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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{ED intro}} |
| : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-08-08 15:36:12 UTC</tt>.<br>
| |
| : The original revision id was <tt>588922354</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">**114edo** is the [[equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit]] it [[tempering out|tempers out]] 2048/2025, in the [[7-limit]] 245/243, in the [[11-limit]] 121/120 and 176/175, in the [[13-limit]] 196/195 and 325/324, in the [[17-limit]] 136/135 and 154/153, in the [[19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic family|shrutar temperament]]; it is in fact the [[optimal patent val]] for [[shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.
| |
|
| |
|
| ===Period of 19-limit Shrutar=== | | == Theory == |
| ||~ Degree ||~ Cents ||
| | In the [[5-limit]] the equal temperament [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]), in the [[7-limit]] [[245/243]], in the [[11-limit]] [[121/120]], [[176/175]] and notably the [[quartisma]], in the [[13-limit]] [[196/195]] and [[325/324]], in the [[17-limit]] [[136/135]] and [[154/153]], in the [[19-limit]] [[286/285]] and [[343/342]]. These commas make for 114edo being an excellent tuning for the [[shrutar]] temperament; it is in fact the [[optimal patent val]] for shrutar in the 11-, 13-, 17-, and 19-limit, as well as the rank-3 [[bisector]] temperament. |
| || 2 || 21.05263 ||
| |
| || 3 || 31.57895 ||
| |
| || 5 || 52.63158 ||
| |
| || 7 || 73.68421 ||
| |
| || 8 || 84.21053 ||
| |
| || 10 || 105.26316 ||
| |
| || 12 || 126.31579 ||
| |
| || 13 || 136.842105 ||
| |
| || 15 || 157.89474 ||
| |
| || 17 || 178.94737 ||
| |
| || 18 || 189.47369 ||
| |
| || 20 || 210.52632 ||
| |
| || 22 || 231.57895 ||
| |
| || 23 || 242.10526 ||
| |
| || 25 || 263.157895 ||
| |
| || 27 || 284.21053 ||
| |
| || 29 || 305.26316 ||
| |
| || 30 || 315.78947 ||
| |
| || 32 || 336.842105 ||
| |
| || 34 || 357.89474 ||
| |
| || 35 || 368.42105 ||
| |
| || 37 || 389.47368 ||
| |
| || 39 || 410.52632 ||
| |
| || 40 || 421.05263 ||
| |
| || 42 || 442.10526 ||
| |
| || 44 || 463.157895 ||
| |
| || 45 || 473.68421 ||
| |
| || 47 || 494.73684 ||
| |
| || 49 || 515.78947 ||
| |
| || 50 || 526.31579 ||
| |
| || 52 || 547.36842 ||
| |
| || 54 || 568.42105 ||
| |
| || 55 || 578.94737 ||</pre></div>
| |
| <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>114edo</title></head><body><strong>114edo</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into 114 parts, each of 10.52632 <a class="wiki_link" href="/cent">cent</a>s. In the <a class="wiki_link" href="/5-limit">5-limit</a> it <a class="wiki_link" href="/tempering%20out">tempers out</a> 2048/2025, in the <a class="wiki_link" href="/7-limit">7-limit</a> 245/243, in the <a class="wiki_link" href="/11-limit">11-limit</a> 121/120 and 176/175, in the <a class="wiki_link" href="/13-limit">13-limit</a> 196/195 and 325/324, in the <a class="wiki_link" href="/17-limit">17-limit</a> 136/135 and 154/153, in the <a class="wiki_link" href="/19-limit">19-limit</a> 286/285 and 343/342. These commas make for 114edo being an excellent tuning for <a class="wiki_link" href="/Diaschismic%20family">shrutar temperament</a>; it is in fact the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for <a class="wiki_link" href="/shrutar">shrutar</a> in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Period of 19-limit Shrutar"></a><!-- ws:end:WikiTextHeadingRule:0 -->Period of 19-limit Shrutar</h3>
| |
|
| |
|
| |
|
| <table class="wiki_table">
| | === Odd harmonics === |
| <tr>
| | {{Harmonics in equal|114}} |
| <th>Degree<br />
| |
| </th>
| |
| <th>Cents<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>21.05263<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>31.57895<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>52.63158<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>73.68421<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>84.21053<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>105.26316<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>126.31579<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>136.842105<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>157.89474<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17<br />
