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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:xenwolf|xenwolf]] and made on <tt>2014-08-25 04:13:45 UTC</tt>.<br>
| |
| : The original revision id was <tt>519523810</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">
| |
| <span style="font-size: 18px; line-height: 27px;">**10 Equal Divisions of the Tritave**</span>
| |
|
| |
|
| || Degrees || Cents || Approximate Ratios ||
| | == Theory == |
| || 0 || 0 || <span style="color: #660000;">[[1_1|1/1]]</span> ||
| | 10edt most notably inherits the [[5/4]] from 5edt, and introduces some new harmonic elements, such as the 571-cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord. |
| || 1 || 190.196 || [[10_9|10/9]], [[28_25|28/25]] ||
| |
| || 2 || 380.391 || <span style="color: #660000;">[[5_4|5/4]]</span> ||
| |
| || 3 || 570.587 || [[7_5|7/5]] ||
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| || 4 || 760.782 || <span style="color: #660000;">[[14_9|14/9]]</span> ||
| |
| || 5 || 950.978 || [[19_11|19/11]]? ||
| |
| || 6 || 1141.173 || <span style="color: #660000;">[[27_14|27/14]]</span> ||
| |
| || 7 || 1331.369 || [[15_7|15/7]] ([[15_14|15/14]] plus an octave) ||
| |
| || 8 || 1521.564 || [[12_5|5/5]] (<span style="color: #660000;">[[6_5|6/5]]</span> plus an octave) ||
| |
| || 9 || 1711.760 || [[27_10|27/10]] ||
| |
| || 10 || 1901.955 || [[3_1|3/1]] ||
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|
| |
|
| | 10edt also splits the 5/4 in half, categorizing this tuning as a fringe variety of meantone. |
| | |
| | 10edt can serve as the generator chain for the [[pocus]] temperament, a [[temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup. |
|
| |
|
| 10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.</pre></div>
| | === Harmonics === |
| <h4>Original HTML content:</h4>
| | {{Harmonics in equal|10|3|1|columns=11}} |
| <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>10edt</title></head><body><br />
| | {{Harmonics in equal|10|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 10edt (continued)}} |
| <span style="font-size: 18px; line-height: 27px;"><strong>10 Equal Divisions of the Tritave</strong></span><br />
| |
| <br />
| |
|
| |
|
| | == Intervals == |
| | {| class="wikitable center-1 right-2 right-3" |
| | |- |
| | ! # |
| | ! [[Cent]]s |
| | ! [[Hekt]]s |
| | ! Approximate ratios |
| | |- |
| | | 0 |
| | | 0 |
| | | 0 |
| | | [[1/1]] |
| | |- |
| | | 1 |
| | | 190 |
| | | 130 |
| | | [[10/9]], [[28/25]] |
| | |- |
| | | 2 |
| | | 380 |
| | | 260 |
| | | [[5/4]] |
| | |- |
| | | 3 |
| | | 571 |
| | | 390 |
| | | [[7/5]] |
| | |- |
| | | 4 |
| | | 761 |
| | | 520 |
| | | [[14/9]] |
| | |- |
| | | 5 |
| | | 951 |
| | | 650 |
| | | [[26/15]], [[45/26]] |
| | |- |
| | | 6 |
| | | 1141 |
| | | 780 |
| | | [[27/14]] |
| | |- |
| | | 7 |
| | | 1331 |
| | | 910 |
| | | [[15/7]] |
| | |- |
| | | 8 |
| | | 1522 |
| | | 1040 |
| | | [[12/5]] |
| | |- |
| | | 9 |
| | | 1712 |
| | | 1170 |
| | | [[27/20|27/10]] |
| | |- |
| | | 10 |
| | | 1902 |
| | | 1300 |
| | | [[3/1]] |
| | |} |
|
| |
|
| <table class="wiki_table">
| | [[Category:Macrotonal]] |
| <tr>
| |
| <td>Degrees<br />
| |
| </td>
| |
| <td>Cents<br />
| |
| </td>
| |
| <td>Approximate Ratios<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td><span style="color: #660000;"><a class="wiki_link" href="/1_1">1/1</a></span><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>190.196<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/28_25">28/25</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>380.391<br />
