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M U S I C T H E O R Y O N L I N E
A Publication of the
Society for Music Theory
Copyright (c) 1994 Society for Music Theory
+-------------------------------------------------------------+
| Volume 0, Number 10 September, 1994 ISSN: 1067-3040 |
+-------------------------------------------------------------+
All queries to: mto-editor@husc.harvard.edu
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AUTHOR: Robison, Brian
TITLE: Modifying Interval-Class Vectors of Large Collections
to Reflect Registral Proximity Among Pitches
KEYWORDS: harmony, set theory
Brian Robison
Music Department
Lincoln Hall
Cornell University
Ithaca, New York 14853
bcr2@cornell.edu
ABSTRACT: The twelve-tone operations of transposition and
inversion reduce all 12-note collections to one set-class,
all 11-note collections to one set-class, and all ten-note
collections to one of six set-classes. 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 interval-class vector to preserve some
indication of registral proximity among pitches, assigning
fractional counts to intervals greater than six semitones
in order to derive interval-class vectors which distinguish
different sets of a given set-class.
ACCOMPANYING FILES:
mto.94.0.10.robison1.gif
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[1] The following work-in-progress attempts to bridge an
analytic gap between order and disorder: between symmetric
pitch matrices or interval cycles on the one hand, and
asymmetric, all-interval 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 interval-class (ic) vectors
for individual pitch sets (as opposed to pitch-class sets)
requires computer implementation; Appendix 2 lists the
structured BASIC program used to generate the ic vectors
given in Appendix 1.
SETS AND SET-CLASSES
[2] Pitch-class (pc) set theory provides powerful
abstractive tools for relating pitch collections by
operations such as transposition and inversion. However,
these same operations reduce all 12-note collections to one
set-class, all 11-note collections to one set-class, and all
ten-note collections to one of six set-classes distinguished
by the interval-class of the excluded pc pair.(1)
Especially when expressed 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.
==============================
1. Specifically, these "saturated" set-classes are:
12-1 [12 12 12 12 12 6]
11-1 [10 10 10 10 10 5]
10-1 [9 8 8 8 8 4]
10-2 [8 9 8 8 8 4]
10-3 [8 8 9 8 8 4]
10-4 [8 8 8 9 8 4]
10-5 [8 8 8 8 9 4]
10-6 [8 8 8 8 8 5]
Note that, among set-classes 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).
==============================
[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 chord-tones, especially in vertically symmetrical
arrangement, as in the music of the late Witold
Lutoslawski(2) Olivier Messiaen(3), and others(4), to
"all-interval" constructions, typically asymmetrical, as is
often the case in verticalizations of 12-tone rows.
==============================
2. Steven Stucky, *Lutoslawski and his music* (Cambridge:
Cambridge University Press, 1981), pp. 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
*Turangalila-symphonie*), 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 12-note 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).
==============================
[4] Previously proposed functions which distinguish
different members of a given set-class, such as Chapman's
"above-bass" 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.
==============================
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.
==============================
A GEOMETRIC ANALOGY
[5] Many of the difficulties described above result from the
conventional mapping of pitch space to pitch-class 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 D-1. For example, a cube can project
onto a plane as any of the following:
1) a single square,
2) two intersecting squares, with corresponding sides
parallel and corresponding vertices connected, or
3) a regular hexagon with opposing vertices connected.(7)
==============================
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.
==============================
[6] Each representation has its merits and limitations. The
first preserves right angles and equal lengths of some edges,
but effectively obscures the original's three-dimensional
character. The second preserves all parallelisms and most
angles and lengths, but distorts others. The third
preserves all parallelisms and represents 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 preserves registral position but distorts
relative distances (e.g. some semitones appear longer than
others). As mentioned previously, the circular projection
preserves interval-class relationships but eliminates
all information regarding registral position.
INTERVAL-CLASS 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)
==============================
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 set-class 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 pitch-classes).
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 set-class membership for sets widely
dispersed in register, but only lesser membership in the
fuzzy set of *closely-spaced 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: Prentice-Hall, 1988), pp. 1-21.
==============================
[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 non-zero 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 expressing 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.
==============================
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].
==============================
VARIATION SETS
[10] Example 1 provides ten different voicings of 8-24,
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 present several all-interval 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
pitch-classes.
==============================
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).
==============================
[11] The exponential schemes present more dramatic
distinctions than linear ones, as the fractional ic vectors
(Appendix 1: Tables 2a, 2b, 2c) for the 12-note 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 12-note 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 pc-set, but their vectors are still closer to each
other than they are to those of other pc-sets, 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 pc-set-classes but which
share certain adjacency structures could generate vectors
which are closer to each other than to those of other
members of their own pc-set-classes, as measured by
numerical methods such as Isaacson's IcVSIM function.(15)
==============================
15. Isaacson, Eric. "Similarity of interval-class content
between pitch-class sets: The IcVSIM function." *Journal of
music theory* 34 (1990), pp. 1-28.
==============================
[13] However, I do not presume to suggest any standard
scheme, since the preferred 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 interval-class 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 interval-class vectors of
the 10 voicings of 8-24 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 interval-class 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 interval-class vectors
of the 15 voicings of 12-1
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 interval-class vectors
of the 15 voicings of 12-1
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 interval-class vectors
of the 15 voicings of 12-1
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 interval-class 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 interval-class vectors
' using fractional interval-class 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 pitch-class (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 (pitch-class, 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 interval-class 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
' interval-class, 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
+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+
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