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Message: 4375 - Contents - Hide Contents

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]


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Message: 4376 - Contents - Hide Contents

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? :)
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Message: 4377 - Contents - Hide Contents

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?
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Message: 4378 - Contents - Hide Contents

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.
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Message: 4379 - Contents - Hide Contents

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.
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Message: 4380 - Contents - Hide Contents

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 - Type Ok * [with cont.] (Wayb.) > > 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.
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Message: 4381 - Contents - Hide Contents

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]]


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Message: 4382 - Contents - Hide Contents

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 * [with cont.] (Wayb.)>.
from there you'll want to go to http://www-math.cudenver.edu/~jstarret/22ALL.pdf - Type Ok * [with cont.] (Wayb.) 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
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Message: 4383 - Contents - Hide Contents

Date: Tue, 26 Mar 2002 21:17:11

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

From: Herman Miller

On Mon, 25 Mar 2002 15:06:17 -0000, "dkeenanuqnetau" <d.keenan@xx.xxx.xx>
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?
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.
>So far I stand by my earlier proposal that minimises (positive) >generator sizes while having zeros in the lower left triangle of the >matrix (with columns corresponding to increasing primes from left to >right). Each generator is less than half the size of the preceeding >one. This gives maximum information about the melodic structure of the >scale.
I think this makes sense.
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Message: 4384 - Contents - Hide Contents

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
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Message: 4385 - Contents - Hide Contents

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.


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Message: 4386 - Contents - Hide Contents

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.
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Message: 4387 - Contents - Hide Contents

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 - Type Ok * [with cont.] (Wayb.) > > 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
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Message: 4389 - Contents - Hide Contents

