Xenharmonic Wiki

240edo: Difference between revisions

← Older edit
VisualWikitext
Yourmusic Productions (talk | contribs)
autopatrolled, emailconfirmed
1,861 edits
m fix broken link
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"
 
(31 intermediate revisions by 9 users not shown)
Line 1: Line 1:
The 240edo divides the octave into 240 steps of exactly five cents each. One important use for it is in tuning marvel temperament and marvel's extension to spectacle temperament.
{{Infobox ET}}
{{ED intro}}


If we round off to the nearest five cents, we end up with a [[Vals_and_Tuning_Space|val]] (mapping to primes) for 240edo of <240 380 557 674|. This tempers out the [http://en.wikipedia.org/wiki/Septimal_kleisma septimal kleisma] of 225/224, with low resultant errors (two cents flat for the fifth, a little over a cent flat and sharp, respectively, for the major third and the 7/4.) Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning. However [[197edo|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.
== 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]].  


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. We have:
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.  


3 ~ 2 (11/9)^2
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.


5 = 2^2 (5/4)
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]].)


7 ~ 2 (11/9)^4 (5/4)^2
=== Odd harmonics ===
{{Harmonics in equal|240}}


11 ~ 2^2 (11/9)^5
=== 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.


13 ~ 2^3 (11/9)^(-2) (5/4)^4
== Interval table ==
See [[Table of 240edo intervals]].


17 ~ 2^4 (11/9)^(-3) (5/4)^3
== 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 }}
| {{Mapping| 240 380 557 }}
| 0.5998
| 0.5044
| 10.09
|}


It should be noted that the exponents of 5/4 above are all positive and go no higher than 4.
=== 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 ==
; 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
* 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


Here are some examples of scales retuned to 240edo and hence exhibiting marvel temperament.
=== 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


! duodene.scl
== Instruments ==
A [[Lumatone mapping for 240edo]] is now available.


!
== 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''".)


Ellis's Duodene : genus [33355]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/Nu-xBrsd8_o ''microtonal improvisation in 240edo''] (2025)


12
== 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 [[just-noticeable difference]] between pitches.


!
[[Category:Compton]]
 
[[Category:Marvel]]
16/15
 
9/8
 
6/5
 
5/4
 
4/3
 
45/32
 
3/2
 
8/5
 
5/3
 
9/5
 
15/8
 
2/1
 
! duodene240.scl
 
!
 
Ellis's Duodene : genus [33355] retuned to 240edo
 
12
 
!
 
115.
 
200.
 
315.
 
385.
 
500.
 
585.
 
700.
 
815.
 
885.
 
1015.
 
1085.
 
1200.
 
! lumma5.scl
 
!
 
Carl Lumma's scale, 5-limit just version, TL 19-2-99
 
! Also diadie1, prism, Fokker 12-tone just
 
12
 
!
 
16/15
 
9/8
 
75/64
 
5/4
 
4/3
 
45/32
 
3/2
 
8/5
 
5/3
 
225/128
 
15/8
 
2/1
 
! lumma5_240.scl
 
!
 
Carl Lumma's scale aka diadie1, 240edo version
 
12
 
!
 
115.
 
200.
 
270.
 
385.
 
500.
 
585.
 
700.
 
815.
 
885.
 
970.
 
1085.
 
1200.
 
! marvel chords
 
! [-1, -1, 2]-&gt;[-1, 0, -2]||[0, -1, -1]-&gt;[0, 0, -1]-&gt;[0, 0, 0]-&gt;[0, 0, 1]-&gt;[0, 0, 2]
 
! pum14.scl
 
pum14 scale
 
14
 
!
 
25/24
 
16/15
 
10/9
 
75/64
 
5/4
 
4/3
 
64/45
 
3/2
 
25/16
 
8/5
 
5/3
 
16/9
 
15/8
 
2
 
! pum14_240.scl
 
pum14 in 240edo
 
14
 
!
 
70.
 
115.
 
185.
 
270.
 
385.
 
500.
 
615.
 
700.
 
770.
 
815.
 
885.
 
1000.
 
1085.
 
1200.
 
! tetrads [[0, -1, 0], [0, -1, 1], [1, -1, 1], [1, -1, 2], ! [0, 0, 2], [0, -1, -2], [0, 0, 1], [0, -1, -1]]
 
! doubleduo.scl
 
Ellis duodene union 11/9 times the duodene in 240et
 
24
 
!
 
35.
 
115.
 
165.
 
200.
 
235.
 
315.
 
350.
 
385.
 
465.
 
500.
 
550.
 
585.
 
665.
 
700.
 
735.
 
815.
 
850.
 
885.
 
935.
 
1015.
 
1050.
 
1085.
 
1165.
 
1200.
 
==Links==
[[Shaahin_Mohajeri|Shaahin Mohajeri]], an Iranian Tombak player and composer, calls his personal [http://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.
Retrieved from "https://en.xen.wiki/w/240edo"