Volume 0, Number 10, September 1994
Copyright © 1994 Society for Music Theory
Modifying IntervalClass Vectors of Large Collections to Reflect Registral Proximity Among PitchesBrian RobisonKEYWORDS: harmony, set theory ABSTRACT: The twelvetone operations of transposition and inversion reduce all 12note collections to one setclass, all 11note collections to one setclass, and all tennote collections to one of six setclasses. Yet throughout this century, composers have organized large collections in different ways to produce distinctly characteristic intervallic content. I propose here a modified version of Allen Forte’s intervalclass vector to preserve some indication of registral proximity among pitches, assigning fractional counts to intervals greater than six semitones in order to derive intervalclass vectors which distinguish different sets of a given setclass.

[1] The following workinprogress attempts to bridge an analytic gap between order and disorder: between symmetric pitch matrices or interval cycles on the one hand, and asymmetric, allinterval structures (often labeled as “ad hoc”) on the other. Although this discussion treats only large collections, the technique proposed is sufficiently general to apply to sets of any cardinality. The calculation of modified intervalclass (ic) vectors for individual pitch sets (as opposed to pitchclass sets) requires computer implementation; Appendix 2 lists the structured BASIC program used to generate the ic vectors given in Appendix 1. Sets and SetClasses [2] Pitchclass (pc) set theory provides powerful abstractive tools for relating pitch collections by operations such as transposition and inversion. However, these same operations reduce all 12note collections to one setclass, all 11note collections to one setclass, and all tennote collections to one of six setclasses distinguished by the intervalclass of the excluded pc pair.^{(1)}Especially when expssed as chords in close or relatively even spacing, such large collections often do not lend themselves to perceptually clear segmentation into subsets with more distinctive ic vectors. [3] Throughout this century, composers have organized large collections in different ways to produce distinctly characteristic intervallic content. These range from the consistent use of one or two ic’s between vertically adjacent chordtones, especially in vertically symmetrical arrangement, as in the music of the late Witold Lutoslawski^{(2)} Olivier Messiaen^{(3)}, and others^{(4)}, to “allinterval” constructions, typically asymmetrical, as is often the case in verticalizations of 12tone rows. [4] pviously proposed functions which distinguish different members of a given setclass, such as Chapman’s “abovebass” ic vector^{(5)} and Morris’s INT(n) function^{(6)}, provide detailed information about relationships among pitches; however, as set cardinality increases, this detail proliferates geometrically to become unwieldy in practice. A Geometric Analogy [5] Many of the difficulties described above result from the conventional mapping of pitch space to pitchclass space, by which one completely collapses the helix of the former onto the circle of the latter. However, for most objects of dimension D, one can choose among multiple projections to a space of dimension D1. For example, a cube can project onto a plane as any of the following:
