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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:Sarzadoce|Sarzadoce]] and made on <tt>2011-08-16 14:31:35 UTC</tt>.<br>
| |
| : The original revision id was <tt>246279673</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">=21 Equal Divisions of the Tritave=
| |
|
| |
|
| || Degrees || Cents || Approximate Ratio || | | == Intervals == |
| || 0 || 0 || [[1_1|1/1]] || | | {| class="wikitable" |
| || 1 || 90.569 || [[21_20|21/20]], [[135_128|135/128]] ||
| | |- |
| || 2 || 181.139 || [[10_9|10/9]] ||
| | ! Degrees |
| || 3 || 271.708 || [[7_6|7/6]] ||
| | ! Cents |
| || 4 || 362.277 || [[16_13|16/13]] ||
| | ! Hekts |
| || 5 || 452.846 || [[13_10|13/10]] ||
| | ! Approximate Ratio |
| || 6 || 543.416 || [[15_11|15/11]], [[11_8|11/8]] ||
| | |- |
| || 7 || 633.985 || [[13_9|13/9]] ||
| | | colspan="3" style="text-align: center;" | 0 |
| || 8 || 724.554 || 35/23 ||
| | | [[1/1|1/1]] |
| || 9 || 815.124 || [[8_5|8/5]] ||
| | |- |
| || 10 || 905.693 || 27/16 ||
| | | 1 |
| || 11 || 996.262 || 16/9 ||
| | | 90.569 |
| || 12 || 1086.831 || [[15_8|15/8]] ||
| | | 61.905 |
| || 13 || 1177.401 || 69/35 ||
| | | [[21/20|21/20]], [[135/128|135/128]] |
| || 14 || 1267.970 || 27/13 ||
| | |- |
| || 15 || 1358.539 || 11/5 ([[11_10|11/10]] plus an octave) ||
| | | 2 |
| || 16 || 1449.109 || 30/13 ([[15_13|15/13]] plus an octave) ||
| | | 181.139 |
| || 17 || 1539.678 || 39/16 ||
| | | 123.81 |
| || 18 || 1630.247 || 18/7 ([[9_7|9/7]] plus an octave) ||
| | | [[10/9|10/9]] |
| || 19 || 1720.816 || 27/10 ||
| | |- |
| || 20 || 1811.386 || 20/7 ||
| | | 3 |
| || 21 || 1901.955 || 3/1 ||
| | | 271.708 |
| | | 185.714 |
| | | [[7/6|7/6]] |
| | |- |
| | | 4 |
| | | 362.277 |
| | | 247.619 |
| | | [[16/13|16/13]] |
| | |- |
| | | 5 |
| | | 452.846 |
| | | 309.524 |
| | | [[13/10|13/10]] |
| | |- |
| | | 6 |
| | | 543.416 |
| | | 371.429 |
| | | [[15/11|15/11]], [[11/8|11/8]] |
| | |- |
| | | 7 |
| | | 633.985 |
| | | 433.333 |
| | | [[13/9|13/9]] |
| | |- |
| | | 8 |
| | | 724.554 |
| | | 495.238 |
| | | 35/23 |
| | |- |
| | | 9 |
| | | 815.124 |
| | | 557.143 |
| | | [[8/5|8/5]] |
| | |- |
| | | 10 |
| | | 905.693 |
| | | 619.048 |
| | | 27/16 |
| | |- |
| | | 11 |
| | | 996.262 |
| | | 680.952 |
| | | 16/9 |
| | |- |
| | | 12 |
| | | 1086.831 |
| | | 742.857 |
| | | [[15/8|15/8]] |
| | |- |
| | | 13 |
| | | 1177.401 |
| | | 804.762 |
| | | 69/35 |
| | |- |
| | | 14 |
| | | 1267.97 |
| | | 866.667 |
| | | 27/13 |
| | |- |
| | | 15 |
| | | 1358.539 |
| | | 928.571 |
| | | 11/5 ([[11/10|11/10]] plus an octave), 24/11 (12/11 plus an octave) |
| | |- |
| | | 16 |
| | | 1449.109 |
| | | 990.476 |
| | | 30/13 ([[15/13|15/13]] plus an octave) |
| | |- |
| | | 17 |
| | | 1539.678 |
| | | 1052.381 |
| | | 39/16 |
| | |- |
| | | 18 |
| | | 1630.247 |
| | | 1114.286 |
| | | 18/7 ([[9/7|9/7]] plus an octave) |
| | |- |
| | | 19 |
| | | 1720.816 |
| | | 1076.19 |
| | | 27/10 |
| | |- |
| | | 20 |
| | | 1811.386 |
| | | 1238.095 |
| | | 20/7, 128/45 |
| | |- |
| | | 21 |
| | | 1901.955 |
| | | 1300 |
| | | 3/1 |
| | |} |
|
| |
|
| | | 21edt contains 6 intervals from [[7edt|7edt]] and 2 intervals from [[3edt|3edt]], meaning that it introduces 12 new intervals not available in lower edt's. These new intervals allow for construction of strange chords like 9:10:13:16:22:27:30... |
