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    Chapter XVII

    THE PITCH SET AS A LEVEL OFDESCRIPTION FOR STUDYINGMUSICAL PITCH PERCEPTION

    Gerald J. Balzano

    Department of Music

    University of California at San Diego

    La Jolla, California, CJSA

    INTRODUCTION

    How shall we defin e the musical stimulus ? Traditionall y, music

    theorists interested in pitch phenomena have sought the definition in terms

    of ratio s of who le numbe rs. Such rati os can provid e a des cri ptio n of mo st , if

    not all, of the musical intervals in use today, and can be translated readily

    into ratios of physically realizable tone freque ncies . Psycho acousti cians,following the dictates of reductionism, have sought a finer grain of analysis

    than this, pointing out that the essential constituent of an interval is a tone,

    and that the study of music perception must ultimately refer to the

    perc epti on of sing le tones and their freque ncy com pon ents . Accor dingly , a

    great deal of scientific effort has gone into studying the perception of

    isolated tones.

    In this paper I will argue that both of th es e le vel s of description are to o

    fine -gra ined for rep res ent ing music. I would propose instead the pitch set as

    a more re ali sti c level of analysis appropriate to the study of music. Strea ms

    of pitches arrayed over time that do not generate a determinate pitch set,

    such as the intonation contours of normal human speech, do not generally

    elicit a state of the listener commonly associated with perceiving music.

    Given this observation, it is quite possible that properties of pitch sets,

    while not to be found at the level of single tones or ratios, are nonethelessdire ctly ope rati ve in music pe rcepti on. Without a desc ripti on at this higher

    level, many of the most distinctively musical phenomena may be left out of

    account.

    I will introduce a means of description different from those commonly

    rec eiv ed: I shall ch ar ac ter iz e pitch set structu re in te rm s of the langua ge of

    ma the ma ti ca l group theo ry. In music of the Western world today , the

    universal pitch set is the twelve-tone chromatic scale, and the group

    associated with it is Ci2a

    cy cl ic group of order 12. Fel ix Klein showed

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    Euclide an, affine, and topo logi cal space s in ge om et ry . I will use the

    structure of C j 2t o repr esen t pitch sp ace . Pitch set s will be vie wed in

    terms of their 'shape'-like properties vis-a-vis three isomorphic spaces

    Inherent in the structure of Cj 2*

    In describing at the leve l of pitch set we do not nece ssar ily fore golower leveis of analysis. A two-tone interval is a pitch set, as is, for that

    mat ter , a single tone. Pitch se ts with more than two membe rs may be

    ref err ed to as triad s or chor ds, and pitch se ts tha t are large r than this may

    be known as s cal es . Whenever the ter m 'sc ale' or 'triad* is used in this paper ,

    I refer only to a pitch se t, without further conn otat ions such as si mul tane ity

    in time or frequent musical usage.

    Any study of the 'musical stimulus' addressing the level of pitch sets

    must com e to grips with the mos t historicall y enduring of all pitch se ts , the

    major sc al e. When vie wed from the leve l of sing le ton es , ther e appears to

    be nothing spec ial about this sc ale . When vie wed fro m the leve l of ra tio s,

    however, the major scale does appear to possess a number of unusual

    proper ties (Hel mho ltz, 1885; Schenker, 1906; Benade , 1960), and this has

    been taken by some as a decisive advantage for the ratio-based approach.

    One sta rtl ing, but sati sfyi ng conclusio n we will co me to from our pit ch- se tpurview is that the major scale is not merely special, but unique, even in

    re fer enc e to a rati o-ind epen dent set of prop erti es. And as we will se e, the

    major sc ale is lite rall y embedded in the str uctu re of C j 2 .

    The followin g is a ske tch of an approach . Other fac et s of the a pproach

    may be found in Balzano (1978) and Balzano (in press).*

    THE GROUP C 1 2 AN D I T S SUBGROUPS

    The ele men ts of C j 2m a v

    ^e taken as the twelve basic pitch classes,

    re pr es en ted by the int ege rs 0, 1, 2, . . . ,11' modulo 12. Depe nding on th e

    tuning syst em (e. g. 3ust Intonati on, Pythago rean Intonation, Equal

    Temperament), differences among these integers translate either

    approximate ly or exac tly into log frequency diff ere nces . The ele men ts of

    the group are the twelve transformations - musical intervals - that generate

    the pitch clas se s. In situa tions where we assign a give n pitch clas s to the

    'origin' or zero-element of the group, the distinction between intervals and

    pitch cla ss es bec om es som ewh at blurred. When we wish to emp has ize the

    trans forma tional nature of t he group ele me nt s, the notat ion TQ, T I , . . . ,

    T H will be em pl oy ed . "I", for "identity", will be a syno nym for "TQ". In

    The following relev ant ref ere nce s, in alp habe tic al order: Babbitt (1965),

    Boretz (1970), Budden (1572), Chalmers (1975), Fuller (1975), Gamer

    (1967), Lakner (1960), Lewin (1959, 1960, 1977), O'Connell (1962),

    Regener (1973), and Rothenberg (1978a, b), except Budden and

    Rothen berg are from mus ic- the ore tic source s. Babbit (1965) was animportant stimulus for my own work, but otherwise the ideas presented

    in this chapter were developed largely independently of the sources

    cited above. For an exce llen t discussion of the intimate conne ctions

    between group theory and perception, see also Cassirer (1944).

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    Table 1. Group Table For C j 2

    * 0 1 2 3 4 5 6 7OO 9 10 11

    0 0 1 2 3 4 5 6 7 8 9 10 111 1 2 3 4 5 6 7 8 9 10 11 0

    2 2 3 4 5 6 7 8 9 10 11 0 1

    3 3 4 5 6 7 8 9 10 11 0 1 2

    4 4 5 6 7 8 9 10 11 0 1 2 3

    5 5 6 7 8 9 10 11 0 1 2 3 4

    6 6 7 8 9 10 11 0 1 2 3 4 5

    7 7 8 9 10 11 0 1 2 3 4 5 6

    OO

    8 9 10 11 0 1 2 3 4 5 6 7

    9 9 10 11 0 1 2 3 4 5 6 7 8

    10 10 11 0 1 2 3 4 5 6 7 8 9

    11 11 0 1 2 3 4 5 6. 7 8 9 10

    gene ral, w e will us e th e language of i nte rva ls- as- trans fer mat ions when it is

    more natural to speak in active terms, and the language of pitch classes

    when we are describing group elements in terms of passive, spatialized

    pla ces . Every sta tem ent we make may be formulated either way without

    affecting its truth value (Holland, 1972, pp.217-219).

    A group consists of a set of elements and an operation defined over

    pairs of ele me nt s in the se t. In order to cons tit ute a group, the set must

    cont ain an identi ty el em en t, one whos e operatio n has no ef fe ct on any of t he

    group el em en ts . For eac h ele men t of the set , the re must be an invers e

    element also in the set, such that the combination of any element and its

    inverse yields the identity ele ment . In addition to thes e constrain ts, a se t of

    elements can constitute a group only if the set is closed under the operation;

    that is, the result of every co mbinatio n of two group el em en ts under the

    operation must yield an element that also belongs to the set.

    It is easy to show that the set (0, 1, . . . , 11) constitutes a group under

    th e operatio n of mod 12 addition. The ident ity e le me nt is evi dent ly "0", and

    for each element g, its inverse g~* is given by

    Thus 1 and 11 are inve rs es , as are 2 and 10, 6 and so fort h. Fi nal ly, si nce

    the sum of two integers is always an integer, mod-12 reduction of the sum

    provides that it will belong to the set (0, 1, . . . , 11) and closure is thereby

    assured.

    With respect to their effect on pitch class, the octave (p8) and unison

    (pi) are indistinguishable; both behave like an identity element in that theydo not af fe ct the pitch clas s they ope rat e upon. In sym bols , for any interval

    g"f1

    =(12-g) mod 12 (D

    p8 + x = x (2)

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    The ef fect on pitch class of any interval is the same as the combined effect

    of that interval plus the octave. Th e most natural mapping of the remaining

    Il integers/intervals is to associate each interval with i ts integral number

    of semitones, so that m 2 1, M2-2, . . . , M 7 1 1 . We adopt the

    convention of thinking of intervals as upward pitch transformations (thoughnone of the results would be affected by assuming the opposite) . Note th e

    unusual congruence between mathematicians' and musicians' terminology:

    both call the relation between m2 and M 7 by the same name, inverse.

    Table 1 shows the complete group table. The "i_" notation is used

    instead of the "T" notation for the sake of visibility, but it is helpful to

    conceive that the information in the table represents combined effects of

    transformations. The star (*) is used for the rule of combination instead of

    the plus sign of Equation 2. Thus 5 * 9 = 2 represents the fact that the

    combined effect of T5 and T9 on any pitch class is the same as the

    e f f ec t of T 2 alone. To express this relation as a function on pitch class

    places, we would write T5(9) = 2:T^ transforms pitch class 9 into pitch

    class 2 . With the latter notation we can also express the e f f ec t of a

    transformation g (i.e. Tp on an entire pitch set S, writing g(S). For

    example, T (O, 3, 7) = (4, 7, 11) , and in general for a set of s i ze m, T1(SQ, Si, . . . , S m _ j ) = (So + i, Si + i, . . . , S m _i +i). Note

    that, while we use the * operator for combinations of transformations, we

    will often use the more intuitive plus sign for combining of pitch class

    places.

