34edo: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-03-22 11:25:11 UTC</tt>.<br>

: The original revision id was <tt>497558464</tt>.<br>

34edo divides the octave into 34 equal steps of approximately 35.29412 [[xenharmonic/cent|cent]]s. 34edo contains two [[xenharmonic/17edo|17edo]]'s and the half-octave tritone of 600 cents. It excels as a 5-limit system, with tuning even more accurate than [[31edo]], but with a sharp fifth rather than a flat one, and supports hanson, srutal, tetracot, würschmidt and vishnu temperaments. It does less well in the 7-limit, with two mappings possible for 7/4: a flat one from the patent val, and a sharp one from the 34d val. By way of the patent val 34 supports keemun temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit pajara temperament. In the 11-limit, 34de supports 11-limit pajaric, and in fact is quite close to the POTE tuning; it adds 4375/4374 to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports semaphore on the 2.3.7 subgroup.

*The sharpness of ~13 cents of 11/8 can fit somewhat with the 9/8 and 13/8 which both are about 7 cents sharp. Likewise the 20-cent sharpness or flatness of either approximation to 7/4 isn't impossible. The ability to tolerate these errors may depend on subtle natural changes in mood. [[68edo]], double 34, has both these intervals in more perfect form.

|| degrees of 34edo || solfege || cents || approx. ratios of

|| 1 || di || 35.294 ||  ||   || C ^ ||

|| 2 || rih || 70.588 ||   ||  || Db = C ^^ = B# ||

|| 3 || ra || 105.882 || 17/16, 18/17, 16/15 || 15/14 || C#v = Db^ ||

|| 4 || ru || 141.176 || 13/12 || 14/13, 12/11 || C# ||

|| 5 || reh || 176.471 || 10/9 || 11/10 || C#^ = Dv ||

|| 6 || re || 211.765 || 9/8, 17/15 || 8/7 || D ||

|| 7 || raw || 247.059 || 15/13 ||   || D^ ||

|| 8 || meh || 282.353 || 20/17, 75/64 || 7/6, 13/11 || Eb ||

|| 9 || me || 317.647 || 6/5 || 17/14 || D#v ||

|| 10 || mu || 352.941 || 16/13 || 11/9 || D# ||

|| 11 || mi || 388.235 || 5/4 ||  ||  ||

|| 12 || maa || 423.529 || 51/40, 32/25 || 14/11, 9/7 || E ||

|| 13 || maw || 458.823 || 13/10, 17/13 || 22/17 || E^ = Fv ||

|| 14 || fa || 494.118 || 4/3 ||  || F ||

|| 15 || fih || 529.412 ||  || 15/11 || F^ = E#v ||

|| 16 || fu || 564.706 || 18/13 || 11/8 || Gb ||

|| 17 || fi/se || 600 || 17/12, 24/17 || 7/5, 10/7 || Gb^ ||

|| 18 || su || 635.294 || 13/9 || 16/11 || F# ||

|| 19 || sih || 670.588 ||  || 22/15 || F#^ ||

|| 20 || sol || 705.882 || 3/2 ||  || G ||

|| 21 || saw || 741.176 || 20/13, 26/17 || 17/11 || G^ ||

|| 22 || leh || 776.471 || 25/16, 80/51 || 14/9 || Ab ||

|| 23 || le || 811.765 || 8/5 ||  || Ab^ ||

|| 24 || lu || 847.059 || 13/8 || 18/11 || G# ||

|| 25 || la || 882.353 || 5/3 || 28/17 || Av ||

|| 26 || laa || 917.647 || 17/10 || 12/7, 22/13 || A ||

|| 27 || law || 952.941 || 26/15 ||   || A^ = Bbv =G## ||

|| 28 || teh || 988.235 || 16/9, 30/17 || 7/4 || Bb ||

|| 29 || te || 1023.529 || 9/5 || 20/11 || Bb^ ||

|| 30 || tu || 1058.823 || 24/13 || 13/7, 11/6 || A# ||

|| 31 || ti || 1094.118 || 32/17, 17/9, 15/8 || 28/15 || A#^ = Bv ||

|| 32 || taa || 1129.412 ||   ||  || B ||

|| 33 || da || 1164.706 ||  ||   || B^ = A##v ||

34edo divides the octave into 34 equal steps of approximately 35.29412 &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/cent"&gt;cent&lt;/a&gt;s. 34edo contains two &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo"&gt;17edo&lt;/a&gt;'s and the half-octave tritone of 600 cents. It excels as a 5-limit system, with tuning even more accurate than &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, but with a sharp fifth rather than a flat one, and supports hanson, srutal, tetracot, würschmidt and vishnu temperaments. It does less well in the 7-limit, with two mappings possible for 7/4: a flat one from the patent val, and a sharp one from the 34d val. By way of the patent val 34 supports keemun temperament, and 34d is an excellent alternative to &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; for 7-limit pajara temperament. In the 11-limit, 34de supports 11-limit pajaric, and in fact is quite close to the POTE tuning; it adds 4375/4374 to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports semaphore on the 2.3.7 subgroup.&lt;br /&gt;

*The sharpness of ~13 cents of 11/8 can fit somewhat with the 9/8 and 13/8 which both are about 7 cents sharp. Likewise the 20-cent sharpness or flatness of either approximation to 7/4 isn't impossible. The ability to tolerate these errors may depend on subtle natural changes in mood. &lt;a class="wiki_link" href="/68edo"&gt;68edo&lt;/a&gt;, double 34, has both these intervals in more perfect form.&lt;br /&gt;

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         &lt;td&gt;8/7&lt;br /&gt;

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         &lt;td&gt;7/4&lt;br /&gt;

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