Some 31-limit temperaments

Here are some linear (2D) 31-limit temperaments, and a 5D one tossed in for the hell of it. It doesn't have much to do with the challenge, I suppose, which would need to start from commas.

[[1, 2, 2, -3, 3, -1, 1, 6, 6, 2, 5],
[0, 9, -7, -126, -10, -102, -67, 38, 32, -62, 1]]

[[1, 2, 12, -3, 13, -1, 11, 16, 16, -8, -5],
[0, 3, 70, -42, 69, -34, 50, 85, 83, -93, -72]]

[[1, 5, -36, -45, -36, -35, -57, -17, -19, 17, 41],
[0, 9, -101, -126, -104, -102, -161, -56, -62, 32, 95]]

[[2, 0, -26, -99, -122, -56, -51, -38, -85, 15, -7],
[0, 3, 29, 99, 122, 60, 56, 44, 89, -5, 16]]

[[1, 8, -23, 18, 46, 1, -29, 11, 42, 16, 61],
[0, 19, -75, 45, 126, -8, -98, 20, 111, 33, 166]]

[[1, 1, -2, -3, 4, 1, -1, -3, 7, 9, 5],
[0, 13, 96, 129, -12, 60, 113, 161, -55, -92, -1]]

[[2, 18, 11, 84, 62, 30, 11, 89, 38, -15, 89],
[0, 21, 9, 111, 78, 32, 4, 114, 41, -35, 112]]

[[1, 0, 0, 0, 0, 5, 4, -6, 2, 11, 0],
[0, 1, 0, 0, 0, -2, -2, 5, 0, -3, 4],
[0, 0, 1, 0, 0, 0, 1, 1, 0, -1, -1],
[0, 0, 0, 3, 0, 2, 1, 0, -1, 1, 1],
[0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0]]

genewardsmith wrote:

> Here are some linear (2D) 31-limit temperaments, and a 5D one tossed in
> for the hell of it. It doesn't have much to do with the challenge, I
> suppose, which would need to start from commas.

Indeed, Dave's is far from being the only 31-limit linear temperament
worth bothering with. It may well be that *no* such temperaments are of
any use, but whatever. Here's my top 10:

19/236, 96.6 cent generator

basis:
(1.0, 0.0805347131516)

mapping by period and generator:
[(1, 0), (4, -30), (2, 4), (2, 10), (7, -44), (7, -41), (9, -61), (9,
-59), (10, -68), (8, -39), (6, -13)]

mapping by steps:
[(149, 87), (236, 138), (346, 202), (418, 244), (515, 301), (551, 322),
(609, 356), (633, 370), (674, 394), (724, 423), (738, 431)]

highest interval width: 100
complexity measure: 100 (149 for smallest MOS)
highest error: 0.004177 (5.012 cents)

7/304, 27.7 cent generator

basis:
(1.0, 0.0230419838041)

mapping by period and generator:
[(1, 0), (2, -18), (2, 14), (2, 35), (3, 20), (4, -13), (6, -83), (6,
-76), (6, -64), (7, -93), (5, -2)]

mapping by steps:
[(217, 87), (344, 138), (504, 202), (609, 244), (751, 301), (803, 322),
(887, 356), (922, 370), (982, 394), (1054, 423), (1075, 431)]

highest interval width: 128
complexity measure: 128 (130 for smallest MOS)
highest error: 0.002637 (3.164 cents)

23/499, 55.3 cent generator

basis:
(1.0, 0.0460921348233)

mapping by period and generator:
[(1, 0), (2, -9), (2, 7), (-3, 126), (3, 10), (-1, 102), (1, 67), (6,
-38), (6, -32), (2, 62), (5, -1)]

mapping by steps:
[(282, 217), (447, 344), (655, 504), (792, 609), (976, 751), (1044, 803),
(1153, 887), (1198, 922), (1276, 982), (1370, 1054), (1397, 1075)]

highest interval width: 164
complexity measure: 164 (217 for smallest MOS)
highest error: 0.001778 (2.134 cents)

47/504, 111.9 cent generator

basis:
(1.0, 0.0932479134579)

mapping by period and generator:
[(1, 0), (0, 17), (4, -18), (-4, 73), (-4, 80), (12, -89), (-3, 76), (-2,
67), (9, -48), (-4, 95), (14, -97)]

mapping by steps:
[(311, 193), (493, 306), (722, 448), (873, 542), (1076, 668), (1151, 714),
(1271, 789), (1321, 820), (1407, 873), (1511, 938), (1541, 956)]

highest interval width: 192
complexity measure: 192 (193 for smallest MOS)
highest error: 0.001537 (1.845 cents)

