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:

1. c is greater than 1 but less than the smallest interval between any two members of S.
2. 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.

List of essential tempering commas by odd limit

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

5-odd-limit

128/125.

7-odd-limit

64/63, 126/125.

9-odd-limit

126/125, 225/224, 245/243.

11-odd-limit

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

13-odd-limit

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

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

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

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

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, 1127/1125, 1156/1155, 1197/1196, 1216/1215, 1275/1274, 1288/1287, 1311/1309, 1331/1330, 1445/1444, 1496/1495, 1521/1520, 1540/1539, 1575/1573, 1617/1615, 1701/1700, 1729/1728, 1771/1768, 1862/1859, 1863/1862, 2024/2023, 2025/2023, 2025/2024, 2058/2057, 2080/2079, 2093/2090, 2185/2184, 2200/2197, 2299/2295, 2300/2299, 2431/2430, 2432/2431, 2530/2527, 2601/2600, 2646/2645, 2720/2717, 2737/2736, 2912/2907, 2926/2925, 3060/3059, 3136/3135, 3179/3174, 3213/3211, 3381/3380, 3520/3519, 3553/3549, 3705/3703, 3762/3757, 3888/3887, 3971/3969, 3978/3971, 4096/4095, 4200/4199, 4352/4347, 4394/4389, 4693/4692, 4761/4760, 4807/4800, 4914/4913, 5083/5082, 5175/5168, 5415/5408, 5491/5488, 5985/5984, 6080/6069, 6144/6137, 6864/6859, 7942/7935, 9261/9248, 12168/12167.

25-odd-limit (incomplete)

676/675, 715/714, 729/728, 736/735, 760/759, 833/832, 875/874, 897/896, 936/935, 969/968, 1001/1000, 1089/1088, 1105/1104, 1156/1155, 1197/1196, 1216/1215, 1225/1224, 1275/1274, 1288/1287, 1311/1309, 1331/1330, 1445/1444, 1496/1495, 1521/1520, 1540/1539, 1575/1573, 1617/1615, 1701/1700, 1729/1728, 1863/1862, 2024/2023, 2025/2023, 2025/2024, 2058/2057, 2080/2079, 2093/2090, 2185/2184, 2200/2197, 2277/2275, 2300/2299, 2376/2375, 2431/2430, 2432/2431, 2500/2499, 2530/2527, 2601/2600, 2646/2645, 2720/2717, 2737/2736, 2875/2873, 2926/2925, 3025/3024, 3060/3059, 3128/3125, 3136/3135, 3179/3174, 3213/3211, 3250/3249, 3381/3380, 3520/3519, 3553/3549, 3705/3703, 3762/3757, 3888/3887, 3971/3969, 4096/4095, 4200/4199, 4225/4224, 4352/4347, 4394/4389, 4693/4692, 4761/4760, 4807/4800, 4914/4913, 5083/5082, 5175/5168, 5415/5408, 5491/5488, 5776/5775, 5985/5984, 6144/6137, 6175/6174, 6864/6859, 6877/6875, 7942/7935, 8625/8624, 9261/9248, 10626/10625, 12168/12167.

27-odd-limit (incomplete)

736/735, 760/759, 833/832, 875/874, 897/896, 936/935, 969/968, 1001/1000, 1089/1088, 1105/1104, 1156/1155, 1197/1196, 1216/1215, 1225/1224, 1275/1274, 1288/1287, 1331/1330, 1445/1444, 1496/1495, 1521/1520, 1540/1539, 1575/1573, 1617/1615, 1701/1700, 1729/1728, 1863/1862, 2024/2023, 2025/2023, 2025/2024, 2058/2057, 2080/2079, 2185/2184, 2187/2185, 2200/2197, 2277/2275, 2300/2299, 2376/2375, 2431/2430, 2432/2431, 2500/2499, 2530/2527, 2601/2600, 2646/2645, 2720/2717, 2737/2736, 2875/2873, 2926/2925, 3025/3024, 3060/3059, 3128/3125, 3136/3135, 3213/3211, 3250/3249, 3381/3380, 3520/3519, 3553/3549, 3705/3703, 3762/3757, 3888/3887, 3971/3969, 4096/4095, 4200/4199, 4225/4224, 4352/4347, 4375/4374, 4394/4389, 4693/4692, 4761/4760, 4914/4913, 5083/5082, 5175/5168, 5415/5408, 5491/5488, 5776/5775, 5985/5984, 6144/6137, 6175/6174, 6864/6859, 6877/6875, 7942/7935, 8075/8073, 8625/8624, 10626/10625, 12168/12167.