[6] Each repsentation has its merits and limitations. The first pserves right angles and equal lengths of some edges, but effectively obscures the original’s threedimensional character. The second pserves all parallelisms and most angles and lengths, but distorts others. The third pserves all parallelisms and repsents all edges as equal in length, but distorts all angles (some become 60 degrees, others 120 degrees). [7] Returning to the helical model of the pitch continuum, there are likewise several possible approaches. Viewed from the side, the helix appears as a sinusoid. This sinusoidal projection pserves registral position but distorts relative distances (e.g. some semitones appear longer than others). As mentioned pviously, the circular projection pserves intervalclass relationships but eliminates all information regarding registral position. IntervalClass Vectors with Fractional Values [8] I propose here a modified version of Allen Forte’s ic vector^{(8)} which provides some indication of registral proximity among pitches; in the geometric analogy above, it corresponds roughly to viewing the helix from above, but with some element of depth perception. By assigning fractional counts to intervals greater than six semitones, we can derive ic vectors which distinguish different voicings of any given pc collection.^{(9)} [9] Within each ic, the smallest interval receives a value of one; successively larger intervals are counted at successively smaller values, as shown below for three decrement schemes: one linear, one moderately exponential, and one strongly exponential. The linear scheme uses a steady decrement of .05, which maintains nonzero values for intervals within a limit of ten octaves (corresponding to the conventional range of human hearing). The exponential schemes are based on reciprocals of e^{(10)}raised to 1/5 and 1/2 the index of transposition (see table below). In all of these, the conventional ic vector is taken as expssing dissonant maxima^{(11)}, which the composer can emphasize or attenuate by means of registral disposition.^{(12)} Fractional interval counts  Decrement schemes  Interval (a) (b) (c) in Index of Linear Moderately Strongly semitones transpos. exp. exp.      1  6 0 1.0 1.0 1.0 7  12 1 .95 .819 .607 13  18 2 .9 .670 .368 19  24 3 .85 .549 .223 25  30 4 .8 .449 .135 31  36 5 .75 .368 .082 37  42 6 .7 .301 .050 43  48 7 .65 .247 .030 etc. etc. etc. etc. etc. Variation Sets [10] Example 1 provides ten different voicings of 824, displaying varying degrees of symmetry, varying numbers of ic’s as adjacencies, and varying ic’s emphasized as adjacencies. The first three are Messiaen’s accord de resonance and two of its inversions.^{(13)} 1d and 1e emphasize ic’s 1 and 2; 1f emphasizes perfect fourths, while 1g manifests augmented triads in close spacing. 1h, 1i, and 1j psent several allinterval constructions. For 1a and 1c, the fractional ic vectors (Appendix 1: Tables 1a, 1b, 1c) reflect the tertial structure’s relative emphasis by proximity of ic3 over ic1.^{(14)} Like conventional vectors, they fail to distinguish 1a from 1c. However, unlike conventional vectors, and unlike Chapman’s and Morris’s functions, they do distinguish between 1d and 1e, which contain identical vertical sequences of pitchclasses. [11] The exponential schemes psent more dramatic distinctions than linear ones, as the fractional ic vectors (Appendix 1: Tables 2a, 2b, 2c) for the 12note sonorities in Example 2 illustrate. Again, the voicings vary in regard to symmetry and restriction of ic’s as adjacencies. Note that the greater decrements for highly compound intervals produce a wider range of possible values in any given position in the vector. For example, compare the ic6 values in the vectors for the first five sonorities alone (vertically symmetrical 12note chords with only ic3 and ic5 as adjacencies, spanning comparable registers). The linear scheme produces values between 5 and 6, the moderately exponential scheme produces values between 