This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
Contents Hide Contents S 54000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950
4350 - 4375 -
Message: 4375 Date: Mon, 25 Mar 2002 03:44:09 Subject: Hermite normal form version of "25 best" From: genewardsmith Here's the same list, this time using Hermite normal form. The idea of this is to have a standard form which generalizes to higher dimension temperaments and could allow us to measure badness for them. It also is conceptually not wedded to octave-equivalence, but works well in that context. The disadvantage is that you might not like it! I also changed the weighted "g" measure to one which is a weighted mean, since Dave complained bitterly that my adjustment wasn't one. 135/128 (3)^3*(5)/(2)^7 map [[1, 0, 7], [0, 1, -3]] generators 1200. 1877.137655 badness 302.8580950 rms 18.07773392 g_w 2.558772839 ets [2, 7, 9, 11, 16, 23] 256/243 (2)^8/(3)^5 map [[5, 8, 0], [0, 0, 1]] generators 240.0000000 2795.336214 badness 534.3548699 rms 12.75974144 g_w 3.472662942 ets [5, 10, 15, 20, 25, 30] 25/24 (5)^2/(2)^3/(3) map [[1, 1, 2], [0, 2, 1]] generators 1200. 350.9775007 badness 117.6842391 rms 28.85189698 g_w 1.597771402 ets [3, 4, 6, 7, 10, 13, 17, 20] 648/625 (2)^3*(3)^4/(5)^4 map [[4, 0, 3], [0, 1, 1]] generators 300.0000000 1894.134357 badness 467.8848249 rms 11.06006024 g_w 3.484393186 ets [4, 8, 12, 16, 24, 28, 36, 40, 52, 64] 16875/16384 (3)^3*(5)^4/(2)^14 map [[1, 2, 2], [0, 4, -3]] generators 1200. -126.2382718 badness 624.5682202 rms 5.942562596 g_w 4.719203505 ets [1, 9, 10, 19, 20, 28, 29, 38, 47, 48, 57, 76] 250/243 (2)*(5)^3/(3)^5 map [[1, 2, 3], [0, 3, 5]] generators 1200. -162.9960265 badness 317.2740642 rms 7.975800816 g_w 3.413658644 ets [7, 8, 15, 22, 29, 30, 37, 44, 51, 59, 66] 128/125 (2)^7/(5)^3 map [[3, 0, 7], [0, 1, 0]] generators 400.0000000 1908.798145 badness 172.7173147 rms 9.677665780 g_w 2.613294890 ets [3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42] 3125/3072 (5)^5/(2)^10/(3) map [[1, 0, 2], [0, 5, 1]] generators 1200. 379.9679493 badness 321.4409273 rms 4.569472316 g_w 4.128050871 ets [3, 6, 16, 19, 22, 25, 35, 38, 41, 44, 57, 60, 63, 66, 76, 79, 82, 85, 104, 107] 20000/19683 (2)^5*(5)^4/(3)^9 map [[1, 1, 1], [0, 4, 9]] generators 1200. 176.2822703 badness 493.1367768 rms 2.504205191 g_w 5.817894303 ets [7, 27, 34, 41, 48, 61, 68, 75, 82, 95, 102, 109, 116, 136, 143, 150, 177, 184, 191, 218, 225, 259] 81/80 (3)^4/(2)^4/(5) map [[1, 0, -4], [0, 1, 4]] generators 1200. 1896.164845 badness 70.66006887 rms 4.217730828 g_w 2.558772839 ets [5, 7, 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, 55, 57, 62, 67, 69, 74, 76, 81, 86, 88, 93, 98, 100, 105, 117, 129] 2048/2025 (2)^11/(3)^4/(5)^2 map [[2, 0, 11], [0, 1, -2]] generators 600.0000000 1905.446531 badness 145.9438883 rms 2.612821643 g_w 3.822598772 ets [2, 10, 12, 14, 20, 22, 24, 32, 34, 36, 44, 46, 54, 56, 58, 66, 68, 70, 78, 80, 90, 92, 102, 112, 114, 124, 126, 136, 148, 160] 78732/78125 (2)^2*(3)^9/(5)^7 map [[1, 6, 8], [0, 7, 9]] generators 1200. -757.0207028 badness 359.5309133 rms 1.157498409 g_w 6.772337791 ets [8, 19, 27, 38, 46, 57, 65, 73, 76, 84, 92, 103, 111, 122, 130, 141, 149, 157, 168, 176, 187, 195, 214, 233, 241, 252, 260, 279, 298, 306, 317, 325, 344, 363, 382, 390, 409, 428, 447, 474, 493, 539, 558, 623] 393216/390625 (2)^17*(3)/(5)^8 map [[1, 7, 3], [0, 8, 1]] generators 1200. -812.1803271 badness 325.6115779 rms 1.071949828 g_w 6.722154036 ets [3, 6, 28, 31, 34, 37, 62, 65, 68, 71, 93, 96, 99, 102, 127, 130, 133, 136, 158, 161, 164, 167, 192, 195, 198, 201, 223, 226, 229, 232, 257, 260, 263, 266, 288, 291, 294, 297, 322, 325, 328, 331, 353, 356, 359, 362, 365, 387, 390, 393, 421, 452] 2109375/2097152 (3)^3*(5)^7/(2)^21 map [[1, 0, 3], [0, 7, -3]] generators 1200. 