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Message: 9500 - Contents - Hide Contents Date: Thu, 29 Jan 2004 03:57:39 Subject: Re: 114 7-limit temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> But you agree it's a perfectly valid interpretation, due to the >> obvious generalisation of the normalising of weighted MAD, RMS etc. > to>> the normalising of weighted minimax? >> It's hard to say, because of course it's inconceivable for all of the > errors to be equal.In our application perhaps, but not in general. Here's another way to check the normalisation. Scaling all the weights by the same factor (e.g. using a diffferent log base for the errors and their weights) should not change the result. Anyway, the main thing is that we're all happy with the scaling of the error, irrespective of how we choose to interpret it.>> Yes. Presumably any reasonable generating scheme should have some > such>> interprtetation in terms of the TOP complexity, > > Possibly. >>> otherwise what use is it? >> See above. This seems like a more direct definition of 'complexity' > to me. If I take Gene's confirmation to heart, it's the affine- > geometrical size measure (length, area, volume, etc.) of the portion > of the lattice sufficient, under the relevant temperament, to > represent the entire lattice.That sounds pretty good, in a hand-waving sort of way, but when I try to pin down what it actually means in terms of how big a scale we might need to make good use of some temperament, my brain just keeps sliding off it.>> I suspect one can still specify reasonable (maybe even optimal) >> generating schemes with multiple generators. In fact since we're >> tempering the octave, we apparently need to find such schemes > already>> for so-called linear temperaments. >> I don't think we need to do any of this, nor should we want to for > the purposes of a paper brief enough to be published.So we're not goint to talk about how to obtain a finite scale from a temperament. Now that the octave is just another generator (and the pair of generators is no longer unique), don't we have some explaining to do, about why we should iterate one particular generator modulo the other?>> In any case, I'd be happy to hear _any_ such interpretation for this >> new complexity measure, >> See Yahoo groups: /tuning-math/message/8781 * [with cont.] and > Yahoo groups: /tuning-math/message/8806 * [with cont.] . . .Thanks. I may be some time ... :-)>> even if it only works for linear >temperaments. >> The whole point is that it works for equal, linear, planar . . . etc. > > Oh, you said _such_ . . . Sorry. Well, I suspect that's still > possible, especially since Gene himself explained the lower > complexity of schismic vs. miracle in terms of generators-per-prime.Yes. I meant "even if some particular _interpretation_ only works for linear". I understand the _complexity_measure_ is intended to work for all, and as such it sounds brilliant.

Message: 9501 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:57:51 Subject: Re: What the numbers mean From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> Here is a no doubt long overdue discussion of what the numbers in the >> wedgie for a linear temperament mean. > ... >> That was a good explanation. Thanks heaps Gene! >>> If we take absolute values and normalize each by dividing by the log >> base 2 of the two primes involved, we get >> >> [6/p3 7/p5 2/p7 25/(p3p5) 20/(p3p7) 15/(p5p7)] = >> >> [3.785578521, 3.014735907, .7124143742, 6.793166368, 4.494834256, >> 2.301151281] >> >> The complexity is defined by the weighted value for 5/3, which is the >> worst case (as, in fact, it clearly is.) >> You are dividing by the _product_ of the two logs, shouldn't you be > dividing by their _sum_ (the log of the product of the primes)?Product makes a lot more sense to me than sum, in my vague intuitive understanding of these things. You probably see that Gene's defining p2 as 1. Now the wedgie represents a bivector in 4-dimensional space, which means the relevant basis elements are *directed areas* in 4D, of which there are of course 6 "multilinearly-independent" ones. Now, the unit lengths in the lattice are scaled by (p2,) p3, p5, and p7, so the unit areas would have to be scaled by the 6 possible *products*, not *sums*, of these -- (p2)p3, (p2)p5, (p2)p7, p3p5, p3p7, and p5p7.

Message: 9502 - Contents - Hide Contents Date: Thu, 29 Jan 2004 04:08:22 Subject: Re: Attn: Gene 2 From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > '>> geometry, where angles are left undefined . . . Anyhow, since both of >> these methods could be used to address a 3-limit TOP temperament, in >> 5-limit could they be still both be expressible in a single form in a >> general enough framework, say exterior algebra? >> That was my suggestion. You normalize by dividing each coordinate by > log2(p) and take the wedge product up to the (normalized) multival, > and then measure complexity by taking the max of the absolute values > of the coefficients. If you start from the monzo side, you get the > same normalized coefficients up to a constant factor, but now you > might rather take the sum of the absolute values (L1 vs L infinity.) > > The goal, of course, is to produce>> complexity vs. TOP error graphs for 7-limit linear temperaments, >> something I currently don't know how to do. >> Why not use the formula I gave before: > > || <<w1 w2 w3 w4 w5 w6|| || = max(|w1|/p3, |w2|/p5, |w3|/p7, > |w4|/p3p5, |w5|/p3p7, |w6|/p5p7)This formula looks like a good argument for using square brackets << ]] instead of vertical bars << || for multivectors. I though we had agreed to use square brackets at one stage.

