This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

- Contents - Hide Contents - Home - Section 11

Previous Next

10000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500 10550 10600 10650 10700 10750 10800 10850 10900 10950

10300 - 10325 -



top of page bottom of page up down


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?
top of page bottom of page up down


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: 1528.082200
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]
top of page bottom of page up down


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]
top of page bottom of page up down


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.
top of page bottom of page up down


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. :)
top of page bottom of page up down


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.]
top of page bottom of page up down


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!
top of page bottom of page up down


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.
top of page bottom of page up down


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


top of page bottom of page up down


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


top of page bottom of page up down


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.
top of page bottom of page up down


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.
top of page bottom of page up down


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.)


top of page bottom of page up down


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}


top of page bottom of page up down


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).
top of page bottom of page up down


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


top of page bottom of page up down


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.


top of page bottom of page up down


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.
top of page bottom of page up down


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}
top of page bottom of page up down


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.)


top of page bottom of page up down


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.


top of page bottom of page up down


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.)
top of page bottom of page up down


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 intervals
These 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
top of page bottom of page up down


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.)
top of page bottom of page up down


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
top of page bottom of page up

Previous Next

10000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500 10550 10600 10650 10700 10750 10800 10850 10900 10950

10300 - 10325 -

top of page