Date: Tue, 26 Mar 2002 19:44:51

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >>>> >
> OK. So you have gone outside of one-comma-per-prime and > one-(sub)symbol-per-prime. But you have given fair reasons for doing > so in the case of 11 and 13.
Gene had a comment about this, however: --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> If we have more than one comma per prime, we lose the very desireable > property of uniqueness, and automatic translation becomes harder.
I assume that JI-to-ET, ET-to-JI, or ET-to-ET translation by computer software is what is being referred to here. I think that the process has a similarity to conversion of computer image files: In general, conversion of a high-resolution image to one of lower resolution will be more successful than the reverse; likewise, translation of music from JI or a large-number ET (finer resolution) to a lower-number ET (coarser resolution) will be more successful, or at least more straightforward. (Of course, there will exceptions to this if the music is simple enough, but I am stating a general principle.) The only direction where the lack of primal uniqueness is likely to pose a problem is in going from coarser to finer resolution, i.e., the direction which we would not expect to be very successful in the first place. 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; but where dual commas have been introduced, it was for the purpose of *clarifying* the melodic function of the intervals involved to the human reader of the notation. Whoever might get involved in writing the algorithm for any sort of translation would need to be aware of these things and would have to consider providing appropriate menu entries that would govern the translation. For what it's worth, each comma (or more accurately, diesis) in the pairs of 11-and-13-defining intervals is the apotome's complement of the other, and it should be a simple enough matter to follow the principle of using the one that doesn't alter in the opposite direction in combination with a sharp or flat. Likewise, the two 17 commas are the Pythagorean comma's complement of each other, and the proper one can be clearly determined by whether it is associated with a sharp or flat (or natural). And only 2176:2187 is used in combination with other defining commas in the notation, so that restricts the situations in which the two 17 commas pose a problem in translation.
>> I am also outlining a 23-limit approach; I went for the >> 19 limit and got 23 as a bonus when I found that I could approximate >> it using a very small comma. The two approaches could be combined, >> in which case you could have the 11-13 semiflat varieties along with >> the 19 or 23 limit, but the symbols may get a bit complicated -- more >> about that below. >
> You don't actually give more below about combining these approaches.
I meant more about the symbols, not about how to combine the approaches, which I didn't get to until the weekend.
> But I had fun working it out for myself. I'll give my solution later. >
>> I thought more about this and now realize that the problem with 27-ET >> is not as formidable as it seemed. ... The same could be said about >> 50-ET. >
> You're absolutely right.
I'm glad you agree. It looks like we've gotten somewhere with the ET's.
>> Are there any troublesome divisions above 100 that we should >> be concerned about in this regard? >
> Not that I can find on a cursory examination.
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 anticipate that you believe that the JI purists would still want to >> have this distinction, so we should go for 19. >
> Correct. I'll skip the 183-ET based one. Agreed!
>>> I wouldn't spend too much time on the symbols yet. I expect serious >>> problems with the semantics. >>
>> I don't know what problems you are anticipating, ... >
> Well none have materialised yet. :-)
I imagined that it would have been the gap between the sum of the 19 and 17 commas (~12.1 cents) and the 5 comma (~21.5 cents) in a binary sequence, but even that is no problem in 181, 193, and 217 (to give a few examples). 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!)
>> *23-LIMIT APPROACH* >> >> And here is the 23-limit arrangement, which correlates well with >> 217-ET (apotome of 21 degrees): >
> I don't think you can call this a 23-limit notation, since 217-ET is > not 23-limit consistent. But it is certainly 19-prime-limit.
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: --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote [main tuning list, #33699]:
> But for a small fraction of a cent 46-EDO misses 17-limit consistency > for a single pair of intervals (15/13 & 26/15)! I don't think this > precludes using it for 17-limit harmony (no EDO of lower number can > compete with it), and I know from experience that 15-limit harmonies > can be successfully employed in 31-EDO without any disorientation > whatsoever (with 13 being implied, much more successfully, in my > opinion, than 7 is in 12-EDO), even if you have a couple of pairs of > intervals that go over the boundary of consistency by a couple of > cents or so. (The same can be said for 19/13 and 26/19 in 72- EDO.) > I would compare this to briefly driving a car very slightly onto the > shoulder, but not far enough off the road to lose control.
After all, 23 is inconsistent only in combination with two other odd numbers in the 27 limit. But, as you say, we do have the 19-prime limit consistency, and, in addition, 21, 25, and 27 are all consistent with that.
>> Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (4
degrees of 217)
>> Straight right flag (sR): 54:55, ~31.8 cents (6deg217) >> Convex left flag (vL): 4131:4096 (3^5*17:2^12, the 17-as-flat
comma), ~14.7 cents deg217)
>> Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (5deg217) >> Concave left flag (vL): 2187:2176 (3^7:2^7*17, the 17-as-sharp
comma), ~8.7 cents deg217)
>> Concave right flag (vR): 512:513 (the 19-as-flat comma), ~3.4 cents (1deg217) >> >> With the above used in combination, the following useful intervals >> are available: >> >> sL+sR: 32:33 (the 11-as-semisharp comma), ~53.3 cents (10deg217) >> sL+xR: 35:36, ~48.8 cents, which approximates >> ~sL+xR: 1024:1053 (the 13-as-semisharp comma), ~48.3 cents (9deg217) >> vL+sR: 4352:4455 (2^8*17:3^4*5*11), ~40.5 cents, which approximates >> ~vL+sR: 16384:16767 (2^14:3^6*23, the 23 comma), ~40.0 cents (8deg217) >> xL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates >> ~xL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents (8deg217) >> >> The vL+sR approximation of the 23 comma deviates by 3519:3520 (~0.492 cents). >> >> All of the above provide a continuous range of intervals in 217- ET, >> which I selected because it is consistent to the 21-limit and >> represents the building blocks of the notation as approximate >> multiples of 5.5 cents. >
