240edo: Difference between revisions
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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
240edo | 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 | 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. | ||
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 | 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 === | === Odd harmonics === | ||
{{Harmonics in equal|240}} | {{Harmonics in equal|240}} | ||
=== Subsets and supersets === | |||
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. | |||
== Interval table == | == Interval table == | ||
| Line 28: | Line 26: | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 36: | Line 34: | ||
| 2.3.5 | | 2.3.5 | ||
| 531441/524288, {{monzo| -29 -11 20 }} | | 531441/524288, {{monzo| -29 -11 20 }} | ||
| {{ | | {{Mapping| 240 380 557 }} | ||
| 0.5998 | | 0.5998 | ||
| 0.5044 | | 0.5044 | ||
| Line 46: | Line 44: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 65: | Line 63: | ||
|- | |- | ||
| 12 | | 12 | ||
| 77\240<br>(3\240) | |||
| 385.00<br>(15.00) | |||
| 5/4<br>(81/80) | |||
| 77\240<br | |||
| 385.00<br | |||
| 5/4<br | |||
| [[Compton]] | | [[Compton]] | ||
|} | |} | ||
<nowiki />* [[Normal | <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 == | ||
| Line 88: | Line 80: | ||
* 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] | * 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 | * 23 31 80 23 83 – [[Indonesian|Balinese]] pentatonic [[pelog]] scale; [[Tolgahan Çoğulu]]'s tuning | ||
== Instruments == | |||
A [[Lumatone mapping for 240edo]] is now available. | |||
== Music == | == Music == | ||
; [[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) | |||
== | == Trivia == | ||
[[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 [[ | [[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:Compton]] | [[Category:Compton]] | ||
[[Category:Marvel]] | [[Category:Marvel]] | ||
Latest revision as of 13:30, 13 March 2026
| ← 239edo | 240edo | 241edo → |
240 equal divisions of the octave (abbreviated 240edo or 240ed2), also called 240-tone equal temperament (240tet) or 240 equal temperament (240et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 240 equal parts of exactly 5 ¢ each. Each step represents a frequency ratio of 21/240, or the 240th root of 2.
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 is not well approximated, meaning it is a dual-fifth system; its alternative mapping for 3/2 is the 705 ¢ sharp fifth inherited from 80edo.
Although no longer consistent 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.
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 43 & 197 temperament, which 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 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 music.)
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) | |
Subsets and supersets
240edo is the 12th highly composite edo, with subset 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.
Interval table
See Table of 240edo intervals.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 531441/524288, [-29 -11 20⟩ | [⟨240 380 557]] | 0.5998 | 0.5044 | 10.09 |
Rank-2 temperaments
| 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 |
* Octave-reduced form, reduced to the first half-octave, and 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 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]
- 23 31 80 23 83 – Balinese pentatonic pelog scale; Tolgahan Çoğulu's tuning
Instruments
A Lumatone mapping for 240edo is now available.
Music
- 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".)
Trivia
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.