Proper Scale Search: Take Two

OK, I lied, I'm still awake. Looks like I fubared the propriety check
before, so half of the scales I posted weren't even proper. Now I'm
trying again. Feel free to delete the other post.

The problem, for the moment, is that this damn thing takes forever to
run. I don't have an efficient algorithm to calculate all of the
subsets of N size from a set. If I did, this would be a lot faster. So
far I've run this on 12-tet and 15-tet, and I'll leave 22-tet running
while I sleep and see what it's spit out tomorrow.

Here's the 12-tet list:
1 note:
12

2 notes:
1 11
2 10
3 9
4 8
5 7
6 6

3 notes:
5 1 6
4 2 6
3 3 6
2 4 6
1 5 6
5 2 5
4 3 5
3 4 5
4 4 4

4 notes:
2 4 1 5
2 3 2 5
1 5 1 5
1 4 2 5
4 3 1 4
4 2 2 4
4 1 3 4
3 4 1 4
3 3 2 4
3 2 3 4
2 4 2 4
2 3 3 4
3 3 3 3

5 notes - meantone[5] near-MOS's here
2 2 3 1 4
2 2 2 2 4
1 3 3 1 4
1 3 2 2 4
3 3 2 1 3
3 3 1 2 3
3 2 3 1 3
3 2 2 2 3
3 1 3 2 3
2 3 2 2 3

6 notes - Augmented[6] near-MOS's!
2 2 2 2 1 3
2 2 2 1 2 3
2 2 1 3 1 3
2 2 1 2 2 3
2 1 3 2 1 3
2 1 3 1 2 3
2 1 2 3 1 3
2 1 2 2 2 3
1 3 1 3 1 3
1 3 1 2 2 3
1 2 3 1 2 3
1 2 2 2 2 3
2 2 2 2 2 2

7 notes - meantone[7] near-MOS's
1 2 2 1 2 1 3
1 2 1 2 2 1 3
2 2 2 2 1 1 2
2 2 2 1 2 1 2
2 2 1 2 2 1 2

8 notes - Diminished[8] near-MOS's! These are sweet!
2 1 2 1 2 1 1 2
2 1 2 1 1 2 1 2
2 1 1 2 2 1 1 2
2 1 1 2 1 2 1 2
1 2 1 2 1 2 1 2

9 notes - Augmented[9] near-MOS's! These are even more sweet!
1 2 1 1 2 1 1 1 2
1 2 1 1 1 2 1 1 2
1 1 2 1 1 2 1 1 2

10 notes - Paul's decatonic scales show up here
1 1 1 2 1 1 1 1 1 2
1 1 1 1 2 1 1 1 1 2

11 notes - this is the 19-limit extension of Ripple temperament I
posted about before
1 1 1 1 1 1 1 1 1 1 2

12 notes:
1 1 1 1 1 1 1 1 1 1 1 1

-Mike

Here's the 15-tet list - standouts include Porcupine[7], Porcupine[8],
and Blackwood[10] near-MOS's.

1 note:
15

2 notes:
1 14
2 13
3 12
4 11
5 10
6 9
7 8

3 notes:
7 1 7
6 2 7
5 3 7
4 4 7
3 5 7
2 6 7
6 3 6
5 4 6
4 5 6
5 5 5

4 notes:
1 6 1 7
3 5 1 6
3 4 2 6
3 3 3 6
2 6 1 6
2 5 2 6
2 4 3 6
1 5 3 6
5 4 1 5
5 3 2 5
5 2 3 5
5 1 4 5
4 5 1 5
4 4 2 5
4 3 3 5
4 2 4 5
3 5 2 5
3 4 3 5
3 3 4 5
2 4 4 5
4 4 3 4

5 notes:
2 3 4 1 5
2 3 3 2 5
1 4 3 2 5
3 4 3 1 4
3 4 2 2 4
3 4 1 3 4
3 3 4 1 4
3 3 3 2 4
3 3 2 3 4
3 2 4 2 4
3 2 3 3 4
2 4 2 3 4
2 3 3 3 4
3 3 3 3 3

