193edo: Difference between revisions

193edo is consistent to the 11-odd-limit, and almost consistent to the 23-odd-limit, the only failure being 13/11 and its octave complement. This makes it a strong 23-limit system.

It provides the optimal patent val for the sqrtphi temperament in the 13-, 17- and 19-limit, and for the 13-limit minos and vish temperaments.

Approximation of prime harmonics in 193edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.64	-0.82	+1.12	+2.05	-1.15	+0.74	+0.93	-0.30	+2.55	-0.99
Error	Relative (%)	+0.0	+10.2	-13.2	+18.1	+33.0	-18.5	+12.0	+15.0	-4.7	+41.0	-16.0
Steps (reduced)		193 (0)	306 (113)	448 (62)	542 (156)	668 (89)	714 (135)	789 (17)	820 (48)	873 (101)	938 (166)	956 (184)

193edo is the 44th prime edo.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3	[306 -193⟩	[⟨193 306]]	-0.2005	0.2005	3.23
2.3.5	15625/15552, [50 -33 1⟩	[⟨193 306 448]]	-0.0158	0.3084	4.96
2.3.5.7	5120/5103, 15625/15552, 16875/16807	[⟨193 306 448 542]]	-0.1118	0.3146	5.06
2.3.5.7.11	540/539, 1375/1372, 4375/4356, 5120/5103	[⟨193 306 448 542 668]]	-0.2080	0.3408	5.48
2.3.5.7.11.13	325/324, 364/363, 540/539, 625/624, 4096/4095	[⟨193 306 448 542 668 714]]	-0.1216	0.3662	5.89
2.3.5.7.11.13.17	325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095	[⟨193 306 448 542 668 714 789]]	-0.1302	0.3397	5.46
2.3.5.7.11.13.17.19	325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215	[⟨193 306 448 542 668 714 789 820]]	-0.1414	0.3191	5.13

193et has a lower relative error in the 23-limit than any previous equal temperaments, past 190g and followed by 217.
193et is also notable in the 19-limit, where it has a lower absolute error than any previous equal temperaments, past 190g and followed by 212gh.

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperament
1	16\193	99.48	18/17	Quintakwai / quintakwoid
1	18\193	111.92	16/15	Vavoom
1	39\193	242.49	147/128	Septiquarter
1	51\193	317.10	6/5	Countercata (7-limit)
1	56\193	348.19	11/9	Eris
1	61\193	379.28	56/45	Marthirds
1	67\193	416.58	14/11	Sqrtphi
1	79\193	491.19	3645/2744	Fifthplus
1	80\193	497.41	4/3	Kwai

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Approximation of sqrt (π): 159\193 (988.60104 cents), and of φ: 134\193 (833.16062 cents), both inside in the superdiatonic scale: 25 25 25 9 25 25 25 25 9

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 == Theory ==
-edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra|vish]] temperaments.
+edo is [[consistent]] to the [[11-odd-limit]], and almost consistent to the [[23-odd-limit]], the only failure being [[13/11]] and its [[octave complement]]. This makes it a strong 23-limit system.
+As an equal temperament, 193et [[tempering out|tempers out]] the [[15625/15552|kleisma]] in the 5-limit; [[5120/5103]] and [[16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], 4375/4356 in the 11-limit; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], [[4096/4095]] in the 13-limit; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[529/528]] in the 23-limit.
+It provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra|vish]] temperaments.
 === Prime harmonics ===