16edo: Difference between revisions
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<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="display: block; text-align: right;">Other languages: [[xenharmonie/16edo|Deutsch]] | <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="display: block; text-align: right;">Other languages: [[xenharmonie/16edo|Deutsch]] | ||
</span> | </span> | ||
[[toc | [[toc]] | ||
**16-EDO** is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7_4|7/4]] which is six cents sharp, and a [[5_4|5/4]] which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of [[12edo|12-EDO]], and a diminished triad on each scale step. | **16-EDO** is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7_4|7/4]] which is six cents sharp, and a [[5_4|5/4]] which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of [[12edo|12-EDO]], and a diminished triad on each scale step. | ||
=Intervals= | |||
|| Degree || Cents ||= Approximate | || Degree || Cents ||= Approximate | ||
Ratios* || Interval Name || | Ratios* || Interval Name || | ||
| Line 36: | Line 37: | ||
|| 16 || 1200 ||= 2/1 || Octave || | || 16 || 1200 ||= 2/1 || Octave || | ||
*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible. | *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible. | ||
==Intervals by patent val error== | |||
|| Interval || Error || | |||
|| 12/11 || -0.637 || | |||
|| 13/10 || -4.214 || | |||
|| 8/7 || -6.174 || | |||
|| 13/11 || 10.790 || | |||
|| 5/4 || -11.314 || | |||
|| 13/12 || 11.427 || | |||
|| 15/11 || -11.951 || | |||
|| 11/10 || -15.004 || | |||
|| 16/13 || 15.528 || | |||
|| 6/5 || -15.641 || | |||
|| 7/5 || 17.488 || | |||
|| 14/13 || 21.702 || | |||
|| 15/13 || -22.741 || | |||
|| 11/8 || -26.318 || | |||
|| 4/3 || 26.955 || | |||
|| 11/9 || 27.592 || | |||
|| 14/11 || 32.492 || | |||
|| 7/6 || 33.129 || | |||
|| 16/15 || 38.269 || | |||
|| 18/13 || -38.382 || | |||
|| 10/9 || 42.596 || | |||
|| 15/14 || -44.443 || | |||
|| 9/8 || -53.910 || | |||
|| 9/7 || -60.084 || | |||
[[media type="custom" key="24810096"]] | [[media type="custom" key="24810096"]] | ||
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<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>16edo</title></head><body><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/16edo">Deutsch</a><br /> | <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>16edo</title></head><body><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/16edo">Deutsch</a><br /> | ||
</span><br /> | </span><br /> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:21:&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:21 --><!-- ws:start:WikiTextTocRule:22: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><div style="margin-left: 2em;"><a href="#Intervals-Intervals by patent val error">Intervals by patent val error</a></div> | ||
<!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 1em;"><a href="#Hexadecaphonic Octave Theory">Hexadecaphonic Octave Theory</a></div> | |||
<!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 1em;"><a href="#Hexadecaphonic Notation">Hexadecaphonic Notation</a></div> | |||
<!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 1em;"><a href="#Rank two temperaments">Rank two temperaments</a></div> | |||
<!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 1em;"><a href="#Commas">Commas</a></div> | |||
<!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 1em;"><a href="#Armodue Theory (4-line staff)">Armodue Theory (4-line staff)</a></div> | |||
<!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 1em;"><a href="#Books/Literature">Books/Literature</a></div> | |||
<!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 1em;"><a href="#Compositions">Compositions</a></div> | |||
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --></div> | |||
<!-- ws:end:WikiTextTocRule:31 --><br /> | |||
<strong>16-EDO</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into sixteen narrow chromatic semitones each of 75 <a class="wiki_link" href="/cent">cent</a>s exactly. It is not especially good at representing most low-integer musical intervals, but it has a <a class="wiki_link" href="/7_4">7/4</a> which is six cents sharp, and a <a class="wiki_link" href="/5_4">5/4</a> which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of <a class="wiki_link" href="/12edo">12-EDO</a>, and a diminished triad on each scale step.<br /> | <strong>16-EDO</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into sixteen narrow chromatic semitones each of 75 <a class="wiki_link" href="/cent">cent</a>s exactly. It is not especially good at representing most low-integer musical intervals, but it has a <a class="wiki_link" href="/7_4">7/4</a> which is six cents sharp, and a <a class="wiki_link" href="/5_4">5/4</a> which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of <a class="wiki_link" href="/12edo">12-EDO</a>, and a diminished triad on each scale step.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc0"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:3 -->Intervals</h1> | |||
