600edo: Difference between revisions

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== Theory ==

600edo has good approximations to [[harmonic]]s [[3/1|3]], [[5/1|5]], [[13/1|13]], and [[23/1|23]], on top of which we may consider adding [[7/1|7]] or [[19/1|19]]. The equal temperament [[tempering out|tempers out]] the [[~~tricot~~ comma]], {{monzo| 39 -29 3 }}, and the [[mutt comma]], {{monzo| -44 -3 21 }} in the 5-limit, [[65625/65536]], [[250047/250000]], 5250987/5242880, and 28824005/28697814 in the 7-limit, [[support]]ing septimal [[mutt]]. The [[patent val]] is also a tuning for [[atomic]], ~~whereas~~ the 600e [[val]] is a tuning for atomic in the 11-limit~~. In addition, 600e val is a tuning~~ for 20th-octave temperament [[soviet ferris wheel]] as high as the 19-limit.

600edo has good approximations to [[harmonic]]s [[3/1|3]], [[5/1|5]], [[13/1|13]], and [[23/1|23]], on top of which we may consider adding [[7/1|7]] or [[19/1|19]]. The equal temperament [[tempering out|tempers out]] the [[alphatricot comma]], {{monzo| 39 -29 3 }}, and the [[mutt comma]], {{monzo| -44 -3 21 }} in the [[5-limit]], [[65625/65536]], [[250047/250000]], 5250987/5242880, and 28824005/28697814 in the [[7-limit]], [[support]]ing septimal [[mutt]]. The [[patent val]] is also a tuning for [[atomic]].

In the 11-limit, the 600e [[val]], {{val| 600 951 1393 1684 '''2075''' }}, is better tuned than the patent val. 600e is a tuning for atomic in the 11-limit as well as for the 20th-octave temperament [[soviet ferris wheel]] as high as the 19-limit.

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 600 factors into {{~~factorization~~|~~600~~}}, 600edo has subset edos {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 24, 25, 30, 40, 60, 100, 200, and 300 }}. Since 600 = 6 × 100, one step of 600edo is named a ''centitone'', being a relative cent of [[6edo]], otherwise known as the whole tone scale. In addition, because the step size is 2 cents, it is the smallest edo where every possible interval is tuned within 1 cent of error using direct approximation.

Since 600 factors into primes as {{nowrap| 23 × 3 × 52 }}, 600edo has subset edos {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 24, 25, 30, 40, 60, 100, 200, and 300 }}. Since {{nowrap|600 {{=}} 6 × 100}}, one step of 600edo is named a ''centitone'', being a relative cent of [[6edo]], otherwise known as the whole tone scale. In addition, because the step size is 2 cents, it is the smallest edo where every possible interval is tuned within 1 cent of error using direct approximation.

== Regular temperament properties ==

=== Rank-2 temperaments ===

{| 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

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| 566.00

| 81920/59049

| [[~~Tricot~~]]

| [[Alphatricot]]

|-

| 3

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| [[Minutes]] (600e)

|}

<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 lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct

== Music ==

; [[Francium]]

* "So Thankful For You" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/3pKy6BFu0NuQLlkacXFIPg Spotify] | [https://francium223.bandcamp.com/track/so-thankful-for-you Bandcamp] | [https://www.youtube.com/watch?v=cRHgCe_xdGI YouTube] – tricot in 600edo tuning

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|600}}
+{{ED intro}}
 == Theory ==
-edo has good approximations to [[harmonic]]s [[3/1|3]], [[5/1|5]], [[13/1|13]], and [[23/1|23]], on top of which we may consider adding [[7/1|7]] or [[19/1|19]]. The equal temperament [[tempering out|tempers out]] the [[tricot comma]], {{monzo| 39 -29 3 }}, and the [[mutt comma]], {{monzo| -44 -3 21 }} in the 5-limit, [[65625/65536]], [[250047/250000]], 5250987/5242880, and 28824005/28697814 in the 7-limit, [[support]]ing septimal [[mutt]]. The [[patent val]] is also a tuning for [[atomic]], whereas the 600e [[val]] is a tuning for atomic in the 11-limit. In addition, 600e val is a tuning for 20th-octave temperament [[soviet ferris wheel]] as high as the 19-limit.
+edo has good approximations to [[harmonic]]s [[3/1|3]], [[5/1|5]], [[13/1|13]], and [[23/1|23]], on top of which we may consider adding [[7/1|7]] or [[19/1|19]]. The equal temperament [[tempering out|tempers out]] the [[alphatricot comma]], {{monzo| 39 -29 3 }}, and the [[mutt comma]], {{monzo| -44 -3 21 }} in the [[5-limit]], [[65625/65536]], [[250047/250000]], 5250987/5242880, and 28824005/28697814 in the [[7-limit]], [[support]]ing septimal [[mutt]]. The [[patent val]] is also a tuning for [[atomic]].
+In the 11-limit, the 600e [[val]], {{val| 600 951 1393 1684 '''2075''' }}, is better tuned than the patent val. 600e is a tuning for atomic in the 11-limit as well as for the 20th-octave temperament [[soviet ferris wheel]] as high as the 19-limit.
 === Prime harmonics ===
@@ Line 9: / Line 11: @@
 === Subsets and supersets ===
-Since 600 factors into {{factorization|600}}, 600edo has subset edos {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 24, 25, 30, 40, 60, 100, 200, and 300 }}. Since 600 = 6 × 100, one step of 600edo is named a ''centitone'', being a relative cent of [[6edo]], otherwise known as the whole tone scale. In addition, because the step size is 2 cents, it is the smallest edo where every possible interval is tuned within 1 cent of error using direct approximation.
+Since 600 factors into primes as {{nowrap| 2<sup>3</sup> × 3 × 5<sup>2</sup> }}, 600edo has subset edos {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 24, 25, 30, 40, 60, 100, 200, and 300 }}. Since {{nowrap|600 {{=}} 6 × 100}}, one step of 600edo is named a ''centitone'', being a relative cent of [[6edo]], otherwise known as the whole tone scale. In addition, because the step size is 2 cents, it is the smallest edo where every possible interval is tuned within 1 cent of error using direct approximation.
 == Regular temperament properties ==
 === Rank-2 temperaments ===
 {| 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
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
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 | 566.00
 | 81920/59049
-| [[Tricot]]
+| [[Alphatricot]]
 |-
 | 3
@@ Line 51: / Line 54: @@
 | [[Minutes]] (600e)
 |}
-<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 lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
+== Music ==
+; [[Francium]]
+* "So Thankful For You" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/3pKy6BFu0NuQLlkacXFIPg Spotify] | [https://francium223.bandcamp.com/track/so-thankful-for-you Bandcamp] | [https://www.youtube.com/watch?v=cRHgCe_xdGI YouTube] – tricot in 600edo tuning