15edo: Difference between revisions
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:k9assassin|k9assassin]] and made on <tt>2016-04-21 16:07:23 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>580837891</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<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> | <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">[[toc]] | <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">[[toc]] | ||
=15- | =15-Tone Equal Temperament (5-EDO)= | ||
<span style="display: block; text-align: right;">[[15平均律|日本語]] | |||
==Summary== | |||
15 Equal or 15 EDO is a tuning which divides the octave into 15 equally spaced pitches. It can be thought of as three sets of 5-EDO which do not connect by fifths. The fifth at 720 cents is quite wide yet still useable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. The perfect fifth of 15 EDO returns to the octave if stacked five times which is radically different than a meantone system. | |||
==Sagittal Notation:== | |||
15 EDO can be notated in sagittal as a chain of fifths thus 5 EDO would correspond to a pentatonic scale. However, because a chain of five fifths returns to the octave, the semitone is tempered out in 5 EDO equating E with F, B with C, D with Eb and so on. In this way, 5 EDO is the naturals on the staff. EDIT IN PROGRESS | |||
<span style="display: block; text-align: right;"> | |||
</span><span style="display: block; text-align: right;">[[15平均律|日本語]] | |||
</span> | </span> | ||
From [[@http://en.wikipedia.org/wiki/15_equal_temperament|Wikipedia]]: | From [[@http://en.wikipedia.org/wiki/15_equal_temperament|Wikipedia]]: | ||
"In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the [[octave]] into 15 equal steps. Each step represents a frequency ratio of 2^(1/15), or 80 [[cent]]s. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of [[5edo|5 equal divisions of the octave]] (or five scales of [[3edo]])." | "In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the [[octave]] into 15 equal steps. Each step represents a frequency ratio of 2^(1/15), or 80 [[cent]]s. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of [[5edo|5 equal divisions of the octave]] (or five scales of [[3edo]])." | ||
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The following table shows how [[Just-24|some prominent just intervals]] are represented in 15edo (ordered by absolute error). | The following table shows how [[Just-24|some prominent just intervals]] are represented in 15edo (ordered by absolute error). | ||
|| **Interval, complement** || **Error (abs., in [[cent|cents]])** || | || **Interval, complement** || **Error (abs., in [[cent|cents]])** || | ||
||= [[18_13|18/13]], [[13_9|13/9]] | ||= [[18_13|18/13]], [[13_9|13/9]] ||= 3.382 || | ||
||= [[6_5|6/5]], | ||= [[6_5|6/5]], [[5_3|5/3]] ||= 4.359 || | ||
||= [[11_10|11/10]], [[20_11|20/11]] ||= | ||= [[11_10|11/10]], [[20_11|20/11]] ||= 5.004 || | ||
||= [[15_13|15/13]], [[26_15|26/15]] ||= | ||= [[15_13|15/13]], [[26_15|26/15]] ||= 7.741 || | ||
||= [[11_8|11/8]], | ||= [[11_8|11/8]], [[16_11|16/11]] ||= 8.682 || | ||
||= [[8_7|8/7]], | ||= [[8_7|8/7]], [[7_4|7/4]] ||= 8.826 || | ||
||= [[12_11|12/11]], [[11_6|11/6]] | ||= [[12_11|12/11]], [[11_6|11/6]] ||= 9.363 || | ||
||= [[5_4|5/4]], | ||= [[5_4|5/4]], [[8_5|8/5]] ||= 13.686 || | ||
||= [[14_11|14/11]], [[11_7|11/7]] | ||= [[14_11|14/11]], [[11_7|11/7]] ||= 17.508 || | ||
||= [[4_3|4/3]], | ||= [[4_3|4/3]], [[3_2|3/2]] ||= 18.045 || | ||
||= [[13_12|13/12]], [[24_13|24/13]] ||= 21.427 || | ||= [[13_12|13/12]], [[24_13|24/13]] ||= 21.427 || | ||
||= [[10_9|10/9]], | ||= [[10_9|10/9]], [[9_5|9/5]] ||= 22.404 || | ||
||= [[7_5|7/5]], | ||= [[7_5|7/5]], [[10_7|10/7]] ||= 22.512 || | ||
||= [[15_11|15/11]], [[22_15|22/15]] ||= 23.049 || | ||= [[15_11|15/11]], [[22_15|22/15]] ||= 23.049 || | ||
||= [[13_10|13/10]], [[20_13|20/13]] ||= 25.786 || | ||= [[13_10|13/10]], [[20_13|20/13]] ||= 25.786 || | ||
||= [[7_6|7/6]], | ||= [[7_6|7/6]], [[12_7|12/7]] ||= 26.871 || | ||
||= [[11_9|11/9]], | ||= [[11_9|11/9]], [[18_11|18/11]] ||= 27.408 || | ||
||= [[13_11|13/11]], [[22_13|22/13]] ||= 30.790 || | ||= [[13_11|13/11]], [[22_13|22/13]] ||= 30.790 || | ||
||= [[14_13|14/13]], [[13_7|13/7]] | ||= [[14_13|14/13]], [[13_7|13/7]] ||= 31.702 || | ||
||= [[16_15|16/15]], [[15_8|15/8]] | ||= [[16_15|16/15]], [[15_8|15/8]] ||= 31.731 || | ||
||= [[9_7|9/7]], | ||= [[9_7|9/7]], [[14_9|14/9]] ||= 35.084 || | ||
||= [[9_8|9/8]], | ||= [[9_8|9/8]], [[16_9|16/9]] ||= 36.090 || | ||
||= [[15_14|15/14]], [[28_15|28/15]] ||= 39.443 || | ||= [[15_14|15/14]], [[28_15|28/15]] ||= 39.443 || | ||
