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 due to how aggressively it rejects commas that are too large. This does come with the benefit of discarding absurdly large commas. For example, 125/108 is a comma whose essentially tempered chord is a stack of three 6/5's closing at the octave. Assume pure octaves, it requires each 6/5 to be tuned sharper than the just value of 5/4, and is rejected, in fact leaving 128/125 as the only 5-odd-limit essential tempering comma. However, some commas are reasonably sized but still rejected, such as 121/120 in the 11-odd-limit (→ biyatismic chords).
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 using the derivation above:
- 5-odd-limit (complete)
- 128/125.
- 7-odd-limit (complete)
- 64/63, 126/125.
- 9-odd-limit (complete)
- 126/125, 225/224, 245/243.
- 11-odd-limit (complete)
- 126/125, 176/175, 225/224, 243/242, 245/243, 385/384, 441/440, 540/539, 896/891.
- 13-odd-limit (complete)
- 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.
- 15-odd-limit (complete)
- 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.
- 17-odd-limit (complete)
- 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.
- 19-odd-limit (complete)
- 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.
- 21-odd-limit (complete)
- 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.