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Message: 10300 - Contents - Hide Contents Date: Sat, 14 Feb 2004 21:23:56 Subject: Re: A modest proposal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> What about using Herman's phenomenon as a high-error cutoff?Works for linears -- but for other kinds of temperament?
Message: 10301 - Contents - Hide Contents Date: Sat, 14 Feb 2004 15:32:55 Subject: Re: A symmetric-based 7-limit temperament list From: Herman Miller This ordering seems to be good at keeping similar/related temperaments together. It's missing pelogic, injera, and dicot, though. I can understand why pelogic and dicot might be missing, but injera [2, 8, 8, 8, 7, -4] is a good enough temperament that it should have made the list. Gene Ward Smith wrote:> 11: [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]] > rms: 43.659491 symcom: 35.000000 symbad: 1528.082200Number 13 Father TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]> 12: [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]] > rms: 43.142169 symcom: 44.000000 symbad: 1898.255432 Number 62TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105] Audibly very similar to Number 13, and has a simpler mapping.> 13: [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]] > rms: 41.524693 symcom: 35.000000 symbad: 1453.364254 Number 57TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203] Another member of the father temperament family, but the 7:1 approximation is worse than Number 13, and the 7:4 is unrecognizable.> 14: [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] > rms: 34.566097 symcom: 20.000000 symbad: 691.321943Number 4 Beep TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]> 15: [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]] > rms: 23.945252 symcom: 32.000000 symbad: 766.248055Number 32 Decimal TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]> 16: [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] > rms: 20.163282 symcom: 75.000000 symbad: 1512.246136Number 7 Dominant Seventh TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]> 17: [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]] > rms: 19.136993 symcom: 48.000000 symbad: 918.575644Number 17 Diminished TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]> 18: [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]] > rms: 18.042924 symcom: 108.000000 symbad: 1948.635783 Number 85TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166] Would make a good 12-note keyboard mapping. There aren't many temperaments based on 1/6-octave periods; this is the first one I've seen.> 19: [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]] > rms: 16.786584 symcom: 99.000000 symbad: 1661.871769 Number 75TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936] No simpler than Augmented, but sounds a bit more warped.> 20: [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] > rms: 16.598678 symcom: 99.000000 symbad: 1643.269152Number 5 Augmented TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]> 21: [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]] > rms: 15.815352 symcom: 75.000000 symbad: 1186.151431Number 14 Blackwood TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]> 22: [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]] > rms: 12.690078 symcom: 155.000000 symbad: 1966.962143Number 24 Hemifourths TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]> 23: [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]] > rms: 12.273810 symcom: 84.000000 symbad: 1031.000003Number 27 Kleismic TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]> 24: [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]] > rms: 12.188571 symcom: 107.000000 symbad: 1304.177049Number 28 Negri TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]> 25: [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] > rms: 10.903177 symcom: 108.000000 symbad: 1177.543168Number 6 Pajara TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]> 26: [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]] > rms: 10.132266 symcom: 144.000000 symbad: 1459.046340 Number 92TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574] Seems to be an alternate 22-ET-type temperament, not as good as Pajara.> 27: [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] > rms: 8.100679 symcom: 171.000000 symbad: 1385.216092Number 31 Tripletone TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]> 28: [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]] > rms: 6.808962 symcom: 276.000000 symbad: 1879.273474Number 42 Porcupine TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]> 29: [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]] > rms: 6.410458 symcom: 280.000000 symbad: 1794.928214Number 34 Superpythagorean TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]> 30: [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]] > rms: 6.245316 symcom: 283.000000 symbad: 1767.424344Number 79 Beatles TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]> 31: [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]] > rms: 5.052932 symcom: 355.000000 symbad: 1793.790776Number 15 Semisixths TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]> 32: [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] > rms: 4.139051 symcom: 356.000000 symbad: 1473.502082Number 3 Magic TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]> 33: [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]] > rms: 4.006991 symcom: 436.000000 symbad: 1747.048215Not in the 114 list. Seems overly complex to be of much use.> 34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] > rms: 3.665035 symcom: 243.000000 symbad: 890.603432Almost goes without saying, but.... Number 2 Meantone TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]> 35: [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]] > rms: 3.579262 symcom: 420.000000 symbad: 1503.290125Number 35 Supermajor seconds TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]> 36: [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]] > rms: 3.443812 symcom: 571.000000 symbad: 1966.416662Number 84 Squares TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656] Sounds practically identical to Number 35, but with a more complex mapping.> 37: [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] > rms: 3.320167 symcom: 244.000000 symbad: 810.120816Number 29 Nonkleismic TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]> 38: [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]] > rms: 3.065962 symcom: 339.000000 symbad: 1039.361092Number 30 Quartaminorthirds TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]> 39: [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]] > rms: 2.859338 symcom: 603.000000 symbad: 1724.180520Number 8 Schismic TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]> 40: [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] > rms: 2.589237 symcom: 344.000000 symbad: 890.697699Number 10 Orwell TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]> 41: [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]] > rms: 2.469727 symcom: 756.000000 symbad: 1867.113518Now we're starting to get into temperaments that are mostly too complex to be of much interest. This is Number 66 from the big list, and doesn't seem to be enough better than Orwell to justify its complexity. I'll skip most of the rest.> 45: [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] > rms: 1.637405 symcom: 347.000000 symbad: 568.179603Number 9 Miracle TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756] TOP generators [1200.631014, 116.7206423] bad: 29.119472 comp: 6.793166 err: .631014> 49: [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]] > rms: .875363 symcom: 611.000000 symbad: 534.846775Number 11 Hemiwuerschmidt TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]> 76: [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]] > rms: .130449 symcom: 1539.000000 symbad: 200.760896Number 1 Ennealimmal TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