271.5895996 badness 297.1369199 rms .8004099292 g_w 7.187006703 ets [9, 13, 22, 31, 40, 44, 53, 62, 66, 75, 84, 93, 97, 106, 115, 119, 128, 137, 146, 150, 159, 168, 172, 181, 190, 199, 203, 212, 221, 225, 234, 243, 252, 256, 265, 274, 278, 287, 296, 305, 309, 318, 327, 340, 349, 358, 371, 380, 402, 411, 424, 433, 455, 464, 486, 517, 570] 4294967296/4271484375 (2)^32/(3)^7/(5)^9 map [[1, 2, 2], [0, 9, -7]] generators 1200. -55.27549315 badness 599.5982250 rms .4831084292 g_w 10.74662038 ets [1, 21, 22, 43, 44, 64, 65, 66, 86, 87, 108, 109, 129, 130, 131, 151, 152, 173, 174, 195, 196, 216, 217, 218, 238, 239, 260, 261, 282, 283, 303, 304, 325, 326, 347, 348, 369, 390, 391, 412, 413, 434, 456, 477, 478, 499, 521, 543, 564, 565, 586, 608, 630, 651, 673, 695, 716, 738, 760, 803, 825, 890, 977] 15625/15552 (5)^6/(2)^6/(3)^5 map [[1, 0, 1], [0, 6, 5]] generators 1200. 317.0796753 badness 127.9730255 rms 1.029625097 g_w 4.990527341 ets [4, 15, 19, 23, 30, 34, 38, 49, 53, 57, 68, 72, 76, 83, 87, 91, 102, 106, 110, 121, 125, 136, 140, 144, 155, 159, 163, 174, 178, 189, 193, 197, 208, 212, 227, 231, 242, 246, 250, 261, 265, 280, 284, 295, 299, 314, 318, 333, 337, 348, 352, 367, 371, 386, 401, 405, 420, 424, 439, 454, 458, 473, 492, 507, 526, 545, 560, 579, 613, 632, 666, 719] 1600000/1594323 (2)^9*(5)^5/(3)^13 map [[1, 3, 6], [0, 5, 13]] generators 1200. -339.5088256 badness 220.2346413 rms .3831037874 g_w 8.314887839 ets [7, 39, 46, 53, 60, 92, 99, 106, 113, 145, 152, 159, 166, 198, 205, 212, 244, 251, 258, 265, 297, 304, 311, 318, 350, 357, 364, 371, 403, 410, 417, 424, 449, 456, 463, 470, 502, 509, 516, 523, 555, 562, 569, 576, 608, 615, 622, 629, 654, 661, 668, 675, 707, 714, 721, 728, 760, 767, 774, 781, 813, 820, 827, 834, 866, 873, 880, 919, 926, 933, 972, 979, 986] 1224440064/1220703125 (2)^8*(3)^14/(5)^13 map [[1, 5, 6], [0, 13, 14]] generators 1200. -315.2509133 badness 433.8313410 rms .2766026501 g_w 11.61862841 ets [19, 38, 42, 57, 61, 76, 80, 99, 118, 137, 156, 160, 175, 179, 194, 198, 217, 236, 255, 274, 293, 297, 316, 335, 354, 373, 392, 411, 415, 434, 453, 472, 491, 510, 529, 533, 552, 571, 590, 609, 628, 647, 651, 670, 689, 708, 727, 746, 765, 769, 788, 807, 826, 845, 864, 887, 906, 925, 944, 963, 982] 10485760000/10460353203 (2)^24*(5)^4/(3)^21 map [[1, 0, -6], [0, 4, 21]] generators 1200. 475.5422333 badness 384.8802232 rms .1537673823 g_w 13.57752022 ets [5, 48, 53, 58, 106, 111, 159, 164, 212, 217, 265, 270, 275, 318, 323, 328, 371, 376, 381, 424, 429, 434, 482, 487, 535, 540, 588, 593, 598, 641, 646, 651, 694, 699, 704, 747, 752, 757, 805, 810, 858, 863, 911, 916, 964, 969, 974] 6115295232/6103515625 (2)^23*(3)^6/(5)^14 map [[2, 4, 5], [0, 7, 3]] generators 600.0000000 -71.14606343 badness 273.0155936 rms .1940180530 g_w 11.20594372 ets [16, 18, 34, 50, 68, 84, 100, 102, 118, 134, 136, 152, 168, 186, 202, 220, 236, 252, 254, 270, 286, 304, 320, 338, 354, 370, 372, 388, 404, 422, 438, 456, 472, 488, 490, 506, 522, 524, 540, 556, 574, 590, 606, 608, 624, 640, 642, 658, 674, 692, 708, 726, 742, 758, 760, 776, 792, 810, 826, 844, 860, 876, 878, 894, 910, 928, 944, 962, 978, 994, 996] 19073486328125/19042491875328 (5)^19/(2)^14/(3)^19 map [[19, 0, 14], [0, 1, 1]] generators 63.15789474 1902.029094 badness 475.0683684 rms .1047837215 g_w 16.55086763 ets [19, 38, 57, 76, 95, 114, 133, 152, 171, 190, 209, 228, 266, 285, 304, 323, 342, 361, 380, 399, 418, 437, 456, 475, 494, 513, 532, 551, 570, 589, 608, 627, 646, 665, 684, 703, 722, 760, 779, 798, 817, 836, 855, 874, 893, 931, 950, 969, 988] 32805/32768 (3)^8*(5)/(2)^15 map [[1, 0, 15], [0, 1, -8]] generators 1200. 