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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>
| | '''33ed4''' is the [[ed4|Equal Divisions of the Double Octave]] into 33 narrow chromatic semitones each of 72.727 [[cent]]s. It takes out every second step of [[33edo]] and falls between [[16edo]] and [[17edo]]. So even degree 16 or degree 17 can play the role of the [[octave]], depending on the actual melodic or harmonic situation in a given composition. So it can be seen as a kind of '''<span style="color: #080;">Equivocal Tuning</span>'''. |
| : This revision was by author [[User:jauernig|jauernig]] and made on <tt>2015-01-09 19:03:35 UTC</tt>.<br>
| |
| : The original revision id was <tt>536807394</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">**33ed4** is the [[ED4|Equal Divisions of the Double Octave]] into 33 narrow chromatic semitones each of 72.727 [[xenharmonic/cent|cent]]s. It takes out every second step of [[33edo]] and falls between [[16edo]] and [[17edo]]. So even degree 16 or degree 17 can play the role of the [[octave]], depending on the actual melodic or harmonic situation in a given composition. So it can be seen as a kind of equivocal tuning.
| |
|
| |
|
| It has a [[9_5|9/5]] which is 0.6 cents sharp, a [[7_5|7/5]] which is 0.7 cents flat, and a [[9_7|9/7]] which is 1.3 cents sharp. Therefore it is closely related to [[13edt]], the [[Bohlen-Pierce]] scale, although it has no pure [[3_1|3/1]], which is 11.1 cents flat. | | It has a [[9/5]] which is 0.6{{c}} sharp, a [[7/5]] which is 0.7{{c}} flat, and a [[9/7]] which is 1.3{{c}} sharp. Therefore it is closely related to [[13edt]], the [[Bohlen–Pierce scale]], although it has no pure [[3/1]], which is 11.1 cents flat. The lack of a [[3/2|pure fifth]] makes it also interesting. |
|
| |
|
| Furthermore it has some [[11-limit]], [[13-limit]], [[17-limit]] and even [[23-limit]] which are very close (most of them under or nearby 1 cent). | | Furthermore it has some [[11-limit]], [[13-limit]], [[17-limit]] and even [[23-limit]] which are very close (most of them under or nearby 1{{c}}). |
|
| |
|
| ===Intervals===
| | == Intervals == |
| ||~ degree ||~ in cents ||~ nearest JI | | {| class="wikitable right-all mw-collapsible" |
| interval ||~ in cents ||~ difference
| | |+ style="font-size: 105%;" | Intervals of 33ed4 |
| in cents ||
| | |- |
| || 1 || 72,7 || 24/23 || 73,7 || -1,0 ||
| | ! Degree |
| || 2 || 145,5 || 25/23 || 144,4 || 1,1 ||
| | ! Cents |
| || 3 || 218,2 || 17/15 || 216,6 || 1,6 ||
| | ! Nearest JI<br />interval |
| || 4 || 290,9 || 13/11 || 289,2 || 1,7 ||
| | ! Cents |
| || 5 || 363,6 || 16/13 || 359,5 || 4,1 ||
| | ! Difference<br />in cents |
| || 6 || 436,4 || 9/7 || 435,1 || 1,3 ||
| | |- |
| || 7 || 509,1 || 51/38 || 509,4 || -0,3 ||
| | | 1 |
| || 8 || 581,8 || 7/5 || 582,5 || -0,7 ||
| | | 72.7 |
| || 9 || 654,5 || 19/13 || 657,0 || -2,5 ||
| | | 24/23 |
| || 10 || 727,3 || 35/23 || 726,9 || 0,4 ||
| | | 73.7 |
| || 11 || 800,0 || 27/17 || 800,9 || -0,9 ||
| | | −1.0 |
| || 12 || 872,7 || 53/32 || 873,5 || -0,8 ||
