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Message: 10925 - Contents - Hide Contents Date: Sat, 08 May 2004 00:27:54 Subject: Re: 10+16 (continued from tuning) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> 7-limit now . . . >> >> val for 10: >> >> <10 16 23 28] >> >> val for 16: >> >> <16 25 37 45] >> >> wedgie: >> >> <<-6 2 2 17 20 -1]] >> >> According to Gene's file, this has TOP error of 3.740932 and L1 >> complexity of 14.626943. Not too bad. It just barely, by a hair, >> falls outside the bound in my paper. Not too late to change that, >> though . . . >> What is the bound of your paper at the moment?It was going to be, for both 5-limit and 7-limit, (error/16.6667)^(2/3) + (complexity/23.5)^(2/3) < 1 since the bound falls in a nice wide moat in both cases. As of right now, though, I'm planning to use, error/10 + complexity/23.5 < 1 partly due to Dave pleading against exponents smaller than 1. No more moats, which is OK since I don't really have the space for those graphs anyway. This change only adds two to the 7-limit list and one to the 5-limit list, without taking any away. The units are error = max. over all ratios of (cents error)/lg2(n*d), complexity = L1 Tenney multival norm in the 7-limit case, but L1 Tenney multival norm multiplied by 2 in the 5-limit case (I can't really justify that, but the multival has twice as many entries in the 7-limit case). This would appear to penalize complex 5-limit temperaments, but Orwell and Amity still make it in.
Message: 10926 - Contents - Hide Contents Date: Sat, 08 May 2004 00:40:14 Subject: Re: Request for Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Are you giving names to all the temperaments you plan on tabulating, > and if so, which names?I guess this is all up for grabs at the moment. I'm using "semifourths" instead of "hemifourths" because generating by "semifourths" was already mentioned in an XH article. The 5-limit temperament where the Pythagorean comma vanishes will probably be called Compton since, thanks to Carl, Compton's patent is the earliest we know of . . . I'm planning to use Hanson only in the 5- limit, to be true to Larry's intentions . . .
Message: 10927 - Contents - Hide Contents Date: Sat, 08 May 2004 04:33:15 Subject: Re: Request for Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> Are you giving names to all the temperaments you plan on tabulating, >> and if so, which names? >> I guess this is all up for grabs at the moment. I'm > using "semifourths" instead of "hemifourths" because generating > by "semifourths" was already mentioned in an XH article. The 5- limit > temperament where the Pythagorean comma vanishes will probably be > called Compton since, thanks to Carl, Compton's patent is the > earliest we know of . . . I'm planning to use Hanson only in the 5- > limit, to be true to Larry's intentions . . .The hanson thing doesn't matter since catakleismic/hanson7 isn't being discussed anyway. I could give what I presently have down as names for these 25 temperaments, including switching to compton if you like. What theory are you operating under regarding the point beyond which increasing accuracy of tuning no longer makes a practical difference? Another question: are you stopping at the 7-limit? It's a fact of life that more and more temperaments are going to crop up as you go uplimit.
Message: 10928 - Contents - Hide Contents Date: Sat, 08 May 2004 05:13:54 Subject: Re: Request for Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" > <gwsmith@s...> >> wrote: >>>>> Are you giving names to all the temperaments you plan on > tabulating,>>> and if so, which names? >>>> I guess this is all up for grabs at the moment. I'm >> using "semifourths" instead of "hemifourths" because generating >> by "semifourths" was already mentioned in an XH article. The 5- > limit>> temperament where the Pythagorean comma vanishes will probably be >> called Compton since, thanks to Carl, Compton's patent is the >> earliest we know of . . . I'm planning to use Hanson only in the 5- >> limit, to be true to Larry's intentions . . . >> The hanson thing doesn't matter since catakleismic/hanson7 isn't > being discussed anyway. I could give what I presently have down as > names for these 25 temperaments, including switching to compton if > you like. Sure. > What theory are you operating under regarding the point beyond >which > increasing accuracy of tuning no longer makes a practical >difference? None. > Another question: are you stopping at the 7-limit?Yes; you said you couldn't handle the 11-limit case, so I'll save that, as well as {2,3,7}, {3,5,7}, etc., for part 2 and future supplements. This paper is part 1.
