666edo: Difference between revisions

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~~'''666 EDO''' divides the octave into steps of 1.801 cents each.~~

== Theory ==

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 ===

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. 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~~.

=== 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.

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

== Regular temperament properties ==

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | 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)

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

~~666edo provides good approximations for: [[15/11]], [[16/11]], [[16/15]], [[13/3|13/12]], [[13/10]], [[22/15]],~~ [~~[23/~~14]].

== Scales ==

* Factor9Grid[14]: 39 38 36 35 66 62 59 55 52 49 46 44 42 41

~~666 is divisible by {{EDOs|9, 18, 37, 74, 111, 222, and 333}}~~.

== 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

~~666edo also approximates the "Factor 9 Grid", or the just intonation mystical scale deconstructed and debunked by Adam Neely.~~

[[Category:Listen]]

@@ Line 1: / Line 1: @@
-'''666 EDO''' divides the octave into steps of 1.<span style="text-decoration: overline">801</span> cents each.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-{{primes in edo|666|columns=14}}
+edo 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.
+edo 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}}
-edo 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. 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.
+=== 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.
-Using the 666c val, it tempres out [[2401/2400]], [[4375/4374]], and [[9801/9800]] in the 11-limit.
+== 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
-edo provides good approximations for: [[15/11]], [[16/11]], [[16/15]], [[13/3|13/12]], [[13/10]], [[22/15]], [[23/14]].
+== Scales ==
+* Factor9Grid[14]: 39 38 36 35 66 62 59 55 52 49 46 44 42 41
-is divisible by {{EDOs|9, 18, 37, 74, 111, 222, and 333}}.
+== 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
-edo also approximates the "Factor 9 Grid", or the just intonation mystical scale deconstructed and debunked by Adam Neely.
+[[Category:Listen]]