    From th e universal chromatic set of twelve pitch classes one can

    choose 2 * 2 - 1 = 4095 different pitch sets, including the full chromatic

    itself and the twelve trivial single-note pitch sets. Of the remaining sets,

    let us distinguish those that are transpositionally related to one ariother

    from those that are not Two pitch sets S and S' are transpositionally related

    if and only if S' = g(S) for some transformation g in C j 2 . We will cal lWts

    like S and S' members of the same family, for example, the F major scale^

    and C major scale belong to the same family, and so do the two se t s (0, 3 , 7)

    and (4, 7, 11) treated above. If members of the same scale family are not

    distinguished, this results in a reduction to 351 distinct sets, 349 without thefull chromatic and the single-note set .*

    A number of subsets of the full chromatic are special in that they

    correspond to subgroups of C ] 2 . For example, the se t (0, 2, 4, 6, 8, 10) is

    closed under mod 12 addition, contains the identity, and contains the

    inverses of all of its elements. It is therefore a group conta ined within

    * This number is different from that obtained by Forte (1964). In

    essence, Forte treats both transpositions and inversions as producing

    'equivalent1

    scales . In so doing, he is implicitly using a larger group as a

    basis for invariance, namely D ] 2 , the dihedral group of order 24.

    Using this group instead of C ] 2 leads to a number of results that are

    questionable from the present point of view. For example, underD ] 2 , the major triad (0, 4, 7) and the minor triad (0, 3, 7) are not

    distinguished.

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    C ] 2 ;a

    subgroup known as C^, th e cyc li c group of order six . The

    corresponding pitch set is commoniy known in music as the whole-tone

    scale, and its family consists of only two distinct members, (0, 2, 4, 6, 8, 10)

    and (I , 3, 5, 7, 9, 11). In a simila r fas hion , it can be shown tha t (0 , 3, 6, 9) ,(0 , 4, 8), and (0, 6) are als o subgroups of C ] 2 , termed C^, C3 and

    C 2 , res pect ive ly. Each of the corresponding pitch set famil ies already has

    a name in standard music al termin ology : the C^ set corres ponds to the

    diminished-7th chord, the C3 to the augmented triad, and C 2 to the

    interval of a trito ne. All subsets of C ] 2 associated with subgroups have

    the property that their family is incomplete, consisting of fewer than 12

    dis tinc t se ts . Thus the re are only thr ee disti nct dimin ishe d-7th chords, four

    distinct augmented tr iads, and six distinc t tritone s.

    With this background, we are ready to develop a number of basic

    prope rtie s of pitch sets more fully. The three main properties we will

    devel op, together with information specific to the structure of C ] 2 ar e

    more than enough to dem ons tra te the uniqueness of the major sca le .

    SOME DISTINGUISHING PROPERTIES OF PITCH SETS

    We should be cl ea r at ~the ou ts et t hat our cur ren t lev el of de scr ipti on is

    more general than the term 'major scale 1 - it corresponds more closely to

    that of 'diatonic scale family', where by diatonic scale we mean nothing

    mor e than the pitch se t (0, 2, 4, 5, 7, 9, II ). The dia ton ic scal e fam ily is, as

    defin ed, th e se t of 12 group -gene rated transpo sitio ns of the set above . The

    major mode of the diatonic scale is represented by this same pitch set, but

    with th e '0' el em en t individuated as a re fer enc e or 'tonic' pitch. No te ,

    however, that there is nothing in the theory of sets that provides for

    individuating a particular element of a set in this way, and we should like to

    se e how far we can ge t without reco urs e to such a mo ve . The fa ct that, of

    the seven possible tonics in this seven-note scale, fully six had seen

    extensive musical use until about the 17th century, recommends that weleave open the possibility that a tonic may be determined mainly be

    temp oral -con text ual factor s, and not by intrinsic set-s tructu ral properties.

    The 'Uniqueness' or 'Dynamic Quality* Property

    We can consider what properties of a set might be conducive to the

    em er ge nc e of a 'tonic' el em en t. It must be possible to indivi duate th e

    el em en ts of a se t by virtue of their rel ati ons with one another . Acco rding

    to Zuckerkandi (1956), the chromatic scale possesses "no dynamic relations;

    eve ry ton e is as good as eve ry other" (pp.38-39). Furthe r, Zuckerkandi

    (1971) rema rks about th e dynam ic quali ties of pitc hes as membe rs of pitch

    sets:

    The dynamic quality of a tone is part of the immediate

    se ns at io n. We hear it just as we hear pitch or to ne

    color but not under ail cir cum sta nce s. A ton e must

    belong to a musical context in order to have dynamic

    quali ty. Within a musica l co nt ex t no tone will be

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    without its proper dynami c quality . Outside th e

    musical context, however - for instance, in the

    laboratory - ton es have no dynam ic quali ties . Thus,

    the dynamic quality of a tone is its musical qualityproper. It distin guishe s the musical from the phys ical

    phenomenon (pp. 19-20 , original empha sis).

    It is quite possible that these dynamic qualities reside in higher-level - but

    no less obje ctiv ely spe cifi abl e - rela tions among membe rs of a pitch s et ,

    each of whose el em en ts has a unique se t of relati ons with the other s and

    the ref ore has the pot ent ial ity for a unique musical 'role' or 'dynamic quality' .

    More formally, let a pitch set (scale) of cardinality (size) m be

    represented by S = (SQ, Sj 5 . . . , s m _j) with S1 a distinct e lem ent

    (pitch class) of Ci 2 * In acc ord anc e with what we have said earli er, th e

    initial ele men t SQ may be any member of the s et , but it will be c onve nien t

    if we stipulate that remaining set members will be written in 'ascending

    form', such that each si is written in standard numerical order, mod 12.

    For exa mpl e (2, 4, 5, 7, 9, I I, 0) would be a per fec tly ac ce pta bl e way towrite a diaton ic scal e. For each e le me nt of the set, define a ve cto r of

    relations V(sp = Vj = (VJQ, VJJ, . . . , Vi ( m _ j ) ) such that

    whe re the subscrip t "i+j" is to be take n mod m. In th e diato nic s cal e a bov e,

    V 2 = V(5) = (5- 5, 7 -5 , 9- 5, . . . , 4-5) = (0, 2, 4, 6, 7, 9, 11). Now w e can

    say that a set satisfies Uniqueness if and only if

    V1 = V1. i = i' (4)

    That is, the vector of relations associated with each set element must be

    distinct .

    The chromatic scale and the whole-tone scale - in fact, all of the setsassoc iated with subgroups of C ] 2 - fail to satisfy Uniqueness, since all the

    V] are the same for every set el em en t. A quick chec k will verif y that

    Vj for ever y mem ber of a whole to ne sc al e is just (0, 2, 4, 6, 8, 10). Other

    sets that fail Uniqueness in less drastic ways are (0, 1, 4, 5, 8, 9), (0, 1, 3, 4,

    6, 7, 9, 10) and (0, 2, 3, 5, 6, 8, 9, 11). In all cas es it is the very sy mm et ry

    and apparent el eg an ce of the set tha t is its undoing with resp ec t to

    Uniqueness . On the other hand, the uneven looking diaton ic scal e does

    sati sfy Unique ness. It is not diffi cult to show that the Uniqueness property

    entai ls and is ent ail ed by co mpl ete nes s of a set's fami ly.

    I should like to cl aim th at Un ique nes s is not just an abs tra ct pr oper ty of

    a pitch set , but one with so me perceptu al con ten t. By hypothe sis, a melod y

    based on a scale satisfying Uniqueness should be easier for a perceiver to

    deal with, because the notes of the melody are individuated not only by their

    particular frequency locations, but by their interrelations with one another.However, it turns out that many more sets satisfy Uniqueness than fail it, so

    more than this property is nece ssa ry to allow thorough differ entiat ion

    among potential scales.

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    Scalestep-Semitone Coherence

    All scales/sets partake of distance relationships based on the semitone

    unit of C|2 These se mi ton e dist ances are in fac t the cons tit uent s of the

    ve cto r of rel atio ns Vj defined in the precedi ng subse ctio n. Any set short

    of the full chr oma tic als o give s rise to a new l eve l of dista nce-r eckon ing,

    which we will cal l th e scal est ep. The number of semi ton es containe d in a

    distance of one scalestep is given by the V jj element of the Vj

    relations vector, and can be different at different points in the scale: a

    diatonic scale has some scalesteps that are 2 semitones wide, and others

    that ar e 1 se mit one wid e. More generall y, VJJ des cri bes the number of

    sem iton es contai ned in a distance of j sca les teps from the given scale

    me mbe r Sj. For ex am pl e, in the diato nic s et (2 , 4, 5, 7, 9, 11, 0), S 2 = 5,

    V 2 = (0, 2, 4, 6, 7, 9, 11), and we can read fro m th e vec to r tha t a dis tan ce

    of (say) three scalesteps from scale element S 2 contains 6 semitones

    ( v 2 3 = 6).

    In general, greater numbers of scalesteps correspond to greater

    numbers of semitones, but there is nothing that forces this to holdev ery whe re in the sca le . For exa mpl e, in the se t (0, 2, 4, 9) a dis tan ce of

    tw o sca le st eps from th e ele me nt "0" corr eson ds to four sem ito nes , and a

    dis tan ce of one sc al es te p from the el em en t "4" corresponds to five

    se mi ton es . Thus a larger number of sc al es tep s is ass oci ate d here with a

    sma lle r number of sem ito ne s. We will say tha t, for sc ale s like the se, th e

    relation between scalesteps and semitones is not coherent.

    Symbolically, we can say that a set satisfies coherence if for any pair

    of scal e el eme nts Sj and SJ>

    j < k ~ VJJ < vj. k (5)

    where j and k are scal es tep -co unt ing indic es, and tak e on values from 0 to

    m-1 incl usiv e. Equation 5 st ate s that larger numbers of sc ale ste ps are

    alway s ass oci ate d with larger numbers of se mi ton es in a coher ent sc ale . Forsets satisfying this equation, scalesteps are a monotone increasing function

    of se mi ton es . No ti ce that the ent ail men t in Equation 5 applies in one

    dire ctio n only. The con ve rs e would require tha t all rep res ent ati ves of j

    scalesteps contain an identical number of semitones, and this, as we saw in

    the case of the whole-tone scale, would lead to a failure of Uniqueness.

