31edo: Difference between revisions

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//Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It is consistent through the 11-limit, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablance in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

For more encyclopedic info, see [[http://en.wikipedia.org/wiki/31_equal_temperament|Wikipedia's article]].

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&lt;em&gt;Thirty-one tone equal temperament&lt;/em&gt;, also called &lt;em&gt;31-tET&lt;/em&gt;, &lt;em&gt;31-EDO&lt;/em&gt;, &lt;em&gt;31-et&lt;/em&gt;, or &lt;em&gt;tricesimoprimal meantone temperament&lt;/em&gt;, is the scale derived by dividing the octave into 31 &lt;a class="wiki_link" href="/equal"&gt;equally&lt;/a&gt; large steps. The term 'Tricesimoprimal' was first used by &lt;a class="wiki_link" href="/Adriaan%20Fokker"&gt;Adriaan Fokker&lt;/a&gt;. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 &lt;a class="wiki_link" href="/cents"&gt;cents&lt;/a&gt;. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It is consistent through the 11-limit, but is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablance in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.&lt;br /&gt;

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For more encyclopedic info, see &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/31_equal_temperament" rel="nofollow"&gt;Wikipedia's article&lt;/a&gt;.&lt;br /&gt;