666edo

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666edo
Prime factorization 2 × 32 × 37
Step size 1.8018¢
Fifth 390\666 (702.702¢) (→195\333)
Major 2nd 114\666 (205.405¢)
Semitones (A1:m2) 66:48 (118.92¢:86.49¢)

The 666 equal divisions of the octave (666edo), or the 666-tone equal temperament (666tet), 666 equal temperament (666et) when viewed from a regular temperament perspective, divides the octave into 666 equal parts of about 1.8 cents each.

Theory

Approximation of odd harmonics in 666edo
Harmonic 3 5 7 9 11 13 15 17 19 21
Error absolute (¢) +0.748 -0.728 +0.543 -0.306 +0.033 -0.888 +0.020 -0.451 -0.216 -0.511
relative (%) +41 -40 +30 -17 +2 -49 +1 -25 -12 -28
Steps
(reduced)
1056
(390)
1546
(214)
1870
(538)
2111
(113)
2304
(306)
2464
(466)
2602
(604)
2722
(58)
2829
(165)
2925
(261)

666edo is appropriate for use with the 2.11.19.41.43 subgroup. If significant errors are allowed, 666edo can be used with 2.7.11.17.19.23. Harmonics from 2 to 17 for 666edo all land on even numbers, meaning its contorted order 2 and they ultimately derive from 333edo. As such, 666edo provides the optimal patent val for novemkleismic temperament just as 333edo does. 666edo provides good approximations for: 15/11, 16/11, 16/15, 13/12, 13/10, 22/15, 23/14. Its 11/8 ultimately derives from 37edo, and 7/6 from 9edo.

Using the 666c val, it tempres out 2401/2400, 4375/4374, and 9801/9800 in the 11-limit.

666 is divisible by 9, 18, 37, 74, 111, 222, and 333.

666edo also approximates the "Factor 9 grid", or the just intonation esoteric scale deconstructed and debunked by Adam Neely. The best rank two temperament for this scale is 495 & 666. A more general 23-limit version of this temperament can be described also, which results in a temperament with period 1/9 octave.

Rank two temperaments by generator

Periods

per octave

Generator Cents Associated

ratio

Temperaments
9 35\666 63.063 ~28/27 Factor 9 Grid

Music

References