    I will not dem ons tra te it here, but it can be shown than any sc al e of odd

    cardinality containing at r it on e will fail to sati sfy Equation 5 stric tly. The

    failu re, howe ver , wi ll be loc aliz ed to just the tr ito ne interva l (VJ; = 6) and

    can be remedied by relaxing the strict inequality on the right of Equation 5

    for that interval:

    j < k => Vjj < Vjf . for v^j ? 6

    j < k = Vjj ^ V]I^ for Vj j = 6 (5a)

    Perhaps the perceptual content of the Coherence property is obvious; if

    not, a few rema rks are in order. Sem iton es are (approxi mately) a linear

    function of log frequency, and there is evidence, both (a) in the everyday

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    fac t of perce ptual invariance of melo dies under con sta nt log freque ncy

    shi fts, and (b) in the emp irical liter ature (A ttn eav e and Olson, 1971), th at

    we do perc eive musica l pitch in ter ms of log frequ ency. No w, sc ale st eps

    cannot in turn be a linear function of semitones if the scale is to satisfy

    Uni que nes s (rec all the (0, 2, 4, . . . , 10) whole tone sc al e) . But unle ssscalesteps are at least a monotone increasing function of semitones,

    perception mediated by log frequency will fail to yield consistent results in

    ter ms of sc ale ste ps. As a con seq uenc e, perce ptio n would be 'stalled* at the

    se mi to ne level . Indeed , just this sort of thing happens in the c as e of a

    tri ton e. When we hear an inter val four sem ito nes wi de, in or out of co nte xt ,

    we can safe ly identify that inter val as a major 3rd, where its "thirdness" is a

    sca les tep- lev el property. But when we hear an interval six sem ito nes wide,

    we can oft en do no bet ter than to ident ify it by the sc ale -neu tral ter m

    'tritone', since it may function either as an augmented 4th or a diminished

    5th in a diatonic sc al e co nte xt. If all inter vals we re like the t rit one ,

    scalestep-level perception could never occur outside the specific context of

    a spe cif ic p iec e, and within this cont ext only if the piec e weren't changing

    key s to o rapidly. Since learning to rec ogn ize inte rval s may require at leas t

    a temporary isolation of the interval from context, it is hard to see howlearning to perce ive scal est ep-l eve l qualities could occur unless

    scalestep-semitone coherence were satisf ied.

    Cohe renc e may at first sight appear to be a rather superfici al property

    of a sc al e. But it imposes powerful cons trai nts on sc al es so that fe w s ets

    conform to it. Of the 66 esse ntiall y different 5-n ote scale s (i.e. scal e

    fam ili es) , only four satis fy Coh ere nce , one of which is the familiar (0, 2, 4,

    7, 9) pentatonic sc ale . Of the 80 essenti ally different 6-note sca les , only

    two are coher ent , one of which is the whole ton e sca le , and both of which

    fail Uniquen ess. And of the 66 ess enti ally differ ent 7-not e sc al es , only one,

    the diatonic scale, satisfies Coherence.

    Simplicity of Scale Family

    Our first two properties, Uniqueness and Coherence, were concerned

    with features of a scale determined by relational properties of its individual

    ele men ts. Uniqueness is determi ned by relations of sca le el eme nts to one

    another, Coherence by relations of scale elements to the embedding system

    of sem it on e dista nce s. Our third and most powerful property, which we will

    call Simplicity, concerns relations between members of a scale's family.

    Since all family members of a given set have the same cardinality and the

    sa me ens emb le of Vjj rela tions , the only thing that distingui shes the se

    transpositionally related set s is the pitch class es they contain. Let us

    therefore begin by defining the overlap of two members of a scale family as

    the cardinality of their intersection:

    Ov(S,g(S)) = I S H g(S) | (6)

    whe re g(S) = g( s 0 , S 1 , . . . , s m _ j ) = (sQ +S

    s

    l + 8> * * >s

    m - i+

    g) in acc ord ance with the definit ion of a sca le fa mily . Obviously Ov is asy mm et ri c relation. Since S may be any membe r of th e sca le family wit hout

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    Table 2. Matrix of Overlap Bet wee n Pairs of Pen ta ton ic

    Scale Family Members

    Family Member 0 1 2 3 4 5 6 7 OO

    9 10 110 5 0 3 2 1 4 0 4 1 2 3 0

    1 0 5 0 3 2 1 4 0 4 1 2 3

    2 3 0 5 0 3 2 1 4 0 4 1 2

    3 2 3 0 5 0 3 2 1 4 0 4 1

    4 1 2 3 0 5 0 3 2 1 4 0 4.

    5 4 1 2 3 0 5 0 3 2 1 4 0

    6 0 4 1 2 3 0 5 0 3 2 1 4

    7 4 0 4 1 2 3 0 5 0 3 2 1

    8 1 4 0 4 1 2 3 0 5 0 3 2

    9 2 1 4 0 4 1 2 3 0 5 0 3

    10 3 2 1 4 0 4 1 2 3 0 5l

    0

    11 0 3 2 1 4 0 4 1 2 3 0 5

    affecting overlap, we may write Ov^g] , the overlap asso ciate d with

    tran sfor mati on g for scale S. And when it is cle ar what sc al e is being

    dis cus sed , we will simpl ify the nota tion stil l further by dropping the "S" and

    writt ing Ov[g]. Since S is the same as g""*(g(S)), it can easily be shown by

    substitution in Equation 6 and by the symmetry of Ov that Ov[g] = Ov[g~*]

    for all g. For ex amp le , any tw o diatonic sc ale s a p5 apart (g = Tj) share

    six of seven notes, and so do any two diatonic scales a p4 apart (g = T 5 =

    ( T 7 ) -1 ) .

    The basic idea behind the Sim plicity prope rty is that spatia l i nformat ion

    can be deduced from patterns of overlap among scale family members,

    ! he r e are a number of alt ern ati ve ways to proc eed from this notion. Onewould be to trea t values of Ov[g] as simi larity va lue s be twe en pairs of sc ale

    fam ily m emb ers rela ted by g. Given a matrix of pairwise simi lari ty valu es

    such as that shown for the (0, 2, 4, 7, 9) pentatonic scale in Table 2, an

    implied spatial configuration of elements (scales) can be recovered by

    multi dimen sion al sca ling meth ods (e.g. Shepard 1962a , b; Kruskai, 1964a, b).

    Maximum Simplicity of scale family would correspond to minimum

    dime nsio nali ty of the spac e of sc ale s. A diffe rent m eth od, though one that

    leads to quite similar results, would be to define further predicates in terms

    of set o pera tion s, such as was done for Ov, and to dev elop th e implie d

    spa tial rela tion s direc tly . Goodman (1966) is a good exa mpl e of this

    appro ach. It is the latte r path tha t we shall take her e.

    A first attempt along these lines would be to observe that the closest

    possible relation among scale family members obtains when they share all

    el em en ts but one. Let us call this rela tion adjac ency and sy mbo liz e it "A".Formally,

    A(R, S) ^ Ov( R, S) = m-1 (7)

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    where m is, as usual, the number of elements in the scale under

    consideration.

    The proble m wit h "A" is that it is too stro ng a property t o be of much

    use for mos t sca les . Scal es having any family me mbe rs R, S sati sfyi ng A(R,S) are few ana far bet we en . For ex am pl e, it can be shown eit her by

    exhaustive enumeration (Forte, 1964) or by formal proof that there are only

    three size -7 sca les (out of 66) containing scal e family members that satis fy

    "A": (0, 1, 2, 4, 6, 8, 10), (0, 1, 2, 3, 4, 5, 6) , and (0 , 2, 4 , 5, 7, 9, II ). * The

    group ele me nts (transformations) ass ociate d with adjacency in each cas e are

    (T2, T^, T 6 , T 8 , T 1 0 ) , (T] , T n ) , (T5, T7), respectively.

    In the case of (0, 1, 2, 4, 6, 8, 10), the transformation T^(S) yields (4, 5, 6,

    8, 10, 0, 2) = (0, 2, 4, 6, 8, 10), and Ov (S, T^ (S)) = m- 1. In ge ner al it c an

    be seen that all of the transformations (T 2 , T^, . . . , TJQ) carr y th e

    subset (0, 2, 4, 6, 8, 10) into itself and the 'odd' element is the only

    nonoverlapping one . Similarly, inspec tion reveal s that adding or subtracting

    1 - transfo rming by T] or T] j res pec tiv el y - for each el em en t in {0, 1,

    2, 3, 4, 5, 6), yield s a se t containing 6 not es in com mon wi th the original

    se t. Nei the r of thes e first tw o scale s is coh ere nt, and neithe r has se en anymusical usa ge in any cultur e, as far as I am aware . The third sc ale , (0, 2, 4,

    5, 7, 9, 11), is of course the diatonic scale.

    A more general ly useful pred icat e can be defin ed over triple s of sca le

    family mem ber s. In th e spirit of Goodman's (1966) 'betwixt' rela tion amo ng

    manors of qualia, le t us define a bet wee nne ss r elatio nship among three sca le

    family members, X, Y, Z as follows:

    X / Y / Z = [ X Pl Z) C (X Pl Y) ] A [ (X Pl Z) C (Y H Z)] (8)

    wher e "X /Y/Z " is to be read as "Y is be tw ee n X and Z". The ab ove

    definiti on is sta ted in a form close ly analogo us to Goodman's ' betwi xtnes s'.

    From the definition, it follows that (a) no scale is between itself and any

    other sc al e ( ~ X / X / Y ), (b) betw eenn ess is symm etri cal with resp ect to its

    first and third arguments (if X/Y/Z, then Z/Y/X), and (c) other than thesituation just described, betweenness is asymmetrical: for any three scales,

    at most on e is be tw ee n the other tw o (if X/ Y/ Z, then Y/ X/ Z an d~-

    X/Z/Y) .

