323edo: Difference between revisions

Revision as of 14:05, 10 December 2022

323et tempers out the vulture comma, [24 -21 4⟩ and the luna comma, [38 -2 -15⟩, in the 5-limit; 4375/4374, 589824/588245 and 703125/702464 in the 7-limit, supporting 7-limit vulture, lunatic, enneadecal, and gamera. In the 11-limit, the 323e val and the patent val are comparable in errors. 1375/1372, 5632/5625, 14641/14580, and 19712/19683 are tempered out in the patent val; 540/539, 6250/6237, 12005/11979, and 16384/16335 are tempered out in the 323e val. It provides the optimal patent val for the rank-5 temperament tempering out 1573/1568, the lambeth comma, as well as deuteromere, the 2.3.5.11 subgroup temperament tempering out 14641/14580.

323 = 17 × 19, and shares the excellent approximations of 25/24 in 17edo and of the 28/27 and the 6/5 in 19edo.

323edo is an excellent tuning when considered in the no-11 subgroup, with errors of 25% or less all the way into the 31-limit.

Approximation of prime harmonics in 323edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.21	+0.06	+0.83	-1.47	-0.90	-0.93	-0.30	-0.41	-0.48	-0.76
Error	Relative (%)	+0.0	+5.7	+1.7	+22.4	-39.6	-24.2	-25.0	-8.1	-11.1	-12.8	-20.5
Steps (reduced)		323 (0)	512 (189)	750 (104)	907 (261)	1117 (148)	1195 (226)	1320 (28)	1372 (80)	1461 (169)	1569 (277)	1600 (308)

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3	[512 -323⟩	[⟨323 512]]	-0.0669	0.0669	1.80
2.3.5	[24 -21 4⟩, [38 -2 -15⟩	[⟨323 512 750]]	-0.0538	0.0577	1.55
2.3.5.7	4375/4374, 589824/588245, 703125/702464	[⟨323 512 750 907]]	-0.1146	0.1165	3.14
2.3.5.7.13	676/675, 4096/4095, 4375/4374, 16848/16807	[⟨323 512 750 907 1195]]	-0.0431	0.1770	4.76
2.3.5.7.13.17	442/441, 676/675, 2500/2499, 4096/4095, 4375/4374	[⟨323 512 750 907 1195 1320]]	+0.0020	0.1905	5.13
2.3.5.7.11	540/539, 4375/4374, 12005/11979, 16384/16335	[⟨323 512 750 907 1118]] (323e)	-0.2213	0.2375	6.39
2.3.5.7.11.13	364/363, 540/539, 676/675, 4096/4095, 4375/4374	[⟨323 512 750 907 1118 1195]] (323e)	-0.1440	0.2773	7.47
2.3.5.7.11	1375/1372, 4375/4374, 5632/5625, 14641/14580	[⟨323 512 750 907 1117]] (323)	-0.0066	0.2399	6.46
2.3.5.7.11.13	676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374	[⟨323 512 750 907 1117 1195]] (323)	+0.0350	0.2380	6.40

Table of rank-2 temperaments by generator
Periods per 8ve	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	26\323	96.59	200/189	Hemiluna (323)
1	30\323	111.46	16/15	Stockhausenic (323)
1	31\323	115.17	77/72	Semigamera (323)
1	52\323	193.19	352/315	Luna / lunatic (323e)
1	62\323	230.34	8/7	Gamera
1	128\323	475.54	320/243	Vulture
17	134\323 (9\323)	248.92 (33.44)	[-23 5 9 -2⟩ (100352/98415)	Chlorine
19	134\323 (2\323)	497.83 (7.43)	4/3 (225/224)	Enneadecal

@@ Line 45: / Line 45: @@
 |-
 | 2.3.5.7.13
-| 676/675, 4375/4374, 4096/4095, 16848/16807
+| 676/675, 4096/4095, 4375/4374, 16848/16807
-| [{{val| 323 512 750 907 1195}}]
+| [{{val| 323 512 750 907 1195 }}]
 | -0.0431
 | 0.1770
@@ Line 52: / Line 52: @@
 |-
 | 2.3.5.7.13.17
-| 442/441, 1275/1274, 2500/2499, 4096/4095, 4375/4374
+| 442/441, 676/675, 2500/2499, 4096/4095, 4375/4374
-| [{{val| 323 512 750 907 1195 1320}}]
+| [{{val| 323 512 750 907 1195 1320 }}]
 | +0.0020
 | 0.1905
@@ Line 64: / Line 64: @@
 | style="border-top: double;" | 0.2375
 | style="border-top: double;" | 6.39
+|-
+| 2.3.5.7.11.13
+| 364/363, 540/539, 676/675, 4096/4095, 4375/4374
+| [{{val| 323 512 750 907 1118 1195 }}] (323e)
+| -0.1440
+| 0.2773
+| 7.47
 |-
 | style="border-top: double;" | 2.3.5.7.11
@@ Line 71: / Line 78: @@
 | style="border-top: double;" | 0.2399
 | style="border-top: double;" | 6.46
+|-
+| 2.3.5.7.11.13
+| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374
+| [{{val| 323 512 750 907 1117 1195 }}] (323)
+| +0.0350
+| 0.2380
+| 6.40
 |}