29edo: Difference between revisions
Wikispaces>MasonGreen1 **Imported revision 610044611 - Original comment: ** |
Wikispaces>hstraub **Imported revision 614182467 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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:hstraub|hstraub]] and made on <tt>2017-06-05 06:31:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>614182467</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[xenharmonic/The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[xenharmonic/petrmic triad|petrmic triad]], a 13-limit [[xenharmonic/Dyadic chord|essentially tempered dyadic chord]]. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[xenharmonic/k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[xenharmonic/k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas. | Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[xenharmonic/The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[xenharmonic/petrmic triad|petrmic triad]], a 13-limit [[xenharmonic/Dyadic chord|essentially tempered dyadic chord]]. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[xenharmonic/k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[xenharmonic/k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas. | ||
29edo could be thought of as 12edo's "twin", since the 5-limit error for both is almost exactly the same, but in the opposite direction. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively). | 29edo could be thought of as 12edo's "twin", since the 5-limit error for both is almost exactly the same, but in the opposite direction. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively). | ||
A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone. | A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone. | ||
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=Commas= | =Commas= | ||
29 EDO tempers out the following commas. (Note: This assumes the val < [[tel | 29 EDO tempers out the following commas. (Note: This assumes the val < [[tel/29 46 67 81 100 107|29 46 67 81 100 107]] |, cent values rounded to 5 digits.) | ||
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 || | ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 || | ||
||= 16875/16384 || | -14 3 4 > ||> 51.120 ||= Negri Comma ||= Double Augmentation Diesis || | ||= 16875/16384 || | -14 3 4 > ||> 51.120 ||= Negri Comma ||= Double Augmentation Diesis || | ||
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||= 91/90 || | -1 -2 -1 1 0 1 > ||> 19.130 ||= Superleap ||= || | ||= 91/90 || | -1 -2 -1 1 0 1 > ||> 19.130 ||= Superleap ||= || | ||
=The Tetradecatonic System= | |||
A variant of porcupine supported in 29edo is [[xenharmonic/nautilus|nautilus]], which splits the porcupine generator in half (tempering out 50:49 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine. | A variant of porcupine supported in 29edo is [[xenharmonic/nautilus|nautilus]], which splits the porcupine generator in half (tempering out 50:49 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine. | ||
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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>29edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/29edo">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>29edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/29edo">Deutsch</a><br /> | ||
</span><br /> | </span><br /> | ||
<!-- ws:start:WikiTextTocRule:22:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><a href="#x29 tone equal temperament">29 tone equal temperament</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Intervals and linear temperaments">Intervals and linear temperaments</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Nicetone">Nicetone</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <!-- ws:start:WikiTextTocRule:22:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><a href="#x29 tone equal temperament">29 tone equal temperament</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Intervals and linear temperaments">Intervals and linear temperaments</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#The Tetradecatonic System">The Tetradecatonic System</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Nicetone">Nicetone</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | ||
<!-- ws:end:WikiTextTocRule:34 --><hr /> | <!-- ws:end:WikiTextTocRule:34 --><hr /> | ||
<br /> | <br /> | ||
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Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Archipelago">barbados triad</a> 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/petrmic%20triad">petrmic triad</a>, a 13-limit <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Dyadic%20chord">essentially tempered dyadic chord</a>. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">3*29 subgroup</a> 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">2*29 subgroup</a> 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.<br /> | Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Archipelago">barbados triad</a> 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/petrmic%20triad">petrmic triad</a>, a 13-limit <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Dyadic%20chord">essentially tempered dyadic chord</a>. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">3*29 subgroup</a> 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">2*29 subgroup</a> 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.<br /> | ||
<br /> | <br /> | ||
29edo could be thought of as 12edo's &quot;twin&quot;, since the 5-limit error for both is almost exactly the same, but in the opposite direction. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively). <br /> | 29edo could be thought of as 12edo's &quot;twin&quot;, since the 5-limit error for both is almost exactly the same, but in the opposite direction. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively).<br /> | ||
<br /> | <br /> | ||
A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone.<br /> | A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt; | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="The Tetradecatonic System"></a><!-- ws:end:WikiTextHeadingRule:8 -->The Tetradecatonic System</h1> | ||
<br /> | <br /> | ||
A variant of porcupine supported in 29edo is <a class="wiki_link" href="http://xenharmonic.wikispaces.com/nautilus">nautilus</a>, which splits the porcupine generator in half (tempering out 50:49 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine.<br /> | A variant of porcupine supported in 29edo is <a class="wiki_link" href="http://xenharmonic.wikispaces.com/nautilus">nautilus</a>, which splits the porcupine generator in half (tempering out 50:49 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine.<br /> | ||