240edo: Difference between revisions

240edo notably provides the [[optimal patent val]] for the 5-limit [[compton]] temperament, the rank-2 temperament associated with the [[Pythagorean comma]]. However, it is only [[consistent]] in the [[5-odd-limit]]. Its mapping for [[harmonic]] [[3/1|3]] is not well approximated, meaning it is a [[dual-fifth system]]; its alternative mapping for 3/2 is the 705{{c}} sharp fifth inherited from [[80edo]].  

Although no longer consistent to the higher limits, 240edo's [[patent val]] [[tempering out|tempers out]] the [[225/224]] in the 7-limit, [[support]]ing [[marvel]] with harmonics 3, [[5/1|5]], [[7/1|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.  

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 [[rank-3 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]].)

240edo is the 12th [[highly composite edo]], with subset edos {{EDOs| 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 expressible in integer cents.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

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

! colspan="2" | Tuning error

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3.5

| 531441/524288, {{monzo| -29 -11 20 }}

| 0.5998

| 0.5044

| 10.09

 

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods<br>per 8ve

! Associated<br>ratio*

! Temperaments

| 35.00

| 45/44

| [[Gammy]]

| 104976/78125

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

| 385.00<br>(15.00)

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

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

 

; Scales derived from marvel and spectacle temperaments

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

* 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

 

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

 

A [[Lumatone mapping for 240edo]] is now available.

 

; [[Chris Charles]] (via [https://www.youtube.com/@microtonalguitar Microtonal Guitar - Tolgahan Çoğulu])

* [https://www.youtube.com/watch?v=6GoGlj5IyZc ''Balinese Gamelan Music on Microtonal Guitar - Chris Charles''] (2017) (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''".)

 

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/Nu-xBrsd8_o ''microtonal improvisation in 240edo''] (2025)

 

[[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]] between pitches.