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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
An '''essential tempering comma''' is a [[comma]] that induces [[Dyadic chord #Essentially tempered dyadic chords|essentially tempered chords]].  
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-19 16:39:05 UTC</tt>.<br>
: The original revision id was <tt>287490140</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Suppose S is a set of JI intervals i including 1 and 2 with 1 &lt;= i &lt;= 2 such
that if i is in S, so is 2/i. 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
== Finding small essential tempering commas ==
members of S
[[Gene Ward Smith]] derived the following method to find 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:


2. There are three intervals i, j, and k in S such that c = i*j/k
# ''c'' is not the unison, but it is smaller than any interval between two consonances.
# 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>.


For various odd limit diamonds, we get the following essential tempering commas:
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]]).


5: 128/125
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.


7: 126/125, 64/63
== 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:


9: 225/224, 126/125, 245/243
; 5-odd-limit
: 128/125.


11: 540/539, 441/440, 385/384, 243/242, 225/224, 896/891, 176/175, 126/125,
; 7-odd-limit
245/243
: 64/63, 126/125.


13: 1001/1000, 2200/2197, 729/728, 540/539, 441/440, 847/845, 385/384, 364/363,
; 9-odd-limit
352/351, 351/350, 325/324, 1573/1568, 243/242, 1188/1183, 225/224, 640/637,
: 126/125, 225/224, 245/243.
196/195, 1287/1280, 896/891, 176/175


15: 1001/1000, 1575/1573, 2200/2197, 729/728, 676/675, 540/539, 441/440,
; 11-odd-limit
847/845, 385/384, 364/363, 352/351, 351/350, 325/324, 1573/1568, 3388/3375,
: 126/125, 176/175, 225/224, 243/242, 245/243, 385/384, 441/440, 540/539, 896/891.
243/242, 1188/1183


17: 2601/2600, 2431/2430, 1275/1274, 1156/1155, 1089/1088, 2025/2023, 1001/1000,
; 13-odd-limit
936/935, 833/832, 1575/1573, 2200/2197, 729/728, 715/714, 676/675, 595/594,
: 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.
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,
; 15-odd-limit
1729/1728, 1540/1539, 1521/1520, 1445/1444, 6864/6859, 1331/1330, 1275/1274,
: 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.
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,
; 17-odd-limit
2432/2431, 2431/2430, 2080/2079, 2058/2057, 3971/3969, 5491/5488, 1729/1728,
: 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.
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


We don't need to use the full q-limit diamond; from Diamond([1,3,5,7,9,11,15])
; 19-odd-limit
we get: 540/539, 441/440, 385/384, 3388/3375, 243/242
: 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.
</pre></div>
 
<h4>Original HTML content:</h4>
; 21-odd-limit
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Essential tempering commas&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Suppose S is a set of JI intervals i including 1 and 2 with 1 &amp;lt;= i &amp;lt;= 2 such&lt;br /&gt;
: 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.
that if i is in S, so is 2/i. S is intended to represent a set of pitch classes&lt;br /&gt;
 
defining &amp;quot;consonance&amp;quot;. A JI interval c is an essential tempering comma for S if:&lt;br /&gt;
; 23-odd-limit (incomplete)
&lt;br /&gt;
: 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, 5819/5814, 5824/5819, 5985/5984, 6080/6069, 6144/6137, 6864/6859, 7942/7935, 9261/9248, 12168/12167.
1. c is greater than 1 but less than the smallest interval between any two&lt;br /&gt;
 
members of S&lt;br /&gt;
; 25-odd-limit (incomplete)
&lt;br /&gt;
: 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, 1377/1375, 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, 2128/2125, 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, 3328/3325, 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, 5819/5814, 5824/5819, 5985/5984, 6144/6137, 6175/6174, 6864/6859, 6877/6875, 7942/7935, 8625/8624, 9261/9248, 10626/10625, 12168/12167.
2. There are three intervals i, j, and k in S such that c = i*j/k&lt;br /&gt;
 
&lt;br /&gt;
; 27-odd-limit (incomplete)
For various odd limit diamonds, we get the following essential tempering commas:&lt;br /&gt;
: 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, 3328/3325, 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, 5819/5814, 5824/5819, 5985/5984, 6144/6137, 6175/6174, 6864/6859, 6877/6875, 7942/7935, 8075/8073, 8625/8624, 10626/10625, 12168/12167.
&lt;br /&gt;
 
5: 128/125&lt;br /&gt;
; 29-odd-limit (incomplete)
&lt;br /&gt;
: 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, 3328/3325, 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, 5684/5681, 5776/5775, 5819/5814, 5824/5819, 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.
7: 126/125, 64/63&lt;br /&gt;
 