Message: 10302 - Contents - Hide Contents Date: Sat, 14 Feb 2004 21:41:16 Subject: Re: A symmetric-based 7-limit temperament list From: Paul Erlich Injera involves two long chains of fifths, and fifths are just as long as any other consonance in the symmetric lattice Gene used here. In a 5-limit version of this list, 2187;2048 would surely score quite poorly because of the long chain of fifths it involves. --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> This ordering seems to be good at keeping similar/related temperaments > together. It's missing pelogic, injera, and dicot, though. I can > understand why pelogic and dicot might be missing, but injera [2, 8, 8, > 8, 7, -4] is a good enough temperament that it should have made the list. > > Gene Ward Smith wrote: >>> 11: [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]] >> rms: 43.659491 symcom: 35.000000 symbad: 1528.082200 >> Number 13 Father > TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477] >>> 12: [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]] >> rms: 43.142169 symcom: 44.000000 symbad: 1898.255432 > > Number 62> TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105] > Audibly very similar to Number 13, and has a simpler mapping. >>> 13: [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]] >> rms: 41.524693 symcom: 35.000000 symbad: 1453.364254 > > Number 57> TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203] > Another member of the father temperament family, but the 7:1 > approximation is worse than Number 13, and the 7:4 is unrecognizable. >>> 14: [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] >> rms: 34.566097 symcom: 20.000000 symbad: 691.321943 >> Number 4 Beep > TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548] >>> 15: [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]] >> rms: 23.945252 symcom: 32.000000 symbad: 766.248055 >> Number 32 Decimal > TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757] >>> 16: [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] >> rms: 20.163282 symcom: 75.000000 symbad: 1512.246136 >> Number 7 Dominant Seventh > TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933] >>> 17: [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]] >> rms: 19.136993 symcom: 48.000000 symbad: 918.575644 >> Number 17 Diminished > TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404] >>> 18: [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]] >> rms: 18.042924 symcom: 108.000000 symbad: 1948.635783 > > Number 85> TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166] > Would make a good 12-note keyboard mapping. There aren't many > temperaments based on 1/6-octave periods; this is the first one I've seen. >>> 19: [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]] >> rms: 16.786584 symcom: 99.000000 symbad: 1661.871769 > > Number 75> TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936] > No simpler than Augmented, but sounds a bit more warped. >>> 20: [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] >> rms: 16.598678 symcom: 99.000000 symbad: 1643.269152 >> Number 5 Augmented > TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546] >>> 21: [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]] >> rms: 15.815352 symcom: 75.000000 symbad: 1186.151431 >> Number 14 Blackwood > TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698] >>> 22: [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]] >> rms: 12.690078 symcom: 155.000000 symbad: 1966.962143 >> Number 24 Hemifourths > TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166] >>> 23: [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]] >> rms: 12.273810 symcom: 84.000000 symbad: 1031.000003 >> Number 27 Kleismic > TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000] >>> 24: [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]] >> rms: 12.188571 symcom: 107.000000 symbad: 1304.177049 >> Number 28 Negri > TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000] >>> 25: [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] >> rms: 10.903177 symcom: 108.000000 symbad: 1177.543168 >> Number 6 Pajara > TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174] >>> 26: [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]] >> rms: 10.132266 symcom: 144.000000 symbad: 1459.046340 > > Number 92> TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574] > Seems to be an alternate 22-ET-type temperament, not as good as Pajara. >>> 27: [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] >> rms: 8.100679 symcom: 171.000000 symbad: 1385.216092 >> Number 31 Tripletone > TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420] >>> 28: [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]] >> rms: 6.808962 symcom: 276.000000 symbad: 1879.273474 >> Number 42 Porcupine > TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888] >>> 29: [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]] >> rms: 6.410458 symcom: 280.000000 symbad: 1794.928214 >> Number 34 Superpythagorean > TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608] >>> 30: [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]] >> rms: 6.245316 symcom: 283.000000 symbad: 1767.424344 >> Number 79 Beatles > TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226] >>> 31: [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]] >> rms: 5.052932 symcom: 355.000000 symbad: 1793.790776 >> Number 15 Semisixths > TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748] >>> 32: [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] >> rms: 4.139051 symcom: 356.000000 symbad: 1473.502082 >> Number 3 Magic > TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877] >>> 33: [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]] >> rms: 4.006991 symcom: 436.000000 symbad: 1747.048215 >> Not in the 114 list. Seems overly complex to be of much use. >>> 34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] >> rms: 3.665035 symcom: 243.000000 symbad: 890.603432 >> Almost goes without saying, but.... > Number 2 Meantone > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328] >>> 35: [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]] >> rms: 3.579262 symcom: 420.000000 symbad: 1503.290125 >> Number 35 Supermajor seconds > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099] >>> 36: [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]] >> rms: 3.443812 symcom: 571.000000 symbad: 1966.416662 >> Number 84 Squares > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656] > Sounds practically identical to Number 35, but with a more complex mapping. >>> 37: [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] >> rms: 3.320167 symcom: 244.000000 symbad: 810.120816 >> Number 29 Nonkleismic > TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085] >>> 38: [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]] >> rms: 3.065962 symcom: 339.000000 symbad: 1039.361092 >> Number 30 Quartaminorthirds > TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770] >>> 39: [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]] >> rms: 2.859338 symcom: 603.000000 symbad: 1724.180520 >> Number 8 Schismic > TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750] >>> 40: [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] >> rms: 2.589237 symcom: 344.000000 symbad: 890.697699 >> Number 10 Orwell > TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834] >>> 41: [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]] >> rms: 2.469727 symcom: 756.000000 symbad: 1867.113518 >> Now we're starting to get into temperaments that are mostly too complex > to be of much interest. This is Number 66 from the big list, and doesn't > seem to be enough better than Orwell to justify its complexity. I'll > skip most of the rest. >>> 45: [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] >> rms: 1.637405 symcom: 347.000000 symbad: 568.179603 >> Number 9 Miracle > TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756] > TOP generators [1200.631014, 116.7206423] > bad: 29.119472 comp: 6.793166 err: .631014 >>> 49: [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]] >> rms: .875363 symcom: 611.000000 symbad: 534.846775 >> Number 11 Hemiwuerschmidt > TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143] >>> 76: [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]] >> rms: .130449 symcom: 1539.000000 symbad: 200.760896 >> Number 1 Ennealimmal > TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
Message: 10303 - Contents - Hide Contents Date: Sat, 14 Feb 2004 00:00:25 Subject: Re: The same page From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Erlich magic L1 norm; if >>>> >>>> <<a1 a2 a3 a4 a5 a6|| >>>> >>>> is the wedgie, then complexity is >>>> >>>> |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7| >>>>>> Where wedgie is val-wedgie. But apparently there's a monzo- wedgie >>> formualation... >>>> Simply reverse the order of the entries. >> Not sure what you're saying. > > monzo-wedgie = reverse(val-wedgie)Up to some of the signs, yes. Since the above expression for complexity takes the absolute values anyway, you don't have to worry about the signs. The point is that the complexity you end up calculating is the same. We could write If ||a6 a5 a4 a3 a2 a1>> is the monzo-wedgie, then the complexity is |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7| and we'd be getting the same answer as above.
