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.
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[edit]
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[edit]
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.