34edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 315490454 - Original comment: ** |
Wikispaces>guest **Imported revision 322034968 - 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:guest|guest]] and made on <tt>2012-04-18 07:23:34 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>322034968</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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<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="color: #991785; font-family: 'Times New Roman',Times,serif; font-size: 113%;">34 tone equal temperament</span>= | <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="color: #991785; font-family: 'Times New Roman',Times,serif; font-size: 113%;">34 tone equal temperament</span>= | ||
//**34edo**// divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. | //**34edo**// divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two [[xenharmonic/17edo|17edo]]'s and the half-octave tritone of 600 cents. | ||
===Approximations to Just Intonation=== | ===Approximations to Just Intonation=== | ||
Like [[17edo]], 34edo contains good approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11. 34edo adds ratios of 5 into the mix -- including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions -- as well as 17 -- including 17/16, 17/18, 17/12, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the syntonic comma of 81/80, from 21.5 cents to 35.3 cents), it is suitable for 5-limit JI. It is not a [[meantone|meantone ]]system. | Like [[xenharmonic/17edo|17edo]], 34edo contains good approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11. 34edo adds ratios of 5 into the mix -- including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions -- as well as 17 -- including 17/16, 17/18, 17/12, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the syntonic comma of 81/80, from 21.5 cents to 35.3 cents), it is suitable for 5-limit JI. It is not a [[xenharmonic/meantone|meantone ]]system. | ||
//Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.// ([[http://en.wikipedia.org/wiki/34_equal_temperament|Wikipedia]]) | //Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.// ([[http://en.wikipedia.org/wiki/34_equal_temperament|Wikipedia]]) | ||
===34edo and phi=== | ===34edo and phi=== | ||
As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and 140625/140608. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. | As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[xenharmonic/MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and 140625/140608. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. | ||
===Linear temperaments=== | ===Linear temperaments=== | ||
[[xenharmonic/List of 34edo rank two temperaments by badness|List of 34edo rank two temperaments by badness]] | |||
||~ Periods | ||~ Periods | ||
per octave ||~ Generator ||~ Cents ||~ Linear temperaments || | per octave ||~ Generator ||~ Cents ||~ Linear temperaments || | ||
|| 1 || 1\34 || 35.294 || || | || 1 || 1\34 || 35.294 || || | ||
|| || 3\34 || 105.882 || || | || || 3\34 || 105.882 || || | ||
|| || 5\34 || 176.471 || [[Tetracot]]/[[Bunya]]/[[Monkey]] || | || || 5\34 || 176.471 || [[xenharmonic/Tetracot|Tetracot]]/[[xenharmonic/Bunya|Bunya]]/[[xenharmonic/Monkey|Monkey]] || | ||
|| || 7\34 || 247.059 || [[Immunity]] || | || || 7\34 || 247.059 || [[xenharmonic/Immunity|Immunity]] || | ||
|| || 9\34 || 317.647 || [[Hanson]]/[[Keemun]] || | || || 9\34 || 317.647 || [[xenharmonic/Hanson|Hanson]]/[[xenharmonic/Keemun|Keemun]] || | ||
|| || 11\34 || 388.235 || [[Wuerschmidt]]/[[Worschmidt]] || | || || 11\34 || 388.235 || [[xenharmonic/Wuerschmidt|Wuerschmidt]]/[[xenharmonic/Worschmidt|Worschmidt]] || | ||
|| || 13\34 || 458.824 || || | || || 13\34 || 458.824 || || | ||
|| || 15\34 || 529.412 || || | || || 15\34 || 529.412 || || | ||
|| 2 || 1\34 || 35.294 || || | || 2 || 1\34 || 35.294 || || | ||
|| || 2\34 || 70.588 || [[Vishnu]] || | || || 2\34 || 70.588 || [[xenharmonic/Vishnu|Vishnu]] || | ||
|| || 3\34 || 105.882 || [[Srutal]]/[[Pajara]]/[[Diaschismic]] || | || || 3\34 || 105.882 || [[xenharmonic/Srutal|Srutal]]/[[xenharmonic/Pajara|Pajara]]/[[xenharmonic/Diaschismic|Diaschismic]] || | ||
|| || 4\34 || 141.176 || [[Fifive]] || | || || 4\34 || 141.176 || [[xenharmonic/Fifive|Fifive]] || | ||
|| || 5\34 || 176.471 || || | || || 5\34 || 176.471 || || | ||
|| || 6\34 || 211.765 || || | || || 6\34 || 211.765 || || | ||
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===Intervals:=== | ===Intervals:=== | ||
|| degrees of 34edo || cents || approx. ratios of | || degrees of 34edo || cents || approx. ratios of | ||
2.3.5.13.17 [[subgroup]] || additional ratios | 2.3.5.13.17 [[xenharmonic/subgroup|subgroup]] || additional ratios | ||
of the full [[17-limit]] || | of the full [[xenharmonic/17-limit|17-limit]] || | ||
|| 0 || 0.0 || 1/1 || || | || 0 || 0.0 || 1/1 || || | ||
|| 1 || 35.294 || || || | || 1 || 35.294 || || || | ||
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==Links== | ==Links== | ||
* [[http://www.microstick.net/34guitararticle.htm|34 Equal Guitar]] by [[Larry Hanson]]</pre></div> | * [[http://www.microstick.net/34guitararticle.htm|34 Equal Guitar]] by [[xenharmonic/Larry Hanson|Larry Hanson]]</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>34edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x34 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #991785; font-family: 'Times New Roman',Times,serif; font-size: 113%;">34 tone equal temperament</span></h1> | <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>34edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x34 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #991785; font-family: 'Times New Roman',Times,serif; font-size: 113%;">34 tone equal temperament</span></h1> | ||
<br /> | <br /> | ||
<em><strong>34edo</strong></em> divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two <a class="wiki_link" href="/17edo">17edo</a>'s and the half-octave tritone of 600 cents. <br /> | <em><strong>34edo</strong></em> divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo">17edo</a>'s and the half-octave tritone of 600 cents.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x34 tone equal temperament--Approximations to Just Intonation"></a><!-- ws:end:WikiTextHeadingRule:2 -->Approximations to Just Intonation</h3> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x34 tone equal temperament--Approximations to Just Intonation"></a><!-- ws:end:WikiTextHeadingRule:2 -->Approximations to Just Intonation</h3> | ||