Message: 10304 - Contents - Hide Contents Date: Sat, 14 Feb 2004 00:15:03 Subject: Re: The same page From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> I was planning on getting around to it. >> You were? Oh, sorry . . .I guess I was according it about the same urgency you give to my requests. :)
Message: 10305 - Contents - Hide Contents Date: Sat, 14 Feb 2004 00:15:22 Subject: Re: The same page From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> How am I ever going to find these posts of Dave's to get to > > | a b c > ~= | -b c a > > > or whatever? Try: Yahoo groups: /tuning-math/message/7852 * [with cont.]
Message: 10306 - Contents - Hide Contents Date: Sat, 14 Feb 2004 00:18:08 Subject: Re: The same page From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> How am I ever going to find these posts of Dave's to get to >> >> | a b c > ~= | -b c a > >> >> or whatever? > > Try: > Yahoo groups: /tuning-math/message/7852 * [with cont.]Thanks Dave, for looking in even after you've gone away!
Message: 10307 - Contents - Hide Contents Date: Sat, 14 Feb 2004 00:21:21 Subject: Re: Still another From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I don't believe in Dave's > quantization of agony theory,That's "quantification", not "quantization". These are very different things.
Message: 10308 - Contents - Hide Contents Date: Sat, 14 Feb 2004 04:04:31 Subject: A symmetric-based 7-limit temperament list From: Gene Ward Smith This is not a proposed list for a paper, nor even a starting point for such a list, so I used a complexity bound (set high) and a badness bound. The starting point was my big list of 32000-odd wedgies; the complexity bound was a symmetric complexity squared of 15000 and the badness bound was a symmetric badness of 2000. The results are sorted by rms error, and no error bound was set, so you might want to skip down to about number 15 to get an idea of how things worked. I list rms error, squared symmetric complexity, and symmetric badness. Since there seemed no point in taking the square root just to square it again, the badness is just the rms error times the squared symmetric complexity, which is an integer. This complexity measure, or else whatever we would get as the dual to Hahn taxicab distance, seem to be the logical ones to use when we are using symmetric, octave equivalent rms error. Since that has a history going back to Woolhouse and 7/26 comma meantone, it seems to me to be of interest. 1: [1, 1, 0, -1, -3, -3] [[1, 2, 3, 3], [0, -1, -1, 0]] rms: 225.884103 symcom: 4.000000 symbad: 903.536412 2: [1, 2, 1, 1, -1, -3] [[1, 2, 3, 3], [0, -1, -2, -1]] rms: 157.889659 symcom: 8.000000 symbad: 1263.117274 3: [1, -1, 1, -4, -1, 5] [[1, 2, 2, 3], [0, -1, 1, -1]] rms: 154.263172 symcom: 11.000000 symbad: 1696.894891 4: [1, -1, 0, -4, -3, 3] [[1, 2, 2, 3], [0, -1, 1, 0]] rms: 142.097096 symcom: 8.000000 symbad: 1136.776766 5: [1, -1, -2, -4, -6, -2] [[1, 2, 2, 2], [0, -1, 1, 2]] rms: 65.953083 symcom: 20.000000 symbad: 1319.061657 6: [0, 0, 3, 0, 5, 7] [[3, 5, 7, 9], [0, 0, 0, -1]] rms: 61.312549 symcom: 27.000000 symbad: 1655.438815 7: [2, -1, 1, -6, -4, 5] [[1, 2, 2, 3], [0, -2, 1, -1]] rms: 59.930923 symcom: 20.000000 symbad: 1198.618460 8: [0, 2, 2, 3, 3, -1] [[2, 3, 5, 6], [0, 0, -1, -1]] rms: 59.723378 symcom: 16.000000 symbad: 955.574045 9: [2, 1, -1, -3, -7, -5] [[1, 1, 2, 3], [0, 2, 1, -1]] rms: 53.747748 symcom: 20.000000 symbad: 1074.954969 10: [2, 1, 3, -3, -1, 4] [[1, 1, 2, 2], [0, 2, 1, 3]] rms: 48.926006 symcom: 20.000000 symbad: 978.520120 11: [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]] rms: 43.659491 symcom: 35.000000 symbad: 1528.082200 12: [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]] rms: 43.142169 symcom: 44.000000 symbad: 1898.255432 13: [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]] rms: 41.524693 symcom: 35.000000 symbad: 1453.364254 14: [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] rms: 34.566097 symcom: 20.000000 symbad: 691.321943 15: [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]] rms: 23.945252 symcom: 32.000000 symbad: 766.248055 16: [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] rms: 20.163282 symcom: 75.000000 symbad: 1512.246136 17: [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]] rms: 19.136993 symcom: 48.000000 symbad: 918.575644 18: [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]] rms: 18.042924 symcom: 108.000000 symbad: 1948.635783 19: [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]] rms: 16.786584 symcom: 99.000000 symbad: 1661.871769 20: [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] rms: 16.598678 symcom: 99.000000 symbad: 1643.269152 21: [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]] rms: 15.815352 symcom: 75.000000 symbad: 1186.151431 22: [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]] rms: 12.690078 symcom: 155.000000 symbad: 1966.962143 23: [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]] rms: 12.273810 symcom: 