||= [[16_13|16/13]], [[13_8|13/8]] | ||= [[16_13|16/13]], [[13_8|13/8]] ||= 39.472 || | ||
= = | |||
=Notation= | =Alternative Notation= | ||
There are a variety of ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the "main focus" of 15-edo composition. | There are a variety of other ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the "main focus" of 15-edo composition. | ||
==Blackwood Notation== | ==Blackwood Notation== | ||
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[[media type="custom" key="23473468"]]</pre></div> | [[media type="custom" key="23473468"]]</pre></div> | ||
<h4>Original HTML content:</h4> | <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>15edo</title></head><body><!-- ws:start:WikiTextTocRule: | <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>15edo</title></head><body><!-- ws:start:WikiTextTocRule:36:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 1em;"><a href="#x15-Tone Equal Temperament (5-EDO)">15-Tone Equal Temperament (5-EDO)</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 2em;"><a href="#x15-Tone Equal Temperament (5-EDO)-Summary">Summary</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 2em;"><a href="#x15-Tone Equal Temperament (5-EDO)-Sagittal Notation:">Sagittal Notation:</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 1em;"><a href="#Images">Images</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 2em;"><a href="#Intervals-Selected just intervals by error">Selected just intervals by error</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 1em;"><a href="#toc6"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 1em;"><a href="#Alternative Notation">Alternative Notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#Alternative Notation-Blackwood Notation">Blackwood Notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#Alternative Notation-Porcupine Notation">Porcupine Notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#Alternative Notation-Porcupine[8] Based Notation">Porcupine[8] Based Notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 2em;"><a href="#Alternative Notation-Interval names for Porcupine[8]:">Interval names for Porcupine[8]:</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 1em;"><a href="#Rank two temperaments">Rank two temperaments</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 1em;"><a href="#Commas">Commas</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 1em;"><a href="#Theory">Theory</a></div> | ||
< | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 1em;"><a href="#Practical Theory / Books">Practical Theory / Books</a></div> | ||
<span style="display: block; text-align: right;"><a class="wiki_link" href="/15%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br /> | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 1em;"><a href="#Compositions">Compositions</a></div> | ||
<!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --></div> | |||
<!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc0"><a name="x15-Tone Equal Temperament (5-EDO)"></a><!-- ws:end:WikiTextHeadingRule:2 -->15-Tone Equal Temperament (5-EDO)</h1> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc1"><a name="x15-Tone Equal Temperament (5-EDO)-Summary"></a><!-- ws:end:WikiTextHeadingRule:4 -->Summary</h2> | |||
15 Equal or 15 EDO is a tuning which divides the octave into 15 equally spaced pitches. It can be thought of as three sets of 5-EDO which do not connect by fifths. The fifth at 720 cents is quite wide yet still useable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. The perfect fifth of 15 EDO returns to the octave if stacked five times which is radically different than a meantone system.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc2"><a name="x15-Tone Equal Temperament (5-EDO)-Sagittal Notation:"></a><!-- ws:end:WikiTextHeadingRule:6 -->Sagittal Notation:</h2> | |||
15 EDO can be notated in sagittal as a chain of fifths thus 5 EDO would correspond to a pentatonic scale. However, because a chain of five fifths returns to the octave, the semitone is tempered out in 5 EDO equating E with F, B with C, D with Eb and so on. In this way, 5 EDO is the naturals on the staff. EDIT IN PROGRESS <br /> | |||
<br /> | |||
<br /> | |||
<span style="display: block; text-align: right;"><br /> | |||
</span><span style="display: block; text-align: right;"><a class="wiki_link" href="/15%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br /> | |||
</span><br /> | </span><br /> | ||
<br /> | |||
<br /> | |||
