Callender ContinuousHarmonicSpaces

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Continuous Hmonic Spces

Clifton Callender

Abstract This article generalizes Ian Quinns recent harmonic characterization of pitch-class sets in equal tem-

pered spaces to chords drawn from continuous pitch and pitch-class spaces. Using the Fourier transform, chords

of any real-valued pitches or pitch classes are represented by their spectraand located in a harmonic space of all

possible chord spectra. Euclidean and angular distance metrics defined on chord spectra correlate strongly with

common interval-based similarity measures such as IcVSIM and ANGLE. Thus, we can approximate these com-

mon measures of harmonic similarity in continuous environments, applying the corresponding harmonic intuitions

to all possible chords of pitches and pitch classes in all possible tuning systems. This Fourier-based approach

to harmony is used to compare the properties of twelve-note chords in Witold Lutosawski and Elliot Carter, to

analyze the opening section of Grard Griseys Partiels, and to investigate the structural properties underlying the

Z-relation (part of ongoing research with Rachel Hall).

1. Introduction

Suppose we wish to compe sonoities in the just intontion systems of

Hy Ptch nd Lou Hison, the vious equl tempeed systems of Es-

ley Blckwood, the spectl music of Tistn Muil nd Gd Gisey, o

the moe feely composed nontempeed wok of Gygy Ligeti nd othes.

In othe wods, we wish to compe chods dwn fom continuous pitch o

pitch-clss spce the thn some fom of discete tuning. Tditionl simil-

ity mesues bsed on intevl vectos o subset embedding e of little help

fo the simple eson tht continuous spces contin n infinite numbe of

intevls. Fo exmple, C-mjo tids dwn fom twelve-tone equl tem-pement, {0, 4, 7}, nd just intontion, {0, 12log254, 12log232} < {0, 3.86, 7.02},

would be judged by tditionl mesues s mximlly dissimil, becuse they

contin no shed intevls.1 Ntully, we would like mens of comping

these sonoities tht coesponds to ou peception of these chods s being

quite simil nd, in sense, eqully deseving of the lbel C-mjo tid.

277

Journal of Music Theory 51:2, Fll 2007

DOI 10.1215/00222909-2009-004 2009 by Yle Univesity

1 The most natural unit of distance in continuous pitch-

class space is the octave, so the interval of an octave is 1,

a perfect fifth is 7/12, and so forth. However, given readers

familiarity with intervals measured as multiples of a semi-tone, throughout this article the unit of distance in pitch

and pitch-class spaces will be the semitone. Thus, the pitch

class 12log254is approximately 3.86 semitones higher than

C or 14 cents lower than E, the pitch class 12log232

is

approximately 2 cents higher than G, and so forth.


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278 J O U r N a L o f M U S I C T H E O r Y

One wy to compe the two mjo tids is to mesue the smoothness

of the voice leding between them, ccoding to some voice-leding metic.2

The smoothest voice leding consists of one voice (the thid) moving 14/100

of semitone fom 4 to 3.86 nd nothe (the fifth) moving by only 2/100 of

semitone fom 7 to 7.02, while the oot is held s common tone. By ny

esonble metic, this voice leding is extemely smooth, nd the two tids

should be vey close within ny voice-leding spce.

Q R S

Obviously, such minuscule petubtions of chod will hve only lim-

ited effect on its hmonic content. Tht is, chods tht e vey close in

voice-leding spce will tend to be close in hmonic spce, s well. (Indeed,

in the exteme cse of infinitesiml voice leding between two chods, the

distnce between these two chods in eithe voice-leding o hmonic spce

must lso be infinitesiml.) Howeve, the convese is not necessily tue, sdemonstted by the thee chods shown in Figue 1. although Qnd Re

not elted by tnsposition o invesion, the two chods sound quite simil.

This simility is esy to explin: both chods e stcks of pefect fifths with

one of the fifths divided into mjo nd mino thid. Howeve, the voice

leding between Qnd R, indicted by lines in Figue 1, is not t ll smooth.

Even llowing fo splitting nd fusing (in the sense of Stus 2003 nd Lewin

1998) {D4}$

{B3, D4} nd {F4, a4}$

{a4}, the voice leding equies totl

of six semitones of motion. Thus, while Qnd Rshould be close in hmonic

spce unde ny esonble metic of hmonic distnce, these two chods e

not pticully close in voice-leding spce. Compe the eltion between

chods Rnd Sin which single voice moves by semitone. Despite the smooth

voice leding, the two chods sound quite diffeentcetinly moe diffeent

thn Qnd R.3 In one cse, eltively lge voice leding yields hmoniclly

Figure 1. Qand Rare closer in

harmonic space than Rand S;

Rand Sare closer in voice-leading

space than Qand R

2 Roeder (1987), Lewin (1997), Cohn (1998), Straus (2003,

2005), Callender (2004), Tymoczko (2006), Hall and Tymoc-

zko (2007), and Callender, Quinn, and Tymoczko (2008) all

propose comparing sets on the basis of their distance from

one another in a voice-leading space.

3 Sdoes not contain an extended stack of fifths, is not a

subset of a diatonic collection, substitutes an embedded

augmented triad, {G, B, E}, for a major triad, and contains

two dissonant intervals not in Rinterval classes 1 and 6. In

contrast, Qpreserves the stack of perfect fifths, substitutes

one major triad for another, is a subset of a diatonic collec-

tion, and does not possess a tritone. As pitch-class sets, Q

and Rare considered to be significantly more similar thaneither Rand Sor Qand Sby standard similarity measures,

including IcVSIM (Isaacson 1990), ISIM2 (Isaacson 1996),


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279Clifton Callender Continuous Hmonic Spces

simil chods, nd in the othe smll voice leding yields eltively dissimil

hmonic objects. Thus, thee is no simple coeltion between distnces in

voice-leding nd hmonic spces.4 Though elted, these two types of spces

e distinct.

In the eminde of this ticle, I focus exclusively on ppoching con-

tinuous hmonic spces vi the Fouie tnsfom mthemticl tech-

nique fo discoveing peiodicities in wide nge of phenomen.5 Lewin

(1959) ws the fist to note the connection between the Fouie tnsfom

nd chods hmonic content, connection to which he etuned vey

biefly in 1987 (1034) nd moe fully in 2001. Since Dvid Lewins fist ti-

cle, othe theoists hve incopoted this mthemticl technique into thei

wok, including Vuz (1993), Quinn (2006 nd 2007), nd amiot (2007). The

following extension of this wok into continuous domin dws substntilly

upon In Quinns wok, which links Lewins wok to viety of ppochesto chod qulity, yielding well-developed theoy of chod pototypes nd

fuzzy hmonic ctegoies. Using the metpho of Fouie blnces, Quinn

employs these mthemticl tools in n intuitive mnne, such tht even those

with little mthemticl bckgound cn follow this Fouie-bsed ppoch

to music theoy. (Indeed, it is quite possible fo one to fully digest his wok

without being we of the sophisticted mthemticl concepts lying just

beneth the sufce.) While Lewin nd Quinn e ceful to keep the undely-

ing mechnics of the Fouie tnsfom in the bckgound, it will be necessy

to confont some of the detils hee, given the inheently moe complicted

ntue of continuous vesus discete spces. Howeve, we will do so only to

the extent bsolutely necessy, nd concete, clifying exmples ppe fe-

quently thoughout.6

2. The Fourier transform

2.1 Interval cycles

Undelying Quinns wok with the Fouie tnsfom is pticul mens

of comping chods with vious intevl cycles, which povides kind of

hmonic bluepint by which we my chcteize chod nd compe it toothes. Fo exmple, of the thee chods discussed in the intoduction, Qnd

ANGLE (Scott and Isaacson 1998), SIM (Morris 1979), and

SATSIM2 (Buchler 2000). As pitch sets, Qand Rare also

considered to be more similar by PM (Morris 1995) and

PSATSIM (Buchler 1997), two of the few measures of pitch-

set similarity.

4 Tymoczko 2008 explores the relation between voice-

leading spaces and harmonic spaces based on the Fourier

transform. See note 17 for further discussion.

5 Most musicians come into contact with the Fourier trans-

form in the context of analyzing or modifying audio. Complex

sound waves, represented as time-varying changes of inten-

sity, can be transformed into frequency spectra, which can

then be analyzed and/or modified and converted back into

sound. A good reference for audio applications of the Fourier

transform is Smith 2008.

6 My thanks to Michael Buchler, Rachel Hall, Evan Jones,

Ian Quinn, two anonymous readers for JMT, and especially

Dmitri Tymoczko, whose insightful and insistent questions

motivated me to think more deeply about numerous sec-tions, especially 6.2, and the more general matter of the

relationship between voice-leading and Fourier spaces.


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Rech contin fou membes belonging to the cycle of pefect fifths { . . . , G3,

D4, a4, E5, . . . }, while Scontins only thee membes belonging to this cycle.

Using set intesection s cude mesue of simility, we could clim thtQ

nd Re moe simil to this pefect fifths cycle thn is S. But wht bout the

setU5 {G3, F4, F4, a4, E5}? Though Ulso hs thee membes belonging

to the intevl cycle in question, is it s simil to this cycle s S, even though

the ltte just misses hving fou membes in the cycle by single semitone

(substituting E4 fo D4)? One of Quinns centl insights is tht the Fouie

tnsfom povides mens of fuzzifiction of membeship in (o simility

to) n intevl cycle. In the eminde of this section, we get feel fo how

this woks nd the musicl elevnce of such n ppoch, befoe delving into

gete specifics in 3.

By n l-cycle we will men ny set of the fom { . . . , x2l, x, x1l, . . . }.

The l-cycle tht contins 0 will be designted zl5 { . . . , 2l, 0, l, . . . ,}. Conside set consisting of single pitch, p, nd the intevl cycle z15 { . . . , 21, 0,

1, . . . }. How well doespfit o coelte with this semitone cycle? We might

intepet the question s sking how in tunepis with the twelve-tone equl

tempeed (12-tet) scle contining pitch 0. The closepis to ny intege, the

moe in tune it is, o the bette it fits, with the 12-tet scle. In the cse thtp

is n intege, then pcoeltes mximlly with z1. The fthe p is fom ny

intege, the less in tune it is, o the less well it fits, with the 12-tet scle. In the

cse thtpis exctly hlfwy between two consecutive integes (1.5, 2.5, 3.5,

etc.), thenpcoeltes minimlly with z1.

2 1 1 2

(p, cos(2p))

y = cos(2x)

Figure 2. The correlation of pwith z15 { . . . , 1, 0, 1, . . . } is cos(2p)

The sitution is epesented in Figue 2 by cosine wve with peiod of

1 epesenting the coeltion ofpwith z1. The peks of the cosine wve co-

espond to the membes of the intevl cycle (with coeltion of 1), nd the

toughs coespond to vlues tht lie exctly hlfwy between the membes of

the cycle (with coeltion of21). Fo ny vlue ofp, the coeltion ofpwith

z1 is given by cos(2p).7 It is impotnt not to confuse this cosine wve with the

intevl cycle itself. The intevl cycle is discete set, while the cosine wve

7 In principle, we could measure this correlation with a

number of different periodic functions. For example, wecould use the triangle wave, which connects the appropri-

ate maxima and minima in a linear fashion. However, as we

shall see, using the Fourier transform yields deep connec-

tions with existing methods of measuring similarity basedon interval content.


