Essential tempering comma
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Original Wikitext content:
Suppose S is a set of JI intervals i including 1 and 2 with 1 <= i <= 2 such that if i is in S, so is 2/i. S is intended to represent a set of pitch classes defining "consonance". A JI interval c is an essential tempering comma for S if: 1. c is greater than 1 but less than the smallest interval between any two members of S 2. There are three intervals i, j, and k in S such that c = i*j/k For various odd limit diamonds, we get the following essential tempering commas: 5: 128/125 7: 126/125, 64/63 9: 225/224, 126/125, 245/243 11: 540/539, 441/440, 385/384, 243/242, 225/224, 896/891, 176/175, 126/125, 245/243 13: 1001/1000, 2200/2197, 729/728, 540/539, 441/440, 847/845, 385/384, 364/363, 352/351, 351/350, 325/324, 1573/1568, 243/242, 1188/1183, 225/224, 640/637, 196/195, 1287/1280, 896/891, 176/175 15: 1001/1000, 1575/1573, 2200/2197, 729/728, 676/675, 540/539, 441/440, 847/845, 385/384, 364/363, 352/351, 351/350, 325/324, 1573/1568, 3388/3375, 243/242, 1188/1183 17: 2601/2600, 2431/2430, 1275/1274, 1156/1155, 1089/1088, 2025/2023, 1001/1000, 936/935, 833/832, 1575/1573, 2200/2197, 729/728, 715/714, 676/675, 595/594, 561/560, 540/539, 442/441, 441/440, 847/845, 2880/2873, 2028/2023, 385/384, 375/374, 364/363, 352/351, 351/350, 4928/4913, 2295/2288, 325/324, 1573/1568 19: 4200/4199, 3136/3135, 2926/2925, 2601/2600, 2432/2431, 2431/2430, 5491/5488, 1729/1728, 1540/1539, 1521/1520, 1445/1444, 6864/6859, 1331/1330, 1275/1274, 1216/1215, 1156/1155, 1089/1088, 2025/2023, 1001/1000, 969/968, 936/935, 2720/2717, 6144/6137, 833/832, 1575/1573, 5415/5408, 3762/3757, 2200/2197, 729/728, 715/714, 676/675, 1862/1859, 595/594, 2912/2907, 2299/2295, 3978/3971, 561/560, 540/539, 513/512, 495/494, 476/475, 2304/2299, 456/455, 442/441, 441/440, 4704/4693, 847/845, 1235/1232, 2880/2873, 2057/2052, 2028/2023, 400/399, 385/384, 375/374, 364/363 21: 5985/5984, 4914/4913, 4200/4199, 4096/4095, 3136/3135, 2926/2925, 2601/2600, 2432/2431, 2431/2430, 2080/2079, 2058/2057, 3971/3969, 5491/5488, 1729/1728, 1701/1700, 3213/3211, 1540/1539, 1521/1520, 1445/1444, 6864/6859, 1331/1330, 1275/1274, 1216/1215, 1156/1155, 1089/1088, 2025/2023, 1001/1000, 969/968, 936/935, 2720/2717, 3553/3549, 4394/4389, 6144/6137, 833/832, 1617/1615, 1575/1573, 5415/5408, 3762/3757, 2200/2197, 729/728, 715/714, 9261/9248, 676/675, 1862/1859, 595/594, 2912/2907, 2299/2295, 3978/3971, 561/560, 6080/6069, 540/539, 513/512, 495/494, 476/475, 2304/2299, 456/455 We don't need to use the full q-limit diamond; from Diamond([1,3,5,7,9,11,15]) we get: 540/539, 441/440, 385/384, 3388/3375, 243/242
Original HTML content:
