3GMP scales that minimize average pairwise harmonic entropy

Here are the results of a search for local minima of mean pairwise harmonic entropy for scales that satisfy the N=3 generalized Myhill's property (no more than 3 specific intervals per generic interval):

https://spreadsheets.google.com/spreadsheet/ccc?key=0AtCMyeCjAMQGdGhBbWNHcE5yaG5lVTlNUVRjcEdvVGc&hl=en_US

https://spreadsheets.google.com/spreadsheet/ccc?key=0AtCMyeCjAMQGdFNTY1VoOE0wSThKczFtZ05CbUNkd0E&hl=en_US

The first column is the mean harmonic entropy (simply the mean, no trimming involved). The second is the scale pattern, and next are the values in cents of the three steps, a, b, and c, next are descriptions I've given to some of them. Column G is a non-trivial relation between the steps; if there's something here, that means the scale isn't actually rank-3, but is a rank-2 scale or subset of an equal temperament popping up. You can see there are a lot of these. The last column is the number of notes per octave.

This list was derived by picking a lot (1000s) of random starting points for two of the steps (a and b), and using each of those pairs as the initial guess for a numerical optimization algorithm (scipy.optimize.fmin). If there are enough random points, every significant local minimum should eventually be found, but it's likely some are missing. The more "important" minima, however (in the sense of being deeper and having a larger basin of attraction), are more likely to be found.

(I made an "extra fine" spreadsheet too, but in that case it was clear that it *was* missing a bunch of important minima, so maybe that will come later after I give it more number crunching time.)

Here are the highlights so far:

The overall best rank-3 scale so far is the 7-note marvel scale of the form ssLsmsL which has these as a few of its modes:
1/1 16/15 5/4 4/3 3/2 8/5 15/8 2/1
1/1 9/8 6/5 7/5 3/2 8/5 15/8 2/1
1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1
This scale is not new, and goes by many names, as Gene pointed out. It's incredibly beautiful and versatile, and I urge you to try it out.

The scale in the 2.3.5.11 temperament that tempers out 121/120 (does this temperament have a name?) with the pattern sssLssm where s ~= 158 cents, L ~= 229 cents, and m ~= 183 cents. This scale is interesting because, just as the diatonic scale has all thirds either minor (6/5) or major (5/4), this scale has all thirds either minor (6/5), neutral (11/9), or major (5/4). However, in my opinion it suffers from a lack of 3/2s. Note that this scale is not mirror-symmetric, so it's actually two different scales, sssLssm and sssmssL. (You could also say it simply has 14 different modes rather than 7.)

Finally, the scale in the 50/49 planar temperament (jublismic?) with the pattern smsLsmsL is just like an octatonic scale, except about as in-tune as an octatonic scale can get. Half of its "thirds" are 7/6 and the other half 6/5. Half of the "fourths" are 4/3, and the rest are 5/4 or 9/7.

Please contribute by giving descriptions or names to the still-undescribed scales! Which abstract temperament does each belong to?