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is continuous function of the coeltion of pitches with n intevl cycle of

the sme peiod nd phse.

We cn genelize the sitution bit by consideing how well the pitch

pcoeltes with the l-cycle contining 0, zl. The sitution is epesented in

Figue 3 by cosine with peiod oflepesenting the coeltion ofpwith

zl. Fo ny vlue ofp, the coeltion ofpwith zl is given by cos(2p/l). Theclosepis to ny multiple ofl, the bette it fits o coeltes with zl; convesely,

the fthepis fom ny multiple ofl(o the close it is to 2l/2, l/2, 3l/2, etc.),

the wose it fits o coeltes with zl.

Genelizing the sitution futhe, we cn conside how well set of

multiple pitches fits with given intevl cycle by simply dding up the co-

eltions of the individul membes with this intevl cycle. Fo exmple, con-

side the setS5 {21,p, 2} nd the intevl cycle of semitones z1. (agin, we

cn intepet the fit ofSwith z1 s n indiction of how in tune Sis with the

12-tet scle.) The pitches 21 nd 2 ech hve coeltion of 1 with z1, nd,

s we sw bove, the coeltion ofpwith z1 is cos(2p). So, the totl fit ocoeltion ofSwith z1 is 2 1 cos(2p). The closepis to n intege, the close

this fit fo setSis to 3; the fthepis fom n intege, the close this fit is to

1. (See Figue 4.)

Figure 3. The correlat ion of pwith zl5 { . . . , l, 0, l, . . . } is cos(2p/l)

Figure 4. The correlation ofS= {1, p, 2} with z1 is 1 + 1 + cos(2p)

2l l l 2l

(p, cos(2p/l))

y = cos(2x/l)

2 1 1 2

(p, cos(2p))

(1, 1) (2, 1)y = cos(2x)

We cn lso compe set not only with given intevl cycle, but lso

with ll possible tnspositions of this cycle. Fo exmple, conside the set

P5 {21.5, 2.5, 1.5}. The coeltion between Pnd z1 is s low s possible,

23, since evey pitch is s out of tune s possible. Howeve, it is cle tht

Pis mximlly in tune with the 12-tet scle T.5(z1) 5 { . . . , 2.5, .5, 1.5, . . . ),

the tnsposition ofz1 up (o down) by qute tone. The sitution is epe-

sented in Figue 5, whee the cosine wve hs been shifted to the ight by .5.


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Thus, the mximum coeltion between Pnd some tnsposition ofz1 is 3;

we will sy thtPhs magnitudeof 3 with espect to 1-cycle. Moe genelly,

the mgnitude of ny set with espect to n l-cycle is the mximum coeltion

between the set nd Tx(zl) s xvies continuously. Intuitively, we simply slide

the cosine wve to the ight until the sum of the coeltions of individul

pitches with the coesponding intevl cycle is mximized. (See 2.3 below.)

2.2 Chord spectra

Figure 5. The correlation of P= {1.5, .5, 1.5} with Th(z1) is maximized when h= .5 (mod 1);

the correlation for individual members of Pis given by cos(2(p+ h))

Figure 6. The magnitude of P = {0, 5, 15} with respect to a 5-cycle is 1 + 1 + 1 = 3

2 1 1 2

(1.5, 1) (.5, 1) (1.5, 1)y = cos(2(x + .5))

(0, 1) (5, 1) (15, 1)

0 5 15

Now conside the pitch setP5 {0, 5, 15} o {C4, F4, E5}, which obviously

belongs to the 5-cycle z5.8 The sitution is depicted gphiclly in Figue 6,

whee the coeltion of individul membes ofPis epesented by cosine

wve with peiod of 5. Since the vlues ofpcoesponding to the membes of

Pcoincide with the cests of the cosine wve, by the eckoning of the peced-

ing section, Phs mgnitude of 3 with espect to 5-cycle. In compison, P

coeltes less well with 4- nd 6-cycles, shown in Figue 7. In the fist cse, thecoeltions of individul pitches ofPwith z4 e 1, 0, nd 0, yielding totl

mgnitude of only 1. (No othe tnsposition ofz4 yields highe coeltion

with P.) In the second cse, the coeltions of individul pitches with the

6-cycle in Figue 7 e .5, 1, nd .5, which lso yield totl mgnitude of 1.

(In this cse, the sum of the coeltions is mximized when the cosine wve of

peiod 6 is shifted to the left by 1/6 of peiod. See 2.3 below fo detils.)

8 Throughout this article, pitch sets will be labeled P, Q, R,

and so forth. The corresponding pitch-class sets generally

will be labeled PO, QO, RO, and so forth whenever it is unclearif a set is of pitches or pitch classes. (The subscript O

indicates the set modulo octave equivalence.) Additionally,

the term setsas used in this article is synonymous with the

term multisets. Multisets are unordered sets in which mul-

tiple occurrences of an element are counted separately. Forexample, the multiset {a, a, b} is not the same as {a, b}.


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Since we e woking in continuous spce, thee is no need to limit ou

investigtion to cycles of n intege numbe of semitones. Using the intuitive

method outlined bove, we cn find the mgnitude ofPwith espect to cycles

of 5.1 semitones, 2 semitones, o x semitones fo ny vlue of x. Figue 8

gphs the mgnitude ofPwith espect to x-cycles s xvies continuously. The

mgnitudes ofPwith espect to x-cycles fo ll possible vlues ofxfom the

spectraofP. (Fo moe on chod spect, including the deivtion of gphssuch s Figue 8, see 3.2.)

Figure 7. The magnitude of P= {0, 5, 15} (a) with respect to a 4-cycle is 1 + 0 + 0 = 1;

(b) with respect to a 6-cycle, the magnitude is .5 + 1 .5 = 1

Figure 8. Magnitude of P= {0, 5, 15} with respect to x-cycles

(0, 1) (5, 0) (15, 0)

0 5 15

(a)

(0, .5) (5, 1)

(15,.5)

0 5

15

x-cycles552

53

magnitude

as expected, since Phs mximl mgnitude with espect to 5-cycle,

thee is pek tx5 5 whee the gph is equl to 3. (Since thee e thee

membes of the set nd the cest of the cosine wve is 1, no othe pek cn

be highe thn 3, nd this vlue is obtined only when the cests of the cosine

wve e pefectly ligned with vlues coesponding to the membes of the

set. This is the sense in which set hs mximl mgnitude with espect to n

intevl cycle.) Thee e, howeve, numbe of othe peks, locted tx5 52nd x5 53, tht e eqully high. Since z55 { . . . , 5, 0, 5, . . . } is subset of

the 52 - cycle { . . . , 5, 52 , 0,

52 , 5, . . . }, Pobviously hs mximl mgnitude

(a)

(b)


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with espect to 52 -cycle. Likewise, since z5 is subset of the 53 -cycle { . . . , 25,

2103 , 253 , 0,

53 ,

103 , 5, . . . }, Pwill lso hve mximl mgnitude with espect to

53-cycle. Moe genelly, if set hs mximl mgnitude with espect to n

x-cycle, it will lso hve mximl mgnitude with espect to n xk -cycle, whee

kis ny intege. Thus, continuing the gph of Figue 8 to smlle vlues ofx,

thee will be identicl peks tx5 54 , x5 55 5 1, nd so on.

(0, 1) (5, 1) (15, 1)

0 5 15

y = cos(2x/) = 1

Figure 9. Magnitude of P= {0, 5, 15} with respect to the `-cycle

Figure 10. Magnitude of P9 = {0, 6, 15} with respect to x-cycles

Thee is one othe cycle with espect to which any set hs mximl

mgnitudethe `-cycle. a cosine with n infinitely long peiod is simply

hoizontl line ty5 1, shown in Figue 9. It is cle tht the mgnitude of

set with espect to the `-cycle will be equl to the cdinlity of the set.

Thee e othe (lesse) peks in the gph ofPtx< 3, x< 323, ndnumeous othe vlues fo x. The pek ne x5 3 is esy to undestnd by is-

ing the middle pitch fom F4 to F4 nd noting tht the esulting chod,P95 {0,

6, 15}, belongs to mino-thid cycle. The closeness ofPnd P9 in voice-ledingspce ccounts fo the eltively stong mgnitude of P with espect to

3-cycle s well s the mgnitude ofP9 with espect to 5-cycle. (See Figue 10

fo the gph ofP9.)

x-cycles3

magnitude

The pek tx< 323 in Figue 8 is only slightly moe complicted. Fist,note tht tking evey thid pitch in 323 - o

113 -cycle yields cycle of mjo

sevenths: ( . . . , 0, 323, 713, 11, . . . ). Loweing the middle pitch ofPfom F4 to

E4 foms mjo seventh with E5 nd mjo thid with C4, which is nearly

equl to 323 semitones. rising C4 by 13 of semitone yields P05 {

13, 4, 15},

which belongs to the 323 -cycle { . . . , 13, 4, 7

23, . . . } nd is gphed in Figue 11.


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agin, the closeness ofPnd P0 in voice-leding spce ccounts fo the el-

tively stong peks ne x5 323 in the gph ofPnd x5 5 in the gph ofP0.

Ce must be tken when intepeting these gphs to distinguish

between intevls within chod nd the mgnitude of this chod with espect

to vious intevl cycles. Fo exmple, since thee is one mino seventh in Pnd the vlue of the gph in Figue 8 tx5 10 is 1, we might be tempted to

sset n exct coespondence between the two. Howeve, this is not the cse,

s shown by the gph of {0, 5, 10, 15} in Figue 12. The ddition of pitch 10 to

chod Pcetes second mino seventh, but the vlue of the coesponding

gph tx5 10 is now zeo. This is becuse the mino sevenths e septed

by pefect fouthn intevl tht bisects mino seventh. accodingly, if

we shift cosine wve with peiod of 10 so tht its cests fll on one of the

mino sevenths, its toughs will fll on the othe mino seventh, nd the two

will cncel one nothe. Some edes my find this lck of diect coespon-dence between intevls nd mgnitudes with espect to intevl cycles discon-

ceting, but it is pecisely this popety tht llows us to employ n ppoch

bsed on intevl content in continuous spces. as we shll see (pticully

in 6.3), thee is vey stong connection between intevl content nd the

Fouie tnsfom.

x-cycles323

magnitude

Figure 11. Magnitude of P0 = {1/3, 4, 15} with respect to x-cycles

x-cycles

magnitude

10

Figure 12. Magnitude of {0, 5, 10, 15} with respect to x-cycles

2.3 Transposition and phase

In 2.1 we mesued the coeltion of pitch with zl by cosine wve with

peiod ofl. Suppose we wish to similly mesue the coeltion of pitch

with the tnsposition of this intevl cycle byxsemitones, Tx(zl). In this cse,

the coeltion is given by the sme cosine wve shifted to the rightbyx. In othe


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wods, tnsposition of n intevl cycle coesponds to phase shiftof the cosine

wve indicting the coeltion of pitch with the intevl cycle. Fo exmple,

conside the coeltion of pitch with the intevl cycle T1(z5) 5 { . . . , 1,

6, 11, . . . }. In this cse, coeltion with the intevl cycle is mesued by

cosine wve with peiod of 5 shifted by 1/5 of cycle (the tnsposition fcto

divided by the geneting intevl of n intevl cycle), which yields phse of

53608

5 728 o 2/5 dins (Figue 13). (2 dins is equivlent to 3608. I will

use dins insted of degees fo the eminde of the ticle.) Tnsposing z5byxsemitones coesponds to shift of the elevnt cosine wve byx/5 cycles

o phse of 2x/5. Moe genelly, tnsposing zl byxcoesponds to shift

of the elevnt cosine wve byx/lcycles o phse of 2x/l(Figue 13b).