| 21edt contains 6 intervals from [[7edt]] and 2 intervals from [[3edt]], meaning that it introduces 12 new intervals not available in lower edt's. These new intervals allow for construction of strange chords like 9:10:13:16:22:27:30... | |
|
| |
|
| 21edt contains a 7L7s MOS similar to Whitewood, which I call Ivory. It has a period of 1/7 of the tritave and the generator is one step. The major scale is LsLsLsLsLsLsLs, and the minor scale is sLsLsLsLsLsLsL. | | 21edt contains a 7L7s MOS similar to Whitewood, which I call Ivory. It has a period of 1/7 of the tritave and the generator is one step. The major scale is LsLsLsLsLsLsLs, and the minor scale is sLsLsLsLsLsLsL. |
|
| |
|
| | 21edt also contains a 4L5s MOS similar to [[BP|BP]], with a 4:1 ratio of large to small; quite exaggerated from the optimal 2:1. Although the 7/3 is a little off, the 4L+5s BP scale is pretty. However, one of the star scales in 21edt is the 3L+6s (ssLssLssL and modes thereof) which is very harmonically rich, the cornerstone of which is the approximate 9:13:19 chord (which is just the [[3edt|3edt]] essentially tempered chord). |
|
| |
|
| 21edt also contains a 4L5s MOS similar to [[BP]], with a 4:1 ratio of large to small; quite exaggerated from the optimal 2:1. Not the best approximations but all within 20 cents, it has 5th (+20c), 7th(-16c), 10th (+2c), 11th (+15c), 13th (-3c), 17th (-14c), 19th (-24 c), 23rd (+6 c), and 37th (-2c) harmonics. For a lower division of the tritave that's quite a constellation! The chord is a little out of tune but it works, you can really sink into it.</pre></div>
| | == Harmonics== |
| <h4>Original HTML content:</h4>
| | Not the best approximations but all within 20 cents: it has 5th (+20¢), 7th (−16¢), 10th (+2¢), 11th (+15¢), 13th (−3¢), 17th (−14¢), 23rd (+6¢), and 37th (−2¢) harmonics. For a lower division of the tritave that's quite a constellation! The chord is a little out of tune but it works, you can really sink into it. |
| <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>21edt</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x21 Equal Divisions of the Tritave"></a><!-- ws:end:WikiTextHeadingRule:0 -->21 Equal Divisions of the Tritave</h1>
| |
| <br />
| |
| | |
|
| |
|
| <table class="wiki_table">
| | {{Harmonics in equal |
| <tr>
| | | steps = 21 |
| <td>Degrees<br />
| | | num = 3 |
| </td>
| | | denom = 1 |
| <td>Cents<br />
| | | intervals = integer |
| </td>
| | }} |
| <td>Approximate Ratio<br />
| | {{Harmonics in equal |
| </td>
| | | steps = 21 |
| </tr>
| | | num = 3 |
| <tr>
| | | denom = 1 |
| <td>0<br />
| | | start = 12 |
| </td>
| | | collapsed = 1 |
| <td>0<br />
| | | intervals = integer |
| </td>
| | }} |
| <td><a class="wiki_link" href="/1_1">1/1</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>90.569<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/21_20">21/20</a>, <a class="wiki_link" href="/135_128">135/128</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>181.139<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/10_9">10/9</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>271.708<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/7_6">7/6</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>362.277<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/16_13">16/13</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>452.846<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/13_10">13/10</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>543.416<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/15_11">15/11</a>, <a class="wiki_link" href="/11_8">11/8</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>633.985<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/13_9">13/9</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>724.554<br />
