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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) 1995 Society for Music Theory
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| Volume 1, Number 3 May, 1995 ISSN: 1067-3040 |
+-------------------------------------------------------------+
All queries to: mto-editor@boethius.music.ucsb.edu or to
mto-manager@boethius.music.ucsb.edu
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AUTHOR: Hermann, Richard
TITLE: Towards a New Analytic Method for Post-Tonal Music: A
Response to Thomas R. Demske
KEYWORDS: similarity, atonal, post-tonal analysis, REL, set-theory,
ASIM, ATEMB, contour theory, multidimensional similarity
REFERENCE: mto.95.1.2.demske.art
Richard Hermann
University of New Mexico
Department of Music
Albuquerque, NM 87131-1411
harhar@unm.edu
ABSTRACT: In an article entitled "Relating Sets: On
Considering a Computational Model of Similarity Analysis,"
*Music Theory Online* 1.2 (1995), Thomas Demske criticizes
some older published similarity relations and points to some
general problems of analysis in post-tonal music. This
response sketches a new analytical method for post-tonal
music that places those similarity relations and other
theoretical tools of the recent past in the context of some
recent research and, in so doing, replies to some of the
issues Demske raises.
INTRODUCTION
[1] In his article "Relating Sets: On Considering a
Computational Model of Similarity Analysis," *Music Theory
Online* 1.2 (1995), Thomas R. Demske seeks to use techniques of
cluster analysis upon similarity relations found between pc-
sets "abstracted from post-tonal analysis." He finds that the
evaluation component that would define the boundary values for
cluster analysis (which group similarity function return
values between sets into rough equivalence classes) most
difficult to find, that "the similarity relationship is too
abstract to imply guidelines for its own application," and
that "other potential criteria resist formal implementation."
He further states that "other more commonly used tools in
post-tonal analysis are susceptible to the concerns raised
here."
[2] Demske raises some important issues about the inherent
nature and limitations of similarity relations, computational
models, and typically used analytical techniques for post-
tonal music. This response briefly reviews some his concerns
but will not alleviate them. Instead, this response places
similarity relationships into broader contexts and then
suggests how they might reasonably be used with other post-
tonal theoretical tools of fairly recent vintage. While my
suggested approach might ease his discomfort somewhat, other
interesting issues arise. Unfortunately, due to the broad
issues summoned by Demske and due to the limitations of space
in this forum, my response can only broadly sketch the
approach. It is hoped that this response might spark
continued responses upon the issues surrounding post-tonal
theory; recent composition; history, analysis, perception and
perhaps even the sociology of post-tonal music that Demske
reopens or implies.
FOUR OF DEMSKE'S DISSATISFACTIONS WITH ABSTRACT SIMILARITY
RELATIONS
[3] 1) The author seems to find fault with the fact that
similarity relations lack transitivity, although he does not
mention the transitive property in his essay. See his
footnote three where he writes "Blind subset polling is a
basic source of such barriers. Two REL calls with the same
pivot may yield identical results, and yet differ with respect
to the types of subsets counted. Ignoring the degree of this
difference when comparing REL value spreads strikes me as
questionable." Similar thoughts are found in paragraph [10].
[4] By definition, similarity relations lack transitivity.
Thus, if transitivity is valued so highly that its lack
becomes a standard for rejection of a theory, then much of
value will be lost to post-tonal analysis. For instance,
similarity relations might just be the class of tools best
used to describe how change is accomplished from one process
or segment to another differing process or segment within a
work. Similarity relations might also be of analytical use in
describing interesting instances of variation such as might
occur in Schoenberg's concept of developing variations. Other
compelling uses will be discussed shortly.
