240edo: Difference between revisions

{{EDO intro|240}}

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]] [[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 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]].)

 

 

==Regular temperament properties==

! rowspan="2" | Optimal<br />8ve stretch (¢)

| {{mapping| 240 380 557 }}

! Periods<br />per 8ve

! Associated<br />ratio*

| 77\240<br />(3\240)

| 385.00<br />(15.00)

| 5/4<br />(81/80)

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

* 23 17 23 14 23 17 23 23 14 26 14 23 &ndash; [[Alexander Ellis|Ellis]]'s [[Duodene]] genus [33355] retuned to 240edo

* 23 17 14 23 23 17 23 23 14 17 23 23 &ndash; [[Carl Lumma]]'s scale

* 14 9 14 17 23 23 23 17 14 9 14 23 17 23 &ndash; 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 &ndash; Ellis duodene union [[11/9]] times the duodene

* 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 &ndash; [[Compton]][24]

* 23 31 80 23 83 &ndash; [[Indonesian|Balinese]] pentatonic [[pelog]] scale; [[Tolgahan Çoğulu]]'s tuning

The video [https://www.youtube.com/watch?v=6GoGlj5IyZc ''Balinese Gamelan Music on Microtonal Guitar - Chris Charles''] on the YouTube channel [https://www.youtube.com/@microtonalguitar Microtonal Guitar - Tolgahan Çoğulu] uses a 5-tone subset of 240edo for all three pieces performed in the recording. As explained in the video description: "''The scale we used in the piece: Pelog Selisir: D, Eb +30 F -15 A -30 Bb -15''".

== Links ==

[[Shaahin Mohajeri]], an [[Arabic, Turkish, Persian music|Iranian]] Tombak player and composer, calls his personal [https://sites.google.com/site/240edo/ Google site] "240edo", where he makes the point that five cents is a size close to the [[Just-noticeable_difference|just noticeable difference]] between pitches.