240edo
| ← 239edo | 240edo | 241edo → |
Theory
240edo's patent val tempers out the 225/224 in the 7-limit, 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. One important use for it is in tuning marvel temperament and marvel's extension to spectacle temperament.
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, [-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 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.
- 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
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.