Tuning-Math Digests messages 4375 - 4399

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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]


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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? :)


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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?


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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.


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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.


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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.


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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]]


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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


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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


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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.


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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.


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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


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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.


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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.


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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.


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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.


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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


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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


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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.


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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.


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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.


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