600edo: Difference between revisions

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The '''600 equal divisions of the octave''' ('''600edo'''), or the '''1200(-tone) equal temperament''' ('''600tet''', '''600et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 600 [[equal]] parts of exactly 2 [[cent]]s each. Because the step size is 2 cents, it is the smallest EDO where every possible interval is tuned within 1 cent of error.

== 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 [[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 ===

600edo has the ~~same mapping as~~ [[~~1200edo~~]] ~~in the~~ 5-~~limit~~.

=== Subsets and supersets ===

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"

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 283\600

| 566.00

| 81920/59049

| [[Alphatricot]]

|-

| 3

| 193\600 (7\600)

| 386.00 (14.00)

| 5/4 (126/125)

| [[Mutt]]

|-

| 12

| 249\600 (1\600)

| 498.00 (2.00)

| 4/3 (32805/32768)

| [[Atomic]] (600e)

|-

| 20

| 283\600 (13\600)

| 566.00 (26.00)

| 238/165 (?)

| [[Soviet ferris wheel]] (600e)

|-

| 60

| 249\600 (1\600)

| 498.00 (2.00)

| 4/3 (32805/32768)

| [[Minutes]] (600e)

|}

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

[[~~Category~~:~~Equal divisions of the octave~~|~~####]] <!~~-- 4-~~digit number -->~~

== 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}}
-The '''600 equal divisions of the octave''' ('''600edo'''), or the '''1200(-tone) equal temperament''' ('''600tet''', '''600et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 600 [[equal]] parts of exactly 2 [[cent]]s each. Because the step size is 2 cents, it is the smallest EDO where every possible interval is tuned within 1 cent of error.
+{{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 [[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 ===
 {{Harmonics in equal|600}}
-edo has the same mapping as [[1200edo]] in the 5-limit.
+=== Subsets and supersets ===
+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"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperaments
+|-
+| 1
+| 283\600
+| 566.00
+| 81920/59049
+| [[Alphatricot]]
+|-
+| 3
+| 193\600<br>(7\600)
+| 386.00<br>(14.00)
+| 5/4<br>(126/125)
+| [[Mutt]]
+|-
+| 12
+| 249\600<br>(1\600)
+| 498.00<br>(2.00)
+| 4/3<br>(32805/32768)
+| [[Atomic]] (600e)
+|-
+| 20
+| 283\600<br>(13\600)
+| 566.00<br>(26.00)
+| 238/165<br>(?)
+| [[Soviet ferris wheel]] (600e)
+|-
+| 60
+| 249\600<br>(1\600)
+| 498.00<br>(2.00)
+| 4/3<br>(32805/32768)
+| [[Minutes]] (600e)
+|}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
+== 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