240edo: Difference between revisions

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~~==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~~.

== Theory ==

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]].

~~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.

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 ~~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.

240edo is similar to [[197edo]] in terms of intonation in the 7-limit. 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 {{nowrap| 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 [[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]].)

=== Odd harmonics ===

=== 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.

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.

~~In addition, as every fifth step of~~ [[~~1200edo~~]]~~, it is the largest highly composite EDO compatible with cents~~.

== Interval table ==

See [[Table of 240edo intervals]].

=== ~~Odd harmonics~~ ===

== Regular temperament properties ==

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

|-

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

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

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.3.5

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

| {{Mapping| 240 380 557 }}

| 0.5998

| 0.5044

| 10.09

|}

==~~Scales~~==

=== Rank-2 temperaments ===

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

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 7\240

| 35.00

| 45/44

| [[Gammy]]

|-

| 1

| 101\240

| 505.00

| 104976/78125

| [[Countermeantone]]

|-

| 12

| 77\240 (3\240)

| 385.00 (15.00)

| 5/4 (81/80)

| [[Compton]]

|}

<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 ==

; 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 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 - Carl Lumma's scale

* 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

* 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

* 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]

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

~~; Other scales~~

== Instruments ==

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

* 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]

== Music ==

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

; [[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''".)

~~==Music==~~

; [[Bryan Deister]]

~~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''".~~

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

==~~Links~~==

== Trivia ==

[[~~Shaahin_Mohajeri|~~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.

[[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.

~~[[Category:Equal divisions of the octave|###]] ~~

[[Category:Compton]]

[[Category:Marvel]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|240}}
+{{ED intro}}
-==Theory==
-edo 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.
+== Theory ==
+edo 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]].
-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.
+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 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.
+edo is similar to [[197edo]] in terms of intonation in the 7-limit. 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 {{nowrap| 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 [[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]].)
+=== Odd harmonics ===
+{{Harmonics in equal|240}}
 === Subsets and supersets ===
-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.
+edo 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.
-In addition, as every fifth step of [[1200edo]], it is the largest highly composite EDO compatible with cents.
+== Interval table ==
+See [[Table of 240edo intervals]].
-=== Odd harmonics ===
+== Regular temperament properties ==
-{{Harmonics in equal|240}}
+{| 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 }}
+| {{Mapping| 240 380 557 }}
+| 0.5998
+| 0.5044
+| 10.09
+|}
-==Scales==
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperaments
+|-
+| 1
+| 7\240
+| 35.00
+| 45/44
+| [[Gammy]]
+|-
+| 1
+| 101\240
+| 505.00
+| 104976/78125
+| [[Countermeantone]]
+|-
+| 12
+| 77\240<br>(3\240)
+| 385.00<br>(15.00)
+| 5/4<br>(81/80)
+| [[Compton]]
+|}
+<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 ==
 ; 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 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 - Carl Lumma's scale
+* 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
+* 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
+* 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]
+* 23 31 80 23 83 – [[Indonesian|Balinese]] pentatonic [[pelog]] scale; [[Tolgahan Çoğulu]]'s tuning
-; Other scales
+== Instruments ==
+A [[Lumatone mapping for 240edo]] is now available.
-* 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]
+== Music ==
-* 23 31 80 23 83 - [[Indonesian|Balinese]] pentatonic [[pelog]] scale; [[Tolgahan Çoğulu]]'s tuning
+; [[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''".)
-==Music==
+; [[Bryan Deister]]
-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''".
+* [https://www.youtube.com/shorts/Nu-xBrsd8_o ''microtonal improvisation in 240edo''] (2025)
-==Links==
+== Trivia ==
-[[Shaahin_Mohajeri|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.
+[[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.
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Compton]]
 [[Category:Marvel]]