6 notes - I'm not sure how to categorize some of these...
3 1 3 3 1 4
2 3 2 3 1 4
2 3 2 2 2 4
2 3 1 4 1 4
2 3 1 3 2 4
2 2 3 3 1 4
2 2 3 2 2 4
2 2 2 4 1 4
2 2 2 3 2 4
1 4 1 4 1 4
1 4 1 3 2 4
1 3 3 3 1 4
1 3 3 2 2 4
1 3 3 1 3 4
1 3 2 3 2 4
3 3 3 2 1 3
3 3 3 1 2 3
3 3 2 3 1 3
3 3 2 2 2 3
3 3 1 3 2 3
3 2 3 3 1 3
3 2 3 2 2 3
3 2 2 3 2 3
2 3 2 3 2 3

7 notes - Some of these are definitely Porcupine[7] near-MOS's! Are all of them?
3 1 2 3 2 1 3
2 2 3 2 2 1 3
2 2 3 2 1 2 3
2 2 3 1 3 1 3
2 2 3 1 2 2 3
2 2 2 3 2 1 3
2 2 2 3 1 2 3
2 2 2 2 3 1 3
2 2 2 2 2 2 3

8 notes - Porcupine[8] near-MOS's, I think?
2 2 1 2 2 2 1 3
2 1 3 1 2 2 1 3
2 1 2 2 2 2 1 3
2 1 2 2 2 1 2 3
1 3 1 3 1 2 1 3
1 3 1 2 2 2 1 3
1 2 3 1 2 2 1 3
1 2 2 2 2 2 1 3
1 2 2 2 2 1 2 3
1 2 2 2 1 2 2 3
2 2 2 2 2 2 1 2

9 notes - near-MOS's of the LLsLLsLLs scale, whatever temperament that is...
1 2 1 2 2 1 2 1 3
2 2 1 2 2 2 1 1 2
2 2 1 2 2 1 2 1 2
2 2 1 2 1 2 2 1 2
2 1 2 2 1 2 2 1 2

10 notes- Blackwood near-MOS's! These are AWESOME
2 1 2 1 2 1 2 1 1 2
2 1 2 1 2 1 1 2 1 2
2 1 2 1 1 2 2 1 1 2
2 1 2 1 1 2 1 2 1 2
2 1 1 2 2 1 1 2 1 2
2 1 1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2 1 2

11 notes - I don't know
1 2 1 1 2 1 2 1 1 1 2
1 2 1 1 2 1 1 2 1 1 2

12 notes - Looking like Augmented[12] near-MOS's
1 1 2 1 1 1 2 1 1 1 1 2
1 1 2 1 1 1 1 2 1 1 1 2
1 1 1 2 1 1 1 2 1 1 1 2

13 notes:
1 1 1 1 1 2 1 1 1 1 1 1 2

14 notes:
1 1 1 1 1 1 1 1 1 1 1
1 1 2

15 notes:
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1

-Mike

I don't think the 22-tet one will ever finish; I need to work more on
this algorithm. Here's some 16-tet 7-note proper scales, many of which
I'm sure are one chromatic unison vector away from Mavila[7]. Load
some sines up and play with these - they are amazing! 2 3 1 3 2 2 3 in
particular. 2 3 2 2 2 2 3 is anti-melodic minor, in the same sense
that Mavila[7] is anti-diatonic. Seriously, I need to go to bed.

7 notes:
2 2 2 2 3 1 4
2 2 2 2 2 2 4
1 3 2 2 3 1 4
1 3 2 2 2 2 4
3 2 2 3 2 1 3
3 2 2 3 1 2 3
3 2 2 2 3 1 3
3 2 2 2 2 2 3
3 2 1 3 3 1 3
3 2 1 3 2 2 3
3 1 3 3 1 2 3
3 1 3 2 3 1 3
3 1 3 2 2 2 3
3 1 2 3 2 2 3
2 3 2 3 2 1 3
2 3 2 3 1 2 3
2 3 2 2 3 1 3
2 3 2 2 2 2 3
2 3 1 3 2 2 3
2 2 3 2 2 2 3

-Mike

Even better work!