1901.727514 badness 34.18600169 rms .1616933186 g_w 5.957335766 ets [12, 17, 24, 29, 36, 41, 53, 65, 77, 82, 89, 94, 101, 106, 118, 130, 135, 142, 147, 154, 159, 171, 183, 195, 200, 207, 212, 219, 224, 236, 248, 253, 260, 265, 272, 277, 289, 301, 313, 318, 325, 330, 342, 354, 366, 371, 378, 383, 390, 395, 407, 419, 424, 431, 436, 443, 448, 460, 472, 484, 489, 496, 501, 508, 513, 525, 537, 542, 549, 554, 561, 566, 578, 590, 602, 607, 614, 619, 626, 631, 643, 655, 660, 667, 672, 679, 684, 696, 708, 720, 725, 732, 737, 744, 749, 761, 773, 778, 785, 790, 797, 802, 814, 826, 838, 843, 850, 855, 862, 867, 879, 891, 896, 903, 908, 915, 920, 932, 944, 956, 961, 968, 973, 985, 997] 274877906944/274658203125 (2)^38/(3)^2/(5)^15 map [[1, 4, 2], [0, 15, -2]] generators 1200. -193.1996149 badness 155.7009575 rms .6082244804e-1 g_w 13.67967551 ets [25, 31, 56, 62, 87, 93, 112, 118, 143, 149, 174, 180, 205, 211, 230, 236, 261, 267, 292, 298, 323, 329, 348, 354, 379, 385, 410, 416, 441, 447, 466, 472, 497, 503, 528, 534, 559, 584, 590, 615, 621, 646, 652, 671, 677, 702, 708, 733, 739, 764, 770, 789, 795, 820, 826, 851, 857, 882, 888, 907, 913, 938, 944, 969, 975, 1000] 7629394531250/7625597484987 (2)*(5)^18/(3)^27 map [[9, 1, 1], [0, 2, 3]] generators 133.3333333 884.3245134 badness 177.0527789 rms .2559250891e-1 g_w 19.05445924 ets [27, 45, 72, 99, 126, 144, 171, 198, 243, 270, 297, 315, 342, 369, 414, 441, 468, 486, 513, 540, 567, 585, 612, 639, 684, 711, 738, 756, 783, 810, 855, 882, 909, 927, 954, 981] 9010162353515625/9007199254740992 (3)^10*(5)^16/(2)^53 map [[2, 1, 6], [0, 8, -5]] generators 600.0000000 162.7418923 badness 101.3097955 rms .1772520822e-1 g_w 17.87941745 ets [22, 44, 52, 66, 74, 96, 118, 140, 162, 170, 184, 192, 206, 214, 236, 258, 280, 288, 302, 310, 324, 332, 354, 376, 398, 406, 420, 428, 442, 450, 472, 494, 516, 538, 546, 560, 568, 590, 612, 634, 656, 664, 678, 686, 708, 730, 752, 774, 782, 796, 804, 826, 848, 870, 892, 900, 914, 922, 936, 944, 966, 988]
Message: 4376 Date: Mon, 25 Mar 2002 20:45:22 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > - it's john negri's system in 19-equal (looks a little better in the > > 7-limit). > > If you keep this one (tertiathirds) you're also gonna have to keep the > other one you asked about, septathirds (4294967296/4271484375), since > septathirds is better than tertiathirds by Gene's badness. > > But I agree that it's extremely boring melodically, being essentially > 22-tET. hey, how could i argue with any reference to 22-equal? :)
Message: 4377 Date: Mon, 25 Mar 2002 21:34:49 Subject: Re: Hermite normal form From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > Do you want this if half-fourth doesn't work? Hermite form seems to > allow twin meantone and schismic, and half-fifth meantome and >schismic, but not the half-fourth versions. that seems bizarre. is there an intuitive explanation of why this should be the case?
Message: 4378 Date: Mon, 25 Mar 2002 05:48:12 Subject: Re: Hermite normal form From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > > Would you be so kind as to give as examples, the HNF matrices and > > consequent generators for Meantone, Diaschismic and Augmented, as > > 5-limit linear temperaments, Twin meantone and Half meantone fifth > > as degenerate 5-limit whatevers, and Starling as the 7-limit planar > > temperament where the 125:126 vanishes? > > Do you want this if half-fourth doesn't work? Hermite form seems to > allow twin meantone and schismic, and half-fifth meantome and schismic, but not the half-fourth versions. That's odd, but I'm still interested in seeing the others.