Message: 9503 - Contents - Hide Contents Date: Thu, 29 Jan 2004 22:01:11 Subject: Re: Maximum Error in TOP From: Kalle Aho --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote: >> Hi! >>>> How do I get the maximum Tenney-weighted error in a TOP > temperament?>> Is it the same as the maximum weighted error of the primes > > Yes.Thanks, Paul. Can you explain why this is so? Kalle

Message: 9504 - Contents - Hide Contents Date: Thu, 29 Jan 2004 04:13:41 Subject: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> One might try > > k*sqrt(comp) + sqrt(err) < x > > for starters.All right, just for you . . . This time I'll insist Waage just makes it in despite its high complexity. Then k=1.75 gives the most inclusive cutoff which is still more stringent than Gene's, so culling that list will work again. Number 1 Meantone (Huygens) [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328] TOP generators [1201.698520, 504.1341314] bad: 4.6061 comp: 3.562072 err: 1.698521 Number 2 Magic [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877] TOP generators [1201.276744, 380.7957184] bad: 4.7480 comp: 4.274486 err: 1.276744 Number 3 Pajara [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174] TOP generators [598.4467109, 106.5665459] bad: 4.7881 comp: 2.988993 err: 3.106578 Number 4 Dominant Seventh [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933] TOP generators [1195.228951, 495.8810151] bad: 4.9260 comp: 2.454561 err: 4.771049 Number 5 Augmented [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546] TOP generators [399.9922103, 107.3111730] bad: 4.9876 comp: 2.147741 err: 5.870879 Number 6 Semisixths [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]] TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748] TOP generators [1198.389531, 443.1602931] bad: 5.0349 comp: 4.630693 err: 1.610469 Number 7 Schismic [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]] TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750] TOP generators [1200.760624, 498.1193303] bad: 5.1036 comp: 5.618543 err: .912904 Number 8 Injera [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]] TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835] TOP generators [600.8889070, 93.60982493] bad: 5.1411 comp: 3.445412 err: 3.582707 Number 9 Orwell [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834] TOP generators [1199.532657, 271.4936472] bad: 5.1530 comp: 5.706260 err: .946061 Number 10 Hemifourths [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]] TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166] TOP generators [1203.668841, 252.4803582] bad: 5.1637 comp: 3.445412 err: 3.668842 Number 11 Kleismic [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]] TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000] TOP generators [1203.187309, 317.8344609] bad: 5.1902 comp: 3.785579 err: 3.187309 Number 12 Negri [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]] TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000] TOP generators [1203.187309, 124.8419629] bad: 5.1986 comp: 3.804173 err: 3.187309 Number 13 Diminished [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]] TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404] TOP generators [298.5321149, 101.4561401] bad: 5.2032 comp: 2.523719 err: 5.871540 Number 14 Tripletone [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420] TOP generators [399.0200131, 92.45965769] bad: 5.2344 comp: 4.045351 err: 2.939961 Number 15 Blackwood [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]] TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698] TOP generators [239.1786927, 83.83059859] bad: 5.2708 comp: 2.173813 err: 7.239629 Number 16 Superpythagorean [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]] TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608] TOP generators [1197.596121, 489.4271829] bad: 5.3047 comp: 4.602303 err: 2.403879 Number 17 Miracle [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756] TOP generators [1200.631014, 116.7206423] bad: 5.3555 comp: 6.793166 err: .631014 Number 18 Porcupine [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]] TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888] TOP generators [1196.905960, 162.3176609] bad: 5.3860 comp: 4.295482 err: 3.094040 Number 19 Zeta Reticuli Trisex Porky [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]] TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174] TOP generators [598.4467109, 162.3159606] bad: 