34edo

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=<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. As a Fibonacci number, 34edo contains a 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.


===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.

//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]])

===Linear temperaments=== 
||~ Periods
per octave ||~ Generator ||~ Cents ||~ Linear temperaments ||
|| 1 || 1\34 || 35.294 ||   ||
||   || 3\34 || 105.882 ||   ||
||   || 5\34 || 176.471 || [[Tetracot]]/[[Bunya]]/[[Monkey]] ||
||   || 7\34 || 247.059 || [[Immunity]] ||
||   || 9\34 || 317.647 || [[Hanson]]/[[Keemun]] ||
||   || 11\34 || 388.235 || [[Wuerschmidt]]/[[Worschmidt]] ||
||   || 13\34 || 458.824 ||   ||
||   || 15\34 || 529.412 ||   ||
|| 2 || 1\34 || 35.294 ||   ||
||   || 2\34 || 70.588 || [[Vishnu]] ||
||   || 3\34 || 105.882 || [[Srutal]]/[[Pajara]]/[[Diaschismic]] ||
||   || 4\34 || 141.176 || [[Fifive]] ||
||   || 5\34 || 176.471 ||   ||
||   || 6\34 || 211.765 ||   ||
||   || 7\34 || 247.059 ||   ||
||   || 8\34 || 282.353 ||   ||
|| 17 || 1\34 || 35.294 ||   ||
===Intervals:=== 
|| degrees of 34edo || cents || approx. ratios of
2.3.5.13.17 [[subgroup]] || additional ratios
of the full [[17-limit]] ||
|| 0 || 0.0 || 1/1 ||   ||
|| 1 || 35.294 ||   ||   ||
|| 2 || 70.588 ||   ||   ||
|| 3 || 105.882 || 17/16, 18/17, 16/15 || 15/14 ||
|| 4 || 141.176 || 13/12 || 14/13, 12/11 ||
|| 5 || 176.471 || 10/9 || 11/10 ||
|| 6 || 211.765 || 9/8, 17/15 || 8/7 ||
|| 7 || 247.059 || 15/13 ||   ||
|| 8 || 282.353 || 20/17 || 7/6, 13/11 ||
|| 9 || 317.647 || 6/5 || 17/14 ||
|| 10 || 352.941 || 16/13 || 11/9 ||
|| 11 || 388.235 || 5/4 ||   ||
|| 12 || 423.529 ||   || 9/7 ||
|| 13 || 458.823 || 13/10, 17/13 || 22/17 ||
|| 14 || 494.118 || 4/3 ||   ||
|| 15 || 529.412 ||   || 15/11 ||
|| 16 || 564.706 || 18/13 || 11/8 ||
|| 17 || 600 || 17/12, 24/17 || 7/5, 10/7 ||
|| 18 || 635.294 || 13/9 || 16/11 ||
|| 19 || 670.588 ||   || 22/15 ||
|| 20 || 705.882 || 3/2 ||   ||
|| 21 || 741.176 || 20/13, 26/17 || 17/11 ||
|| 22 || 776.471 ||   || 14/9 ||
|| 23 || 811.765 || 8/5 ||   ||
|| 24 || 847.059 || 13/8 || 18/11 ||
|| 25 || 882.353 || 5/3 || 28/17 ||
|| 26 || 917.647 || 17/10 || 12/7, 22/13 ||
|| 27 || 952.941 || 26/15 ||   ||
|| 28 || 988.235 || 16/9, 30/17 || 7/4 ||
|| 29 || 1023.529 || 9/5 || 20/11 ||
|| 30 || 1058.823 || 24/13 || 13/7, 11/6 ||
|| 31 || 1094.118 || 32/17, 17/9, 15/8 || 28/15 ||
|| 32 || 1129.412 ||   ||   ||
|| 33 || 1164.706 ||   ||   ||


==Listen== 
* [[@http://www.archive.org/details/Ascension_105|Ascension]]
Drums Bass and 34-tone guitar

==Links== 
* [[http://www.microstick.net/34guitararticle.htm|34 Equal Guitar]] by [[Larry Hanson]]

Original HTML content:

<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 />
<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. As a Fibonacci number, 34edo contains a 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.<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>
 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 />
<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 />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x34 tone equal temperament--Linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Linear temperaments</h3>
 

