Essential tempering comma
An essential tempering comma is a comma that induces essentially tempered chords.
Gene's derivation
Suppose S is a set of JI ratios r including 1 and 2 with 1 ≤ r ≤ 2 such that if r is in S, so is 2/r. S is intended to represent a set of pitch classes defining "consonance". A JI interval c is an essential tempering comma for S if:
- c is greater than 1 but less than the smallest interval between any two members of S.
- There are three ratios r1, r2, and r3 in S such that c = r1r2/r3.
This derivation is a sufficient and not necessary condition for identifying essentially tempered triads. For example, 121/120 in the 11-odd-limit is rejected, but it induces essentially tempered triads (→ biyatismic chords). One of them is 1-11/8-5/3 with steps 11/8-11/9-6/5.
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.
Examples
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
It is not necessary 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