84.000000 symbad: 1031.000003 24: [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]] rms: 12.188571 symcom: 107.000000 symbad: 1304.177049 25: [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] rms: 10.903177 symcom: 108.000000 symbad: 1177.543168 26: [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]] rms: 10.132266 symcom: 144.000000 symbad: 1459.046340 27: [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] rms: 8.100679 symcom: 171.000000 symbad: 1385.216092 28: [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]] rms: 6.808962 symcom: 276.000000 symbad: 1879.273474 29: [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]] rms: 6.410458 symcom: 280.000000 symbad: 1794.928214 30: [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]] rms: 6.245316 symcom: 283.000000 symbad: 1767.424344 31: [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]] rms: 5.052932 symcom: 355.000000 symbad: 1793.790776 32: [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] rms: 4.139051 symcom: 356.000000 symbad: 1473.502082 33: [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]] rms: 4.006991 symcom: 436.000000 symbad: 1747.048215 34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] rms: 3.665035 symcom: 243.000000 symbad: 890.603432 35: [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]] rms: 3.579262 symcom: 420.000000 symbad: 1503.290125 36: [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]] rms: 3.443812 symcom: 571.000000 symbad: 1966.416662 37: [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] rms: 3.320167 symcom: 244.000000 symbad: 810.120816 38: [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]] rms: 3.065962 symcom: 339.000000 symbad: 1039.361092 39: [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]] rms: 2.859338 symcom: 603.000000 symbad: 1724.180520 40: [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] rms: 2.589237 symcom: 344.000000 symbad: 890.697699 41: [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]] rms: 2.469727 symcom: 756.000000 symbad: 1867.113518 42: [8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]] rms: 2.064340 symcom: 667.000000 symbad: 1376.914655 43: [15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]] rms: 1.731230 symcom: 952.000000 symbad: 1648.130712 44: [13, -10, 6, -46, -27, 42] [[1, 2, 2, 3], [0, -13, 10, -6]] rms: 1.678518 symcom: 1139.000000 symbad: 1911.832046 45: [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] rms: 1.637405 symcom: 347.000000 symbad: 568.179603 46: [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]] rms: 1.610555 symcom: 1091.000000 symbad: 1757.115994 47: [23, -1, 13, -55, -44, 33] [[1, 9, 2, 7], [0, -23, 1, -13]] rms: 1.226222 symcom: 1571.000000 symbad: 1926.394008 48: [24, 20, 16, -24, -42, -19] [[4, 6, 9, 11], [0, 6, 5, 4]] rms: .881659 symcom: 1328.000000 symbad: 1170.842682 49: [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]] rms: .875363 symcom: 611.000000 symbad: 534.846775 50: [5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]] rms: .845880 symcom: 1931.000000 symbad: 1633.393513 51: [12, 34, 20, 26, -2, -49] [[2, 4, 7, 7], [0, -6, -17, -10]] rms: .600319 symcom: 2444.000000 symbad: 1467.178486 52: [2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]] rms: .585156 symcom: 1592.000000 symbad: 931.569106 53: [34, 29, 23, -33, -59, -28] [[1, -7, -5, -3], [0, 34, 29, 23]] rms: .404751 symcom: 2708.000000 symbad: 1096.066600 54: [20, 52, 31, 36, -7, -74] [[1, 3, 6, 5], [0, -20, -52, -31]] rms: .345464 symcom: 5651.000000 symbad: 1952.215876 55: [17, 35, -21, 16, -81, -147] [[1, -1, -3, 6], [0, 17, 35, -21]] rms: .255750 symcom: 6859.000000 symbad: 1754.190470 56: [22, -5, 3, -59, -57, 21] [[1, 3, 2, 3], [0, -22, 5, -3]] rms: .253343 symcom: 1672.000000 symbad: 423.589817 57: [52, 56, 41, -32, -81, -62] [[1, -21, -22, -15], [0, 52, 56, 41]] rms: .244554 symcom: 7883.000000 symbad: 1927.817350 58: [23, -13, 42, -74, 2, 134] [[1, 11, -3, 20], [0, -23, 13, -42]] rms: .239309 symcom: 7144.000000 symbad: 1709.625905 59: [20, -30, -10, -94, -72, 61] [[10, 16, 23, 28], [0, -2, 3, 1]] rms: .228948 symcom: 5200.000000 symbad: 1190.529406 60: [38, -3, 8, -93, -94, 27] [[1, -7, 3, 1], [0, 38, -3, 8]] rms: .228693 symcom: 4219.000000 symbad: 964.856656 61: [1, -8, 39, -15, 59, 113] [[1, 2, -1, 19], [0, -1, 8, -39]] rms: .223412 symcom: 5320.000000 symbad: 1188.553383 62: [40, 22, 21, -58, -79, -13] [[1, 21, 13, 13], [0, -40, -22, -21]] rms: .222189 symcom: 3211.000000 symbad: 713.449285 63: [26, -37, -12, -119, -92, 76] [[1, -1, 6, 4], [0, 26, -37, -12]] rms: .221987 symcom: 8227.000000 symbad: 1826.286511 64: [21, 3, -36, -44, -116, -92] [[3, 5, 7, 8], [0, -7, -1, 12]] rms: .221824 symcom: 6840.000000 symbad: 1517.273890 65: [2, -57, -28, -95, -50, 95] [[1, 1, 19, 11], [0, 2, -57, -28]] rms: .201747 symcom: 9259.000000 symbad: 1867.972277 66: [56, 24, 26, -92, -116, -7] [[2, 4, 5, 6], [0, -28, -12, -13]] rms: .187109 symcom: 6316.000000 symbad: 1181.780562 