5.3943 comp: 4.306766 err: 3.106578 Number 20 Supermajor seconds [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]] TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099] TOP generators [1201.698520, 232.5214630] bad: 5.4159 comp: 5.522763 err: 1.698521 Number 21 Flattone [1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]] TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278] TOP generators [1202.536419, 507.1379663] bad: 5.4700 comp: 4.909123 err: 2.536420 Number 22 Nonkleismic [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085] TOP generators [1198.828458, 309.8926610] bad: 5.4781 comp: 6.309298 err: 1.171542 Number 23 [6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]] TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076] TOP generators [1202.659696, 82.97467050] bad: 5.4973 comp: 4.306766 err: 3.480440 Number 24 [0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]] TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906] TOP generators [99.80617249, 24.58395811] bad: 5.5130 comp: 4.295482 err: 3.557008 Number 25 Decimal [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]] TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757] TOP generators [603.8288989, 250.6116362] bad: 5.5474 comp: 2.523719 err: 7.657798 Number 26 Quartaminorthirds [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]] TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770] TOP generators [1199.792743, 77.83315314] bad: 5.5686 comp: 6.742251 err: 1.049791 Number 27 [3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]] TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070] TOP generators [1202.900537, 570.4479508] bad: 5.5734 comp: 4.891080 err: 2.900537 Number 28 [3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]] TOP tuning [1202.624742, 1900.726787, 2792.408176, 3361.457323] TOP generators [1202.624742, 569.0491468] bad: 5.5985 comp: 5.168119 err: 2.624742 Number 29 Pelogic [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]] TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957] TOP generators [1209.734056, 532.9412251] bad: 5.6088 comp: 2.022675 err: 9.734056 Number 30 Dicot [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]] TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113] TOP generators [1204.048159, 356.3998255] bad: 5.6237 comp: 2.137243 err: 9.396316 Number 31 [5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]] TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528] TOP generators [1194.335372, 99.13879319] bad: 5.6284 comp: 3.445412 err: 5.664628 Number 32 Catakleismic [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]] TOP tuning [1200.536356, 1901.438376, 2785.068335, 3370.331646] TOP generators [1200.536355, 316.9063960] bad: 5.6313 comp: 7.836558 err: .536356 Number 33 [3, -5, -6, -15, -18, 0] [[1, 3, 0, 0], [0, -3, 5, 6]] TOP tuning [1195.486066, 1908.381352, 2796.794743, 3356.153692] TOP generators [1195.486066, 559.3589487] bad: 5.6577 comp: 4.075900 err: 4.513934 Number 34 [8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]] TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814] TOP generators [1201.135545, 387.5841360] bad: 5.6663 comp: 6.411729 err: 1.525246 Number 35 Beatles [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]] TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226] TOP generators [1197.104145, 354.7203384] bad: 5.6780 comp: 5.162806 err: 2.895855 Number 36 Formerly Number 82 [6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]] TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030] TOP generators [601.7004928, 230.8749260] bad: 5.6954 comp: 4.619353 err: 3.740932 Number 37 [3, 5, 9, 1, 6, 7] [[1, 2, 3, 4], [0, -3, -5, -9]] TOP tuning [1193.415676, 1912.390908, 2789.512955, 3350.341372] TOP generators [1193.415676, 158.1468146] bad: 5.6994 comp: 3.205865 err: 6.584324 Number 38 [3, 0, 9, -7, 6, 21] [[3, 5, 7, 9], [0, -1, 0, -3]] TOP tuning [1193.415676, 1912.390908, 2784.636577, 3350.341372] TOP generators [397.8052253, 76.63521863] bad: 5.7070 comp: 3.221612 err: 6.584324 Number 39 {28/27, 50/49} [2, 6, 6, 5, 4, -3] [[2, 3, 4, 5], [0, 1, 3, 3]] TOP tuning [1191.599639, 1915.269258, 2766.808679, 3362.608498] TOP generators [595.7998193, 127.8698005] bad: 5.7115 comp: 2.584059 err: 8.400361 Number 40 [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]] TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936] TOP generators [399.8000105, 155.5708520] bad: 5.7136 comp: 