    The character of betweenness may be more intuitively appreciated

    from an immediate consequence of the above definition:

    X/ Y/ Z = (X PlZ) C Y (8a)

    In word s, if Y is bet we en X and Z, the n all el em en ts share d by X and Z are

    con tai ned in Y. Figur e 1 displ ays Venn diag rams of se ts th at do and do not

    satis fy the definiti on of bet wee nne ss . Loos ely, only se ts X, Y, Z sati sfyin g

    X/Y/Z can be represented 'on a line1

    with Y repr ese nted ' betwee n' X and Z.

    The comp le te proof would take us too far afie ld here; int ere ste d re aders

    should write to the author.

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    Most scale family members contain many triples in a betweenness

    rela tion . Considering the three se ts just exa min ed in the discuss ion of

    adjacency, we find the following:

    (I) In (0, 1, 2, 3, 4, 5, 6), it is eas y to show th at T n / T i / T 2 . In

    addition, T0/T1/T3, T Q / T i / T ^ , and in general Tg/Tg +j / T g +j , so

    long as ic) and j

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    Table 3. Besi denes s Rel atio ns for Fami ly Members of

    Harmonic Minor and Diatonic Scales

    Family Member Harmonic Minor Diatonic

    () 3 9 5,7

    1 4,10 6,8

    2 5,11 7,9

    3 0,6 8,10

    4 1,7 9,11

    5 2,8 0,10

    6 3,9 1,11

    7 4,10 0,2

    8 5,11 1,3

    9 0,6 2,4

    10 1,7 3,5

    11 2,8 4,6

    Pursuing the spatial analogy, it is proposed that the simp les t sca le

    families have a large number of betweenness relations and a small number

    of besideness relations.

    The simp lest kinds of besidene ss rela tion s occu r when each sc ale family

    memb er is besi de exac tly t wo others (as points on a line, for examp le) . This

    situation can occur in two rather different ways, exemplified by the

    har mon ic minor sc al e and th e major sca le in Table 3. In the form er, TQ is

    beside T3, T3 beside T 6 , Tg besi de T9, and T9 bes ide TQ , but

    there are no besideness relations connecting these four scales with the other

    eight family members. Scales asso ciate d with (T j , T^, T7, TJ Q)

    and ( T 2 , T

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    Fig . 2. Map of bes id ene ss rel ati ons for th e sc al e (0, 1, 2, 3, 4, 5, 6). (b)

    Map of bes ide nes s rela tions for the diat onic sca le . The figure

    also depic ts isomorphic repres entatio ns of the group C j 2 referred

    to in the text as "the cycle of semitones" (a) and "the cycle of

    fifths" (b).

    GROUP GENE RATORS AN D THE ISOMORPHISMS OF C ] 2

    Semitone Space and Fifths Space

    The subgroups of C ] 2 have been pres ente d in an earl ier sec tio n. The

    large st subgroup of C ] 2 is C 6 , which contains the ele men ts ( 0 , 2, 4 , 6, 8,

    1 0 ) . Transformationally, the subgroup would be written {TQ 5 T 2 , TZ^ T 6 , Tg,

    T [ 0 )=

    ^2t T4, T 6 , Tg, T]Q) Not e that ever y ele me nt in the subgroup is

    expressible as a power of T 2 . Thus T 2 * T 2 = ( T 2 )2

    = T 4, T 2 * T 2 * T 2 = ( T 2 P -

    T 6 , and so fort h. We say that the group el em en t T 2 is of period six, since

    ( T 2 ) ^ = 1, and that T 2 thereby generates the subgroup C 6 , The sa me is true

    of T]QJt n e

    inverse of T 2 , which generates the same six elements in reverse

    order.

    Similarly, the subgroup C4 consists of (Tn, T3, T 6 , T9) = (I, T3, T%,

    T%) = (I, T % T%, T9). In th is case we would say t ha t T3 and T9 are of

    peri od four, and ea ch ge ne ra te s C4.. In like fas hion , T^fTg) is of per iod

    three, generating the C3 subgroup, and T 6 is of period two , generat ing C 2 .The only el em en ts tha t do not appear in any of th e subgroups of C ] 2

    are T] and Tj (and their inve rse s). Both of thes e el em en ts are in fa ct of

    period Yl- and gen erat e all of C ] 2 . If a subgroup of C ] 2 contained T7, it

    would also have to con tai n all the pow ers of T7 in order to satis fy the

    clos ure property for groups. Since the pow ers of T7 exha ust C ] 2 ,

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    33 4 CH APTERXVlI

    no proper subgroup (analogous to 'proper subset ') of C ] 2c a n

    contain T7.

    On the other hand, elements that are contained in proper subgroups of

    C j 2 cannot 'reach' all of the ele me nts in C j 2 ;or

    >ly elements that

    gener ate all of C ] 2 can do this. Thus, we cannot exp res s the relatio n

    between every pair of group elements in terms of, say T 2 or T3. In

    terms of T 2 , the r elation b et we en group el em en ts "4" (T^) and "5"

    (T5) is ind ete rm ina te. In es se nc e, only Tj and T7 can ser ve as acom ple te basis for a spa ce of sc ale rel ation s as discuss ed in the previous

    se cti on. Graphical repr ese ntati ons of the stru ctur es genera ted by Tj and

    T7 - independently of any considerations of scales - are precisely those

    given in Figure 2. The said stru cture s are both direc tly implied by C j 2 :

    indeed, they are C ] 2 . We will call Figur e 2a by th e nam e "se mit one

    spac e" or "the cy cl e of semito nes" , Figure 2b will be referred to as "fifths

    spac e" or "the cy cl e of fifths."

    Unlike the versions of C 6 generated by T 2 and T JQ , Figure 2a

    and 2b are not simply rela ted by mirror rev ers al. But the y are no net hel es s

    fully isomorphic representations of C ] 2 - What this mea ns is tha t th e

    mapping A - B: i(mod 12) 7 i (mod 12) is o ne -t o- on e and

    structure -preserving . Any true sta tem ent about ele men ts in sys tem A is

    true of their ima ges in syst em B. What is diffe rent a bout the two sy st em s

    lie s in the proximity relations among ele me nt s. In sy st em A (Figure 2a),el em en ts "2" and "3" are clo se tog eth er, "2" and "9" far apart, whil e th e

    rev ers e is the ca se in System B (Figure 2b). But both of thes e isomo rphism s

    are on an equal footing in the sense that neither is logically prior to the

    other.

    It may be argued, however, that the two spaces in Figure 2 are not on

    an equal percept ual footing. With or without music al co nte xt, it might be

    said that the normative sense of tioseness' of two pitches corresponds to

    that depict ed in Figure 2a, not 2b. Additional const raint s of some sort

    would be necessary to render the proximity structure of fifths space (Figure

    2b) perceptually available information.

    Furnishing such constr aint would appear to be exact ly the func tion

    serv ed by a diaton ic sc ale . The bes ide nes s and adjac ency r elat ions held by

    members of the diatonic scal e line up precis ely with the T7 -gener ated

    isomorphism of C i 2 call ed fifths sp ace . If we exami ne the stru ctur e of a

    diatonic scale with respect to fifths space, we can see just how this is so.

    As Figure 3 shows, a diatonic scale corresponds to a connected region of

    fifth s spa ce. Transfor ming the sc ale by a per fec t fifth correspon ds to the

    minimal rotation of the region, such that m-1 elements are common to any

    two adjacent sc al es . We have shown how unusual this property is. The set

    (0, 1, 2, 3, 4, 5, 6) has similar properties, but with respect to the

    T j-generated space.

    From Figure 3, it can perhaps be appreciated that scales of any

    cardin ality repr esen table as region s in se mi to ne spac e or fifths sp ac e, and

    only those scales, have besideness relations fully determined by Tj or

    T7, resp ect ive ly. Thus the sc ale s (0, 1, 2, 3, 4, 5), (0, 1, 2, 3, 4), (0, 2, 4,

    5, 7, 9), and (0, 2, 4, 7, 9) are all sim ilar in th at r es pe ct . If we re st ri ct our

    view to scales exhibiting Coherence, however, we are left with only two

    scales: (0, 2, 4, 5, 7, 9, 11), and (0, 2, 4, 7, 9). The forme r is the diat onic

    scale, and the latter is none other than the well-known and culturally

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    8

    6

    Fig. 3 The diat oni c sca le repre sente d as a conn ecti on region of fifths

    spa ce. Transforming the sc al e by T 2 or T5 leads to a scale

    with six (out of seven) overlapping elements.

    widesp read pe nta toni c sc al e. Our inquiry into the struc ture of scale s and of

    the embedding group C j 2 has led us to the two most nearly universal

    sc ale s in the history of music. But our desc ripti on of C j 2 isn o t

    comple t e

    until we consider one more isomorphic representation.

    Thirds Space

    Both of our previous isomorphi sms of C j 2 have been 'one-dimensional'

    in the sense that a single generator - like a basis vector in the theory of

    vector spac es - spans ail of the ele men ts under consi deration.

    Higher-dimensional vector spaces can be realized by forming so-called

    Cartesian products, and in an analogous fashion there exist product groups,

    where each element is an n-tuple, each component of which is a member ofso me group. For exa mpl e, the product group C 2 x C3 consists of

    2-tupies of the form (x,y,) where x is an element of C 2 and y is an

    el em en t of C 3. We use the int eg er s mod n as a mod el for C n .

    Combination of 2-tuples proceeds componentwise, so in C 2 x C3

    ( X J , y j ) * ( x 2 , y 2 ) = (x ]+ x 2 mod 2, yj+y 2 mod 3) (10)

    It is easy to verify that there are six such elements and that the

    requirements for a group are satisfied.