67: [41, 14, 60, -73, -20, 100] [[1, -14, -3, -20], [0, 41, 14, 60]] rms: .186938 symcom: 8683.000000 symbad: 1623.186237 68: [4, -32, -15, -60, -35, 55] [[1, 1, 7, 5], [0, 4, -32, -15]] rms: .183810 symcom: 3211.000000 symbad: 590.213786 69: [58, 49, 39, -57, -101, -47] [[1, -13, -10, -7], [0, 58, 49, 39]] rms: .182983 symcom: 7828.000000 symbad: 1432.388742 70: [74, 51, 44, -91, -138, -41] [[1, -25, -16, -13], [0, 74, 51, 44]] rms: .154407 symcom: 11491.000000 symbad: 1774.294552 71: [3, -24, -54, -45, -94, -58] [[3, 5, 5, 4], [0, -1, 8, 18]] rms: .146908 symcom: 8379.000000 symbad: 1230.939666 72: [14, 59, 33, 61, 13, -89] [[1, -3, -17, -8], [0, 14, 59, 33]] rms: .143876 symcom: 7828.000000 symbad: 1126.265088 73: [19, 19, 57, -14, 37, 79] [[19, 30, 44, 53], [0, 1, 1, 3]] rms: .140199 symcom: 6859.000000 symbad: 961.625009 74: [59, 41, 78, -72, -42, 66] [[1, 4, 4, 6], [0, -59, -41, -78]] rms: .137131 symcom: 13300.000000 symbad: 1823.841297 75: [42, -35, -7, -153, -129, 82] [[7, 9, 18, 20], [0, 6, -5, -1]] rms: .132906 symcom: 12152.000000 symbad: 1615.071114 76: [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]] rms: .130449 symcom: 1539.000000 symbad: 200.760896 77: [60, -8, 11, -152, -151, 48] [[1, 19, 0, 6], [0, -60, 8, -11]] rms: .129643 symcom: 11171.000000 symbad: 1448.244216 78: [78, 19, 29, -151, -173, 14] [[1, 29, 9, 13], [0, -78, -19, -29]] rms: .126772 symcom: 13268.000000 symbad: 1682.017076 79: [15, 51, 72, 46, 72, 24] [[3, 3, 1, 0], [0, 5, 17, 24]] rms: .077212 symcom: 12996.000000 symbad: 1003.453443 80: [37, 46, 75, -13, 15, 45] [[1, 15, 19, 30], [0, -37, -46, -75]] rms: .070153 symcom: 11476.000000 symbad: 805.075750
Message: 10309 - Contents - Hide Contents Date: Sat, 14 Feb 2004 04:44:39 Subject: Re: A symmetric-based 7-limit temperament list From: Gene Ward Smith Here is an abridged version of the aame list, where the badness is less than 900. This gets rid of everything with higher error than beep without setting an error bound. The three temperaments between hemiwuerschmidt and ennealimmal have come up before, but because of the widespread disdain for high complexity I've not named them. They are all well-convered by 171; ennealimmal is also but unlike with them, a continued fraction finds 441-et instead when using the rms generators. Beep 14: [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] rms: 34.566097 symcom: 20.000000 symbad: 691.321943 Decimal 15: [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]] rms: 23.945252 symcom: 32.000000 symbad: 766.248055 Meantone 34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] rms: 3.665035 symcom: 243.000000 symbad: 890.603432 Nonkleismic 37: [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] rms: 3.320167 symcom: 244.000000 symbad: 810.120816 Orwell 40: [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] rms: 2.589237 symcom: 344.000000 symbad: 890.697699 Miracle 45: [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] rms: 1.637405 symcom: 347.000000 symbad: 568.179603 Hemiwuerschmidt 49: [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]] rms: .875363 symcom: 611.000000 symbad: 534.846775 {2401/2400, 65625/65536} 11/171 56: [22, -5, 3, -59, -57, 21] [[1, 3, 2, 3], [0, -22, 5, -3]] rms: .253343 symcom: 1672.000000 symbad: 423.589817 {2401/2400, 48828125/48771072} 83/171 62: [40, 22, 21, -58, -79, -13] [[1, 21, 13, 13], [0, -40, -22, -21]] rms: .222189 symcom: 3211.000000 symbad: 713.449285 {2401/2400, 32805/32768} 25/171 68: [4, -32, -15, -60, -35, 55] [[1, 1, 7, 5], [0, 4, -32, -15]] rms: .183810 symcom: 3211.000000 symbad: 590.213786 Ennealimmal 76: [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]] rms: .130449 symcom: 1539.000000 symbad: 200.760896 {4375/4374, 52734375/52706752} 62/171 80: [37, 46, 75, -13, 15, 45] [[1, 15, 19, 30], [0, -37, -46, -75]] rms: .070153 symcom: 11476.000000 symbad: 805.075750
Message: 10310 - Contents - Hide Contents Date: Sat, 14 Feb 2004 04:46:27 Subject: Re: A symmetric-based 7-limit temperament list From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> squared symmetric >> complexity, which is an integer. This complexity measure, or else >> whatever we would get as the dual to Hahn taxicab distance, >> What is the difference between symmetric complexity and Hahn > taxicab distance?They aren't measuring the same thing. You need to compare symmetric distance and Hahn distance, or symmetric complexity and Hahn-dual complexity. Since the former are similar, the latter will be also.
Message: 10311 - Contents - Hide Contents Date: Sat, 14 Feb 2004 04:54:47 Subject: Re: Dicot and "Number 56" From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> I'm wondering why "number 56" and "hexidecimal" are so far down the > list, if they're clearly better than the ones labeled "dicot" and > "pelogic"?They are more complex. What happens is that for these low-complexity temperaments you don't go out very far before running into the terminus of the Miller-consistent region.
