Computer-generated list of 65 7-note scales with at least one "quarter-tone", according to simple rules.

Because it's generally better to share work rather than keep it to one's self, I'm posting this here.

I *am* interested in people pointing out typos and other errors of mechanics. But let's keep any discussion limited to that kind of thing. Some of these are reminiscent of maqams, but the overlap is only partial. I've added a few names, and I'd like to add more, but I don't wish to argue. I'd rather start practicing. If anyone else wishes to do a similar project, I'd be interested to look at it. The rules are designed to be restrictive-enough to generate less than, say, a hundred scales. You can't practice, or conceive of, 2,000 scales.

I have checked these over extensively, but there may still be some typos and other inconsistencies. For example, my spelling of enharmonics (D# or Eb) is indifferent. I don't really care too much about that. Only a little.

*The project here is to define a simple set of rules, and have the computer generate the list.*

This approach could then be combined with any other approach to make a more complete list of quarter-tone scales, should anyone be so inclined.

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

Below are:

-The rules, with a link.

-The scales, with a link to a more-complete document in my Box account. There are 65, in 10 groups. The groups are related by interval content and by inversion. So they are modal shifts and inversions of each other, within a group. The ordering of the groups is arbitrary, except that the computer followed a rule of right-most packing going to left-most packing.

-The 5-or-6-note "no-wolves" list, in 5 groups. (These have no 650-cent or 750-cent intervals).

-A link to the computer program, by W.D Clinger. It's just a brute-force generate-and-test program.

-A list of "conventional" scales, for completeness.

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

-The rules, with a link.

Link: http://forums.randi.org/showthread.php?postid=9883286#post9883286

1) Scales are 7 notes.

2) Scales are *modular* (mod 24).

3) There would be 24 possible pitches (numbered 0 to 23), but pitches 1, 23, 13, and 15 are not allowed, so there are 20 possible pitches. (This is the principle of avoiding pitches too close to the "root" and "fifth" of the scale.)

4) Pitch 0 is always the first pitch, and it is never omitted.

5) Any 3 pitches can occupy a region no smaller than 7. That is, the pitch sequences 0,2,3, or 0,2,4 or 3,4,5 or 3,5,7 etc. are not allowed. This is also a rule for the highest numbers where they combine with the lowest, because the system is modular. That is, 21 & 22 are not allowed together, nor are 20 & 22, nor are 22 & 2, because that would make the combination 21,22,0 or 22,0,2. Note: This is a space two higher than where I started, I think. See examples.

6) An interval of 1 is not allowed between any pitches. Up to two intervals of 13, or 15 (mod 24) are allowed anywhere between any pitches.

7) Scales must have at least 1 odd value (a "quarter-tone").

-The scales, with a link to a more-complete document in my Box account.

Link: https://app.box.com/s/sp97ixl1u7lnnlbmukzm (You may need to re-format this, or resize the window. It looks ok in TextEdit.)

A,C-,D,Eb,F,F#,G#
A,B,C,D#-,F,F#,G#
A,B,C,D,Eb,F#-,G#
A,Bb,C#-,D#,E,F#,G
A,Bb,C,C#,D#,E,G-

A,C-,C#,D#,F,F#,G#
A,B,C,D#-,E,F#,G#
A,B,C#,D,E,F,G#-
A,Bb,C#-,D,E,F#,G
A,B,C,D,Eb,F#-,G
------------------------------------------
A,B,C#,D#-,F,F#,G#
A,B,C,D,E,F#-,G#
A,B,C#-,D#,E,F#,G
A,B-,C#,D,E,F,G
A,Bb,C,Db,Eb,F,G-

A,C-,C#,D#,E,F#,G#
A,B,C,D,E,F,G#-
A,B,C,D#-,E,F#,G
A,Bb,C,D,Eb,F#-,G
A,Bb,C#-,D,E,F,G
-----------------
A,B,C#-,D#,E,F#,G#
A,B,C#,D,E,F#-,G#
A,B-,C#,D,E,F#,G
A,Bb,C,D-,E,F,G
A,Bb,C,D,Eb,F,G-

A,B,C#,D#-,E,F#,G#
A,B,C#,D,E,F#,G#- Jiharkah/Ajam
A,B,C,D,E,F#-,G
------------------------
A,B,C#-,D,E,F#,G Rast
A,B-,C,D,E,F,G

