378edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''378 equal division''' divides the octave into 378 equal parts of 3.175 cents each. It tempers out 32805/32768 in the 5-limit and 3136/3125 in the 7-limit, so that it [[support]]s [[Schismatic_family#Bischismatic|bischismatic temperament]] and in fact provides the [[Optimal_patent_val|optimal patent val]]. It tempers out 441/440 and 8019/8000 in the 11-limit and 729/728 and 1001/1000 in the 13-limit so that it supports 11 and 13 limit bischismatic, and it also gives the optimal patent val for 13-limit bischismatic.
{{EDO intro|378}}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
The equal temperament [[tempering out|tempers out]] 32805/32768 ([[schisma]]) in the 5-limit and [[3136/3125]] in the 7-limit, so that it [[support]]s [[bischismic]], and in fact provides the [[optimal patent val]]. It tempers out [[441/440]] and [[8019/8000]] in the 11-limit and [[729/728]] and [[1001/1000]] in the 13-limit so that it supports 11- and 13-limit bischismatic, and it also gives the optimal patent val for 13-limit bischismic.
 
=== Prime harmonics ===
{{Harmonics in equal|378}}
 
=== Subsets and supersets ===
Since 378 factors into {{factorization|378}}, 378edo has subset edos {{EDOs| 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, and 189 }}.
 
[[Category:Bischismic]]

Revision as of 14:39, 9 November 2023

← 377edo 378edo 379edo →
Prime factorization 2 × 33 × 7
Step size 3.1746 ¢ 
Fifth 221\378 (701.587 ¢)
Semitones (A1:m2) 35:29 (111.1 ¢ : 92.06 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

The equal temperament tempers out 32805/32768 (schisma) in the 5-limit and 3136/3125 in the 7-limit, so that it supports bischismic, and in fact provides the optimal patent val. It tempers out 441/440 and 8019/8000 in the 11-limit and 729/728 and 1001/1000 in the 13-limit so that it supports 11- and 13-limit bischismatic, and it also gives the optimal patent val for 13-limit bischismic.

Prime harmonics

Approximation of prime harmonics in 378edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.37 +0.99 -0.57 +1.06 +0.74 -0.19 +0.90 +0.30 -1.01 +1.00
Relative (%) +0.0 -11.6 +31.1 -18.0 +33.5 +23.4 -6.1 +28.3 +9.4 -31.7 +31.4
Steps
(reduced) 		378
(0) 		599
(221) 		878
(122) 		1061
(305) 		1308
(174) 		1399
(265) 		1545
(33) 		1606
(94) 		1710
(198) 		1836
(324) 		1873
(361)

Subsets and supersets

Since 378 factors into 2 × 33 × 7, 378edo has subset edos 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, and 189.

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