Message: 4379 Date: Mon, 25 Mar 2002 21:33:31 Subject: Re: Hermite normal form version of "25 best" From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > Here's the same list, this time using Hermite normal form. again, i'm wondering why you're not putting these in order of g_w. p.s. is graham going to veto this weighting business? i hope not.
Message: 4380 Date: Mon, 25 Mar 2002 21:10:38 Subject: Re: Decatonics From: paulerlich --- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote: > I've drawn up a little diagram: > > http://www.argonet.co.uk/users/mark.gould/images/C22_Decatonic.jpg - Ok * > > hopefully it explains itself. > > Mark 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. (note that there is no 'equal' in the title of my paper). 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.
Message: 4381 Date: Mon, 25 Mar 2002 04:48:24 Subject: Starling example From: genewardsmith 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]]
Message: 4382 Date: Mon, 25 Mar 2002 22:33:34 Subject: Re: Diatonics From: paulerlich --- In tuning-math@y..., graham@m... wrote: > I have it as <A few items from Paul Erlich *>. from there you'll want to go to http://www-math.cudenver.edu/~jstarret/22ALL.pdf - Ok * carl lumma or someone inserted the word 'equal' into the title -- at no time did any version of this document residing on my computer contain the word 'equal' in the title. note that page 20 is all wrong -- just ignore it. also, the table of key signatures at the end is wrong -- i'll put up the corrected version if anyone cares to see it. cheers, paul
Message: 4384 Date: Tue, 26 Mar 2002 21:09:22 Subject: Re: Hermite normal form version of "25 best" From: Carl Lumma >How about dropping the g_w cutoff to 13 for the best 20 like we >agreed, or even to 10 for the best 17? Who wants the 20 and who wants >the 17? Paul, Graham, Carl, Herman, anyone? Thanks for asking, Dave, but I'm afraid I've had a hard time keeping up with events around here. I did get a chance to play around with your spreadsheet. I looked at rankings by (complexity^n)(error), where n was 1, 1.5, 2, 3, and 4. I found that I couldn't get porcupine and diminished high enough and fourth-thirds low enough at the same time to suit me. So I tried rounding the error to the nearest multiple of 1, and then to the nearest multiple of 3. In this latter case I found that an n of about 2 with a complexity cutoff of 10 produced a ranking I was about as happy with as the one given by your badness measure. Unfortunately, I never caught the derivation of your measure, so I can't endorse it. Anyway, I think the problem is not in n, but in the error function. Ideally, it would only really start going up above 2 or 3 cents, then get only slightly higher from 3 to 10, respectibly higher from 10 to 20, and astronomically higher above that. All this would give pelogic, limmal, fourth-thirds very bad numbers indeed. And as temperaments, I say fine. However, with key-limiting Wilson-like full comma jumps, many of these spring to life. Maybe that means they are properly planar temperaments, and should be ranked poorly as linear temperaments, I don't know. I gather that g_w is weighted complexity, which I don't endorse at all. However, for the 5-limit, an unweighted complexity cutoff of 10 is fine by me, since we've already searched a huge slice of temperament space for good temperaments of any g. Take from this what you will. -Carl