From <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/15_equal_temperament" rel="nofollow" target="_blank">Wikipedia</a>:<br /> | From <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/15_equal_temperament" rel="nofollow" target="_blank">Wikipedia</a>:<br /> | ||
&quot;In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the <a class="wiki_link" href="/octave">octave</a> into 15 equal steps. Each step represents a frequency ratio of 2^(1/15), or 80 <a class="wiki_link" href="/cent">cent</a>s. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of <a class="wiki_link" href="/5edo">5 equal divisions of the octave</a> (or five scales of <a class="wiki_link" href="/3edo">3edo</a>).&quot;<br /> | &quot;In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the <a class="wiki_link" href="/octave">octave</a> into 15 equal steps. Each step represents a frequency ratio of 2^(1/15), or 80 <a class="wiki_link" href="/cent">cent</a>s. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of <a class="wiki_link" href="/5edo">5 equal divisions of the octave</a> (or five scales of <a class="wiki_link" href="/3edo">3edo</a>).&quot;<br /> | ||
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A recommended method to the notation of 15-edo by some is a system based on porcupine[8] in which eight nominals form the base diatonic scale. In this sense, the &quot;quill&quot; is the name given to the two step interval (160c) of 15-edo while the &quot;small quill&quot; (80c) is the chroma of 15-edo. This produces a very consistent notation for both porcupine[8] and Blackwood[10] and seems to work much better than attempting to put 15-edo into a seven nominal based framework.<br /> | A recommended method to the notation of 15-edo by some is a system based on porcupine[8] in which eight nominals form the base diatonic scale. In this sense, the &quot;quill&quot; is the name given to the two step interval (160c) of 15-edo while the &quot;small quill&quot; (80c) is the chroma of 15-edo. This produces a very consistent notation for both porcupine[8] and Blackwood[10] and seems to work much better than attempting to put 15-edo into a seven nominal based framework.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc3"><a name="Images"></a><!-- ws:end:WikiTextHeadingRule:8 -->Images</h1> | ||
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<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1073:&lt;img src=&quot;/file/view/15edo%20wheel.png/516943384/225x225/15edo%20wheel.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 225px; width: 225px;&quot; /&gt; --><img src="/file/view/15edo%20wheel.png/516943384/225x225/15edo%20wheel.png" alt="15edo wheel.png" title="15edo wheel.png" style="height: 225px; width: 225px;" /><!-- ws:end:WikiTextLocalImageRule:1073 --><!-- ws:start:WikiTextLocalImageRule:1074:&lt;img src=&quot;/file/view/15edo%20wheel%2002.png/516943402/250x250/15edo%20wheel%2002.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 250px; width: 250px;&quot; /&gt; --><img src="/file/view/15edo%20wheel%2002.png/516943402/250x250/15edo%20wheel%2002.png" alt="15edo wheel 02.png" title="15edo wheel 02.png" style="height: 250px; width: 250px;" /><!-- ws:end:WikiTextLocalImageRule:1074 --><!-- ws:start:WikiTextLocalImageRule:1075:&lt;img src=&quot;/file/view/15edo%20wheel%2003.png/516943418/220x220/15edo%20wheel%2003.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 220px; width: 220px;&quot; /&gt; --><img src="/file/view/15edo%20wheel%2003.png/516943418/220x220/15edo%20wheel%2003.png" alt="15edo wheel 03.png" title="15edo wheel 03.png" style="height: 220px; width: 220px;" /><!-- ws:end:WikiTextLocalImageRule:1075 --><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc4"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:10 -->Intervals</h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc5"><a name="Intervals-Selected just intervals by error"></a><!-- ws:end:WikiTextHeadingRule:12 -->Selected just intervals by error</h2> | ||
The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 15edo (ordered by absolute error).<br /> | The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 15edo (ordered by absolute error).<br /> | ||
| Line 614: | Line 641: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, | <td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/5_3">5/3</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">4.359<br /> | <td style="text-align: center;">4.359<br /> | ||
| Line 632: | Line 659: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, | <td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, <a class="wiki_link" href="/16_11">16/11</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">8.682<br /> | <td style="text-align: center;">8.682<br /> | ||
