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Message: 4450 - Contents - Hide Contents 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 largerthat 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

Message: 4451 - Contents - Hide Contents 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

Message: 4452 - Contents - Hide Contents 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]]

Message: 4453 - Contents - Hide Contents 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?

Message: 4454 - Contents - Hide Contents 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.

Message: 4455 - Contents - Hide Contents 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.

Message: 4456 - Contents - Hide Contents 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) > uniqueThis 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?

Message: 4457 - Contents - Hide Contents 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.

Message: 4458 - Contents - Hide Contents 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

Message: 4459 - Contents - Hide Contents 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)?

Message: 4460 - Contents - Hide Contents 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 - Type Ok * [with cont.] (Wayb.) 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 * [with cont.] (Wayb.)

Message: 4461 - Contents - Hide Contents 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?

Message: 4462 - Contents - Hide Contents 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 1i don't like this kind of constraint because it makes 11/7 seem as complex as 77/64.

Message: 4463 - Contents - Hide Contents 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.

Message: 4464 - Contents - Hide Contents 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.

Message: 4465 - Contents - Hide Contents 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 doingso I may> have 'broken' some 'rules'. As for my own 'rules': I am happy break those > too. > > Markmake 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'!

Message: 4466 - Contents - Hide Contents 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.

Message: 4467 - Contents - Hide Contents 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, thenI 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!

Message: 4468 - Contents - Hide Contents 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:

Message: 4469 - Contents - Hide Contents 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? > > -Carlsure -- 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.

Message: 4470 - Contents - Hide Contents 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.

Message: 4471 - Contents - Hide Contents 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.

Message: 4472 - Contents - Hide Contents 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.

Message: 4473 - Contents - Hide Contents 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. >

Message: 4474 - Contents - Hide Contents 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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