Optimal patent vals

Given a temperament and an error measure, one can ask for the patent
val giving the least error; this is likely to come pretty close to the
optimal tuning. Below I give optimal patent vals for corank one
(single comma) temperaments, using minimax error. Since the val tuning
can't be better than the minimax tuning, I can bound the search by
going up only so far as makes the step sizes smaller than the minimax
error.

The five-limit stuff has the familiar (41 as a way of tempering out
3125/3072) the less familiar (23 for 135/128, 59 for 250/243) the
unfamiliar (2129-et as a way of doing schismatic?) and the totally off
the wall (9-et is the system for tempering out 27/25--except, it does
a *perfect* 27/25!)

Seven gives cool things like 55, the tuning of Telemann and Mozart, as
the king of 1029/1000. The eleven limit tells us that 99 is not just a
great 7-limit temperament, it specializes in tempering out 121/120.
111 isn't just a high-limit system, but a way of tempering out
176/175. The same with 436 and 1375/1372. Plus you never know when
knowing things like 296 is good for 540/539 stuff will turn out
useful, etc.

Below I give the comma, the et, and the max error in cents.

5-limit

27/25 9 48.974620
16/15 8 63.686286
135/128 23 23.694131
25/24 17 37.299889
648/625 40 15.641287
250/243 59 9.909406
128/125 42 13.686286
3125/3072 41 6.309933
81/80 81 5.658705
2048/2025 114 3.308157
78732/78125 382 1.505161
393216/390625 359 1.435159
2109375/2097152 517 1.013907
15625/15552 458 1.389281
1600000/1594323 205 .484024
1224440064/1220703125 1199 .378568
6115295232/6103515625 1636 .238853
19073486328125/19042491875328 1653 .148187
32805/32768 2129 .217096
582076609134674072265625/581595589965365114830848 11305 .042153
274877906944/274658203125 7174 .081603
50031545098999707/50000000000000000 14087 .031241
7629394531250/7625597484987 2718 .032002
2475880078570760549798248448/2474715001881122589111328125 25989 .018965
9010162353515625/9007199254740992 3414 .021974

7-limit

1029/1000 55 15.239465
36/35 33 24.732196
525/512 45 12.980381
49/48 29 18.595044
50/49 26 17.487807
686/675 83 6.629942
64/63 59 9.909406
875/864 138 6.097843
3125/3087 225 4.358713
2430/2401 146 4.361971
245/243 177 3.129745
126/125 247 4.710113
4000/3969 230 2.619583
1728/1715 142 3.551630
1029/1024 252 2.159240
225/224 228 2.103188
19683/19600 659 .892785
16875/16807 327 1.037570
10976/10935 654 .860816
3136/3125 384 1.262193
5120/5103 531 .869858
6144/6125 412 1.090259
65625/65536 1623 .361773
703125/702464 2912 .187340
420175/419904 4355 .143007
2401/2400 2749 .191349
4375/4374 9058 .057543
250047/250000 5787 .037347
78125000/78121827 101240 .004703

11-limit

77/75 39 28.666039
45/44 26 21.822829
55/54 51 17.996176
56/55 36 19.258726
245/242 91 8.716755
99/98 127 9.887245
100/99 104 9.326789
121/120 99 8.014001
1331/1323 251 3.209167
176/175 111 4.147067
896/891 145 3.714861
243/242 346 2.881077
385/384 284 2.022168
441/440 320 1.410002
1375/1372 436 .992613
6250/6237 899 .868673
540/539 296 1.300311
4000/3993 998 .927635
5632/5625 1210 .463146
43923/43904 2715 .171594
3025/3024 3207 .177332
9801/9800 18870 .044619
151263/151250 33541 .025087
3294225/3294172 241732 .003991

Gene Ward Smith wrote:
> Given a temperament and an error measure, one can ask for the patent
> val giving the least error; this is likely to come pretty close to the
> optimal tuning. Below I give optimal patent vals for corank one
> (single comma) temperaments, using minimax error. Since the val tuning
> can't be better than the minimax tuning, I can bound the search by
> going up only so far as makes the step sizes smaller than the minimax
> error.