[5] 2) Demske sees as flawed a situation where some
collection of set-classes can be grouped together in different
ways by the similarity relation REL depending upon which pc-
set is selected as the "pivot" set-class. Again, see his
third footnote and paragraph [10]. The pivot set-class is the
set-class which is held as a constant in measuring the
similarity relation with each of the other set-classes.(1)
================================
1. Lewin has noted that Demske's use of REL is not completely
in accord with Lewin's definition. Instead of a single pivot
set-class, Lewin uses a collection of set-classes called TEST
selected from the local context. REL measures similarity of
other analytically interesting set-classes, collectively
called COMPARE, with those from TEST. See Lewin's mto-talk
message of 22 Mar 1995 and especially his "A Response to a
Response: On PCSet Relatedness," *Perspectives of New Music* 18.1-2
(Fall-Winter 1979, Spring-Summer 1980): 498-502 where the
definition of REL is found.
================================
[6] The flaw described in paragraph [5] above could be seen as
a virtue. For instance, when a member of set-class 3-11 [037]
is found at a temporal posterior border of an octatonic
collection and as a temporal anterior border of an abutting
diatonic collection, it seems reasonable and even desirable
for formal and abstract similarity relations to yield
different return values between the octatonic collection and
the member of set-class 3-11 and between the diatonic
collection and the 3-11 member in that context. The passage
of time through the passage does change the "color" of the
member of 3-11.
[7] 3) Demske finds that intuition can be mightily strained
in attending, in the abstract, to all of the possible
similarity relations that occur between all of the set-
classes. See his paragraph [12].
[8] While we need to be aware of the potentialities of
similarity in the abstract, we only need in analysis to attend
to those reasonable relationships pertaining to the piece or
segment under consideration.
[9] 4) The author seems to resist context sensitive criteria
such as a selected collection of set-classes to be used as the
metaphoric yardstick such as Lewin's TEST from which to
measure similarity. See his paragraph [13] and footnote
number four in that regard.
[10] Context sensitive criteria seem reasonable when the
repertoire under study has already shown that individual
pieces inhabit the post-tonal pc set world in very different
ways. Also, certain segments at various levels of formal
design within the same piece may also inhabit that world in
significantly different ways.
TWO PROBLEMS ENCOUNTERED IN THE ESSAY
[11] 1) At times Demske appears to veer between
formalist theoretical statements or claims and
phenomenological or perceptual observations and desires. He
judges one by the values of the other. In paragraph [16] he
writes: "The idea that any formal evaluation procedure could
embrace all of the [analytical segmentational] possibilities
seems untenable. On what bases would a partial set of
possibilities be selected for implementation? Since the
different criteria may address different---and possibly
conflicting--aspects of perception, how would the application
of one criterion be coordinated with that of another?"
[12] All analysts must make decisions about what strikes
them as the most salient or important features of the piece
and then select the appropriate "formal evaluation procedure"
designed to address those features. Certainly to run all
possible theoretical tools at the piece in an analysis would
quickly overwhelm the analyst with a plethora of analytical
observations upon the data: most observations are likely to be
true, but many may be of little significance, aural or
otherwise. Thus, the selection and use of theoretical tools
for analysis acts like a set of filters upon the analyst and
the piece at hand. It seems then that Demske's disagreement may
well lie with the selection of tools--that is, with
what is to be considered perceptually important, rather than
with the nature of the formalist tool itself.(2) If a
carpenter selects a hammer to cut a board, then poor results
can be expected.
==================================
2. In his mto-talk message of 30 March 1995, David Lewin
writes: "While the word [similarity] is suggestive, it might
be a good idea to stop using it in formal theoretical
discourse, because the intuitions it invokes are not all that
reliable. (Except we probably can't stop using it at this
stage of matters, ...."
I suggest that we retain the use of the word *similarity*
for formal relations that possess reflexivity and symmetry, but
lack transitivity as mathematicians would have it: see Seymour
Lipschutz, *Discrete Mathematics*, (New York: McGraw-Hill,
1976): 28. We then might use Robert D. Morris's phrase *aural
similitude*--from his "A Similarity Index for Pitch-Class
Sets," *Perspectives of New Music* 18/2 (1979-80): 445-60, as Demske
acknowledges in his paragraph 5--when we wish to discuss
perceptual matters. I also propose that, following a
suggestion of Morris's, we use the term *resemblance relation*
when we discuss relations that model inclusion relations
whether or not they are formally similarity relations. These
relations then potentially model some modest sense of "aural
similitude" in the abstract. Whether or not these relations
actually do model perception in a specific passage depends upon
whether the passage exhibits its materials in such a way that
encourages its perception with that tool by a reasonably
experienced listener.