| Line 638: | Line 665: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, | <td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, <a class="wiki_link" href="/7_4">7/4</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">8.826<br /> | <td style="text-align: center;">8.826<br /> | ||
| Line 650: | Line 677: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, | <td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/8_5">8/5</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">13.686<br /> | <td style="text-align: center;">13.686<br /> | ||
| Line 662: | Line 689: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, | <td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/3_2">3/2</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">18.045<br /> | <td style="text-align: center;">18.045<br /> | ||
| Line 674: | Line 701: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, | <td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/9_5">9/5</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">22.404<br /> | <td style="text-align: center;">22.404<br /> | ||
| Line 680: | Line 707: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, | <td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, <a class="wiki_link" href="/10_7">10/7</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">22.512<br /> | <td style="text-align: center;">22.512<br /> | ||
| Line 698: | Line 725: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, | <td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/12_7">12/7</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">26.871<br /> | <td style="text-align: center;">26.871<br /> | ||
| Line 704: | Line 731: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, | <td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, <a class="wiki_link" href="/18_11">18/11</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">27.408<br /> | <td style="text-align: center;">27.408<br /> | ||
| Line 728: | Line 755: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, | <td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, <a class="wiki_link" href="/14_9">14/9</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">35.084<br /> | <td style="text-align: center;">35.084<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, | <td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, <a class="wiki_link" href="/16_9">16/9</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">36.090<br /> | <td style="text-align: center;">36.090<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc6"><!-- ws:end:WikiTextHeadingRule:14 --> </h1> | ||
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc7"><a name="Alternative Notation"></a><!-- ws:end:WikiTextHeadingRule:16 -->Alternative Notation</h1> | |||
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There are a variety of ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the &quot;main focus&quot; of 15-edo composition.<br /> | There are a variety of other ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the &quot;main focus&quot; of 15-edo composition.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc8"><a name="Alternative Notation-Blackwood Notation"></a><!-- ws:end:WikiTextHeadingRule:18 -->Blackwood Notation</h2> | ||
<ul><li><strong>Decimal Version:</strong> Using the nominals 1-0 (with 0 representing &quot;10&quot;), one of the three chains of 5-edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).</li><li><strong>Guitar Version:</strong> On a 15-edo guitar, because the &quot;perfect fourth&quot; comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or &quot;perfect fourths&quot; and &quot;perfect fifths&quot;) that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.</li></ul><br /> | <ul><li><strong>Decimal Version:</strong> Using the nominals 1-0 (with 0 representing &quot;10&quot;), one of the three chains of 5-edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).</li><li><strong>Guitar Version:</strong> On a 15-edo guitar, because the &quot;perfect fourth&quot; comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or &quot;perfect fourths&quot; and &quot;perfect fifths&quot;) that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.</li></ul><br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc9"><a name="Alternative Notation-Porcupine Notation"></a><!-- ws:end:WikiTextHeadingRule:20 -->Porcupine Notation</h2> | ||