3.804173 err: 5.291448 Number 41 Superkleismic [9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]] TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245] TOP generators [1201.371918, 322.3731369] bad: 5.7153 comp: 6.742251 err: 1.371918 Number 42 [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]] TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574] TOP generators [599.2769413, 272.3123381] bad: 5.7395 comp: 5.047438 err: 3.268439 Number 43 Diaschismic [2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]] TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311] TOP generators [599.3662015, 103.7870123] bad: 5.7450 comp: 6.966993 err: 1.267597 Number 44 Beep [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548] TOP generators [1194.642673, 254.8994697] bad: 5.7543 comp: 1.292030 err: 14.176105 Number 45 [1, -8, -2, -15, -6, 18] [[1, 2, -1, 2], [0, -1, 8, 2]] TOP tuning [1195.155395, 1894.070902, 2774.763716, 3382.790568] TOP generators [1195.155395, 496.2398890] bad: 5.7634 comp: 4.075900 err: 4.974313 Number 46 [6, 0, 15, -14, 7, 35] [[3, 5, 7, 9], [0, -2, 0, -5]] TOP tuning [1197.060039, 1902.856975, 2793.140092, 3360.572393] TOP generators [399.0200131, 46.12154491] bad: 5.7697 comp: 5.369353 err: 2.939961 Number 47 [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]] TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166] TOP generators [199.0788921, 88.83392059] bad: 5.7715 comp: 3.820609 err: 5.526647 Number 48 [0, 12, 12, 19, 19, -6] [[12, 19, 28, 34], [0, 0, -1, -1]] TOP tuning [1198.015473, 1896.857833, 2778.846497, 3377.854234] TOP generators [99.83462277, 16.52294019] bad: 5.7717 comp: 5.168119 err: 3.215955 Number 49 Supersupermajor [3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]] TOP tuning [1200.231588, 1903.372996, 2784.236389, 3366.314293] TOP generators [1200.231587, 234.3804692] bad: 5.7926 comp: 7.670504 err: .894655 Number 50 [11, 13, 17, -5, -4, 3] [[1, 3, 4, 5], [0, -11, -13, -17]] TOP tuning [1198.514750, 1899.600936, 2789.762356, 3371.570447] TOP generators [1198.514750, 154.1766650] bad: 5.8290 comp: 6.940227 err: 1.485250 Number 51 {1728/1715, 4000/3993} [11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]] TOP tuning [1199.083445, 1901.293958, 2784.185538, 3371.399002] TOP generators [1199.083445, 45.17026643] bad: 5.8298 comp: 7.752178 err: .916555 Number 52 {25/24, 81/80} Jamesbond? [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]] TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906] TOP generators [172.7759159, 86.69241190] bad: 5.8344 comp: 2.493450 err: 9.431411 Number 53 [5, 3, 7, -7, -3, 8] [[1, 1, 2, 2], [0, 5, 3, 7]] TOP tuning [1192.540126, 1890.131381, 2803.635005, 3361.708008] TOP generators [1192.540126, 139.5182509] bad: 5.8395 comp: 3.154649 err: 7.459874 Number 54 [9, -2, 14, -24, -3, 38] [[1, 3, 2, 5], [0, -9, 2, -14]] TOP tuning [1201.918556, 1904.657347, 2781.858962, 3363.439837] TOP generators [1201.918557, 189.0109248] bad: 5.8541 comp: 6.521440 err: 1.918557 Number 55 Octacot [8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]] TOP tuning [1199.031259, 1903.490418, 2784.064367, 3366.693863] TOP generators [1199.031259, 88.05739491] bad: 5.8567 comp: 7.752178 err: .968741 Number 56 Wizard [12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]] TOP tuning [1200.639571, 1900.941305, 2784.828674, 3368.342104] TOP generators [600.3197857, 216.7702531] bad: 5.8788 comp: 8.423526 err: .639571 Number 57 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]] TOP tuning [1197.060039, 1897.499011, 2793.140092, 3360.572393] TOP generators [133.0066710, 35.40561749] bad: 5.8950 comp: 5.706260 err: 2.939961 Number 58 Squares [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]] TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656] TOP generators [1201.698520, 426.4581630] bad: 5.8971 comp: 6.890825 err: 1.698521 Number 59 [2, 1, -4, -3, -12, -12] [[1, 1, 2, 4], [0, 2, 1, -4]] TOP tuning [1204.567524, 1916.451342, 2765.076958, 3394.502460] TOP generators [1204.567524, 355.9419091] bad: 5.8982 comp: 2.696901 err: 9.146173 Number 60 Waage [0, 12, 24, 19, 38, 22] [[12, 19, 28, 34], [0, 0, -1, -2]] TOP tuning [1200.617051, 1900.976998, 2785.844725, 3370.558188] TOP generators [100.0514209, 16.55882096] bad: 5.9023 comp: 8.548972 err: .617051> I'd really like to see these on a chart.I think Gene may be using the wrong norm to get his complexity values. I'll wait until I'm sure they're right or corrected.

