240edo: Difference between revisions

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Although no longer consistent to to the higher limits, 240edo's [[patent val]] [[tempering out|tempers out]] the [[225/224]] in the 7-limit, supporting [[marvel]] temperaments with harmonics 3, 5, 7 having less than two cents of error. Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning.

From a regular temperament theory perspective in the 7-limit, 240edo is similar to [[197edo]]. The main difference is that 197edo, despite a flatter third, gives generally better results and may be preferred, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the 197 ~~& 240~~ temperament, ~~whhich~~ has a comma basis {225/224, {{monzo|-49 19 -10 15}}} in the 7-limit.

From a regular temperament theory perspective in the 7-limit, 240edo is similar to [[197edo]]. The main difference is that 197edo, despite a flatter third, gives generally better results and may be preferred, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the 43 & 197 temperament, which has a comma basis {225/224, {{monzo| -49 19 -10 15 }}} in the 7-limit.

For higher limits, 240edo tempers out [[243/242]] in the 11-limit, [[351/350]] in the 13-limit, and [[375/374]] in the 17-limit, and adding these to the mix converts marvel temperament into [[spectacle]] temperament. This is still a planar temperament, but more complex as two undecimal neutral thirds of [[11/9]] make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 is the 350-cent interval often employed in [[24edo]] versions of [[Arabic, Turkish, Persian music|Arabic music]].) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. 3/2 is equated with two 11/9 as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.

For higher limits, 240edo tempers out [[243/242]] in the 11-limit, [[351/350]] in the 13-limit, and [[375/374]] in the 17-limit, and adding these to the mix converts marvel temperament into [[spectacle]] temperament. This is still a planar temperament, but more complex as two undecimal neutral thirds of [[11/9]] make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 is the 350-cent interval often employed in [[24edo]] versions of [[Arabic, Turkish, Persian music|Arabic music]].)

=== Subsets and supersets ===

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 Although no longer consistent to to the higher limits, 240edo's [[patent val]] [[tempering out|tempers out]] the [[225/224]] in the 7-limit, supporting [[marvel]] temperaments with harmonics 3, 5, 7 having less than two cents of error. Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning.
-From a regular temperament theory perspective in the 7-limit, 240edo is similar to [[197edo]]. The main difference is that 197edo, despite a flatter third, gives generally better results and may be preferred, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the 197 & 240 temperament, whhich has a comma basis {225/224, {{monzo|-49 19 -10 15}}} in the 7-limit.
+From a regular temperament theory perspective in the 7-limit, 240edo is similar to [[197edo]]. The main difference is that 197edo, despite a flatter third, gives generally better results and may be preferred, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the 43 & 197 temperament, which has a comma basis {225/224, {{monzo| -49 19 -10 15 }}} in the 7-limit.
-For higher limits, 240edo tempers out [[243/242]] in the 11-limit, [[351/350]] in the 13-limit, and [[375/374]] in the 17-limit, and adding these to the mix converts marvel temperament into [[spectacle]] temperament. This is still a planar temperament, but more complex as two undecimal neutral thirds of [[11/9]] make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 is the 350-cent interval often employed in [[24edo]] versions of [[Arabic, Turkish, Persian music|Arabic music]].) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. 3/2 is equated with two 11/9 as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.
+For higher limits, 240edo tempers out [[243/242]] in the 11-limit, [[351/350]] in the 13-limit, and [[375/374]] in the 17-limit, and adding these to the mix converts marvel temperament into [[spectacle]] temperament. This is still a planar temperament, but more complex as two undecimal neutral thirds of [[11/9]] make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 is the 350-cent interval often employed in [[24edo]] versions of [[Arabic, Turkish, Persian music|Arabic music]].)
 === Subsets and supersets ===