Hermite normal form version of "25 best"

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]

--- 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 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!
>

You're right. I don't like it. Generators bigger than an octave (in
one case bigger than two octaves) and negative generators. This is not
a form that allows a temperament's musical ramifications to be easily
understood. However, I like the fact that the octave-related (or
lowest-prime-related) generator comes first. If it facilitates badness
calculations, well and good, but I wouldn't publish a list of
temperaments in this form in a pink fit (unless it was not intended to
be read by musicians).

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 also changed the weighted "g" measure to one which is a weighted
mean, since Dave complained bitterly that my adjustment wasn't one.
>

Thanks for doing that.

--- 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.

--- 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.

On Mon, 25 Mar 2002 15:06:17 -0000, "dkeenanuqnetau" <d.keenan@uq.net.au>
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.

>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

--- 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.

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

that would be fine with me (though it's not really the 'best 20' by
any complete criterion, just the 'first 20' in complexity to pass a
certain condition).

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > I think we should leave room for various preferences in this
> > department, and don't see why we can't have a best 20.
>
> that would be fine with me (though it's not really the 'best 20' by
> any complete criterion, just the 'first 20' in complexity to pass a
> certain condition).

Sure. If any one of us wants the 20, I'm happy to go with it.

Lets move on to the full 7-limit. When you've got time Gene, could you
set your limits somewhat wide to start with. Say 35 cents rms error.
Weighted complexity of 30 gens, badness to give about 40 temperaments?

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

> Lets move on to the full 7-limit. When you've got time Gene, could
you
> set your limits somewhat wide to start with. Say 35 cents rms
error.
> Weighted complexity of 30 gens, badness to give about 40
temperaments?

I've been thinking of doing 7-limit planars first.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Lets move on to the full 7-limit. When you've got time Gene,
could
> you
> > set your limits somewhat wide to start with. Say 35 cents rms
> error.
> > Weighted complexity of 30 gens, badness to give about 40
> temperaments?
>
> I've been thinking of doing 7-limit planars first.

wow -- that could be a long list. the nice thing is that my heuristic
should work well for these, since there's only one unison vector.

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

let's see how the heuristic for complexity matches up with the g_w
measure. i'll leave out the heuristic for error, since we appear to
have settled on an unweighted error measure.

> 135/128 (3)^3*(5)/(2)^7
>
> g_w 2.558772839

log(135)/2.558772839 = 1.9170

> 256/243 (2)^8/(3)^5
>
> g_w 3.472662942

log(243)/3.472662942 = 1.5818

> 25/24 (5)^2/(2)^3/(3)
>
> g_w 1.597771402

log(25)/1.597771402 = 2.0146

> 648/625 (2)^3*(3)^4/(5)^4
>
> g_w 3.484393186

log(625)/3.484393186 = 1.8476

> 16875/16384 (3)^3*(5)^4/(2)^14
>
> g_w 4.719203505

log(16875)/4.719203505 = 2.0625

> 250/243 (2)*(5)^3/(3)^5
>
> g_w 3.413658644

log(250)/3.413658644 = 1.6175

> 128/125 (2)^7/(5)^3
>
> g_w 2.613294890

log(125)/2.613294890 = 1.8476

> 3125/3072 (5)^5/(2)^10/(3)
>
> g_w 4.128050871

log(3125)/4.128050871 = 1.9494

> 20000/19683 (2)^5*(5)^4/(3)^9
>
> g_w 5.817894303

log(19683)/5.817894303 = 1.6995

> 81/80 (3)^4/(2)^4/(5)
>
> g_w 2.558772839

log(81)/2.558772839 = 1.7174

> 2048/2025 (2)^11/(3)^4/(5)^2
>
> g_w 3.822598772

log(2025)/3.822598772 = 1.9917

> 78732/78125 (2)^2*(3)^9/(5)^7
>
> g_w 6.772337791

log(78125)/6.772337791 = 1.6635

> 393216/390625 (2)^17*(3)/(5)^8
>
> g_w 6.722154036

log(390625)/6.722154036 = 1.9154

> 2109375/2097152 (3)^3*(5)^7/(2)^21
>
> g_w 7.187006703

log(2109375)/7.187006703 = 2.0261

> 4294967296/4271484375 (2)^32/(3)^7/(5)^9
>
> g_w 10.74662038

log(4271484375)/10.74662038 = 2.0635

> 15625/15552 (5)^6/(2)^6/(3)^5
>
> g_w 4.990527341

log(15625)/4.990527341 = 1.9350

> 1600000/1594323 (2)^9*(5)^5/(3)^13
>
> g_w 8.314887839

log(1594323)/8.314887839 = 1.7176

> 1224440064/1220703125 (2)^8*(3)^14/(5)^13
>
> g_w 11.61862841

log(1220703125)/11.61862841 = 1.8008

> 10485760000/10460353203 (2)^24*(5)^4/(3)^21
>
> g_w 13.57752022

log(10460353203)/13.57752022 = 1.6992

> 6115295232/6103515625 (2)^23*(3)^6/(5)^14
>
> g_w 11.20594372

log(6103515625)/11.20594372 = 2.0107

> 19073486328125/19042491875328 (5)^19/(2)^14/(3)^19
>
> g_w 16.55086763

log(19073486328125)/16.55086763 = 1.8476

> 32805/32768 (3)^8*(5)/(2)^15
>
> g_w 5.957335766

log(32805)/5.957335766 = 1.7455

> 274877906944/274658203125 (2)^38/(3)^2/(5)^15
>
> g_w 13.67967551

log(274658203125)/13.67967551 = 1.9254

> 7629394531250/7625597484987 (2)*(5)^18/(3)^27
>
> g_w 19.05445924

log(7625597484987)/19.05445924 = 1.5567

> 9010162353515625/9007199254740992 (3)^10*(5)^16/(2)^53
>
> g_w 17.87941745

log(9010162353515625)/17.87941745 = 2.0547

the ratios fall between 1.5567 and 2.0635. so it's good as a rough
guesstimate . . .

--- 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.

--- 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.

--- 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)?

--- 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.

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

huh? let me try this again. forget g_w. if the comma is what you're
starting from, why can't you sort by the size of the numbers in the
comma, instead of the size of the comma?

>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.

true -- i'm just being anal in that i don't like wading through all
kinds of complicated temperaments before even seeing schismic.