415edo: Difference between revisions

Revision as of 14:32, 6 November 2023

If harmonic 5 is used, 415edo tends very sharp. In the 5-limit the equal temperament tempers out the parakleisma, [8 14 -13⟩; in the 7-limit 3136/3125 and 4375/4374, so that it supports parakleismic, the 99 & 316 temperament, and provides the optimal patent val. In the 11-limit it tempers out 12005/11979, 16384/16335, and 41503/41472; and in the 13-limit, 676/675, 1001/1000, 2080/2079, 3584/3575, and 10648/10647.

Approximation of odd harmonics in 415edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.70	+1.16	-0.15	+1.39	+0.97	+0.92	-1.04	-0.86	+0.32	+0.54	-0.80
Error	Relative (%)	+24.1	+40.0	-5.2	+48.1	+33.6	+31.8	-36.0	-29.7	+11.0	+18.8	-27.8
Steps (reduced)		658 (243)	964 (134)	1165 (335)	1316 (71)	1436 (191)	1536 (291)	1621 (376)	1696 (36)	1763 (103)	1823 (163)	1877 (217)

Since 415 factors into 5 × 83, 415edo contains 5edo and 83edo as subsets.

@@ Line 2: / Line 2: @@
 {{EDO intro|415}}
-If [[5/1|harmonic 5]] is used, 415edo tends very sharp. In the 5-limit the equal temperament tempers out the [[parakleisma]], {{monzo| 8 14 -13 }}; in the 7-limit [[3136/3125]] and [[4375/4374]], so that it [[support]]s the [[parakleismic]] temperament and provides the [[optimal patent val]].
+If [[5/1|harmonic 5]] is used, 415edo tends very sharp. In the 5-limit the equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}; in the 7-limit [[3136/3125]] and [[4375/4374]], so that it [[support]]s [[parakleismic]], the 99 & 316 temperament, and provides the [[optimal patent val]]. In the 11-limit it tempers out 12005/11979, [[16384/16335]], and [[41503/41472]]; and in the 13-limit, [[676/675]], [[1001/1000]], [[2080/2079]], 3584/3575, and [[10648/10647]].
 === Odd harmonics ===
@@ Line 8: / Line 8: @@
 === Subsets and supersets ===
-edo has subset edos {{EDOs| 5 and 83 }}.
+Since 415 factors into {{factorization|415}}, 415edo contains [[5edo]] and [[83edo]] as subsets.
 [[Category:Parakleismic]]