| |
| </td>
| |
| <td><span style="color: #660000;"><a class="wiki_link" href="/5_4">5/4</a></span><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>570.587<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/7_5">7/5</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>760.782<br />
| |
| </td>
| |
| <td><span style="color: #660000;"><a class="wiki_link" href="/14_9">14/9</a></span><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>950.978<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/19_11">19/11</a>?<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>1141.173<br />
| |
| </td>
| |
| <td><span style="color: #660000;"><a class="wiki_link" href="/27_14">27/14</a></span><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>1331.369<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/15_7">15/7</a> (<a class="wiki_link" href="/15_14">15/14</a> plus an octave)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>1521.564<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/12_5">5/5</a> (<span style="color: #660000;"><a class="wiki_link" href="/6_5">6/5</a></span> plus an octave)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>1711.760<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/27_10">27/10</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>1901.955<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/3_1">3/1</a><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
| | |
| <br />
| |
| <br />
| |
| 10edt, like <a class="wiki_link" href="/5edt">5edt</a>, has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of &quot;meantone&quot; temperament.</body></html></pre></div>
| |
| Prime factorization
|
2 × 5
|
| Step size
|
190.196 ¢
|
| Octave
|
6\10edt (1141.17 ¢) (→ 3\5edt)
|
| Consistency limit
|
3
|
| Distinct consistency limit
|
3
|
10 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 10edt or 10ed3), is a nonoctave tuning system that divides the interval of 3/1 into 10 equal parts of about 190 ¢ each. Each step represents a frequency ratio of 31/10, or the 10th root of 3.
Theory
10edt most notably inherits the 5/4 from 5edt, and introduces some new harmonic elements, such as the 571-cent tritone, which can function as 7/5. We can use this to readily construct chords such as 4:5:7:12, although the 7/4, being 18 cents flat, does considerable damage to the consonance of this chord.
10edt also splits the 5/4 in half, categorizing this tuning as a fringe variety of meantone.
10edt can serve as the generator chain for the pocus temperament, a merge of sensamagic and 2.3.5.7.13 catakleismic, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.
Harmonics
Approximation of harmonics in 10edt
| Harmonic
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
| Error
|
Absolute (¢)
|
-58.8
|
+0.0
|
+72.5
|
+66.6
|
-58.8
|
+54.7
|
+13.7
|
+0.0
|
+7.8
|
+33.0
|
+72.5
|
| Relative (%)
|
-30.9
|
+0.0
|
+38.1
|
+35.0
|
-30.9
|
+28.8
|
+7.2
|
+0.0
|
+4.1
|
+17.3
|
+38.1
|
Steps
(reduced)
|
6
(6)
|
10
(0)
|
13
(3)
|
15
(5)
|
16
(6)
|
18
(8)
|
19
(9)
|
20
(0)
|
21
(1)
|
22
(2)
|
23
(3)
|
Approximation of harmonics in 10edt (continued)
| Harmonic
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
24
|
| Error
|
Absolute (¢)
|
-66.0
|
-4.1
|
+66.6
|
-45.1
|
+40.1
|
-58.8
|
+37.8
|
-51.0
|
+54.7
|
-25.8
|
+87.4
|
+13.7
|
| Relative (%)
|
-34.7
|
-2.2
|
+35.0
|
-23.7
|
+21.1
|
-30.9
|
+19.9
|
-26.8
|
+28.8
|
-13.6
|
+46.0
|
+7.2
|
Steps
(reduced)
|
23
(3)
|
24
(4)
|
25
(5)
|
25
(5)
|
26
(6)
|
26
(6)
|
27
(7)
|
27
(7)
|
28
(8)
|
28
(8)
|
29
(9)
|
29
(9)
|
Intervals