[1] A number of recent studies have considered
the measurement of
similarity between pcset classes.
[2] It is the concern with the effects of context on similarity judgments that motivates this paper. The paper begins first, however, with a critical examination of the state of research in the area of set-class similarity. Specifically, it articulates some unresolved issues surrounding this narrowly focussed aspect of similarity. After this close-up shot on set-class similarity, the paper begins to zoom out, considering next how we might take musical context into account when considering similarity of this sort. Zooming further out, it will look at how set class similarity might interact with similarity in other dimensions through an analysis of Schoenberg's Op. 19, No. 4. Finally, the paper looks at some more general issues relating to context- sensitive similarity, and briefly considers the place of these fairly limited notions of similarity in the larger context of human cognition.
[4] There are some unresolved questions pertaining to such
measures, however, and we begin by exploring these. First, what
does it mean for two set classes to be maximally or minimally
similar? This is important because the various
similarity
measures described in the literature do not fully agree about
what constitutes maximum and minimum similarity. For example,
under Lewin's REL, Rahn's ATMEMB, and Castren's RECREL,
[5] At the other end of the scale, does it make sense for non- identical set classes to be judged maximally similar? All measures which base their similarity rating only on interval- class content necessarily judge Z-related set classes to be maximally similar. My IcVSIM also considers sets to be maximally similar whenever their ic-vector entries differ by a constant value, the vectors thus having the same "contour." Set classes 5-11 {0,2,3,4,7} [222220] and 8Z15 {0,1,2,3,4,6,8,9} [555553] form such a pair, since adding 3 to each entry of 5-11's ic vector yields the vector for 8Z15. Measures based on the proportional distribution of the vectors' contained elements, such as Castren's RECREL, find pairs like the whole-tone set class 6-35 {0,2,4,6,8,10} [060603] and its lone pentachordal subset 5-33 {0,2,4,6,8} [040402] to be maximally similar. Lewin's REL_2_ function--that is, REL using the dyads as the TEST sets--also finds these set classes maximally similar. REL where TEST contains all set classes does not, however, because 6-35 contains an instance of 6-35 which 5-33 does not. The REL value of 0.97 (99.1%) reflects the subtle dissimilarity of these set classes. There is no question that this latter pair of set classes is particularly closely related, but it would seem desirable to judge non-identical set classes to be less than maximally similar. This is not possible when considering only ic content.
[6] Another question is how similarity should correlate with
inclusion. In their study of musical contour,
Marvin and Laprade
assert that "one of the most intuitively satisfying ways of
judging similarity in csegs [contour segments] of differing
cardinalities is to count the number of times the smaller cseg is
embedded in the larger".
[7] Metaphorically, we might consider how the relationship between a set and its (perhaps multiply embedded) subset parallels the relationship between a daughter and her mother. To judge the similarity of a daughter and her mother, we compare their features: hair color, nose shape, laugh, temperament, and so on. Though similarity in any of these features may result from their genetic lineage, lineage is not the same as similarity. After all, there are certainly instances in which one pair of unrelated people is more similar in appearance than another pair with shared genes (consider Jay Leno's Dancing Itos, or the world's overabundance of Elvis look-alikes, for example!). Likewise, multiple embedding of set class X in set class Y may make X and Y similar, but not necessarily more similar than Y and a third set, Z. Thus, while the similarity between musical objects may correlate with an embedding relationship--and often does to a large, if not perfect extent--it need not.
[8] One such case is illustrated in Table 1. The table shows the relationship between the embedding number (EMB) and two similarity measures, REL (with all set classes in TEST) and RECREL (the former of which is based explicitly on the embedding number), for set class 7-28 and the tetrachords. The choice of similarity functions here is insignificant; other functions yield similar results. For each function, the tetrachords are sorted according to their similarity to 7-28. Nineteen four-note set classes are subsets of 7-28, with four set classes (4-12, 4-13, 4-18, and 4-27) occurring as many as three times. Ten tetrachordal set classes are not subsets of 7-28. If the embedding number and the similarity values were maximally correlated, the embedding numbers would decrease as one moves down the list. While those set classes occurring as subsets of 7-28 three times are among those set classes most similar to 7-28, a number of 7-28 subsets are found far down both lists, including set-class 4-28, which is the tetrachord least similar to 7-28 according to both functions. On the other hand, set classes not found in 7-28 can be found as high as the thirteenth position in the REL list and tenth in the RECREL list.