3.28 and 6, and the strongly exponential scheme produces values between 1.57 and 6. Fractional vectors derived with strongly exponential decrements thus approach a vector calculated by counting adjacencies alone. Preliminary Implications for Analysis [12] To ensure meaningful comparison, it’s important that an analyst apply either the same scheme or closely comparable ones to all collections in a given context. Conservative linear decrement schemes distinguish different voicings of a given pcset, but their vectors are still closer to each other than they are to those of other pcsets, and even to those of voicings of other sets which display highly similar adjacency structures. In contrast, the exponential schemes produce vectors in which the intervallic cardinality (i.e. the sum of the terms of the ic vector) of a chord distributed over a wide range approximates that of a much smaller conventional collection. In extreme cases, pitch sets which belong to different pcsetclasses but which share certain adjacency structures could generate vectors which are closer to each other than to those of other members of their own pcsetclasses, as measured by numerical methods such as Isaacson’s IcVSIM function.^{(15)} [13] However, I do not psume to suggest any standard scheme, since the pferred size and nature of intervallic decrements will vary among individuals and among repertoires. To return to the geometric analogy, these variations correspond to differences among individual perceptions of the relative spacing of the helix’s coils and the resulting degree to which aural “perspective” attenuates the dissonance of highly compound intervals. In addition, aspects of timbre and orchestration often motivate different perceptual distances; consider the difference in relative dissonance of any one of the sonorities above as performed on a harpsichord, clavichord, piano, or organ, or as played by bowed strings, muted, ppp, or brass choir, fff. As with any other feature of analysis, one should note explicitly the information which one chooses to sacrifice for the sake of brevity and clarity. Appendix 1: Tables Table 1a Fractional intervalclass vectors of the 10 voicings of 8–24 in Example 1 (linear decrements) 4 6 4 7 4 3 a 3.6 5.65 3.95 6.6 3.75 2.9 b 3.7 5.6 4 6.5 3.7 3 c 3.6 5.65 3.95 6.6 3.75 2.9 d 4 5.95 4 6.9 3.9 3 e 3.8 5.35 3.7 5.8 3.2 2.5 f 3.8 5.1 3.3 6.0 4 3 g 3.5 5.45 3.5 6.9 3.5 2.8 h 3.7 5.65 3.8 6.3 3.6 2.7 i 3.35 5.05 3.6 6.2 3.8 2.5 j 3.75 5.4 3.55 6.3 3.75 2.9 ============================== Table 1b Fractional intervalclass vectors of the 10 voicings of 824 in Example 1 (moderately exponential decrements) 4 6 4 7 4 3 a 2.71 4.80 3.82 5.71 3.19 2.67 b 2.98 4.71 4 5.29 3.10 3 c 2.71 4.80 3.82 5.71 3.19 2.67 d 4 5.82 4 6.64 3.64 3 e 3.34 4.26 3.01 3.75 1.86 1.57 f 3.27 3.49 2.00 4.09 4 3 g 2.44 4.20 2.54 6.64 2.59 2.34 h 3.04 4.80 3.37 4.85 2.86 2.12 i 2.24 3.49 2.86 4.67 3.34 1.75 j 3.16 4.20 2.67 4.85 3.19 2.67 ==================== Table 1c Fractional intervalclass vectors of the 10 voicings of 8–24 in Example 1 (strongly exponential decrements) 4 6 4 7 4 3 a 1.57 3.56 3.61 4.57 2.44 2.37 b 1.95 3.57 4 3.53 2.45 3 c 1.57 3.56 3.61 4.57 2.44 2.37 d 4 5.61 4 6.21 3.21 3 e 2.74 3.13 2.10 1.72 .67 .64 f 2.43 1.84 .72 2.00 4 3 g 1.18 2.54 1.48 6.21 1.67 1.74 h 2.20 3.56 2.83 3.16 2.06 1.50 i 1.16 2.10 2.06 3.02 2.74 1.19 j 2.34 2.79 1.73 3.16 2.44 2.37 ============================== Table 2a Fractional intervalclass vectors of the 15 voicings of 121 provided in Example 2 (linear decrements) 12 12 12 12 12 6 a 10.35 9.7 10.55 11.2 10.35 5.0 b 10.3 10.5 11.05 10.4 10.8 5.4 c 10.45 9.7 10.85 11.2 10.55 5.0 d 10.65 10.1 11.05 11.2 10.75 