36edo: Difference between revisions

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36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly ~~33.333...~~ cents.

<b>36edo</b>, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33⅓ cents.

36 is a highly composite number, factoring into ~~2x2x3x3~~. Since 36 is divisible by 12, it contains the overly-familiar [[12edo|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|18edo]] ("third tones") and [[9edo|9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo|6edo]] whole tone scale, [[4edo|4edo]] full-diminished seventh chord, and the [[3edo|3edo]] augmented triad, all of which are present in 12edo.

36 is a highly composite number, factoring into 2×2×3×3. Since 36 is divisible by 12, it contains the overly-familiar [[12edo|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|18edo]] ("third tones") and [[9edo|9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo|6edo]] whole tone scale, [[4edo|4edo]] full-diminished seventh chord, and the [[3edo|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']). Three 12edo instruments could play the entire gamut.

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Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.

=~~'''~~Relation to 12edo~~'''~~=

=Relation to 12edo=

For people accustomed to 12edo, 36edo is one of the easiest (if not ''the'' easiest) higher edo to become accustomed to. This is because one way to envision it is as an extended 12edo to which [https://en.wikipedia.org/wiki/Blue_note blue notes] (which are a sixth-tone lower than normal) and "red notes" (a sixth-tone higher) have been added.

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 __FORCETOC__
-edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents.
+<b>36edo</b>, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33⅓ cents.
-is a highly composite number, factoring into 2x2x3x3. Since 36 is divisible by 12, it contains the overly-familiar [[12edo|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|18edo]] ("third tones") and [[9edo|9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo|6edo]] whole tone scale, [[4edo|4edo]] full-diminished seventh chord, and the [[3edo|3edo]] augmented triad, all of which are present in 12edo.
+is a highly composite number, factoring into 2×2×3×3. Since 36 is divisible by 12, it contains the overly-familiar [[12edo|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|18edo]] ("third tones") and [[9edo|9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo|6edo]] whole tone scale, [[4edo|4edo]] full-diminished seventh chord, and the [[3edo|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']). Three 12edo instruments could play the entire gamut.
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 Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.
-='''Relation to 12edo'''=
+=Relation to 12edo=
 For people accustomed to 12edo, 36edo is one of the easiest (if not ''the'' easiest) higher edo to become accustomed to. This is because one way to envision it is as an extended 12edo to which [https://en.wikipedia.org/wiki/Blue_note blue notes] (which are a sixth-tone lower than normal) and "red notes" (a sixth-tone higher) have been added.