See Richard Hermann, "A General Measurement for Similarity
Relations: A Heuristic for Constructing or Evaluating Aspects
of Possible Musical Grammars," Ph.D. Dissertation, Eastman
School of Music, University of Rochester, 1994: 1-119 for a
classification system for resemblance relations and a
mathematical and historical evaluation of published
resemblance relations. See pages 123-36 for discussion and
classification of all possible classes of resemblance
relations.
===================================
[13] 2) In the essay, the reader may get the sense that Demske expects
too much from any given class of theoretical tools such as the here
discussed similarity relations. For example, in traditional tonal
music the theory of harmony fails in explaining supertonic expansions
through voice- exchange and in progressions found within some
sequences. Consider, for example, Mozart's *Piano Concerto* number 9
in E-flat major, K. 271/II at measures 28 through 30 as an instance of
supertonic expansion, where a supertonic harmony at measure 28 is
followed by a tonic in measure 29 (an "illegal" harmonic elision); or
Bach's *Little Prelude*in C major, BWV 924, at measures one through
three, as an instance of a sequence where the harmonies go "backward"
along the circle of fifths (I-V-II- VI-III, "illegal" harmonic
retrogressions). These can be found respectively on pages 132-33 and
253-254 in Aldwell and Schachter's *Harmony and Voice-Leading*, 2nd
ed. New York: Harcourt Brace Jovanovich, 1989. The problem here is not
so much with the theory of harmony--although it does have its well-known
problems--but rather with knowing when the use of the theory of
harmony is appropriate. In these tonal instances, the effects of
structural outer voice counterpoint, form, rhythm and so forth inform
how the theory of harmony--*Stufen* in this case--is reasonably to be
employed. For instance in the case of the Mozart, analysts need to
realize that the tonic harmony is not functional but rather is the
result of harmonizing a deeper layer passing tone with an incomplete
neighbor--a contrapuntal relation--between the outer voices.
DEMSKE'S EXAMPLE OF ABSTRACT SIMILARITY RELATION FAILURE IN A
PASSAGE FROM THE FIRST MOVEMENT, *LITURGE DE CRISTAL*, OF
MESSIAEN'S *QUATUOR POUR LA FIN DU TEMPS*
[14] In paragraphs [14] through [17] and their accompanying figures,
Demske notes that--within the repeating sequence of 29 chords--his
intuitions of aural similitude run counter to the rough agreement
found in the analytical results of Lewin's REL, Rahn's ATEMB, and
Morris's ASIM similarity relations upon this chordal sequence. Their
results are elegantly displayed in Demske's Figure 5.(3) In the
context he has shown, Demske's complaint that these abstract
similarity relations yield poor results from the vantage of aural
similitude is clearly on target: the master's hammer was the wrong
tool indeed. Brian C. Robison, in responding to Demske's intuitions,
brings his own more appropriate tool to bear upon the passage: it
deals more directly with pitches of the score and gives a plausible
explanation for Demske's intuitions.(4) Clearly, ASIM, ATEMB, and REL
are too "coarsely grained" for this particular situation while
Robison's more "finely grained" work is here more suitable. Perhaps
in some other post-tonal music with frequent octave duplications the
more coarsely grained tools would better reflect aural similitude than
Robison's tool.
===================================
3. See John Rahn, "Relating Sets," *Perspectives of New Music*
18.2 (1979-80): 488-97 for information on his ATEMB. See footnote 1
above for information on Lewin's REL and footnote 2 above for
information on Morris's ASIM.
4. See Robison's mto-talk postings of 29, 30, and 31 March
1995 in this regard. For a more in depth look at his
theoretical work employed in those postings, see his
"Modifying Interval-Class Vectors of Large Collections to
Reflect Registral Proximity Among Pitches," *Music Theory
Online* 0.10 (1994).
