Tuning-Math Digests messages 4450 - 4474

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Message: 4450

Date: Sat, 30 Mar 2002 06:42:14

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > Find the lowest dimensioned 31-limit temperament that has no 
unison 
> > vector larger than 0.5 cent. 
> 
> There will always be 31-limit temperaments for each dimension up to 
> pi(31)=11 such that they have a basis consisting of commas no larger 
that half a cent, there will never be one such that all the commas are 
less than half a cent. This does not seem to be a well-defined 
question, so I think I'll just go ponder some 31-limit temperaments.
>

Thanks for convincing me that I haven't supplied enough constraints.

I think this is what we want:

Find the lowest dimensioned 31-limit temperament having a basis 
consisting of commas no larger that half a cent, where the absolute 
value of the exponent of each prime in each comma of the basis is no 
greater than:

Prime Exponent limit
---------------------
2     unbounded (but because of the other constraints it won't be 
bigger than 57)
3     12  (because of Pythagorean-12 based notation)
5     2   (because 25 is in the 31 odd-limit)
7     1
11    1
13    1
17    1
19    1
23    1
29    1
31    1


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Message: 4451

Date: Sat, 30 Mar 2002 07:46:26

Subject: 31-limit ets

From: genewardsmith

Here are 31-limit log-flat badness scores for every "standard" hn which scored less than 1.25 up to 612:

7   1.086959041
10   1.180005939
12   1.201543288
19   1.063482905
31   1.157990819
34   1.180070372
46   1.246127210
72   1.127555928
87   1.078433731
111   1.229732367
118   1.168950295
140   1.201387377
149   1.054063461
159   1.167498770
193   1.140668095
217   .9985489320
277   1.247460140
282   .9833666225
296   1.223223277
311   .8596486701
323   1.208026520
388   .8872894036
422   .9335227872
487   1.158392401


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Message: 4452

Date: Sat, 30 Mar 2002 08:19:29

Subject: Some 31-limit temperaments

From: genewardsmith

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]]


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Message: 4453

Date: Sat, 30 Mar 2002 00:22:39

Subject: Re: Starling example

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here's a matrix of 3 et columns giving "starling":
> 
> [[31, 49, 72, 87], [46, 73, 107, 129], [50, 79, 116, 140]]
> 
> Here's a unimodular transformation matrix:
> 
> [[-15, -4, 13], [-14, 4, 5], [16, -1, -9]]
> 
> Here's the final result, the Hermite normal form for starling:
> 
> [[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]]

And what are the 3 generators implied by that?


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Message: 4454

Date: Sat, 30 Mar 2002 11:41 +0

Subject: Re: Some 31-limit temperaments

From: graham@xxxxxxxxxx.xx.xx

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


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Message: 4455

Date: Sat, 30 Mar 2002 01:14:39

Subject: Re: Decatonics

From: genewardsmith

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Is the entry for val in monz's tuning dictionary the only version
> there is?

There's one in my contribution to the paper project, which I haven't finished.


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Message: 4456

Date: Sat, 30 Mar 2002 13:10:34

Subject: Re: Some 31-limit temperaments

From: dkeenanuqnetau

--- 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?


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Message: 4457

Date: Sat, 30 Mar 2002 01:47:27

Subject: Re: Starling example

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > Here's the final result, the Hermite normal form for starling:
> > 
> > [[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]]
> 
> And
what are the 3 generators implied by that?

From the fact that the top part of the matrix is the 3x3 identity it
follows that the generators are what Hermite form produces if it
can--generators approximating 2,3, and 5. If we assume pure octaves
and take 7-limit rms values, we get

a = 1200.000
b = 1899.984
c = 2789.270

as generators, giving us an approximate 7 of 3367.841 cents. The fifth
is the 12-et fifth, but of course the third and 7 are not close to
12-et. 31, 46 and 77 give fairly decent but not more than that
versions of starling.


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Message: 4458

Date: Sat, 30 Mar 2002 15:13 +0

Subject: Re: Some 31-limit temperaments

From: graham@xxxxxxxxxx.xx.xx

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


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Message: 4459

Date: Sat, 30 Mar 2002 02:15:16

Subject: Re: Hermite normal form version of "25 best"

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > again, i'm wondering why you're not putting these in order of g_w.
> 
> Because sorting a list of commas by size and then computing from 
>that is easier.

why is that easier than sorting by the size of the numbers in the 
commas (which is my heuristic approximation for g_w)?


