240edo
| ← 239edo | 240edo | 241edo → |
Theory
240edo, being a small multiple of 12, tempers out the Pythagorean comma. Alternate mapping is the 705-cent sharp fifth inherited from 80edo.
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.
Divisors and multipliers
240edo is the 12th highly composite EDO, with subset EDOs 1, 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 compatible with cents.
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) | |
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.