<html><head><title>Essential tempering commas</title></head><body>Suppose S is a set of JI intervals i including 1 and 2 with 1 <= i <= 2 such<br /> that if i is in S, so is 2/i. S is intended to represent a set of pitch classes<br /> defining "consonance". A JI interval c is an essential tempering comma for S if:<br /> <br /> 1. c is greater than 1 but less than the smallest interval between any two<br /> members of S<br /> <br /> 2. There are three intervals i, j, and k in S such that c = i*j/k<br /> <br /> For various odd limit diamonds, we get the following essential tempering commas:<br /> <br /> 5: 128/125<br /> <br /> 7: 126/125, 64/63<br /> <br /> 9: 225/224, 126/125, 245/243<br /> <br /> 11: 540/539, 441/440, 385/384, 243/242, 225/224, 896/891, 176/175, 126/125,<br /> 245/243<br /> <br /> 13: 1001/1000, 2200/2197, 729/728, 540/539, 441/440, 847/845, 385/384, 364/363,<br /> 352/351, 351/350, 325/324, 1573/1568, 243/242, 1188/1183, 225/224, 640/637,<br /> 196/195, 1287/1280, 896/891, 176/175<br /> <br /> 15: 1001/1000, 1575/1573, 2200/2197, 729/728, 676/675, 540/539, 441/440,<br /> 847/845, 385/384, 364/363, 352/351, 351/350, 325/324, 1573/1568, 3388/3375,<br /> 243/242, 1188/1183<br /> <br /> 17: 2601/2600, 2431/2430, 1275/1274, 1156/1155, 1089/1088, 2025/2023, 1001/1000,<br /> 936/935, 833/832, 1575/1573, 2200/2197, 729/728, 715/714, 676/675, 595/594,<br /> 561/560, 540/539, 442/441, 441/440, 847/845, 2880/2873, 2028/2023, 385/384,<br /> 375/374, 364/363, 352/351, 351/350, 4928/4913, 2295/2288, 325/324, 1573/1568<br /> <br /> 19: 4200/4199, 3136/3135, 2926/2925, 2601/2600, 2432/2431, 2431/2430, 5491/5488,<br /> 1729/1728, 1540/1539, 1521/1520, 1445/1444, 6864/6859, 1331/1330, 1275/1274,<br /> 1216/1215, 1156/1155, 1089/1088, 2025/2023, 1001/1000, 969/968, 936/935,<br /> 2720/2717, 6144/6137, 833/832, 1575/1573, 5415/5408, 3762/3757, 2200/2197,<br /> 729/728, 715/714, 676/675, 1862/1859, 595/594, 2912/2907, 2299/2295, 3978/3971,<br /> 561/560, 540/539, 513/512, 495/494, 476/475, 2304/2299, 456/455, 442/441,<br /> 441/440, 4704/4693, 847/845, 1235/1232, 2880/2873, 2057/2052, 2028/2023,<br /> 400/399, 385/384, 375/374, 364/363<br /> <br /> 21: 5985/5984, 4914/4913, 4200/4199, 4096/4095, 3136/3135, 2926/2925, 2601/2600,<br /> 2432/2431, 2431/2430, 2080/2079, 2058/2057, 3971/3969, 5491/5488, 1729/1728,<br /> 1701/1700, 3213/3211, 1540/1539, 1521/1520, 1445/1444, 6864/6859, 1331/1330,<br /> 1275/1274, 1216/1215, 1156/1155, 1089/1088, 2025/2023, 1001/1000, 969/968,<br /> 936/935, 2720/2717, 3553/3549, 4394/4389, 6144/6137, 833/832, 1617/1615,<br /> 1575/1573, 5415/5408, 3762/3757, 2200/2197, 729/728, 715/714, 9261/9248,<br /> 676/675, 1862/1859, 595/594, 2912/2907, 2299/2295, 3978/3971, 561/560,<br /> 6080/6069, 540/539, 513/512, 495/494, 476/475, 2304/2299, 456/455<br /> <br /> We don't need to use the full q-limit diamond; from Diamond([1,3,5,7,9,11,15])<br /> we get: 540/539, 441/440, 385/384, 3388/3375, 243/242</body></html>