At 05:13 AM 3/13/2011, you wrote:
>OK, I lied, I'm still awake. Looks like I fubared the propriety check
>before, so half of the scales I posted weren't even proper. Now I'm
>trying again. Feel free to delete the other post.
>
>The problem, for the moment, is that this damn thing takes forever to
>run. I don't have an efficient algorithm to calculate all of the
>subsets of N size from a set. If I did, this would be a lot faster. So
>far I've run this on 12-tet and 15-tet, and I'll leave 22-tet running
>while I sleep and see what it's spit out tomorrow.
>
>Here's the 12-tet list:
>1 note:
> 12
>
>2 notes:
> 1 11
> 2 10
> 3 9
> 4 8
> 5 7
> 6 6
>
>3 notes:
> 5 1 6
> 4 2 6
> 3 3 6
> 2 4 6
> 1 5 6
> 5 2 5
> 4 3 5
> 3 4 5
> 4 4 4
>
>4 notes:
> 2 4 1 5
> 2 3 2 5
> 1 5 1 5
> 1 4 2 5
> 4 3 1 4
> 4 2 2 4
> 4 1 3 4
> 3 4 1 4
> 3 3 2 4
> 3 2 3 4
> 2 4 2 4
> 2 3 3 4
> 3 3 3 3
>
>5 notes - meantone[5] near-MOS's here
> 2 2 3 1 4
> 2 2 2 2 4
> 1 3 3 1 4
> 1 3 2 2 4
> 3 3 2 1 3
> 3 3 1 2 3
> 3 2 3 1 3
> 3 2 2 2 3
> 3 1 3 2 3
> 2 3 2 2 3
>
>6 notes - Augmented[6] near-MOS's!
> 2 2 2 2 1 3
> 2 2 2 1 2 3
> 2 2 1 3 1 3
> 2 2 1 2 2 3
> 2 1 3 2 1 3
> 2 1 3 1 2 3
> 2 1 2 3 1 3
> 2 1 2 2 2 3
> 1 3 1 3 1 3
> 1 3 1 2 2 3
> 1 2 3 1 2 3
> 1 2 2 2 2 3
> 2 2 2 2 2 2
>
>7 notes - meantone[7] near-MOS's
> 1 2 2 1 2 1 3
> 1 2 1 2 2 1 3
> 2 2 2 2 1 1 2
> 2 2 2 1 2 1 2
> 2 2 1 2 2 1 2
>
>8 notes - Diminished[8] near-MOS's! These are sweet!
> 2 1 2 1 2 1 1 2
> 2 1 2 1 1 2 1 2
> 2 1 1 2 2 1 1 2
> 2 1 1 2 1 2 1 2
> 1 2 1 2 1 2 1 2
>
>9 notes - Augmented[9] near-MOS's! These are even more sweet!
> 1 2 1 1 2 1 1 1 2
> 1 2 1 1 1 2 1 1 2
> 1 1 2 1 1 2 1 1 2
>
>10 notes - Paul's decatonic scales show up here
> 1 1 1 2 1 1 1 1 1 2
> 1 1 1 1 2 1 1 1 1 2
>
>11 notes - this is the 19-limit extension of Ripple temperament I
>posted about before
> 1 1 1 1 1 1 1 1 1 1 2
>
>12 notes:
> 1 1 1 1 1 1 1 1 1 1 1 1
>
>
>-Mike
>

Mike wrote:
>The problem, for the moment, is that this damn thing takes forever to
>run. I don't have an efficient algorithm to calculate all of the
>subsets of N size from a set. If I did, this would be a lot faster.

Gene presumably found an efficient method. Since I can't find
Manuel's list from the '90s on the web any more, I've posted
it here

/tuning-math/files/carl/ProperScalesInETs.txt

-C.