<ul><li>See the main <a class="wiki_link" href="/Porcupine%20Notation">porcupine notation</a> page.</li></ul><!-- ws:start:WikiTextHeadingRule: | <ul><li>See the main <a class="wiki_link" href="/Porcupine%20Notation">porcupine notation</a> page.</li></ul><!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc10"><a name="Alternative Notation-Porcupine[8] Based Notation"></a><!-- ws:end:WikiTextHeadingRule:22 -->Porcupine[8] Based Notation</h2> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1076:&lt;img src=&quot;/file/view/porcupine%20alphabet.png/561701315/400x81/porcupine%20alphabet.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 81px; width: 400px;&quot; /&gt; --><img src="/file/view/porcupine%20alphabet.png/561701315/400x81/porcupine%20alphabet.png" alt="porcupine alphabet.png" title="porcupine alphabet.png" style="height: 81px; width: 400px;" /><!-- ws:end:WikiTextLocalImageRule:1076 --><br /> | ||
William Lynch proposes notating 15 EDO based on an eight nominal system using the old english letters: b c d ð e f þ æ as the alphabet.<br /> | William Lynch proposes notating 15 EDO based on an eight nominal system using the old english letters: b c d ð e f þ æ as the alphabet.<br /> | ||
Porcupine[7] can also be notated by omitting &quot;ð&quot;<br /> | Porcupine[7] can also be notated by omitting &quot;ð&quot;<br /> | ||
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alphabet with ups and downs: b b^ c c^ d ð ð^ e e^ f f^ þ þ^ æ æ^<br /> | alphabet with ups and downs: b b^ c c^ d ð ð^ e e^ f f^ þ þ^ æ æ^<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc11"><a name="Alternative Notation-Interval names for Porcupine[8]:"></a><!-- ws:end:WikiTextHeadingRule:24 -->Interval names for Porcupine[8]:</h2> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc12"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:26 -->Rank two temperaments</h1> | ||
<a class="wiki_link" href="/List%20of%2015et%20rank%20two%20temperaments%20by%20badness">List of 15et rank two temperaments by badness</a><br /> | <a class="wiki_link" href="/List%20of%2015et%20rank%20two%20temperaments%20by%20badness">List of 15et rank two temperaments by badness</a><br /> | ||
<a class="wiki_link" href="/List%20of%20edo-distinct%2015et%20rank%20two%20temperaments">List of edo-distinct 15et rank two temperaments</a><br /> | <a class="wiki_link" href="/List%20of%20edo-distinct%2015et%20rank%20two%20temperaments">List of edo-distinct 15et rank two temperaments</a><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc13"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:28 -->Commas</h1> | ||
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15 EDO <a class="wiki_link" href="/tempering%20out">tempers</a> out the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the val &lt; 15 24 35 42 52 56 |.)<br /> | 15 EDO <a class="wiki_link" href="/tempering%20out">tempers</a> out the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the val &lt; 15 24 35 42 52 56 |.)<br /> | ||
| Line 1,342: | Line 1,371: | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc14"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:30 -->Theory</h1> | ||
<a class="wiki_link_ext" href="http://sonic-arts.org/darreg/dar35.htm" rel="nofollow">The 15-Tone Scale System</a> by <a class="wiki_link" href="/Ivor%20Darreg">Ivor Darreg</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xZyzKBEW" rel="nofollow">Permalink</a><br /> | <a class="wiki_link_ext" href="http://sonic-arts.org/darreg/dar35.htm" rel="nofollow">The 15-Tone Scale System</a> by <a class="wiki_link" href="/Ivor%20Darreg">Ivor Darreg</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xZyzKBEW" rel="nofollow">Permalink</a><br /> | ||
<a class="wiki_link_ext" href="http://www.inteas.com/Penta01.htm" rel="nofollow">The Pentadecaphonic System</a><br /> | <a class="wiki_link_ext" href="http://www.inteas.com/Penta01.htm" rel="nofollow">The Pentadecaphonic System</a><br /> | ||
<a class="wiki_link_ext" href="http://home.comcast.net/%7Ebrentishere/15noteequaltempermenttutorial.html" rel="nofollow">15-EDO Tutorial</a> by <a class="wiki_link" href="/Brent%20Carson">Brent Carson</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeJYBsDg" rel="nofollow">Permalink</a><br /> | <a class="wiki_link_ext" href="http://home.comcast.net/%7Ebrentishere/15noteequaltempermenttutorial.html" rel="nofollow">15-EDO Tutorial</a> by <a class="wiki_link" href="/Brent%20Carson">Brent Carson</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeJYBsDg" rel="nofollow">Permalink</a><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc15"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:32 -->Practical Theory / Books</h1> | ||