Message: 10312 - Contents - Hide Contents Date: Sat, 14 Feb 2004 05:07:23 Subject: A modest proposal From: Gene Ward Smith What about using Herman's phenomenon as a high-error cutoff? That way we get something we can rationally justify, and would only need a rational justification for a badness exponent and cutoff to settle the whole thing. One way to get such an exponent would be using the convex hull. The cutoff justification could be moats, or how many items we want on the list. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10313 - Contents - Hide Contents Date: Sun, 15 Feb 2004 16:43:04 Subject: Symmetric complexity of 7-limit commas From: Gene Ward Smith Here is the list of 7-limit commas with relative error < 0.06 and epimericity < 0.5, sorted by what shell they belong to--or in other words, symmetric lattice error. Two commas belonging to the same shell can be geometrically isomorphic, and will be if two other invariants are equal. I've mentioned how this leads to isomorphisms of planar temperaments, but it also leads to automorphisms of the associated linear temperaments. Linear temperaments with such automorphisms include: Beep: 49/48 ^ 36/35 shell 3 Tripletone: 126/125 ^ 64/63 shell 7 Blackwood: 64/63 ^ 28/27 shell 7 Dominant seventh 256/245 ^ 64/63 shell 7 Diminished (torsional): (126/125 ^ 360/343)/2 "Number 59": 28/27 ^ 126/125 shell 7 "Number 92": 1728/1715 ^ 875/864 shell 10 Kleismic (torsional): (875/864 ^ 1029/1000)/2 shell 10 Supermajor seconds: 1029/1024 ^ 81/80 shell 13 Jamesbond: 81/80 ^ 135/128 shell 13 <5 -4 -10 -18 -30 -12| 3136/3125 ^ 3125/3087 shell 19 <9 15 19 3 5 2| 3125/3087 ^ 250/243 shell 19 <6 10 25 2 23 30| 250/243 ^ 3136/3125 shell 19 3: {36/35, 49/48} 4: {50/49} 7: {28/27, 360/343, 64/63, 126/125, 256/245} 9: {200/189, 392/375, 128/125, 225/224} 10: {1029/1000, 875/864, 1728/1715} 11: {2401/2400, 525/512} 13: {81/80, 135/128, 686/675, 1029/1024} 15: {405/392, 6144/6125} 16: {648/625} 17: {245/243} 19: {250/243, 3125/3087, 4000/3969, 3136/3125} 21: {3125/3072} 23: {2430/2401} 25: {256/243} 27: {16875/16807} 28: {2048/2025} 31: {15625/15552} 35: {4375/4374} 37: {5120/5103} 38: {65625/65536} 42: {10976/10935} 45: {250047/250000} 47: {420175/419904} 49: {703125/702464} 57: {19683/19600} 73: {32805/32768} 149: {78125000/78121827}
Message: 10314 - Contents - Hide Contents Date: Sun, 15 Feb 2004 14:58:23 Subject: Re: A symmetric-based 7-limit temperament list From: Herman Miller Gene Ward Smith wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote: >> Number 62>> TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105] >> Audibly very similar to Number 13, and has a simpler mapping. > >> It has a period of 1/2 octave, which cuts down on how simple the > mapping is. Maybe it should be Bigamist. The numbers game has it as a > little more complex. > > TM commas: {16/15, 50/49}Oops, I should have noticed that! Well, it still has a 16/15 comma, so that explains why it sounds like the other father temperaments. I have mixed feelings about partial octave temperaments in general. On the one hand, they have higher complexity because they need more notes per octave. But they're also more symmetrical and allow for more transposition. An extreme example would be something like Ennealimmal, where even with the minimum complexity, each chord has 9 possible transpositions. So even though this is slighly more complex than Father [1, -1, 3, -4, 2, 10], it could still be of interest (if any of the father temperaments are of interest).
Message: 10315 - Contents - Hide Contents Date: Sun, 15 Feb 2004 23:04:28 Subject: Another lattice-chord scale From: Gene Ward Smith The Canasta alternative I gave came from symmetry around a lattice point of the tetrad lattice, which means it is also symmetrical around a tetrad in the note-class lattice, which is a shallow hole. The deep holes are hexanies, and corresponing to them are the holes of the cubic lattice, the centers of the cubes. If we look at shells around the holes, we get shells of size 8, 24, 24, 32, 48, 24 .... The 8 chord shell gives us the stellated hexany. The first two together give us 32 chords, using 38 notes. The smallest four commas arising from approximate 7-limit consonaces are 2401/2400, 6144/6125, 225/224, 1029/1024. The first and second together give hemiwuerschmidt, the first and third, first and fourth, third and fourth all miralce, the second and third orwell, the second and fourth valentine. It seems to be a good candidate for both hemiwuerschmidt and miracle. It has no 2401/2400 steps, but does have 225/224, so miracle will boil it down. Here a the list of the step sizes: [225/224, 1728/1715, 126/125, 875/864, 64/63, 50/49, 49/48, 128/125, 36/35] Here is the scale itself: [49/48, 25/24, 21/20, 15/14, 35/32, 9/8, 147/128, 7/6, 75/64, 6/5, 49/40, 5/4, 245/192, 9/7, 21/16, 75/56, 175/128, 7/5, 45/32, 10/7, 35/24, 3/2, 49/32, 25/16, 63/40, 45/28, 105/64, 5/3, 12/7, 7/4, 25/14, 9/5, 175/96, 147/80, 15/8, 245/128, 63/32, 2] Here it is tempered by TOP tuned hemiwuerschmidt: ! hemball.scl Ball 2 around tetrad lattice hole, TOP hemiwuerschmidt tempered 38 ! 36.757438 73.514875 83.550113 120.307550 157.064988 203.857663 240.615101 267.337304 277.372538 314.129979 350.887417 387.644854 424.402292 434.437529 471.194967 507.952404 544.709842 581.467283 591.502517 618.224720 654.982158 701.774833 738.532271 775.289708 785.324946 822.082383 858.839821 885.562024 932.354699 969.112137 1005.869574 1015.904812 1042.627012 1052.662250 1089.419687 1126.177125 1172.969800 1199.692003
Message: 10316 - Contents - Hide Contents Date: Sun, 15 Feb 2004 17:48:37 Subject: Number 64 as a 19-note well temperament From: Herman Miller From the big list of 114 7-limit temperaments: Number 64 [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: 69.388565 comp: 4.891080 err: 2.900537 This scale is very close to 19-ET with a generator of about 9/19 (an augmented fourth). It has two sizes of recognizable fifths (694.457 cents and 706.863 cents). Because of the slightly sharpened octaves, the "wolf" fifths are actually slightly better than the normal fifths. The "wolf" major thirds are still in the reasonable range, at 372.028 cents, and the regular major thirds are within a couple cents of just at 384.434 cents. The "wolf" 7th harmonic at 942.476 cents is worse than the 19-ET equivalent, but the regular 7:4 approximation is reasonable at 954.882 cents. So this looks like a good candidate for a 19-note well-tempered scale.