FT1(5) = 2/5

-4 1 6-9

(a)

FTx(l) = 2x/l

0x l x x + l(b)

Figure 13. Transposition corresponds to a phase shift: (a) T1(z5), (b) Tx(zl)

Dividing by the geneting intevl, l, of n intevl cycle s in the pe-

vious pgph is the sme s multiplying by 1/l, which is the intevl cycle

frequency. The fequency is the numbe of cycles pe semitone. Fo exmple,

semitone spns 1/5 of 5-cycle nd two complete cycles of qutetone

cycle; thus, these cycles hve fequencies of 1/5 nd 2, espectively. Usingz5 1/lfo the fequency, we cn ewite the phse of the cosine wve mesu-

ing coeltion with Tx(zl) s simply 2xz.

We cn combine the mgnitude nd phse of set with espect to given

fequency by vectos in two-dimensionl plne, whee the mgnitude coe-

sponds to the length of the vecto nd the phse coesponds to the ngle of

clockwise ottion fom due noth.9 Fo exmple, with espect to 5-cycle,

P5 {0, 5, 15} hs mgnitude of 3 nd phse of 0. We cn epesent the

(b)

(a)

9 The association in this article of zero phase with due north

and a positivechange of phase with clockwise rotation (orrightward shift) runs contrary to mathematical convention.

By convention, zero phase is associated with the positive

x-axis and a positive change of phase with counterclockwise

rotation (or leftward shift). The contrary association adoptedin this article is in agreement with the clock-face diagrams

in Quinn 2007.


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sitution by vecto with length of thee pointing due noth (see Figue

14). The vecto coesponding to T1(P) with espect to 5-cycle is shown in

Figue 14b, whee the vecto in Figue 14 hs been otted clockwise by 1/5.

recll fom Figue 7b tht with espect to 6-cycle Phs mgnitude of 1. also

note tht the cosine wve tht gives the highest coeltion is shifted to the left

by one semitone o one-sixth of cycle, yielding phse of22/6 52/3.

The coesponding vecto, with length of 1 nd otted counteclockwise

by 1/6, is shown in Figue 14c.

Combining mgnitudes nd phses yields the Fourier transformof setP

with espect to fequencyz5 1/l, witten FP(z) o, in bbevited fom, FP.10

The mgnitude is indicted by|FP| nd the phse by/FP.11 Fo exmple, theFouie tnsfom of setPwith espect to 6-cycle is FP(1/6), whee the mg-

nitude is |FP(1/6)|5 1 nd the phse is /FP (1/6) 52/3.The Fouie tnsfom tkes set fom the domin of pitches o pitch

clsses into the domin of intevl fequencies whee the undelying peiod-

icities of set e explicitly epesented. It is in the intevl-fequency domintht the infinitely mny sets in continuous pitch o pitch-clss spce cn be

nlyzed, comped, nd tnsfomed on equl footing. Fist, we must con-

side how to clculte the Fouie tnsfom.12

(a) (b) (c)

25

3

(a) (b) (c)

FP

15

FT1(P)

15

FP

16

magnitude = 3 magnitude = 3 magnitude = 1

phase = 0 phase = 2/5 phase = 2/6 = /3

Figure 14. Representing magnitudes and phases of (a) P= {0, 5, 15} with respect to a 5-cycle,

(b) T1(P) with respect to a 5-cycle, and (c) Pwith respect to a 6-cycle

10 Strictly speaking, we take the Fourier transform of the

characteristic function of set P. (See 8.) Nonetheless, for

our purposes the slight abuse of mathematical language is

convenient and not particularly problematic.

11 The magnitude and phase can be combined into the sin-gle complex number, FP(z) 5reiq, where ris the magnitude,

q is the phase, and i521.

12 Quinn 2007 (part 3) demonstrates the relevance of the

Fourier series with respect to equal tempered pitch-class

spaces. Sections 3 and 4 of the present article summarize

Quinns work (especially in the comments on Figure 21),

tease out the underlying mathematics, and both extend it

to continuous pitch-class spaces and connect it with pitch

space and the continuous Fourier transform.


12/56


3. Pitch sets

3.1 Fourier transform of pitch sets

We begin with the Fouie tnsfom of the singleton pitch set {0}. Since

singleton is (tivilly) subset of ll intevl cycles (up to tnsposition), the

mgnitude of its tnsfom will be 1 fo ll vlues of z. Moeove, since the

intevl cycle to which {0} belongs will obviously contin pitch 0, the phse of

the tnsfom will be 0 fo ll vlues ofz. Figue 15 shows F{0}(z) fo ll vluesofz: unit vecto pointing due noth.

Next we conside the Fouie tnsfom of the bity singleton pitch

set {p}. While the mgnitude of the tnsfom will gin be 1 fo ll vlues

ofz, the phse will vy ccoding to the specific vlues ofpnd z. Sincepis

simplyTp(0), we know fom section 2.3 tht the phse will be 2pz. Fo exm-

ple, Figue 16 shows the Fouie tnsfoms of {0}, {5}, nd {15} fo z5 1/6

(o 6-cycle) with mgnitudes of 1 nd phses of 0, 2? 56 5 535 /3, nd

2? 1565 55, espectively.

Figure 15. The Fourier transform F{0}(z) for all values of z

Figure 16. The Fourier transforms of {0}, { 5}, and {15} with respect to a 6-cycle

3

F{0}

16

F{5}

16

F{15}

16

The Fouie tnsfom of pitch set contining multiple pitches is

found by dding togethe the tnsfoms of the individul pitches.13 adding

two vectos togethe is ccomplished by simply plcing the til of one vecto

on the hed of the othe. The esulting vecto tht extends fom the til of

the fist to the hed of the lst is the sum of the vectos. Since ow ddition

is commuttive, we cn dd the ows in ny ode. Fo exmple, to find the

Fouie tnsfom ofP5 {0, 5, 15} with espect to 6-cycle, we dd the vectos

13 This follows from the general principle that FP


13/56


fo {0}, {5}, nd {15} shown in Figue 16. The vecto esulting fom this sum-

mtion, shown in Figue 17, hs mgnitude of 1 nd phse of /3, con-

fiming the infoml esults shown in Figue 7b. In this pticul cse, thee

is mens to simplify the ddition of vectos. Note tht the vectos ssocited

with the tnsfoms of {0} nd {15} with espect to 6-cycle point in opposite

diections with the sme mgnitude. Thus, when these two vectos e dded,

they will cncel ech othe, yielding the zeo vecto ( point) shown in Figue

17b. Since dding ny vecto to the zeo vecto will yield the oiginl vecto,

the tnsfom of {0, 5, 15} with espect to 6-cycle will be the sme s tht fo

{5}, shown in Figue 17c. This cnceling of vectos is n impotnt phenom-

enon tht will become pticully useful in 7.

Figure 17. The Fourier transform of {0, 5, 15} with respect to a 6-cycle

In ode to ccutely mesue the mgnitude nd phse of the Fouie

tnsfom of pitch sets, it is helpful to fist convet the coesponding vectosto Ctesin coodintes. agin, we begin with singleton pitch sets. Given

set {p} nd fequency z, the vecto ssocited with the Fouie tnsfom

F{p}(z) extends fom the oigin to the point (x,y), whee x5 sin(2pz) nd

y5 cos(2pz). Fo exmple, lettingp5 5 nd z5 1/8 (n 8-cycle), the vecto

ssocited with the Fouie tnsfom F{5}(1/8) extends fom the oigin to

the pointx15 (22/2, 22/2), since sin(2? 58) 5 cos(2? 58) 522/2.

Likewise, we cn clculte tht the vectos ssocited with the Fouie tnsfoms

fo {0} nd {15} with espect to n 8-cycle extend fom the oigin to the points

x25

(0, 1) ndx3

5

(2

2/2,

2/2), espectively.as befoe, we cn find the vecto ssocited with the Fouie tnsfom

ofP5 {0, 5, 15} with espect to n 8-cycle by dding the vectos fo the indi-

vidul pitches. Hving conveted the mgnitude nd phse infomtion to

Ctesin coodintes, this is equivlent to dding the xndycoodintes of

ech of the thee points, x1, x2, nd x3, independently. Summing the xcoodi-

ntes, we hve 22/2 1 0 22/2 522; summing theycoodintes, wehve 22/2 1 1 12/2 5 1. Thus, the Fouie tnsfom ofPwith espectto n 8-cycle, FP(1/8), is epesented by the vecto extending fom the oigin

to the point (22, 1).

Moe genelly, fo n bity pitch setQnd fequencyz, the Fouietnsfom FQ(z) is epesented by the vecto extending fom the oigin to the

(a) (b) (c)


14/56


point (x,y), whee x5Sq[Q

sin(2qz) ndy5Sq[Q

cos(2qz). We will cll xthe sine

component of the tnsfom ndythe cosinecomponent, espectively, witten

FxQ nd FyQ.

We cn clculte the length of this vecto, nd thus the mgnitude of the

ssocited Fouie tnsfom, by using the Pythgoen theoem. The distnce

fom the oigin to the point (x,y) is given by x2 y2 . Fo exmple, eclling

tht the vecto fo the Fouie tnsfom ofPwith espect to n 8-cycle extends

fom the oigin to the point (22, 1), the length of the vecto is 2 2 1 5211 53. Thus, the mgnitude of the tnsfom is |F

P(1/8)|53. This

vlue does not necessily tell us much without compisons with othe tids,

which is why gphs such s Figue 8 e pticully helpful.

Moe genelly, the mgnitude of the Fouie tnsfom ofQwith fe-

quencyzis

FQ

(z) qQ

sin(2qz)

2

qQ

cos(2qz) .

