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. The equal temperament 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, 328edo 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*	Cents*	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

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

@@ Line 12: / Line 12: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3.5.7
@@ Line 41: / Line 50: @@
 | 0.174
 | 4.77
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
 Note: 5-limit temperaments supported by 164et are not listed.
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
 |-
 | 1
@@ Line 83: / Line 99: @@
 | 567/512<br />(352/351)
 | [[Hemicountercomp]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:Hemiwürschmidt]]
 [[Category:Semiporwell]]