| |
| </td>
| |
| <td>35/23<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>815.124<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/8_5">8/5</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>905.693<br />
| |
| </td>
| |
| <td>27/16<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>996.262<br />
| |
| </td>
| |
| <td>16/9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>1086.831<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/15_8">15/8</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>1177.401<br />
| |
| </td>
| |
| <td>69/35<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>1267.970<br />
| |
| </td>
| |
| <td>27/13<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>1358.539<br />
| |
| </td>
| |
| <td>11/5 (<a class="wiki_link" href="/11_10">11/10</a> plus an octave)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16<br />
| |
| </td>
| |
| <td>1449.109<br />
| |
| </td>
| |
| <td>30/13 (<a class="wiki_link" href="/15_13">15/13</a> plus an octave)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17<br />
| |
| </td>
| |
| <td>1539.678<br />
| |
| </td>
| |
| <td>39/16<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>1630.247<br />
| |
| </td>
| |
| <td>18/7 (<a class="wiki_link" href="/9_7">9/7</a> plus an octave)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19<br />
| |
| </td>
| |
| <td>1720.816<br />
| |
| </td>
| |
| <td>27/10<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>1811.386<br />
| |
| </td>
| |
| <td>20/7<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21<br />
| |
| </td>
| |
| <td>1901.955<br />
| |
| </td>
| |
| <td>3/1<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | [[Category:tritave]] |
| <br />
| | {{todo|expand|improve synopsis}} |
| 21edt contains 6 intervals from <a class="wiki_link" href="/7edt">7edt</a> and 2 intervals from <a class="wiki_link" href="/3edt">3edt</a>, meaning that it introduces 12 new intervals not available in lower edt's. These new intervals allow for construction of strange chords like 9:10:13:16:22:27:30...<br />
| |
| <br />
| |
| 21edt contains a 7L7s MOS similar to Whitewood, which I call Ivory. It has a period of 1/7 of the tritave and the generator is one step. The major scale is LsLsLsLsLsLsLs, and the minor scale is sLsLsLsLsLsLsL.<br />
| |
| <br />
| |
| <br />
| |
| 21edt also contains a 4L5s MOS similar to <a class="wiki_link" href="/BP">BP</a>, with a 4:1 ratio of large to small; quite exaggerated from the optimal 2:1. Not the best approximations but all within 20 cents, it has 5th (+20c), 7th(-16c), 10th (+2c), 11th (+15c), 13th (-3c), 17th (-14c), 19th (-24 c), 23rd (+6 c), and 37th (-2c) harmonics. For a lower division of the tritave that's quite a constellation! The chord is a little out of tune but it works, you can really sink into it.</body></html></pre></div>
| |
| Prime factorization
|
3 × 7
|
| Step size
|
90.5693 ¢
|
| Octave
|
13\21edt (1177.4 ¢)
|
| Consistency limit
|
4
|
| Distinct consistency limit
|
4
|
21 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 21edt or 21ed3), is a nonoctave tuning system that divides the interval of 3/1 into 21 equal parts of about 90.6 ¢ each. Each step represents a frequency ratio of 31/21, or the 21st root of 3.