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*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.<br /> | *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc1"><a name="Intervals-Intervals by patent val error"></a><!-- ws:end:WikiTextHeadingRule:5 -->Intervals by patent val error</h2> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>Interval<br /> | |||
</td> | |||
<td>Error<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>12/11<br /> | |||
</td> | |||
<td>-0.637<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/10<br /> | |||
</td> | |||
<td>-4.214<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>8/7<br /> | |||
</td> | |||
<td>-6.174<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/11<br /> | |||
</td> | |||
<td>10.790<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>5/4<br /> | |||
</td> | |||
<td>-11.314<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/12<br /> | |||
</td> | |||
<td>11.427<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/11<br /> | |||
</td> | |||
<td>-11.951<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/10<br /> | |||
</td> | |||
<td>-15.004<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>16/13<br /> | |||
</td> | |||
<td>15.528<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>6/5<br /> | |||
</td> | |||
<td>-15.641<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7/5<br /> | |||
</td> | |||
<td>17.488<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14/13<br /> | |||
</td> | |||
<td>21.702<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/13<br /> | |||
</td> | |||
<td>-22.741<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/8<br /> | |||
</td> | |||
<td>-26.318<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>4/3<br /> | |||
</td> | |||
<td>26.955<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/9<br /> | |||
</td> | |||
<td>27.592<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14/11<br /> | |||
</td> | |||
<td>32.492<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7/6<br /> | |||
</td> | |||
<td>33.129<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>16/15<br /> | |||
</td> | |||
<td>38.269<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>18/13<br /> | |||
</td> | |||
<td>-38.382<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>10/9<br /> | |||
</td> | |||
<td>42.596<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/14<br /> | |||
</td> | |||
<td>-44.443<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9/8<br /> | |||
</td> | |||
<td>-53.910<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9/7<br /> | |||
</td> | |||
<td>-60.084<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | <br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc2"><a name="Hexadecaphonic Octave Theory"></a><!-- ws:end:WikiTextHeadingRule:7 -->Hexadecaphonic Octave Theory</h1> | ||
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The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent &quot;3/4-tone&quot; equal division of the traditional 300-cent minor third.<br /> | The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent &quot;3/4-tone&quot; equal division of the traditional 300-cent minor third.<br /> | ||
| Line 360: | Line 555: | ||
<br /> | <br /> | ||
Another interesting approach can include two interwoven <a class="wiki_link" href="/8edo">8-EDO</a> scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's <strong>Mavila</strong> as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (<a class="wiki_link" href="/19_16">19/16</a>, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).<br /> | Another interesting approach can include two interwoven <a class="wiki_link" href="/8edo">8-EDO</a> scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's <strong>Mavila</strong> as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (<a class="wiki_link" href="/19_16">19/16</a>, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).<br /> | ||
<!-- ws:start:WikiTextRemoteImageRule: | <!-- ws:start:WikiTextRemoteImageRule:721:&lt;img src=&quot;http://www.ronsword.com/161928%20copy.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 179px; width: 316px;&quot; /&gt; --><img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 179px; width: 316px;" /><!-- ws:end:WikiTextRemoteImageRule:721 --><br /> | ||
<br /> | <br /> | ||
