36edo: Difference between revisions

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== Theory ==

~~{{Primes in edo|36}}~~

Since 36 is divisible by 12, it contains the overly-familiar [[12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal steps of approximately 33 cents, which could be called "sixth tones." 36edo also contains [[18edo]] ("third tones") and [[9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo]] whole tone scale, [[4edo]] full-diminished seventh chord, and the [[3edo]] augmented triad, all of which are present in 12edo.

~~'''36edo''', by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33⅓ cents.~~

~~36 is the 7th [[Highly melodic EDO|highly composite]] number, factoring into 2×2×3×3.~~ Since 36 is divisible by 12, it contains the overly-familiar [[12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal steps of approximately 33 cents, which could be called "sixth tones." 36edo also contains [[18edo]] ("third tones") and [[9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo]] whole tone scale, [[4edo]] full-diminished seventh chord, and the [[3edo]] augmented triad, all of which are present in 12edo.

That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']{{Dead link}}). Three 12edo instruments could play the entire gamut.

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The 36edo patent val, like 12, tempers out 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.

As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is <36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9~~>~~ is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.

As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is <36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" {{monzo|29 0 -9}} is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.

=== Prime harmonics ===

=== Divisors ===

36edo is the 7th [[highly composite EDO]], with subset edos {{EDOs|1, 2, 3, 4, 6, 9, 12, 18}}.

== Intervals ==

{| class="wikitable center-all right-2"

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 {{Infobox ET}}
+{{EDO intro|36}}
 == Theory  ==
-{{Primes in edo|36}}
+Since 36 is divisible by 12, it contains the overly-familiar [[12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal steps of approximately 33 cents, which could be called "sixth tones." 36edo also contains [[18edo]] ("third tones") and [[9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo]] whole tone scale, [[4edo]] full-diminished seventh chord, and the [[3edo]] augmented triad, all of which are present in 12edo.
-'''36edo''', by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33⅓ cents.
-is the 7th [[Highly melodic EDO|highly composite]] number, factoring into 2×2×3×3. Since 36 is divisible by 12, it contains the overly-familiar [[12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal steps of approximately 33 cents, which could be called "sixth tones." 36edo also contains [[18edo]] ("third tones") and [[9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo]] whole tone scale, [[4edo]] full-diminished seventh chord, and the [[3edo]] augmented triad, all of which are present in 12edo.
 That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']{{Dead link}}). Three 12edo instruments could play the entire gamut.
@@ Line 15: / Line 13: @@
 The 36edo patent val, like 12, tempers out 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.
-As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is &lt;36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9&gt; is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.
+As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is &lt;36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" {{monzo|29 0 -9}} is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.
+=== Prime harmonics ===
+{{harmonics in equal|36}}
+=== Divisors ===
+edo is the 7th [[highly composite EDO]], with subset edos {{EDOs|1, 2, 3, 4, 6, 9, 12, 18}}.
 == Intervals  ==
 {| class="wikitable center-all right-2"