For other theoretical work capable of addressing Demske's
concerns in this chordal sequence, see Robert D. Morris,
"Equivalence and Similarity in Pitch and their Interaction
with Pcset Theory," an unpublished mss. delivered at the
Society for Music Theory Annual Conference held at
Tallahassee, November, 1995 and Richard Hermann, "Theories of
Chordal Shape, Aspects of Linguistics, and their Roles in
Structuring Berio's *Sequenza IV for Piano*," an essay from
*Concert Music, Rock, and Jazz since 1945, Essays and
Analytical Studies,* Elizabeth West Marvin and Richard
Hermann, eds. Rochester, New York: University of Rochester
Press, forthcoming.
===================================
TWO FURTHER COMPLAINTS OF MINE ABOUT EXISTING RESEMBLANCE
RELATIONS
[15] 1) Until quite recently, resemblance relations
have typically concerned themselves with resemblance between
pitch-sets through the powerfully reductive concepts of pitch-
class and set-class. Resemblance relations that address
musical dimensions such as pitch-space, time, timbre, sound
source direction, and so forth have just recently begun to
appear.
[16] 2) Apart from some work on serial music, I am
aware of no work yet in print that simultaneously addresses
resemblance relations in more than one musical dimension. In
order to get a better fit between formal models and aural
similitude for some important pieces in the repertoire,
multidimensional resemblance relations need to be
investigated.(5)
===================================
5. For some first steps in that direction, see Larry
Polansky, "Morphological Metrics: An Introduction to a Theory
of Formal Distances," in Proceedings of the International
Computer Music Conference (San Francisco: Computer Music
Association, 1987) and Richard Hermann, "A General Measurement
for Similarity Relations:..": pp. 120-78 and "An Approach to
Multidimensional and Multisubdimensional Similarity for Post-
Tonal Music" delivered November 1995 at the Society for Music
Theory annual conference, Tallahassee, Florida. Subdimensions
can informally be understood through some examples:
subdimensions from the dimension of time are duration, metric
position, attack-point position, and so forth.
===================================
TOWARDS AN ANALYTICAL METHOD FOR POST-TONAL MUSIC: A
PROVISIONAL SKETCH
[17] Another way of looking at this situation is in speculating that
some sort of unknown or partially known "grammar" may well dictate
when and where various existing post-tonal analytical tools are to be
best employed. If that grammar is even partially known, it might even
suggest where gaps in our knowledge need to be filled. It may be
possible now to start the discussion of how some recent tools might
fit together with older ones within an overall sketch of a new
post-tonal analytical method. That we are likely to disagree on this
sketched method is highly probable; note how difficult it is/has been
to obtain general agreement on a precisely specified teaching
methodology for tonal analysis. And that lack of agreement is over a
repertoire that has enjoyed several hundred years of intense
theoretical contemplation and analytical study. Nonetheless, we gain
insight into what is lacking by evaluating how various methods of
combining tools fare. As pieces inhabit the post-tonal worlds in
different ways, I suspect that multiple methods of combining and
selecting tools will be necessary.
[18] Discussion of Demske's complaints about some abstract similarity
relations has shown that in order to get formal analytical results
that correlate with our sense of aural similitude, our tools must be
employed at the correct level of precision. In the Messiaen example,
the pitch-class/set-class similarity approach was counter-intuitive
while a pitch-space approach was more successful. Recent developments
in contour theory and investigations into different kinds of musical
spaces now suggest that post-tonal analysis can have a variety of
spaces available to it ranging from the most diffuse on up to the most
precise.(6) On the most diffuse extreme, spaces whose elements lack
identifiable intervals between them--called "preintervallic"--have
been investigated and those investigations have yielded interesting
insights into form and instrumental/relative register assignments.(7)
More precise are spaces inhabited by the relative or ordinally based
intervals found in the various contour theories. Here, only
inequalities can rank the elements within the space. Note that Marvin
and Hermann have also extended contour theory to musical dimensions
other than pitch.(8) Next in precision are those intervals such as the
familiar interval-classes that are collapsed from an infinite space
onto a finite space through modulo arithmetic. And yet greater in
precision are the absolute intervals, such as the distances between
equal-tempered pitches. Certainly other kinds of intervals lurk in
between these: see the Morris and Lewin writings of footnote six
above. Recent theoretical developments in post-tonal theory occurring
since ASIM, ATEMB, and REL have widened the scope for development of
resemblance relations in other musical dimensions and subdimensions
and have even started to show how they can be coordinated. See
footnotes two and five above. With research continuing in the fields
of musical spaces, resemblance relations in other musical dimensions,
and multidimensional or multisubdimensional similarity, analysts will
soon have a greater variety of tools to select in order to best model
their perceptions of aural similitude.