    In so me c as es di rec t product groups can be isomorphic to simple r groups

    whose ele me nt s are 1-tuples. It turns out that C ] 2 is isomorphic to the

    dir ec t pro duct of tw o of its subgroup s, C3XC4. El em en ts of C3XC4

    are 2-tuples consisting of an element of the set (0, 1, 2) followed by an

    el em en t of the se t (0, I, 2, 3). The rule of comb ina tio n for such (x, y) pairsis analogous to C 2 xC3 Thus C3XC4,

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    It is not hard to see that there are 12 elements in this group, but to

    dem ons tra te an isomorphism with the group C j 2w e

    rnust show a one-one

    structure-preserving mapping of the twelve 2-tuples to the elements in the

    se t (0 , 1, . . . , 11). He re it is:

    C^xC^ ^12

    (0,0) 0

    (0,1) 3

    (0,2) 6

    (0,3) 9(1.0) 4

    (1.1) 7(1.2) 10

    (1.3) 1

    (2.0) 8

    (2.1) 11(2.2) 2

    (2.3) 5

    In gen era l (a,b) - -[ 4a + 3b ] ] 2 (where the subscript "12" mer ely indi cate s

    th at the right side is to be take n mod 12). The reader may readily ver ify

    tha t the 2-tuples on the lef t play analogous structural roles to their C i 2im age s. For exa mpl e, (1,1) is a genera tor of C3 x C^; (1,1 )*(! ,!)* (1,1 )

    = ( 0,3 ), 7*7*7 = 9 and 9 is the ima ge of (0,3) (se e above ).

    Graphically, C3 x C4 would require two orthogonal axes, one

    ge ne ra te d by major thirds (C3: (0, 4, 8) [(0, 0), (1,0), (0,2)]) and th e

    oth er minor thirds (C^: (0, 3, 6, 9) [(0, 0), (1,0 ), (0,2), (0,3)]. A doubly

    cyclic structure such as this one is representabie on the surface of a torus in

    th re e dime nsi ons. In Figure 4 we cut and unroll th e torus and lay s eve ral

    torii nex t to one another to provide a more rea dable planar equival ent of the

    repres entatio n. To faci lita te comparison with the other isomorphisms, the

    points of C3XC4 have been labelled with their C j 2 images rather than

    th e 2-tu pl es . The points are lab elle d with "i" rathe r than "T", mainl y

    eliminate to redundancy, but also to remind that the figure depicts a

    str uctu re both of plac es and of trans forma tions be twe en place s. We will

    call this final isomorphism "thirds space."

    In fifths s pace, the simplest , maximally co mpac t 'shapes' were intervals

    of the fif th itse lf and pitch se ts consist ing of chains of fift hs. In thirds

    sp ac e, the situat ion is simi lar with rega rd to chai ns of major and minor

    thirds. But beca use thirds spac e is two- dime nsio nal, there are also simple ,

    co mp ac t shapes to be found that are not merely chain s. The simpl est of

    these is a unit right triangle with a major third and a minor third

    con sti tut ing the two perpendicular si des. If both sides are trace d out in a

    positive-going direction, there are two possible kinds of triangles,

    rep res enta bie as (0, 4, 7) and (0, 3, 7). Thes e two pitch se ts are musicall ywell-known as the major triad and the minor triad, respectively.

    From here we can ask about the ef fe ct o f 'chaining' tog eth er thes e

    higher-order structures, just as we did for the lower-order generating

    inte rval s. The result can be see n outlined in Figure 4. When we s we ep out a

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    0 4 8 0 4 8 0 4

    9 1 5 9 1 5 9 1

    6 IO 2 6 10 .2 6 10

    3 7 11 3 . 7/ / 3 7

    / \ /0 4 8 ,04 8 0 4

    -/ x

    y

    9 1 3 - - 9 1 5 9 1

    6 10 2 6 10 2 6 10

    3 7 11 3 7 I I 3 7

    0 4 8 0 4 8 0 4

    Fig. 4 Thirds spa ce repre sent ed on a plane. The horiz ontal ax is is in

    major-3rd (4 mod 12) units, the vertical axis in minor-3rd (3 mod

    12) units. This spa ce is isomor phic to se mit one and fifths spa ce.

    The parall elogr am repr esen ts the region of a diato nic, sc al e. Thetriangular constituents of the parallelogram correspond to the

    regio ns of major and minor triads. The bol dfac e numbers contain

    the basic module of thirds space; adjoining modules are duplications.

    region of thirds space by adjoining positive-going triangles together, we

    reach the point from which we began after obtaining six triangles

    int erc onn ect ing seven pitch plac es. And the patte rn of tho se sev en pitch

    pla ces , forming a con vex , spac e-fil ling regio n of thirds spa ce, is the pattern

    of a diatonic s ca le. *

    Other scales do not fare so well in attaining a simple structure vis-a-vis

    thirds spa ce. The pen tato nic scal e, on an equal footing with the diat onic in

    fifth s s pac e, is not a co mpa ct region in thirds sp ac e. It con tain s five note s,

    but only two triad s. Other fift h-ge ner ated ch ains suffer simila r de fe ct s inthirds spac e. Our se mi ton e- spa ce dia toni c anal og (0, 1, 2, 3, 4, 5, 6), can be

    found in thirds spa ce if we alter the 'handedness' of the s pac e and co nst ruc t

    tri ang les with one posi ti ve- and one neg ati ve- goi ng side: adjoining six such

    triangles in the manner of our former diatonic scale construction yields (0,

    1, 2, . . . , 6) or a member of its fami ly. The con sti tue nt trian gles

    th em se lv es are set s of the form (0, 1, 4) and (0, 3, 4). Like the parent sc ale ,

    these pitch sets have seen virtually no musical use.

    With the addition of thirds space we exhaust the structure of C] 2 J the

    cycle of semitones and the cycle of fifths are the only single-generator

    sp ac es poss ible , and our C 3 x thirds sp ac e is th e only produc t group

    The reader who wonders where the seventh triad had gone may find it in

    the 'seam* form ed by joining the (11 , 2) side of th e uppe rmo st tri ang lewith the (2, 5) side of the lower mos t tri angle . This triad, the "viin" in

    major, is itself neither a major nor a minor triad, but a diminished triad

    belonging to the family (0, 3, 6) - a size-3 chain along the minor-3rds

    axis.

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    isomorphic to C j 2 . In

    particular, C 2 x Cg, a group with twelve

    ele men ts, is nor isomorphic to Cj 2 - F r o n e

    thing, there are no elements

    in this group having period twelve, and therefore there are no group

    el em en ts to correspond to sem ito nes or fifths . In addition, there are thr ee

    C 2 x C 6 elements of period two, so there are too many elements thatpoten tiall y correspond to a trit one. Higher-ord er product groups such as

    C 2 x C 2 x C3 also fail to be isomorphic to C j 2 .

    The reader is invited to consider the isomorphic semitone, thirds and

    fifths spaces as bases for melodic, harmonic, and key relations in music,

    res pec tiv ely . We shall have more to say about thi s apparent corr espo ndenc e

    in the closing sec tio n of the chap ter. For now, note that the struc tural

    properties of C j 2 and of its scale s have all been develop ed without

    rec ours e to the con cep t of a frequenc y ratio. Indeed, all the propert ies of

    our pitch system that have been the subject of the last two sections are

    logically independent of ratio concerns, including the special nature of the

    diat onic sc ale , the major and minor triads , and the cyc le of fifths.

    The present theoretical framework provides a role for several kinds of

    experiments in musical perception, many of which have yet to be

    perfor med. In the next sectio n I will review the results of sev era lexperiments that are congenial to the present framework, in an attempt to

    sketch out the kind of experimental tradition that is in the spirit developed

    here.

    EMPIRICAL RESEARCH

    From the psychophysical and ratio-based traditions of pitch perception

    research, we have inherited studies on the perception of such things as

    beat s, harmo nics and combinati on ton es . On the whole, it must be

    concluded that human perceptual sensitivity to these phenomena is rather

    low . Consider: (a) Many the ore tic all y possible comb inat ion tones can not be

    heard at all, and tho se that can are audi ble only under r es tr ic te d

    fre quen cy-a mpl itud e cond itio ns (Plomp, 1976). (b) It has long been knownthat detecting harmonics requires a mode of listening anathemic to the

    perc epti on of a running musical co nte xt (He lmho ltz , 1885), and eve n under

    the best conditions humans seem to possess no significant sensitivity to

    harm onics beyond the sixth or sev enth (P lomp, 1964). (c) The pre se nce of

    beats, besides being difficult to distinguish from vibrato, is apparently not

    even functionally related to the perception of 'in-tuneness', contrary to

    many commonly held beliefs (Corso, 1954).

    By way of contrast, musicians and nonmusicians alike show ample

    evi den ce of perceptual sensitivi ty to pitch set structure . The many studies

    that show so me kind of adva ntag e - or any kind of diffe rent ial perfo rmanc e

    - under 'tonal1

    versu s 'atonal' pitch cont ex ts ail exem plif y this basic

    findi ng.* For exa mpl e, the ability to de te ct a changed note in a mel odic

    sequence is greater when that sequence is tonal than when it is atonal

    In the cases to be reported, the 'tonal' contexts corresponded to pitch

    sets representabie as regions in fifths space, usually the size-7 region

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    (Fr ance s, 1958). In gener al, mem ory for the pitch of a given to ne is bett er

    when th e la tte r is embedded in a tonal co nt ex t ( Dewar, Cuddy dc Mewhort,

    1977; Dew ar, 1977; Krumhansi, 1979). The se findings appear to hold up

    equally we ll regardl ess of the degre e of musica lity of the lis tene r. In so me

    cases, it is actually the less musically inclined listener that shows the largereffects (Dewar et al., 1977).