The interval between the 28th &amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &quot;narrow fifth&quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.<br /> | The interval between the 28th &amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &quot;narrow fifth&quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.<br /> | ||
<br /> | <br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc3"><a name="Hexadecaphonic Notation"></a><!-- ws:end:WikiTextHeadingRule:9 -->Hexadecaphonic Notation</h1> | ||
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16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.<br /> | 16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.<br /> | ||
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In 16-EDO diatonic scales are dissonant and &quot;shimmery&quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.<br /> | In 16-EDO diatonic scales are dissonant and &quot;shimmery&quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.<br /> | ||
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unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&quot;<br /> | unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&quot;<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc4"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:11 -->Rank two temperaments</h1> | ||
<a class="wiki_link" href="/List%20of%2016et%20rank%20two%20temperaments%20by%20badness">List of 16et rank two temperaments by badness</a><br /> | <a class="wiki_link" href="/List%20of%2016et%20rank%20two%20temperaments%20by%20badness">List of 16et rank two temperaments by badness</a><br /> | ||
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135/128 (the difference between 16/15 and 9/8) is tempered out....&quot;<br /> | 135/128 (the difference between 16/15 and 9/8) is tempered out....&quot;<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc5"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:13 -->Commas</h1> | ||
16 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes <a class="wiki_link" href="/val">val</a> &lt; 16 25 37 45 55 59 |.)<br /> | 16 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes <a class="wiki_link" href="/val">val</a> &lt; 16 25 37 45 55 59 |.)<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc6"><a name="Armodue Theory (4-line staff)"></a><!-- ws:end:WikiTextHeadingRule:15 --><strong>Armodue Theory (4-line staff)</strong></h1> | ||
<a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow">Armodue</a>: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.<br /> | <a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow">Armodue</a>: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.<br /> | ||
Translations of parts of the Armodue pages can be found <a class="wiki_link" href="/Armodue">here</a> on this wiki.<br /> | Translations of parts of the Armodue pages can be found <a class="wiki_link" href="/Armodue">here</a> on this wiki.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:17:&lt;h1&gt; --><h1 id="toc7"><a name="Books/Literature"></a><!-- ws:end:WikiTextHeadingRule:17 -->Books/Literature</h1> | ||
Sword, Ronald. &quot;Thesaurus of Melodic Patterns and Intervals for 16-Tones&quot; IAAA Press, USA. First Ed: August, 2011<br /> | Sword, Ronald. &quot;Thesaurus of Melodic Patterns and Intervals for 16-Tones&quot; IAAA Press, USA. First Ed: August, 2011<br /> | ||
Sword, Ronald. &quot;Hexadecaphonic Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)<br /> | Sword, Ronald. &quot;Hexadecaphonic Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)<br /> | ||
Sword, Ronald. &quot;Esadekaphonic Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)<br /> | Sword, Ronald. &quot;Esadekaphonic Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:19:&lt;h1&gt; --><h1 id="toc8"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:19 -->Compositions</h1> | ||
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&amp;songID=12370649&amp;showPlayer=true" rel="nofollow" target="_blank">Prenestyna Highway</a> by <a class="wiki_link_ext" href="http://fiale.tk" rel="nofollow" target="_blank">Fabrizio Fulvio Fausto Fiale</a><br /> | <a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&amp;songID=12370649&amp;showPlayer=true" rel="nofollow" target="_blank">Prenestyna Highway</a> by <a class="wiki_link_ext" href="http://fiale.tk" rel="nofollow" target="_blank">Fabrizio Fulvio Fausto Fiale</a><br /> | ||
<a class="wiki_link_ext" href="http://lastsacrament.bandcamp.com/album/enantiodromia" rel="nofollow" target="_blank">Enantiodromia (album)</a> by Last Sacrament<br /> | <a class="wiki_link_ext" href="http://lastsacrament.bandcamp.com/album/enantiodromia" rel="nofollow" target="_blank">Enantiodromia (album)</a> by Last Sacrament<br /> | ||