Message: 9505 - Contents - Hide Contents Date: Thu, 29 Jan 2004 22:18:00 Subject: Re: Maximum Error in TOP From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> > wrote: >>> Hi! >>>>>> How do I get the maximum Tenney-weighted error in a TOP >> temperament?>>> Is it the same as the maximum weighted error of the primes >> >> Yes. >> Thanks, Paul. Can you explain why this is so? > > KalleIt follows from the prime factorization theorem. The maximum Tenney-weighted error being T implies that error(p)/log(p) <= T for all the primes p. If the factors in a chosen ratio are 2^a 3^b 5^c . . . (each exponent may either be positive or negative), then the error of the chosen ratio cannot be greater than T*(|a|/log(2) + |b|/log(3) + |c|/log(5) . . .) since the errors in the primes that make up the chosen ratio, at worst, add up without cancellation to the error in the chosen ratio. The complexity of the ratio, meanwhile, is exactly (|a|/log(2) + |b|/log(3) + |c|/log(5) . . .) So the Tenney-weighted error in the ratio cannot be greater than the second-to-last expression divided by the last expression, i.e., T.

Message: 9506 - Contents - Hide Contents Date: Thu, 29 Jan 2004 00:28:39 Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Number 15 Porcupine >> >> [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]] >> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888] >> TOP generators [1196.905960, 162.3176609] >> bad: 8.9144 comp: 4.295482 err: 3.094040 >> >> Number 16 >> >> [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]] >> TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174] >> TOP generators [598.4467109, 162.3159606] >> bad: 8.9422 comp: 4.306766 err: 3.106578 >> These two are related, and very close in terms of error and > complexity. Bi porcupine makes me think of porcupine sex, and I'm not > sure I want to, but some sort of porky name seems apt. >>> Number 25 Miracle >> >> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] >> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756] >> TOP generators [1200.631014, 116.7206423] >> bad: 9.8358 comp: 6.793166 err: .631014 >> This seems absurdly far down the list. I think mine was better.I don't think this is necessarily too far down the list for every conceivable user, including possibly Herman. As one would like all the harmonies to come from a single generalized-diatonic scale, some musicians may find the complexity of Miracle nearly prohibitive. However, I was by no means suggesting a ranking like this for our paper, and in fact I still believe we should avoid ranking altogether, simply giving a "survey" (ordered by complexity, perhaps, though almost certainly we would need to start with the most familiar examples before generalizing) for the systems that fall beneath whatever cutoff curve we choose to draw on the graph.

Message: 9507 - Contents - Hide Contents Date: Thu, 29 Jan 2004 04:19:32 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> That sounds pretty good, in a hand-waving sort of way, but when I try > to pin down what it actually means in terms of how big a scale we > might need to make good use of some temperament, my brain just keeps > sliding off it.Might that be, in a way, a necessary result of dropping octave- equivalence?>>> I suspect one can still specify reasonable (maybe even optimal) >>> generating schemes with multiple generators. In fact since we're >>> tempering the octave, we apparently need to find such schemes >> already>>> for so-called linear temperaments. >>>> I don't think we need to do any of this, nor should we want to for >> the purposes of a paper brief enough to be published. >> So we're not goint to talk about how to obtain a finite scale from a > temperament. Now that the octave is just another generator (and the > pair of generators is no longer unique), don't we have some explaining > to do, about why we should iterate one particular generator modulo the > other?For linear temperaments, I agree that we may still want to give the period/generator specification, but there are many ways to justify the attention on this. For example, though we didn't assume octave- equivalence, we may want to assume that the musician will generally be most interested in subsets of the temperament that repeat at the (tempered) octave.

Message: 9508 - Contents - Hide Contents Date: Thu, 29 Jan 2004 22:28:36 Subject: Re: Maximum Error in TOP From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> >> wrote: >>>> Hi! >>>>>>>> How do I get the maximum Tenney-weighted error in a TOP >>> temperament?>>>> Is it the same as the maximum weighted error of the primes >>> >>> Yes. >>>> Thanks, Paul. Can you explain why this is so? >> >> KalleWhoops, I screwed up. Let me try again: It follows from the prime factorization theorem. The maximum Tenney-weighted error being T implies that error(p)/log(p) <= T so error(p) <= T*log(p) for all the primes p. If the factors in a chosen ratio are 2^a 3^b 5^c . . . (each exponent may either be positive or negative), then the error of the chosen ratio cannot be greater than T*(|a|*log(2) + |b|*log(3) + |c|*log(5) . . .) since the errors in the primes that make up the chosen ratio, at worst, add up without cancellation to the error in the chosen ratio. The complexity of the ratio, meanwhile, is exactly (|a|*log(2) + |b|*log(3) + |c|*log(5) . . .) So the Tenney-weighted error in the ratio cannot be greater than the second-to-last expression divided by the last expression, i.e., T.

Message: 9509 - Contents - Hide Contents Date: Thu, 29 Jan 2004 07:10:08 Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments) From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> One might try >> >> k*sqrt(comp) + sqrt(err) < x >> >> for starters. >> All right, just for you . . . This time I'll insist Waage just makes > it in despite its high complexity. Then k=1.75 gives the most > inclusive cutoff which is still more stringent than Gene's, so > culling that list will work again. ...Well I wouldn't want as many as sixty. I think about half that would be plenty.> I think Gene may be using the wrong norm to get his complexity > values. I'll wait until I'm sure they're right or corrected.Well something's wrong. Whether its the badness functions or only the complexity I don't know. But Diaschismic shouldn't be so far down. I don't think Miracle should be so far down either. Sure it gets a hit for having 6 gens to the fifth, but not that much of a hit I would think. And where's Shrutar?