5.2 e 10.5 10.4 11.85 10.4 10.5 6 f 10.55 10.1 10.75 11.2 11.05 5.2 g 10.4 10.55 10.85 10.5 11.05 5.5 h 10.75 10.1 10.65 10 9.45 4.8 i 10.15 9.7 10.75 11 10.45 5 j 10.5 10.6 11.85 10.6 10.5 6 k 10.55 10.7 11.1 10.8 11.05 5.3 l 10.95 10.85 10.9 10.6 11 5.6 m 9.10 8.90 9.15 9.15 8.9 4.4 n 10.5 10.65 11.25 10.6 10.7 5.7 o 10.65 10.6 11.1 10.55 10.5 5.4 ============================== Table 2b Fractional intervalclass vectors of the 15 voicings of 121 provided in Example 2 (moderately exponential decrements) 12 12 12 12 12 6 a 7.30 5.80 8.02 9.23 9.00 3.28 b 7.48 7.41 9.26 7.59 8.40 4.24 c 7.53 5.80 8.99 9.23 8.00 3.28 d 7.97 6.58 9.45 9.23 8.39 3.80 e 7.66 7.20 11.46 7.27 7.67 6 f 7.77 6.58 8.45 9.23 9.36 3.80 g 7.26 7.77 9.00 7.65 8.96 4.57 h 8.47 6.58 8.45 7.15 5.66 2.70 i 6.79 5.92 8.79 8.88 7.53 3.50 j 7.38 7.81 11.46 7.27 7.67 6 k 7.57 8.21 9.26 8.41 9.55 3.80 l 8.77 8.60 8.85 7.83 9.07 4.79 m 7.97 8.00 8.45 7.70 7.30 3.80 n 7.75 7.81 9.86 7.92 8.35 5.12 o 7.88 7.96 9.53 7.76 7.84 4.42 ============================== Table 2c Fractional intervalclass vectors of the 15 voicings of 121 provided in Example 2 (strongly exponential decrements) 12 12 12 12 12 6 a 3.95 2.35 5.58 6.32 7.52 1.57 b 4.65 3.85 7.40 4.64 5.74 3.01 c 4.30 2.35 7.26 6.32 5.62 1.57 d 4.86 2.91 8.04 6.32 5.90 2.54 e 4.54 3.55 10.82 3.86 4.68 6 f 4.73 2.91 6.15 6.32 7.58 2.54 g 3.76 4.63 7.36 4.64 6.19 3.64 h 5.75 2.91 6.65 4.47 2.55 .81 i 3.20 2.53 7.08 6.36 4.28 2.37 j 3.69 4.63 10.82 4.82 4.09 6 k 4.08 5.51 7.15 5.48 8.22 1.97 l 6.12 6.04 6.77 4.62 6.91 3.87 m 4.86 5.02 6.40 4.78 4.74 2.54 n 4.78 4.47 8.41 4.87 5.94 4.50 o 4.62 5.06 7.94 4.73 5.17 3.55 Appendix 2: The FRACTION Program DECLARE SUB EnterPitches () DECLARE SUB FractionalCounts () DECLARE SUB ShowVector () DIM SHARED Quant DIM SHARED IntClass DIM SHARED PitchArray(3, 30) ‘Thirty is an arbitrary value, selected because ‘ BASIC does not allow matrices of variable dimension. ‘If analyzing collections of more than 30 pitches, ‘ substitute a larger value in this dimension statement. DIM SHARED IcMatrix(12, 6) DEF fnPitchClassOrder (p, q) ‘This function enables the FractionalCount routine ‘ to assign fractional counts to their proper positions ‘ in the intervalclass matrix. IF IntClass = 6 THEN SELECT CASE p CASE IS < q fnPitchClassOrder = p CASE IS > q fnPitchClassOrder = q END SELECT EXIT DEF END IF SELECT CASE p CASE IS = (q + IntClass) MOD 12 fnPitchClassOrder = q CASE ELSE fnPitchClassOrder = p END SELECT END DEF ‘Program Fraction ‘This program calculates intervalclass vectors ‘ using fractional intervalclass counts ‘ to reflect emphasis by registral proximity. DO CLEAR CALL EnterPitches CALL FractionalCounts CALL ShowVector PRINT “ ” INPUT “Process another collection”; Response$ IF Response$ = “n” THEN EXIT DO LOOP END SUB EnterPitches ‘This routine accepts pitch data as ordered pairs ‘ of pitchclass (in integer notation) ‘ and register (according to the ASA scheme). ‘It then converts each pair to a position ‘ for calculating intervals. CLS INPUT “Enter the number of pitches in the collection: ”; Quant PRINT “Enter each pitch as an ordered pair (pitchclass, register)” FOR i = 1 TO Quant PRINT “ Pitch No.”; i; INPUT PitchArray(1, i), PitchArray(2, i) PitchArray(3, i) = 12 * PitchArray(2, i) + PitchArray(1, i) NEXT i END SUB SUB FractionalCounts ‘This routine calculates intervals and assigns their ‘ fractional counts to an intervalclass matrix. FOR v = 1 TO Quant  1 FOR w = v + 1 TO Quant Interval = ABS(PitchArray(3, v)  PitchArray(3, w)) IntClass = Interval MOD 12 Index = 2 * ((Interval  IntClass) / 12) IF IntClass > 6 THEN IntClass = 12  IntClass Index = Index + 1 END IF Delta = Index * .05 