240edo: Difference between revisions
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{{EDO intro|240}} | {{EDO intro|240}} | ||
==Theory== | ==Theory== | ||
240edo provides the optimal patent val for the [[compton]] temperament | 240edo is a [[dual-fifth system]]. Nonetheless, it is [[consistent]] in the 5-limit and notably provides the [[optimal patent val]] for the [[compton]] temperament, 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, 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. | Although no longer consistent to to the higher limits, 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. | 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. 3/2 is equated with two 11/9 | 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 as a corollary of 243/242 being 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. | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
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[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Compton]] | |||
[[Category:Marvel]] | |||
Revision as of 19:39, 4 March 2023
| ← 239edo | 240edo | 241edo → |
Theory
240edo is a dual-fifth system. Nonetheless, it is consistent in the 5-limit and notably provides the optimal patent val for the compton temperament, the rank-2 temperament associated with the Pythagorean comma. Alternate mapping for 3/2 is the 705-cent sharp fifth inherited from 80edo.
Although no longer consistent to to the higher limits, 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, [-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. 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 11/9s 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.
Subsets and supersets
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.
In addition, as every fifth step of 1200edo, it is the largest highly composite EDO compatible with cents.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.96 | -1.31 | +1.17 | +1.09 | -1.32 | -0.53 | +1.73 | +0.04 | +2.49 | -0.78 | +1.73 |
| Relative (%) | -39.1 | -26.3 | +23.5 | +21.8 | -26.4 | -10.6 | +34.6 | +0.9 | +49.7 | -15.6 | +34.5 | |
| Steps (reduced) |
380 (140) |
557 (77) |
674 (194) |
761 (41) |
830 (110) |
888 (168) |
938 (218) |
981 (21) |
1020 (60) |
1054 (94) |
1086 (126) | |
Scales
- 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
- 14 9 14 17 23 23 23 17 14 9 14 23 17 23 - Pum[14] scale
- 16 10 7 7 16 7 7 16 7 10 7 16 7 7 16 7 7 10 16 7 7 16 7 - Ellis duodene union 11/9 times the duodene
- Other scales
- 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 - Compton[24]
Links
Shaahin Mohajeri, an Iranian Tombak player and composer, calls his personal Google site "240edo", where he makes the point that five cents is a size close to the just noticeable difference between pitches.