TABLE 1.
Comparison of EMB numbers with REL and RECREL values for
each tetrachord and set class 7-28. Listed within the two major
columns are (1) the Forte number, (2) prime form, (3) interval-
class vector, (4) percentile value for that set class and 7-28
under the given function, and (5) embedding number for that set
class and 7-28. Each major column is sorted on the percentile
column. The percentile value reflects the position of a given
value among the values produced by all pairs involving set
classes of size 2 through 10. This arranges values for both
functions on a continuous scale from 100 (maximum) to 0 (minimum
similarity).
Cardinality 4 in 7-28 {0,1,3,5,6,7,9} [344433]
REL Pctl. EMB RECREL Pctl. EMB 4Z15 {0146} [111111] 54.1 2 4Z15 {0146} [111111] 82.8 2 4Z29 {0137} [111111] 54.1 2 4Z29 {0137} [111111] 82.8 2 4-12 {0236} [112101] 49.5 3 4-12 {0236} [112101] 58.3 3 4-27 {0258} [012111] 49.5 3 4-27 {0258} [012111] 58.3 3 4-13 {0136} [112011] 46.6 3 4-18 {0147} [102111] 51.0 3 4-18 {0147} [102111] 46.6 3 4-13 {0136} [112011] 50.3 3 4-11 {0135} [121110] 40.4 2 4-11 {0135} [121110] 45.7 2 4-5 {0126} [210111] 39.1 1 4-5 {0126} [210111] 42.0 1 4-16 {0157} [110121] 39.1 1 4-16 {0157} [110121] 42.0 1 4-2 {0124} [221100] 33.8 2 4-4 {0125} [211110] 37.9 0 4-22 {0247} [021120] 33.8 2 4-14 {0237} [111120] 37.9 0 4-19 {0148} [101310] 31.2 2 4-2 {0124} [221100] 27.7 2 4-4 {0125} [211110] 30.6 0 4-22 {0247} [021120] 27.7 2 4-14 {0237} [111120] 30.6 0 4-17 {0347} [102210] 26.5 0 4-24 {0248} [020301] 28.6 2 4-3 {0134} [212100] 26.1 0 4-21 {0246} [030201] 27.1 2 4-26 {0358} [012120] 26.1 0 4-6 {0127} [210021] 25.4 1 4-10 {0235} [122010] 25.6 0 4-8 {0156} [200121] 25.4 1 4-7 {0145} [201210] 24.2 0 4-3 {0134} [212100] 23.3 0 4-20 {0158} [101220] 24.2 0 4-26 {0358} [012120] 23.3 0 4-19 {0148} [101310] 23.6 2 4-10 {0235} [122010] 22.6 0 4-8 {0156} [200121] 21.4 1 4-17 {0347} [102210] 22.6 0 4-25 {0268} [020202] 19.8 1 4-25 {0268} [020202] 22.6 1 4-6 {0127} [210021] 18.9 1 4-7 {0145} [201210] 20.8 0 4-21 {0246} [030201] 18.6 2 4-20 {0158} [101220] 20.8 0 4-24 {0248} [020301] 18.2 2 4-9 {0167} [200022] 18.1 1 4-1 {0123} [321000] 14.6 0 4-1 {0123} [321000] 15.5 0 4-23 {0257} [021030] 14.6 0 4-23 {0257} [021030] 15.5 0 4-9 {0167} [200022] 13.8 1 4-28 {0369} [004002] 15.0 1 4-28 {0369} [004002] 9.2 1
[9] Another issue is whether all instances of an interval class
(or other embedded subset) contained in a set class should be
considered equivalent. Existing measures of ic similarity, for
example, treat the fourth instance of an ic as having the same
"value" as the first. Yet the difference
between having one
instance compared to zero of some ic is proportionately more
dramatic than the difference between having four compared to
three.