Message: 4385 Date: Tue, 26 Mar 2002 00:14:42 Subject: _The_ 31-limit temperament? From: dkeenanuqnetau 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. It is consistent with 311-ET and 388-ET, the only two 31-limit-consistent ETs less than 1200. It has an octave period and a generator of MA optimum 405.14866 c 1.177 c max-abs error RMS optimum 405.15025 c 0.528 c rms error The generator is the temperament's approximate enneadecimal major third (19:24). The mapping is prime gens ------------ 3 -19 5 75 7 -45 11 -126 13 8 17 98 19 -20 23 -111 29 -33 31 -166 Weighted rms complexity is 109.2 generators. Max-absolute complexity is 316 generators. All 108 31-limit ratios are represented uniquely.
Message: 4386 Date: Tue, 26 Mar 2002 10:44:32 Subject: Re: Hermite normal form version of "25 best" From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > You're right. I don't like it. Generators bigger than an octave (in > one case bigger than two octaves) and negative generators. We could modify the Hermite form by allowing changes of an octave; since we assume octave equivalence this would not affect the generator count, and we could still get the higher-dimensional generalization I want out of it. There is, I suppose, something to be said for using a well-recognized standard reduction, which Hermite form certainly is.
Message: 4387 Date: Tue, 26 Mar 2002 12:43 +0 Subject: Re: Decatonics From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <B8C4A26C.38C5%mark.gould@xxxxxxx.xx.xx> Mark Gould wrote: > I've drawn up a little diagram: > > http://www.argonet.co.uk/users/mark.gould/images/C22_Decatonic.jpg - Ok * > > hopefully it explains itself. It doesn't follow either of your proposed methods for 7-limit diatonics though. The two pentatonics don't intersect, which is a shame. When looking for something else, I did happen to find a diatonic that fits this pattern from 22-equal: 22 5 14 10 19 15 2 20 7 3 12 8 17 0 It's based on a 7:9:12 chord. I don't know if it fulfils all the right criteria, but it does work with some simpler ones: 1) It's an octave-based MOS 2) Both intervals in the grid are a single step apart in the parent scale (22-equal in the example) 3) It's a "diatonic" rather than "pentatonic", meaning the larger step sizes are more common (9 steps of 2/22 and 4 of 1/22) For (2) to work, you need an odd number of notes in the MOS and the generator has to span an even number of diatonic steps. That's easy, because you always have two generators to choose from, and one of them will always be an even number of steps if there is an odd number to the octave. But you also need the same generator to be an odd number of chromatic steps, which won't always be the case. That is, it needs to be an odd number of steps in the parent scale, which has to be the next step down on the scale tree. For (3) to work, I think the difference between the number of notes in the parent and diatonic scales has to be more than half the number of notes in the diatonic. In this case, 2*(22-13)>13. I also found a 21 from 26 "pentatonic" which is a bit like Blackjack. 26 21 24 16 19 11 14 6 9 1 4 22 25 17 20 12 15 7 10 2 5 0 Call the defining chord 14:15:16 Graham
Message: 4390 Date: Tue, 26 Mar 2002 21:50:04 Subject: Re: Hermite normal form version of "25 best" From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > > Here's the same list, this time using Hermite normal form. > > How about dropping the g_w cutoff to 13 for the best 20 like we > agreed, or even to 10 for the best 17? Who wants the 20 and who wants > the 17? Paul, Graham, Carl, Herman, anyone? i think 17 is enough. i'm still hoping my posts from yesterday show up, especially those in reply to mark gould.