Like <a class="wiki_link" href="/17edo">17edo</a>, 34edo contains good approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11. 34edo adds ratios of 5 into the mix -- including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions -- as well as 17 -- including 17/16, 17/18, 17/12, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the syntonic comma of 81/80, from 21.5 cents to 35.3 cents), it is suitable for 5-limit JI. It is not a <a class="wiki_link" href="/meantone">meantone </a>system.<br /> | Like <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo">17edo</a>, 34edo contains good approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11. 34edo adds ratios of 5 into the mix -- including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions -- as well as 17 -- including 17/16, 17/18, 17/12, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the syntonic comma of 81/80, from 21.5 cents to 35.3 cents), it is suitable for 5-limit JI. It is not a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/meantone">meantone </a>system.<br /> | ||
<br /> | <br /> | ||
<em>Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.</em> (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/34_equal_temperament" rel="nofollow">Wikipedia</a>)<br /> | <em>Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.</em> (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/34_equal_temperament" rel="nofollow">Wikipedia</a>)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x34 tone equal temperament--34edo and phi"></a><!-- ws:end:WikiTextHeadingRule:4 -->34edo and phi</h3> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x34 tone equal temperament--34edo and phi"></a><!-- ws:end:WikiTextHeadingRule:4 -->34edo and phi</h3> | ||
As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates <a class="wiki_link" href="/MOSScales">Moment of Symmetry</a> scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and 140625/140608. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth.<br /> | As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">Moment of Symmetry</a> scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and 140625/140608. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x34 tone equal temperament--Linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 -->Linear temperaments</h3> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x34 tone equal temperament--Linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 -->Linear temperaments</h3> | ||
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/List%20of%2034edo%20rank%20two%20temperaments%20by%20badness">List of 34edo rank two temperaments by badness</a><br /> | |||
<table class="wiki_table"> | <table class="wiki_table"> | ||
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<td>176.471<br /> | <td>176.471<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Tetracot">Tetracot</a>/<a class="wiki_link" href="/Bunya">Bunya</a>/<a class="wiki_link" href="/Monkey">Monkey</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tetracot">Tetracot</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Bunya">Bunya</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monkey">Monkey</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>247.059<br /> | <td>247.059<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Immunity">Immunity</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Immunity">Immunity</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 159: | Line 161: | ||
<td>317.647<br /> | <td>317.647<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Hanson">Hanson</a>/<a class="wiki_link" href="/Keemun">Keemun</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Hanson">Hanson</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Keemun">Keemun</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 169: | Line 171: | ||
<td>388.235<br /> | <td>388.235<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Wuerschmidt">Wuerschmidt</a>/<a class="wiki_link" href="/Worschmidt">Worschmidt</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Wuerschmidt">Wuerschmidt</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Worschmidt">Worschmidt</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 209: | Line 211: | ||
<td>70.588<br /> | <td>70.588<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Vishnu">Vishnu</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vishnu">Vishnu</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>105.882<br /> | <td>105.882<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Srutal">Srutal</a>/<a class="wiki_link" href="/Pajara">Pajara</a>/<a class="wiki_link" href="/Diaschismic">Diaschismic</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Srutal">Srutal</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Pajara">Pajara</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Diaschismic">Diaschismic</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>141.176<br /> | <td>141.176<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Fifive">Fifive</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Fifive">Fifive</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 294: | Line 296: | ||
</td> | </td> | ||
<td>approx. ratios of<br /> | <td>approx. ratios of<br /> | ||
2.3.5.13.17 <a class="wiki_link" href="/subgroup">subgroup</a><br /> | 2.3.5.13.17 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/subgroup">subgroup</a><br /> | ||
</td> | </td> | ||
<td>additional ratios<br /> | <td>additional ratios<br /> | ||
of the full <a class="wiki_link" href="/17-limit">17-limit</a><br /> | of the full <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17-limit">17-limit</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x34 tone equal temperament-Links"></a><!-- ws:end:WikiTextHeadingRule:12 -->Links</h2> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x34 tone equal temperament-Links"></a><!-- ws:end:WikiTextHeadingRule:12 -->Links</h2> | ||
<ul><li><a class="wiki_link_ext" href="http://www.microstick.net/34guitararticle.htm" rel="nofollow">34 Equal Guitar</a> by <a class="wiki_link" href="/Larry%20Hanson">Larry Hanson</a></li></ul></body></html></pre></div> | <ul><li><a class="wiki_link_ext" href="http://www.microstick.net/34guitararticle.htm" rel="nofollow">34 Equal Guitar</a> by <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Larry%20Hanson">Larry Hanson</a></li></ul></body></html></pre></div> | ||