666edo: Difference between revisions

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'''666 EDO''' divides the octave into steps of 1.<span style="text-decoration: overline">801</span> cents each.
{{Infobox ET}}
{{ED intro}}
 
== Theory ==
== Theory ==
{{primes in edo|666|columns=14}}
666edo is [[enfactoring|enfactored]] in the 17-limit, with the same mapping as [[333edo]], but 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. 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, 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 ===
{{Harmonics in equal|666}}
 
=== Subsets and supersets ===
Since 666 factors into {{factorization|666}}, 666edo has subset edos {{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 ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 175\666
| 315.31
| 6/5
| [[Parakleismic]] (666b)
|-
| 9
| 35\666
| 63.063
| 336/323
| [[Enneasoteric]]
|-
| 9
| 175\666<br />(27\666)
| 315.315<br />(48.648)
| 6/5<br />(36/35)
| [[Ennealimmal]] (666c)
|-
| 18
| 138\666<br />(27\666)
| 248.648<br />(48.648)
| 15/13<br />(99/98)
| [[Hemiennealimmal]] (666cf)
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Scales ==
* Factor9Grid[14]: 39 38 36 35 66 62 59 55 52 49 46 44 42 41


666edo is appropriate for use with the 2.11.19.41.43 subgroup, a choice with very large prime harmonics. If significant errors are allowed, 666edo can be used with 2.7.11.17.19.23. The alternations between approxmation make 666edo a good choice for "no-number" subgroups which skip particular harmonics.
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=Y4gVyK9vNkk&t=7s ''Timeline''] (2022) – classical piano
* [https://www.youtube.com/watch?v=KV0-kT5slZc ''Church of Original Sin''] (2022) – as part of the symphonic metal project Mercury Amalgam


666 is divisible by {{EDOs|9, 18, 37, 74, 111, 222, and 333}}.
[[Category:Listen]]

Latest revision as of 22:53, 20 February 2025

← 665edo 666edo 667edo →
Prime factorization 2 × 32 × 37
Step size 1.8018 ¢ 
Fifth 390\666 (702.703 ¢) (→ 65\111)
Semitones (A1:m2) 66:48 (118.9 ¢ : 86.49 ¢)
Dual sharp fifth 390\666 (702.703 ¢) (→ 65\111)
Dual flat fifth 389\666 (700.901 ¢)
Dual major 2nd 113\666 (203.604 ¢)
Consistency limit 3
Distinct consistency limit 3

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, but 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. 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, 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

Approximation of odd harmonics in 666edo
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

Table of rank-2 temperaments by generator
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

Eliora
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