| | |- |
| || 13 || 945,5 || 19/11 || 946,2 || -0,7 ||
| | | 2 |
| || 14 || 1018,2 || 9/5 || 1017,6 || 0,6 ||
| | | 145.5 |
| || 15 || 1090,9 || 15/8 || 1088,3 || 2,6 ||
| | | 25/23 |
| || 16 || 1163,6 || 45/23 || 1161,9 || 1,7 ||
| | | 144.4 |
| || 17 || 1236,4 || 49/24 || 1235,7 || 0,7 ||
| | | 1.1 |
| || 18 || 1309,1 || 32/15 || 1311,7 || -2,6 ||
| | |- |
| || 19 || 1381,8 || 20/9 || 1382,4 || -0,6 ||
| | | 3 |
| || 20 || 1454,5 || 44/19 || 1453,8 || 0,7 ||
| | | 218.2 |
| || 21 || 1527,3 || 29/12 || 1527,6 || -0,3 ||
| | | 17/15 |
| || 22 || 1600,0 || 68/27 || 1599,1 || 0,9 ||
| | | 216.6 |
| || 23 || 1672,7 || 21/8 || 1670,8 || 1,9 ||
| | | 1.6 |
| || 24 || 1745,5 || 52/19 || 1743,0 || 2,5 ||
| | |- |
| || 25 || 1818,2 || 20/7 || 1817,5 || 0,7 ||
| | | 4 |
| || 26 || 1890,9 || 116/39 || 1887,1 || 3,8 ||
| | | 290.9 |
| || 27 || 1963,6 || 28/9 || 1964,9 || -1,3 ||
| | | 13/11 |
| || 28 || 2036,4 || 13/4 || 2040,5 || -4,1 ||
| | | 289.2 |
| || 29 || 2109,1 || 44/13 || 2110,8 || -1,7 ||
| | | 1.7 |
| || 30 || 2181,8 || 60/17 || 2183,3 || -1,5 ||
| | |- |
| || 31 || 2254,5 || 114/31 || 2254,4 || 0,1 ||
| | | 5 |
| || 32 || 2327,3 || 23/6 || 2326,3 || 1,0 ||
| | | 363.6 |
| || 33 || 2400,0 || 4/1 || 2400,0 || 0,0 ||
| | | 16/13 |
| | | 359.5 |
| | | 4.1 |
| | |- style="font-weight: bold" |
| | | 6 |
| | | 436.4 |
| | | 9/7 |
| | | 435.1 |
| | | 1.3 |
| | |- |
| | | 7 |
| | | 509.1 |
| | | 51/38 |
| | | 509.4 |
| | | −0.3 |
| | |- style="font-weight: bold" |
| | | 8 |
| | | 581.8 |
| | | 7/5 |
| | | 582.5 |
| | | −0.7 |
| | |- |
| | | 9 |
| | | 654.5 |
| | | 19/13 |
| | | 657.0 |
| | | −2.5 |
| | |- |
| | | 10 |
| | | 727.3 |
| | | 35/23 |
| | | 726.9 |
| | | 0.4 |
| | |- |
| | | 11 |
| | | 800.0 |
| | | 27/17 |
| | | 800.9 |
| | | −0.9 |
| | |- |
| | | 12 |
| | | 872.7 |
| | | 53/32 |
| | | 873.5 |
| | | −0.8 |
| | |- |
| | | 13 |
| | | 945.5 |
| | | 19/11 |
| | | 946.2 |
| | | −0.7 |
| | |- style="font-weight: bold" |
| | | 14 |
| | | 1018.2 |
| | | 9/5 |
| | | 1017.6 |
| | | 0.6 |
| | |- |
| | | 15 |
| | | 1090.9 |
| | | 15/8 |
| | | 1088.3 |
| | | 2.6 |
| | |- style="font-weight: bold; color: #080" |
| | | 16 |
| | | 1163.6 |
| | | 45/23 |
| | | 1161.9 |
| | | 1.7 |
| | |- style="font-weight: bold; color: #080" |
| | | 17 |
| | | 1236.4 |
| | | 49/24 |
| | | 1235.7 |
| | | 0.7 |
| | |- |
| | | 18 |
| | | 1309.1 |
| | | 32/15 |
| | | 1311.7 |
| | | −2.6 |
| | |- style="font-weight: bold" |
| | | 19 |
| | | 1381.8 |
| | | 20/9 |
| | | 1382.4 |
| | | −0.6 |
| | |- |
| | | 20 |
| | | 1454.5 |
| | | 44/19 |
| | | 1453.8 |
| | | 0.7 |
| | |- |
| | | 21 |
| | | 1527.3 |
| | | 29/12 |
| | | 1527.6 |
| | | −0.3 |
| | |- |
| | | 22 |
| | | 1600.0 |
| | | 68/27 |
| | | 1599.1 |
| | | 0.9 |
| | |- |
| | | 23 |
| | | 1672.7 |
| | | 21/8 |
| | | 1670.8 |
| | | 1.9 |
| | |- |
| | | 24 |
| | | 1745.5 |
| | | 52/19 |
| | | 1743.0 |
| | | 2.5 |
| | |- style="font-weight: bold" |
| | | 25 |
| | | 1818.2 |
| | | 20/7 |
| | | 1817.5 |
| | | 0.7 |
| | |- |
| | | 26 |
| | | 1890.9 |
| | | 116/39 |
| | | 1887.1 |
| | | 3.8 |