Message: 9510 - Contents - Hide Contents Date: Thu, 29 Jan 2004 22:38:05 Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> wrote: >>>>> One might try >>> >>> k*sqrt(comp) + sqrt(err) < x >>> >>> for starters. >>>> All right, just for you . . . This time I'll insist Waage just makes >> it in despite its high complexity. Then k=1.75 gives the most >> inclusive cutoff which is still more stringent than Gene's, so >> culling that list will work again. > ... >> Well I wouldn't want as many as sixty. I think about half that would > be plenty.Of course our paper should have even fewer. I'd like to see Injera but I won't insist on it. But I was just playing along for the purposes of this thread.>> I think Gene may be using the wrong norm to get his complexity >> values. I'll wait until I'm sure they're right or corrected. >> Well something's wrong. Whether its the badness functions or only the > complexity I don't know.Would you take a closer look, then? I think it's important to rethink the problem each time, to allow old prejudices a chance of dissolving.> But Diaschismic shouldn't be so far down.For 7-limit? This is only one of at least 3 possible 7-limit mappings, so it's weird to even use that name.> And where's Shrutar?It fell off the end, with a badness of 6.1221 in this formulation. Doesn't shine as a 7-limit linear temperament . . .

Message: 9511 - Contents - Hide Contents Date: Thu, 29 Jan 2004 07:30:08 Subject: Re: rank complexity explanation updated From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> The interval matrix often, if not typically, has all unisons/octaves > along the diagonal. This one is merely a reshuffling so that the > diagonal becomes a right vertical.That's what I used. Manuel's way is more useful for seeing how the harmony works out in the scale, however.

Message: 9512 - Contents - Hide Contents Date: Thu, 29 Jan 2004 01:17:16 Subject: Re: rank complexity explanation updated From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>>> When a listener hears a melody in a fixed scal... * [with cont.] (Wayb.) >>>>>> >>>>>>> Hi Paul, >>>>>>>>>> I'm afraid I don't know what an "external" interval is. Here's >>>>> the interval matrix of the diatonic scale in 12-equal, as given >>>>> by Scala... >>>>> >>>>> 100.0 : 2 4 5 7 9 11 12 >>>>> 200.0 : 2 3 5 7 9 10 12 >>>>> 400.0 : 1 3 5 7 8 10 12 >>>>> 500.0 : 2 4 6 7 9 11 12 >>>>> 700.0 : 2 4 5 7 9 10 12 >>>>> 900.0 : 2 3 5 7 8 10 12 >>>>> 1100.0: 1 3 5 6 8 10 12 >>>>>>>> Q: Shouldn't the first row be 000.0? >>>>>> It is quite safe to ignore the values before the colon, as they >>> are merely an artifact of scala's output. >>>> What kind of artifact are they, if not an error? >> They indicate the interval between 1/1 and the degree of the > original 'native' mode, on which the mode shown on the particular > line is based.In that case, shouldn't the first entry be 000.0?> In this case they are artifactal only because they are given in > different units than is the interval matrix itself.OK, but more serious would seem to be the actual error.

Message: 9513 - Contents - Hide Contents Date: Thu, 29 Jan 2004 07:28:20 Subject: Re: 114 7-limit temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> That sounds pretty good, in a hand-waving sort of way, but when I > try>> to pin down what it actually means in terms of how big a scale we >> might need to make good use of some temperament, my brain just keeps >> sliding off it. >> Might that be, in a way, a necessary result of dropping octave- > equivalence?It might be, but in any case I'd still like to get on top of it. Without octave equivalence we can still make it finite by limiting it to the range of human hearing, say 10 octaves.

Message: 9514 - Contents - Hide Contents Date: Thu, 29 Jan 2004 07:34:56 Subject: Re: 114 7-limit temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> No particular generator basis is assumed in the TOP complexity > calculations. Instead, it's a direct measure of how much the > tempering simplifies the lattice, and reduces (Gene seems to > imply/agree) to the number of notes per acoustic whatever in the case > of equal (1-dimensional) temperaments.I thought equal temperaments were "0-dimensional"? The complexity measure for an equal temperament pretty much reduces to the division of the octave. You could take the maximum of |(mapping of p)/log2(p)| if you wanted to be precise about it.

