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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
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| Volume 0, Number 10 September, 1994 ISSN: 1067-3040 |
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All queries to: mto-editor@husc.harvard.edu
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AUTHOR: Smoliar, Stephen
TITLE: Mathematical Logic: Response to Jay Rahn
KEYWORDS: mathematical logic
REFERENCE: mto.94.0.9.rahn.art
Stephen W. Smoliar
Institute of Systems Science
National University of Singapore
Heng Mui Keng Terrace
Kent Ridge 0511
SINGAPORE
smoliar@iss.nus.sg
[1] The title of Jay Rahn's article covers a lot of
terminological ground: "From Similarity to Distance; From
Simplicity to Complexity; From Pitches to Intervals; From
Description to Causal Explanation." Ultimately, however,
it seems to boil down to an exercise in the expressiveness
of first-order logic for articulating a theory of pitch.
In paragraph 2.3 he tips his hand by acknowledging that all
that has gone before (and much of what is to follow) is
based on the implicit premise that "the world is, in fact,
truly characterized according to first-order logic").
Since what is actually at stake here is the ability to
express some fundamental properties of auditory perception,
I think that this premise should not be allowed to slip by
without comment.
[2] The issue is not whether the world is "truly
characterized according to first-order logic." Those of us
who are interested in the world must draw upon ways to try
to describe it. This is as true when we engage in dialog
and try to argue out our descriptions and the inferences we
can draw from them as when we think about these matters in
solitude and basically try to describe the world to
ourselves. First-order logic is one way to approach this
descriptive task; but it is not the only way.
Furthermore, it is unclear that there is any standard of
rating which would allow us to conclude that it is the best
way (or that any other way is decisively better). All
approaches to description vary in effectiveness according
to the nature of the situation being described.
[3] So what situation are we trying to describe here? I
think there is a potential confusion lurking here because
*physical* stimuli and *perceived* stimuli are not the same
phenomena. Ultimately, it seems as if Rahn wants to tackle
describing the properties of perceived stimuli; but it
also seems as if he keeps falling back on properties of
physical stimuli.
[4] As a consequence, much of what we know about auditory
perception tends to ride along in the back seat.
Nevertheless, in paragraph 2.1, with the discussion of
Shepard tones, it looks as if perception might even get to
take the wheel; but Rahn backs away with the observation
that the "illusion" of Shepard tones "depends on temporal
succession." This is certainly true if you are only
interested in illusions of infinite ascent or descent; but
Diana Deutsch has demonstrated that there are more to
Shepard tones than this simple illusion. She demonstrated
this during a special "cognitive" session at the Oakland
SMT meeting in 1990, where all of us present got to serve
as subjects for one of her more classical experiments: If
you play the interval of a tritone in Shepard tones, it
turns out that just about any group of listeners will
disagree reasonably evenly as to which of the two pitches
sound higher. Thus, Rahn's AH predicate, while it makes
reasonable sense when applied to physical stimuli, is not
necessarily (and certainly not always) a well-formed
descriptor of how pitch is perceived.
[5] The key problem with Rahn's approach is that it
overlooks two fundamental properties of human perception
(which includes auditory perception). One is that it is
*context-dependent*, and the other is that it is
*subjective*. By virtue of either (or both) of these
properties, two physically identical stimuli may be
perceived as different. Within Rahn's descriptive
formalism, this would mean that, at the perceptual level,
one could have a physical thing be higher than itself, thus
violating one of the most important properties underlying
the predicates of his model.
[6] There are some other premises which also deserve some
questioning. One is in paragraph 1.9: "Behaviorally,
however, it is generally advantageous for a listener that
hears pitchwise to hear with optimum pitch acuity . . . ."
Actually, there seem to be behavioral advantages to
categorical perception. This is where (as is discussed in
Stephen Handel's *Listening* book) our acuity seems to be
deliberately "blunted," enabling us to categorize a rather
broad range of frequencies as all corresponding to a common
pitch. (This seems behaviorally advantageous, since none
of us, as performers, produce pitches with the frequency
acuity of machines; so we would not want to try to
perceive each other's performances as being so machine-
like.)
[7] Bringing the psychology of auditory perception into the
picture also throws a new light on the challenges of
nominalism. Much of what Rahn discusses hinges on being
able to come up with sufficiently accurate descriptors for
quantitative differences. However, during a discussion
session at the recent International Conference for Music
Perception and Cognition, I raised what I called the "One-
Two-Three-Infinity" Hypothesis about the perception of
quantitative differences. (The title is a debt to my
youthful enjoyment of the first anecdote related by George
Gamow in his book of the same name.) The point is that,
along some given metric, we may be able to recognize
perceptual differences of one, two, or even three units;
but, after some difference, we only recognize the distance
as being "a lot" and can no longer resolve it with the same
quantitative accuracy. I do not know of any psychological
experiments which have been designed to test this
hypothesis. However, given the different ways in which we
seem inclined to work with difference metrics, particularly
when comparing pitches or pitch classes, I think it is
about time that some of those experiments be designed and
run.
[8] In spite of the rather negative tone of this response,
I am as concerned as Rahn is with the challenges of
nominalism. While his punch line casts a net over two
millennia of music history, I think that there *is* a real
challenge in trying to identify just what is worth saying
for a theory of pitch. However, I question his use of the
world "else" in paragraph 5.1 because, as I have tried to
argue above, Rahn's AH predicate just does not cut the
mustard. To end on a more positive note, I would argue
that anything worth saying for a theory of pitch should be
grounded in a theory of perceptual categorization, such as
the one I suggested in my "Neuronal" paper in *In Theory
Only* (Smoliar, 1992). Establishing which of those
statements can be so grounded will again require
psychological experiments which have not even been
designed, let alone performed. However, any theory of
pitch must be a theory of how pitches are perceived, rather
than how they are produced; and I believe that a better
understanding of perceptual categorization is our best hope
for such a theoretical approach to pitch.
REFERENCE CITED
Smoliar, Stephen W. 1992. "Elements of a Neuronal Model
of Listening to Music." In Theory Only 12, 3-4: 29-46.
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