| | |- style="font-weight: bold" |
| | | 27 |
| | | 1963.6 |
| | | 28/9 |
| | | 1964.9 |
| | | −1.3 |
| | |- |
| | | 28 |
| | | 2036.4 |
| | | 13/4 |
| | | 2040.5 |
| | | −4.1 |
| | |- |
| | | 29 |
| | | 2109.1 |
| | | 44/13 |
| | | 2110.8 |
| | | −1.7 |
| | |- |
| | | 30 |
| | | 2181.8 |
| | | 60/17 |
| | | 2183.3 |
| | | −1.5 |
| | |- |
| | | 31 |
| | | 2254.5 |
| | | 114/31 |
| | | 2254.4 |
| | | 0.1 |
| | |- |
| | | 32 |
| | | 2327.3 |
| | | 23/6 |
| | | 2326.3 |
| | | 1.0 |
| | |- style="font-weight: bold" |
| | | 33 |
| | | 2400.0 |
| | | 4/1 |
| | | 2400.0 |
| | | 0.0 |
| | |} |
|
| |
|
| ===Music=== | | == Harmonics == |
| [[http://soundcloud.com/ahornberg/sets/equivocal-tuning-33ed4|Equivocal Tuning]] by Ahornberg</pre></div>
| | {{Harmonics in equal |
| <h4>Original HTML content:</h4>
| | | steps = 33 |
| <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>33ed4</title></head><body><strong>33ed4</strong> is the <a class="wiki_link" href="/ED4">Equal Divisions of the Double Octave</a> into 33 narrow chromatic semitones each of 72.727 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/cent">cent</a>s. It takes out every second step of <a class="wiki_link" href="/33edo">33edo</a> and falls between <a class="wiki_link" href="/16edo">16edo</a> and <a class="wiki_link" href="/17edo">17edo</a>. So even degree 16 or degree 17 can play the role of the <a class="wiki_link" href="/octave">octave</a>, depending on the actual melodic or harmonic situation in a given composition. So it can be seen as a kind of equivocal tuning.<br />
| | | num = 4 |
| <br />
| | | denom = 1 |
| It has a <a class="wiki_link" href="/9_5">9/5</a> which is 0.6 cents sharp, a <a class="wiki_link" href="/7_5">7/5</a> which is 0.7 cents flat, and a <a class="wiki_link" href="/9_7">9/7</a> which is 1.3 cents sharp. Therefore it is closely related to <a class="wiki_link" href="/13edt">13edt</a>, the <a class="wiki_link" href="/Bohlen-Pierce">Bohlen-Pierce</a> scale, although it has no pure <a class="wiki_link" href="/3_1">3/1</a>, which is 11.1 cents flat.<br />
| | }} |
| <br />
| | {{Harmonics in equal |
| Furthermore it has some <a class="wiki_link" href="/11-limit">11-limit</a>, <a class="wiki_link" href="/13-limit">13-limit</a>, <a class="wiki_link" href="/17-limit">17-limit</a> and even <a class="wiki_link" href="/23-limit">23-limit</a> which are very close (most of them under or nearby 1 cent).<br />
| | | steps = 33 |
| <br />
| | | num = 4 |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 -->Intervals</h3>
| | | denom = 1 |
|
| | | start = 12 |
| | | collapsed = 1 |
| | }} |
|
| |
|
| <table class="wiki_table">
| | == Music == |
| <tr>
| | * [http://soundcloud.com/ahornberg/sets/equivocal-tuning-33ed4 Equivocal Tuning] — Set of compositions by Ahornberg |
| <th>degree<br />
| |
| </th>
| |
| <th>in cents<br />
| |
| </th>
| |
| <th>nearest JI<br />
| |
| interval<br />
| |
| </th>
| |
| <th>in cents<br />
| |
| </th>
| |
| <th>difference<br />
| |
| in cents<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>72,7<br />