Hmm. Looks like some interesting results; could you describe the procedure in a little more detail?

> The five-limit stuff has the familiar (41 as a way of tempering out
> 3125/3072) the less familiar (23 for 135/128, 59 for 250/243) the
> unfamiliar (2129-et as a way of doing schismatic?) and the totally off
> the wall (9-et is the system for tempering out 27/25--except, it does
> a *perfect* 27/25!)

Well, 23-ET is reasonably well known as a mavila tuning, so this makes sense. I've known about 59-ET, but never considered it a practical tuning due to the number of notes. Still, it sounds nice:

http://www.io.com/~hmiller/midi/porcupine-59.mid

Could this also be a way to set an approximate maximum size for how many notes a temperament can have? It looks like it's easier to define than the "consistency limit" I was looking for, and agrees with my intuitions about bug, father, mavila, and porcupine at least.

> Seven gives cool things like 55, the tuning of Telemann and Mozart, as
> the king of 1029/1000. The eleven limit tells us that 99 is not just a
> great 7-limit temperament, it specializes in tempering out 121/120.
> 111 isn't just a high-limit system, but a way of tempering out
> 176/175. The same with 436 and 1375/1372. Plus you never know when
> knowing things like 296 is good for 540/539 stuff will turn out
> useful, etc.
> > Below I give the comma, the et, and the max error in cents.
> > 5-limit
> > 27/25 9 48.974620

That's what I would have guessed, given my experience with bug temperament.

> 16/15 8 63.686286

And 8-ET is pretty much the usable limit of father temperament.

> 135/128 23 23.694131

Yep. 23-note mavila.

> 25/24 17 37.299889

Now this is interesting. Any possible connection with the theoretical Arabic 17-note scale or the neutral thirds of Arabian music?

> 648/625 40 15.641287

Interesting that it picked out 40-ET out of the various possibilities. How does it compare with 12-ET? Its major thirds are obviously quite a bit better, but its fifths are 12 cents flat.

...

> 50/49 26 17.487807

Why am I not surprised? :-)

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Hmm. Looks like some interesting results; could you describe the
> procedure in a little more detail?

First I calculate the unweighted minimax error, call it e cents, using
Maple's simplex algorithm. Then 1200/e is the largest division I need
consider, since the n-edo error can't get more accurate than the
minimax error. Now I check each (uncontorted) n such that hn(comma) =
0, where hn is the patent val, up to my previously determined limit.
The one with the least minimax error is the winner.

> Could this also be a way to set an approximate maximum size for how
many
> notes a temperament can have?

Seems a bit arbitary.

It looks like it's easier to define than
> the "consistency limit" I was looking for, and agrees with my
intuitions
> about bug, father, mavila, and porcupine at least.

What about shismatic?

> > 27/25 9 48.974620
>
> That's what I would have guessed, given my experience with bug
temperament.

Now that I think about it, I ran into it looking at
bug-ennealimmalotation. It's still weird.

> > 25/24 17 37.299889
>
> Now this is interesting. Any possible connection with the theoretical
> Arabic 17-note scale or the neutral thirds of Arabian music?

Quite possibly, if you believe in a 17 Pythagorean phase to that.

> > 648/625 40 15.641287
>
> Interesting that it picked out 40-ET out of the various possibilities.
> How does it compare with 12-ET? Its major thirds are obviously quite a
> bit better, but its fifths are 12 cents flat.

The minimax error is 15.64 cents, and the simplex method gave a 3 of
1886.3 and a 5 of 2786.3. 40 has the identical minimax error, with a 3
of 1890 and a 5 of 2790. 12 also has the same minimax error, with a 3
of 1900 and a 5 of 2800. So, I should rewrite my routine to deal with
cases where there are muultiple solutions; however 40 is closer to
what the simplex dodad gave, for whatever relevance that has. The
actual list of possible choices goes 12, 40, 52, 64, and also 24 and
36 if you allow contorsion. Is it best to get the whole list or select
the one closest to the midpoint value?