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> Here are the results of a search for local minima of mean pairwise harmonic entropy for scales that satisfy the N=3 generalized Myhill's property (no more than 3 specific intervals per generic interval):
>
> https://spreadsheets.google.com/spreadsheet/ccc?key=0AtCMyeCjAMQGdGhBbWNHcE5yaG5lVTlNUVRjcEdvVGc&hl=en_US
>
> https://spreadsheets.google.com/spreadsheet/ccc?key=0AtCMyeCjAMQGdFNTY1VoOE0wSThKczFtZ05CbUNkd0E&hl=en_US
>
> The first column is the mean harmonic entropy (simply the mean, no trimming involved). The second is the scale pattern, and next are the values in cents of the three steps, a, b, and c, next are descriptions I've given to some of them. Column G is a non-trivial relation between the steps; if there's something here, that means the scale isn't actually rank-3, but is a rank-2 scale or subset of an equal temperament popping up. You can see there are a lot of these. The last column is the number of notes per octave.
>
> This list was derived by picking a lot (1000s) of random starting points for two of the steps (a and b), and using each of those pairs as the initial guess for a numerical optimization algorithm (scipy.optimize.fmin). If there are enough random points, every significant local minimum should eventually be found, but it's likely some are missing. The more "important" minima, however (in the sense of being deeper and having a larger basin of attraction), are more likely to be found.
>
> (I made an "extra fine" spreadsheet too, but in that case it was clear that it *was* missing a bunch of important minima, so maybe that will come later after I give it more number crunching time.)
>
> Here are the highlights so far:
>
> The overall best rank-3 scale so far is the 7-note marvel scale of the form ssLsmsL which has these as a few of its modes:
> 1/1 16/15 5/4 4/3 3/2 8/5 15/8 2/1
> 1/1 9/8 6/5 7/5 3/2 8/5 15/8 2/1
> 1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1
> This scale is not new, and goes by many names, as Gene pointed out. It's incredibly beautiful and versatile, and I urge you to try it out.
>
> The scale in the 2.3.5.11 temperament that tempers out 121/120 (does this temperament have a name?) with the pattern sssLssm where s ~= 158 cents, L ~= 229 cents, and m ~= 183 cents. This scale is interesting because, just as the diatonic scale has all thirds either minor (6/5) or major (5/4), this scale has all thirds either minor (6/5), neutral (11/9), or major (5/4). However, in my opinion it suffers from a lack of 3/2s. Note that this scale is not mirror-symmetric, so it's actually two different scales, sssLssm and sssmssL. (You could also say it simply has 14 different modes rather than 7.)
>
> Finally, the scale in the 50/49 planar temperament (jublismic?) with the pattern smsLsmsL is just like an octatonic scale, except about as in-tune as an octatonic scale can get. Half of its "thirds" are 7/6 and the other half 6/5. Half of the "fourths" are 4/3, and the rest are 5/4 or 9/7.
>
> Please contribute by giving descriptions or names to the still-undescribed scales! Which abstract temperament does each belong to?
>

I know this is a year old, but it struck me that all are roughly forms of the Hungarian/Arabic/Gypsy/Jewish scale. Of course, this would be 12-tET, which is passe
to say the least, but its so interesting because in my set theory this septachord plays
a prominent role, almost a central one. You mention it is not symmetrical, in 12-tET of
course it is though. So I know this isn't the answer you are looking for, however, because
the Hungarian scale is so important, personally, I am interested in its expressions up to the 7-limit to say the least, and in higher temperaments. Since I analyze interval vectors I am interested in that also. (Interval set content of the entire scale).

