240edo: Difference between revisions

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240edo provides the optimal patent val for the [[compton]] temperament in the 5-limit, the rank-2 temperament associated with the [[Pythagorean comma]]. Alternate mapping for 3/2 is the 705-cent sharp fifth inherited from [[80edo]].

240edo's patent val tempers out the [[225/224]] in the 7-limit, with ~~low resultant errors (two cents flat for the fifth~~, ~~a little over a cent flat and sharp~~, ~~respectively, for the major third and the~~ 7/4.) Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning~~. One important use for it is in tuning marvel temperament and marvel's extension to spectacle temperament~~.

240edo's patent val tempers out the [[225/224]] in the 7-limit, supporting [[marvel]] and [[spectacle]] 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.

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 unidecimal 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 music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. ~~We have:~~

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 unidecimal 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 music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. 3/2 is equated with two 11/9 meaning 243/242 is tempered out, 7/4 is equated with a stack of four 11/9s and two 5/4s, 11/8 is equated with a stack of five 11/9s, 13/8 is equated with a stack of two 18/11s and four 5/4s, and 17/16 is equated with three 18/11s and three 5/4s. Every harmonic is reached with help of other intervals at most with three 5/4s.

~~3 ~ 2 (11/9)^2~~

=== Subsets and supersets ===

~~5 = 2^2 (5/4)~~

~~7 ~ 2 (11/9)^4 (5/4)^2~~

~~11 ~ 2^2 (11/9)^5~~

~~13 ~ 2^3 (11/9)^(-2) (5/4)^4~~

~~17 ~ 2^4 (11/9)^(-3) (5/4)^3~~

~~It should be noted that the exponents of 5/4 above are all positive and go no higher than 4.~~

=== ~~Divisors~~ and ~~multipliers~~ ===

240edo is the 12th [[highly composite EDO]], with subset EDOs 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120.

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==Scales==

; Scales ~~associated with tempering out the~~ marvel ~~comma~~

; Scales derived from marvel and spectacle temperaments

* 23 17 23 14 23 17 23 23 14 26 14 23 - Ellis's Duodene genus [33355] retuned to 240edo

* 23 17 14 23 23 17 23 23 14 17 23 23 - Carl Lumma's scale

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 edo provides the optimal patent val for the [[compton]] temperament in the 5-limit, the rank-2 temperament associated with the [[Pythagorean comma]]. Alternate mapping for 3/2 is the 705-cent sharp fifth inherited from [[80edo]].
-edo's patent val tempers out the [[225/224]] in the 7-limit, with low resultant errors (two cents flat for the fifth, a little over a cent flat and sharp, respectively, for the major third and the 7/4.) Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning. One important use for it is in tuning marvel temperament and marvel's extension to spectacle temperament.
+edo's patent val tempers out the [[225/224]] in the 7-limit, supporting [[marvel]] and [[spectacle]] 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.
-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 unidecimal 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 music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. We have:
+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 unidecimal 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 music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. 3/2 is equated with two 11/9 meaning 243/242 is tempered out, 7/4 is equated with a stack of four 11/9s and two 5/4s, 11/8 is equated with a stack of five 11/9s, 13/8 is equated with a stack of two 18/11s and four 5/4s, and 17/16 is equated with three 18/11s and three 5/4s. Every harmonic is reached with help of other intervals at most with three 5/4s.
-~ 2 (11/9)^2
+=== Subsets and supersets ===
-= 2^2 (5/4)
-~ 2 (11/9)^4 (5/4)^2
-~ 2^2 (11/9)^5
-~ 2^3 (11/9)^(-2) (5/4)^4
-~ 2^4 (11/9)^(-3) (5/4)^3
-It should be noted that the exponents of 5/4 above are all positive and go no higher than 4.
-=== Divisors and multipliers ===
 edo is the 12th [[highly composite EDO]], with subset EDOs 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120.
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 ==Scales==
-; Scales associated with tempering out the marvel comma
+; Scales derived from marvel and spectacle temperaments
 * 23 17 23 14 23 17 23 23 14 26 14 23 - Ellis's Duodene genus [33355] retuned to 240edo
 * 23 17 14 23 23 17 23 23 14 17 23 23 - Carl Lumma's scale