A,B,C#-,D,E,F#,G#
A,B,C#,D#,E,F#,G#-
A,B-,C,D,E,F#,G
A,B,C#,D,E,F#-,G
A,Bb,C,D-,Eb,F,G
-----------------
A,B,C,D,E,F#,G#-
A,B,C#,D#-,E,F#,G
A,Bb,C,D,E,F#-,G
A,B,C#-,D,E,F,G
A,B-,C,D,Eb,F,G

A,B,C#-,D,E,F#-,G#
A,B-,C#,D,E,F#-,G
A,Bb,C,D-,Eb,F,G-
------------------------
A,B,C#-,D#,E,F#,G#-
A,Bb,C#-,D,E,F#-,G

A,B,C#-,D,E,F,G#
A,B,C,D#,E,F#,G#-
A,Bb,C#,D,E,F#-,G Hijaz
A,Bb,C,D-,Eb,F,F#
------------------------
A,B-,C,D,Eb,F#,G

A,B,C#,D#-,E,F#,G#-
A,B,C#-,D,E,F#,G#- Rast (Wiki)
A,B-,C#-,D#-,E,F#-,G#- Sikah/Nahawand
A,B,C#-,D,E,F#-,G Bayati
-----------------------
A,B-,C,D,E,F#-,G

A,B,C#-,D#-,E,F#,G#-
A,B,C#-,D#-,E,F#-,G#-
A,B,C#-,D,E,F#-,G#-
--------------------
A,B-,C#-,D,E,F#-,G#-
A,B-,C#-,D,E,F#-,G

A,B,C,D#-,E,F#,G#-
A B-,C#-,D-,E,F#-,G#-
A,B,C#-,D,E,F,G#-
A,B-,C,D,Eb,F#-,G
------------------
A,C-,C#-,D#-,E,F#-,G#-

-The 5-or-6-note "no-wolves" list, in 5 groups. (These have no 650-cent or 750-cent intervals).

A, D, E, F#-, G#: 10,4,3,5,2
A, B-, C#, D, G: 3,5,2,10,4
A, Bb, D#, F, G-: 2,10,4,3,5
A, C-, C#-,F#-, G#-: 5,2,10,4,3
A, B, C#-,D#, E: 4,3,5,2,10
=================================
A, D, D#, F#-, G: 10,2,5,3,4
A, C-, C#, D#, G#: 5,3,4,10,2
A, B, E, F, G#-: 4,10,2,5,3
A, B-, C#-,F#-, G-; 3,4,10,2,5
A, Bb, C#-,D, E: 2,5,3,4,10

A, C-, D, D#, F#-, G#: 5,5,2,5,5,2
A, C-, C#-,D#, F#-,G-: 5,2,5,5,2,5
A, Bb, C#-,D#, E, G-: 2,5,5,2,5,5

A, B, F, F#, G#-: 4,12,2,3,3
A, Bb, C-, C#, D#: 2,3,3,4,12
===============================
A, B-, C-, F#-, G#-: 3,2,12,4,3

A, B, E, F#, G#-: 4,10,4,3,3
A, D, E, F#-, G: 10,4,3,3,4
A, B-, C#-,F#-, G#-: 3,4,10,4,3
A, B-, C, D, G: 3,3,4,10,4
A, B, C#-,D, E: 4,3,3,4,10

A, C#-,E, F#-, G#-: 7,7,3,4,3
A, B, C#-,E, G#-: 4,3,7,7,3
A, B-, D, F#-, G: 3,7,7,3,4
A, C#-,D, E, F#-: 7,3,4,3,7
A, B-, C#-,D, F#-: 3,4,3,7,7

A, Bb, C-, C#-, D#, G-: 2,3,2,5,7,5 these two are against the "rules", but I like them
A, Bb, C-, D, D#: 2,3,5,2,12

-A link to the computer program, by W.D Clinger. It's just a brute-force generate-and-test program.
http://forums.randi.org/showthread.php?postid=9883339#post9883339

-A list of "conventional" scales, for completeness.

Lydian
Ionian
Mixolydian
Dorian
Aeolian
Phrygian
Locrian

Lydian #5
Lydian b7
Ionian b3
Mixo b6
Dor. b2
Aeol b5
Locr. b4

Lydian #2 #5
Lydian b3
Ion. b6 or Har. major
Mixo. b2
Dor. b5
Phryg. b4
Locr. bb7

Lyd. #2
Dor. #4
Aeol. #7 (N7) Har minor
Phryg. b4
Locr. mj6 (N6)
Ion. #5
Locr. b4 bb7

whole-tone

hex -- minor third
hex -- half-step

Oct. -- dim
Oct. -- dom.