666edo: Difference between revisions

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== Theory ==

~~{{Harmonics~~ in ~~equal|666}}~~

666edo is enfactored in the 17-limit, with the same mapping as [[333edo]]. Using precision as a criterion, 2.11.19.41.43 subgroup is great for 666edo. If significant errors are allowed, 666edo can be used with 2.7.11.17.19.23. 666edo provides good direct approximations for: [[15/11]], [[16/11]], [[16/15]], [[13/12]], [[13/10]], [[22/15]], [[23/14]]. 666edo also has a strong approximation for [[11/8]] derived from [[37edo]] and for [[7/6]] derived from 9edo, but on the 2.7/6.11 subgroup it is enfactored, with the same tuning again as 333edo, tempering out the [[37-11-comma]] and the [[septimal ennealimma]]. The 666c val, tempers out [[2401/2400]], [[4375/4374]], and [[9801/9800]] in the 11 limit.

666edo is ~~appropriate for use with the 2.11.19.41.43 subgroup. If significant errors are allowed, 666edo can be~~ used ~~with 2.7.11.17.19.23. Harmonics from 2 to 17 for 666edo all land on even numbers, meaning its contorted order 2 and they ultimately derive from~~ [[~~333edo~~]]~~. As such, 666edo provides~~ the ~~optimal patent val for~~ [[~~novemkleismic~~]] ~~temperament~~ just ~~as 333edo does~~. ~~666edo provides good approximations for: [[15/11]]~~, ~~[[16/11]], [[16/15]], [[13/12]], [[13/10]], [[22/15]], [[23~~/14]]. ~~Its 11/8 ultimately derives from [[37edo]]~~, ~~and [[7/6]] from [[9edo]]~~.

666edo is also used by [[Eliora]] to approximate the "[[Factor 9 grid]]", or the just intonation esoteric scale deconstructed and debunked by Adam Neely. Now, it may be worth noting that the tuning system which truly has an excellent approximation of the Factor 9 grid is 666ed15/14. However, this fact wasn't spotted by Eliora until after first music was composed in 666edo due to the temperament finder layout making it not immediately obvious what is the interval of equivalence, so the Factor 9 grid representation by 666edo still remains notable given that it's a scale for some of the first music composed in this edo.

~~Using the 666c val, it tempres out [[2401/2400]], [[4375/4374]], and [[9801/9800]]~~ in ~~the 11 limit.~~

=== Odd harmonics ===

~~666 is divisible by {{EDOs|9, 18, 37, 74, 111, 222,~~ and ~~333}}.~~

=== Subsets and supersets ===

~~666edo also approximates the "[[Factor~~ 9 ~~grid]]"~~, ~~or the just intonation esoteric scale deconstructed~~ and ~~debunked by Adam Neely. The best rank two temperament for this scale is 495 & 666. A more general 23 limit version~~ of ~~this temperament can be described also,~~ which ~~results in~~ a ~~temperament with period 1/9 octave~~.

666 is divisible by {{EDOs|9, 18, 37, 74, 111, 222, and 333}}, of which 111edo is a notable system due to its large amount of regular temperaments.

== Regular temperament properties ==

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 == Theory ==
-{{Harmonics in equal|666}}
+edo is enfactored in the 17-limit, with the same mapping as [[333edo]]. Using precision as a criterion, 2.11.19.41.43 subgroup is great for 666edo. If significant errors are allowed, 666edo can be used with 2.7.11.17.19.23. 666edo provides good direct approximations for: [[15/11]], [[16/11]], [[16/15]], [[13/12]], [[13/10]], [[22/15]], [[23/14]]. 666edo also has a strong approximation for [[11/8]] derived from [[37edo]] and for [[7/6]] derived from 9edo, but on the 2.7/6.11 subgroup it is enfactored, with the same tuning again as 333edo, tempering out the [[37-11-comma]] and the [[septimal ennealimma]]. The 666c val, tempers out [[2401/2400]], [[4375/4374]], and [[9801/9800]] in the 11 limit.
-edo is appropriate for use with the 2.11.19.41.43 subgroup. If significant errors are allowed, 666edo can be used with 2.7.11.17.19.23. Harmonics from 2 to 17 for 666edo all land on even numbers, meaning its contorted order 2 and they ultimately derive from [[333edo]]. As such, 666edo provides the optimal patent val for [[novemkleismic]] temperament just as 333edo does. 666edo provides good approximations for: [[15/11]], [[16/11]], [[16/15]], [[13/12]], [[13/10]], [[22/15]], [[23/14]]. Its 11/8 ultimately derives from [[37edo]], and [[7/6]] from [[9edo]].
+edo is also used by [[Eliora]] to approximate the "[[Factor 9 grid]]", or the just intonation esoteric scale deconstructed and debunked by Adam Neely. Now, it may be worth noting that the tuning system which truly has an excellent approximation of the Factor 9 grid is 666ed15/14. However, this fact wasn't spotted by Eliora until after first music was composed in 666edo due to the temperament finder layout making it not immediately obvious what is the interval of equivalence, so the Factor 9 grid representation by 666edo still remains notable given that it's a scale for some of the first music composed in this edo.
-Using the 666c val, it tempres out [[2401/2400]], [[4375/4374]], and [[9801/9800]] in the 11 limit.
+=== Odd harmonics ===
+{{Harmonics in equal|666}}
-is divisible by {{EDOs|9, 18, 37, 74, 111, 222, and 333}}.
+=== Subsets and supersets ===
-edo also approximates the "[[Factor 9 grid]]", or the just intonation esoteric scale deconstructed and debunked by Adam Neely. The best rank two temperament for this scale is 495 & 666. A more general 23 limit version of this temperament can be described also, which results in a temperament with period 1/9 octave.
+is divisible by {{EDOs|9, 18, 37, 74, 111, 222, and 333}}, of which 111edo is a notable system due to its large amount of regular temperaments.
 == Regular temperament properties ==