2

(1)

We will efe to this function s the spectrumofQnd use the symbols Q, R,

S, nd so on, to efe to the spect ofQ, R , nd S. We will see lte tht the

squaredmgnitude of the Fouie tnsfom (|FQ|2 o Q2), often clled the

power spectrum, will be even moe useful fo ou puposes.

120 1

2

. . . . . .

(a) FxP(z)

120 1

2

. . . . . .

(b) FyP(z)

1

2

0 12

. . . . . .

(c) |FP(z)|

552

53

(d) |FP(l)|, l = 1/z

Figure 18. Derivation of the spectrum |F{0, 5, 15}| graphed in Figure 8

(a)

(c)

(d)

(b)


15/56


3.2 Graphing chord spectra

a couple of exmples should help to mke the detils of 3.1 cle. We begin

by e-ceting the spectum ofP5 {0, 5, 15}, which ws gphed in Figue 8,

ccoding to the following steps. (See Figue 18.)() Find the sine component of the Fouie tnsfom, FxP:

FxP(z) 5 sin(0) 1 sin(5 3 2z) 1 sin(15 3 2z)

(b) Find the cosine component of the Fouie tnsfom, FyP:

FyP(z) 5 cos(0) 1 cos(5 3 2z) 1 cos(15 3 2z)

(c) Find the mgnitude of the Fouie tnsfom:

F

P(z)

F

xP(z)

2

FyP(z)

2

(d) Since the gph in Figue 8 is plotted s function of intevl size,

l, which is the ecipocl of intevl fequencyz, the finl step is tosubstitute l5 1/zfo znd gph

FP(l) 1cos(10/l)cos(30/l)2sin(10/l)sin(30/l)2

Fo second exmple, we compe the spect of the thee pitch sets

fom the intoduction: Q5 {G3, D4, F4, a4, E4}, R5 {G3, B3, D4, a4, E4}, nd

S5 {G3, B3, E4, a4, E4}. The spect ofQnd Re supeimposed in Figue

19, nd those ofRnd Se supeimposed in Figue 19b. (In the gphs,

the hoizontl xis is l5 1/z, the peiod of ech sinusoid o, equivlently, the

size of the intevls in ech intevl cycle.) It is cle fom visul inspectionof the two spect in Figue 19 thtQnd Re quite simil hmoniclly.

Both contin stong nd vitully identicl peks ne l 5 7 nd ll of its

intege divisos, l5 7/2, l5 7/3, nd so foth, but no othe peks of signifi-

cnt mgnitude. In ddition, ll of the diffeences between the two spect

involve eltively wek components. By compison, the spect of Rnd S

diffe significntly. Thei peks do not lign except fo the somewht weke

pek ne x5 7, nd those potions tht e simil (e.g., the egion between

x5 4 nd x5 4.5) involve wek components. The intuitions bout hmonic

simility discussed in the intoduction e mde mnifest in these gphs ofchod spect. (We will see in 6 how to mke these intuitions moe concete

by mesuing the distnce between these spect.)

4. Pitch-class sets

4.1 Fourier transform of pitch-class sets

Fo the Fouie tnsfom of pitch sets, it is necessy to conside ll possible

intevl cycles. Howeve, fo pitch-clss sets, it is necessy only to conside

those intevl cycles tht divide the octve into n integl numbe of pts,

such s the octve itself, the titone, the ugmented tid, the diminished


16/56


seventh chod, the five-note set-clss {0, 2.4, 4.8, 7.2, 9.6}, nd so foth.14 Divid-

ing the octve into k[Z pts yields 12/k-cycle, which hs coespond-

ing fequency ofz5k/12. Thus, while the Fouie tnsfom of pitch set

is (genelly) continuous function of intevl fequency, the tnsfom

of pitch-clss set is discete seies (t ll fequencies k/12) clled the

Fouie seies.

Knowing the spectum of ny pitch set, P, we cn esily deive the

spectum of its coesponding pitch-clss set, notted PO, in mnne dem-

onstted by Figue 20. Figue 20 shows the spectum of the pitch set

Q5 {0, 5, 15}, with the vlues t fequencies of the fom k/12 mked by veti-cl lines. These veticl lines e the spectum of the pitch-clss setQO, shown

in Figue 20b. In othe wods, the Fouie tnsfom ofQOis sampledvesion

of the tnsfom ofQ, whee the smpling occus t ll fequencies of the

fom k/12. We will efe to these fequencies s the harmonicsof the octve,

with the fequencyk/12 being the kth hmonic. Fomlly, the sine nd cosine

Figure 19. Comparison of the spectra of the three pitch sets from Figure 1: (a) Qand R,

(b) Rand S

x-cycles772

73

magnitude

(a)

x-cycles772

73

magnitude

(b)

(a)

(b)

14 If lis rational, then any l-cycle in pitch space is equivalent

to some equal division of the octave in pitch-class space.

For example, consider an infinite cycle of perfect fifths, a

7-cycle. As is well known, an infinite cycle of perfect fifthsis equivalent to a 12-fold division of the octave, disregarding

order. More generally, any l-cycle in pitch space where lis a

rational number of the form a/bis equivalent to a 12bgcd(a,12) -fold

division of the octave in pitch-class space. Irrational interval

cycles in pitch space such as (. . . , 2 2, 0, 2, . . .) do notyield closed interval cycles in pitch-class space. (See 5.)


17/56


components of the Fouie tnsfom of ny pitch-clss set, PO, e

FxP(k/12)5Sp[P

sin(2pk/12), FyP(k/12) 5Sp[P

cos(2pk/12), k[ Z. (2)

Note tht it does not mtte which epesenttive of ech pitch clss we

use fo Eqution 2. Fo exmple, suppose pitch 15 in Qis eplced by 3 in

Q95 {0, 3, 5}. Even though the spect ofQnd Q9 e diffeent (Figue 20c),

the spect ofQond Q9oe the sme (Figue 20d). (This, of couse, hd

bette be the cse, since Qond Q9oe the sme pitch-clss set.)

4.2 Examples of pitch-class spectra

Fo esons tht will become cle in 5, it is only necessy to conside

finite numbe of hmonics fo the spect of equl tempeed pitch-clss sets.

Specificlly, fo pitch-clss set dwn fom n-tone equl tempement, it is

only necessy to conside hmonics 0 though , the integl pt ofn/2.

Fo exmple, ifnis 12 o 13, the entie infinite spectum of pitch-clss setcn be econstucted fom hmonics 0 though 6 (since 6 is the integl pt

of 13/2).

Figue 21 gphs the spect of the twelve clssicl pitch-clss tichodl

set-clsses (those without pitch-clss duplictions) in twelve-tone equl

tempement. The numbeed comments below coespond to ech of the

hmonic components of these twelve set-clsses:15

120 1

2

. . . . . .

120 1

2

. . . . . .

(a) |F{0,5,15}(z)| (b) |F{0,5,15}O(z)|

12

0 12

. . . . . .

12

0 12

. . . . . .

(c) |F{0,3,5}(z)| (d) |F{0,3,5}O(z)|

Figure 20. Derivation of the transform of {0, 3, 5}O from (a and b) {0, 5, 15} and (c and d) {0, 3, 5}

15 For more on the interpretation of the magnitudes for

various harmonics, see Quinn 2007. Analogous comments

apply to the spectra of set-classes with any number of pitchclasses.

(a)

(c)

(b)

(d)


18/56


(0) as noted befoe, hmonic 0 coesponds to the cdinlity of

given pitch o pitch-clss set. Thus, ech of the spect gphed in

Figue 21 hs mgnitude of 3 fo the zeoth hmonic.

(1) Hmonic 1 is the mgnitude of pitch-clss set with espect to n

octve o unison. The mgnitude of this hmonic is thus me-

sue of how unevenly set divides the octve. Highly uneven sets

such s {0, 1, 2}O hve high mgnitude fo this hmonic, while

sets tht divide the octve into equl divisions such s {0, 4, 8}Ohve

zeo mgnitude.

(2) Hmonic 2 is the mgnitude of set with espect to twofolddivision of the octve, o titone. Not supisingly, the mgnitude

0 1 2 3 4 5 6

|F{0,1,2}O |

1

2

3

0 1 2 3 4 5 6

|F{0,1,3}O |

1

2

3

0 1 2 3 4 5 6

|F{0,1,4}O |

1

2

3

0 1 2 3 4 5 6

|F{0,1,5}O |

1

2

3

0 1 2 3 4 5 6

|F{0,1,6}O |

1

2

3

0 1 2 3 4 5 6

|F{0,2,4}O |

1

2

3

0 1 2 3 4 5 6

|F{0,2,5}O |

1

2

3

0 1 2 3 4 5 6

|F{0,2,6}O |

1

2

3

0 1 2 3 4 5 6

|F{0,2,7}O |

1

2

3

0 1 2 3 4 5 6

|F{0,3,6}O |

1

2

3

0 1 2 3 4 5 6

|F{0,3,7}O |

1

2

3

0 1 2 3 4 5 6

|F{0,4,8}O |

1

2

3

Figure 21. Spectra of twelve-tone equal tempered trichordal pitch-class sets


19/56


of the second hmonic is quite high fo {0, 1, 6}O, while the mg-

nitude vnishes fo {0, 2, 4}O, since this set-clss divides the titone

into thee equl pts.

(3) Hmonic 3 is the mgnitude of set with espect to theefold

division of the octve, o n ugmented tid. Obviously, {0, 4, 8} Ohs mximl mgnitude fo this hmonic, nd, not supisingly,

the mgnitudes fo {0, 1, 4}O nd {0, 1, 5}O e lso quite high.

(4) The mgnitude of the fouth hmonic is mximl fo {0, 3, 6}O,

since this set-clss belongs to single foufold division of the

octve, o diminished seventh chod. The coesponding mgni-

tude vnishes fo {0, 1, 2}O, {0, 1, 5}O, {0, 2, 4}O, {0, 2, 7}O, nd {0, 4, 8}O,

since the membes of these set-clsses e dwn eqully fom the

thee unique diminished seventh chods.

(5) The fifth hmonic is the mgnitude of set with espect to five-fold division of the octve, o the set-clss {0, 2.4, 4.8, 7.2, 9.6}O. The

geneting intevl of this cycle, 2.4, is nely hlfwy in between

n equl tempeed mjo second nd mino thid, but twice this

intevl, 4.8, is vey close to n equl tempeed pefect fouth.

Thus, those set-clsses tht belong to smll segment of 5-cycle,

such s {0, 2, 7}O nd {0, 2, 5}O, hve significnt mgnitudes fo the

fifth hmonic.

(6) Finlly, hmonic 6 is the mgnitude of set with espect to six-

fold division of the octve, o whole-tone collection. The thee

tichodl set-clsses tht e subsets of single whole-tone collec-

tion, {0, 2, 4}O, {0, 2, 6}O, nd {0, 4, 8}O, hve mximl mgnitude fo

this hmonic. Since the emining tichodl set-clsses ll hve

two membes tht belong to one whole-tone collection while the

emining membe belongs to the othe collection, ech of these

set-clsses hs mgnitude of 1 fo this hmonic.