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Message: 4460

Date: Sat, 30 Mar 2002 10:39:26

Subject: Re: 31-limit microtemperament challenge

From: David C Keenan

OK. I've used brute force to find all the sub-half-cent 31-limit commas
with the constarints I mentioned on prime exponents. 363 of them. You will
find them in
http://uq.net.au/~zzdkeena/Music/31LimitHalfCentCommas.xls.zip - Ok *

So what's the lowest dimensioned temperament you can make with them? I only
need a linearly independent set of commas for it (preferably the set with
the most zeros in the vectors).

If it turns out you can get below 6D with them, then I probably need some
other constraint, such as that the minimax error for the temperament is
less than 0.5 c.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 4461

Date: Sat, 30 Mar 2002 02:23:42

Subject: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: dkeenanuqnetau

--- I wrote:
> Just for the record, since I worked it out while looking at notation 
> issues, here's what might be the only 31-limit temperament of any 
> musical interest, and even that interest is extremely doubtful.

That should have been "might be the only 31-limit _linear_ 
temperament".

31-limit rational is 11D (in the sense that meantone is 2D) and George 
Secor, as part of the notation effort, has apparently found an 8D 
31-limit temperament whose unison vectors are all smaller than 0.5 c, 
but I'd like to be sure this is the best we can do.

So here's the challenge:

Find the lowest dimensioned 31-limit temperament that has no unison 
vector larger than 0.5 cent. I think the 0.5 c limit must apply to any 
possible set of unison vectors for the temperament. Is this a coherent 
requirement?


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Message: 4462

Date: Sat, 30 Mar 2002 20:52:56

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Thanks for convincing me that I haven't supplied enough constraints.
> 
> I think this is what we want:
> 
> Find the lowest dimensioned 31-limit temperament having a basis 
> consisting of commas no larger that half a cent, where the absolute 
> value of the exponent of each prime in each comma of the basis is 
no 
> greater than:
> 
> Prime Exponent limit
> ---------------------
> 2     unbounded (but because of the other constraints it won't be 
> bigger than 57)
> 3     12  (because of Pythagorean-12 based notation)
> 5     2   (because 25 is in the 31 odd-limit)
> 7     1
> 11    1
> 13    1
> 17    1
> 19    1
> 23    1
> 29    1
> 31    1

i don't like this kind of constraint because it makes 11/7 seem as 
complex as 77/64.


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Message: 4463

Date: Sat, 30 Mar 2002 02:24:52

Subject: Re: Digest Number 331

From: paulerlich

--- In tuning-math@y..., John Chalmers <JHCHALMERS@U...> wrote:
> Manuel: Thanks for the counter-example to CS equalling strict 
propriety.
> I stand corrected
> 
> As for harmonic and inharmonic vocal timbres. I was apparently 
mistaken.
> What confused me was the fact that outside of the European culture 
area,
> vocal timbres are usually nasal and/or strident and their use may be
> correlated with non-JI (or close approximations) tunings and 
intervals.
> For example, how harmonic is the spectrum of the Indonesian singing
> voice or that of American Indians?

perfectly harmonic, with a certain amount of noise, as always.

> For that matter, how harmonically
> related are the formants of speech in many languages (Khoisan, North
> Caucasian, etc.).

i don't know what you mean by 'harmonically related formants'. 
formants are recognized by their absolute frequency, and of course 
they operate by amplifying harmonics near that frequency. but . . . ?

> It seemed to me that to produce the clear harmonic
> tone of European singing (primarily Church and Italianate styles) 
takes
> a lot of training. Untrained voices often sound less harmonic to 
me, but
> I could be wrong. 

they may contain more noise, but do an fft (or anything like that) 
and you won't find a systematic significant deviation of the partials 
from a harmonic series, in one direction or the other.

> How in tune are the harmonics and are the usual pitches of the vowel
> formants for most speakers actually close to harmonics?

again, not sure what you mean by this.


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Message: 4464

Date: Sat, 30 Mar 2002 22:32:57

Subject: Re: Some 31-limit temperaments

From: dkeenanuqnetau

--- 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.


