===       ===     =============        ====
             ===       ===           ==            ==   ==
            == ==    ====           ==           ==      =
           ==   ==== ===           ==           ==      ==
          ==     ==  ==           ==            =      ==
         ==         ==           ==             ==   == 
        ==         ==           ==               ====
       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
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
Great Britain
[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).
Copyright Statement
[1] Music Theory Online (MTO) as a whole is Copyright (c) 1995,
all rights reserved, by the Society for Music Theory, which is
the owner of the journal.  Copyrights for individual items 
published in (MTO) are held by their authors.  Items appearing in 
MTO may be saved and stored in electronic or paper form, and may be 
shared among individuals for purposes of scholarly research or 
discussion, but may *not* be republished in any form, electronic or 
print, without prior, written permission from the author(s), and 
advance notification of the editors of MTO.
[2] Any redistributed form of items published in MTO must
include the following information in a form appropriate to
the medium in which the items are to appear:
	This item appeared in Music Theory Online
	It was authored by [FULL NAME, EMAIL ADDRESS],
	with whose written permission it is reprinted 
[3] Libraries may archive issues of MTO in electronic or paper 
form for public access so long as each issue is stored in its 
entirety, and no access fee is charged.  Exceptions to these 
requirements must be approved in writing by the editors of MTO, 
who will act in accordance with the decisions of the Society for 
Music Theory.