Message: 4391 Date: Wed, 27 Mar 2002 01:01:15 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: > Something we'll have to keep in mind is how much primal uniqueness > should be traded off against human comprehension of the symbols. I > think that the deciding factor should be in favor of the human, not > the machine -- software can be written to handle all sorts of > complicated situations; I agree. But it is also possible to disambiguate dual purpose flags by say adding a blob to the end of the stroke for one use and not the other. > I had never given much thought to notating divisions above 100, but I > would like to see how well the JI notation will work with these. > Which ones between 94 and 217 would you consider the most important > to be covered by this notation (listed in order of importance)? I don't know order of importance. 96, 105, 108, 111, 113, 121, 130, 144, 149, 152, 159, 166, 171, 183, 190, 198, 212. > And if 217 seems suitable, then we should stick with > it. (Over the weekend I happened to notice that it's 7 times 31 -- > in effect a division built on meantone quarter-commas!) Yes. I noticed that too. But I'm not sure it matters much, since 31-ET will of course _not_ be notated the same as every seventh note of 217-ET. > I thought that, in the event somebody *absolutely must* have 23, one > could allow a little bit of slack if the following were taken into > account: ... > and, in addition, 21, 25, and 27 are all > consistent with that. You're right. I'd even like to see if we can push it to 31-limit, consistent with 311-ET, since we're so close, but this would use additional flags (and/or additional schismas like 4095:4096 and 3519:3520) and not affect the existing 23-limit, 217-ET correspondence. You know I went thru the prime factorisation of all the superparticulars in John Chalmer's list, and you've found the only two that are useful for this purpose. But I was wondering if we can somehow use 1539:1540 which says that the 19-flag is the difference between the 11/5 flag and the 7-flag. Probably not, since it involves a subtraction and this schisma is a whole cent. John's list only goes up to 23-limit. I'd like to see a list of all the 31-limit superparticulars to be sure we're not missing something. Gene? But then there's no guarantee that a useful schisma like that will be superparticular. > > the 5 and 7 commas must correspond to single flags. > > That's correct (referring to the constraint). > > > Neither of them > > can map to a single flag in either 282-ET or 311-ET and so the > mapping > > of commas to arrows is just way too obscure. > > I don't know exactly what you mean by this. As single flags these > would just have to indicate more degrees, and in order to fill in the > gaps I would have to come up with a 4th kind of flag and another > comma to go with it (but to what purpose?). In any case, your > conclusion stands: What I meant was that you can actually cover all the values from 0 steps to the number of steps in the 13-as-semiflat comma in 311-ET using only 6 flags, but if you do that, the 5 and 7 commas cannot be single-flag. As you say, we can always add more flags to fill in the gaps. > I spent some time this past weekend figuring out how all of this was > going to translate into various ET's under 100, and every division I > tried could be notated without any lateral mirroring whatsoever. > (Even 58-ET, which had given me problems before, now looks very good.) That's great. > > 2. Swap the flags for the 7 comma and 11-as-semisharp/5 comma. (xR > and sR) > > This is the one (and only) thing that is different in our efforts at > achieving a combination of my two approaches and is something that I > hadn't considered. It has the effect of making both convex flags > larger than both straight flags. Yes, this was the other thing that recommended it to me. > It didn't take me very long to reach a definite conclusion. I recall > that it was the issue of lateral confusibility that first led to the > adoption of a curved right-hand flag for the 7-comma alteration in > the 72-ET notation. Before that all of the flags were straight. > Making the xR-sR symbol exchange would once again give the 7-comma > alteration a straight flag, which would negate the original reason > for the curved flag. Yes. I was considering putting a blob on the end of the straight 7 flag, but no. I agree with you now. Keep the curved flag for the 7-comma. It is most important to get the 11-limit right. The rest is just icing on the cake, and a little lateral confusability there can be tolerated. > By the way, something else I figured out over the weekend is how to > notate 13 through 20 degrees of 217 with single symbols, i.e., how to > subtract the 1 through 8-degree symbols from the sagittal apotome > (/||\). The symbol subtraction for notation of apotome complements > works like this: > > For a symbol consisting of: > 1) a left flag (or blank) > 2) a single (or triple) stem, and > 3) a right flag (or blank): > 4) convert the single stem to a double (or triple to an X); > 5) replace the left and right flags with their opposites according to > the following: > a) a straight flag is the opposite of a blank (and vice versa); > b) a convex flag is the opposite of a concave flag (and vice versa). You gotta admit this isn't exactly intuitive (particularly 5a). I'm more interested in the single-stem saggitals used with the standard sharp-flat symbols, but it's nice that you can do that. > I will prepare a diagram illustrating the progression of symbols for > JI and for various ET's so we can see how all of this is going to > look. > > Stay tuned! Sure. This is fun.