Message: 10929 - Contents - Hide Contents Date: Sat, 08 May 2004 06:28:56 Subject: Re: Request for Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Yes; you said you couldn't handle the 11-limit case, so I'll save > that, as well as {2,3,7}, {3,5,7}, etc., for part 2 and future > supplements. This paper is part 1.My computer keeps crashing now every 1000 temperaments or so, but I could finish doing 11 limit if your graph was truly crucial; I have in mind buying a new one sometime and seeing if that helps, for that matter. I think however that discussing 5 and 7 limit temperaments is enough for one paper, and that if you go to the 11 limit you should probably continue on to 13. Are you putting much theory in it? ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 10930 - Contents - Hide Contents Date: Mon, 10 May 2004 16:31:29 Subject: Re: Request for Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Yes; you said you couldn't handle the 11-limit case, so I'll save >> that, as well as {2,3,7}, {3,5,7}, etc., for part 2 and future >> supplements. This paper is part 1. >> My computer keeps crashing now every 1000 temperaments or so, but I > could finish doing 11 limit if your graph was truly crucial; I have > in mind buying a new one sometime and seeing if that helps, for that > matter. I think however that discussing 5 and 7 limit temperaments > is enough for one paper, Yes. > and that if you go to the 11 limit you > should probably continue on to 13.Yes, in the next paper or two. Math content is minimal, at John's request. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 10931 - Contents - Hide Contents Date: Tue, 11 May 2004 18:37:27 Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments) From: Paul Erlich Even though we were using a different complexity measure on January 27 (L-inf instead of L1), my current list of 25 is quite close to this one. So the method can't make that much difference -- maybe the Klein constraint Gene was talking about has something to do with this. Here's what made it in and what didn't: IN> Number 1 Meantone > > [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: 6.5251 comp: 3.562072 err: 1.698521 IN > 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: 7.0687 comp: 4.274486 err: 1.276744 IN > 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: 7.1567 comp: 2.988993 err: 3.106578 IN > Number 4 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: 7.8851 comp: 4.630693 err: 1.610469 IN > Number 5 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: 8.0970 comp: 2.454561 err: 4.771049 IN > Number 6 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: 8.2512 comp: 3.445412 err: 3.582707 IN > Number 7 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: 8.3168 comp: 3.785579 err: 3.187309 IN > Number 8 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: 8.3374 comp: 3.445412 err: 3.66884 IN > Number 9 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: 8.3420 comp: 3.804173 err: 3.187309 IN > Number 10 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: 8.4214 comp: 4.045351 err: 2.939961 IN > Number 11 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: 8.5260 comp: 5.618543 err: .912904 IN > Number 12 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: 8.6400 comp: 4.602303 err: 2.403879 IN > Number 13 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: 8.6780 comp: 5.706260 err: .946061 IN > Number 14 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: 8.7811 comp: 2.147741 err: 5.870879 IN > 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 IN > 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 IN > Number 17 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: 9.1819 comp: 5.522763 err: 1.698521 IN > Number 18 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: 9.1883 comp: 4.909123 err: 2.536420 IN > Number 19 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: 9.2912 comp: 2.523719 err: 5.871540 OUT > Number 20 > > [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: 9.3161 comp: 4.306766 err: 3.480440 IN > Number 21 > > [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: 9.3774 comp: 4.295482 err: 3.557008 OUT > Number 22 > > [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: 9.5280 comp: 4.891080 err: 2.900537 OUT > Number 23 > > [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: 9.6275 comp: 