666edo
| ← 665edo | 666edo | 667edo → |
666 equal divisions of the octave (666edo), or 666-tone equal temperament (666tet), 666 equal temperament (666et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 666 equal parts of about 1.8 ¢ each.
Theory
666edo is enfactored in the 17-limit, with the same mapping as 333edo. Using precision as a criterion, 2.11.19.41.43 subgroup is great for 666edo. If significant errors are allowed, 666edo can be used with 2.7.11.17.19.23. 666edo provides good direct approximations for: 15/11, 16/11, 16/15, 13/12, 13/10, 22/15, 23/14. 666edo also has a strong approximation for 11/8 derived from 37edo and for 7/6 derived from 9edo, but on the 2.7/6.11 subgroup it is enfactored, with the same tuning again as 333edo, tempering out the 37-11-comma and the septimal ennealimma. The 666c val, tempers out 2401/2400, 4375/4374, and 9801/9800 in the 11 limit.
666edo is also used by Eliora to approximate the "Factor 9 grid", or the just intonation esoteric scale deconstructed and debunked by Adam Neely. Now, it may be worth noting that the tuning system which truly has an excellent approximation of the Factor 9 grid is 666ed15/14. However, this fact wasn't spotted by Eliora until after first music was composed in 666edo due to the temperament finder layout making it not immediately obvious what is the interval of equivalence, so the Factor 9 grid representation by 666edo still remains notable given that it's a scale for some of the first music composed in this edo.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.748 | -0.728 | +0.543 | -0.306 | +0.033 | -0.888 | +0.020 | -0.451 | -0.216 | -0.511 | +0.554 |
| relative (%) | +41 | -40 | +30 | -17 | +2 | -49 | +1 | -25 | -12 | -28 | +31 | |
| Steps (reduced) |
1056 (390) |
1546 (214) |
1870 (538) |
2111 (113) |
2304 (306) |
2464 (466) |
2602 (604) |
2722 (58) |
2829 (165) |
2925 (261) |
3013 (349) | |
Subsets and supersets
666 has subset edos 1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, of which 111edo is a notable system due to its large amount of regular temperaments.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator | Cents | Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 175\666 | 315.31 | 6/5 | Parakleismic (666b) |
| 9 | 35\666 | 63.063 | 336/323 | Enneasoteric |
| 9 | 175\666 (27\666) |
315.315 (48.648) |
6/5 (36/35) |
Ennealimmal (666c) |
| 18 | 138\666 (27\666) |
248.648 (48.648) |
15/13 (99/98) |
Hemiennealimmal (666cf) |
Scales
- Factor9Grid[14]: 39 38 36 35 66 62 59 55 52 49 46 44 42 41
Music
- Timeline, classical piano
- Church of Original Sin, as part of the symphonic metal project Mercury Amalgam