    The important but often-missed implication of these findings and

    observations is not that they show tonal music to be 'better' or even 'more

    famili ar', but that th ey imply a direct sens itiv ity on the part of the

    percei ver to differenc es in the structure of pitch set s. To atte mpt to

    expl ain aw ay such findings in ter ms of "familiarity" be gs the whole questi on

    of what the deter min ants of perc eived fam ilia rity might be. So when Lundin

    (1953) says, "these responses are part of our cultural attitudes, and even

    though one does not have specific musical training, this does not mean he

    has not acquired the typically cultural reactions", (p. 146) he does not appear

    to real ize tha t his "typical cultural reac tion s" presuppose an ability to

    perfor m a perc eptu al discrimina tion. The com mon property most likely to

    serve as a basis for this discrimination is the structure of the pitch set.

    Perception of Dynamic Qualities

    In this sec ti on w e will consider e xpe rim ent s on the 'dynamic qualit ies'

    (Zuckerkandi, 1956) of the degress (notes) of diatonic scales.

    The use of a diatonic scale in a musical context usually involves the

    se le ct io n of a ton ic or tonal cen ter , which then ac ts as a point of ref ere nce

    or perce ptual origin for other deg ree s of the sca le. If the pitch "0" is

    se le ct ed a s a ton ic for the set (0, 2, 4, 5, 7, 9, II ), we have the major mode

    of th e diat onic s ca le . _ We symb oliz e the corresp onding scal e degre es by

    el em ent s of the set ( I , 2, 3, 4, 5, 6, 7).

    Given the apparent importance of establishing a tonic in a musical

    context, the first question one might ask about the dynamic qualities of

    sc al e degr ees is: how are they perceptual ly related to the toni c de gree?

    Under a strict psychophysical view, we would expect the answer to be that

    perceptual relatedness is determined by differences in tone height, perhaps

    as meas ured by log freq uency . Taking the notion of oc ta ve equiv ale nce into

    account might lead one to predict that differences in tone chroma (see

    Re ve sz , 1954; Bac hem , 1950; Shepard, 1964) would control perc eptu al

    re lat ed nes s. By this sch em e, pitch cl ass es "5" (degree 4) and "7" (degree 5)

    might be equal ly rela ted to "0" (degr ee 1) sin ce the pitch clas s diff ere nce or

    chroma difference is the same in these two cases.

    It turns out that neither of the above notions is sufficient to account

    for the structure of affinities displayed between the tonic and other

    pit che s. Krumhansi and Shepard (1979) put th e question to empir ical te st by

    known as the diatonic sc ale . 'Atonal' co nte xts did not neces sar ily

    cons ist of larger pitch se ts , but pitch set s more thinly spread over fifths

    space.

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    playing the first seven notes of an ascending or descending major scale,

    fol lowe d by a variabl e eighth ton e. The lat ter could be any one of 13

    pitches contained in the octave spanned by the first tone and the tone an

    oc tave abov e (for ascending scales) or below (for descending sca les ). The

    task for subject s was to judge how well the final note compl et ed the

    sequen ce in each ca se.

    Of the 24 subjects in the experiment, only the six subjects with thesmallest amount of musical experience showed anything like an effect of

    tone height . Every subject gav e the highest ratings to the tonic and its

    oc ta ve neighbor. For all subj ects , tones belonging to the sca le r ece ive d

    sign ificantl y higher ratings than ton es outside the scal e. And, for the more

    musical subjects anyway , the pattern of responses could be nicely model ed

    by a combination of distances on the cycle of semitones and the cycle of

    fift hs. That is, the farther away a given final note was from the ton ic , as

    measured by distance in semitone space and fifths space, the less well it was

    judged to co mplet e the sc al e and thus subs ti tu te for the toni c (se e also

    Shepard, in press). There are also a hint in the data tha t third-re iatednes s

    played a significant role in determining judgments, but the authors did not

    specifically examine the data from this point of view.

    In a rela ted study, Krumhansi (1979) looked at judgments of per cei ved

    simil arit y among all possible pairs of pitch class es. Before eac h pair ofpit che s to be judged, subjec ts heard the same diaton ic contex t-i nduc ing

    scale or triad (the triad used was the tonic triad, t - 3 - 5 - 8 ) . The results for

    pitch pairs containing the tonic as a member looked very similar to the

    Krumhansi and Shepard (1979) study. The overal l matrix of rated

    similarities was subjected to multidimensional scaling, with a resulting

    conical configuration resembling that given in Figure 5. Points in the figure

    are label ed with pitch cla ss numbers using "0" as the ton ic ("12" is the pitch

    an oc ta ve above "0"). This configurat ion shows a cle ar role for thi rd- and

    fifth-r elated ness in determining perceived similarity. The pitch clas ses

    most closely related to one another are the members of the tonic triad, (0,

    4, 7), sc al e degrees, 1 , 3 and 3 . At the next leve l in the configuration are

    the remaining diatoni c ton es - 2(2), 5(4), 9(6) and 11(7) - that, t oge the r with

    the member of the tonic triad, const itu te th e diatonic region of fifths and

    thirds spa ce . The five chro mati c ton es outs ide the scal e are displaced t o

    another level of the configuration. Within eac h level of the confi gurat ion,

    the ordering of tones is apparently determined by semit one- spac e relati ons.

    Whether the diatonic con text was induced by the full sca le or the ton ic triad

    did not affect these findings.

    While such studies constitute strong evidence for the importance of

    abstract group-theoretic pitch relations, it might be argued that subjects'

    judgments were somewhat remov ed from direct perceptual experie nce.

    Perhaps, the argument would go, the contribution of abstract pitch relations

    in these experiments is a cognitive effect occuring during the process of

    forming a judgment, and not a true perc eptu al ef fe ct . I will now present

    some new data that address this issue.

    The subjects in the study to be described were all college students in a

    large introductory course in music theory and 'ear training1

    designed for

    non-mus ic majors. Many of thes e student s had had li tt le or noth ing in the

    way of formal musical training.

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    Fig . 5 Multidimensi onal scal ing config urati on depi ctin g judged simil arity

    of pitch pairs in a diatonic co nt ex t. "0" is the ton ic of the implied

    con tex t. The configuration is some what idealize d (after

    Krumhansi, 1979).

    The experimental task, degree discrimination, was not unlike a standard

    pitch discrimination task, except that notes that were octave-related had to

    be tre at ed as equi val ent . Prior to the first trial of th e study, and ad lib

    thereafter, listeners heard two octaves of an ascending major scale, to set

    th e co nt ex t for th e trial s to follo w. Each trial involve d the present atio n of

    a tone belonging to a particular subset of the full sc al e. The subs ets of

    pre se nt concer n are ( 1 , 5) and I, 7). The total en sem ble of tone s for th es e

    sub set s are shown in Figur e 6. The listener's task was to disc rim inat e the

    ton ic de gre e from the other s cal e deg ree in the sub set (e ither 5 or 7), and to

    depre ss one of two buttons as soon as the dec isio n had been m ade. Bothcor re ct nes s and lat enc y of response were reco rded on eac h trial. All stim uli

    were delivered by a PET-2001 microcomputer: tone spectra resembled that

    of a square wav e. The actu al freque ncie s used varied from subje ct to

    subject , but in eac h case approximated equa l-tem pered tuning within the

    frequency resolution of the computer.

    On the whole the subjects were rather good at the task. Mean response

    latency over 49 subjects was 1260 milliseconds (msec), and mean percent

    cor rec t was 87. 4. No subject performed at les s than 74% accur acy, and

    eve n th e long est mean resp onse lat enc y was barely over 3 sec ond s. The 1-7

    discrimination was performed 170 msec faster than the 1-5 discrimination

    (1175 vs 1345 msec, respectively), and 4.6 more accurately (89.7% vs

    85.1%). Both of the se differences were highly significant sta tist ical ly, F(I,

    48) = 8. 71, p < .005 for late nci es, and F(I, 48) = 36. 55, p < .001 for p erce nt

    cor rec t. Besides being reliable over the 49 subje cts, the 1-7 advantage alsogen era liz ed over th e five tone s in the ens emb les , F(I, 4) = 20. 30, p < .001

    for la te nc ie s and F( 1, 4) = 9.2 37, p < .05 for perce nt cor rec t.

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    (1,5 subset} ( t, 7 subset]

    OZLU3OLJ

    tr.

    LUZO

    5

    5

    I

    7

    Fig. 6 Tone ensem bles used in degree expe rime nts.

    Thus we see that, even on a speeded discrimination task, perceived

    similarity of scale degrees is mediated not be frequency separation, but by

    fif th-r ela ted nes s. In ter ms of our group isomo rphism s, we would say that

    perceptual closeness in this task appears to be determined by distance in

    fifths space, and not distance in semitone space.

    Pitch

    Subset

    i - 7

    Table 4. Res ults of Deg ree Discrimi natio n Studies

    First Study Three Months Later

    Latency (msec)

    1345

    1175

    Correct

    ~Q75A

    89.7

    Latency (msec)

    885

    840

    Correct

    90.4

    93.1

    These data were collected right near the beginning of the course,

    before students could have arguably become 'indoctrinated' to music-

    the ore tic beliefs. To the extent that such belie fs or knowledge could aff ect

    one's perception, as is sometimes claimed, we would expect evidence of

    enhanced perceptual similarity of i and 5 to be even stronger for students

    who wer e thus indoc trin ated . But follo w-up meas ure men ts did not confirm

    this idea. Data co ll ec te d on ess enti ally the sam e group of students t hre e

    months later (the course lasts an entire school year) revealed a general

    improvement of about 400 msec in latencies, down to a mean of 863 msec,

    and an impr ove ment of about 4 in cor rec t res ponse s, up to 91.8%. But the

    si ze of the ( 1 , 7) - (1, 3 ) dif fer enc e wa s great ly redu ced, such that the (1, 7)

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    advantage was only 45 msec (as opposed to 170 msec) and 2.7 (as opposed

    to 4.6). These diffe renc es were still stat ist ical ly reliable over subje cts,

    but just barely so for the lat en cie s, F(1, 52) = 3.98 , p < .05 for la ten ci es ,

    and F(I , 52) = 17.39 , p < .001 for perc ent co rre ct. Being incre asing ly

    ind oct rin ate d to the id ea of 1 and 5 being sim ila r appare ntly did not m ake 1and 5 more perce ptua lly conf usable . The tw o se ts of data values are shown

    in Table 4.