[10] Any of the similarity measures based on
difference vectors,
such as Morris's SIM
The first row of graphs shows the ic vectors for the two sets and their difference vector. Since the entries in the difference vector are all 1, IcVSIM judges the sets to be maximally similar (IcVSIM=0.0). The second row of graphs shows the vector entries scaled by the square root function. The difference vector at the lower right shows that the extra ic1 in 7-1 is much less significant (0.21) than the extra ic6 (1.0). The difference vector is no longer level and the similarity value of 0.273 puts these two sets at the 78th percentile of all ISIM2 values, compared to the 100th percentile under IcVSIM.
[11] Another factor not taken into account in the existing
similarity literature is the qualitative differences between the
interval classes. For example, do we hear ic1 and ic2 as equal
in dissimilarity to, say, ic1 and ic5? Might
their similarity be
affected by factors such as relative consonance or dissonance?
(Interval-classes 1, 2, and 6 are relatively dissonant, for
example, while ics 3, 4, and 5 are relatively consonant
[12] Having now backed into the issue of perception, let us now
consider it more explicitly. There is a sometimes unstated, and
always unsubstantiated, claim among the authors of similarity
measures that there is a correlation between theoretical and
actually perceivable similarity. Morris, for
example, suggests
his measure would provide "a rationale for the selection of sets
that insure predictable degrees of aural similitude."
[13] Stepping back a little further, we can now consider this
more interesting and more difficult question: How does musical
context affect our judgment of similarity? Measures of set-
class similarity must assume that all pcs in a set are equally
"connected" and implicitly treat the intervals found between them
as perceptually equivalent. But once a set class is
instantiated, the salience of its members will vary, and the
connection strength between those pitches will likewise vary,
influencing our aural picture of a musical segment's interval-
class content. Factors that affect the relative salience of
pitches include their registral placement, dynamics, duration,
and timbre. Lerdahl includes these and other
salience conditions
in an engaging article in which he proposes an extension of his
and Jackendoff's theory of tonal prolongational structure to the
atonal repertoire.
[14] Consider Examples. 2A and B, which show two
instances of the segment
[15] Consider now Examples 2C-E which show three realizations of the
chromatic hexachord, set class 6-1, whose ic vector is [543210].
The version in Example 2C--the set in its "prime" form--emphasizes
the set class's chromatic features, though I think we would be
inclined to hear the other ics in roughly the same proportion as
they are found in the ic vector.
In Examples 2D and E, rhythmic
differentiation is introduced. Because their attack points are
closer together, we are inclined to group a short note with a
following rather than a preceding long note. Thus, in Example 2D,
the semi-tonal connections, indicated on the example by slurs,
would be heard more strongly than the whole-tone connections.
This is reversed in Example 2E, where the temporally more proximate
pitches are related by whole step. We might thus expect a
contextually derived interval-class vector for Example 2D to weight
ic1 more than a similar vector for Example 2E.
Likewise, Example 2E's
vector would reflect the greater emphasis on ic2, relative to Example
2D. The issue of ic weighting in large vertical
structures is
explored by Robison, who suggests that the greater the registral
distance between two pitches, the less the interval-class between
those pitches should count in the ic vector.
[16] We have been concerned to this point with a narrowly defined
notion of "similarity"--the intervallic or subset similarity of
pitch-class sets. As noted at the beginning of this paper,
however, in most Western music musical meaning derives
substantially from the manipulation of similarity and
dissimilarity in various musical dimensions. Frequently, musical
objects being compared will be similar in some ways and
dissimilar in others. This is clearly central to such concepts
as "motivic development" and "theme and variations," but it plays
a role in many other ways. There are numerous other dimensions
along which we might measure similarity. These would include,
but certainly not be limited to, contour, rhythm, metric
orientation, register, distribution in pitch-space, textural
deployment (vertical versus horizontal), location within the
overall texture, articulation, dynamics, and timbre. Similarity
in some non-pitch parameters has been the formal subject of
recent studies. For example, Marvin and Laprade,
and Morris,
discuss contour similarity, and Marvin discusses rhythmic
contours in the music of Varese.