===================================
6. See Robert D. Morris, *Composition with Pitch-Classes*,
(New Haven: Yale University Press, 1987): pp. 23-7 and David
Lewin, *Generalized Musical Intervals and Transformations*
(New Haven: Yale University Press, 1987): pp. 16-30 for
discussions of interval--of varying kinds--based spaces in
several different musical dimensions.
7. See Richard Hermann, "On 'Preintervallic' Spaces and on
Their Interactions with Some Intervallic Spaces,"unpublished
mss. delivered November 1994 at the Society for Music Theory
annual conference, Montreal, Canada.
8. The following is a bibliography of recent writings on
contour theory by theorists: Michael L. Friedmann, "A
Methodology for the Discussion of Contour: Its Application to
Schoenberg's Music," *Journal of Music Theory* 29, 2 (Spring
1985): 223-248; Morris, *Composition with Pitch-Classes*, 26-
32; Elizabeth West Marvin and Paul A. Laprade, "Relating
Music Contours: Extensions of a Theory for Contour," Journal
of Music Theory 31, 2 (Spring 1987): 225-267; Michael L.
Friedmann,"A Response: My Contour, Their Contour," *Journal of
Music Theory* 31, 2 (Spring 1987): 223-248; 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; Larry Polansky and Richard S. Bassein,
"Possible and Impossible Melodies: Some Formal Aspects of
Contour," *Journal of Music Theory* 36, 2 (Fall 1992): 259-279;
Robert D. Morris, "New Directions in the Theory and Analysis
of Musical Contour," *Music Theory Spectrum* 15, 2 (Fall 1993):
61-78; Richard Hermann,"A General Measurement for Similarity
Relations: ...": 123-43; and Elizabeth West Marvin, "A
Generalization of Contour Theory to Diverse Musical Spaces:
Analytical Applications to the Music of Dallapiccola and
Stockhausen," in *Concert Music, Rock and Jazz since 1945:
Essays and Analytical Studies*.
===================================
[19] When faced with music by composers such as
Peter Maxwell Davies, Morton Feldman, or Ralph Shapey--to give only
a few examples in which there is no known or reasonably
convincing and widely accepted "grammar" to act as a guide in
analysis--how might we more profitably proceed?(9)
===================================
9. Much of the musical "grammar" has been well established
and accepted for the serial works of composers such as
Schoenberg, Webern, Berg, and Stravinsky and for those serial
composers such as Babbitt. So perhaps this methodological
sketch has less import for that galaxy of the post-tonal
universe. For a veritable *summa* of serial technique, see
Robert D. Morris's *Composition with Pitch-Classes*. For an
important and more recent study that greatly extends serial
combinatorial theory through a partitional approach, see Brian
Alegant's *The Seventy-Seven Partitions of the Aggregate:
Analytical and Theoretical Implications*, Ph.D. Dissertation,
Eastman School of Music of the University of Rochester, 1993.
===================================
[20] After the analyst becomes thoroughly familiar with
the music's sound and symbol, the first issue to be faced is
segmentation: into what units should the piece be divided?