| |
| </td>
| |
| <td>24/23<br />
| |
| </td>
| |
| <td>73,7<br />
| |
| </td>
| |
| <td>-1,0<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>145,5<br />
| |
| </td>
| |
| <td>25/23<br />
| |
| </td>
| |
| <td>144,4<br />
| |
| </td>
| |
| <td>1,1<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>218,2<br />
| |
| </td>
| |
| <td>17/15<br />
| |
| </td>
| |
| <td>216,6<br />
| |
| </td>
| |
| <td>1,6<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>290,9<br />
| |
| </td>
| |
| <td>13/11<br />
| |
| </td>
| |
| <td>289,2<br />
| |
| </td>
| |
| <td>1,7<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>363,6<br />
| |
| </td>
| |
| <td>16/13<br />
| |
| </td>
| |
| <td>359,5<br />
| |
| </td>
| |
| <td>4,1<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>436,4<br />
| |
| </td>
| |
| <td>9/7<br />
| |
| </td>
| |
| <td>435,1<br />
| |
| </td>
| |
| <td>1,3<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>509,1<br />
| |
| </td>
| |
| <td>51/38<br />
| |
| </td>
| |
| <td>509,4<br />
| |
| </td>
| |
| <td>-0,3<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>581,8<br />
| |
| </td>
| |
| <td>7/5<br />
| |
| </td>
| |
| <td>582,5<br />
| |
| </td>
| |
| <td>-0,7<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>654,5<br />
| |
| </td>
| |
| <td>19/13<br />
| |
| </td>
| |
| <td>657,0<br />
| |
| </td>
| |
| <td>-2,5<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>727,3<br />
| |
| </td>
| |
| <td>35/23<br />
| |
| </td>
| |
| <td>726,9<br />
| |
| </td>
| |
| <td>0,4<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>800,0<br />
| |
| </td>
| |
| <td>27/17<br />
| |
| </td>
| |
| <td>800,9<br />
| |
| </td>
| |
| <td>-0,9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>872,7<br />
| |
| </td>
| |
| <td>53/32<br />
| |
| </td>
| |
| <td>873,5<br />
| |
| </td>
| |
| <td>-0,8<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>945,5<br />
| |
| </td>
| |
| <td>19/11<br />
| |
| </td>
| |
| <td>946,2<br />
| |
| </td>
| |
| <td>-0,7<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>1018,2<br />
| |
| </td>
| |
| <td>9/5<br />
| |
| </td>
| |
| <td>1017,6<br />
| |
| </td>
| |
| <td>0,6<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>1090,9<br />
| |
| </td>
| |
| <td>15/8<br />
| |
| </td>
| |
| <td>1088,3<br />
| |
| </td>
| |
| <td>2,6<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16<br />
| |
| </td>
| |
| <td>1163,6<br />
| |
| </td>
| |
| <td>45/23<br />
| |
| </td>
| |
| <td>1161,9<br />
| |
| </td>
| |
| <td>1,7<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17<br />
| |
| </td>
| |
| <td>1236,4<br />
| |
| </td>
| |
| <td>49/24<br />
| |
| </td>
| |
| <td>1235,7<br />
| |
| </td>
| |
| <td>0,7<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>1309,1<br />
| |
| </td>
| |
| <td>32/15<br />
| |
| </td>
| |
| <td>1311,7<br />
| |
| </td>
| |
| <td>-2,6<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19<br />
| |
| </td>
| |
| <td>1381,8<br />
| |
| </td>
| |
| <td>20/9<br />
| |
| </td>
| |
| <td>1382,4<br />
| |
| </td>