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Message: 4465

Date: Sat, 30 Mar 2002 02:28:38

Subject: Re: Rules for Diatonics

From: paulerlich

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

> As I stated in my introduction:
> 
> "This paper approaches these and other 'diatonic' scales from the 
viewpoint
> of a composer seeking new materials for creative work, rather than 
trying
> for rigorous mathematical proof."
> 
> I do have an awareness of the mathematical basis for tuning theory, 
but the
> scope of my studies was to derive scales that have _musical_ 
properties that
> bore resemblance to the one diatonic scale that (in whatever form) 
does
> exist. Shape and structure guided what I looked for, and in doing 
so I may
> have 'broken' some 'rules'. As for my own 'rules': I am happy break 
those
> too.
> 
> Mark

make no mistake: i feel exactly the same way you do about this. i 
will take a look at your ideas again when i have an opportunity -- 
meanwhile, i hope you will take a look at my 'gentle introdution to 
fokker periodicity blocks' -- talk about 'shape' and 'structure'!


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Message: 4466

Date: Sat, 30 Mar 2002 22:40:39

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > Thanks for convincing me that I haven't supplied enough 
constraints.
> > 
> > I think this is what we want:
> > 
> > Find the lowest dimensioned 31-limit temperament having a basis 
> > consisting of commas no larger that half a cent, where the 
absolute 
> > value of the exponent of each prime in each comma of the basis is 
> no 
> > greater than:
> > 
> > Prime Exponent limit
> > ---------------------
> > 2     unbounded (but because of the other constraints it won't be 
> > bigger than 57)
> > 3     12  (because of Pythagorean-12 based notation)
> > 5     2   (because 25 is in the 31 odd-limit)
> > 7     1
> > 11    1
> > 13    1
> > 17    1
> > 19    1
> > 23    1
> > 29    1
> > 31    1
> 
> i don't like this kind of constraint because it makes 11/7 seem as 
> complex as 77/64.

Remember that the purpose of this temperament is to make a notation 
with a minimum number of symbols (or sagittal flags) that can notate 
rational scales so even Johnny Reinhard can't tell the difference, and 
notate all ETs below 100-ET and many above it.


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Message: 4467

Date: Sat, 30 Mar 2002 02:39:28

Subject: Re: Digest Number 331

From: paulerlich

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:
> 
> 
> > From: tuning-math@y...
> > Reply-To: tuning-math@y...
> > Date: 29 Mar 2002 16:15:12 -0000
> > To: tuning-math@y...
> > Subject: [tuning-math] Digest Number 331
> > 
> > that's the pentachordal decatonic scale -- hopefully you're also
> > aware of the symmetrical decatonic i proposed. each of the two
> > decatonics can be seen as a pair of interlaced 3/2-generated
> > pentatonics -- in the symmetrical case the separation is 600 cents
> > instead of 109 cents.
> Hence my diagram
> > 
> 
> > (note that there is no 'equal' in the title of my paper).
> Seems that I have a duff titled copy then. Apologies : I will 
correct it
> asap.
> > 
> > it seems you are choosing a mode without a 4/3 over the tonic --
> > nothing inherently wrong with this choice, but i wonder what is
> > motivating it. most likely we have different views about which
> > properties of the diatonic scale are appropriate to keep in the
> > process of generalization -- it would be fun to flesh this out.
> 
> I am off for the next few days, but I will get back to you on this. 
I will
> say that I considered the 5 pentatonic
> 
> as 9 0 2 4 7 9 0 2 4 7
>     3 2 2 3 2 3 2 2 3 etc
> 
> 22 Tone:
> 
> 11 13 16 18 20 0 2 4 7 9 11 13 16
>   2  3  2  2  2 2 2 3 2 2  2  3
> 
> The decatonic, like the pentatonic, has two groups of 2s. Taking 
the smaller
> group and putting 3s around it we get:
> 
>     4 7 9 11 13 16
>      3 2 2  2  3
> 
> Then choosing the top tone : 16, in the same way as the pentatonic:
> 
>      4 7 9 0   Choosing 0.
>       3 2 3
> 
> That was my choice, based purely on shape and symmetry. No maths at 
all.

so analogy based on outward appearance.

i use just as little math in my paper. but i feel i base my choices 
on less arbitrary and more acoustically plausible criteria. i show 
that the 'statically tonal' modes of the pentatonic scale are the 
familiar major and minor pentatonic modes, correctly identify the 
most tonal modes of the diatonic scale, and go on to present choices 
for the decatonic scale which seem to hold up remarkably well in 
continued musical exploration on 22-tone instruments.

> As for the choice of cyclic intervals for the 'generators', or the 
grid
> intervals, these came about by the simple method of searching 
manually until
> the pentatonics arose.

if you draw an actual harmonic lattice of 22-equal, where the 'rungs' 
are the 7-limit consonant intervals, you'll see these pentatonics 
immediately (each living within a single 3-7 plane). no searching 
necessary.

