|
|
| (13 intermediate revisions by 9 users not shown) |
| Line 1: |
Line 1: |
| <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:guest|guest]] and made on <tt>2012-05-09 11:59:51 UTC</tt>.<br>
| |
| : The original revision id was <tt>332528534</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">=6 Equal Divisions of the Tritave=
| |
|
| |
|
| || Degrees || Cents || ApproximateRatios ||
| | == Theory == |
| || 0 || 0 || 1/1 ||
| | Since 6edt contains one interval of [[2edt]] and two intervals of [[3edt]], it introduces 2 new notes unseen in previous edts. These new notes happen to approximate [[6/5]] and [[5/2]] very well, the former being only 1.351 [[cents]] sharp. |
| || 1 || 316.993 || [[6_5|6/5]], 65/54 ||
| | 6edt is therefore the smallest edt other than [[5edt]] to accurately approximate [[5-limit]] harmony, as well as some elements from the [[13-limit]] inherited from [[3edt]]. 6edt allows for construction of chords such as 2:5:6:15:18:26:31:45:54... |
| || 2 || 633.985 || [[13_9|13/9]] ||
| |
| || 3 || 950.978 || 19/11, 26/15 ||
| |
| || 4 || 1276.970 || 27/13 ||
| |
| || 5 || 1584.963 || 5/2 ([[5_4|5/4]] plus an octave) ||
| |
| || 6 || 1901.955 || 3/1 ||
| |
|
| |
|
| Since 6edt contains 1 intervals of [[2edt]] and 2 intervals of [[3edt]], it only introduces 2 new notes. These new notes happen to approximate 6/5 and 5/2 quite well.
| | === Harmonics === |
| 6edt is therefore smallest edt other than [[5edt]] to accurately approximate 5-limit harmony. 6edt allows for construction of chords chords such as 2:5:6:15:18:26:31:45:54...
| | {{Harmonics in equal|6|3|1|columns=16}} |
|
| |
|
| | == Intervals == |
| | {{Interval table}} |
|
| |
|
| ==6n-edt Family:==
| | [[Category:Nonoctave]] |
| [[12edt]] | | [[category:Macrotonal]] |
| [[18edt]] | |
| [[24edt]]
| |
| [[30edt]]</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>6edt</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x6 Equal Divisions of the Tritave"></a><!-- ws:end:WikiTextHeadingRule:0 -->6 Equal Divisions of the Tritave</h1>
| |
| <br />
| |
|
| |
|
| | | [[Category:todo:add sound example]] |
| <table class="wiki_table">
| |
| <tr>
| |
| <td>Degrees<br />
| |
| </td>
| |
| <td>Cents<br />
| |
| </td>
| |
| <td>ApproximateRatios<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td>1/1<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>316.993<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/6_5">6/5</a>, 65/54<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>633.985<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/13_9">13/9</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>950.978<br />
| |
| </td>
| |
| <td>19/11, 26/15<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>1276.970<br />
| |
| </td>
| |
| <td>27/13<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>1584.963<br />
| |
| </td>
| |
| <td>5/2 (<a class="wiki_link" href="/5_4">5/4</a> plus an octave)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>1901.955<br />
| |
| </td>
| |
| <td>3/1<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
| | |
| <br />
| |
| Since 6edt contains 1 intervals of <a class="wiki_link" href="/2edt">2edt</a> and 2 intervals of <a class="wiki_link" href="/3edt">3edt</a>, it only introduces 2 new notes. These new notes happen to approximate 6/5 and 5/2 quite well.<br />
| |
| 6edt is therefore smallest edt other than <a class="wiki_link" href="/5edt">5edt</a> to accurately approximate 5-limit harmony. 6edt allows for construction of chords chords such as 2:5:6:15:18:26:31:45:54...<br />
| |
| <br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x6 Equal Divisions of the Tritave-6n-edt Family:"></a><!-- ws:end:WikiTextHeadingRule:2 -->6n-edt Family:</h2>
| |
| <a class="wiki_link" href="/12edt">12edt</a><br />
| |
| <a class="wiki_link" href="/18edt">18edt</a><br />
| |
| <a class="wiki_link" href="/24edt">24edt</a><br />
| |
| <a class="wiki_link" href="/30edt">30edt</a></body></html></pre></div>
| |
| Prime factorization
|
2 × 3
|
| Step size
|
316.993 ¢
|
| Octave
|
4\6edt (1267.97 ¢) (→ 2\3edt)
|
| Consistency limit
|
7
|
| Distinct consistency limit
|
3
|
| Special properties
|
|
6 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 6edt or 6ed3), is a nonoctave tuning system that divides the interval of 3/1 into 6 equal parts of about 317 ¢ each. Each step represents a frequency ratio of 31/6, or the 6th root of 3.
Theory
Since 6edt contains one interval of 2edt and two intervals of 3edt, it introduces 2 new notes unseen in previous edts. These new notes happen to approximate 6/5 and 5/2 very well, the former being only 1.351 cents sharp.
6edt is therefore the smallest edt other than 5edt to accurately approximate 5-limit harmony, as well as some elements from the 13-limit inherited from 3edt. 6edt allows for construction of chords such as 2:5:6:15:18:26:31:45:54...
Harmonics
Approximation of harmonics in 6edt
| Harmonic
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
16
|
17
|
| Error
|
Absolute (¢)
|
+68
|
+0
|
+136
|
+67
|
+68
|
+118
|
-113
|
+0
|
+135
|
-30
|
+136
|
-3
|
-131
|
+67
|
-45
|
-150
|
| Relative (%)
|
+21.4
|
+0.0
|
+42.9
|
+21.0
|
+21.4
|
+37.3
|
-35.7
|
+0.0
|
+42.5
|
-9.6
|
+42.9
|
-0.8
|
-41.3
|
+21.0
|
-14.2
|
-47.3
|
Steps
(reduced)
|
4
(4)
|
6
(0)
|
8
(2)
|
9
(3)
|
10
(4)
|
11
(5)
|
11
(5)
|
12
(0)
|
13
(1)
|
13
(1)
|
14
(2)
|
14
(2)
|
14
(2)
|
15
(3)
|
15
(3)
|
15
(3)
|
Intervals
| Steps
|
Cents
|
Hekts
|
Approximate ratios
|
| 0
|
0
|
0
|
1/1
|
| 1
|
317
|
216.7
|
6/5, 7/6, 11/9, 13/11
|
| 2
|
634
|
433.3
|
7/5, 10/7, 13/9, 19/13
|
| 3
|
951
|
650
|
7/4, 12/7, 19/11
|
| 4
|
1268
|
866.7
|
15/7, 19/9, 21/10
|
| 5
|
1585
|
1083.3
|
5/2, 18/7
|
| 6
|
1902
|
1300
|
3/1
|