5.168119 err: 2.624742 IN > Number 24 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: 9.7206 comp: 6.309298 err: 1.171542 IN > 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 OUT > Number 26 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: 9.8915 comp: 5.162806 err: 2.895855 IN > Number 27 -- 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: 10.0002 comp: 4.619353 err: 3.740932 OUT > Number 28 > > [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: 10.0368 comp: 4.075900 err: 4.513934 IN > Number 29 > > [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: 10.1077 comp: 5.047438 err: 3.268439 IN > Number 30 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: 10.1851 comp: 2.173813 err: 7.239629 OUT > Number 31 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: 10.1855 comp: 6.742251 err: 1.049791 OUT > Number 32 > > [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: 10.2131 comp: 6.411729 err: 1.525246 OUT > Number 33 > > [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: 10.2154 comp: 5.369353 err: 2.939961 OUT > Number 34 > > [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: 10.2188 comp: 5.168119 err: 3.215955 OUT > Number 35 > > [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: 10.3332 comp: 3.445412 err: 5.664628 OUT > Number 36 > > [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: 10.4461 comp: 3.804173 err: 5.291448 OUT > Number 37 > > [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: 10.4972 comp: 4.075900 err: 4.974313 OUT > Number 38 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: 10.5077 comp: 6.742251 err: 1.371918 OUT > Number 39 > > [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: 10.6719 comp: 5.706260 err: 2.939961 OUT > Number 40 > > [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: 10.7036 comp: 3.820609 err: 5.526647 OUT > Number 41 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: 10.7079 comp: 6.966993 err: 1.267597
Message: 10932 - Contents - Hide Contents Date: Tue, 11 May 2004 22:18:53 Subject: Re: Request for Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Very interesting. Now, how about: > > [3, 12, 11, 12, 9, -8] (Gawel?)TM basis: {81/80, 686/675} commas: [1029/1000, 686/675, 81/80, 177147/175616, 10976/10935]> [6, 10, 3, 2, -12, -21]TM basis: {49/48, 250/243} commas: [282475249/262144000, 250/243, 49/48, 4000/3969]> [2, -9, -4, -19, -12, 16]TM basis: {64/63, 686/675} commas: [524288/492075, 6272/6075, 686/675, 64/63, 2401/2400}> The idea is that I'm moving my boundary out to > error/10 + complexity/24 < 1Moving to a fixed exponent after all!> This adds these three 7-limit temperaments, for a total of 28. The > number of 5-limit temperaments remains at 21. If I include > ennealimmal as a "bonus" temperament, that's 50 altogether -- "Fifty > Temperaments" will make a nice subtitle for the paper. 10, 24, and 50 > are all nice, round numbers. Finality approaches.I think ennealimmal is so striking that discussing it anyway makes sense--while your exponent favors low complexity over low error, there is a school of thought which would favor low error, and even interest itself in microtempering. From that point of view the Heavenly Hemis (hemiwuerschmidt and hemififths) are also interesting, but ennealimmal makes the point that we can get effective JI while making use of useful approximations in scales of Partchian dimensions.
Message: 10933 - Contents - Hide Contents Date: Tue, 11 May 2004 19:29:33 Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> IN>> Number 17 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: 9.1819 comp: 5.522763 err: 1.698521This appears, to the best of my fading recollection, to be the temperament behind Andrzej Gawel's 19-of-36-equal scale. Does anyone have the Mills tuning list archives? Robert Walker only made six or so members' posts public: Mills messages - Contents * [with cont.] (Wayb.) If anyone has more, I'd love to see the results of a search for "Gawel".
Message: 10934 - Contents - Hide Contents Date: Tue, 11 May 2004 22:23:52 Subject: Re: Request for Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> [3, 12, 11, 12, 9, -8] (Gawel?)Gawel is OK by me.> [6, 10, 3, 2, -12, -21]I don't have a name for this one.> [2, -9, -4, -19, -12, 16]This I called beatles, on the highly questionable grounds that 19/64 is a poptimal generator. Does that suit you?