    It might be argued that confusabiiity of pitches related by fifths, as

    found her e, was som eho w due to th e spec tra l composi tion of the ton es . I

    hav e co ll ec te d som e pilot data with pure ton es that sugge st such is not the

    ca se . Als o it should be noted, firs t, that no cla ss ica l pitch di scri minat ion

    experiment using complex tones has found any hint of the pattern of results

    reported here; pitch discrimination, we are told, is mediated by the

    frequency separati on of the fundamentals. Secondly, we should recog nize

    that tones of greater spectral complexity are more representative of what is

    actually found in music, so any result that occurred only with pure tones

    would be of limit ed gene ralit y or inte res t in any ca se . There is really no

    paradox here, as long as it is conceded that discrimination of pitched

    dynamic qualities is not the same thing as, nor is necessarily reducible to,pitch discrimination.

    As a final informal result on the notions of Uniqueness and dynamic

    quality, consider the task of deciding what kind of scale a given melody is

    based on. On an exam inat ion, stud ents in my ear-t raini ng cla ss were g iven

    four mel odie s to lis ten to , and asked to dec ide for eac h me lody whet her it

    was based on a pentatonic scale, a major scale, a harmonic minor scale, a

    whole tone sc ale , or a chromatic sca le. The pentat onic, major, and

    harm onic minor sc al es consi st of 5, 7 and 7 not es, re spe cti vel y, and all

    sat is fy Uniquen ess. The whole ton e and ch rom at ic scal es consi st of 6 and 12

    no te s res pec tiv el y, and both fail Unique ness. Of the four melo dies played on

    the exam ina tio n, one was based on the pent ato nic , one on harmo nic minor,

    one on the wh ole tone , and one on th e chr oma tic sca le . Tabula tion of the 42

    identification errors that occurred showed the following breakdown: fully 23

    errors involved who le-t one/ chr omat ic co nfusions, and 15 errors involvedconfu sion s within the pentato nic/ majo r/mi nor cl ass . Only 4 out of 42 errors

    were bet wee n-c las s errors; in general, sc ales satisfy ing Uniqueness were

    hardly eve r confused with scales failing to sati sfy Un iquen ess. While the

    resu lt is only sug ge sti ve , it supports the idea that abstra ct properti es of

    pitch collections are indeed perceptible.

    Scale-step-level Perception

    When a scale is coherent, the scaiestep-ievel distances it induces on its

    intervals are both a determinate function of semitones and a consistent (i.e.

    mono toni c) function of sem ito ne dist ance s. It was argued earlier tha t only

    coher ent sc ale s are conduc ive to the perceptual learning of sca les tep- lev el

    prope rties of inte rval s. The question to be tre ate d her e is wheth er suchperceptual learning does in fact occur: are higher-level scalestep properties

    of interva ls perc eptibl e? Data from several experi ments appear to sugges t

    that the answer is yes.

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    The global cultural context of diatonic scales has determined the names

    of the twelve intervals from m2 to p8 inclusive: what makes a minor 3rd and

    a major 3rd both thirds is predicated on nothing more or less than their

    function in a diaton ic sca le . But it is anothe r thing enti rel y to sugges t t hat

    this global cont ex t in which interva ls are heard and learned actu ally

    det erm ine s the way they sound to lis ten ers . Yet that se em s to be a

    reasonable description of what actually occurs in the course of learning toidentify the twelve intervals, as we shall see.

    Plomp, Wagenaar, and Mimpen (1973) tested the recognition of

    simulta neous intervals by musicians. To induce errors, thes e inv estiga tors

    played the intervals at very short durations, the longest of which was 120

    ms ec . What they found was that mos t of the confusion errors invo lved

    intervals separated by a semito ne, but of the se , an overwhelm ing majority

    of th e errors were to inter vals that shared the same scal es te p valu e. I will

    refer t o . such interval s as sca les tep -eq uiv ale nt. Thus a major 2nd is a

    semitone away from both a minor 2nd and a major 3rd, but it is

    scalestep-equivalent to only the minor 2nd (since they are both seconds).

    What the data of Plomp et ai. reveal is that scalestep equivalence is a

    powerful determinant of perceptual confusabiiity, even when separation in

    semitones is held constant.

    Killam, Lorton and Schubert (1975) replicated and extended this finding

    to situations involving

    (a) longe r dura tions ,

    (b) sequ enti al (melodic) as well as sim ultan eous (harmonic) inte rval s, and

    (c) a non-e xper t subjec t population, namely a cla ss of stud ents invol ved

    with learning to recognize intervals.

    As in the case of Plomp et al., scalestep-equivalent intervals were the locus

    of many more identification errors than could be accounted for by closeness

    in semitones.

    The result was further replicated and extended by Balzano (1977a,

    1977b), who measured late nci es as well as errors in a slight ly diffe rent

    exper iment al paradigm. Briefly, a trial in thes e experimen ts consiste d of a

    visually presented probe, which was the name of one of the twelve basic

    intervals (m2-p8), followed by stimulus, a harmonic or melodic interval that

    eith er did or did not match the interval na med by the probe. Subj ect s, all

    skilled interval recognizers, responded by pressing one of two keys, labelled

    SAME and DIF FERE NT, as soon as the y thought the y knew the ans wer . On

    DIFFERENT trials, the stimulus could be any one of the eleven intervals not

    named by the probe. The data showed tha t, eve n with se mit one dis tanc e

    between probe and stimulus held constant, latencies were significantly

    longer and errors significantly more frequent when the probe and stimulus

    intervals were scale ste p equivalen t. The latenc y differe nce, in particular,

    was as high as 258 msec for harmonic int erva ls, one of the larges t respon se

    late ncy d iffe renc es I have seen in binary choice experi ments of this general

    kind.

    In a rel ated exp eri men t, B alzano (1977a) found evi den ce that int erval s

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    may be recognized at the scalestep level directly, without meditation

    through th e se mi ton e lev el. By this I mean tha t subject s appeared to

    re co gni ze , s ay, a major 3rd, as a "3rd" dire ctly , rather tha n by a tw o- st ag e

    process of the form "major 3rd, therefore 3rd." The test was designed as

    follows: In addition to the semi ton e-l eve l probes in the other exper iment s, a

    number of sc al es te p- le ve l probes wer e used as wel l. An exa mpl e of the

    latter would be the visual probe "3rd", indicating that the subject shouldrespond SAME if the stimulus is either a minor 3rd or a major 3rd, and

    DIFFERENT othe rwis e. Now ther e is considerable evi denc e that it is more

    diffi cult to look (Sternber g, 1967; Cavanaugh , 1972) or liste n (Clifton &

    Cruse, 1977) for two things than for just one, but that is not what happened

    here . Rather, the sca les tep -le vel probes led to responses that we re

    signif icantl y faster and more acc urat e than sem ito ne- lev el probes. Stated

    slightly differently: for a given interval, say a major 3rd, it was easier and

    faster for subjects to verify this interval as an instance of the higher-level

    ca te gor y "3rd" than th e low er- lev el cat ego ry "major 3rd". It would se em ,

    the re fo re , that the "3rdness" of an interval is perc eptu ally a vail able prior to

    its 'minor-3rdness'.

    It has often be en sugg est ed (e.g. Trotter, 1967) tha t, ev en for naiv e

    listeners, scalestep-level properties are more perceptually salient than

    se mi ton e-l eve l properties . Thus the sense in which adjacent mov es of an

    asc endi ng major sc ale are the 'same' se em s to be more evi den t than the

    se ns e in which they diff er. In a more rigorous co nt ex t, Do wling (1978) has

    dem ons tra ted a variant of this pheno menon . He playe d a standard me lody

    fol low ed by a compar ison me lody and asked subj ect s whether the t wo

    mel odie s wer e the sam e or not. Melodies were based on diatonic scal es, and

    'same' comparison melodies were exact transpositions of the standard, i.e.

    melo dies based on the same scale deg rees of a different member of the

    standard melody's sca le family . On a number of trials, the c ompar ison

    melody was not an exact transposition of the standard to another scale, but

    a scalestep-preserving movement of the melody to a different point in the

    sam e scale . Result s indicated that musical and non-musical subjects alike

    had considerable difficulty in distinguishing the scalestep-equivalent

    nontra nsposi tions from true trans posit ions. So eve n - if not esp eci ally - th e

    lis tene r who is not musical ly trained appears to per cei ve mel odi es in ter ms

    of sc ale ste p-l eve l, and not merely sem iton e-le vel , relations. If it weren't

    for the scalestep-semitone coherence of diatonic scales, this could not

    oc cur , of cours e, but it is non eth ele ss surprising just how pote ntly

    scalestep-level properties enter into the perceptual process.

    Scale Families, Pitch Set Overlap and Key Relatedness

    In the subsection on dynamic qualities we saw some evidence for the

    percep tual reality of fifths-s pace di stanc e relations . Here we address a

    similar issue, this time regarding the behavior of pitch sets under

    transpos ition, i.e. sca le families . Since we generally listen, not to scales per

    se , but to melodies based on scales, the most natural way to approach thisques tion is through stud ies of melod y reco gniti on. Given that a mel ody

    reta ins its ide nti ty under transposition, is there any differenc e in the ext ent

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    3 4 6 CHAPTER XVlI

    to which this is true as a function of key re la te ne ss ? Does a melody

    originally heard in C major somehow retain more of its perceptible identity

    when trans pose d to the adj ace nt key of G major than th e dis tant (in fift hs

    space) key of F# major?