Message: 4393 Date: Wed, 27 Mar 2002 07:49:03 Subject: Re: A common notation for JI and ETs From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > I don't know order of importance. 96, 105, 108, 111, 113, 121, 130, > 144, 149, 152, 159, 166, 171, 183, 190, 198, 212. This list gives me indigestion--what happened to 99, 118 and 140, for starters? > John's list only goes up to 23-limit. I'd like to see a list of > all the 31-limit superparticulars to be sure we're not missing > something. Gene? But then there's no guarantee that a useful schisma > like that will be superparticular. I don't have such a lit, but it would make more sense to look for such up to a size limit, I think.
Message: 4394 Date: Wed, 27 Mar 2002 07:51:32 Subject: Re: Hermite normal form version of "25 best" From: genewardsmith --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote: > The best 16 would be enough for me; I don't see AMT as particularly > interesting as a 5-limit temperament, and the last three just don't seem > all that useful for musical purposes as far as I can tell. I haven't done > as much playing around with these scales as I'd like, but it generally > seems to be the case that the less complex scales are also the ones that > are most musically interesting to me. I think we should leave room for various preferences in this department, and don't see why we can't have a best 20.
Message: 4395 Date: Wed, 27 Mar 2002 12:41 +0 Subject: Re: Decatonics From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <B8C71E03.38EA%mark.gould@xxxxxxx.xx.xx> Mark Gould wrote: > > It doesn't follow either of your proposed methods for 7-limit > > diatonics > > though. The two pentatonics don't intersect, which is a shame. > > > No, but I did say that there are many 3D shapes within a lattice, and I > didn't state that the two I chose for mention were the only ones. You didn't say very much at all. > In any case, the decatonic is the same as Paul's standard one, which was > what I was hoping people would notice. The pentachordal one to be precise. > This one I also found, but the scale 'fifth' is 17 steps wide and has a > very > odd structure in terms of its keyboard" > b b b b b b b b b > W W W W W W W W W W W W W What's odd about it? > I never gave this scale any more investigation. Oh. > This is inconsistent with my rule ii: it contains segments of 3 and more > adjacent PCs > > 0 1 2, 4 5 6 7, 9 10 11 12, etc Yes, that follows from it being a "pentatonic". You could always embed it in 57- or even 47-equal to clear this criterion. > So it will show up as intervallically inchoerent (as defined by > Balzano). Why is that important? There are some 5:6:7 "pentatonics" as well 9 5 8 1 4 0 13 7 11 14 5 8 12 2 6 0 but they become inconsistent if you try to remove this incoherence. The classic pentatonic could be based on 6:7:8. The next I can find for 6:7:8 is 12/29 from 41 which is also a "pentatonic". 7:8:9 is your 11 from 41, 8:9:10 is Magic and 9:10:11 is a neutral third scale. 10:11:12 is getting a bit tight, and 7:9:12 is the same complexity, but you didn't like it. Graham
Message: 4396 Date: Wed, 27 Mar 2002 15:51 +0 Subject: Re: Pitch Class and Generators From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <B8C6611F.38DC%mark.gould@xxxxxxx.xx.xx> Mark Gould wrote: > >From my recollection, I think the C12 group is isomorphic with > relation to > two of its sub groups C3 and C4 I don't know what that means. One thing though, the main significance of 3 and 4 wrt a diatonic is that the fourth and fifth are 3 and 4 diatonic steps respectively. It's a coincidence that they happen to add up to the number of steps to a fifth in 12-equal, and follows from the numbers of steps to the generators also being the numbers of steps to the MOS subsets. As this is also true of Balzano's 11/20 diatonic, perhaps that's what he was doing with the group theory, and you rejected. > like this: > > (a,b) <--> (4a + 3b) mod 12 > > Every interval in C12 can thus be measured as from zero to two major > thirds > (difference of 4, i.e. the C3 cycle), and zero to three minor thirds > (differece of 3, i.e. the C4 cycle) To get every interval of a diatonic scale, with correct spelling, you need a larger range. A semitone is an octave minus two major and one minor thirds, or (4*-2 + 3*-1) mod 12. A tone is an octave minus two minor and one major thirds, or (4*-1 + 3*-2) mod 12. > This is plotted graphically as > > > 0 4 8 0 > 9 1 5 9 > 6 10 2 6 > 3 7 11 3 > 0 4 8 0 (extends in all directions, but is in reality a torus) > > And so we arrive at