<a class="wiki_link_ext" href="http://www.swordguitars.com" rel="nofollow">Sword, Ronald. &quot;Pendecaphonic Scales for Guitar&quot; IAAA Press, UK-USA. First Ed: June 2009.</a> - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.<br /> | <a class="wiki_link_ext" href="http://www.swordguitars.com" rel="nofollow">Sword, Ronald. &quot;Pendecaphonic Scales for Guitar&quot; IAAA Press, UK-USA. First Ed: June 2009.</a> - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc16"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:34 -->Compositions</h1> | ||
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<strong><a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/" rel="nofollow" target="_blank">XA 15-ET Directory</a></strong><br /> | <strong><a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/" rel="nofollow" target="_blank">XA 15-ET Directory</a></strong><br /> | ||
| Line 1,375: | Line 1,404: | ||
<a class="wiki_link_ext" href="http://youtu.be/6TyQ9kDm2fk" rel="nofollow">Ode For Ada</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/5f07ebbdf8b4ba8ef067ed3d00a38357-158.html" rel="nofollow">blog entry</a>)<br /> | <a class="wiki_link_ext" href="http://youtu.be/6TyQ9kDm2fk" rel="nofollow">Ode For Ada</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/5f07ebbdf8b4ba8ef067ed3d00a38357-158.html" rel="nofollow">blog entry</a>)<br /> | ||
Suite in 15-Note Equal Tuning, opus 33 by Easley Blackwood (as well as one of the Twelve Microtonal Etudes, opus 28)<br /> | Suite in 15-Note Equal Tuning, opus 33 by Easley Blackwood (as well as one of the Twelve Microtonal Etudes, opus 28)<br /> | ||
<!-- ws:start:WikiTextFileRule: | <!-- ws:start:WikiTextFileRule:1078:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file/15edo-Chords.ogg?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@15edo-Chords.ogg&quot; title=&quot;File: 15edo-Chords.ogg&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><div class="objectEmbed"><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg" onclick="ws.common.trackFileLink('/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="15edo-Chords.ogg" /></a><div><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg" onclick="ws.common.trackFileLink('/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg');" class="filename" title="15edo-Chords.ogg">15edo-Chords.ogg</a><br /><ul><li><a href="/file/detail/15edo-Chords.ogg">Details</a></li><li><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg">Download</a></li><li style="color: #666">2 MB</li></ul></div></div><!-- ws:end:WikiTextFileRule:1078 --> Some nice sounds I found in 15 EDO<br /> | ||
<!-- ws:start:WikiTextFileRule: | <!-- ws:start:WikiTextFileRule:1079:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file/happenstance15.ogg?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@happenstance15.ogg&quot; title=&quot;File: happenstance15.ogg&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><div class="objectEmbed"><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg" onclick="ws.common.trackFileLink('/file/view/happenstance15.ogg/257380320/happenstance15.ogg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="happenstance15.ogg" /></a><div><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg" onclick="ws.common.trackFileLink('/file/view/happenstance15.ogg/257380320/happenstance15.ogg');" class="filename" title="happenstance15.ogg">happenstance15.ogg</a><br /><ul><li><a href="/file/detail/happenstance15.ogg">Details</a></li><li><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg">Download</a></li><li style="color: #666">6 MB</li></ul></div></div><!-- ws:end:WikiTextFileRule:1079 -->Sonic experiment in 15. Somewhat familiar tonality.<br /> | ||
<span class="long-title"><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=kQECU5ecCd4" rel="nofollow" target="_blank">Portrait of insects with 15-tone equal tempered guitar music </a></span><br /> | <span class="long-title"><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=kQECU5ecCd4" rel="nofollow" target="_blank">Portrait of insects with 15-tone equal tempered guitar music </a></span><br /> | ||
<a class="wiki_link_ext" href="http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et" rel="nofollow">PentadecafoniCoda (15et)</a> by <span class="wiki_link_ext">F.F.F. Fiale</span><br /> | <a class="wiki_link_ext" href="http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et" rel="nofollow">PentadecafoniCoda (15et)</a> by <span class="wiki_link_ext">F.F.F. Fiale</span><br /> | ||