Essential tempering comma: Difference between revisions

Note also that it only identifies ''triads''. There are commas that induce essentially tempered chords whose basic forms are ''tetrads''. For example, 81/80 induces an essentially tempered tetrad (→ [[didymic chords]]), despite that any three of the components are essentially just.

For various odd limit diamonds, we get the following essential tempering commas using the derivation above:

: 128/125.  

: 64/63, 126/125.  

: 126/125, 225/224, 245/243.  

: 126/125, 176/175, 225/224, 243/242, 245/243, 385/384, 441/440, 540/539, 896/891.  

: 176/175, 196/195, 225/224, 243/242, 325/324, 351/350, 352/351, 364/363, 385/384, 441/440, 540/539, 640/637, 729/728, 847/845, 896/891, 1001/1000, 1188/1183, 1287/1280, 1573/1568, 2200/2197.  

: 243/242, 325/324, 351/350, 352/351, 364/363, 385/384, 441/440, 540/539, 676/675, 729/728, 847/845, 1001/1000, 1188/1183, 1573/1568, 1575/1573, 2200/2197, 3388/3375.  

: 325/324, 351/350, 352/351, 364/363, 375/374, 385/384, 441/440, 442/441, 540/539, 561/560, 595/594, 676/675, 715/714, 729/728, 833/832, 847/845, 936/935, 1001/1000, 1089/1088, 1156/1155, 1275/1274, 1573/1568, 1575/1573, 2025/2023, 2028/2023, 2200/2197, 2295/2288, 2431/2430, 2601/2600, 2880/2873, 4928/4913.  

: 364/363, 375/374, 385/384, 400/399, 441/440, 442/441, 456/455, 476/475, 495/494, 513/512, 540/539, 561/560, 595/594, 676/675, 715/714, 729/728, 833/832, 847/845, 936/935, 969/968, 1001/1000, 1089/1088, 1156/1155, 1216/1215, 1235/1232, 1275/1274, 1331/1330, 1445/1444, 1521/1520, 1540/1539, 1575/1573, 1729/1728, 1862/1859, 2025/2023, 2028/2023, 2057/2052, 2200/2197, 2299/2295, 2304/2299, 2431/2430, 2432/2431, 2601/2600, 2720/2717, 2880/2873, 2912/2907, 2926/2925, 3136/3135, 3762/3757, 3978/3971, 4200/4199, 4704/4693, 5415/5408, 5491/5488, 6144/6137, 6864/6859.  

: 456/455, 476/475, 495/494, 513/512, 540/539, 561/560, 595/594, 676/675, 715/714, 729/728, 833/832, 936/935, 969/968, 1001/1000, 1089/1088, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1521/1520, 1540/1539, 1575/1573, 1617/1615, 1701/1700, 1729/1728, 1862/1859, 2025/2023, 2058/2057, 2080/2079, 2200/2197, 2299/2295, 2304/2299, 2431/2430, 2432/2431, 2601/2600, 2720/2717, 2912/2907, 2926/2925, 3136/3135, 3213/3211, 3553/3549, 3762/3757, 3971/3969, 3978/3971, 4096/4095, 4200/4199, 4394/4389, 4914/4913, 5415/5408, 5491/5488, 5985/5984, 6080/6069, 6144/6137, 6864/6859, 9261/9248.  

; 23-odd-limit (incomplete)

: 540/539, 561/560, 576/575, 595/594, 676/675, 715/714, 729/728, 736/735, 760/759, 833/832, 897/896, 936/935, 969/968, 1001/1000, 1089/1088, 1105/1104, 1156/1155, 1197/1196, 1216/1215, 1275/1274, 1288/1287, 1331/1330, 1445/1444, 1496/1495, 1521/1520, 1540/1539, 1575/1573, 1617/1615, 1701/1700, 1729/1728, 1862/1859, 1863/1862, 2024/2023, 2025/2023, 2025/2024, 2058/2057, 2080/2079, 2185/2184, 2200/2197, 2299/2295, 2300/2299, 2431/2430, 2432/2431, 2601/2600, 2646/2645, 2720/2717, 2737/2736, 2912/2907, 2926/2925, 3060/3059, 3136/3135, 3213/3211, 3381/3380, 3520/3519, 3553/3549, 3762/3757, 3888/3887, 3971/3969, 3978/3971, 4096/4095, 4200/4199, 4394/4389, 4693/4692, 4761/4760, 4914/4913, 5083/5082, 5415/5408, 5491/5488, 5985/5984, 6080/6069, 6144/6137, 6864/6859, 9261/9248, 12168/12167.  

It is not necessary to use the full ''q''-odd-limit diamond; from diamond ([1, 3, 5, 7, 9, 11, 15]) we get: 243/242, 385/384, 441/440, 540/539, 3388/3375.