[18] In the opening bars of Op. 19, No. 4 (Ex. 3),
a melodic fragment is interrupted by a short, accented dyad.
These two gestures are dissimilar in many ways: one is melodic,
the other harmonic; one extends over two bars, the other lasts
just a 32nd note; the melodic gesture is marked piano, while the
dyad is forte (dynamics are not shown on the example).
The
gestures are also dissimilar intervallically. The melodic line,
set-class 5Z17 {0,1,3,4,8} [212320], lacks only a tritone; the
dyad with which it is juxtaposed is exactly that interval class,
set-class 2-6. The ISIM2 value for these sets is at just the 5th
percentile, reflecting strong dissimilarity. These contrasts in
various dimensions establish a theme for the rest of the piece.
[19] The following phrase, shown in Example 4, is like
the first in many ways. A melodic line--with more notes than
that of the first phrase, but with an obviously derivative
general contour--is again interrupted by a shorter harmonic
gesture, marked with a contrasting articulation (legato versus
staccato).
The example shows the melodic line divided into two
segments based on the recovery of the initial register upon the
leap up to D5. As in the first phrase, the contrasting parts of
the texture also contrast with respect to their intervallic
similarity. The ISIM2 percentile values for the first melodic
segment compared to the two harmonic segments are 8.0% and 12.7%,
while those for the second melodic segment compared to the
harmonic segments are 0.3% and 0.8%. Consider now, however, the
relation between the two melodic segments. Although they are
clearly similar in many ways--rhythm, register, general contour,
articulation--intervallically they are quite dissimilar (2.9%).
Likewise, the two components of the harmonic gesture at the end
of the phrase are also very dissimilar (2.5%). This becomes a
pattern in this piece. With one exception, adjacent musical
segments which are similar in one or more ways are made up of
distinctly dissimilar set classes.
[20] Examples 5 and 6 show this for two of the three remaining
phrases of the piece. In Example 5,
the chords in m.
6 have an ISIM2 value below the 10th percentile, while the ISIM2
percentile for the two melodic segments is 4.0%.
Finally, in
Example 6, the two chords that accompany the closing
melodic gesture have an ISIM2 value below the 10th percentile.
Each of the phrases shown in these examples seems to contain a
sense of progression through a change in intervallic content
within segments that are otherwise similar in one or more
"surface" ways. Though I am not sure that my hearing isn't being
affected by my analysis, I am certain that I hear the piece going
this way.
[21] The one exception to this pattern occurs in the rapid figure
played between the previous two examples (Ex. 7).
The melody here divides into two segments which are similar in
overall contour, register, rhythm, and, uniquely in this piece,
ic content (87.6% according to ISIM2).
This measure goes by very
quickly, and again I may be imagining this, but I do not hear the
sense of progression or contrast that I hear in the earlier
examples.
[22] This is a natural point to discuss the crucial role
segmentation plays in analysis of this sort. The identification
of musical segments can have a powerful effect on similarity
relations. Adding one note to a five-note segment, for example,
increases the number of interval classes by 50%. Depending on
the circumstances this can sometimes change substantially the
intervallic makeup of that segment and thereby affect the
relations between that and other segments. The extent of the
effect, however, depends on what interval classes are actually
added.
[23] This issue is explored in Example 8, which reconsiders the
melodic line of mm. 3-5, a passage originally examined in Example 4.
As shown in Example 8A, this melodic line includes four non-
overlapping major seconds (set class 2-2), with an additional M2
between the last two notes.
The saturation of this interval
provides considerable unity to the melodic line. Yet combining
these dyads in different ways, or segmenting the notes in ways
which do not preserve this feature, reveals dissimilarities on a
larger-scale. In Example 4, for example, this line was divided into
two segments based on recovery of the higher register, 6-2 and
5-33, strongly dissimilar sets (their ISIM2 value ranks at 2.9%).
[24] Examples 8B-D show three other possible
segmentations.
The apparent dissimilarity between
segments 2 and 3 (21%) is exaggerated by the fact that similarity
ratings involving dyads tend to be quite low for most similarity
functions because of the paucity of interval-classes.