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> >
> > Here are the results of a search for local minima of mean pairwise harmonic entropy for scales that satisfy the N=3 generalized Myhill's property (no more than 3 specific intervals per generic interval):
> >
> > https://spreadsheets.google.com/spreadsheet/ccc?key=0AtCMyeCjAMQGdGhBbWNHcE5yaG5lVTlNUVRjcEdvVGc&hl=en_US
> >
> > https://spreadsheets.google.com/spreadsheet/ccc?key=0AtCMyeCjAMQGdFNTY1VoOE0wSThKczFtZ05CbUNkd0E&hl=en_US
> >
> > The first column is the mean harmonic entropy (simply the mean, no trimming involved). The second is the scale pattern, and next are the values in cents of the three steps, a, b, and c, next are descriptions I've given to some of them. Column G is a non-trivial relation between the steps; if there's something here, that means the scale isn't actually rank-3, but is a rank-2 scale or subset of an equal temperament popping up. You can see there are a lot of these. The last column is the number of notes per octave.
> >
> > This list was derived by picking a lot (1000s) of random starting points for two of the steps (a and b), and using each of those pairs as the initial guess for a numerical optimization algorithm (scipy.optimize.fmin). If there are enough random points, every significant local minimum should eventually be found, but it's likely some are missing. The more "important" minima, however (in the sense of being deeper and having a larger basin of attraction), are more likely to be found.
> >
> > (I made an "extra fine" spreadsheet too, but in that case it was clear that it *was* missing a bunch of important minima, so maybe that will come later after I give it more number crunching time.)
> >
> > Here are the highlights so far:
> >
> > The overall best rank-3 scale so far is the 7-note marvel scale of the form ssLsmsL which has these as a few of its modes:
> > 1/1 16/15 5/4 4/3 3/2 8/5 15/8 2/1
> > 1/1 9/8 6/5 7/5 3/2 8/5 15/8 2/1
> > 1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1
> > This scale is not new, and goes by many names, as Gene pointed out. It's incredibly beautiful and versatile, and I urge you to try it out.
> >
> > The scale in the 2.3.5.11 temperament that tempers out 121/120 (does this temperament have a name?) with the pattern sssLssm where s ~= 158 cents, L ~= 229 cents, and m ~= 183 cents. This scale is interesting because, just as the diatonic scale has all thirds either minor (6/5) or major (5/4), this scale has all thirds either minor (6/5), neutral (11/9), or major (5/4). However, in my opinion it suffers from a lack of 3/2s. Note that this scale is not mirror-symmetric, so it's actually two different scales, sssLssm and sssmssL. (You could also say it simply has 14 different modes rather than 7.)
> >
> > Finally, the scale in the 50/49 planar temperament (jublismic?) with the pattern smsLsmsL is just like an octatonic scale, except about as in-tune as an octatonic scale can get. Half of its "thirds" are 7/6 and the other half 6/5. Half of the "fourths" are 4/3, and the rest are 5/4 or 9/7.
> >
> > Please contribute by giving descriptions or names to the still-undescribed scales! Which abstract temperament does each belong to?
> >
>
>
> I know this is a year old, but it struck me that all are roughly forms of the Hungarian/Arabic/Gypsy/Jewish scale. Of course, this would be 12-tET, which is passe
> to say the least, but its so interesting because in my set theory this septachord plays
> a prominent role, almost a central one. You mention it is not symmetrical, in 12-tET of
> course it is though. So I know this isn't the answer you are looking for, however, because
> the Hungarian scale is so important, personally, I am interested in its expressions up to the 7-limit to say the least, and in higher temperaments. Since I analyze interval vectors I am interested in that also. (Interval set content of the entire scale).

Uh, I think you may be confusing different scales I was talking about.

The marvel heptatonic is always symmetric: it's sLsmsLs.

The scale I said was asymmetric is in 121/120 planar temperament, so it does not exist in 12edo.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> > >
> > > Here are the results of a search for local minima of mean pairwise harmonic entropy for scales that satisfy the N=3 generalized Myhill's property (no more than 3 specific intervals per generic interval):
> > >
> > > https://spreadsheets.google.com/spreadsheet/ccc?key=0AtCMyeCjAMQGdGhBbWNHcE5yaG5lVTlNUVRjcEdvVGc&hl=en_US
> > >
> > > https://spreadsheets.google.com/spreadsheet/ccc?key=0AtCMyeCjAMQGdFNTY1VoOE0wSThKczFtZ05CbUNkd0E&hl=en_US
> > >
> > > The first column is the mean harmonic entropy (simply the mean, no trimming involved). The second is the scale pattern, and next are the values in cents of the three steps, a, b, and c, next are descriptions I've given to some of them. Column G is a non-trivial relation between the steps; if there's something here, that means the scale isn't actually rank-3, but is a rank-2 scale or subset of an equal temperament popping up. You can see there are a lot of these. The last column is the number of notes per octave.
> > >
> > > This list was derived by picking a lot (1000s) of random starting points for two of the steps (a and b), and using each of those pairs as the initial guess for a numerical optimization algorithm (scipy.optimize.fmin). If there are enough random points, every significant local minimum should eventually be found, but it's likely some are missing. The more "important" minima, however (in the sense of being deeper and having a larger basin of attraction), are more likely to be found.
> > >
> > > (I made an "extra fine" spreadsheet too, but in that case it was clear that it *was* missing a bunch of important minima, so maybe that will come later after I give it more number crunching time.)
> > >
> > > Here are the highlights so far:
> > >
> > > The overall best rank-3 scale so far is the 7-note marvel scale of the form ssLsmsL which has these as a few of its modes:
> > > 1/1 16/15 5/4 4/3 3/2 8/5 15/8 2/1
> > > 1/1 9/8 6/5 7/5 3/2 8/5 15/8 2/1
> > > 1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1
> > > This scale is not new, and goes by many names, as Gene pointed out. It's incredibly beautiful and versatile, and I urge you to try it out.
> > >
> > > The scale in the 2.3.5.11 temperament that tempers out 121/120 (does this temperament have a name?) with the pattern sssLssm where s ~= 158 cents, L ~= 229 cents, and m ~= 183 cents. This scale is interesting because, just as the diatonic scale has all thirds either minor (6/5) or major (5/4), this scale has all thirds either minor (6/5), neutral (11/9), or major (5/4). However, in my opinion it suffers from a lack of 3/2s. Note that this scale is not mirror-symmetric, so it's actually two different scales, sssLssm and sssmssL. (You could also say it simply has 14 different modes rather than 7.)
> > >
> > > Finally, the scale in the 50/49 planar temperament (jublismic?) with the pattern smsLsmsL is just like an octatonic scale, except about as in-tune as an octatonic scale can get. Half of its "thirds" are 7/6 and the other half 6/5. Half of the "fourths" are 4/3, and the rest are 5/4 or 9/7.
> > >
> > > Please contribute by giving descriptions or names to the still-undescribed scales! Which abstract temperament does each belong to?
> > >
> >
> >
> > I know this is a year old, but it struck me that all are roughly forms of the Hungarian/Arabic/Gypsy/Jewish scale. Of course, this would be 12-tET, which is passe
> > to say the least, but its so interesting because in my set theory this septachord plays
> > a prominent role, almost a central one. You mention it is not symmetrical, in 12-tET of
> > course it is though. So I know this isn't the answer you are looking for, however, because
> > the Hungarian scale is so important, personally, I am interested in its expressions up to the 7-limit to say the least, and in higher temperaments. Since I analyze interval vectors I am interested in that also. (Interval set content of the entire scale).
>
> Uh, I think you may be confusing different scales I was talking about.
>
> The marvel heptatonic is always symmetric: it's sLsmsLs.
>
> The scale I said was asymmetric is in 121/120 planar temperament, so it does not exist in 12edo.
>
> Keenan
>