Recent work by Polansky and Uno, following work by Tenney, use
principles of Gestalt psychology to create computer-based
formal procedural models for segmentation. These models are
explicitly multidimensional. After Tenney, formal segments are
called temporal gestalt units, and these formal units can nest
within one another several layers deep. Gestalt psychology
investigates the role of shape in perception. Thus, these
shape oriented theories of segmentation are likely to
coordinate well with the new contour space theories because they
are also concerned with shape in various dimensions. In this
light, contour theory can profitably be thought of as a kind
of abstract master dimension. Segmentations that arise from
formalist theories sensitive to the perceptual issues of shape
will go a ways towards eliminating Demske's complaint of
"blind subset polling." Lefkowitz and Taavola, Brinkman, and
Hasty have also made contributions to this developing area of
segmentation in post-tonal theory.(10)
===================================
10. See Larry Polansky, "Morphological Metrics: An
Introduction to a Theory of Formal Distances," in Proceedings of the
International Computer Music Conference, San Francisco: Computer Music
Association; Yayoi Uno, "The Roles of Compositional Aim, Syntax, and
Design in the Assessment of Musical Styles: Analyses of Piano Music by
Boulez, Cage, Babbitt, and Xenakis Circa 1950," (Ph.D.
dissertation, University of Rochester, 1994); David S.
Lefkowitz and Kristen Taavola, "Generalizing Segmentation: A
Multi-Dimensional Approach/Piece-Specific Weighting System,"
unpublished mss. given at the 1993 New England Conference of
Music Theorists; Christopher F. Hasty, "Phrase Formation in
Post-Tonal Music," *Journal of Music Theory* 28, 2 (Fall 1984):
167-190; and Alexander R. Brinkman, *Pascal Programming for
Music Research*, (Chicago: University of Chicago Press, 1990),
783-97.
For earlier foundational work in this area, see James
Tenney, *Meta + Hodas and META Meta + Hodas*, 2nd ed.
(Oakland: Frog Peak Music, 1988); James Tenney and Larry
Polansky, "Temporal Gestalt Perception in Music," *Journal of
Music Theory* 24, 2 (Fall 1980): 205-241 and their
*Hierarchical Temporal Gestalt Perception in Music: A "Metric
Space" Model* (Toronto: York University Press, 1978).
===================================
[21] The next issue to be faced might best be described by the
question: What levels of precision--that is what kinds of musical
space--best capture the intuitions of relatedness and dissimilarity in
the piece? Here, multidimensional or multisubdimensional similarity
relations can act as a heuristic to narrow the number of interesting,
potentially applicable musical spaces to those likely to yield the
best results. Those multidimensional or multisubdimensional
similarity relations yielding probabilistic return values scaled
through standard deviation techniques can do an analysis of the
potentials of the spaces themselves in the abstract, an *a priori*
analysis. Then, those results can be compared with empirically
derived results from the score: an *a posteriori* statistical and
probabilistic analysis. Where the two sets of results do not
significantly correlate, important features of a relevant grammar may
have been identified. Follow up analyses of the segments--selected by
shape based segmentation theories--could then be done using
equivalence class analysis, resemblance relations, and so forth,
designed for those specific musical spaces.(11) Relations, operations,
and transformations that significantly preserve shape are preferred to
those that do not preserve shape. This analytic process seems likely
to reveal a good ratio of "hearable" structures to other less
"hearable" structures. The fruits of those tailored analytical
processes can then be organized as strings of operators, networks of
various kinds, and so forth as is appropriate to the music.(12)
===================================
11. This approach has been employed with Luciano Berio's
*Sequenza IV for Piano solo*. See Hermann, "A General
Measurement for Similarity Relations: ....": pp. 201-34 and
252-54. The use of a redesigned contour theory along with a
probability based similarity relation scaled by statistical
techniques gives the flexibility to be able to deal
simultaneously with various different kinds of musical spaces
of differing cardinalities in an n-dimensional probability
space.
12. See Morris, *Composition with Pitch-Classes* for
technical information on groups of operators and their use in
compositional designs; and David Lewin, *Generalized Musical
Intervals and Transformations*: pp. 157-254 for technical
information on the design of networks for analytic use, and
his *Musical Form and Transformation: 4 Analytic Essays* (New
Haven: Yale University Press, 1993) for extended analyses
using networks.