[000] [001] [002] [003] [004] [005] [006]

[012] [013] [014] [015] [016]

[024] [025] [026] [027]

[036] [037]

[048]

Figure 22. Fundamental region for trichordal set-classes in continuous pitch-c lass space


20/56


While the snpshots povided in Figue 21 povide hmonic bluepint

fo ech of the set-clsses, we cn gin moe complete undestnding of the

undelying topology by consideing the spect of tichods in continuous

pitch-clss spce. We begin by plotting tichod set-clsses in two dimensions

Figure 23. Magnitudes of all trichordal set-classes in twelve-tone equal temperament for the

first six harmonics of the octave


21/56


Figure 23. (cont.)

such tht [0,y, x] is locted t the point (x,y). Fo exmple, ugmented ti-

ds e locted t (8, 4), chomtic tichods t (2, 1), nd tipled unisons t

(0, 0). Fo esons tht lie beyond the scope of this ticle, we will lso use

n oblique coodinte system in which the x- ndy-xes e in 1208 eltion


22/56


the thn 908. Plotting pitch-clss sets in this mnne, ll possible tichod

set-clsses lie within the shded egion of Figue 22. This is the fundamental

regionof tichod set-clsses, since ll possible thee-note set-clsses lie in this

egion nd thee e no duplictionsno two points within the egion ep-

esent the sme set-clss. Set-clsses (including those with pitch-clss duplic-

tions) fom twelve-tone equl tempement e lbeled only s guide fo the

ede; ll points within the egion epesent unique set-clsses, nd none e

to be given conceptul pioity.16

Next, we dd thid dimension tht coesponds to the mgnitude of

ech set-clss with espect to given hmonic, shown in Figue 23. Fo h-

monic k, thee will be globl mxim t those set-clsses tht e subsets of

single k-fold division of the octve. Thus, fo the fist hmonic, thee is

single mximum t the tipled unison, {0, 0, 0}O, since this is the only tichodl

set-clss tht belongs to onefold division of the octve. In fct, since uni-son (with ny numbe of pitch-clss duplictions) is tivilly membe of ny

intevl cycle (up to tnsposition), thee will be mximum t this set-clss fo

ll hmonics. Fo the second hmonic, thee is n dditionl mximum t

{0, 0, 6}O; fo the thid hmonic, thee e dditionl mxim t {0, 0, 4}Ond {0, 4, 8}O nd similly fo the emining hmonics. Since the division of

the octve into five pts does not yield twelve-tone equl tempeed set, the

mxim fo the fifth hmonic do not lie on fmili set-clsses. Insted, thee

e mxim t {0, 0, 0}O, {0, 0, 2.4}O, {0, 0, 4.8}O, {0, 2.4, 4.8}O, nd {0, 2.4, 7.2}O.

Set-clsses lying ne mxim will hve eltively high mgnitudes fo

given hmonic. Fo exmple, {0, 1, 6}O lies ne {0, 0, 6}O nd thus hs

eltively high mgnitude fo the second hmonic, while the mgnitude of

{0, 2, 7}Owith espect to the fifth hmonic is eltively high due to its poxim-

ity to the mximum t {0, 2.4, 7.2}O.17 also note tht the highe the hmonic,

the gete the numbe of mxim nd the steepe the slope wy fom these

high points. Thus, the topology fo the fist hmonic consists of gdul

descending slope fom single mximum t the minimlly even set-clss,

while the lndscpe of the sixth hmonic is studded with seven peks nd

coespondingly steep gdients.

anlogous n-dimensionl stuctues obtin fo the spect ofn-note setsin continuous pitch-clss spce. Despite the difficulties in visulizing the cses

fo sets of moe thn thee notes, the spces e esily descibed: fo the kth

16 For a more detailed explanation of this region and its

derivation, see Callender 2004 and Callender, Quinn, and

Tymoczko 2008.

17 Tymoczko 2008 makes the same point in a much more

rigorous manner, demonstrating a strong inverse relation-

ship between the magnitude of the kth harmonic in a chords

spectrum and the voice-leading distance from that chord to

the nearest subset of a k-fold division of the octave. Thismeans that the location of a chord in voice-leading space

determines the chords spectrum to a high degree of accu-

racy. Conversely, two chords will be close in Fourier space if

the set of their voice-leading distances to the nearest sub-

set of equal divisions of the octave are similar. Of course,

two chords can be similarly distant from these subsets with-

out necessarily being close to one another in voice-leading

space. While Tymoczkos work exposes a deep connection

between voice-leading and Fourier spaces, it also helps

to explain the situation encountered in the introduction in

which chords are close in one space but relatively more dis-

tant in the other.


23/56


hmonic, peks e locted t ll set-clsses of the fom {ai}ni= 1, whee ai[ 0

mod 12/k. The peks (nd toughs) fom equl-spced lttices coespond-

ing to tingles fo thee-note chods, tethed fo fou-note chods, nd

so foth.18

5. Properties of chord spectra

It is ppopite t this point to collect numbe of genel popeties of

chod spect, focusing on those tht e most elevnt fo continuous pitch

nd pitch-clss spces.

(1) Chods with identicl intevl content hve identicl spect. This

is evident by ewiting Eqution 1 in the ltente fom

F

P(z)

i, jcos2(

pi

pj)z.

(3)In this ltente fom the spectum is defined in tems of the

diected intevls fom chod to itself:pipj. Thus, tnsposition-

lly elted, invesionlly elted, nd Z-elted sets hve identicl

spect.

(2) Multiplying chod hs the effect of dilting its spectum;

specificlly, if P 5 Mx(Q), then FP(z) 5FQ(xz) o, equivlently,

FP(z/x) 5FQ(z). Quinn (2007) lbels this popety the multiplica-

tion principle. Fo exmple, since {0, 2, 4} is elted to {0, 1, 2} byM2,

the spectum of {0, 2, 4} with espect to fequencyzis equl to thespectum of {0, 1, 2} with espect to fequency 2z.

Putting this popety togethe with popety (1) bove, if

the spect ofMx(P) nd Mx(Q) e equl t fequencyz, then the

spect ofPnd Qwill be equl t fequencyxz. as n exmple of

the usefulness of this popety, conside the pitch-clss sets PO5

{0, 1, 3}O nd QO 5 {0, 1, 4}O. Multiplying both sets by 2 yields

M2(PO) 5 {0, 2, 6}O nd M2(QO) 5 {0, 2, 8}O. Since these two sets,

M2(PO) nd M2(QO), e elted by invesion, thei spect e iden-

ticl fo ll hmonics. By the multipliction pinciple, this implies

tht the spect ofPO nd QO e equl fo ll hmonics of thefom 2k(whee kis ny intege), which cn be veified by exmin-

ing the even hmonics of the elevnt spect in Figue 21. Moe

genelly, given ny pitch-clss sets POnd QO, ifMx(PO) nd Mx(QO)

e tnspositionlly elted, invesionlly elted, o Z-elted,

then the spect ofPO nd QO will be equl t ll hmonics of the

fom xz. This popety will ply n impotnt ole in the discussion

of Z-elted sets in 7.

18 For more on higher-dimensional analogues of Figure22 and on voice-leading spaces in general, see Callender,

Quinn, and Tymoczko 2008.


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(3) Given pitch setP, if evey membe ofPbelongs to the sme l-cycle,

then the spectum ofPis peiodic with peiod of 1/l:

P #Ty(zl)|FP(z)|5 |FP(z1 1/l)|

Fo exmple, the pitch set {0, 5, 15} belongs to 5-cycle, so its spec-tum, ledy shown in Figue 18c, hs peiod of 1/5. Fo nothe

exmple, the pitch set {0, 3/4, 5/3} belongs to (1/12)-cycle, since

1/12 is the getest common diviso of the sets intevls. Thus, this

sets spectum hs peiod of 12.

The sme pinciple pplies fo sets with itionl intevls,

s long s the ratioof these intevls is tionl. Fo exmple, the

set {0, 2, 32} belongs to cycle of2 semitones, { . . . , 0, 2,22, 32, . . . }, so its spectum hs peiod of 1/2. Howeve,

the set {0, 2, } cnnot belong to n intevl cycle since the tiosof its intevls, such s 2/, e not tionl. Thus, the spectumof this set is peiodic.

(4) The spectum of pitch-clss setPO is peiodic if the spectum of

the coesponding pitch setPhs peiod of the fom k/12, k[ Z.

anothe wy to stte this is tht if pitch-clss set belongs to some

k-tone equl tempeed system, then the spectum of the pitch-

clss set is peiodic with peiod of k/12. Fo exmple, while the

pitch setP5 {0, 5, 15} belongs to 5-cycle nd its spectum hs

peiodicity of 1/5, the spectum of the pitch-clss set PO cnnot

hve peiodicity of 1/5, since 5-cycle does not evenly dividethe octve. Howeve, Plso belongs to 1-cycle, which divides the

octves into the fmili twelve-tone equl tempeed collection,

so the spectum of PO hs peiod of 1. Similly, {0, 3/4, 5/3}O

belongs to 144-fold division of the octve, so its spectum hs

peiod of 12. Howeve, while the pitch set {0, 2, 32} hs pei-odic spectum, this set cn neve be embedded in single equl

tempeed system. Thus, the spectum of the pitch-clss set {0, 2,32}O is peiodic.

(5) Chod spect e symmetic boutz5 0: |FP(z)|5|FP(2z)|. It iscle tht n intevl cycle geneted by l, { . . . , 2l, 0, l, . . . }, willbe the sme s cycle geneted by2l, { . . . , l, 0, 2l, . . . }. Thus the

mgnitude of given set with espect to z5 1/lwill be the sme s

its mgnitude with espect to 2z521/l. (This is lso evident fom

Eqution 3, since cos q5 cos 2q.)

(6) Putting popeties (3), (4), nd (5) togethe, if the spectum of

pitch o pitch-clss set is peiodic with peiod of 1/l, then the

entie spectum cn be geneted by the mgnitudes fo 0 #z#

1/2l. By popeties (3) nd (4), we cn see tht the entie spec-

tum cn be geneted by ny intevl spnning 1/l. In pticu-l, we cn genete the entie spectum by the mgnitudes fo


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21/2l# z# 1/2l. By popety (5), we cn genete the neg-

tive potion of this nge fom the positive fequencies. Fo

pitch-clss set dwn fom k-tone equl tempement, it is thus

only necessy to conside those fequencies in the nge 0 #z#

k/(2 ? 12), which coesponds to hmonics 0 though . Con-

vesely, fo pitch-clss set tht is not dwn fom nyk-tone equl

tempement, it is necessy to conside the infinite numbe of

hmonics, since the spect of these sets e peiodic.