Intervals
| Degrees
|
Cents
|
Hekts
|
Approximate Ratio
|
| 0
|
1/1
|
| 1
|
90.569
|
61.905
|
21/20, 135/128
|
| 2
|
181.139
|
123.81
|
10/9
|
| 3
|
271.708
|
185.714
|
7/6
|
| 4
|
362.277
|
247.619
|
16/13
|
| 5
|
452.846
|
309.524
|
13/10
|
| 6
|
543.416
|
371.429
|
15/11, 11/8
|
| 7
|
633.985
|
433.333
|
13/9
|
| 8
|
724.554
|
495.238
|
35/23
|
| 9
|
815.124
|
557.143
|
8/5
|
| 10
|
905.693
|
619.048
|
27/16
|
| 11
|
996.262
|
680.952
|
16/9
|
| 12
|
1086.831
|
742.857
|
15/8
|
| 13
|
1177.401
|
804.762
|
69/35
|
| 14
|
1267.97
|
866.667
|
27/13
|
| 15
|
1358.539
|
928.571
|
11/5 (11/10 plus an octave), 24/11 (12/11 plus an octave)
|
| 16
|
1449.109
|
990.476
|
30/13 (15/13 plus an octave)
|
| 17
|
1539.678
|
1052.381
|
39/16
|
| 18
|
1630.247
|
1114.286
|
18/7 (9/7 plus an octave)
|
| 19
|
1720.816
|
1076.19
|
27/10
|
| 20
|
1811.386
|
1238.095
|
20/7, 128/45
|
| 21
|
1901.955
|
1300
|
3/1
|
21edt contains 6 intervals from 7edt and 2 intervals from 3edt, meaning that it introduces 12 new intervals not available in lower edt's. These new intervals allow for construction of strange chords like 9:10:13:16:22:27:30...
21edt contains a 7L7s MOS similar to Whitewood, which I call Ivory. It has a period of 1/7 of the tritave and the generator is one step. The major scale is LsLsLsLsLsLsLs, and the minor scale is sLsLsLsLsLsLsL.
21edt also contains a 4L5s MOS similar to BP, with a 4:1 ratio of large to small; quite exaggerated from the optimal 2:1. Although the 7/3 is a little off, the 4L+5s BP scale is pretty. However, one of the star scales in 21edt is the 3L+6s (ssLssLssL and modes thereof) which is very harmonically rich, the cornerstone of which is the approximate 9:13:19 chord (which is just the 3edt essentially tempered chord).
Harmonics
Not the best approximations but all within 20 cents: it has 5th (+20¢), 7th (−16¢), 10th (+2¢), 11th (+15¢), 13th (−3¢), 17th (−14¢), 23rd (+6¢), and 37th (−2¢) harmonics. For a lower division of the tritave that's quite a constellation! The chord is a little out of tune but it works, you can really sink into it.
Approximation of harmonics in 21edt
| Harmonic
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
| Error
|
Absolute (¢)
|
-22.6
|
+0.0
|
-45.2
|
+21.3
|
-22.6
|
-17.8
|
+22.8
|
+0.0
|
-1.3
|
+14.9
|
-45.2
|
| Relative (%)
|
-25.0
|
+0.0
|
-49.9
|
+23.6
|
-25.0
|
-19.6
|
+25.1
|
+0.0
|
-1.4
|
+16.4
|
-49.9
|
Steps
(reduced)
|
13
(13)
|
21
(0)
|
26
(5)
|
31
(10)
|
34
(13)
|
37
(16)
|
40
(19)
|
42
(0)
|
44
(2)
|
46
(4)
|
47
(5)
|
Approximation of harmonics in 21edt
| Harmonic
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
| Error
|
Absolute (¢)
|
-2.6
|
-40.4
|
+21.3
|
+0.2
|
-14.2
|
-22.6
|
-25.6
|
-23.9
|
-17.8
|
-7.7
|
+5.9
|
| Relative (%)
|
-2.9
|
-44.6
|
+23.6
|
+0.2
|
-15.7
|
-25.0
|
-28.3
|
-26.3
|
-19.6
|
-8.5
|
+6.5
|
Steps
(reduced)
|
49
(7)
|
50
(8)
|
52
(10)
|
53
(11)
|
54
(12)
|
55
(13)
|
56
(14)
|
57
(15)
|
58
(16)
|
59
(17)
|
60
(18)
|