> Once I understood your constraints, I spent hours looking at the > problem. I see that you can push it as far as 29-limit in 282-ET if > you want both sets of 11 and 13 commas, and 31-limit in 311-ET if you > can live with only the smaller 11 and 13 commas. But to make these > work you have to violate what is probably an implicit constraint, that > 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:
> 217-ET is definitely the highest ET you can use with the above > additional constraint. > > I notice that left-right confusability has gone out the window.
Not entirely (more comments to follow).
> But maybe that's ok, if we accept that this is not a notation for > sight-reading by performers.
I do want this to be a notation for sight-reading. In order to get all of the combinations required for the JI notation in single symbols, you have to allow some lateral mirroring, but I don't think it's going to occur often enough to cause a lot of trouble. There will be a few instances that may be a bit tricky at first, but once you learn what to watch out for (just as I learned to get the g's and t's right in the word sagittal), it should be easy enough to deal with. 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.)
> However, it is possible to improve the > situation by making the left-right confusable pairs of symbols either > map to the same number of steps of 217-ET or only differ by one step, > so a mistake will not be so disastrous. At the same time as we do this > we can reinstate your larger 11 and 13 commas, so you have both sizes > of these available. The 13 commas will have similar flags on left and > right, while the 11 commas will have dissimilar flags. It seems better > that the 11 commas should be confused with each other than the 13 > commas, since the 11 commas are closer together in size.
Good point! I'm glad to see that you have gotten involved in trying to combine my two approaches. I also spent some time over the weekend on this, and I notice that we did the 17 and 19 commas the same way.
> To do this you make the following changes to your 217-ET-based scheme. > 1. Swap the flags for the 17 comma and 19 comma (vL and vR)
In your subsequent message with the correction, you changed this back to what I originally had. I chose that arrangement because I found that the flags were more useful in combination that way (specifically 2176:2187). In my list of useful combinations I inadvertently omitted my primary choice for 7deg217 as 2176:2187 plus 63:64, but you figured it out (along with the alternate way to get 7 degrees).
> 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. It also has other consequences, which I will address below.
> 3. Make the xL flag the 11-as-semiflat/7 comma instead of the 17-as- flat comma.
This restores a flag that was in my 17-limit approach, and I also did this in the combination of my two previous approaches.
> So we have [according to the second (corrected) posting] > Steps Flags Commas > ------------------------ > 1 vR 19-comma 512:513 > 2 vL 17-comma 2187:2176 > 3 vL+vR 17-as-flat-comma 4131:4096 > 4 sL 5-comma 80:81 > 5 sR 7-comma 63:64 > 6 xL or xR > 7 vL+sR or xL+vR > 8 vL+xR > 9 sL+sR 13-as-semisharp comma 1024:1053 > 10 sL+xR 11-as-semisharp comma 32:33 > 11 xL+sR 11-as-semiflat comma 704:729 > 12 xL+xR 13-as-semiflat comma 26:27 > 21 apotome
I have given this quite a bit of study since yesterday, not only for JI, but also to see how this translates to various ET notations, many of which are affected by the exchange of the xR-sR symbols. This proposal changes things that I have become familiar with over the past couple of months, but that is irrelevant inasmuch as our objective is to make this notation as good as we possibly can. It is best to do it right the first time so that we don't have to change it later. 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. The 72-ET notation could still use curved right flags, but they would no longer symbolize the 7-comma alteration, but the 54:55 alteration instead, which tends to obscure rather than clarify the harmonic relationships. Also, since the JI notation would use straight flags for both the 5 and 7-comma alterations, then lateral confusibility would make it more difficult to distinguish between two of the most important prime factors, and we would be giving this up without receiving anything of comparable benefit in return. Notating the 7-comma with the xR (curved) flag, on the other hand, makes a clear distinction between ratios of 5 and 7 in JI, 72-ET, and anywhere else that 80:81 and 63:64 are a different number of degrees. It also minimizes the use of curved flags in the ET notations, introducing them only as it is necessary or helpful: 1) to avoid lateral confusibility (in 72-ET); 2) to distinguish 32:33 from 1024:1053 (in 46 and 53-ET, *without* lateral confusibility!); and 3) to notate increments smaller than 80:81 (in 94-ET). Lateral confusibility enters the picture only when one goes above the 11 limit: In one instance one must learn to distinguish between 1024:1053 and 26:27 by observing which way the straight flag points (leftward for the smaller ratio and rightward for the larger). Another instance does not come up until the 19 limit, which involves distinguishing the 17-as-sharp flag from the 19 flag. So I think we have enough reasons to stick with the convex curved flag for the 7 comma. (I will also give one more reason below.) 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). This produces a reasonable and orderly progression of symbols (assuming that 63:64 is a curved convex flag; it does not work as well with 63:64 as a straight flag) that is consistent with the manner in which I previously employed the original sagittal symbols for various ET's. 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! --George
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Message: 4390 - Contents - Hide Contents

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.
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Message: 4391 - Contents - Hide Contents

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.
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Message: 4393 - Contents - Hide Contents

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.
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Message: 4394 - Contents - Hide Contents

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.
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Message: 4395 - Contents - Hide Contents

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
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Message: 4396 - Contents - Hide Contents

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
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Message: 4397 - Contents - Hide Contents

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.
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Message: 4398 - Contents - Hide Contents

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.
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Message: 4399 - Contents - Hide Contents

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.
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