Nevertheless, segment 3 is notably more similar to segment 4 than
to segment 2. Considering non-adjacent segments, segments 1 and
3 are as similar as a dyad could be to 3-2, but segments 2 and 4
are decidedly dissimilar (7%), suggesting again a 2+2 grouping of
these segments. Finally, segments 1 and 4 are also quite
dissimilar (15%). Though there are some changes in the
particulars, the segmentation in Example 8B bears out the relations
shown in Example 4.
[25] The three-segment partitioning of the melody in Example 8C is
based on the underlying metric framework, the similarity in
articulation between beats 1 and 2 in m. 3, and the quasi-contour
inversion (high dyad--low dyad -> low dyad--high dyad) in those
same beats.
The resulting melodic set classes, 3-2, 4-23, and
3-6, are again highly dissimilar to one another, with the closest
relation, that between segments 1 and 3, reaching only 15%.
Including the anacrusis C# in the first segment seems justified,
given the parallel with the opening of the movement. Doing so,
however, changes things somewhat. In particular, the new melodic
segment 1 (including the C#) and segment 2 are more closely
related, having reached the middle third of ISIM2 values (36%).
Lewin's RELt function, in fact, rates these sets at the 71st
percentile among RELt values. (This is so because 4-10 (0235)
and 4-23 (0257) each contain two (025) subsets.) On the other
hand, including the C# anacrusis increases the dissimilarity
between segments 1 and 3--in agreement with the other
segmentations considered so far. What segmentation 8C shows that
the others do not is the repetition, not just of set class, but
pitch-class content as well, between melodic segment 2 and the
first left-hand segment of m. 4, 4-23 (see Ex. 4).
[26] One last segmentation, Example 8D, divides the twelve
pitches into three groups of four notes, each beginning with a
16th-note anacrusis.
Though this segmentation is perhaps more
salient visually than aurally, it shares with the others a
notable dissimilarity between the earlier segments and the final
segment--a dissimilarity which persists despite unity within
dimensions such as rhythm, articulation, and small-scale
intervallic organization. It agrees nicely, however, with the
contour relations among the three segments. Segments 1 and 2,
which could be termed "neutrally" similar (47%), share a contour
feature in which their "downbeats" are approached from below and
"resolve" downward in the manner of an appoggiatura. Segment 3,
which is strongly dissimilar to the other segments, is approached
from above and "resolves" upward.
[28] This argues for continuing to consider similarity on an
individual parameter basis, rather than looking for a generalized
similarity index. In the detailed study of individual works, a
generalized index would sacrifice much interesting information.
(Deciding which parameters to include and how to weight them
seems like a impossible task, as well.) But comparing similarity
in two or more selected dimensions as above might be useful
sometimes. Specific types of similarity relationships could be
used in defining musical features.Klangfarbenmelodie, for
example, could be defined as a negative correlation between
textural stratum and timbre. And observations such as those made
above in connection with Op. 19, No. 4, help us understand more
about the musical "vectors" at work in a piece. On the other
hand, there are undoubtedly times when a generalized notion of
similarity would be useful, too. The music history course
exercise of distinguishing the music of Mozart and Haydn is a
categorization problem in which, through a sometimes (but not
always) unconscious comparing of various musical parameters
against remembered features of each composer and his music,
students attribute a piece to one or the other composer.
[29] To these ends, continued work at developing and refining
meaningful similarity measures for different musical parameters
would be useful. Similarity of various sorts is implicated in
such varied musical concepts as motive, phrase, theme, contrast,
variation, development, recapitulation, cadence, meter, form,
pitch class, interval, set class, instrument families, and
register. It would be fruitful to evaluate more systematically
the relative significance of the various parameters for evoking
musical meaning, since it is through the manipulation of
similarity in various of these domains that composers communicate
meaning. In which parameters do similarities and differences
help create structural boundaries in a piece? Which help to
relate sections to one another? Which help us understand more
local organizational levels? Which help to define a particular
composer's style? Do some seem to work in combination with
others? The role of context in listening needs more explicit
attention as well. Goldstone, for example,
discusses five types
of context which can influence experiments on similarity judgment
(not musical similarity specifially): cultural context,
perspectival context, recent (laboratory) context, concurrently
displayed information, and "context that is created by subjects
when there is none."