> I then simply overlaid the necessary transposed
> pentatonic. The tonic derivation I give above. I will look at your 
other
> decatonics in due course. If this is all very unmathematical, then 
I am not
> ashamed to say that the 'maths' is not of concern to me, only 
the 'shapes'.

likewise. i think one can peer a little deeper into the shapes, and 
i'm excited at the opportunity to share what i (and others like gene) 
have discovered with you. hope you'll be patient with my exuberance!


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Message: 4468

Date: Sat, 30 Mar 2002 02:43:43

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I think the 0.5 c limit must apply to any 
> possible set of unison vectors for the temperament. Is this a 
coherent 
> requirement?

no.


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Message: 4469

Date: Sat, 30 Mar 2002 02:48:35

Subject: Re: Digest Number 331

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Would there be any
> point in having the boys over on the main list run some stuff 
through
> their latest?
> 
> -Carl

sure -- or we could ask francois, as he's apparantly done plenty of 
analyses on human voices. i'm quite confident we won't find human 
voices with statistically significantly stretched or contracted 
partials relative to the harmonic series -- the vocal folds simply 
have no way of vibrating in such a manner.


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Message: 4470

Date: Sat, 30 Mar 2002 04:20:07

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > I think the 0.5 c limit must apply to any 
> > possible set of unison vectors for the temperament. Is this a 
> coherent 
> > requirement?
> 
> no.

Ok. Forget it. Maybe one set of unison vectors, all less than 0.5 c 
will do.


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Message: 4471

Date: Sat, 30 Mar 2002 04:24:03

Subject: Re: Hermite normal form version of "25 best"

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> > Because sorting a list of commas by size and then computing from 
> >that is easier.
> 
> why is that easier than sorting by the size of the numbers in the 
> commas
(which is my heuristic approximation for g_w)?

Because the comma is what I use to compute g_w, not the other way
around. I would need to write more code to do it your way, and it
doesn't seem to matter much, given that anyone can arrange things any
way they like.


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Message: 4472

Date: Sat, 30 Mar 2002 04:39:53

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Find the lowest dimensioned 31-limit temperament that has no unison 
> vector
larger than 0.5 cent. 

There will always be 31-limit temperaments for each dimension up to 
pi(31)=11 such that they have a basis consisting of commas no larger
that half a cent, there will never be one such that all the commas are
less than half a cent. This does not seem to be a well-defined
question, so I think I'll just go ponder some 31-limit temperaments.


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Message: 4473

Date: Sun, 31 Mar 2002 06:15:24

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > 
> > > Thanks for convincing me that I haven't supplied enough 
> constraints.
> > > 
> > > I think this is what we want:
> > > 
> > > Find the lowest dimensioned 31-limit temperament having a basis 
> > > consisting of commas no larger that half a cent, where the 
> absolute 
> > > value of the exponent of each prime in each comma of the basis 
is 
> > no 
> > > greater than:
> > > 
> > > Prime Exponent limit
> > > ---------------------
> > > 2     unbounded (but because of the other constraints it won't 
be 
> > > bigger than 57)
> > > 3     12  (because of Pythagorean-12 based notation)
> > > 5     2   (because 25 is in the 31 odd-limit)
> > > 7     1
> > > 11    1
> > > 13    1
> > > 17    1
> > > 19    1
> > > 23    1
> > > 29    1
> > > 31    1
> > 
> > i don't like this kind of constraint because it makes 11/7 seem 
as 
> > complex as 77/64.
> 
> Remember that the purpose of this temperament is to make a notation 
> with a minimum number of symbols (or sagittal flags) that can 
notate 
> rational scales so even Johnny Reinhard can't tell the difference, 
and 
> notate all ETs below 100-ET and many above it.

even so.


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Message: 4474

Date: Sun, 31 Mar 2002 07:00:05

Subject: Another thought about standard forms for temperament mappings

From: genewardsmith

We want (for dimensions above linear temperaments) a standard form of
the mapping so as to be able to calculate generator steps. Perhaps
taking Hermite form first, and then doing a Minkowski reduction on the
non-octave part of the lattice (excluding the first column) would be a
good plan. The reduction could be with regard to the weighted distance
function Paul likes. The problem with it all is that it's hard to
compute; an LLL reduction would be much easier. Anyone care to weigh
in on this?


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