6. Metrics for continuous harmonic space

In this section we will conside numbe of metics defined on chod spect

tht my be intepeted s dissimility mesues in continuous hmonic

spce. Howeve, the pimy im is not to put foth ny metic in pticul, butto demonstte both how to constuct metics on chod spect nd tht these

metics cn coelte vey stongly with simility mesues bsed on intevl

vectos. as discussed in Scott nd Iscson 1998, if two simility mesues coe-

lte vey well, they pobbly will stnd togethe o fll togethe when subjected

to empiicl tests. In this context, if metic on chod spect, SPECTra,

nd simility mesue on intevl vectos, XSIM, coelte stongly with one

nothe, we cn think of SPECTra s ppoximting XSIM in continuous

envionment. The dvntge is tht, to the extent XSIM cptues intuitions

concening hmonic simility, SPECTra pplies these intuitions to ll pos-sible chods in ll possible tuning systems (to within some level of ccucy). In

ode to fcilitte such compisons, we will begin with the spect of pitch-clss

sets dwn fom twelve-tone equl tempement, then move to continuous

pitch-clss spce, nd finish with metics defined on continuous pitch spce.

6.1 Pitch-class sets in twelve-tone equal temperament

recll tht the spect of pitch-clss sets in twelve-tone equl tempement

e detemined by the mgnitudes of the fist six hmonics of the octve (in

ddition to hmonic zeo, which coesponds to cdinlity). Let hmonicsone though six define six-dimensionl Fouie tnsfom spce, F6. Using

the mgnitudes of these six hmonics, we cn locte the spectum of ny

equl tempeed pitch-clss set: P5 (p1, . . . , p6), whee pk is the mgnitude

of the kth hmonic. Fo exmple, the spectum ofP5 {0, 4, 8}, which hs

mgnitude of 3 fo the thid nd sixth hmonics nd zeo mgnitude fo

the othe hmonics, is locted in F6 tP5 (0, 0, 3, 0, 0, 3). a numbe of

fmili simility mesues e defined on diffeent six-dimensionl spce

defined by the intevl vecto, including Teitelbums (1965) s.i. (fo simil-

ity index), Iscsons (1990) IcVSIM, rogess (1999) cos Q, nd Scott nd

Iscsons (1998) aNGLE. We now conside two metics on chod spect tht

e nlogous to these fou simility mesues.


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6.1.1 Euclidean metric in F6 a ntul wy to mesue the distnce

between two spect is to tke the Eucliden distnce between thei coe-

sponding points in F6. The Eucliden distnce between the spect of two

pitch-clss sets, Pnd Q, is

d(P,Q) (pi qi)2, (4)

o simply the length of the diffeences of the two vectos,

d(P,Q) 5|P2Q|. (5)

Fo exmple, the distnce between the spect of {0, 4, 8}, P5 (0, 0, 3, 0, 0, 3),

nd {0, 3, 6},Q5(1, 1, 1, 3, 1, 1), is d(P,Q)5 1 1 4 9 1 4 20< 4.47.This is quite simil to Teitelbums (1965) s.i., which mesues the Eucliden

distnce between two points in the spce defined by the intevl vecto. Both

dnd s.i. use the sme metic; the diffeence is the spce on which this metic

is defined.richd Teitelbum uses s.i. only to compe sets of the sme size,

since the metic tends to give unintuitive esults when comping sets of

diffeent size. (as Iscson [1990] notes, s.i. judges ll seven-note sets to

be moe simil to {0, 1, 2} thn e ll eight-note sets, which e ll moe

simil to the chomtic tichod thn e the nine-note sets.) as might be

expected, dnd s.i. coelte well; comping pitch-clss sets of equl cdi-

nlity (nging fom thee to nine notes), the coeltion coefficient fo d

nd s.i. is r5 .87.19

While this is fily high coeltion, much highe vlue esults fom vint ofd. recll tht the sque of the mgnitude spectum is the powe

spectum, which emphsizes eltively stong components nd deemphsizes

eltively wek ones. Since it is the stong components of spectum tht

most fully chcteize the content of set, it mkes sense to define the

Eucliden metic on the powe the thn the mgnitude spectum. Fo

instnce, wht mttes most in distinguishing {0, 4, 8} nd {0, 3, 6} is tht the

pek mgnitudes of 3 occu in diffeent hmonics; the fct thtQ hs mg-

nitudes of 1 in hmonics whee Phs mgnitude of 0 is of eltively little

consequence. Denoting Eucliden metics defined on the powe spect by

dpow, we hve

dpow(P,Q) 5|P22Q2|. (6)

The coeltion between dpownd s.i. is quite high: comping sets of cdinl-

ity 3 though 9 (limited to sets of equl cdinlity), r5 .96, mening tht 92%

of the vince in the metics is elted (r25 .92).

In ode to ovecome the limittion of s.i. with espect to cdinlity,

Iscson (1990) poposes mesuing the stndd devition of the diffeences

19 All correlations given in this article are statistically signifi-

cant, with p# .0001.


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of two intevl vectos, mesue he temed IcVSIM. Fo two intevl vectos,

a5 (a1, . . . , a6) nd b5 (b1, . . . , b6), let the diffeences of the two vectos be

c5 (a12b1, . . . , a62b6) 5 (c1, . . . , c6). IcVSIM is defined s the stndd

devition ofc:

1

6(c) (ci c)

2, (7)

whee cis the men of the vlues ofc. as Eic Iscson notes, when comping

sets of equl cdinlity, IcVSIM is the sme s s.i. (up to constnt scling fc-

to: IcVSIM 5 s.i. /6). Thus, it should not be supising thtdpownd IcVSIMcoelte stongly; comping ll sets of cdinlities 3 though 6 (not limited

to those of equl cdinlity), r5 .95, while sets of cdinlities 3 though 9

yield slightly lowe coeltion ofr5 .89.

While dpow coeltes stongly with IcVSIM, the fome does not pos-

sess some of the moe idiosynctic fetues of the ltte. In pticul, setsfo which the intevl vectos diffe by constnt e mesued s mximlly

simil by IcVSIM nd thus e locted t the sme point in the coesponding

hmonic spce. Fo instnce, the intevl vectos fo A5 {0, 3, 6} nd B5

{0, 1, 3, 6, 7, 9} e k002001l nd k224223l, espectively, diffeing by constntvlue of 2. While these two sets e clely simil in sound, it is hd to justify

this equivlence, especilly given tht othe, seemingly more simil pis of

sets, such s {0, 3, 6} nd {0, 3, 6, 9}, e consideed less simil. (Both Cstn

[1994] nd Buchle [1997] lso question this equivlence.) In contst, dpowfinds the two sets to be simil, but not equivlent: dpow(A,B) < 5.48, which is

smlle distnce thn tht between 91% of ll possible piings.20

6.1.2 Angular distance inF6 anothe ntul wy to mesue the dis-

tnce between two points in spce is to tke the ngle between the vec-

tos extending fom the oigin to ech of the points. This is vey common

metic, one tht is used instinctively wheneve we compe the loctions of

objects in the sky. We cnnot judge the ctul distnce between two sts by

the nked eye, but we cn ppoximte thei ngul distnce fily esily.

Figue 24 gives exmples of how ngul distnce pplies to intevl vec-

tos. In Figue 24, {0, 2, 4} nd {0, 4, 8} e plotted on Ctesin plne

with the x-xis epesenting the numbe of intevl clss 2 nd the y-xisepesenting the numbe of intevl clss 4. The ngul distnce between

the two vectos is q< 63.58 o < .45 dins. Smlle vlues fo q indicteintevl vectos tht e moe simil with q5 0, indicting mximl simi-

lity. Fo exmple, Figue 24b plots the intevl vectos fo {0, 3, 6} nd

{0, 3, 6, 9} on Ctesin plne with the x-xis epesenting the numbe of

20 One idiosyncratic feature of measuring distance in F6

is that complements have identical magnitudes for each

of these harmonics. Thus, complements are located at thesame point in F6 and are judged to be maximally similar.

One solution would be to incorporate the zeroth harmonic,

which is equal to cardinality, so that chords of different car-

dinalities would not be considered equivalent.


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intevl clss 3 nd the y-xis epesenting the numbe of intevl clss 6.

Since the intevl vectos of these pitch-clss sets e exctly popotionl,

thei vectos point in the sme diection, so the ngle between them is 0.

Mximl dissimility, indicted byq5 908 o /2 dins, occus pecisely

when two sets hve no intevllic content in common.21

Scott nd Iscsons (1998) aNGLE mesues the ngle between vectos

in six-dimensionl intevl-vecto spce, while rogess (1999) cos Q me-sues the cosine of this ngle. The cosine of the ngle q between two vectos,

x5 (x1, . . . , xn) ndy5 (y1, . . . ,yn), is given by

, (8)cos() xy

xy

whee |x| is the mgnitude ofx(o the length fom the oigin to x) nd x?yis the dot poduct ofxndy: x?y5 (xiyi. The vlues of cos Q nge fom 0,

indicting mximl dissimility, to 1, indicting mximl simility. We cn

convet this mesue to metic in hmonic spce with the widely used

cosine distnce, 1 2 cos Q, o by tking the ccos of Eqution 8 yielding q,

which is the sme s aNGLE. While ech of these mesues yields diffeent

sets of numbes, they e ll mesuing pecisely the sme thingngul

distnceso we cn tke aNGLE s the epesenttive fo ll thee.

Eqution 8 cn lso be used to mesue the ngul distnce between

points in Fouie tnsfom spce. as with the Eucliden metics of section

6.1.1, defining ngul distnce on powe spect yields stonge coe-

ltion with aNGLE thn the sme metic defining on mgnitude spect.

{0,4,8}

{0,2,4}=63.5

ic 2

ic 4

{0,3,6,9}

{0,3,6}=0

ic 3

ic 6

Figure 24. Angular distance between interval vectors for (a) {0,2,4} and {0,4,8} and (b) {0,3,6}and {0,3,6,9}

(a) (b)

21 Since the sets in Figure 24a contain only interval classes

2 and 4 and the sets in Figure 24b contain only intervalclasses 3 and 6, all nonzero dimensions are taken into

account, and the graphs give complete information.


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Designting this metic s dq, we hve

d(P2,Q2) arccos .

P2Q2

P2Q2(9)

Fo exmple, ifP5 {0, 3, 6} nd Q5 {0, 3, 6, 9}, then

P25 (1, 1, 1, 9, 1, 1) nd Q 25 (0, 0, 0, 16, 0, 0),

P2Q 25 9 16,

|P2| 5( 1 1 1 92 1 1 86, and

|Q2| 5 16.