Message: 9515 - Contents - Hide Contents Date: Thu, 29 Jan 2004 01:20:18 Subject: Re: 114 7-limit temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > It's the minimax over *all* intervals.You mean the 1/log(product_complexity) weighted minimax over all intervals in the prime limit.>> Obviously you don't do an infinite >> number of calculations. >> No, you only need to set the primes, and the rest falls out correctly. >Yes, It's very clever in that way. But still it flies in the face of Partch's "observation one" (or whatever it is called) that as the complexity of a frequency ratio increases it must be tuned more and more accurately to be perceived as just (or something), until it becomes too complex to hear that way even when tuned precisely. The very fact that TOP cannot distinguish between a good 7-limit temperament and a good 9-limit temperament (nor 8 or 10 limit) should make one suspicious. I suspect that while the results of tenney weighting are quite good for 5-limit, and may be acceptable for 7-limit, we might find it agreeing less and less with our subjective experience when we go to 11 and 13 limits. But then again, maybe not. I'll wait and see (or hear).>> But even so, the normalisation factor may >> still converge. >> Yes, it may. >>> If it's a minimax type of thing then all we need is to divide the >> current result by the maximum weight of any interval. >> This doesn't seem right -- what's your reasoning?I gave it in Yahoo groups: /tuning-math/message/8876 * [with cont.] It's what you get when you generalise the correctly normalised MAD (p=1), RMS (p=2) etc. as p goes to infinity. And in this application the interval with the maximum weight is the one with the lowest product_complexity, namely the octave.>> Let me guess: >> this is lg2(2) = 1 so there would be no change?I've just confirmed that for myself regarding errors (but not complexity yet), by examining a few examples from Gene's list. Although I should have given this maximum weight as 1/lg2(2).> There is no need to use base 2 -- the result is the same regardless > of which base you use in the logarithms. The error is measured using > logarithms, but so is the complexity = log(n*d), so error divided by > complexity, which is what you're minimizing the maximum of, is > insensitive to choice of base.Paul, that's clearly not what Gene has done in this list, otherwise the error figures would be a factor of 1200 smaller than they are. Gene has in fact already correctly normalised the errors and given them in cents. If you scan down the list Yahoo groups: /tuning-math/message/8809 * [with cont.] you will see that in most cases the error figure given for the temperament happens to be the same as the cents error in the octave, and is never less than it. It's just that one now has to think like this: If the minimax tenney-weighted error for the temperament is X cents, then the errors in the following intervals could be as large as: Interval Error(cents) ---------------------- 1:2 X 2:3 2.6 X 4:5 4.3 X 5:6 4.9 X 4:7 4.8 X 5:7 5.1 X 6:7 5.4 X 4:9 5.2 X 5:9 5.5 X 7:9 6.0 X where the multiplication factor is the base-2 log of the product of the two sides of the ratio in lowest terms.>> Is the weighting the same for the complexity? Minimax where gens per >> interval is divided by lg2(product_complexity(interval))? >> No particular generator basis is assumed in the TOP complexity > calculations. Instead, it's a direct measure of how much the > tempering simplifies the lattice, and reduces (Gene seems to > imply/agree) to the number of notes per acoustic whatever in the case > of equal (1-dimensional) temperaments.What's an "acoustic whatever"? Anything that relates to sound??? I'm being a pedant here in case you didn't guess. :-) OK. That sounds alright, but how do I relate it to upper limits (or whatever) on the numbers of notes before I get a particular interval? i.e. in a similar way to what I just did above with the errors.

Message: 9516 - Contents - Hide Contents Date: Thu, 29 Jan 2004 07:38:32 Subject: Re: rank complexity explanation updated From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> They indicate the interval between 1/1 and the degree of the > original 'native' mode, on which the mode shown on the particular > line is based.In which case the first line should have 0.000 cents, as people have been saying.

Message: 9517 - Contents - Hide Contents Date: Thu, 29 Jan 2004 07:40:30 Subject: Re: Graef article on rationalization of scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> That the diatonic scale is a 'good' PB seems like the best example.Is PB a synonym for Fokker block, or is it more general, and if so, how precisely is it defined?

Message: 9518 - Contents - Hide Contents Date: Thu, 29 Jan 2004 01:29:27 Subject: Re: Graef article on rationalization of scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> This raises an interesting question. What is our approved method >>>>> for finding Fokker blocks for an arbitrary irrational scale? >>>>> Such a method would surely make Graf's look silly. >>>>>>>> All such methods are silly, >>>>>> Perhaps you mean Graf's idea of people wanting "just" versions of >>> arbitrary scales is silly. That's for sure. >> >> Yup. >>>>>> but I prefer the hexagonal (rhombic dodecahedral, etc.) or Kees >>>> blocks that result from the min-"odd-limit" criterion. But the >>>> whole idea of rationalizing a tempered scale is completely >>>> backwards and misses the point in a big way. >>>>>> I think you missed my point. >>>> Would you, then, clarify your point, perhaps with examples? >> That the diatonic scale is a 'good' PB seems like the best example.Sure, but when 81:80 is tempered out, it's a good "periodicity strip" (figure 5 in TFoT), which is even better, and doesn't require you to arbitrarily rationalize the pitches.> Since you practically single-handedly launched the 'popular scales > are good PBs' program, I find it highly unusual that you are now > asking me what it is.Now I understand you better. Yet, 'popular scales' will often have a large number of plausible derivations from a PB, in terms of its shape, its position in the lattice, and the unison vectors involved, so going from the scale to the PB still seems like a step backwards, a step from greater generality to lesser generality.