| |
| </td>
| |
| <td>178.94737<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>189.47369<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>210.52632<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22<br />
| |
| </td>
| |
| <td>231.57895<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
| |
| </td>
| |
| <td>242.10526<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>25<br />
| |
| </td>
| |
| <td>263.157895<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>27<br />
| |
| </td>
| |
| <td>284.21053<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>29<br />
| |
| </td>
| |
| <td>305.26316<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>30<br />
| |
| </td>
| |
| <td>315.78947<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>32<br />
| |
| </td>
| |
| <td>336.842105<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>34<br />
| |
| </td>
| |
| <td>357.89474<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>35<br />
| |
| </td>
| |
| <td>368.42105<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>37<br />
| |
| </td>
| |
| <td>389.47368<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>39<br />
| |
| </td>
| |
| <td>410.52632<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>40<br />
| |
| </td>
| |
| <td>421.05263<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>42<br />
| |
| </td>
| |
| <td>442.10526<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>44<br />
| |
| </td>
| |
| <td>463.157895<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>45<br />
| |
| </td>
| |
| <td>473.68421<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>47<br />
| |
| </td>
| |
| <td>494.73684<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>49<br />
| |
| </td>
| |
| <td>515.78947<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>50<br />
| |
| </td>
| |
| <td>526.31579<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>52<br />
| |
| </td>
| |
| <td>547.36842<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>54<br />
| |
| </td>
| |
| <td>568.42105<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>55<br />
| |
| </td>
| |
| <td>578.94737<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| </body></html></pre></div>
| | === Subsets and supersets === |
| | Since 114 factors into {{factorization|114}}, 114 edo has subset edos {{EDOs| 2, 3, 6, 19, 38, and 57 }}. |
| | |
| | == Intervals == |
| | {{Interval table}} |
| | |
| | [[Category:Shrutar]] |
| | [[Category:Bisector]] |
| Prime factorization
|
2 × 3 × 19
|
| Step size
|
10.5263 ¢
|
| Fifth
|
67\114 (705.263 ¢)
|
| Semitones (A1:m2)
|
13:7 (136.8 ¢ : 73.68 ¢)
|
| Consistency limit
|
7
|
| Distinct consistency limit
|
7
|
114 equal divisions of the octave (abbreviated 114edo or 114ed2), also called 114-tone equal temperament (114tet) or 114 equal temperament (114et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 114 equal parts of about 10.5 ¢ each. Each step represents a frequency ratio of 21/114, or the 114th root of 2.
Theory
In the 5-limit the equal temperament tempers out 2048/2025 (diaschisma), in the 7-limit 245/243, in the 11-limit 121/120, 176/175 and notably the quartisma, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for the shrutar temperament; it is in fact the optimal patent val for shrutar in the 11-, 13-, 17-, and 19-limit, as well as the rank-3 bisector temperament.
Odd harmonics
Approximation of odd harmonics in 114edo
| Harmonic
|
3
|
5
|
7
|
9
|
11
|
13
|
15
|
17
|
19
|
21
|
23
|
| Error
|
Absolute (¢)
|
+3.31
|
+3.16
|
-0.40
|
-3.91
|
-3.95
|
+1.58
|
-4.06
|
+0.31
|
-2.78
|
+2.90
|
+3.30
|
| Relative (%)
|
+31.4
|
+30.0
|
-3.8
|
-37.1
|
-37.5
|
+15.0
|
-38.6
|
+2.9
|
-26.4
|
+27.6
|
+31.4
|
Steps
(reduced)
|
181
(67)
|
265
(37)
|
320
(92)
|
361
(19)
|
394
(52)
|
422
(80)
|
445
(103)
|
466
(10)
|
484
(28)
|
501
(45)
|
516
(60)
|
Subsets and supersets
Since 114 factors into 2 × 3 × 19, 114 edo has subset edos 2, 3, 6, 19, 38, and 57.