Message: 10935 - Contents - Hide Contents Date: Tue, 11 May 2004 19:55:55 Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments) From: Carl Lumma>>> >umber 17 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: 9.1819 comp: 5.522763 err: 1.698521 >>This appears, to the best of my fading recollection, to be the >temperament behind Andrzej Gawel's 19-of-36-equal scale. Does >anyone have the Mills tuning list archives? Robert Walker only >made six or so members' posts public: > >Mills messages - Contents * [with cont.] (Wayb.) > >If anyone has more, I'd love to see the results of a search >for "Gawel".Some months back I ganked all the stuff on the mills site I could find. It isn't much, and the string "gawel" apparently doesn't appear within. However, I did find this in my inbox... """>>> One thing your example reminds me of is Andrzej Gawel's >>> 19-of-36-tET scale. Gawel ingeniously took the 7-of-12-tET >>> diatonic scale and divided each of the six instances of the >>> generator, 7/12 oct. = 19/12 oct., into a chain of three >>> sub-generators, 19/36 oct., allowing all six of the ordinary >>> diatonic triads to be completed as 7-limit tetrads, >>> and in fact the scale has 14 7-limit tetrads. >>>> Wow. Paul, is this right? >> 0 4 5 6 7 11 12 13 14 18 19 20 21 26 27 28 33 34 35 > >No, it's >>0 2 4 6 8 10 12 14 16 18 19 21 23 25 27 29 31 33 35 """ -Carl
Message: 10936 - Contents - Hide Contents Date: Tue, 11 May 2004 20:24:38 Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:>>>> Number 17 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: 9.1819 comp: 5.522763 err: 1.698521 >>>> This appears, to the best of my fading recollection, to be the >> temperament behind Andrzej Gawel's 19-of-36-equal scale. Does >> anyone have the Mills tuning list archives? Robert Walker only >> made six or so members' posts public: >> >> Mills messages - Contents * [with cont.] (Wayb.) >> >> If anyone has more, I'd love to see the results of a search >> for "Gawel". >> Some months back I ganked all the stuff on the mills site I could > find. It isn't much, and the string "gawel" apparently doesn't > appear within.No, it's too early.> However, I did find this in my inbox... > > """>>>> One thing your example reminds me of is Andrzej Gawel's >>>> 19-of-36-tET scale. Gawel ingeniously took the 7-of-12-tET >>>> diatonic scale and divided each of the six instances of the >>>> generator, 7/12 oct. = 19/12 oct., into a chain of three >>>> sub-generators, 19/36 oct., allowing all six of the ordinaryHmm . . . so it's not quite the same. Gawel might be this one, though: OUT> Number 23 > > [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: 9.6275 comp: 5.168119 err: 2.624742 Thanks, Carl!>>>> diatonic triads to be completed as 7-limit tetrads, >>>> and in fact the scale has 14 7-limit tetrads. >>>>>> Wow. Paul, is this right? >>> 0 4 5 6 7 11 12 13 14 18 19 20 21 26 27 28 33 34 35 >> >> No, it's >>>> 0 2 4 6 8 10 12 14 16 18 19 21 23 25 27 29 31 33 35 > """ > > -Carl
Message: 10937 - Contents - Hide Contents Date: Tue, 11 May 2004 21:01:12 Subject: Re: Request for Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Thanks again for this, Gene. >> >> >> Would it be too much trouble to also do >> >> [8, 6, 6, -9, -13, -3] >> 390625/373248, 5971968/5764801, 50/49, 875/864, 1728/1715 > >> and >>>> [6, -2, -2, -17, -20, 1] >> 140625/131072, 525/512, 50/49, 1029/1024Very interesting. Now, how about: [3, 12, 11, 12, 9, -8] (Gawel?) [6, 10, 3, 2, -12, -21] [2, -9, -4, -19, -12, 16] The idea is that I'm moving my boundary out to error/10 + complexity/24 < 1 This adds these three 7-limit temperaments, for a total of 28. The number of 5-limit temperaments remains at 21. If I include ennealimmal as a "bonus" temperament, that's 50 altogether -- "Fifty Temperaments" will make a nice subtitle for the paper. 10, 24, and 50 are all nice, round numbers. Finality approaches. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 10938 - Contents - Hide Contents Date: Wed, 12 May 2004 23:52:50 Subject: Re: Adding wedgies? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> What's going on here? > > [5, 13, -17, 9, -41, -76], TOP error 0.27611 > "plus" > [13, 14, 35, -8, 19, 42], TOP error 0.26193 > "equals" > [18, 27, 18, 1, -22, -34], TOP error 0.036378Parakleismic + Amity = Ennealimmal. Both parakleismic and amity have 4375/4374 as a comma, and so does their sum (and difference, for that matter.) I did talk about it before, though I can't recall what I said about it. It is related to the Klein stuff. For 7-limit wedgies, define the Pfaffian as follows: let X = <<x1 x2 x3 x4 x5 x6|| Y = <<y1 y2 y3 y4 y5 y6|| Then Pf(X, Y) = y1x6 + x1y6 - y2x5 - x2y5 + y3x4 + x3y4 It is easily checked that we have the identity Pf(X+Y, X+Y) = Pf(X, X) + 2 Pf(X, Y) + Pf(Y, Y) The Klein condition for the wedgie X is Pf(X, X)=0. If X and Y both satisfy the Klein condition, and if Pf(X, Y)=0, then X+Y also satisfies the Klein condition, and hence is a wedgie. What Pf(X, Y)=0 means is that X and Y are related; they share a comma. Probably, you will not want to talk about this in the paper. :)