I finally got the 22-tet 7-note scales done.

22-tet - 7 notes:
3 3 3 3 4 1 5
3 3 3 3 3 2 5
3 3 3 3 2 3 5
3 3 3 2 5 1 5
3 3 3 2 4 2 5
3 3 3 2 3 3 5
3 3 2 4 4 1 5
3 3 2 4 3 2 5
3 3 2 4 2 3 5
3 3 2 3 5 1 5
3 3 2 3 4 2 5
3 3 2 3 3 3 5
3 2 4 3 4 1 5
3 2 4 3 3 2 5
3 2 4 3 2 3 5
3 2 4 2 5 1 5
3 2 4 2 4 2 5
3 2 4 2 3 3 5
3 2 3 4 4 1 5
3 2 3 4 3 2 5
3 2 3 4 2 3 5
3 2 3 3 5 1 5
3 2 3 3 4 2 5
3 2 3 3 3 3 5
2 4 3 3 4 1 5
2 4 3 3 3 2 5
2 4 3 3 2 3 5
2 4 3 2 5 1 5
2 4 3 2 4 2 5
2 4 3 2 3 3 5
2 4 2 4 4 1 5
2 4 2 4 3 2 5
2 4 2 4 2 3 5
2 4 2 3 5 1 5
2 4 2 3 4 2 5
2 4 2 3 3 3 5
2 3 4 3 4 1 5
2 3 4 3 3 2 5
2 3 4 3 2 3 5
2 3 4 2 5 1 5
2 3 4 2 4 2 5
2 3 4 2 3 3 5
2 3 3 4 4 1 5
2 3 3 4 3 2 5
2 3 3 4 2 3 5
2 3 3 3 5 1 5
2 3 3 3 4 2 5
2 3 3 3 3 3 5
1 4 4 3 3 2 5
1 4 4 3 2 3 5
1 4 4 2 4 2 5
1 4 4 2 3 3 5
1 4 3 4 3 2 5
1 4 3 4 2 3 5
1 4 3 3 4 2 5
1 4 3 3 3 3 5
4 3 3 4 3 1 4
4 3 3 4 2 2 4
4 3 3 4 1 3 4
4 3 3 3 4 1 4
4 3 3 3 3 2 4
4 3 3 3 2 3 4
4 3 3 2 4 2 4
4 3 3 2 3 3 4
4 3 2 4 4 1 4
4 3 2 4 3 2 4
4 3 2 4 2 3 4
4 3 2 3 4 2 4
4 3 2 3 3 3 4
4 3 1 4 4 2 4
4 3 1 4 3 3 4
4 2 4 4 2 2 4
4 2 4 4 1 3 4
4 2 4 3 4 1 4
4 2 4 3 3 2 4
4 2 4 3 2 3 4
4 2 4 2 4 2 4
4 2 4 2 3 3 4
4 2 3 4 4 1 4
4 2 3 4 3 2 4
4 2 3 4 2 3 4
4 2 3 3 4 2 4
4 2 3 3 3 3 4
4 2 2 4 3 3 4
4 1 4 3 4 2 4
4 1 4 3 3 3 4
4 1 3 4 3 3 4
3 4 3 4 3 1 4
3 4 3 4 2 2 4
3 4 3 4 1 3 4
3 4 3 3 4 1 4
3 4 3 3 3 2 4
3 4 3 3 2 3 4
3 4 3 2 4 2 4
3 4 3 2 3 3 4
3 4 2 4 3 2 4
3 4 2 4 2 3 4
3 4 2 3 4 2 4
3 4 2 3 3 3 4
3 4 1 4 3 3 4
3 3 4 3 3 2 4
3 3 4 3 2 3 4
3 3 4 2 4 2 4
3 3 4 2 3 3 4
3 3 3 4 3 2 4
3 3 3 4 2 3 4
3 3 3 3 4 2 4
3 3 3 3 3 3 4

-Mike

PS - what the hell is this 7-note proper scale, taken from 15-tet?