143/745, 230.3 cent generator

basis:
(1.0, 0.191942887125)

mapping by period and generator:
[(1, 0), (6, -23), (10, -40), (3, -1), (43, -206), (49, -236), (30, -135),
(34, -155), (17, -65), (11, -32), (42, -193)]

mapping by steps:
[(422, 323), (669, 512), (980, 750), (1185, 907), (1460, 1117), (1562,
1195), (1725, 1320), (1793, 1372), (1909, 1461), (2050, 1569), (2091,
1600)]

highest interval width: 236
complexity measure: 236 (323 for smallest MOS)
highest error: 0.001207 (1.448 cents)

183/398, 551.8 cent generator

basis:
(1.0, 0.459806624376)

mapping by period and generator:
[(1, 0), (14, -27), (6, -8), (-28, 67), (3, 1), (6, -5), (68, -139), (70,
-143), (62, -125), (49, -96), (-7, 26)]

mapping by steps:
[(311, 87), (493, 138), (722, 202), (873, 244), (1076, 301), (1151, 322),
(1271, 356), (1321, 370), (1407, 394), (1511, 423), (1541, 431)]

highest interval width: 210
complexity measure: 210 (224 for smallest MOS)
highest error: 0.001538 (1.846 cents)

71/460, 185.2 cent generator

basis:
(1.0, 0.154344259681)

mapping by period and generator:
[(1, 0), (11, -61), (-2, 28), (15, -79), (28, -159), (25, -138), (1, 20),
(-1, 34), (9, -29), (1, 25), (19, -91)]

mapping by steps:
[(311, 149), (493, 236), (722, 346), (873, 418), (1076, 515), (1151, 551),
(1271, 609), (1321, 633), (1407, 674), (1511, 724), (1541, 738)]

highest interval width: 239
complexity measure: 239 (311 for smallest MOS)
highest error: 0.001203 (1.444 cents)

73/528, 165.9 cent generator

basis:
(1.0, 0.138261156028)

mapping by period and generator:
[(1, 0), (2, -3), (12, -70), (-3, 42), (13, -69), (-1, 34), (11, -50),
(16, -85), (16, -83), (-8, 93), (-5, 72)]

mapping by steps:
[(311, 217), (493, 344), (722, 504), (873, 609), (1076, 751), (1151, 803),
(1271, 887), (1321, 922), (1407, 982), (1511, 1054), (1541, 1075)]

highest interval width: 233
complexity measure: 233 (311 for smallest MOS)
highest error: 0.001283 (1.539 cents)

33/733, 54.0 cent generator

basis:
(1.0, 0.0450217950553)

mapping by period and generator:
[(1, 0), (1, 13), (-2, 96), (-3, 129), (4, -12), (1, 60), (-1, 113), (-3,
161), (7, -55), (9, -92), (5, -1)]

mapping by steps:
[(422, 311), (669, 493), (980, 722), (1185, 873), (1460, 1076), (1562,
1151), (1725, 1271), (1793, 1321), (1909, 1407), (2050, 1511), (2091,
1541)]

highest interval width: 284
complexity measure: 284 (311 for smallest MOS)
highest error: 0.000963 (1.155 cents)
unique

19/335, 34.0 cent generator

basis:
(0.5, 0.0283513085743)

mapping by period and generator:
[(2, 0), (3, 3), (3, 29), (0, 99), (0, 122), (4, 60), (5, 56), (6, 44),
(4, 89), (10, -5), (9, 16)]

mapping by steps:
[(388, 282), (615, 447), (901, 655), (1089, 792), (1342, 976), (1436,
1044), (1586, 1153), (1648, 1198), (1755, 1276), (1885, 1370), (1922,
1397)]

highest interval width: 127
complexity measure: 254 (282 for smallest MOS)
highest error: 0.001214 (1.457 cents)

The search is slow running, so I won't enable it on the web.

Dave's temperament, by comparison

236/699, 405.1 cent generator

basis:
(1.0, 0.337623880246)

mapping by period and generator:
[(1, 0), (8, -19), (-23, 75), (18, -45), (46, -126), (1, 8), (-29, 98),
(11, -20), (42, -111), (16, -33), (61, -166)]

mapping by steps:
[(388, 311), (615, 493), (901, 722), (1089, 873), (1342, 1076), (1436,
1151), (1586, 1271), (1648, 1321), (1755, 1407), (1885, 1511), (1922,
1541)]

highest interval width: 316
complexity measure: 316 (388 for smallest MOS)
highest error: 0.000981 (1.177 cents)

gets beaten on both complexity, smallest MOS and highest error by one of
those in my list. It doesn't seem to be 31-limit unique, either.

