his dualrelationshipbylisting the csegclass'sordinal numbertwice.

33. The maximumpossiblevalueforCSIM(A,B)betweencseg A with repeatednotesandcseg B without, is equal to sigma(n) - r, where r is the total numberof cp repeti-

sigma (n)tions. Such a comparisoncannotthereforereturn a value of "1."

34. Such a patternwill alwaysresultbetweeninversionally-relatedsegs in whichadjacentcps add to an odd indexnumber, n this case, 5. Otherpatternsof invariancebetween

inversionally-related ontours may be predictedusing theTnI

cycles. See Daniel

Starr,"Sets,Invariance,andPartitions," ournalof Music Theory22 (1978):1-42, fora detailedexaminationof this subject.

relating musical contours - extensions of a theory for contour

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