fig 5 from Balzano. > > Thus the C3xC4 group contains generators of C12. That must be a different definition of "generator" from > >> A generator for Cn is an N^i, such that successive applications of > >> N^i to the starting tone will generate the full Cn. > >> > >> For any n i=1 is a generator, as is i = -1 > >> > >> For C12, i = 5 and its mod 12 complement 7 are generators. because neither 3 nor 4 will generate the full 12. It's also different to the way Gene talks of a pair of generators giving a linear temperament. The tone is exactly half a major third, and so can't be got by combining major and minor thirds. You have to add the octave to the full list of generators, which gives a planar temperament. Or rather a planar scale, because this is 5-limit JI with no tempering. The tone and semitone also generate a diatonic scale. Carey and Clampitt show how to transform [5 7] to [1 2]. Graham
Message: 4397 Date: Wed, 27 Mar 2002 20:50:32 Subject: Re: Pitch Class and Generators From: genewardsmith --- In tuning-math@y..., graham@m... wrote: > In-Reply-To: <B8C6611F.38DC%mark.gould@a...> > Mark Gould wrote: > > >From my recollection, I think the C12 group is isomorphic with > > relation to > > two of its sub groups C3 and C4 > I don't know what that means. C12 is isomorphic to the direct product C3 x C4. It can be expressed in terms of a single generator of order 12, but its subgroups can be expressed in terms of generators of degree 3 and 4 respectively, and these also generate C12. In fact, since 3 and 4 are relatively prime, any integer can be expressed as a linear combination of 3s and 4s, so any 2^(k/12) can be expressed as a product of major and minor 12-et thirds, without the use of octaves.
Message: 4398 Date: Wed, 27 Mar 2002 21:09:30 Subject: Re: Decatonics From: paulerlich --- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote: > This is inconsistent with my rule ii: it contains segments of 3 and more > adjacent PCs > > 0 1 2, 4 5 6 7, 9 10 11 12, etc > > So it will show up as intervallically inchoerent (as defined by Balzano). this is what we call 'rothenberg improper'. but i don't think that's a good reason to throw it out. the diatonic scale in pythagorean tuning is rothenberg improper! i sure hope my other responses to you show up, mark. but basically, i wish there was some text accompanying your decatonic diagram. i have no idea why you are using 4 (218¢) and 5 (273¢) as your 'generators', for example.
Message: 4399 Date: Wed, 27 Mar 2002 21:22:46 Subject: Re: Pitch Class and Generators From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., graham@m... wrote: > > In-Reply-To: <B8C6611F.38DC%mark.gould@a...> > > Mark Gould wrote: > > > > >From my recollection, I think the C12 group is isomorphic with > > > relation to > > > two of its sub groups C3 and C4 > > > I don't know what that means. > > C12 is isomorphic to the direct product C3 x C4. It can be expressed > in terms of a single generator of order 12, but its subgroups can be > expressed in terms of generators of degree 3 and 4 respectively, and > these also generate C12. In fact, since 3 and 4 are relatively prime, > any integer can be expressed as a linear combination of 3s and 4s, so > any 2^(k/12) can be expressed as a product of major and minor 12-et > thirds, without the use of octaves. yes, but to claim (as balzano did) that the fundamental importance of the diatonic scale hinges on this fact is to pull the wool over the eyes of the numerically inclined reader. the fact is that around the time of the emergence of tonality in diatonic music, many musicians advocated a 19- or 31-tone system in which to embed the diatonic scale, and 12 won out only because of convenience. it is only with the work of late 19th century russian composers that the cycle-3 and cycle-4 aspects of C12 became musically important. in fact, the diatonic scale emerged over and over again around the world without any 'chromatic universe' whatsoever, let alone an equal- tempered one. the important properties of the diatonic scale must, i feel, be found in the scale itself, in whatever tuning it may be found (with reasonable allowances for the ear's ability to accept small errors) -- any 'chromatic totality' considerations should wait until, and be completely dependent upon, the establishment of the fundamental 'diatonic' entity upon which the music is to be based. this was my approach in my paper, and more recently, in my adaptation of fokker's periodicity block theory.
4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950
4350 - 4375 -