> > 50/49 26 17.487807
>
> Why am I not surprised? :-)

You've obviously dealt with 26 more than me; I tend to associate 50/49
with 22, actually. This is another of those deals with multiple
entries, though. 22, 26, 38, 48 and 54 are all contenders, with 44
contorted.

Here is a revised list; I trimmed a few of the most complex commas on
the grounds that they take the longest to compute and even I wouldn't
try to write music in them. We struck gold here with the 19-comma,
where 494 is optimal!

5-limit

27/25 9 48.974620
16/15 8 63.686286
135/128 23 23.694131
25/24 17 37.299889
648/625 [12, 40, 52, 64] 15.641287
250/243 59 9.909406
128/125 [27, 39, 42] 13.686286
3125/3072 41 6.309933
81/80 81 5.658705
2048/2025 [114] 3.308157
78732/78125 382 1.505161
393216/390625 359 1.435159
2109375/2097152 517 1.013907
15625/15552 458 1.389281
1600000/1594323 205 .484024
1224440064/1220703125 1199 .378568
6115295232/6103515625 [1636] .238853
19073486328125/19042491875328 [494, 817, 1159, 1311, 1653, 1805, 1824,
1957, 2128, 2147, 2299, 2489, 2641, 2774, 2793, 2812, 2945, 2983,
3097, 3116, 3135, 3154, 3287, 3439, 3591, 3629, 3762, 3781, 3800,
3819, 3971, 4104, 4123, 4142, 4275, 4313, 4408, 4427, 4465, 4579,
4617, 4750, 4769, 4788, 4921, 5073, 5092, 5111, 5130, 5263, 5453,
5605, 5738, 5757, 5776, 5909, 6080, 6099, 6251, 6403, 6593, 6745,
7068, 7239, 7733] .148187
32805/32768 2129 .217096
582076609134674072265625/581595589965365114830848 [11305] .042153
274877906944/274658203125 7174 .081603

7-limit

1029/1000 55 15.239465
36/35 33 24.732196
525/512 45 12.980381
49/48 29 18.595044
50/49 [22, 26, 38, 48] 17.487807
686/675 83 6.629942
64/63 59 9.909406
875/864 138 6.097843
3125/3087 225 4.358713
2430/2401 146 4.361971
245/243 177 3.129745
126/125 247 4.710113
4000/3969 230 2.619583
1728/1715 142 3.551630
1029/1024 252 2.159240
225/224 228 2.103188
19683/19600 659 .892785
16875/16807 327 1.037570
10976/10935 654 .860816
3136/3125 384 1.262193
5120/5103 531 .869858
6144/6125 412 1.090259
65625/65536 1623 .361773
703125/702464 2912 .187340
420175/419904 4355 .143007
2401/2400 2749 .191349
4375/4374 9058 .057543
250047/250000 [5787] .037347

11-limit

77/75 39 28.666039
45/44 26 21.822829
55/54 51 17.996176
56/55 36 19.258726
245/242 91 8.716755
99/98 127 9.887245
100/99 104 9.326789
121/120 99 8.014001
1331/1323 251 3.209167
176/175 111 4.147067
896/891 145 3.714861
243/242 346 2.881077
385/384 284 2.022168
441/440 320 1.410002
1375/1372 436 .992613
6250/6237 899 .868673
540/539 296 1.300311
4000/3993 998 .927635
5632/5625 1210 .463146
43923/43904 2715 .171594
3025/3024 3207 .177332
9801/9800 [18870] .044619
151263/151250 33541 .025087

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Could this also be a way to set an approximate maximum size for how
many
> notes a temperament can have? It looks like it's easier to define than
> the "consistency limit" I was looking for, and agrees with my
intuitions
> about bug, father, mavila, and porcupine at least.

What about 1/minimax_error (log2) or 1200/minimax_error (cents) as a
maximum size? So, the maximum size of meantone (5 or 7 limit) is 223,
etc. Of course, this is a seriously maximal maximum, but 205 was
already in my mind as kind of the outer limit of meantones as it was.