Many of the issues raised in this response have been
commented upon by Jay Rahn. See his "From Similarity to
Distance; From Simplicity to Complexity; From Pitches to
Intervals; From Description to Causal Explanation," *Music
Theory Online* 0.9 (1994). Other pertinent works of his are
found in that article's reference list.
===================================
CONCLUSION: A WISH
[22] As these formalist techniques are all easily
amenable to computer implementation--and many have been so
implemented--I wish that these tools could all be found in one
big sophisticated suite of software. Pieces could be encoded
and loaded into database-like structures so that analysts
could then follow their intuitions and call up the needed
software tools designed for the specific kinds of musical
spaces desired and use them on the work.(13)
===================================
13. A database structure for electronically "holding" encoded
scores is already available. See Alexander R. Brinkman,
*Pascal Programming for Music Research*, (Chicago: University
of Chicago Press, 1990): pp. 137-154, 751-812, and 825-915.
===================================
[23] Although much theoretical work has been done since ASIM, ATEMB,
and REL made their first appearances in 1980--and only a small amount
of the work could be cited here--much more yet remains to be done in
understanding the various musical spaces and designing appropriate
equivalence class and resemblance relation tools for them. One
benefit of continuing this line of research is that "updating" ASIM,
ATEMB, and REL for use in other dimensions and for use in
multidimensional and multisubdimensional analysis is possible. Marvin
and Laprade have already updated ATEMB for use with their contour
theory (see footnote number eight). While other quite important
contributions and issues such as timbre theory, atonal voice-leading,
and the influence of feminist thought, among others, upon analysis of
this repertoire can only be barely mentioned or imagined here, it is
time to start the discussion of how and when it is appropriate to use
various combinations of these theoretical entities. Demske's reminder
about these issues deserves our serious attention, and calls for
discussion to begin.
REFERENCES CITED
Aldwell, Edward and Carl Schachter. 1989. *Harmony and Voice-
Leading*. 2nd ed. New York: Harcourt Brace Jovanovich.
Alegant, Brian. 1993. *The Seventy-Seven Partitions of the
Aggregate: Analytical and Theoretical Implications*. Ph.D.
Dissertation, Eastman School of Music, University of
Rochester.
Bach, Johann Sebastian. *Little Prelude* in C major. BWV 924.
Berio, Luciano. 1967. *Sequenza IV for Piano solo*. London:
Universal Editions.
Brinkman, Alexander R. 1990. *Pascal Programming for Music
Research*. Chicago: University of Chicago Press.
Demske, Thomas R. 1995. "Relating Sets: On Considering a
Computational Model of Similarity Analysis." *Music Theory
Online* 1.2.
Friedmann, Michael L. 1985. "A Methodology for the Discussion
of Contour: Its Application to Schoenberg's Music." Journal of
Music Theory 29, 2: 223-248.
Friedmann, Michael L. 1987. "A Response: My Contour, Their
Contour." *Journal of Music Theory* 31, 2: 223-248.
Hasty, Christopher F. 1984. "Phrase Formation in Post-Tonal
Music." *Journal of Music Theory* 28, 2: 167-190.
Hermann, Richard. 1994. "A General Measurement for Similarity
Relations: A Heuristic for Constructing or Evaluating Aspects
of Possible Musical Grammars." Ph.D. Dissertation, Eastman
School of Music, University of Rochester.
Hermann, Richard. 1994. "On "Preintervallic" Spaces and on
Their Interactions with Some Intervallic Spaces," unpublished
mss. delivered at the Society for Music Theory annual
conference, Montreal, Canada.
Hermann, Richard. 1995. "An Approach to Multidimensional and
Multisubdimensional Similarity for Post-Tonal Music"
unpublished mss. delivered at the Society for Music Theory
annual conference, Tallahassee, Florida.
Hermann, Richard. forthcoming. "Theories of Chordal Shape,
Aspects of Linguistics, and their Roles in Structuring Berio's
*Sequenza IV for Piano*." an essay from *Concert Music, Rock,
and Jazz since 1945, Essays and Analytical Studies.* Elizabeth
West Marvin and Richard Hermann, eds. Rochester, New York:
University of Rochester Press.