Message: 10939 - Contents - Hide Contents Date: Wed, 12 May 2004 05:22:41 Subject: Re: Request for Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> Very interesting. Now, how about: >> >> [3, 12, 11, 12, 9, -8] (Gawel?) >> TM basis: {81/80, 686/675} > commas: [1029/1000, 686/675, 81/80, 177147/175616, 10976/10935] >>> [6, 10, 3, 2, -12, -21] >> TM basis: {49/48, 250/243} > commas: [282475249/262144000, 250/243, 49/48, 4000/3969] >>> [2, -9, -4, -19, -12, 16] >> TM basis: {64/63, 686/675} > commas: [524288/492075, 6272/6075, 686/675, 64/63, 2401/2400} >>> The idea is that I'm moving my boundary out to > >> error/10 + complexity/24 < 1 >> Moving to a fixed exponent after all!Huh? How does changing 23.5 to 24 make it a fixed exponent?>> This adds these three 7-limit temperaments, for a total of 28. The >> number of 5-limit temperaments remains at 21. If I include >> ennealimmal as a "bonus" temperament, that's 50 altogether -- "Fifty >> Temperaments" will make a nice subtitle for the paper. 10, 24, and 50 >> are all nice, round numbers. Finality approaches. >> I think ennealimmal is so striking that discussing it anyway makes > sense--while your exponent favors low complexity over low error,I have no idea what you're talking about. What do you mean?
Message: 10940 - Contents - Hide Contents Date: Wed, 12 May 2004 06:38:33 Subject: Re: Request for Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Huh? How does changing 23.5 to 24 make it a fixed exponent?Sorry, I was thinking logs, but you mean the no logs.
Message: 10941 - Contents - Hide Contents Date: Wed, 12 May 2004 20:17:15 Subject: Re: Request for Gene From: Paul Erlich How do you calculate if a given comma vanishes in the temperament represented by a given wedgie?
Message: 10942 - Contents - Hide Contents Date: Wed, 12 May 2004 20:55:50 Subject: Adding wedgies? From: Paul Erlich What's going on here? [5, 13, -17, 9, -41, -76], TOP error 0.27611 "plus" [13, 14, 35, -8, 19, 42], TOP error 0.26193 "equals" [18, 27, 18, 1, -22, -34], TOP error 0.036378 I think Gene talked about this before but I didn't quite catch on then. BTW, these are the three most accurate 7-limit temperaments with L1 multival complexity less than 40 (though it's over 38 for each of the three).
Message: 10943 - Contents - Hide Contents Date: Wed, 12 May 2004 21:09:24 Subject: Re: Adding wedgies? From: Carl Lumma>What's going on here? > >[5, 13, -17, 9, -41, -76], TOP error 0.27611 >"plus" >[13, 14, 35, -8, 19, 42], TOP error 0.26193 >"equals" >[18, 27, 18, 1, -22, -34], TOP error 0.036378 > >I think Gene talked about this before but I didn't quite catch on >then. > >BTW, these are the three most accurate 7-limit temperaments with >L1 multival complexity less than 40 (though it's over 38 for each >of the three).Boy, Paul, I am sure looking forward to your paper!! This reminds me that my 'tuning-math forms' project is in limbo. I have all the materials collected... just gotta find time to do it... (if anybody wants to take a stab I'll make the materials available...) -Carl
Message: 10944 - Contents - Hide Contents Date: Wed, 12 May 2004 21:15:47 Subject: Re: Adding wedgies? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:>> What's going on here? >> >> [5, 13, -17, 9, -41, -76], TOP error 0.27611 >> "plus" >> [13, 14, 35, -8, 19, 42], TOP error 0.26193 >> "equals" >> [18, 27, 18, 1, -22, -34], TOP error 0.036378 >> >> I think Gene talked about this before but I didn't quite catch on >> then. >> >> BTW, these are the three most accurate 7-limit temperaments with >> L1 multival complexity less than 40 (though it's over 38 for each >> of the three). >> Boy, Paul, I am sure looking forward to your paper!!Other than ennealimmal, which is a "bonus" temperament, my paper stays below complexity < 24 -- and I doubt I'll be talking about adding wedgies. So don't get your hopes up.> This reminds me that my 'tuning-math forms' project is in limbo. What's that?
