666edo
| ← 665edo | 666edo | 667edo → |
666 equal divisions of the octave (abbreviated 666edo or 666ed2), also called 666-tone equal temperament (666tet) or 666 equal temperament (666et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 666 equal parts of about 1.8 ¢ each. Each step represents a frequency ratio of 21/666, or the 666th root of 2.
Theory
666edo is enfactored in the 17-limit, with the same mapping as 333edo which it doubles where it tempers out exactly the same commas and inherits its strong approximations for 11/8 and 7/6. Hence on the patent vals up to 17-limit it doesn't provide much new apart from characteristics of what it doubles.
Nonetheless, the 2.9.15.21.11.17.19 subgroup is great for 666edo. Alternatively, it can be used with 2.9.15.7.11.19.23. Additionally, the 2.9.15.11.39.23/7 subgroup is useful as a fractional subgroup, representing direct approximations for 15/11, 16/11, 16/15, 13/12, 13/10, 22/15, 23/14.
For non-patent vals, 666ceefg val ⟨666 1056 1547 1870 2305 2465 2723] is recommended as it is the best tuned in the 17-limit (better than the patent val). A comma basis for the 17-limit is {936/935, 3584/3575, 2431/2430, 4375/4374, 2401/2400, 34496/34425}. This val supports the ennealimmal temperament in the 7-limit and a lesser known extension of it into the 13-limit as 270 & 666ceef.
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, approximately equivalent to 6691edo. However, this fact was not 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 is 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 (%) | +41.5 | -40.4 | +30.2 | -17.0 | +1.9 | -49.3 | +1.1 | -25.0 | -12.0 | -28.3 | +30.8 | |
| 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
Since 666 factors into 2 × 32 × 37, 666edo has subset edos 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, of which 111edo is a notable system due to its accuracy relative to its size.
Regular temperament properties
Rank-2 temperaments
Note: temperaments supported by 333edo are not included.
| 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) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Scales
- Factor9Grid[14]: 39 38 36 35 66 62 59 55 52 49 46 44 42 41
Music
- Timeline (2022) – classical piano
- Church of Original Sin (2022) – as part of the symphonic metal project Mercury Amalgam