| |
| <td>-0,6<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>1454,5<br />
| |
| </td>
| |
| <td>44/19<br />
| |
| </td>
| |
| <td>1453,8<br />
| |
| </td>
| |
| <td>0,7<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21<br />
| |
| </td>
| |
| <td>1527,3<br />
| |
| </td>
| |
| <td>29/12<br />
| |
| </td>
| |
| <td>1527,6<br />
| |
| </td>
| |
| <td>-0,3<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22<br />
| |
| </td>
| |
| <td>1600,0<br />
| |
| </td>
| |
| <td>68/27<br />
| |
| </td>
| |
| <td>1599,1<br />
| |
| </td>
| |
| <td>0,9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
| |
| </td>
| |
| <td>1672,7<br />
| |
| </td>
| |
| <td>21/8<br />
| |
| </td>
| |
| <td>1670,8<br />
| |
| </td>
| |
| <td>1,9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>24<br />
| |
| </td>
| |
| <td>1745,5<br />
| |
| </td>
| |
| <td>52/19<br />
| |
| </td>
| |
| <td>1743,0<br />
| |
| </td>
| |
| <td>2,5<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>25<br />
| |
| </td>
| |
| <td>1818,2<br />
| |
| </td>
| |
| <td>20/7<br />
| |
| </td>
| |
| <td>1817,5<br />
| |
| </td>
| |
| <td>0,7<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>26<br />
| |
| </td>
| |
| <td>1890,9<br />
| |
| </td>
| |
| <td>116/39<br />
| |
| </td>
| |
| <td>1887,1<br />
| |
| </td>
| |
| <td>3,8<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>27<br />
| |
| </td>
| |
| <td>1963,6<br />
| |
| </td>
| |
| <td>28/9<br />
| |
| </td>
| |
| <td>1964,9<br />
| |
| </td>
| |
| <td>-1,3<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>28<br />
| |
| </td>
| |
| <td>2036,4<br />
| |
| </td>
| |
| <td>13/4<br />
| |
| </td>
| |
| <td>2040,5<br />
| |
| </td>
| |
| <td>-4,1<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>29<br />
| |
| </td>
| |
| <td>2109,1<br />
| |
| </td>
| |
| <td>44/13<br />
| |
| </td>
| |
| <td>2110,8<br />
| |
| </td>
| |
| <td>-1,7<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>30<br />
| |
| </td>
| |
| <td>2181,8<br />
| |
| </td>
| |
| <td>60/17<br />
| |
| </td>
| |
| <td>2183,3<br />
| |
| </td>
| |
| <td>-1,5<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>31<br />
| |
| </td>
| |
| <td>2254,5<br />
| |
| </td>
| |
| <td>114/31<br />
| |
| </td>
| |
| <td>2254,4<br />
| |
| </td>
| |
| <td>0,1<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>32<br />
| |
| </td>
| |
| <td>2327,3<br />
| |
| </td>
| |
| <td>23/6<br />
| |
| </td>
| |
| <td>2326,3<br />
| |
| </td>
| |
| <td>1,0<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>33<br />
| |
| </td>
| |
| <td>2400,0<br />
| |
| </td>
| |
| <td>4/1<br />
| |
| </td>
| |
| <td>2400,0<br />
| |
| </td>
| |
| <td>0,0<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | [[Category:Equal-step tuning]] |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--Music"></a><!-- ws:end:WikiTextHeadingRule:2 -->Music</h3>
| | {{todo|expand}} |
| <a class="wiki_link_ext" href="http://soundcloud.com/ahornberg/sets/equivocal-tuning-33ed4" rel="nofollow">Equivocal Tuning</a> by Ahornberg</body></html></pre></div>
| |