3 1 2 3 2 1 3

Is it a hobbit of something? I can't figure it out.

-Mike

On Sun, Mar 13, 2011 at 4:04 PM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
> >The problem, for the moment, is that this damn thing takes forever to
> >run. I don't have an efficient algorithm to calculate all of the
> >subsets of N size from a set. If I did, this would be a lot faster.
>
> Gene presumably found an efficient method. Since I can't find
> Manuel's list from the '90s on the web any more, I've posted
> it here
>
> /tuning-math/files/carl/ProperScalesInETs.txt

This is very useful. This list seems to include scales more than once
as different modal permutations of one another. Perhaps it would be
useful for me to just import the list and prune modes when they appear
twice.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> The problem, for the moment, is that this damn thing takes forever to
> run. I don't have an efficient algorithm to calculate all of the
> subsets of N size from a set.

Have you checked Knuth for a next combination of n things taken m at a time algorithm?

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> Gene presumably found an efficient method.

Yes, I ran a preexisting Maple routine. But I think this is in Knuth, and if coded in C, the damned thing would run a lot faster than in Maple.

Yes. It's section 7.2.1.3. It's Volume 4A, fascicle 3 if they are still
available now that the whole Volume 4A is out. Other sections of Volume 4A
cover generating all permutations, n-tuples, partitions, set partitions and
trees, so you might want to get the whole Volume 4A or try to find it in the
library. If you aren't fussy about the order the combinations appear, then
you might check out the loopless algorithm in CACM 19(1976) 517-521.

David Bowen

On Sun, Mar 13, 2011 at 6:50 PM, genewardsmith
<genewardsmith@sbcglobal.net>wrote:

>
>
>
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> > Gene presumably found an efficient method.
>
> Yes, I ran a preexisting Maple routine. But I think this is in Knuth, and
> if coded in C, the damned thing would run a lot faster than in Maple.
>
>
>

On Sun, Mar 13, 2011 at 9:08 PM, David Bowen <dmb0317@gmail.com> wrote:
>
> Yes. It's section 7.2.1.3. It's Volume 4A, fascicle 3 if they are still available now that the whole Volume 4A is out. Other sections of Volume 4A cover generating all permutations, n-tuples, partitions, set partitions and trees, so you might want to get the whole Volume 4A or try to find it in the library. If you aren't fussy about the order the combinations appear, then you might check out the loopless algorithm in CACM 19(1976) 517-521.

Thanks for the reference, hopefully something in there will do. One
thing I want to avoid is taking combinations that actually yield
different modes of the same scale. That is, I don't want to generate
combinations that are just linear translations or circularly shifted
versions of one another. Hopefully there's something in there to cover
that.

-Mike

On Mon, Mar 14, 2011 at 6:57 AM, Mike Battaglia <battaglia01@gmail.com>wrote:

>
>
> On Sun, Mar 13, 2011 at 9:08 PM, David Bowen <dmb0317@gmail.com> wrote:
> >
> > Yes. It's section 7.2.1.3. It's Volume 4A, fascicle 3 if they are still
> available now that the whole Volume 4A is out. Other sections of Volume 4A
> cover generating all permutations, n-tuples, partitions, set partitions and
> trees, so you might want to get the whole Volume 4A or try to find it in the
> library. If you aren't fussy about the order the combinations appear, then
> you might check out the loopless algorithm in CACM 19(1976) 517-521.
>
> Thanks for the reference, hopefully something in there will do. One
> thing I want to avoid is taking combinations that actually yield
> different modes of the same scale. That is, I don't want to generate
> combinations that are just linear translations or circularly shifted
> versions of one another. Hopefully there's something in there to cover
> that.
>
> -Mike
>
> Mike,

I don't recall seeing specific code for that. For scales, you could make
sure that C was always included and then use the generation algorithm to
handle the remaining pitches. That won't catch all of the translations, but
you can define a normal form (say earliest lexicographical order) and then
reject any scale that isn't in normal form. Let me know if you have
additional questions.

Dave Bowen