Here's the top 5 containing a consistent pair of ETs and a worst error
(however it's being measured) no greater than 0.1 cents

367/8881, 49.6 cent generator

basis:
(1.0, 0.0413241658842)

mapping by period and generator:
[(1, 0), (42, -978), (11, -210), (60, -1384), (79, -1828), (48, -1072),
(80, -1837), (4, 6), (123, -2867), (38, -802), (105, -2421)]

mapping by steps:
[(4501, 4380), (7134, 6942), (10451, 10170), (12636, 12296), (15571,
15152), (16656, 16208), (18398, 17903), (19120, 18606), (20361, 19813),
(21866, 21278), (22299, 21699)]

highest interval width: 2940
complexity measure: 2940 (3049 for smallest MOS)
highest error: 0.000061 (0.074 cents)
unique

313/5783, 64.9 cent generator

basis:
(1.0, 0.0541241847005)

mapping by period and generator:
[(1, 0), (9, -137), (25, -419), (-78, 1493), (-80, 1542), (68, -1188),
(20, -294), (-62, 1224), (-45, 915), (31, -483), (-64, 1274)]

mapping by steps:
[(5395, 388), (8551, 615), (12527, 901), (15146, 1089), (18664, 1342),
(19964, 1436), (22052, 1586), (22918, 1648), (24405, 1755), (26209, 1885),
(26728, 1922)]

highest interval width: 2730
complexity measure: 2730 (3067 for smallest MOS)
highest error: 0.000077 (0.092 cents)
unique

1321/5706, 277.8 cent generator

basis:
(1.0, 0.231510667232)

mapping by period and generator:
[(1, 0), (-85, 374), (166, -707), (135, -571), (-116, 516), (-191, 841),
(265, -1127), (167, -703), (-183, 810), (-191, 846), (-293, 1287)]

mapping by steps:
[(5395, 311), (8551, 493), (12527, 722), (15146, 873), (18664, 1076),
(19964, 1151), (22052, 1271), (22918, 1321), (24405, 1407), (26209, 1511),
(26728, 1541)]

highest interval width: 2701
complexity measure: 2701 (2907 for smallest MOS)
highest error: 0.000081 (0.097 cents)
unique

1915/9896, 232.2 cent generator

basis:
(1.0, 0.193512518442)

mapping by period and generator:
[(1, 0), (25, -121), (173, -882), (182, -926), (220, -1119), (-220, 1156),
(-220, 1158), (298, -1518), (-129, 690), (-65, 361), (-68, 377)]

mapping by steps:
[(5395, 4501), (8551, 7134), (12527, 10451), (15146, 12636), (18664,
15571), (19964, 16656), (22052, 18398), (22918, 19120), (24405, 20361),
(26209, 21866), (26728, 22299)]

highest interval width: 2922
complexity measure: 2922 (3607 for smallest MOS)
highest error: 0.000076 (0.091 cents)
unique

187/6462, 34.7 cent generator

basis:
(1.0, 0.0289383892511)

mapping by period and generator:
[(1, 0), (-13, 504), (22, -680), (18, -525), (-17, 707), (-29, 1130), (40,
-1241), (23, -648), (-30, 1193), (-29, 1170), (-41, 1588)]

mapping by steps:
[(6151, 311), (9749, 493), (14282, 722), (17268, 873), (21279, 1076),
(22761, 1151), (25142, 1271), (26129, 1321), (27824, 1407), (29881, 1511),
(30473, 1541)]

highest interval width: 2948
complexity measure: 2948 (3041 for smallest MOS)
highest error: 0.000075 (0.090 cents)
unique

> [[1, 2, 2, -3, 3, -1, 1, 6, 6, 2, 5],
> [0, 9, -7, -126, -10, -102, -67, 38, 32, -62, 1]]

That's my number 3

> [[1, 2, 12, -3, 13, -1, 11, 16, 16, -8, -5],
> [0, 3, 70, -42, 69, -34, 50, 85, 83, -93, -72]]

That's my number 8

> [[1, 5, -36, -45, -36, -35, -57, -17, -19, 17, 41],
> [0, 9, -101, -126, -104, -102, -161, -56, -62, 32, 95]]
>
> [[2, 0, -26, -99, -122, -56, -51, -38, -85, 15, -7],
> [0, 3, 29, 99, 122, 60, 56, 44, 89, -5, 16]]

That's my number 10

> [[1, 8, -23, 18, 46, 1, -29, 11, 42, 16, 61],
> [0, 19, -75, 45, 126, -8, -98, 20, 111, 33, 166]]
>
> [[1, 1, -2, -3, 4, 1, -1, -3, 7, 9, 5],
> [0, 13, 96, 129, -12, 60, 113, 161, -55, -92, -1]]

That's my number 9, and the one that beats Dave's 388&311 by all measures.