Message: 10317 - Contents - Hide Contents Date: Sun, 15 Feb 2004 03:01:15 Subject: Re: A symmetric-based 7-limit temperament list 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: >>> This complexity measure, or else >> whatever we would get as the dual to Hahn taxicab distance, >> Wouldn't the complexity measure here be the Hahn taxicab distance > itself?That's a comma measure; I need a wedgie measure. Or at least a Euclidean version of it? Where does duality> come into play?It's octave-equivalent, so the wedgie measure ends up being for the first half of the wedgie, which is a mapping, or octave-equivalent val dual to octave-equivalent monzos. Tossing out 2 makes the 7-limit in some ways like the 5-limit.>> seem to be >> the logical ones to use when we are using symmetric, octave > equivalent >> rms error. >> Which ones -- taxicab and Euclidean? Both.
Message: 10318 - Contents - Hide Contents Date: Sun, 15 Feb 2004 04:27:12 Subject: Re: A symmetric-based 7-limit temperament list From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> This ordering seems to be good at keeping similar/related temperaments > together. It's missing pelogic, injera, and dicot, though. I can > understand why pelogic and dicot might be missing, but injera [2, 8, 8, > 8, 7, -4] is a good enough temperament that it should have made the list.If I raised the badness cutoff to 2300, it would be on the list; other things no doubt would be also.>> 11: [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]] >> rms: 43.659491 symcom: 35.000000 symbad: 1528.082200 >> Number 13 Father > TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]TM commas: {16/15, 28/27}>> 12: [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]] >> rms: 43.142169 symcom: 44.000000 symbad: 1898.255432 > > Number 62> TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105] > Audibly very similar to Number 13, and has a simpler mapping.It has a period of 1/2 octave, which cuts down on how simple the mapping is. Maybe it should be Bigamist. The numbers game has it as a little more complex. TM commas: {16/15, 50/49}>> 13: [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]] >> rms: 41.524693 symcom: 35.000000 symbad: 1453.364254 > > Number 57> TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203] > Another member of the father temperament family, but the 7:1 > approximation is worse than Number 13, and the 7:4 is unrecognizable.Logically we ought to be looking at the rms tuning now, not the TOP tuning. That gives a decent 7/4 of 981 cents, and of course no longer has super-flat octaves. TM commas: {16/15, 49/45} (= {16/15, 49/48})>> 18: [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]] >> rms: 18.042924 symcom: 108.000000 symbad: 1948.635783 > > Number 85> TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166] > Would make a good 12-note keyboard mapping. There aren't many > temperaments based on 1/6-octave periods; this is the first one I've seen.It's come up before, and I think I even mentioned it as a 12-et tuning; it's two 6-equals 83 cents apart in the rms tuning. TM commas: {50/49, 128/125}> Number 6 Pajara > TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174] >>> 26: [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]] >> rms: 10.132266 symcom: 144.000000 symbad: 1459.046340 > Number 92> TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574] > Seems to be an alternate 22-ET-type temperament, not as good as Pajara.It's the 22 and 26 system; pajara is 22 and 12. TM commas: {50/49, 875/864}>> 33: [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]] >> rms: 4.006991 symcom: 436.000000 symbad: 1747.048215 >> Not in the 114 list. Seems overly complex to be of much use.It's not that bad; two generators give a 7/5 and three a 5/3. It might make more sense as a 13-limit temperament, with a generator of ~13/11 and TM basis {100/99, 196/195, 275/273, 385/384}. TM commas: {875/864, 2401/2400}>> 34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] >> rms: 3.665035 symcom: 243.000000 symbad: 890.603432 >> Almost goes without saying, but.... > Number 2 Meantone > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328] >>> 35: [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]] >> rms: 3.579262 symcom: 420.000000 symbad: 1503.290125 >> Number 35 Supermajor seconds > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099] >>> 36: [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]] >> rms: 3.443812 symcom: 571.000000 symbad: 1966.416662 >> Number 84 Squares > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656] > Sounds practically identical to Number 35, but with a more complex mapping.They both are 81/80 temperaments, and have the same 5-limit TOP tuning as meantone.>> 41: [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]] >> rms: 2.469727 symcom: 756.000000 symbad: 1867.113518 >> Now we're starting to get into temperaments that are mostly too complex > to be of much interest.I'm suspicious of such claims at this level of complexity. I just finished an ennealimmal piece, and ennealimmal doesn't seem to be too complex, though it does seem to have reached the area of diminishing returns so far as tuning goes--as Dave would no doubt point out if he were officially here, it isn't as much better, earwise, than miracle or hemiwuerschmidt as the numbers say it should be, because our ears just aren't that good. This one has some useful commas and two generators get us to 12/7; again we could push the prime limit on this one, as the generator is quite close to 17/13, and the fact that 80 and 111 cover it strongly suggest putting the limit higher anyway. Here is the mapping for the 19-limit version of 80&111: [<1 12 6 12 20 -11 -10 -8|, <0 17 6 15 27 -24 -23 -20|] TM commas: {1728/1715, 3136/3125}
