600edo: Difference between revisions
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{{Infobox ET}} | |||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
600edo tempers out the [[ | 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 === | === Prime harmonics === | ||
{{Harmonics in equal|600}} | {{Harmonics in equal|600}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 600 factors into 2<sup>3</sup> × 3 × 5<sup>2</sup>, | 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. | ||
Since 600 = 6 | |||
== Regular temperament properties == | == Regular temperament properties == | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| 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 | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 27: | Line 28: | ||
| 566.00 | | 566.00 | ||
| 81920/59049 | | 81920/59049 | ||
| [[ | | [[Alphatricot]] | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 45: | Line 46: | ||
| 566.00<br>(26.00) | | 566.00<br>(26.00) | ||
| 238/165<br>(?) | | 238/165<br>(?) | ||
| [[Soviet | | [[Soviet ferris wheel]] (600e) | ||
|- | |- | ||
| 60 | | 60 | ||
| Line 53: | Line 54: | ||
| [[Minutes]] (600e) | | [[Minutes]] (600e) | ||
|} | |} | ||
<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 | |||
Latest revision as of 05:15, 5 June 2025
| ← 599edo | 600edo | 601edo → |
600 equal divisions of the octave (abbreviated 600edo or 600ed2), also called 600-tone equal temperament (600tet) or 600 equal temperament (600et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 600 equal parts of exactly 2 ¢ each. Each step represents a frequency ratio of 21/600, or the 600th root of 2.
Theory
600edo has good approximations to harmonics 3, 5, 13, and 23, on top of which we may consider adding 7 or 19. The equal temperament tempers out the alphatricot comma, [39 -29 3⟩, and the mutt comma, [-44 -3 21⟩ in the 5-limit, 65625/65536, 250047/250000, 5250987/5242880, and 28824005/28697814 in the 7-limit, supporting septimal mutt. The patent val is also a tuning for atomic.
In the 11-limit, the 600e 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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.045 | -0.314 | -0.826 | +0.682 | -0.528 | -0.955 | +0.487 | -0.274 | +0.423 | +0.964 |
| Relative (%) | +0.0 | +2.2 | -15.7 | -41.3 | +34.1 | -26.4 | -47.8 | +24.3 | -13.7 | +21.1 | +48.2 | |
| Steps (reduced) |
600 (0) |
951 (351) |
1393 (193) |
1684 (484) |
2076 (276) |
2220 (420) |
2452 (52) |
2549 (149) |
2714 (314) |
2915 (515) |
2973 (573) | |
Subsets and supersets
Since 600 factors into primes as 23 × 3 × 52, 600edo has subset 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.
Regular temperament properties
Rank-2 temperaments
| 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) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct