source file: mills2.txt
Date: Sat, 2 Mar 1996 05:51:19 -0800

Subject: Reply to Andrew Milne

From: COUL@ezh.nl (Manuel Op de Coul)

In response to Andrew Milne's question about propriety I can only 
point to the proper article, which I haven't read myself yet:

Balzano, Gerald J. "The group theoretic description of 12-fold and microtonal
pitch systems", Computer Music Journal vol. 4 no. 4, pp. 66-84, 1980.

Proper modes are I think not considered to somehow have more
validity, but to have cognitive advantages over nonproper modes.
For an exposition about it on the Web, go to Ken Overton's pages:
http://music.dartmouth.edu/~kov/lerdahl/
Then follow the links "Lerdahl's proposed cognitive constraints",
"Tuning and Temperament", "The argument for equal temperament",
"14. Uniqueness, Coherence and Simplicity", "The invocation of
Balzano".
The term used there is coherence. Below some text from these pages.
       
Manuel Op de Coul    coul@ezh.nl
-----

   13. Division of the octave into equal parts facilitates transposition
   and reduces memory load.
   
   This is one of the constraints that Lerdahl acknowledges "... can be
   variously jettisoned." (p. 114), which certainly seems plausible given
   that people presumably did enjoy music before equal temperament came
   into practice.
   
   So when should one jettison this constraint? It would seem plausible
   that someone who wishes different key areas to have distinct timbral
   qualities may find various Just tunings valuable. On the other hand,
   if a composer desires to write tonal music in which modulations occur
   often without any structural importance, this constraint may be heeded
   to good effect.
   
   14. Assume pitch sets of n-fold equal divisions of the octave. Then
   subsets that satisfy uniqueness, coherence, and simplicity will
   facilitate location within the overall pitch space.
   
   This is entirely derived from an article by Gerald Balzano in which he
   uses set-theory to find various relationships between different scale
   types. This is a fundamentally different approach to the cognition of
   pitch space which assumes that our scales are derived from tuning
   theory and the harmonicity of fundamental frequencies.
   
----
                           THE INVOCATION OF BALZANO
                                       
   The words Uniqueness, Coherence, and Simplicity come from a paper by
   Gerald Balzano published in MUSIC, MIND AND BRAIN, a collection of
   articles on the neuropsychology of music. It was edited by Manfred
   Clynes and published by Plenum Press (New York) in 1982.
   
   Balzano proposed the study of pitch sets as an alternative to the
   study of timbre or harmonicity for psychoacousticians. The bulk of his
   analysis was devoted to the 12-note collection: " ... depending on the
   tuning system (e.g. just, pythagorean, equal temperament), differences
   among these integers translate either approximately or exactly into
   log frequency differences." (p. 322) His analysis and discussion was
   devoted to various subsets of this 12-note collection, with extended
   discussion of the diatonic subsets.
   
   Below are the general definitions of each word. If you would like to
   read a more complete discussion of a word, click on that word.
   
     * Uniqueness: Uniqueness contributes to the emergence of a 'tonic'
       element. It refers to the ability to individuate the elements of a
       set by virtue of their relations with one another. If more than
       one element in a set have the same intervallic relationship to all
       the other members of that set, it is NOT unique.
       
     * Scalestep-Semitone Coherence: This requires that the same distance
       is not traversed in one part of the scale by one step and in
       another part of the scale by more than one step.
       
     * Simplicity of Scale Family: Simplicity is a criterion which
       guarantees transpositional equivalence by comparing patterns of
       overlap among scale family members and their vectors of relations.

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