Okay, I reread it. So the 7-note marvel temperament scale, which has 3 modes, although not necessarily 12-tET, is symmetrical, and to my ears, would be Hungarian in all 3 cases.
The first form, which is 5-limit, at least could be 12-tET very easily, without hardly any effort. In any case, it shows that this scale, which was used by the Arabs, Gypsies, Armenians and Hungarians, (with different starting notes), might have indeed been tuned like one of these, by one of those groups, at least on the violin, one would suspect. I think the first mode, C, Db, E, F, G Ab, B, C is one of them (Arabic?) I would have to look it up.

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
> Okay, I reread it. So the 7-note marvel temperament scale, which has 3 modes, although not necessarily 12-tET, is symmetrical, and to my ears, would be Hungarian in all 3 cases.
> The first form, which is 5-limit, at least could be 12-tET very easily, without hardly any effort. In any case, it shows that this scale, which was used by the Arabs, Gypsies, Armenians and Hungarians, (with different starting notes), might have indeed been tuned like one of these, by one of those groups, at least on the violin, one would suspect. I think the first mode, C, Db, E, F, G Ab, B, C is one of them (Arabic?) I would have to look it up.

If you're talking about the traversals I posted,

1/1 16/15 5/4 4/3 3/2 8/5 15/8 2/1
1/1 9/8 6/5 7/5 3/2 8/5 15/8 2/1
1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1 ,

these are different traversals of different modes of exactly the same scale, because that's how marvel temperament works. I could give a purely 5-limit traversal of the third mode as

1/1 75/64 5/4 45/32 3/2 225/128 15/8 2/1 ,

but in marvel temperament that's the same thing as

1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1 .

I simply posted the simplest ratio in each case, which happens to be a different traversal for each different mode. But the tempered scale works as all three of these at once.

This scale is symmetrical and, as you say, perfectly consistent with 12-equal, so unfortunately it doesn't sound very xenharmonic. The cool thing about the sound is that it's this familiar "hungarian"/"gypsy"/"double harmonic" scale, but there are all these quite accurate 7-limit intervals and chords hidden in it.

Here is a "blank canvas" sharecode for http://outlash.co.uk/grid.html if anyone wants to jam in it:

0,7,23,30,42,49,65,72,79,95,102,114,121,137,144,151,2,72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

Keenan