240edo: Difference between revisions
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240edo is [[consistent]] in the [[5-odd-limit]] and notably provides the [[optimal patent val]] for the 5-limit [[compton]] temperament, the rank-2 temperament associated with the [[Pythagorean comma]]. However, its mapping for 3 is not well approximated, meaning it is a [[dual-fifth system]], with alternate mapping for 3/2 is the 705-cent sharp fifth inherited from [[80edo]]. | 240edo is [[consistent]] in the [[5-odd-limit]] and notably provides the [[optimal patent val]] for the 5-limit [[compton]] temperament, the rank-2 temperament associated with the [[Pythagorean comma]]. However, its mapping for 3 is not well approximated, meaning it is a [[dual-fifth system]], with 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]] [[ | Although no longer consistent to to the higher limits, 240edo's [[patent val]] [[tempers out]] the [[225/224]] in the 7-limit, supporting [[marvel]] 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 43 & 197 temperament, which 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. | ||
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See [[Table of 240edo intervals]]. | See [[Table of 240edo intervals]]. | ||
==Regular temperament properties== | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
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| [[Compton]] | | [[Compton]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
; Scales derived from marvel and spectacle temperaments | ; Scales derived from marvel and spectacle temperaments | ||
* 23 17 23 14 23 17 23 23 14 26 14 23 | * 23 17 23 14 23 17 23 23 14 26 14 23 – [[Alexander Ellis|Ellis]]'s [[Duodene]] genus [33355] retuned to 240edo | ||
* 23 17 14 23 23 17 23 23 14 17 23 23 | * 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 | * 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 | * 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 === | === 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 | * 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] | ||
* 23 31 80 23 83 | * 23 31 80 23 83 – [[Indonesian|Balinese]] pentatonic [[pelog]] scale; [[Tolgahan Çoğulu]]'s tuning | ||
== Music == | == Music == | ||