328edo: Difference between revisions

328edo is enfactored in the 5-limit, with the same tuning as 164edo, but the approximation of higher harmonics are much improved. It has a sharp tendency, with harmonics 3 through 17 all tuned sharp. It tempers out 2401/2400, 3136/3125, and 6144/6125 in the 7-limit, 9801/9800, 16384/16335 and 19712/19683 in the 11-limit, 676/675, 1001/1000, 1716/1715 and 2080/2079 in the 13-limit, 936/935, 1156/1155 and 2601/2600 in the 17-limit, so that it supports würschmidt and hemiwürschmidt, and provides the optimal patent val for 7-limit hemiwürschmidt, 11- and 13-limit semihemiwür, and 13-limit semiporwell.

Approximation of prime harmonics in 328edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.48	+1.49	+0.69	+1.12	+0.94	+1.14	-1.17	+0.99	-1.53	+0.09
Error	Relative (%)	+0.0	+13.2	+40.8	+18.8	+30.6	+25.6	+31.2	-32.0	+27.2	-41.8	+2.4
Steps (reduced)		328 (0)	520 (192)	762 (106)	921 (265)	1135 (151)	1214 (230)	1341 (29)	1393 (81)	1484 (172)	1593 (281)	1625 (313)

Since 328 factors into 2³ × 41, it has subset edos 2, 4, 8, 41, 82, and 164.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma list	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3.5.7	2401/2400, 3136/3125, 589824/588245	[⟨328 520 762 921]]	-0.298	0.229	6.27
2.3.5.7.11	2401/2400, 3136/3125, 9801/9800, 19712/19683	[⟨328 520 762 921 1135]]	-0.303	0.205	5.61
2.3.5.7.11.13	676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647	[⟨328 520 762 921 1135 1214]]	-0.295	0.188	5.15
2.3.5.7.11.13.17	676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125	[⟨328 520 762 921 1135 1214 1341]]	-0.293	0.174	4.77

Note: 5-limit temperaments supported by 164et are not listed.

Table of rank-2 temperaments by generator
Periods per 8ve	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	53\328	193.90	28/25	Hemiwürschmidt
1	117\328	428.05	2800/2187	Osiris
2	17\328	62.20	28/27	Eagle
2	111\328 (53\328)	406.10 (193.90)	495/392 (28/25)	Semihemiwürschmidt
8	136\328 (13\328)	497.56 (47.56)	4/3 (36/35)	Twilight
41	49\328 (1\328)	179.27 (3.66)	567/512 (352/351)	Hemicountercomp

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[164edo]]. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].
+edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[164edo]], but the approximation of higher harmonics are much improved. It has a sharp tendency, with [[harmonic]]s 3 through 17 all tuned sharp. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].
 === Prime harmonics ===
-{{Harmonics in equal|328|columns=11}}
+{{Harmonics in equal|328|intervals=prime|columns=11}}
 === Divisors ===