Thus,dq(P2,Q2) 5 ccos 9 16

86 16< ccos .97 < 13.958 o < .24 dins,

indicting tht ccoding to dq the two sets e quite simil (close thn 99

pecent of set-clss pis), but not equivlent s is the cse fo aNGLE.The coeltion between dq in F6 nd aNGLE is not pticully high,

with r5 .7 fo cdinlities 3 though 6 nd r5 .52 fo cdinlities 3 though

9. a much highe coeltion obtins between aNGLE nd dq in F12, the

twelve-dimensionl spce defined by the mgnitudes fo hmonics 1 though

12. The coeltion between dq in F12 nd aNGLE is r5 .94 nd r5 .93 fo

cdinlities 3 though 6 nd 3 though 9, espectively. dq in F12 lso coeltes

stongly with ISIM2 (Scott nd Iscson 1998), r5 .88, nd rECrEL (Cstn

1994), .81 # r# .99, fo cdinlities 3 though 9. We will conside the eson

fo this significntly highe coeltion in 6.3.

6.2 Pitch-class sets in continuous space

as we sw in 5, it is necessy to conside n infinite numbe of hmonics of

the octve in ode to fully cptue itionl pitch-clss sets such s {0, 2, }tht do not belong to ny equl tempeed system. The spectum of given

pitch-clss set is single point within n infinite dimensionl Fouie spce,

F`, whee ech dimension coesponds to unique hmonic of the octve.

In theoy, then, it should be necessy fo metic on the Fouie tnsfom

of continuous pitch-clss spce to be defined on F`, but thee e esons whywe should not nd, in fct, cnnot do so, t lest not if we wish fo the esults

to be meningful.

Lets conside thee hmonics of fou pitch-clss sets: P 5 {0, 0.46,

0.95, 1.41}, Q5 {0, 1.01, 6, 7.01}, R5 {0, 3, 6, 9}, nd slight vition ofR,

R95 {0, 3.005, 6, 9.005}. The mgnitudes of the fist hmonic ep1< 3.83,q1< 0, r15 0, nd r91< 0. This hmonic divides the fou pitch-clss sets intotwo ctegoies: the unblnced set, P, nd the blnced sets Q, R, nd R9.

The mgnitudes of the second hmonic ep2< 3.41, q2< 3.41, r25 0, ndr92< 0.01. This hmonic divides the fou pitch-clss sets into two diffeentctegoies: those tht coelte stongly with twofold division of the octve,

Pnd Q, nd those tht do not, Rnd R9.


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These two hmonics povide diffeent levels of esolution with which

to investigte these pitch-clss sets nd thus give diffeent types of infom-

tion bout thei hmonic content. While the fist hmonic successfully dis-

tinguishes P fom Q, its esolution is too low to disciminte between two

diffeent wys of obtining blnce within pitch-clss set: mximlly even

distibution of ll pitch clsses in R, nd mximlly even distibution of

pi of uneven dyds in Q. Likewise, while the second hmonic success-

fully distinguishes between Qnd R, its esolution is too high to disciminte

two diffeent wys of coelting stongly with titone, by clusteing ound

single point, P, o by hving n equl numbe of pitch clsses clusteed

ound two points titone pt, Q. Obviously, neithe hmonic is successful

t distinguishing Rnd R9. In ode to mke this type of fine distinction, it is

necessy to use vey high fequency of esolution. Jumping (f) hed to

the 1200th hmonic, the mgnitudes ep12005q12005r12005 4 nd r912005 0.The mximl mgnitudes tz5 1200 fo P, Q, nd Result fom the fct tht

in ech cse ll fou pitch clsses belong to the equl division of the octve

into 1200 pts, o cent scle. Since R9 contins two pitch clsses fom one

cent scle nd nothe two fom the opposing cent scle, it hs mgnitude of

zeo. (Opposed cent scles e septed by tnsposition of one-hlf cent.)

This hmonic successfully distinguishes R9 fom Rbut povides f too high

esolution to distinguish P, Q, nd Ro ny of the unique pitch-clss sets in

1200-tone equl tempement.

This exmple is puposely exteme in ode to dive home the min

point tht diffeent hmonics povide diffeent levels of infomtion bout

hmonic content. It is not tht some hmonics e inheently supeio to

othes, but tht, depending on the ppliction t hnd, some hmonics will be

moe usefulthn othes. If, fo whteve eson, one wishes to investigte the

harmonicdiffeence between Rnd R9, then it is necessy to use n extemely

high hmonic nd coespondingly high fequency of esolution.

The pticul ppliction t hnd in this section is to mesue the dis-

tnce between pitch-clss sets in continuous hmonic spce. If distnces

e tken between points in F` nd ll hmonics e weighted eqully, then

the diffeence between r1200 nd r91200, which is 4, will hve slightly geteeffect on the mesued distnce thn the diffeence betweenp1 nd r1, which

is ppoximtely 3.83. Howeve, the implicit ssumption is tht we wish fo

the mesued distnce to eflect ou intuitions of peceived hmonic simil-

ity. Since Rnd R9 e peceptully identicl, the lge diffeence between

r1200 nd r91200 is meningless s fcto in mesuing peceived distnce.

In othe wods, hmonic 1200 povides f too high esolution fo this

pticul ppliction. at the othe exteme, it is cle tht hmonics 1 nd 2

togethe do not povide enough infomtion to seve s the sole bsis fo me-

suements of peceived distnceugmented tids nd diminished seventh

chods (which e cetinly esily distinguishble sonoities!) both hve zeo


31/56


mgnitude fo hmonics 1 nd 2. The tick is to find set of hmonics tht

e both necessy nd sufficient to seve s bsis fo esonble metic of

hmonic distnce in continuous pitch-clss spce.22

Thee e two common solutions to this poblem. The fist is to weight

dimensions in descending mnne. Fo exmple, suppose we scle the nth

hmonic by 1/n: pn/n. Then diffeences in the fist hmonic will be 1200

times moe impotnt in detemining distnce thn diffeences in the 1200th

hmonic. Fo instnce, ccoding to this method of weighting dimensions,

the weighted diffeence between r1200 nd r91200 is only 4/1200 < 0.0033, whichwould hve the desied negligible impct on the ovell distnce. anothe

solution is to simply tke the fistnhmonics s the bsis fo the Fouie

spce Fn, selecting ns ppopite. We will dopt the second ppoch, since

it coesponds moe closely with ou ppoch to pitch-clss sets in twelve-

tone equl tempement.In ode to get feel fo metics in continuous pitch-clss spce, we will

compe distnces between two pitch-clss sets, Pnd Q, whee Pis some given

tichod nd Qvies mong ll possible tichods. recll fom 4.2 tht the

fundmentl egion of tichod set-clsses is the tingul egion gphed in

Figue 22. To this two-dimensionl egion we dd thid dimension tht co-

esponds to normalized similarity, which vies fom 0 to 1, indicting miniml

nd mximl simility, espectively. The highe point within the egion is,

the moe simil its ssocited set-clss, /Q/, is to the given compison set-

clss, /P/. Convesely, lowe points coespond to set-clsses tht e eltively

dissimil to /P/. Fo exmple, Figue 25 plots the nomlized simility

between the ugmented tid nd ll tichod set-clsses using the metic dpowdefined on the fist six hmonics of the octve, o the spce F6. Note tht the

plot contins globl mximum t the ugmented tid, indicting mximl

simility to {0, 4, 8}, nd globl minimum t the tipled unison, indicting

miniml simility. Note lso the locl peks t o ne {0, 0, 4}, {0, 2, 6}, nd {0,

2, 4}, nd the slight idge extending fom {0, 0, 4} to both {0, 4, 8} nd {0, 2, 4}.

The cente of this idge coesponds to ll tichodl set-clsses tht contin

mjo thidthose set-clsses of the fom /{0, x, x1 4}/. Thus, we cn see the

influence tht the mjo thid, the defining intevl of the ugmented tid,exets ove the simility contou fo the continuous spce of tichods.

Howeve, not ll tichods contining mjo thid e judged to be

eqully simil to the ugmented tid. The simility contou tkes into

22 There is also a strictly mathematical problem. Consider

the pitch-class sets P5 {0, 4, 8}, Q5 {0, 4 1, 8 1 3},

and R5 {0, , 3}, where 5 1/p, p[Z. Taking p to be

arbitrarily large, and thus to be arbitrarily small, in F12p

dpow(P,Q) 5dpow(P,R). At the very least, we should want

a set that is indistinguishable from {0, 4, 8} (Q) to be closer

to the maximally even set than a set that is indistinguish-

able from the maximally uneven set (R)! This is because all

three sets belong to 12p-tone equal temperament and their

respective interval functions are maximally dissimilar. See

Equations 11 and 36 with their accompanying discussions

for more details. (The same problem exists for the angular

distance in F`.)


32/56


Figure 25. Plot of normalized similarity between {0, 4, 8} and all other trichordal set-classes

based on dpow in (a) F6

, (b) F12

, and (c) F48

(a)

(b)

(c)


33/56


ccount othe impotnt fetues of {0, 4, 8}, including its mximl fit with

whole-tone nd mjo-thid cycles nd its mximlly even distibution of pitch

clsses. Fo instnce, the pek t {0, 0, 4}, which lso belongs to both whole-

tone nd mjo-thid cycles, is highe thn the othe locl peks t o ne

{0, 2, 4} nd {0, 2, 6}. Of the two smlle peks, the one ne {0, 2, 6} is highe,

since this set-clss hs moe even distibution of pitch clsses thn {0, 2, 4}.

Mesues of hmonic simility tht e bsed on the Fouie tnsfom thus

cptue mny fetues of given chod.

also note tht the simility contou is smooth, such tht set-clsses lying

ne one nothe in the fundmentl domin e judged to be oughly equl

in thei simility to the ugmented tid. This confoms to ou expecttion

tht simility judgments should not be getly ffected by vey smll displce-

ments of chod membes. This fetue is diectly elted to the numbe of h-

monics tken into ccount. Fo exmple, Figue 25b is the contou mp thtobtins fom distnces bsed on the fist twelvehmonics, F12, the thn

just the fist six. This contou is noticebly moe jgged thn Figue 25, while

the idge coesponding to set-clsses contining mjo thid hs become

less diffeentited. These tendencies e even moe ponounced in Figue

25c, which is bsed on distnces in the Fouie spce defined by the fist 48

hmonics, F48. In the limit cse, the nomlized simility bsed on dpow in Fn,

whee ngoes to infinity, would yield contou of simility between {0, 4, 8}

nd ll othe tichods tht consists of single impulse t the ugmented tid

nd n lmost entiely undiffeentited, infinitesimlly now idge long

set-clsses contining mjo thid.23 In othe wods, the highe the vlue of

n(nd the gete numbe of hmonics tken into considetion), the moe

distnces bsed on chod spect become like those bsed solely on intevl

content, losing thei unique nd dvntgeous fetues long the wy.

