666edo: Difference between revisions

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== 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 [[enfactoring|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.

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. 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.

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 {{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 ===

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=== 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 ===

Note: temperaments supported by [[333edo]] are not included.

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

|+Table of rank-2 temperaments by generator

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

! Periods per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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|-

| 9

| 175\666 (27\666)

| 175\666 (27\666)

| 315.315 (48.648)

| 315.315 (48.648)

| 6/5 (36/35)

| 6/5 (36/35)

| [[Ennealimmal]] (666c)

|-

| 18

| 138\666 (27\666)

| 138\666 (27\666)

| 248.648 (48.648)

| 248.648 (48.648)

| 15/13 (99/98)

| 15/13 (99/98)

| [[Hemiennealimmal]] (666cf)

|}

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

<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Scales ==

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* [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

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|666}}
+{{ED intro}}
 == Theory ==
-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 [[enfactoring|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.
-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. 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.
+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 {{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.
+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 ===
@@ Line 11: / Line 15: @@
 === 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 ===
+Note: temperaments supported by [[333edo]] are not included.
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 36: / Line 44: @@
 |-
 | 9
-| 175\666<br>(27\666)
+| 175\666<br />(27\666)
-| 315.315<br>(48.648)
+| 315.315<br />(48.648)
-| 6/5<br>(36/35)
+| 6/5<br />(36/35)
 | [[Ennealimmal]] (666c)
 |-
 | 18
-| 138\666<br>(27\666)
+| 138\666<br />(27\666)
-| 248.648<br>(48.648)
+| 248.648<br />(48.648)
-| 15/13<br>(99/98)
+| 15/13<br />(99/98)
 | [[Hemiennealimmal]] (666cf)
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Scales ==
@@ Line 56: / Line 64: @@
 * [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
+[[Category:Listen]]

666edo: Difference between revisions

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