Message: 10319 - Contents - Hide Contents Date: Sun, 15 Feb 2004 07:47:34 Subject: A cubic alternative to Canasta From: Gene Ward Smith If we take the 7-limit lattice of tetrads and considers shells of tetrads around a major (or inverting, a minor) tetrad, we get shells of size 1,6,12,8,6,24,24,0, ... (Note we have moats again, from the well-known theorem that numbers of the form 8n-1 cannot be represented as a sum of three squares.) If we take shells 0 to 2, we get 19 tetrads and 28 notes, adding shell 3 gives us four more notes, 32, but eight more tetrads, for a total of 27 in a 3x3x3 cube. If we look at what ratios between scale degrees can approximate, we find 2401/2400, 225/224, 1029/1024 as the three smallest commas giving near 7-limit consonances, making these scales excellent candidates for tempering by miracle. Doing so also exterminates an unwanted scale interval of 2401/2400, so the 32 notes of the JI cube boil down to 31 for the miracle cube, the same number as Canasta. While I have seen no great rush to compose in Canasta, this would be an alternative for anyone thinking about it--me, for instance. Here is the miracle version of the 19 tetrad scale, in terms of secors: Shell 2 scale: [-20, -15, -14, -13, -12, -10, -9, -8, -7, -6, -5, -4, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 19] Three more notes (not four, because of the tempering) gives us the Miracle Cube Scale: [-21, -20, -15, -14, -13, -12, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 18, 19] This doesn't beat Canasta in terms of quantity of chords, but it's nearly as good and nicely arraged. It has an inverted version which differs, so you get two versions of the scale. Here is the major-tetrad centered version in TOP tuning: ! mircube.scl Major harmonic cube of 27 tetrads in TOP miracle tuning 31 ! 33.424588 66.849178 83.296052 116.720642 150.145230 200.016694 233.441286 266.865874 316.737338 350.161928 383.586516 433.457980 466.882568 500.307160 583.603212 617.027802 650.452390 700.323854 733.748442 767.173032 817.044498 850.469084 883.893676 900.340550 933.765140 967.189728 1000.614318 1017.061192 1083.910370 1150.759550 1200.631010 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10320 - Contents - Hide Contents Date: Mon, 16 Feb 2004 01:03:53 Subject: Re: Number 64 as a 19-note well temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote: Herman, this is great. I've tried to cook up 19-tone well- temperaments, and never got one I thought was worth posting.
Message: 10321 - Contents - Hide Contents Date: Mon, 16 Feb 2004 03:55:26 Subject: Re: Number 64 as a 19-note well temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> From the big list of 114 7-limit temperaments: > > Number 64 > > [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: 69.388565 comp: 4.891080 err: 2.900537 > > This scale is very close to 19-ET with a generator of about 9/19 (an > augmented fourth). It has two sizes of recognizable fifths (694.457 > cents and 706.863 cents). Because of the slightly sharpened octaves, the > "wolf" fifths are actually slightly better than the normal fifths.You do have some rather flat major thirds along with really good ones, but it seems promising. Since the generator is a 7/5 not a 3/2, when used as a meantone it's a little odd. The TM basis for the temperament, in case we are interested, is {225/224, 1029/1000}. ! miller19.scl Herman Miller circulating based on {225/224, 1029/1000} 19 ! 62.004635 124.009271 186.013906 248.018542 322.429409 384.434045 446.438680 508.443315 570.447951 632.452586 694.457222 756.461857 818.466492 880.471128 954.881995 1016.886631 1078.891266 1140.895902 1202.900537 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10322 - Contents - Hide Contents Date: Tue, 17 Feb 2004 08:06:00 Subject: Re: rank complexity explanation updated From: Carl Lumma Welcome back, Manuel.>Carl wrote on 29-1:>> This happened because I wanted to give the interval matrix in >> 'steps of 12-tET' units. Unfortunately (and one of my biggest >> desired features) Scala does not offer 'degrees of n-ET' units. >>Fortunately it does, use >set attribute et_step <steps/oct> > >To see the intervals in terms of these units, do >show/attribute intervalsThese don't look like units to me, but some secondary abstraction. Can I author a scl file using them? Does Scala display all its output in them? No, it still displays cents, with these in a separate column. -Carl
Message: 10323 - Contents - Hide Contents Date: Tue, 17 Feb 2004 12:00:04 Subject: Re: rank complexity explanation updated From: Manuel Op de Coul Carl wrote on 29-1:>This happened because I wanted to give the interval matrix in >'steps of 12-tET' units. Unfortunately (and one of my biggest >desired features) Scala does not offer 'degrees of n-ET' units.Fortunately it does, use set attribute et_step <steps/oct> To see the intervals in terms of these units, do show/attribute intervals Manuel ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10324 - Contents - Hide Contents Date: Tue, 17 Feb 2004 17:26:54 Subject: Re: rank complexity explanation updated From: Manuel Op de Coul Carl wrote:>These don't look like units to me, but some secondary abstraction.Ah, if you mean you expected non-integer numbers, that's also possible. Then the command is: set attribute <steps/oct> For example: set attribute 12.0>Does Scala display all its output in them?No, that's not possible. Note that you can use the input command to enter a scale in any logarithmic unit if you convert it afterwards with mult/abs 2/1, assuming 2/1 is the period you want.>No, it still displays cents, with these in a >separate column.It gave me an idea for an enhancement though, showing the attributes for the intervals of the interval matrix. I'll put that in the next version under the command show/attribute/line intervals. Manuel
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