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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
+-------------------------------------------------------------+
| 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: Parncutt, Richard
TITLE: Response to Demske: Relating sets
KEYWORDS: similarity, pc-sets, timbre, pitch commonality
REFERENCE: mto.95.1.2.demske.art
Richard Parncutt
Depts of Psychology and Music
Keele University
Keele, Staffordshire
ST5 5BG
Great Britain
r.parncutt@keele.ac.uk
[1] Thomas Demske's thorough treatment of the similarity of pc-sets
demonstrates how problematical it can be to derive a general and yet
musically relevant similarity function. The essay highlights inherent
weaknesses in the pc-set paradigm for post-tonal analysis, and
suggests that perceptually based approaches may be more appropriate
than approaches based purely on pc-sets.
[2] First, there seems to be a tacit problem of definition. What
exactly does it mean for two pc-sets to be similar? If we are to speak
meaningfully about the similarity of two different pc-sets, then we
must first of all satisfy ourselves that one and the same pc-set is
highly similar to itself.  Unfortunately, even this apparently trivial
condition is not satisfied. To take a simple example: A melodic
statement of {01369} sounds entirely different from the same tones
heard as a sonority. And different voicings (inversions, spacings) of
that sonority can sound more different from each other than similar
voicings of different sets. It's familiar stuff: Octave equivalence
ain't always valid.
[3] *Perceptual* similarity may be a more promising starting point for
a theory of similar pitch structures in atonal music. Demske makes
several references to perception in his essay. Perceptual similarity
is easy to define: It is the average subjective judgment of global
similarity by a representative group of listeners. Theorists may be
included as one of the groups. Of course, the results depend on
musical expertise and experience -- as does the perception of
listeners in the concert hall.
[4] Unlike a pc-set-based theory of similarity, a perceptual theory
must account for effects of voicing, onset asynchrony, spectral
envelope, temporal envelope, and so on. Consider first an isolated
pair of steady-state complex sonorities of the same loudness and
duration. Their global similarity breaks down into similarity of pitch
and similarity of timbre. Similarity of pitch in turn breaks down into
two parts, depending upon whether individual pitches are perceived to
fall in the same category (chromatic scale degree) or different
categories. These two parts may be called pitch commonality and pitch
distance (respectively); tentative algorithms are given in my book
*Harmony: A Psychoacoustical Approach*, Springer 1989, and in my
recent article in PNM. Different listeners emphasize different aspects
of pitch similarity in their responses, depending on their orientation
and experience.
[5] Alternatively, we might look at the similarity of two melodic
fragments.  That depends on the similarity of their contours and of
their underlying scales; for details see papers by Annabel Cohen, Jay
Dowling, Marilyn Boltz, Mari Riess Jones, Lola Cuddy. This is quite a
different affair from the similarity of sonorities, and needs to be
treated independently.
[6] For an appropriate set of stimuli for a perceptual experiment on
similarity, we need not look past the piano chords in Messiaen's
*Quatuor pour le fin du temps* analyzed by Demske. A possible
experimental paradigm might involve presenting the chords in pairs to
listeners and asking them to rate their global similarity. Then, model
the results as a linear combination of pitch commonality and pitch
distance. Finally, wonder about the effect of context on similarity
judgments. Analogous effects in tonal music have been studied in some
detail (see Carol Krumhansl, *Cognitive Foundations of Musical Pitch*,
OUP 1990).
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END OF MTO ITEM