456edo: Difference between revisions

Revision as of 07:20, 3 November 2023

456edo is enfactored in the 5-limit, with the same tuning as 152edo, defined by tempering out 1600000/1594323 (amity comma) and [32 -7 -9⟩ (escapade comma), as well as [23 6 -14⟩ (vishnuzma), [41 -20 -4⟩ (undim comma), and [-14 -19 19⟩ (enneadeca). In the 7-limit, it tempers out 10976/10935, 235298/234375, and 134217728/133984375, providing the optimal patent val for the chromat temperament.

Approximation of odd harmonics in 456edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.68	+0.53	-0.40	-1.28	+1.31	-1.05	+1.20	+0.31	-0.14	+0.27	+0.67
Error	Relative (%)	+25.7	+20.1	-15.4	-48.6	+49.9	-40.1	+45.8	+11.7	-5.5	+10.3	+25.6
Steps (reduced)		723 (267)	1059 (147)	1280 (368)	1445 (77)	1578 (210)	1687 (319)	1782 (414)	1864 (40)	1937 (113)	2003 (179)	2063 (239)

Since 456 factors into 2³ × 3 × 19, 456edo has subset edos 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, and 228.

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''456edo''' is the [[EDO|equal division of the octave]] into 456 parts of 2.63158 cents each. It is [[contorted]] in the 5-limit, tempering out 1600000/1594323 (amity comma) and 4294967296/4271484375 (escapade comma), as well as 6115295232/6103515625 (vishnuzma), 2199023255552/2179240250625 (undim comma), and 19073486328125/19042491875328 (enneadeca). In the 7-limit, it tempers out 10976/10935, 235298/234375, and 134217728/133984375, providing the [[optimal patent val]] for the [[chromat]] temperament.
+{{EDO intro|456}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[152edo]], defined by [[tempering out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 32 -7 -9 }} ([[escapade comma]]), as well as {{monzo| 23 6 -14 }} ([[vishnuzma]]), {{monzo| 41 -20 -4 }} (undim comma), and {{monzo| -14 -19 19 }} ([[enneadeca]]). In the 7-limit, it tempers out [[10976/10935]], 235298/234375, and 134217728/133984375, providing the [[optimal patent val]] for the [[chromat]] temperament.
+=== Odd harmonics ===
+{{Harmonics in equal|456}}
+=== Subsets and supersets ===
+Since 456 factors into {{factorization|456}}, 456edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, and 228 }}.
+[[Category:Chromat]]