2. Robert Morris comments on the compositional potential
of
similarity relations in "A Similarity Index for Pitch-class
Sets,"Perspectives of New Music 18 (1979-80): 446. In his
dissertation, James Bennighof bases a composition on John Rahn's
similarity function, ATMEMB. See James M. Bennighof, "A Theory
of Harmonic Areas Defined by Pitch-class Sets" (Ph.D.
dissertation, University of Iowa, 1984); and John Rahn, "Relating
Sets," Perspectives of New Music 18 (1979-80): 483-497.
3. Thomas R. Demske, "Relating Sets: On Considering a
Computational Model of Similarity Analysis,"Music Theory Online
1.2 (1995). Lewin's remarks were broadcast to mto-talk, the
electronic discussion forum associated with Music Theory
Online, on 30 March 1995. The comments are available in the MTO
archive under the filename mto-talk.march95
(ftp://societymusictheory.org/pub/mto/mto-talk/mto-talk.march95).
4. See
http://cognitrn.psych.indiana.edu/rgoldsto/papers.html for
a list of Goldstone's papers.
5. John Rahn, "Toward a Theory of Chord
Progression."In Theory
Only 11/1-2 (1989): 9.
6. David Lewin, "A Response to a Response: On Pcset
Relatedness,"
Perspectives of New Music 18 (1979-80): 498-502; John Rahn,
"Relating Sets,"Perspectives of New Music 18 (1979-80): 483-497;
and Castren, "RECREL," pp. 101-125.
7. Throughout this paper, similarity values are often given with
a percentile figure. The percentile indicates where that
particular value falls in the context of all values produced by
that similarity function. For all functions, a percentile of 0
indicates minimum similarity, while a 100 indicates maximum
similarity.
8. Having invoked intuition, we must acknowledge that
this is a
sticky area, since what we call intuition is largely subjective.
There will always be situations where people's intuitions differ,
sometimes because of differences in musical experience, sometimes
because of different choices from among multiple possible
hearings. But while it seems somewhat slippery to insist, "Well,
that's how I hear it," absent some objective measure--whatever
that would be--it will have to do.
9. Elizabeth West Marvin and Paul Laprade, "Relating
Musical
Contours," Journal of Music Theory31 (1987): 237.
10. David Lewin, "Forte's Interval Vector, My Interval
Function,
and Regener's Common-note Function," Journal of Music Theory 21
(1977): 194-237. The embedding number counts the number of
members of an equivalence class contained in some specific set
and forms the basis for Lewin's REL function. For any set
classes X and Y, REL(X,Y) is based on the suitably scaled
summation, as Z ranges over the sets in TEST, of SQRT(EMB(/Z/,X)
times EMB(/Z/,Y)), where TEST might be the dyad classes to
measure intervallic similarity, or all set classes to measure
total subset similarity (Lewin, "A Response to a Response").
11. This point was raised by Michael Friedman in
response to a
paper I read at the 1992 meeting of the Society for Music Theory. Return to text
12. Robert Morris, "A Similarity Index for Pitch-class
Sets."
13. This and other ISIM measures using different scaling
functions are included in my WinSIMS and DosSIMS computer
programs. These programs are available for download from my
World-Wide Web home page
(http://ezinfo.ucs.indiana.edu/~isaacso/). Descriptions of the
scaling functions are also available there.
14. David Huron, "Interval-Class Content in Equally
Tempered
Pitch-Class Sets: Common Scales Exhibit Optimum Tonal
Consonance,"Music Perception 11 (1994): 289-305.
15. Morris, "A Similarity Index for Set Classes," p.
446.
16. Isaacson, "Similarity of Interval-Class Content,"
Ph.D.
diss., p. 251.
17. Castren, "RECREL," p. 148.
18. Fred Lerdahl, "Atonal Prolongational
Structure,"Contemporary
Music Review 4 (1989): 65-88; Fred Lerdahl and Ray Jackendoff,A
Generative Theory of Tonal Music (Cambridge: MIT Press,
1983).
