129edo: Difference between revisions

Revision as of 17:23, 27 May 2024

129edo is inconsistent to the 5-odd-limit and both harmonics 3 and 5 are about halfway between its steps. It is the last patent val that tempers out 81/80 so as to support meantone and its higher-limit expansions. It also tempers out 1029/1024 and 1728/1715 in the 7-limit; 176/175 and 540/539 in the 11-limit; 507/500, 676/675 and 847/845 in the 13-limit; 221/220 in the 17-limit; 171/170 and 286/285 in the 19-limit. It provides the optimal patent val for the 11-limit rank-3 clio temperament.

Approximation of odd harmonics in 129edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-4.28	+4.38	-1.38	+0.74	-2.48	-3.32	+0.10	-2.63	+0.16	+3.64	+4.28
Error	Relative (%)	-46.0	+47.1	-14.9	+8.0	-26.7	-35.7	+1.1	-28.3	+1.7	+39.1	+46.1
Steps (reduced)		204 (75)	300 (42)	362 (104)	409 (22)	446 (59)	477 (90)	504 (117)	527 (11)	548 (32)	567 (51)	584 (68)

Since 129 factors into 3 × 43, 129edo contains 3edo and 43edo as its subsets.

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 {{EDO intro|129}}
-edo provides the [[optimal patent val]] for the 11-limit rank-3 [[clio]] temperament. It is the last [[patent val]] that [[tempering out|tempers out]] 81/80 so as to support [[meantone]] and its higher-limit expansions. It also tempers out [[1029/1024]] and [[1728/1715]] in the [[7-limit]]; [[176/175]] and [[540/539]] in the [[11-limit]]; [[507/500]], [[676/675]] and [[847/845]] in the [[13-limit]]; [[221/220]] in the [[17-limit]]; [[171/170]] and [[286/285]] in the [[19-limit]].
+edo is in[[consistent]] to the [[5-odd-limit]] and both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. It is the last [[patent val]] that [[tempering out|tempers out]] [[81/80]] so as to [[support]] [[meantone]] and its higher-limit expansions. It also tempers out [[1029/1024]] and [[1728/1715]] in the [[7-limit]]; [[176/175]] and [[540/539]] in the [[11-limit]]; [[507/500]], [[676/675]] and [[847/845]] in the [[13-limit]]; [[221/220]] in the [[17-limit]]; [[171/170]] and [[286/285]] in the [[19-limit]]. It provides the [[optimal patent val]] for the 11-limit rank-3 [[clio]] temperament.
-The factorization of 129 is [[3edo|3]] and [[43edo|43]].
 === Odd harmonics ===
 {{Harmonics in equal|129}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+=== Subsets and supersets ===
+Since 129 factors into {{factorization|129}}, 129edo contains [[3edo]] and [[43edo]] as its subsets.
 [[Category:Clio]]