Figue 26 is contou mp of similities bsed on distnces in F6 me-

sued bydq the thn dpow. a compison of the gphs in 25 nd 26 shows

how simil the topogphies esulting fom dpownd dq e. Both contin the

sme idge nd thee locl peks discussed bove. at lest when limited to

comping sets of the sme cdinlities, dpownd dq (nd thei intevl-vecto

bsed cousins, IcVSIM nd aNGLE) coelte quite stongly. The sole excep-tion occus t the tipled unison, which is globl minimum ccoding to the

Eucliden metic but the stong pek ccoding to ngul distnce.24

23 More precisely, the normalized similarity between

{0, 4, 8} and all other trichords based on dpow in F` yields a

contour map consisting of a single maximal spike of 1 at

{0, 4, 8}, a ridge of .78 at all sets of the form {0, x, x1 4}

with the exception of a slight spike of .89 at {0, 0, 4} and

slight dips of .74 and .7 at {0, 2, 6} and {0, 2, 4}, and a broad

plain of .54 for all other sets except for cliffs of .48 and .4at sets of the form {0, x, 2x} and {0, 0, x} and a downward

spike to 0 at {0, 0, 0}.

24 The spectrum of the maximally uneven set {0, . . . , 0}

contains maximal magnitude in every harmonic. Thus, the

vector associated with this spectrum points directly in the

middle of that part of Fn defined by nonnegative values

for each harmonic. This means that the maximum angular

distance between the maximally uneven set and any other

pitch-class set is 458 or /4 radians.


34/56


as discussed bove, pt of the stength nd ttction of Fouie spces

is the flexibility to conside only those hmonics tht e elevnt to specific

context. Fo instnce, most edes hve been conditioned to he nything

belonging to twelve-tone equl tempement s in tune nd (most) evey-

thing else s moe o less out of tune. This suggests tht, ll othe things

being equl, twelve-tone equl tempeed chods will be peceived s moe

simil to one nothe thn the neighboing out of tune chods. Since the

twelfth hmonic mesues the extent to which set belongs to twelvefold

division of the octve, we cn use this hmonic to help model this phenom-enon, mesuing distnces in seven-dimensionl Fouie spce defined by

hmonics 16 nd 12. (Note tht this spce does notcontin hmonics 711.)

Figue 27 once gin shows the nomlized simility between tichods nd

the ugmented tid, but hee distnces e mesued with espect to this

seven-dimensionl Fouie spce. The esulting contou mp contins locl

mxim t ech set-clss ssocited with twelve-tone equl tempeed chods,

including those with pitch-clss duplictions. (Thus, the peks in Figue 27

coespond to the points lbeled in Figue 22.)

6.3 More on Euclidean and angular distance on chord spectra and interval content

We hve seen two metics on chod spect tht coelte stongly with met-

ics on intevl content. It ws suggested tht this stong coeltion llows us

to pply the intuitions bout hmonic simility tht e implicit in pomi-

nent simility mesues to continuous pitch-clss spces. But wht e these

intuitions? Wht exctly is the eltionship between these metics on chod

spect nd intevl content?

To undestnd the connection between these metics, we must conside

Lewins (1987) intevl function (wht mthemticins cll the Pttesonfunction). Fo given setP, the intevl function, which we will wite s DP, is

the set of ll diected intevls between membes ofP: DP5 {pi2pj mod 12},

Figure 26. Plot of normalized similarity between {0, 4, 8 } and other trichordal set-classes

based on dq in F6


35/56


fo llpi,pj[P.25 Fo exmple, in the setQ5 {0, 1, 3} thee e thee wys

of moving by 0 (fom ny membe ofQto itself) nd one wy of moving by

intevls of 1, 2, 3, 9, 10, nd 11 semitones. Thus, the intevl function ofQis

DQ5 {0, 0, 0, 1, 2, 3, 9, 10, 11}. We could lso wite this set s vecto whee

the ith enty (beginning with i5 0) coesponds to the multiplicity ofiin the

intevl function; fo exmple, DQ5 (3, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1).

The elevnce of the intevl function is tht, when mesuing distnce

between twelve-tone equl tempeed sets in F12 the thn F6, dq nd dpowcn

ech be ewitten in tems of the intevl function. (See 8.) Fo exmple, wecn ewite Eqution 9 s

d(P,Q) arccos .

PQ

PQ(10)

In othe wods, mesuing ngul distnce inF12 is identicl to mesuing ngu-

l distnce in the twelve-dimensionl spce defined by the intevl function.26

retuning to pevious exmple, ifP5 {0, 3, 6} nd Q5 {0, 3, 6, 9}, then

DP5 (3, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0),

DQ5 (4, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0),

DP?DQ5 3 ? 4 1 2 ? 4 1 2 ? 4 1 2 ? 4 5 36,|DP|5 32 3 22 21,nd |DQ|5 4 42 8.

Thus, mesuing ngul distnce in twelve-dimensionl intevl function

spce, we hve

aNGLE12(P,Q) 5 ccos21 8

36 < ccos .982 < 10.98 o < .19 dins.

Figure 27. Plot of normalized similarity between {0, 4, 8} and all other trichordal set-classes

based on dpow in the Fourier space defined by the magnitudes of harmonics 16 and 12

25 Lewin's interval function is actually defined on two sets,

Pand Q, and is the set of all directed intervals from mem-

bers of Pto members of Q: IFUNC(P,Q) 5 {p2qmod 12}for all p[Pand q[Q. We will write DPas an abbreviation

for IFUNC(P,P).

26 More generally, for n-tone equal tempered sets, dq in Fn

is identical to measuring angular distance in n-dimensional

interval-function space.


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Likewise, mesuing ngul distnce in F12, we hve

|FDP|5 (1, 1, 1, 9, 1, 1, 1, 9, 1, 1, 1, 9),|FDQ|5 (0, 0, 0, 16, 0, 0, 0, 16, 0, 0, 0, 16),

|FDP|?|FDQ|5 3 ? 9 ? 16,FP

3 92 9 16 7 , nd

FQ

3 162 16 3 ,

which yields the sme esult:

dF

P,F

P 5 ccos

6 7 16 3

3 9 16 < 10.98 o < .19 dins.

It is this identity between ngul distnce mesued on powe spect nd the

twelve-dimensionl intevl function tht explins the high coeltion noted

bove between dq, mesued in F12, nd aNGLE, bsed on the six-dimensionl

intevl vecto, becuse aNGLE coeltes stongly with ngul distnce onintevl functions.

Similly, Eucliden-bsed metics on chod spect cn be expessed in

tems of the intevl function. Conside dpow defined in F12 the thn F6. as

mesued in the twelve-dimensionl spce of powe spect, when pplied to

twelve-tone equl tempeed sets Eqution 6 my be simplified s

dpow(P,Q) 5 12 |DP DQ|. (11)

Tht is, Eucliden distnce between powe spect in F12 is identicl (up to

constnt scling fcto) to Eucliden distnce between intevl functions.27

Defined in F12, dpow coeltes vey stongly with Teitelbums s.i (o theEucliden distnce between intevl vectos), with r5 .997. (Indeed, the two

mesues e nely identicl.) To the extent tht this wy of mesuing dis-

tnce cptues something of ou sense of hmonic simility, dpow (in F12)

llows us to pply this intuition in continuous pitch-clss spce.

The coeltion between s.i. nd IcVSIM is fily wek (r5 .41 fo cdi-

nlities 3 though 9), so the stong coeltion between dpowin F12 nd s.i. cn-

not be the eson fo the stong coeltion between dpow in F6 nd IcVSIM.28

The intepettion ofdpow in eithe F6 o F12 is vey simil, with the fome

diffeing fom the ltte in giving dditionl weight to the degee to which the

elevnt sets embed within single whole-tone collection. LetOXndEXbe the

numbe of odd nd even intevls in the intevl function ofX. The quntity

OXEX epesents the degee to which the membes ofXbelong to opposing

whole-tone collections.29 Eqution 11 fo dpow in F12 cn be modified in the

27 Again more generally, for n-tone equal tempered sets, dpow

in Fn is a scaled version of s.i. defined on the n-dimensional

interval function rather than the -dimensional interval

vector.

28 The correlation between dpow in F6 and a variant of IcVSIM

defined on the interval function rather than the interval vec-

tor is similarly strong (r5 .93 for cardinalities 3 through 9).

29 For example, consider three tetrachords: in X all four

members belong to the same whole-tone collection, in Y

there are three members in one collection and one member

in the other, and in Zthere are two members in each collec-

tion. Thus, we have OXEX5 0 16 5 0, OYEY5 6 3 10 5 60,

and OZEZ5 8 5 8 5 64, with the values increasing as thepitch classes are distributed more evenly between the two

whole-tone collections.


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following wy to yield the eqution fo dpow in F6

:dpow

(P,Q) 6PQ 2(OPE

P

OQE

Q

(O

PE

Q

OQE

P)) (12)

Essentilly, the expession OPEP1OQEQ(OPEQ1OQEP) epesents the degee

to which Pnd Qe similly divided between whole-tone collections. If the

two sets e divided between whole-tone collections to the sme degeetht

is, OP5OQ nd EP5EQthen the vlue of the expession is 0. The mxi-

mum vlue fo the expession occus when one set is dwn fom single

whole-tone collection while the othe is s evenly divided between whole-tone

collections s possible. While the diffeence between s.i. nd IcVSIM is due

to the mnne in which ech mesue dels with compisons between setsof diffeent cdinlities, it is suggestive tht the coesponding metics on

chod spect, dpow in F12 nd F6, diffe not in tems of cdinlity diectly but

in tems of whole-tone embedding.

6.4 Pitch sets

Defining distnce between spect is much moe complicted fo pitch sets

thn fo pitch-clss sets, becuse the spect of pitch sets e continuous. To

bette undestnd the difficulties, lets etun to the thee sets fom the into-

duction: Q5 {G3, D4, F4, a4, E4}, R5 {G3, B3, D4, a4, E4}, nd S5 {G3, B3,

E4, a4, E4}. Musicl intuition suggests tht the spect ofRshould be moe

Figure 28. Defining distance between pitch spectra in terms of area

(a)

(c)

(b)

(d)

abs(FQ

2

F

R2) abs(F

R2

F

S2)

FQ

2 and FR

2 FR

2 and FS2


38/56


simil to tht ofQthn to tht ofS, implying thtQnd Rshould lie close

togethe in hmonic spce thn Rnd S. But how cn we mesue this dis-

tnce? Since the spect of pitch sets e continuous, we cnnot use discete

summtion s with pitch-clss sets. Insted, we must ely on the nlogous

geometic concept of e. The spect ofQnd Re supeimposed in Fig-

ue 28, while the spect ofRnd Se supeimposed in Figue 28b. In both

cses, the egion bounded by the two spect nd veticl boundies tx5 0

nd x5 1/2 is shded. Equivlently, we cn tke the bsolute vlue of the dif-

feence of two spect nd shde the egion bounded by the esulting cuve,

the x-xis, nd the sme veticl boundies, s shown in Figue 28, (c) nd

(d). The shded egions in () nd (c) e equl, s e the egions in (b) nd

(d). If we wee to mesue the e of the shded egions, we would obseve

tht the e of the egion in (c) is less thn tht in (d). We

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