Reply to Andrew Milne

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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John C.:
Thanks for the clarification on positive systems. Thanks also for the
elucidation on "doubly positive." I had suspected that was the connection, but
because Bosanquet didn't use that terminology, I wasn't absolutely sure.
Digging up a copy of XH 2, I see now that in Erv Wilson's classification of
tonal systems, he goes up to "triply positive" systems; however, he lists no
corresponding "doubly negative" (e.g. 50-tET) or "triply negative" (e.g.
33-tET) systems.

As far as using systems other than 12-tET to determine positive- and
negativeness, I need to study Wilson's article in XH 3 in much greater depth
before I can venture a coherent comment.

John

---
John G. Pusey
xen@tiac.net



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> John Chalmers said :
>
> Paul: I've had some discussion with Brian about the precise meaning of
> "Positive" system. I prefer to follow Bosanquet and call any regular cyclic
> system whose fifth is sharper than 700 cents (12-tet value) "positive," but
> Brian claims that there is a widespread and growing convention of reserving
> positive only for systems whose fifths are sharper than 3/2.

Maybe this is the same thing, but I though positive simply meant that the
"3/2" of the temperment was "above" of the "5/4" after four trips, rather
than dead-one as in a meantone system.

Or is this the same thing after you throw some math at it?

Bob Valentine

------------------------------

End of TUNING Digest 1465
*************************

Robert C Valentine wrote:
> Maybe this is the same thing, but I though positive simply meant that the
> "3/2" of the temperment was "above" of the "5/4" after four trips, rather
> than dead-one as in a meantone system.

Uhmmm no, or not as I understand it anyway, but it is that concept
relative to the Pythagorean comma rather than the syntonic comma. In
other words, a positive system where a stack of 12 of whose fifths is
larger than 7 octaves.