Fraction = 1  Delta ‘The above two lines implement the linear decrement ‘ scheme discussed in the article text. ‘For moderately exponential decrements, replace them ‘ with the single line ‘ Fraction = 1/(2.718281828# ^ (Index/5)) ‘For strongly exponential decrements, replace them ‘ with the single line ‘ Fraction = 1/(2.718281828# ^ (Index/2)) r = fnPitchClassOrder(PitchArray(1, v), PitchArray(1, w)) IF Fraction > IcMatrix(r + 1, IntClass) THEN IcMatrix(r + 1, IntClass) = Fraction END IF NEXT w NEXT v END SUB SUB ShowVector ‘This routine sums the fractional counts for each ‘ intervalclass, producing the fractional IcVector. PRINT “Fractional IcVector = ”; DIM sum(6) FOR m = 1 TO 6 sum(m) = 0 FOR n = 1 TO 12 sum(m) = sum(m) + IcMatrix(n, m) NEXT n PRINT sum(m); NEXT m END SUB
Brian Robison Footnotes1. Specifically, these “saturated” setclasses are:
Note that, among setclasses of cardinality 10, the vectors
show a higher value for the “excluded” pc pair (i.e. by
reflexivity, the excluded pair removes only one instance
of its own ic, but two instances of every other ic). 2. Steven Stucky, Lutoslawski and his music (Cambridge:
Cambridge University Press, 1981), 114–119. 3. Some of these collections arise from interval cycles
(as in the chord of alternating fourths and tritones which
repeatedly appears in the first movement of his
Turangalilasymphonie), others from the inherent
symmetries of his modes of limited transposition (Messiaen:
Technique de mon langage musical, Paris: A. Leduc, 1944.). 4. Two prominent early collections of limited adjacencies
are the 12note chords that accompany the deaths of the
title characters in Alban Berg’s Wozzeck (ic’s 3 and 4,
symmetric) and Lulu (ic’s 1, 5, and 6, asymmetric). 5. Chapman, Alan. “Some intervallic aspects of pitch
class set relations.” Journal of music theory 25 (1981):
275–290. 6. Morris, Robert. Composition with pitch classes:
A theory of compositional design. New Haven: Yale
University Press, 1987. 7. For the sake of simplicity, these examples assume
parallel perspective in projection (i.e. no vanishing
point!). Of course, the introduction of one or more
vanishing points further multiplies the variety of possible
projections. 8. Because the system outlined by Forte in The structure
of atonal music (New Haven: Yale University Press, 1973)
has become a lingua franca among theorists, it provides a
valuable framework in which to build. Thus, the counting
schemes outlined below all produce ic vectors of six values
ranging from [0 0 0 0 0 0] for the empty setclass to
[12 12 12 12 12 6] for the complete aggregate with all 66
possible intervals occurring in their most compact forms
(due to duplication of pitchclasses). 9. Although these fractional counts resemble membership
functions of fuzzy sets, as set forth by Lotfi Zadeh in his
article “Fuzzy sets,” Information and control 8 (1965),
pp. 338–353, I do not mean them to imply a reduced degree
of ic membership for compound intervals! Rather, they
denote degrees of membership in the fuzzy set of close
intervals, summed by ic as six scalar cardinalities to
produce an ic vector. By extension, the fractional vectors
do not imply reduced setclass membership for sets widely
dispersed in register, but only lesser membership in the
fuzzy set of closelyspaced sonorities. For a lucid
review of basic concepts of traditional (“crisp”) set theory
and fuzzy set theory, see George J. Klir and Tina A. Folger:
Fuzzy sets, uncertainty, and information (Englewood
Cliffs: PrenticeHall, 1988), 1–21. 10. e is the base of the system of natural logarithms,
approximately 2.718 in value. The choice of e as a basis
for these exponential schemes is purely arbitrary.