&lt;br /&gt;
; 31-odd-limit (incomplete)
9: 225/224, 126/125, 245/243&lt;br /&gt;
: 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, 3328/3325, 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, 4125/4123, 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, 5244/5239, 5425/5423, 5491/5488, 5643/5642, 5684/5681, 5776/5775, 5797/5796, 5819/5814, 5824/5819, 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.
&lt;br /&gt;
 
11: 540/539, 441/440, 385/384, 243/242, 225/224, 896/891, 176/175, 126/125,&lt;br /&gt;
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.
245/243&lt;br /&gt;
 
&lt;br /&gt;
== See also ==
13: 1001/1000, 2200/2197, 729/728, 540/539, 441/440, 847/845, 385/384, 364/363,&lt;br /&gt;
* [[List of essentially tempered scales]]
352/351, 351/350, 325/324, 1573/1568, 243/242, 1188/1183, 225/224, 640/637,&lt;br /&gt;
* [[Dyadic chord #Essentially tempered dyadic chords]]
196/195, 1287/1280, 896/891, 176/175&lt;br /&gt;
 
&lt;br /&gt;
[[Category:Regular temperament theory]]
15: 1001/1000, 1575/1573, 2200/2197, 729/728, 676/675, 540/539, 441/440,&lt;br /&gt;
[[Category:Comma]]
847/845, 385/384, 364/363, 352/351, 351/350, 325/324, 1573/1568, 3388/3375,&lt;br /&gt;
243/242, 1188/1183&lt;br /&gt;
&lt;br /&gt;
17: 2601/2600, 2431/2430, 1275/1274, 1156/1155, 1089/1088, 2025/2023, 1001/1000,&lt;br /&gt;
936/935, 833/832, 1575/1573, 2200/2197, 729/728, 715/714, 676/675, 595/594,&lt;br /&gt;
561/560, 540/539, 442/441, 441/440, 847/845, 2880/2873, 2028/2023, 385/384,&lt;br /&gt;
375/374, 364/363, 352/351, 351/350, 4928/4913, 2295/2288, 325/324, 1573/1568&lt;br /&gt;
&lt;br /&gt;
19: 4200/4199, 3136/3135, 2926/2925, 2601/2600, 2432/2431, 2431/2430, 5491/5488,&lt;br /&gt;
1729/1728, 1540/1539, 1521/1520, 1445/1444, 6864/6859, 1331/1330, 1275/1274,&lt;br /&gt;
1216/1215, 1156/1155, 1089/1088, 2025/2023, 1001/1000, 969/968, 936/935,&lt;br /&gt;
2720/2717, 6144/6137, 833/832, 1575/1573, 5415/5408, 3762/3757, 2200/2197,&lt;br /&gt;
729/728, 715/714, 676/675, 1862/1859, 595/594, 2912/2907, 2299/2295, 3978/3971,&lt;br /&gt;
561/560, 540/539, 513/512, 495/494, 476/475, 2304/2299, 456/455, 442/441,&lt;br /&gt;
441/440, 4704/4693, 847/845, 1235/1232, 2880/2873, 2057/2052, 2028/2023,&lt;br /&gt;
400/399, 385/384, 375/374, 364/363&lt;br /&gt;
&lt;br /&gt;
21: 5985/5984, 4914/4913, 4200/4199, 4096/4095, 3136/3135, 2926/2925, 2601/2600,&lt;br /&gt;
2432/2431, 2431/2430, 2080/2079, 2058/2057, 3971/3969, 5491/5488, 1729/1728,&lt;br /&gt;
1701/1700, 3213/3211, 1540/1539, 1521/1520, 1445/1444, 6864/6859, 1331/1330,&lt;br /&gt;
1275/1274, 1216/1215, 1156/1155, 1089/1088, 2025/2023, 1001/1000, 969/968,&lt;br /&gt;
936/935, 2720/2717, 3553/3549, 4394/4389, 6144/6137, 833/832, 1617/1615,&lt;br /&gt;
1575/1573, 5415/5408, 3762/3757, 2200/2197, 729/728, 715/714, 9261/9248,&lt;br /&gt;
676/675, 1862/1859, 595/594, 2912/2907, 2299/2295, 3978/3971, 561/560,&lt;br /&gt;
6080/6069, 540/539, 513/512, 495/494, 476/475, 2304/2299, 456/455&lt;br /&gt;
&lt;br /&gt;
We don't need to use the full q-limit diamond; from Diamond([1,3,5,7,9,11,15])&lt;br /&gt;
we get: 540/539, 441/440, 385/384, 3388/3375, 243/242&lt;/body&gt;&lt;/html&gt;</pre></div>
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