Intervals
| Steps
|
Cents
|
Approximate ratios
|
Ups and downs notation
|
| 0
|
0
|
1/1
|
D
|
| 1
|
10.5
|
|
^D, v6E♭
|
| 2
|
21.1
|
|
^^D, v5E♭
|
| 3
|
31.6
|
|
^3D, v4E♭
|
| 4
|
42.1
|
40/39, 41/40, 42/41, 43/42
|
^4D, v3E♭
|
| 5
|
52.6
|
33/32, 34/33
|
^5D, vvE♭
|
| 6
|
63.2
|
|
^6D, vE♭
|
| 7
|
73.7
|
24/23
|
v6D♯, E♭
|
| 8
|
84.2
|
21/20, 43/41
|
v5D♯, ^E♭
|
| 9
|
94.7
|
37/35
|
v4D♯, ^^E♭
|
| 10
|
105.3
|
17/16
|
v3D♯, ^3E♭
|
| 11
|
115.8
|
31/29, 46/43, 47/44
|
vvD♯, ^4E♭
|
| 12
|
126.3
|
14/13, 43/40
|
vD♯, ^5E♭
|
| 13
|
136.8
|
13/12, 40/37
|
D♯, ^6E♭
|
| 14
|
147.4
|
37/34
|
^D♯, v6E
|
| 15
|
157.9
|
23/21, 34/31
|
^^D♯, v5E
|
| 16
|
168.4
|
32/29, 43/39
|
^3D♯, v4E
|
| 17
|
178.9
|
41/37
|
^4D♯, v3E
|
| 18
|
189.5
|
29/26, 48/43
|
^5D♯, vvE
|
| 19
|
200
|
37/33, 46/41
|
^6D♯, vE
|
| 20
|
210.5
|
26/23, 35/31
|
E
|
| 21
|
221.1
|
42/37
|
^E, v6F
|
| 22
|
231.6
|
8/7
|
^^E, v5F
|
| 23
|
242.1
|
23/20
|
^3E, v4F
|
| 24
|
252.6
|
22/19, 37/32
|
^4E, v3F
|
| 25
|
263.2
|
|
^5E, vvF
|
| 26
|
273.7
|
34/29, 41/35, 48/41
|
^6E, vF
|
| 27
|
284.2
|
33/28, 46/39
|
F
|
| 28
|
294.7
|
|
^F, v6G♭
|
| 29
|
305.3
|
31/26, 37/31
|
^^F, v5G♭
|
| 30
|
315.8
|
6/5
|
^3F, v4G♭
|
| 31
|
326.3
|
29/24, 35/29
|
^4F, v3G♭
|
| 32
|
336.8
|
17/14
|
^5F, vvG♭
|
| 33
|
347.4
|
|
^6F, vG♭
|
| 34
|
357.9
|
16/13, 43/35
|
v6F♯, G♭
|
| 35
|
368.4
|
26/21, 47/38
|
v5F♯, ^G♭
|
| 36
|
378.9
|
46/37
|
v4F♯, ^^G♭
|
| 37
|
389.5
|
|
v3F♯, ^3G♭
|
| 38
|
400
|
29/23
|
vvF♯, ^4G♭
|
| 39
|
410.5
|
|
vF♯, ^5G♭
|
| 40
|
421.1
|
37/29
|
F♯, ^6G♭
|
| 41
|
431.6
|
|
^F♯, v6G
|
| 42
|
442.1
|
31/24, 40/31
|
^^F♯, v5G
|
| 43
|
452.6
|
13/10, 48/37
|
^3F♯, v4G
|
| 44
|
463.2
|
17/13
|
^4F♯, v3G
|
| 45
|
473.7
|
46/35
|
^5F♯, vvG
|
| 46
|
484.2
|
37/28, 41/31
|
^6F♯, vG
|
| 47
|
494.7
|
|
G
|
| 48
|
505.3
|
|
^G, v6A♭
|
| 49
|
515.8
|
31/23, 35/26
|
^^G, v5A♭
|
| 50
|
526.3
|
42/31
|
^3G, v4A♭
|
| 51
|
536.8
|
|
^4G, v3A♭
|
| 52
|
547.4
|
48/35
|
^5G, vvA♭
|
| 53
|
557.9
|
29/21, 40/29
|
^6G, vA♭
|
| 54
|
568.4
|
25/18, 43/31
|