Empirical research in the perceived distances of compound
intervals may suggest alternate decrement functions. 11. Here “dissonant” refers loosely to the acoustical
phenomenon of critical bandwidth, corresponding to the
colloquial notion that a minor second is “crunchier” than a
major seventh, and so on. 12. Thus, in order to preserve the bounds described in
note (7) above, in collections containing more than one
pitch of a given pc, the algorithm implemented in the
Fraction program (Appendix 2) always chooses the more
compact form of a given pc pair, e.g. for the collection
{C4,D4,B4,C5} under the linear scheme above, the ic vector
is [1 1 .95 0 0 0]. 13. Messiaen, Technique de mon langage musical, Vol. 2,
Exx. 208, 209, 210. I’ve transposed Messiaen’s Ex. 209
so that it shares the same pitch classes with the
collections of Ex. 1a and 1d–1j. 14. That is, these two sonorities each (enharmonically)
contain fairly compact versions of ic3 (three minor thirds
and one major sixth), vs. compound versions of ic1 (no minor
seconds, two major sevenths, one minor ninth, and one major
fourteenth). 15. Isaacson, Eric. “Similarity of intervalclass content
between pitchclass sets: The IcVSIM function.” Journal of
music theory 34 (1990), 1–28. Specifically, these “saturated” setclasses are:
Note that, among setclasses of cardinality 10, the vectors show a higher value for the “excluded” pc pair (i.e. by reflexivity, the excluded pair removes only one instance of its own ic, but two instances of every other ic). Steven Stucky, Lutoslawski and his music (Cambridge:
Cambridge University Press, 1981), 114–119. Some of these collections arise from interval cycles
(as in the chord of alternating fourths and tritones which
repeatedly appears in the first movement of his
Turangalilasymphonie), others from the inherent
symmetries of his modes of limited transposition (Messiaen:
Technique de mon langage musical, Paris: A. Leduc, 1944.). Two prominent early collections of limited adjacencies
are the 12note chords that accompany the deaths of the
title characters in Alban Berg’s Wozzeck (ic’s 3 and 4,
symmetric) and Lulu (ic’s 1, 5, and 6, asymmetric). Chapman, Alan. “Some intervallic aspects of pitch
class set relations.” Journal of music theory 25 (1981):
275–290. Morris, Robert. Composition with pitch classes:
A theory of compositional design. New Haven: Yale
University Press, 1987. For the sake of simplicity, these examples assume
parallel perspective in projection (i.e. no vanishing
point!). Of course, the introduction of one or more
vanishing points further multiplies the variety of possible
projections. Because the system outlined by Forte in The structure
of atonal music (New Haven: Yale University Press, 1973)
has become a lingua franca among theorists, it provides a
valuable framework in which to build. Thus, the counting
schemes outlined below all produce ic vectors of six values
ranging from [0 0 0 0 0 0] for the empty setclass to
[12 12 12 12 12 6] for the complete aggregate with all 66
possible intervals occurring in their most compact forms
(due to duplication of pitchclasses). Although these fractional counts resemble membership
functions of fuzzy sets, as set forth by Lotfi Zadeh in his
article “Fuzzy sets,” Information and control 8 (1965),
pp. 338–353, I do not mean them to imply a reduced degree
of ic membership for compound intervals! Rather, they
denote degrees of membership in the fuzzy set of close
intervals, summed by ic as six scalar cardinalities to
produce an ic vector. By extension, the fractional vectors
do not imply reduced setclass membership for sets widely
dispersed in register, but only lesser membership in the
fuzzy set of closelyspaced sonorities. For a lucid
review of basic concepts of traditional (“crisp”) set theory
and fuzzy set theory, see George J. Klir and Tina A. Folger:
Fuzzy sets, uncertainty, and information (Englewood
Cliffs: PrenticeHall, 1988), 1–21. e is the base of the system of natural logarithms,
approximately 2.718 in value. The choice of e as a basis
for these exponential schemes is purely arbitrary.
Empirical research in the perceived distances of compound
intervals may suggest alternate decrement functions. Here “dissonant” refers loosely to the acoustical
phenomenon of critical bandwidth, corresponding to the
colloquial notion that a minor second is “crunchier” than a
major seventh, and so on. Thus, in order to preserve the bounds described in
note (7) above, in collections containing more than one
pitch of a given pc, the algorithm implemented in the
Fraction program (Appendix 2) always chooses the more
compact form of a given pc pair, e.g. for the collection
{C4,D4,B4,C5} under the linear scheme above, the ic vector
is [1 1 .95 0 0 0]. Messiaen, Technique de mon langage musical, Vol. 2,
Exx. 208, 209, 210. I’ve transposed Messiaen’s Ex. 209
so that it shares the same pitch classes with the
collections of Ex. 1a and 1d–1j. That is, these two sonorities each (enharmonically)
contain fairly compact versions of ic3 (three minor thirds
and one major sixth), vs. compound versions of ic1 (no minor
seconds, two major sevenths, one minor ninth, and one major
fourteenth). Isaacson, Eric. “Similarity of intervalclass content
between pitchclass sets: The IcVSIM function.” Journal of
music theory 34 (1990), 1–28.
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