Message: 10945 - Contents - Hide Contents Date: Wed, 12 May 2004 22:01:06 Subject: Re: Adding wedgies? From: Carl Lumma>> >oy, Paul, I am sure looking forward to your paper!! >>Other than ennealimmal, which is a "bonus" temperament, my paper >stays below complexity < 24 -- and I doubt I'll be talking about >adding wedgies. So don't get your hopes up.Oh, I suspected as much. I'm looking forward to it for other reasons!>> This reminds me that my 'tuning-math forms' project is in limbo. > >What's that?Remember, the one where I show how to calculate this stuff on paper, 'long division' style? By "materials" I just meant the relevant posts from you, Gene, and Dave. If Gene answers this thread it will probably get included as well. -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 10946 - Contents - Hide Contents Date: Thu, 13 May 2004 21:01:13 Subject: Re: 22 7-limit temperaments in the upper uv quadrant From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> We were so busy arguing that we didn't notice how very close our >> lists were. >> It's hardly the case that I didn't notice that.Well, that makes a lot more sense.> Do you began to > understand why I became so frustrated? I still can't figure out why > it all blew up the way it did; normally, we communicate better than > this. >>> So what exactly led to your results here? I can't understand how > you>> arrived at them. >> My idea was to get something closer to what people seemed to want, > two exponents could be used; this could be smoothed out if we used a > hyperbolic boundry in the log-log plane. In order to accomodate two > exponents, making them the vertical and horizontal axis seemed like a > good plan. This posting follows up a previous one which explains all > of that.I still don't see the intuition behind it. Could you draw some pictures to help?
Message: 10947 - Contents - Hide Contents Date: Thu, 13 May 2004 00:23:25 Subject: Re: Adding wedgies? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> What's going on here? >> >> [5, 13, -17, 9, -41, -76], TOP error 0.27611 >> "plus" >> [13, 14, 35, -8, 19, 42], TOP error 0.26193 >> "equals" >> [18, 27, 18, 1, -22, -34], TOP error 0.036378 >> Parakleismic + Amity = Ennealimmal. Both parakleismic and amity have > 4375/4374 as a comma, and so does their sum (and difference, for that > matter.) > > I did talk about it before, though I can't recall what I said about > it. It is related to the Klein stuff. For 7-limit wedgies, define the > Pfaffian as follows: let > > X = <<x1 x2 x3 x4 x5 x6|| > Y = <<y1 y2 y3 y4 y5 y6|| > > Then > > Pf(X, Y) = y1x6 + x1y6 - y2x5 - x2y5 + y3x4 + x3y4 > > It is easily checked that we have the identity > > Pf(X+Y, X+Y) = Pf(X, X) + 2 Pf(X, Y) + Pf(Y, Y) > > The Klein condition for the wedgie X is Pf(X, X)=0. If X and Y both > satisfy the Klein condition, and if Pf(X, Y)=0, then X+Y also > satisfies the Klein condition, and hence is a wedgie. What > Pf(X, Y)=0 means is that X and Y are related; they share a comma.It seems below that the unshared commas, correspondingly, "add" (actually multiply). But how do you simply check whether a particular wedgie eats a particular comma? 81:80 shared: Meantone + DominantSevenths = Injera 126:125 "+" 35:36 "=" 49:50 Meantone + Catler = DominantSevenths 126:125 "+" 125:128 "=" 63:64 Meantone + Flattone = Semifourths 225:224 "+" 512:525 "=" 48:49 126:125 "+" 4375:4374 "=" 245:243 126:125 "+" 875:846 "=" 49:48 Injera + Flattone = SupermajorSeconds 50:49 "+" 512:525 "=" 1024:1029 50:49 "+" 864:875 "=" 1728:1715 etc.
Message: 10949 - Contents - Hide Contents Date: Thu, 13 May 2004 03:40:28 Subject: Re: Vanishing tratios From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> Hmm . . . the two formulae seem to give the same result as long as gcd >> (a,b,c) = 1, which was the case for all the tratios in question. >> Right? > > Right.So how can we express the tratio, or the lcm, in terms of the wedgie?
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