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 [[xenharmonic/17edo|17edo]]'s and the half-octave tritone of 600 cents. ===Approximations to Just Intonation=== 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]]) ===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 [[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=== [[xenharmonic/List of 34edo rank two temperaments by badness|List of 34edo rank two temperaments by badness]] ||~ Periods per octave ||~ Generator ||~ Cents ||~ Linear temperaments || || 1 || 1\34 || 35.294 || || || || 3\34 || 105.882 || || || || 5\34 || 176.471 || [[xenharmonic/Tetracot|Tetracot]]/[[xenharmonic/Bunya|Bunya]]/[[xenharmonic/Monkey|Monkey]] || || || 7\34 || 247.059 || [[xenharmonic/Immunity|Immunity]] || || || 9\34 || 317.647 || [[xenharmonic/Hanson|Hanson]]/[[xenharmonic/Keemun|Keemun]] || || || 11\34 || 388.235 || [[xenharmonic/Wuerschmidt|Wuerschmidt]]/[[xenharmonic/Worschmidt|Worschmidt]] || || || 13\34 || 458.824 || || || || 15\34 || 529.412 || || || 2 || 1\34 || 35.294 || || || || 2\34 || 70.588 || [[xenharmonic/Vishnu|Vishnu]] || || || 3\34 || 105.882 || [[xenharmonic/Srutal|Srutal]]/[[xenharmonic/Pajara|Pajara]]/[[xenharmonic/Diaschismic|Diaschismic]] || || || 4\34 || 141.176 || [[xenharmonic/Fifive|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 [[xenharmonic/subgroup|subgroup]] || additional ratios of the full [[xenharmonic/17-limit|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 [[xenharmonic/Larry Hanson|Larry Hanson]]
Original HTML content:
<html><head><title>34edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><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="http://xenharmonic.wikispaces.com/17edo">17edo</a>'s and the half-octave tritone of 600 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><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="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 />
<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:<h3> --><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="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 />
<!-- ws:start:WikiTextHeadingRule:6:<h3> --><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">
<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="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>
</tr>
<tr>
<td><br />
</td>
<td>7\34<br />
</td>
<td>247.059<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/Hanson">Hanson</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/Wuerschmidt">Wuerschmidt</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/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="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>
</tr>
<tr>
<td><br />
</td>
<td>4\34<br />
</td>
<td>141.176<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/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:8:<h3> --><h3 id="toc4"><a name="x34 tone equal temperament--Intervals:"></a><!-- ws:end:WikiTextHeadingRule:8 -->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="http://xenharmonic.wikispaces.com/subgroup">subgroup</a><br />
</td>
<td>additional ratios<br />
of the full <a class="wiki_link" href="http://xenharmonic.wikispaces.com/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:10:<h2> --><h2 id="toc5"><a name="x34 tone equal temperament-Listen"></a><!-- ws:end:WikiTextHeadingRule:10 -->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:12:<h2> --><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="http://xenharmonic.wikispaces.com/Larry%20Hanson">Larry Hanson</a></li></ul></body></html>