v6G♯, A♭
|
| 55
|
578.9
|
|
v5G♯, ^A♭
|
| 56
|
589.5
|
|
v4G♯, ^^A♭
|
| 57
|
600
|
41/29
|
v3G♯, ^3A♭
|
| 58
|
610.5
|
37/26, 47/33
|
vvG♯, ^4A♭
|
| 59
|
621.1
|
|
vG♯, ^5A♭
|
| 60
|
631.6
|
36/25
|
G♯, ^6A♭
|
| 61
|
642.1
|
29/20, 42/29
|
^G♯, v6A
|
| 62
|
652.6
|
35/24
|
^^G♯, v5A
|
| 63
|
663.2
|
|
^3G♯, v4A
|
| 64
|
673.7
|
31/21
|
^4G♯, v3A
|
| 65
|
684.2
|
46/31
|
^5G♯, vvA
|
| 66
|
694.7
|
|
^6G♯, vA
|
| 67
|
705.3
|
|
A
|
| 68
|
715.8
|
|
^A, v6B♭
|
| 69
|
726.3
|
35/23
|
^^A, v5B♭
|
| 70
|
736.8
|
26/17
|
^3A, v4B♭
|
| 71
|
747.4
|
20/13, 37/24
|
^4A, v3B♭
|
| 72
|
757.9
|
31/20, 48/31
|
^5A, vvB♭
|
| 73
|
768.4
|
39/25
|
^6A, vB♭
|
| 74
|
778.9
|
|
v6A♯, B♭
|
| 75
|
789.5
|
41/26
|
v5A♯, ^B♭
|
| 76
|
800
|
46/29
|
v4A♯, ^^B♭
|
| 77
|
810.5
|
|
v3A♯, ^3B♭
|
| 78
|
821.1
|
37/23
|
vvA♯, ^4B♭
|
| 79
|
831.6
|
21/13
|
vA♯, ^5B♭
|
| 80
|
842.1
|
13/8
|
A♯, ^6B♭
|
| 81
|
852.6
|
|
^A♯, v6B
|
| 82
|
863.2
|
28/17
|
^^A♯, v5B
|
| 83
|
873.7
|
48/29
|
^3A♯, v4B
|
| 84
|
884.2
|
5/3
|
^4A♯, v3B
|
| 85
|
894.7
|
47/28
|
^5A♯, vvB
|
| 86
|
905.3
|
|
^6A♯, vB
|
| 87
|
915.8
|
39/23
|
B
|
| 88
|
926.3
|
29/17, 41/24
|
^B, v6C
|
| 89
|
936.8
|
43/25
|
^^B, v5C
|
| 90
|
947.4
|
19/11
|
^3B, v4C
|
| 91
|
957.9
|
40/23
|
^4B, v3C
|
| 92
|
968.4
|
7/4
|
^5B, vvC
|
| 93
|
978.9
|
37/21
|
^6B, vC
|
| 94
|
989.5
|
23/13
|
C
|
| 95
|
1000
|
41/23
|
^C, v6D♭
|
| 96
|
1010.5
|
43/24
|
^^C, v5D♭
|
| 97
|
1021.1
|
|
^3C, v4D♭
|
| 98
|
1031.6
|
29/16
|
^4C, v3D♭
|
| 99
|
1042.1
|
31/17, 42/23
|
^5C, vvD♭
|
| 100
|
1052.6
|
|
^6C, vD♭
|
| 101
|
1063.2
|
24/13, 37/20
|
v6C♯, D♭
|
| 102
|
1073.7
|
13/7
|
v5C♯, ^D♭
|
| 103
|
1084.2
|
43/23
|
v4C♯, ^^D♭
|
| 104
|
1094.7
|
32/17
|
v3C♯, ^3D♭
|
| 105
|
1105.3
|
|
vvC♯, ^4D♭
|
| 106
|
1115.8
|
40/21
|
vC♯, ^5D♭
|
| 107
|
1126.3
|
23/12
|
C♯, ^6D♭
|
| 108
|
1136.8
|
|
^C♯, v6D
|
| 109
|
1147.4
|
33/17
|
^^C♯, v5D
|
| 110
|
1157.9
|
39/20, 41/21
|
^3C♯, v4D
|
| 111
|
1168.4
|
|
^4C♯, v3D
|
| 112
|
1178.9
|
|
^5C♯, vvD
|
| 113
|
1189.5
|
|
^6C♯, vD
|
| 114
|
1200
|
2/1
|
D
|