    The evi den ce of this question is mixed but gene rall y positiv e. Cuddy,

    Cohen and Miller (1979) used a forced-choice melody recognition task,

    wnere a standard melody was followed by two comparison melodies, one

    tar ge t and one foil . Both tar get s and foils wer e transposed versions of the

    standard; foils had a sing le note alte red by a se mi to ne . Transpositio ns we re

    so me ti me s by a fifth, i .e. to an adjacent key on the c yc le of fif ths and

    so me ti me s by a trit one, the larg est possible dis tanc e in fifths spac e. The

    results showed a small but reasonably consistent advantage (about k%

    overall) for the fifth transpositions.

    Bart lett and Dowling (1980) perf ormed a simi lar ' study, with two

    differences:

    (a) The task was yes -no rather than for ced -ch oic e; subj ects heard only one

    comparison melody on a trial, and decided it was the 'same' as the

    standard or not.

    (b) The cons tructi on of foils was differe nt; in partic ular, many of the f oils

    were melodies that had been both transposed to a new key andtranslated along the scale to a new scale degree.

    The se foils, like those in Dowling (1978), preserved sc ale ste p-l eve l relations

    but violated semi ton e-l eve l relations. The results showed a reliable

    key -di sta nce eff ec t, not on the targ ets , but on the foils . When a foil was

    transposed to a distant key, it was easier to distinguish from a correct

    transposition (i.e. recognize as a foil) than when the foil was transposed to a

    near key. Both adults and grammar school children show thi s ef fe ct . It is

    not simply due to a res pon se bias for sayin g "no" to fa r-k ey trans posi tions ; if

    it were , tar get s would have showed the sa me eff ec t. But for our presen t

    concerns, it would not be diastrous if response bias were involved anyway,

    for even a response bias that is sensitive to key distance is evidence of a

    perceptual ability to respond to that variable.

    To what extent are results like these a function of pitch overlap?Diatonically-based melodies, as we have seen, will have more pitch overlap

    with tran spositi ons to near than to far keys , so the only way to ex am in e this

    question is to use non-tonal melodies - more specifi cally , melodi es whose

    pitc h cont ent is thinly spread through all of fifths spac e. Cohen (1977;

    Cohen, Cuddy & Mewhort, 1977) performed several experiments using

    melodies based on both diatonic pitch sets and pitch sets satisfying the

    abo ve descript ion. The task was basi cally th e same as Cuddy et al (1979),

    and as th er e, the p5 (near) and tt (far) tra nsp osi tio ns wer e use d. What

    Cohen found was an interaction between the factors of pitch set and

    trans posi tion. Diat onic melo dies showed an adva ntag e for the p5

    transpositions, while atonal melodies showed a tritone advanta ge. The

    atonal pitch sets employed by Cohen were (0, 1, 2, 6, 7, 10) and the whole

    scale, both of which exhibit greater pitch class overlap under Tg (tt) than

    Ty (p5). So, for the se exper ime nts anyway, the ability to de te ct a cor rec t

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    transposition of a melody appeared to be a direct function of overlap (Ov) as

    previously defined.

    The final set of experiments to be described in this subsection, those of

    Balzano (1977a), are quite different from those we have just discussed, in

    that melody recoginition was not studied, but interval recognition by skilled

    musicians. These experiments nonethe less fit into the present c onte xt

    because they document a perceptual sensitivity to key-relatedness as

    reckoned by distance in fifths space, moreover one that cannot have been

    med iat ed by pitch class overlap per se . In th es e interv al rec ogni tionexperiments, the base tone (i.e. the tone lower in frequency) of the intervals

    te st ed wa s res tri cte d to a very sma ll set of pit che s. In thr ee of the

    experiments, the base tone was always an E, an A, or a C# (pitch set (0, 4,

    7)), eac h one on one-th ird of the trials; in tw o other expe rim ent s, th e base

    tone was either a G or a C# (pitch set (0, 6)), each on one-half of the trials.

    The reasoning ran as follows (s ee Balza no, 1977a, for a fuller discussion):

    With the former base-tone set, 2/3 of the intervals presented in an

    exp eri men tal se ssi on we re based on E or A, adjacen t keys on the cy cl e of

    fifth s. The othe r 1/3 of the inter vals wer e based on a pitch more r emo te

    from the first two in fift hs spa ce. It was expe ct ed that the overa ll ef fe ct of

    this manipula tion would place the subj ect in a perc eptu al mode similar to

    that for the key of A major, and therefore that recognition would be

    significantly worse to intervals based on the tone (C#) that was more

    re mo te from A in fifths spa ce. The G-C # base tone se t, on the other hand,exhibits a purely symmetrical structure on the cycle of fifths, so neither

    to ne should show any percept ual adva ntag e over the othe r; in particu lar, t he

    disadvantage for C# should no longer occur.

    The res ults confi rme d thes e ide as. While only one of th e three

    exper iment s using the E- A-C # bas e-t one set showed a significant advantag e

    for intervals based on A over intervals based on E, in all three experiments,

    intervals based on A and E were responded to more accurately and rapidly

    than those based on C# . In the two exper ime nts using th e G-C# ens emb le

    there were essentially no base-tone effects; if anything it was C# that led to

    fast er and more acc ur ate responding. Pit ch ove rla p canno t ac cou nt for

    these results, since all of the basic intervals were tested on each base tone;

    that is to say, the upper tone of the intervals was free to vary, and was not

    restricted, for example, to intervals from the base tone's major scale.

    Summing up the results of this subsection, we can tentatively say that

    perceptual distance between transposed melodies is a function of pitch set

    overl ap. For the case of diatonic mel odie s, this direc tly impl icat es the

    cycle of fifths as a basis for perceptual distance, but not necessarily in the

    cas e of atonal melodi es. In the abs enc e of pitch se t constraints , overlap

    becomes nonfunctional, and here fifth-space distance appears to act as a

    per cept ual defa ult . It is evid ent tha t much more res ear ch rema ins to be

    done to clarify these matters.

    CONCLUDING REMARKS

    Experiments were performed long before the notion of a

    psychoacoustical theory ever arose, and they continued to be performed

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    throughout history, not by scientists but by persons mainly attempting to

    disco ver the types of exp eri enc es pitch had to off er. What they hit upon,

    indepe ndentl y and all over th e anci ent worlds, was that diffe rent pitc h

    colle ction s, sets of pitches known as sca les , each presented a pote ntiall y

    unique medium for embedding pitch se que nce s cal led melo die s. Just as the

    desired movement of a melody over time shaped the character of these

    pitch set materials, so it was found that the character of the pitch set in

    turn shaped the percep tible quality of its melo dies ; diff eren t pitch sets th us

    bec ame assoc iated with different mean ings. That this could occur

    constituted a very important discovery about pitch perception.

    An account of the distinctive properties of pitch sets and their

    ele men ts has not been forthcoming from st ate -of -th e-a rt auditory theory . I

    have attempted to provide the beginnings of such an account in this chapter

    by using the language of set and group theory to develop a number of basic

    prope rtie s of pitch se ts . The group C ] 2n a s D e e n

    treated as a 'ground' for

    the musical 'figures' that are pitch se ts , and thre e isom orphi c

    representations of C ] 2 have been presented as alternative 'grounds* for

    pitch sets and events.

    The properties of pitch arrays that have been the subject of this

    chapte r are not found at the lev el of tone se nsa tion s. But 100 year s a fte r

    Helmholtz, it is still far from clear that the sensations of tone truly are a

    basis, physiol ogica l or other, for the theor y of music. It may or may not be

    true that "music stands in a much closer connection with pure sensation than

    any of the other arts " (Hel mhol tz, 188.5), but the prese nt theory is founded

    on the idea that it is the perception of patterns and relations the tones enter

    into and not the sensations of tone themselves that accounts for our ability

    to app rec iat e music. Given that such patte rns and rela tions are made

    available in music through the medium of pitch sets, a theory outlining

    prope rties of pitch set s would appear indispensab le to an eff ort to

    understand music perception in con tex t. To the exte nt that the properties

    of the embedding pitch system (C] 2 )D

    th constrain and interpret

    prope rtie s of it sub set s, a theor y of the pitch syst em would also s ee m a

    nec ess ity . I have tried to provide a foundation for the se basic nece ss iti es,

    but more important than the spec ifi c detail s pres ented here is the fac t that

    a logically coherent, empirically supportable alternative to Helmholtz and

    the psychophysi cal tradit ion is indeed possible . We have seen in this cha pte r

    that a pitch-set level of description grants easy access to an order of

    phenom ena that are both emi nen tly m usica l in natur e and at the sam e ti me

    difficult t o provide a theo reti cal rendering of in lowe r-le vel ter ms. Perhapseven more importantly, human musical behavior itself yields readily to an

    analy sis in the higher- leve l langu age proposed here . From a pitc h-s et

    perceptive, we may well be led to ask different kinds of questions and to

    sear ch for new kinds of phenomena . It may even turn out that the mos t

    fascinating properties of human pitch perception still remain to be

    discovered.

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    SUMMARY

    Properties of single pitches and properties of pitch sets belong to two

    diff ere nt grains of analy sis of the music al pitch sti mulu s. It is sug ges ted

    that musically important properties of pitch are more directly a function of

    pre dica tes defined at the latt er grain. Starti ng from the obse rvati on tha t

    the global pitch system in Western music exhibits the structure of the

    mathematical group C]2>a

    number of systematic properties of pitch sets

    - subsets of C12 'a r e

    formally develope d. At the sam e tim e, three

    isomorphic spatial representations of the inner structure of C12 a r e

    presented, and the familiar diatonic scale is revealed to have unique

    properties both with regard to the abstract description of pitch sets andwith regard to the spatial character of C\2- Empirical evidence is

    rev iew ed and prese nted in support of the percep tual rea lity of the se

    formally developed properties, and the general claim is made that listeners

    are direct ly sensit ive to such pitch se t properties. The sensat ions of tone

    may well be a function of frequency and frequency ratios, but the

    per cep tio n of music need not be, and indeed appears inste ad to be a functio n

    of higher-order properties of pitch sets that are independent of ratios.

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