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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
Suppose S is a set of JI intervals i including 1 and 2 with 1 ≤ i ≤ 2 such
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: The original revision id was <tt>615831315</tt>.<br>
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<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 ≤ i ≤ 2 such
that if i is in S, so is 2/i. S is intended to represent a set of pitch classes
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:
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
1. c is greater than 1 but less than the smallest interval between any two
members of S
members of S


Line 24: Line 20:


11: 540/539, 441/440, 385/384, 243/242, 225/224, 896/891, 176/175, 126/125,
11: 540/539, 441/440, 385/384, 243/242, 225/224, 896/891, 176/175, 126/125,
245/243
245/243


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


15: 1001/1000, 1575/1573, 2200/2197, 729/728, 676/675, 540/539, 441/440,
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,
847/845, 385/384, 364/363, 352/351, 351/350, 325/324, 1573/1568, 3388/3375,
243/242, 1188/1183
243/242, 1188/1183


17: 2601/2600, 2431/2430, 1275/1274, 1156/1155, 1089/1088, 2025/2023, 1001/1000,
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,
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,
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
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,
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,
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,
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,
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,
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,
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,
441/440, 4704/4693, 847/845, 1235/1232, 2880/2873, 2057/2052, 2028/2023,
400/399, 385/384, 375/374, 364/363
400/399, 385/384, 375/374, 364/363


21: 5985/5984, 4914/4913, 4200/4199, 4096/4095, 3136/3135, 2926/2925, 2601/2600,
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,
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,
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,
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,
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,
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,
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
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])
We don't need 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
we get: 540/539, 441/440, 385/384, 3388/3375, 243/242


=See also=
=See also=
[[Essentially tempered scales]]
[[Essentially_tempered_scales|Essentially tempered scales]]
[[Dyadic chord#Essentially tempered dyadic chords|Essentially tempered dyadic chords]]</pre></div>
 
<h4>Original HTML content:</h4>
[[Dyadic_chord#Essentially tempered dyadic chords|Essentially tempered dyadic chords]]
<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 ≤ i ≤ 2 such&lt;br /&gt;
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;
&lt;br /&gt;
1. c is greater than 1 but less than the smallest interval between any two&lt;br /&gt;
members of S&lt;br /&gt;
&lt;br /&gt;
2. There are three intervals i, j, and k in S such that c = ij/k&lt;br /&gt;
&lt;br /&gt;
For various odd limit diamonds, we get the following essential tempering commas:&lt;br /&gt;
&lt;br /&gt;
5: 128/125&lt;br /&gt;
&lt;br /&gt;
7: 126/125, 64/63&lt;br /&gt;
&lt;br /&gt;
9: 225/224, 126/125, 245/243&lt;br /&gt;
&lt;br /&gt;
11: 540/539, 441/440, 385/384, 243/242, 225/224, 896/891, 176/175, 126/125,&lt;br /&gt;
245/243&lt;br /&gt;
&lt;br /&gt;
13: 1001/1000, 2200/2197, 729/728, 540/539, 441/440, 847/845, 385/384, 364/363,&lt;br /&gt;
352/351, 351/350, 325/324, 1573/1568, 243/242, 1188/1183, 225/224, 640/637,&lt;br /&gt;
196/195, 1287/1280, 896/891, 176/175&lt;br /&gt;
&lt;br /&gt;
15: 1001/1000, 1575/1573, 2200/2197, 729/728, 676/675, 540/539, 441/440,&lt;br /&gt;
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;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="See also"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;See also&lt;/h1&gt;
&lt;a class="wiki_link" href="/Essentially%20tempered%20scales"&gt;Essentially tempered scales&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/Dyadic%20chord#Essentially tempered dyadic chords"&gt;Essentially tempered dyadic chords&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
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