29-odd-limit (incomplete)

875/874, 897/896, 936/935, 969/968, 1001/1000, 1015/1014, 1045/1044, 1089/1088, 1105/1104, 1156/1155, 1197/1196, 1216/1215, 1225/1224, 1275/1274, 1276/1275, 1288/1287, 1331/1330, 1445/1444, 1450/1449, 1496/1495, 1521/1520, 1540/1539, 1596/1595, 1625/1624, 1683/1682, 1701/1700, 1729/1728, 1863/1862, 2001/2000, 2002/2001, 2024/2023, 2025/2023, 2025/2024, 2058/2057, 2080/2079, 2176/2175, 2185/2184, 2187/2185, 2205/2204, 2262/2261, 2277/2275, 2300/2299, 2376/2375, 2431/2430, 2432/2431, 2465/2464, 2500/2499, 2530/2527, 2601/2600, 2640/2639, 2646/2645, 2720/2717, 2737/2736, 2755/2754, 2784/2783, 2875/2873, 2926/2925, 3025/3024, 3060/3059, 3128/3125, 3136/3135, 3213/3211, 3249/3248, 3250/3249, 3381/3380, 3451/3450, 3510/3509, 3520/3519, 3553/3549, 3705/3703, 3888/3887, 3971/3969, 4096/4095, 4200/4199, 4225/4224, 4352/4347, 4375/4374, 4394/4389, 4641/4640, 4693/4692, 4761/4760, 4785/4784, 4901/4900, 4914/4913, 5083/5082, 5104/5103, 5491/5488, 5776/5775, 5888/5887, 5985/5984, 6144/6137, 6175/6174, 6670/6669, 6864/6859, 6877/6875, 7425/7424, 7942/7935, 8075/8073, 8625/8624, 8671/8670, 9251/9248, 10557/10556, 10626/10625, 10935/10933, 11340/11339, 12168/12167, 12673/12672, 13225/13224, 13312/13311.

31-odd-limit (incomplete)

969/968, 1001/1000, 1015/1014, 1024/1023, 1045/1044, 1054/1053, 1089/1088, 1105/1104, 1156/1155, 1197/1196, 1210/1209, 1216/1215, 1225/1224, 1275/1274, 1276/1275, 1288/1287, 1331/1330, 1365/1364, 1426/1425, 1445/1444, 1450/1449, 1496/1495, 1519/1518, 1520/1519, 1521/1520, 1540/1539, 1596/1595, 1625/1624, 1683/1682, 1701/1700, 1729/1728, 1768/1767, 1860/1859, 1863/1862, 1955/1953, 2001/2000, 2002/2001, 2016/2015, 2024/2023, 2025/2023, 2025/2024, 2058/2057, 2080/2079, 2176/2175, 2185/2184, 2187/2185, 2205/2204, 2233/2232, 2262/2261, 2277/2275, 2300/2299, 2376/2375, 2431/2430, 2432/2431, 2465/2464, 2500/2499, 2601/2600, 2640/2639, 2646/2645, 2737/2736, 2755/2754, 2784/2783, 2875/2873, 2926/2925, 2945/2944, 2976/2975, 3025/3024, 3060/3059, 3128/3125, 3136/3135, 3213/3211, 3249/3248, 3250/3249, 3381/3380, 3451/3450, 3510/3509, 3520/3519, 3565/3564, 3627/3625, 3705/3703, 3751/3750, 3876/3875, 3888/3887, 3969/3968, 3971/3968, 3971/3969, 4096/4095, 4186/4185, 4200/4199, 4225/4224, 4375/4374, 4641/4640, 4693/4692, 4761/4760, 4785/4784, 4807/4805, 4901/4900, 4914/4913, 4960/4959, 4992/4991, 5083/5082, 5104/5103, 5425/5423, 5491/5488, 5643/5642, 5776/5775, 5797/5796, 5888/5887, 5985/5984, 6076/6075, 6138/6137, 6175/6174, 6293/6292, 6325/6324, 6480/6479, 6670/6669, 6728/6727, 6864/6859, 6877/6875, 7425/7424, 7657/7656, 7905/7904, 7936/7935, 7942/7935, 8075/8073, 8092/8091, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9251/9248, 9425/9424, 10557/10556, 10626/10625, 10881/10880, 10935/10933, 11340/11339, 12122/12121, 12168/12167, 12673/12672, 13225/13224, 13312/13311, 15625/15624, 17577/17576, 19344/19343, 21142/21141, 24025/24024, 29792/29791.

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.

See also

• List of essentially tempered scales
• Dyadic chord #Essentially tempered dyadic chords