> [[2, 18, 11, 84, 62, 30, 11, 89, 38, -15, 89],
> [0, 21, 9, 111, 78, 32, 4, 114, 41, -35, 112]]

Graham

--- In tuning-math@y..., graham@m... wrote:
> Indeed, Dave's is far from being the only 31-limit linear
temperament
> worth bothering with. It may well be that *no* such temperaments
are of
> any use, but whatever. Here's my top 10:
...

Thanks for that.

> 33/733, 54.0 cent generator
>
> basis:
> (1.0, 0.0450217950553)
>
> mapping by period and generator:
> [(1, 0), (1, 13), (-2, 96), (-3, 129), (4, -12), (1, 60), (-1, 113),
(-3,
> 161), (7, -55), (9, -92), (5, -1)]
>
> mapping by steps:
> [(422, 311), (669, 493), (980, 722), (1185, 873), (1460, 1076),
(1562,
> 1151), (1725, 1271), (1793, 1321), (1909, 1407), (2050, 1511),
(2091,
> 1541)]
>
> highest interval width: 284
> complexity measure: 284 (311 for smallest MOS)
> highest error: 0.000963 (1.155 cents)
> unique

This is significant if it's the least complex one that is 31-limit
unique. But I'm worried about your uniqueness tester because of what
it says about the 311&388 temperament.

> Dave's temperament, by comparison
>
> 236/699, 405.1 cent generator
>
> basis:
> (1.0, 0.337623880246)
>
> mapping by period and generator:
> [(1, 0), (8, -19), (-23, 75), (18, -45), (46, -126), (1, 8), (-29,
98),
> (11, -20), (42, -111), (16, -33), (61, -166)]
>
> mapping by steps:
> [(388, 311), (615, 493), (901, 722), (1089, 873), (1342, 1076),
(1436,
> 1151), (1586, 1271), (1648, 1321), (1755, 1407), (1885, 1511),
(1922,
> 1541)]
>
> highest interval width: 316
> complexity measure: 316 (388 for smallest MOS)
> highest error: 0.000981 (1.177 cents)
>
> gets beaten on both complexity, smallest MOS and highest error by
one of
> those in my list. It doesn't seem to be 31-limit unique, either.

So which intervals does it conflate. I can't find any. I just checked
again.

> Here's the top 5 containing a consistent pair of ETs and a worst
error
> (however it's being measured) no greater than 0.1 cents
...

Why would anyone be interested in these, with complexities around
3000?

dkeenanuqnetau wrote:

> This is significant if it's the least complex one that is 31-limit
> unique. But I'm worried about your uniqueness tester because of what
> it says about the 311&388 temperament.

It may not be the least complex, because I can't be sure all the
significant temperaments were included in the search.

> So which intervals does it conflate. I can't find any. I just checked
> again.

29:28 and 28:27 which are both 20 steps in 388-equal and 16 steps in
311-equal.

> > Here's the top 5 containing a consistent pair of ETs and a worst
> error
> > (however it's being measured) no greater than 0.1 cents
> ...
>
> Why would anyone be interested in these, with complexities around
> 3000?

You might be able to find some higher-dimensioned temperaments by turning
common jumps into new generators, or removing a few unison vectors if you
can work out the unison vectors.

Graham

--- In tuning-math@y..., graham@m... wrote:
> dkeenanuqnetau wrote:
>
> > This is significant if it's the least complex one that is 31-limit
> > unique. But I'm worried about your uniqueness tester because of
what
> > it says about the 311&388 temperament.
>
> It may not be the least complex, because I can't be sure all the
> significant temperaments were included in the search.
>
> > So which intervals does it conflate. I can't find any. I just
checked
> > again.
>
> 29:28 and 28:27 which are both 20 steps in 388-equal and 16 steps in
> 311-equal.

Oh yes. They are both 12 gens wide. I wasn't taking absolute values
before comparing numbers of gens. Duh!

Thanks.

> > > Here's the top 5 containing a consistent pair of ETs and a worst
> > error
> > > (however it's being measured) no greater than 0.1 cents
> > ...
> >
> > Why would anyone be interested in these, with complexities around
> > 3000?
>
> You might be able to find some higher-dimensioned temperaments by
turning
> common jumps into new generators, or removing a few unison vectors
if you
> can work out the unison vectors.

OK. Thanks. But I don't know how to. Anyway, I've worked out all the
commas now. I just need to generate all the possible temperaments from
them and find those with low dimensionality (and possibly minimax
error < 0.5 c) I'm hoping Gene or you have something already that you
can use to do that.