Contest!

>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).

> Has anyone seen the following tuning before?
Yes, in just form it is the same as Fokker's 7-limit 12-tone just scale:
15/14 9/8 7/6 5/4 4/3 45/32 3/2 45/28 5/3 7/4 15/8 2/1
which you found in that manual. It is the same mode as yours from Fokker's
31-tone
just scale. In another key (on E) the scale has also been invented by Gary
David in 1967:
16/15 9/8 6/5 9/7 4/3 7/5 3/2 8/5 12/7 9/5 28/15 2/1

Manuel Op de Coul coul@ezh.nl

On Fri, 16 Oct 1998, Dave Keenan wrote:
> Hi, I'm new to this list. Greetings from Brisbane, Australia.

Welcome.

> Has anyone seen the following tuning before?
> [snip]
>
> C Db D D# E F F# G Ab A A# B C
> 3 2 2 3 3 2 3 3 2 2 3 3 (number of 31-TET steps between notes)

I've been working with various 9-out-of-31 scales lately. One of the
most interesting is

3-4-3-3-5-3-4-3-3

which is a subset of Dave's scale. It has 22 7-limit intervals (the
maximum possible with 9-out-of-31) and 3 complete tetrads. If you
invert the lower pentachord, giving

3-3-4-3-5-3-4-3-3

, the result has only 21 7-limit intervals but 4 complete tetrads, the
maximum possible with 9-out-of-31 and the only way to do so. (By
contrast, several 9-out-of-31 scales have 22 7-limit intervals, such as

3-5-2-5-3-3-4-3-3

, which I posted here some weeks ago.)

--pH http://library.wustl.edu/~manynote
O
/\ "Foul? What the hell for?"
-\-\-- o "Because you are chalking your cue with the 3-ball."

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