19. Robert D. Morris, "New Directions in the Theory and
Analysis
of Musical Contour,"Music Theory Spectrum 15/2 (1993):
205-228.
20. Brian Robison, "Modifying Interval-Class Vectors of
Large
Collections to Reflect Registral Proximity Among Pitches,"Music
Theory Online 0.10 (1994).
21. Marvin and Laprade, "Relating Musical Contours";
Robert
Morris,Composition with Pitch-Classes (New Haven: Yale
University Press, 1987), esp. Ch. 2; Elizabeth West Marvin, "The
Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the
Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78.
22. Keith R. Orpen and David Huron, "The Measurement of
Similarity in Music: A Quantitative Approach for Non-parametric
Representations,"Computers in Music Research 4 (1991):
1-44.
23. An alternate segmentation associating the
anacrusis with the
downbeat chord rather than the following melodic line, and
dividing that melodic line into two 5-note segments, yields very
similar results, except that the melodic segments are more
similar to the chords than in the segmentation used here. I have
not shown the inter-textural relations to reduce visual clutter
in the example, except the relation between the second melodic
segment, 7-34, and the chord which is heard in m. 8, 4Z29. This
is the only pair of sets from different dimensions of the texture
that are closely related in this piece. The question of
alternate segmentations is addressed below.
24. The effects of various musical factors on grouping,
the
resulting possibility of multiple segmentations, and choosing
among these, is explored effectively in Christopher Hasty,
"Segmentation and Process in Post-Tonal Music," Music Theory
Spectrum 3 (1981): 54-73.
25. Robert. L. Goldstone, "Mainstream and Avant-garde
Similarity,"Psychologica Belgica 35 (1995): 145-165.
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ANALYTIC IMPLICATIONS
[17] We will focus here specifically on the interaction between
pcset similarity and similarity in other dimensions. Though
means of quantifying similarity in some of these other dimensions
either exist or could be proposed, my approach is sufficiently
informal to allow a more informal approach. Examples 3 through 8
are excerpts from Schoenberg's Kleine Klavierstuck, Op. 19, No.
4. In each example, musical segments are labeled with Forte
numbers, while similarity ratings of various segment pairs are
shown using the ISIM2 function described above. The ISIM2 value
is followed by the percentile ranking placing that value in the
context of ISIM2 values for all set-class. The percentile
ranking makes it unnecessary to interpret the ISIM2 values
(which, incidentally, range from 0 to 1.54). I should note
further that, although the various similarity measures do not
always agree so nicely, all the points I will be making hold,
save one or two, which are noted, if Lewin's REL or Castren's
RECREL functions are used instead.
CONCLUSIONS
[27] In the examples cited above, musical segments with very
similar surface features were often quite dissimilar in terms of
their intervallic makeup. On the other hand, it is equally
likely that in a given composition musical segments which are
similar in an abstract sense may be realized in very dissimilar
ways in the music. Just as people have features which may or may
not be similar (mouth and nose shape, eye and hair color, height
and weight, sense of humor, style of laughter, political
leanings, ethnic background, and so on), the similarity of
musical segments will vary across various musical features. So
how should we deal with the conflicts found in the various
musical parameters with regard to similarity? We should probably
revel in those conflicts since the tension between similarity and
contrast is central to the way much western art music works.
Isaacson, Eric J.
Indiana University
Department of Music Theory
Music Department
Bloomington
Indiana 47405
isaacso@indiana.edu
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References
1. Critical reviews of this literature appear in the
most recent
of these: see Eric Isaacson, "Similarity of Interval-class
Content Between Pitch-class Sets: The IcVSIM Relation,"Journal
of Music Theory 34 (1990): 1-28; Isaacson, "Similarity of
Interval-class Content Between Pitch-class Sets: The IcVSIM
Relation and Its Application" (Ph.D. diss., Indiana University,
1992), esp. pp. 12-135; and Marcus Castren, "RECREL: A Similarity
Measure for Pitch-classes" (Ph.D. diss., Sibelius Academy
(Helsinki), 1994), esp. pp. 16-100.
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3/3/96