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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
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
[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-
[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 
[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.
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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