source file: m1555.txt
Date: Fri, 16 Oct 1998 17:44:18 -0400

Subject: Contest!

From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>

>I developed it after reading Paul Erlich's criteria for 7-limit
generalised-diatonic scales (in which he finds that only a 10 of 22-TET
is suitable)

I haven't PROVED that something significantly different from 10 of 22
can't work, but I doubt it. Also I know of no 9-limit or 11-limit
generalized-diatonic scales, but they might exist (I don't know how
important that would be, since the 9-limit and especially 11-limit
analogues of the minor chord sound pretty dissonant to me, despite
Partch's excellent use of them . . .). I hereby offer a prize of
unending praise and worship, and a promise to get a guitar capable of
playing it and to learn to play it, to anyone who discovers any new
generalized-diatonic scale. First some definitions:

Q in the below means the approximate 3:2 (Q is for quint, the latin =
term
for fifth).
A "characteristic dissonance" is an interval which is formed from the
same number of scale steps as a consonant interval, but is not
consonant.
For the n-limit, all intervals n:m, where m is less than n, and their
octave equivalents, are considered consonant.
The root of a chord is determined by the Q in it: the root is the note
representing 2 in the approximate ratio 3:2. If a chord has more than
one Q in it, you are free to use whichever root you want.

Now the rules:

(0) Octave equivalence:

There is a basic scale which repeats itself exactly at the octave,
extending infinitely both upwards and downwards in pitch.

(1) Scale structure:

EITHER a or b must be satisfied.

(Version a - distributional evenness): The basic scale has two step
sizes, and given these step sizes, the notes are arranged in as close =
as
possible an approximation of an equal tuning with only as many notes =
per
octave as the basic scale.

(Version b - tetrachordality): The basic scale has a structure
emphasizing similarity at the Q. In particular, there is a
"tetrachordal" structure, that is, within any octave span, the pattern
of steps within one approximate 4:3 are replicated in another
approximate 4:3, with the remaining "leftover" interval spanned using
patterns of step sizes (often just one step) found in the "tetrachord."

(2) Chord structure:=20

There exists a pattern of intervals (defined by number of scale steps,
not specific as to exact size) which produces a complete, consonant
chord (containing all non-equivalent consonant interval) on most scale
degrees.

(3) Chord relationships:

The majority of the consonant chords have a root that lies a Q away =
from
the root of another consonant chord.

(4) Key coherence:

A chord progression of no more than three consonant chords is required
to cover the entire scale.

(5) Tonicity:

The notes of the scale are ordered, increasing in pitch, so that the
first note is the root of a complete consonant chord, defined hereafter
as the "tonic chord."

The remaining rules come in "strong" and "weak" versions.
=A0
Strong Version
---------------------

(6) Homophonic stability:

All characteristic dissonances are disjoint from the tonic chord, with
the following possible exception: A characteristic dissonance may share
a note with the tonic chord if, when played together, they form a
consonant chord of the next higher limit (3 =DE 5, 5=DE 7, 7=DE 9).

(7) Melodic guidance:

The rarest step sizes are only found adjacent to notes of the tonic
chord.

Weak Version
--------------------

(6) Homophonic stability:

At least one characteristic dissonance either is disjoint from the =
tonic
chord, or shares a note with the tonic chord such that, when played
together, they form a consonant chord of the next highest limit (3=DE =
5,
5=DE 7, 7=DE 9).