<table class="wiki_table">
    <tr>
        <th>Periods<br />
per octave<br />
</th>
        <th>Generator<br />
</th>
        <th>Cents<br />
</th>
        <th>Linear temperaments<br />
</th>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>1\34<br />
</td>
        <td>35.294<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3\34<br />
</td>
        <td>105.882<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>5\34<br />
</td>
        <td>176.471<br />
</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>
    </tr>
    <tr>
        <td><br />
</td>
        <td>7\34<br />
</td>
        <td>247.059<br />
</td>
        <td><a class="wiki_link" href="/Immunity">Immunity</a><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>9\34<br />
</td>
        <td>317.647<br />
</td>
        <td><a class="wiki_link" href="/Hanson">Hanson</a>/<a class="wiki_link" href="/Keemun">Keemun</a><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>11\34<br />
</td>
        <td>388.235<br />
</td>
        <td><a class="wiki_link" href="/Wuerschmidt">Wuerschmidt</a>/<a class="wiki_link" href="/Worschmidt">Worschmidt</a><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>13\34<br />
</td>
        <td>458.824<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>15\34<br />
</td>
        <td>529.412<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>1\34<br />
</td>
        <td>35.294<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>2\34<br />
</td>
        <td>70.588<br />
</td>
        <td><a class="wiki_link" href="/Vishnu">Vishnu</a><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3\34<br />
</td>
        <td>105.882<br />
</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>
    </tr>
    <tr>
        <td><br />
</td>
        <td>4\34<br />
</td>
        <td>141.176<br />
</td>
        <td><a class="wiki_link" href="/Fifive">Fifive</a><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>5\34<br />
</td>
        <td>176.471<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>6\34<br />
</td>
        <td>211.765<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>7\34<br />
</td>
        <td>247.059<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>8\34<br />
</td>
        <td>282.353<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>1\34<br />
</td>
        <td>35.294<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x34 tone equal temperament--Intervals:"></a><!-- ws:end:WikiTextHeadingRule:6 -->Intervals:</h3>
 

<table class="wiki_table">
    <tr>
        <td>degrees of 34edo<br />
</td>
        <td>cents<br />
</td>
        <td>approx. ratios of<br />
2.3.5.13.17 <a class="wiki_link" href="/subgroup">subgroup</a><br />
</td>
        <td>additional ratios<br />
of the full <a class="wiki_link" href="/17-limit">17-limit</a><br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0.0<br />
</td>
        <td>1/1<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>35.294<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>70.588<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>105.882<br />
</td>
        <td>17/16, 18/17, 16/15<br />
</td>
        <td>15/14<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>141.176<br />
</td>
        <td>13/12<br />
</td>
        <td>14/13, 12/11<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>176.471<br />
</td>
        <td>10/9<br />
</td>
        <td>11/10<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>211.765<br />
</td>
        <td>9/8, 17/15<br />
</td>
        <td>8/7<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>247.059<br />
</td>
        <td>15/13<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>282.353<br />
</td>
        <td>20/17<br />
</td>
        <td>7/6, 13/11<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>317.647<br />
</td>
        <td>6/5<br />
</td>
        <td>17/14<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>352.941<br />
</td>
        <td>16/13<br />
</td>
        <td>11/9<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>388.235<br />
</td>
        <td>5/4<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>423.529<br />
</td>
        <td><br />
</td>
        <td>9/7<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>458.823<br />
</td>
        <td>13/10, 17/13<br />
</td>
        <td>22/17<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>494.118<br />
</td>
        <td>4/3<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>529.412<br />
</td>
        <td><br />
</td>
        <td>15/11<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>564.706<br />
</td>
        <td>18/13<br />
</td>
        <td>11/8<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>600<br />
</td>
        <td>17/12, 24/17<br />
</td>
        <td>7/5, 10/7<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>635.294<br />
</td>
        <td>13/9<br />
</td>
        <td>16/11<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>670.588<br />
</td>
        <td><br />
</td>
        <td>22/15<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>705.882<br />
</td>
        <td>3/2<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>741.176<br />
</td>
        <td>20/13, 26/17<br />
</td>
        <td>17/11<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>776.471<br />
</td>
        <td><br />
</td>
        <td>14/9<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>811.765<br />
</td>
        <td>8/5<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>847.059<br />
</td>
        <td>13/8<br />
</td>
        <td>18/11<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>882.353<br />
</td>
        <td>5/3<br />
</td>
        <td>28/17<br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>917.647<br />
</td>
        <td>17/10<br />
</td>
        <td>12/7, 22/13<br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>952.941<br />
</td>
        <td>26/15<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>988.235<br />
</td>
        <td>16/9, 30/17<br />
</td>
        <td>7/4<br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>1023.529<br />
</td>
        <td>9/5<br />
</td>
        <td>20/11<br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>1058.823<br />
</td>
        <td>24/13<br />
</td>
        <td>13/7, 11/6<br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>1094.118<br />
</td>
        <td>32/17, 17/9, 15/8<br />
</td>
        <td>28/15<br />
</td>
    </tr>
    <tr>
        <td>32<br />
</td>
        <td>1129.412<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>1164.706<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x34 tone equal temperament-Listen"></a><!-- ws:end:WikiTextHeadingRule:8 -->Listen</h2>
 <ul><li><a class="wiki_link_ext" href="http://www.archive.org/details/Ascension_105" rel="nofollow" target="_blank">Ascension</a></li></ul>Drums Bass and 34-tone guitar<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x34 tone equal temperament-Links"></a><!-- ws:end:WikiTextHeadingRule:10 -->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>