240edo
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.
| ← 239edo | 240edo | 241edo → |
If we round off to the nearest five cents, we end up with a val (mapping to primes) for 240edo of <240 380 557 674|. This tempers out the 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, 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.
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:
3 ~ 2 (11/9)^2
5 = 2^2 (5/4)
7 ~ 2 (11/9)^4 (5/4)^2
11 ~ 2^2 (11/9)^5
13 ~ 2^3 (11/9)^(-2) (5/4)^4
17 ~ 2^4 (11/9)^(-3) (5/4)^3
It should be noted that the exponents of 5/4 above are all positive and go no higher than 4.
Scales
Here are some examples of scales retuned to 240edo and hence exhibiting marvel temperament.
! duodene.scl
!
Ellis's Duodene : genus [33355]
12
!
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]->[-1, 0, -2]||[0, -1, -1]->[0, 0, -1]->[0, 0, 0]->[0, 0, 1]->[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, 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.