Message: 9519 - Contents - Hide Contents Date: Thu, 29 Jan 2004 07:42:41 Subject: Re: rank complexity explanation updated From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> I also see I had the wrong definition of interval matrix; >> That's pretty incredible considering that you've got the > Rothenberg papers on the matter.Paul now tells us I had a correct definition. Did Rothenberg invent the idea of an interval matrix?

Message: 9520 - Contents - Hide Contents Date: Thu, 29 Jan 2004 01:32:27 Subject: Re: rank complexity explanation updated From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> The interval matrix often, if not typically, has all unisons/octaves >> along the diagonal. This one is merely a reshuffling so that the >> diagonal becomes a right vertical. >> No, the interval matrix is always written as above (the values > after the colons, at least).Where do you get *always*?? BTW, In matrix algebra, which is where you find characteristic polynomials and the like, the *diagonal*, not the *right vertical*, represents the relations between like elements.

Message: 9521 - Contents - Hide Contents Date: Thu, 29 Jan 2004 02:01:30 Subject: Re: Attn: Gene 2 From: Carl Lumma>> >hy not use the formula I gave before: >> >> || <<w1 w2 w3 w4 w5 w6|| || = max(|w1|/p3, |w2|/p5, |w3|/p7, >> |w4|/p3p5, |w5|/p3p7, |w6|/p5p7) >>This formula looks like a good argument for using square brackets ><< ]] instead of vertical bars << || for multivectors. I though we >had agreed to use square brackets at one stage.I moved in favor of square brackets for this very reason, but nobody paid any attention. -Carl

Message: 9522 - Contents - Hide Contents Date: Thu, 29 Jan 2004 02:02:56 Subject: Re: 114 7-limit temperaments From: Carl Lumma>For linear temperaments, I agree that we may still want to give the >period/generator specification, but there are many ways to justify >the attention on this. For example, though we didn't assume octave- >equivalence, we may want to assume that the musician will generally >be most interested in subsets of the temperament that repeat at the >(tempered) octave.As long as the temperament maps the octave, will it not be periodic at said octave? -Carl

Message: 9523 - Contents - Hide Contents Date: Thu, 29 Jan 2004 21:51:46 Subject: Re: What the numbers mean From: Herman Miller On Thu, 29 Jan 2004 11:20:37 -0000, "Gene Ward Smith" <gwsmith@xxxxx.xxx> wrote:>Here is a no doubt long overdue discussion of what the numbers in the >wedgie for a linear temperament mean. > >The wedgie for miracle is <<6 -7 -2 -25 -20 15||. Normally we would >content ourselves with saying the mapping to primes is given by >[<1 1 3 3|, <0 6 -7 -2|], but that is a 2-centric way of putting it; >we have other maps for 3 and generator, 5 and generator, and 7 and >generator: > >3&g: [<1 1 3 3|, <-6 0 -25 -20|] g ~ 7/72 ~ 11/114 > >5&g: [<-2 -8 1 4|, <7 25 0 15|] g ~ 58/72 ~ 135/167 > >7&g: [<1 7 -4 1|, <-2 -20 15 0|] g ~ 65/72 ~ 182/202That's useful to know. I can see where the second part of the maps come from, but how do you get the first part? It's clear that the element corresponding to the period is always 1 in this example, which makes sense, but is there any easy way to get the other three numbers other than trying a few until you find one that works? In other words, it's easy to determine that [<1 x y z|, <0 6 -7 -2|] is a possible mapping of miracle, as is [<x 1 y z|, <-6 0 -25 -20|], but I don't know how to get x, y, and z. I've been trying to find something like this in the archives, but I don't know where to look. -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Message: 9524 - Contents - Hide Contents Date: Thu, 29 Jan 2004 02:07:00 Subject: Re: Graef article on rationalization of scales From: Carl Lumma>> >hat the diatonic scale is a 'good' PB seems like the best example. >>Is PB a synonym for Fokker block, or is it more general, and if so, >how precisely is it defined?My impression was that PB is weaker than Fokker, the later requiring epimorphism and monotonicity (neither of which I have a solid understanding of) and that the former requires, well, nothing more than the correct number of commas that, when all of them are tempered out, gives an equal temperament. -Carl

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