Essential tempering comma: Difference between revisions

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how did this not use the terms "octave complement", "unison", "octave", or "octave-reduced"
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An '''essential tempering comma''' is a [[comma]] that induces [[Dyadic chord #Essentially tempered dyadic chords|essentially tempered chords]].  
An '''essential tempering comma''' is a [[comma]] that induces [[Dyadic chord #Essentially tempered dyadic chords|essentially tempered chords]].  


== Gene's derivation ==
== Finding small essential tempering commas ==
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:
Consider a set of "consonances" composed of octave-reduced JI ratios, including the unison and octave, such that any interval and its octave complement are both "consonances". A JI interval ''c'' is an essential tempering comma for this set of consonances if:


# ''c'' is greater than 1 but less than the smallest interval between any two members of ''S''.  
# ''c'' is not the unison, but it is smaller than any interval between two consonances.
# There are three ratios ''r''<sub>1</sub>, ''r''<sub>2</sub>, and ''r''<sub>3</sub> in ''S'' such that ''c'' = ''r''<sub>1</sub>''r''<sub>2</sub>/''r''<sub>3</sub>.  
# There are three consonances ''r''<sub>1</sub>, ''r''<sub>2</sub>, and ''r''<sub>3</sub> such that ''c'' = ''r''<sub>1</sub>''r''<sub>2</sub>/''r''<sub>3</sub>.  


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]]).  
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]]).  
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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.
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 ==
== List of small essential tempering commas by odd limit ==
For various odd limit diamonds, we get the following essential tempering commas using the derivation above:
For various odd limit diamonds, we get the following essential tempering commas using the derivation above:


Revision as of 22:47, 19 August 2025

An essential tempering comma is a comma that induces essentially tempered chords.

Finding small essential tempering commas

Consider a set of "consonances" composed of octave-reduced JI ratios, including the unison and octave, such that any interval and its octave complement are both "consonances". A JI interval c is an essential tempering comma for this set of consonances if:

  1. c is not the unison, but it is smaller than any interval between two consonances.
  2. There are three consonances r1, r2, and r3 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 small 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

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