Lefkowitz , David S. and Kristen Taavola. 1993. "Generalizing
Segmentation: A Multi-Dimensional Approach/Piece-Specific
Weighting System." unpublished mss. delivered at the New
England Conference of Music Theorists.
Lewin, David. mto-talk postings of 22 and 30 March 1995.
Lewin, David. 1980. "A Response to a Response: On PCSet
Relatedness." *Perspectives of New Music* 18, 2: 498-502.
Lewin, David. 1987. *Generalized Musical Intervals and
Transformations*. New Haven: Yale University Press.
Lewin, David. 1993. *Musical Form and Transformation: 4
Analytic Essays*. New Haven: Yale University Press.
Lipschutz, Seymour. 1976. *Discrete Mathematics*. New York:
McGraw-Hill.
Marvin, Elizabeth West and Paul A. Laprade. 1987. "Relating
Music Contours: Extensions of a Theory for Contour." Journal
of Music Theory 31, 2: 225-267.
Marvin, Elizabeth West. 1991. "The Perception of Rhythm in
Non-Tonal Music: Rhythmic Contours in the Music of Edgard
Varese." *Music Theory Spectrum* 13: 61-78.
Marvin, Elizabeth West. forthcoming. "A Generalization of
Contour Theory to Diverse Musical Spaces: Analytical
Applications to the Music of Dallapiccola and Stockhausen." in
*Concert Music, Rock and Jazz since 1945: Essays and
Analytical Studies*. Rochester, New York: University of
Rochester Press.
Messiaen, Olivier. 1957. *Quatuor pour la Fin du Temps*.
Paris: Editions Durand and Co.
Morris, Robert D. 1980. "A Similarity Index for Pitch-Class
Sets." *Perspectives of New Music* 18, 2: 445-60.
Morris, Robert D. 1987. *Composition with Pitch-Classes*. New
Haven: Yale University Press.
Morris, Robert D. 1993. "New Directions in the Theory and
Analysis of Musical Contour," *Music Theory Spectrum* 15, 2: 61-
78
Morris, Robert D. 1995. "Equivalence and Similarity in Pitch
and their Interaction with Pcset Theory," an unpublished mss.
delivered at the Society for Music Theory Annual Conference,
Tallahassee, Florida.
Mozart, Wolfgang Amadeus. *Piano Concerto* no. 9 in E-flat
major. K. 271, 2nd mvt.
Polansky, Larry. 1987. "Morphological Metrics: An Introduction
to a Theory of Formal Distances." in Proceedings of the
International Computer Music Conference. San Francisco:
Computer Music Association.
Polansky, Larry and Richard S. Bassein. 1992. "Possible and
Impossible Melodies: Some Formal Aspects of Contour." Journal
of Music Theory 36, 2: 259-279.
Rahn, Jay. 1994. "From Similarity to Distance; From Simplicity
to Complexity; From Pitches to Intervals; From Description to
Causal Explanation." *Music Theory Online* 0.9.
Rahn, John. 1980. "Relating Sets." *Perspectives of New Music*
18, 2: 488-97.
Robison, Brian C. 1994. "Modifying Interval-Class Vectors of
Large Collections to Reflect Registral Proximity Among
Pitches." *Music Theory Online* 0.10.
Robison, Brian C. mto-talk postings of 29, 30, and 31 March
1995.
Tenney, James and Larry Polansky. 1978. *Hierarchical Temporal
Gestalt Perception in Music: A "Metric Space" Model*. Toronto:
York University Press.
Tenney, James and Larry Polansky. 1980. "Temporal Gestalt
Perception in Music." *Journal of Music Theory* 24, 2: 205-41.
Tenney, James. 1988. *Meta + Hodas and META Meta + Hodas*.
2nd ed. Oakland: Frog Peak Music.
Uno, Yayoi . 1994. "The Roles of Compositional Aim, Syntax,
and Design in the Assessment of Musical Styles: Analyses of
Piano Music by Boulez, Cage, Babbitt, and Xenakis Circa 1950."
Ph.D. dissertation, Eastman School of Music, University of
Rochester.
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