36edo: Difference between revisions

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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.

In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent ~~mmajor~~ third. ~~In the 7-limit, however~~, it excels. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

36edo also offers a good approximation to the frequency ratio phi, as 25\36.

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 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.
-In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent mmajor third. In the 7-limit, however, it excels. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
+In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
 edo also offers a good approximation to the frequency ratio phi, as 25\36.