666edo: Difference between revisions

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== Theory ==

~~{{Harmonics~~ in ~~equal|666}}~~

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 ~~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]]~~.

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.

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

=== Odd harmonics ===

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

=== 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.

~~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~~.

== Regular temperament properties ==

===Rank-2 temperaments===

=== Rank-2 temperaments ===

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

~~!Periods per 8ve~~

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

~~!Generator~~

~~!Cents~~

~~!Associated ratio~~

~~!Temperaments~~

|-

|9

! Periods per 8ve

|35\666

! Generator*

|63.063

! Cents*

|~~~28~~/27

! Associated ratio*

|~~Enneasoteric~~

! 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

== Scales ==

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

== 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

* [https://www.youtube.com/watch?v=Y4gVyK9vNkk&t=7s Timeline] by [[User:Eliora|Eliora]]

[[Category:Listen]]

~~== References ==~~

* [http://x31eq.com/cgi-bin/rt.cgi?key=666_1289_1874_2426_2948_3443_3914_4363_4792_5203_5597_5976_6340_6691&ets=666&limit=15%2F14_16%2F14_17%2F14_18%2F14_19%2F14_20%2F14_21%2F14_22%2F14_23%2F14_24%2F14_25%2F14_26%2F14_27%2F14_28%2F14 Approximation of the Factor 9 grid in 666edo]

* [https://www.youtube.com/watch?v=ghUs-84NAAU&t=203s Testing 432 Hz Frequencies and Temperaments Adam Neely]

[[Category:~~Equal divisions of the octave|###~~]] ~~~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|666}}
+{{ED intro}}
 == Theory ==
-{{Harmonics in equal|666}}
+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 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]].
+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.
-Using the 666c val, it tempres out [[2401/2400]], [[4375/4374]], and [[9801/9800]] in the 11 limit.
+=== Odd harmonics ===
+{{Harmonics in equal|666}}
-is divisible by {{EDOs|9, 18, 37, 74, 111, 222, and 333}}.
+=== 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.
-edo 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.
 == Regular temperament properties ==
-===Rank-2 temperaments===
+=== Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-!Periods <br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-!Generator
-!Cents
-!Associated <br>ratio
-!Temperaments
 |-
-|9
+! Periods<br />per 8ve
-|35\666
+! Generator*
-|63.063
+! Cents*
-|~28/27
+! Associated<br />ratio*
-|Enneasoteric
+! 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
 == 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
-* [https://www.youtube.com/watch?v=Y4gVyK9vNkk&t=7s Timeline] by [[User:Eliora|Eliora]]
+[[Category:Listen]]
-== References ==
-* [http://x31eq.com/cgi-bin/rt.cgi?key=666_1289_1874_2426_2948_3443_3914_4363_4792_5203_5597_5976_6340_6691&ets=666&limit=15%2F14_16%2F14_17%2F14_18%2F14_19%2F14_20%2F14_21%2F14_22%2F14_23%2F14_24%2F14_25%2F14_26%2F14_27%2F14_28%2F14 Approximation of the Factor 9 grid in 666edo]
-* [https://www.youtube.com/watch?v=ghUs-84NAAU&t=203s Testing 432 Hz Frequencies and Temperaments   Adam Neely]
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->