Some 13-limit microtemperaments

I took all the 13-limit superparticular commas with numerator greater
than 2000 four at a time, giving 15 linear temperaments, listed below
in TOP/Graham badness order. The first temperament on the list seems
to be a standout, it has TM basis {1716/1715, 2080/2079, 3025/2024,
4096/4095}, a period of 1/2 octave and a 44/39 generator. Ets are 224,
270, and 494.

[22, 48, -38, -34, -54, 25, -122, -130, -167, -223, -245, -303, 36,
-11, -61]
[[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]] 173.898312

[72, 108, 72, 36, -54, 4, -88, -192, -352, -136, -290, -525, -148,
-418, -320]
[[18, 28, 41, 50, 62, 67], [0, 4, 6, 4, 2, -3]] 330.983530

[36, 54, 36, 18, 108, 2, -44, -96, 38, -68, -145, 51, -74, 170, 307]
[[18, 28, 41, 50, 62, 65], [0, 2, 3, 2, 1, 6]] 353.702340

[16, 84, 46, 98, 108, 96, 28, 100, 112, -129, -63, -60, 116, 133, 11]
[[2, 4, 9, 8, 12, 13], [0, -8, -42, -23, -49, -54]] 366.221040

[18, 27, 18, -126, -81, 1, -22, -262, -195, -34, -386, -288, -416,
-294, 186]
[[9, 15, 22, 26, 26, 30], [0, -2, -3, -2, 14, 9]] 366.529508

[2, -57, -28, 46, 81, -95, -50, 66, 121, 95, 304, 399, 226, 331, 110]
[[1, 1, 19, 11, -10, -20], [0, 2, -57, -28, 46, 81]] 376.528284

[34, 111, 64, -28, 27, 97, 6, -162, -83, -163, -449, -348, -300, -161,
197]
[[1, 7, 20, 13, -1, 8], [0, -34, -111, -64, 28, -27]] 388.395969

[92, 78, 62, -44, -54, -90, -160, -388, -426, -75, -372, -414, -338,
-381, -24]
[[2, 26, 24, 21, -4, -6], [0, -46, -39, -31, 22, 27]] 402.962307

[20, -30, -10, -80, 0, -94, -72, -196, -74, 61, -82, 111, -190, 37, 296]
[[10, 16, 23, 28, 34, 37], [0, -2, 3, 1, 8, 0]] 442.288853

[90, 135, 90, -90, -135, 5, -110, -454, -547, -170, -676, -813, -564,
-712, -134]
[[45, 71, 104, 126, 156, 167], [0, 2, 3, 2, -2, -3]] 568.498130

[38, -3, 8, 64, -81, -93, -94, -30, -269, 27, 159, -177, 152, -257, -517]
[[1, -7, 3, 1, -11, 22], [0, 38, -3, 8, 64, -81]] 576.552838

[18, 27, 18, 144, 189, 1, -22, 166, 233, -34, 241, 339, 342, 464, 121]
[[9, 15, 22, 26, 37, 41], [0, -2, -3, -2, -16, -21]] 623.344226

[36, 54, 36, 18, -162, 2, -44, -96, -390, -68, -145, -576, -74, -588,
-627]
[[18, 28, 41, 50, 62, 69], [0, 2, 3, 2, 1, -9]] 650.602169

[56, 24, 26, -62, -162, -92, -116, -292, -464, -7, -227, -465, -264,
-551, -331]
[[2, 4, 5, 6, 6, 5], [0, -28, -12, -13, 31, 81]] 715.033013

[54, 81, 54, -108, -243, 3, -66, -358, -585, -102, -531, -864, -490,
-882, -441]
[[27, 43, 63, 76, 93, 99], [0, -2, -3, -2, 4, 9]] 872.189210

Gene Ward Smith wrote:
> I took all the 13-limit superparticular commas with numerator greater
> than 2000 four at a time, giving 15 linear temperaments, listed below
> in TOP/Graham badness order. The first temperament on the list seems
> to be a standout, it has TM basis {1716/1715, 2080/2079, 3025/2024,
> 4096/4095}, a period of 1/2 octave and a 44/39 generator. Ets are 224,
> 270, and 494.

I take it you must have started with 6 commas, then. Could you tell us what they are? Then I could try duplicating the result. Or you could even past them straight into the box at

http://x31eq.com/temper/vectors.html

It'd help, anyway, if you were to paste commas into your messages instead of typing them. It took me a while to work out that 3025/2024 was supposed to be 3025/3024.

Oh yes, the standout temperament was already top of my 13- and 15-limit microtemperament lists.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:

> I take it you must have started with 6 commas, then. Could you
tell us
> what they are? Then I could try duplicating the result.

2401/2400, 4375/4374, 3025/3024, 9801/9800, 2080/2079, 4096/4095,
4225/4224, 6656/6655, 10648/10647, 123201/123200; a total of ten
commas.

> Oh yes, the standout temperament was already top of my 13- and 15-
limit
> microtemperament lists.

It's come up before, I know. You want to name it?