328edo: Difference between revisions

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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 3136/3125, 589824/588245
| 2401/2400, 3136/3125, 589824/588245
| {{mapping| 328 520 762 921 }}
| {{Mapping| 328 520 762 921 }}
| −0.298
| −0.298
| 0.229
| 0.229
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3136/3125, 9801/9800, 19712/19683
| 2401/2400, 3136/3125, 9801/9800, 19712/19683
| {{mapping| 328 520 762 921 1135 }}
| {{Mapping| 328 520 762 921 1135 }}
| −0.303
| −0.303
| 0.205
| 0.205
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647
| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647
| {{mapping| 328 520 762 921 1135 1214 }}
| {{Mapping| 328 520 762 921 1135 1214 }}
| −0.295
| −0.295
| 0.188
| 0.188
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125
| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125
| {{mapping| 328 520 762 921 1135 1214 1341 }}
| {{Mapping| 328 520 762 921 1135 1214 1341 }}
| −0.293
| −0.293
| 0.174
| 0.174
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 2
| 2
| 111\328<br />(53\328)
| 47\328
| 406.10<br />(193.90)
| 171.95
| 495/392<br />(28/25)
| 243/220
| [[Semiosiris]]
|-
| 2
| 53\328
| 193.90
| 28/25
| [[Semihemiwürschmidt]]
| [[Semihemiwürschmidt]]
|-
|-
| 8
| 8
| 136\328<br />(13\328)
| 13\328
| 497.56<br />(47.56)
| 47.56
| 4/3<br />(36/35)
| 36/35
| [[Twilight]]
| [[Twilight]]
|-
|-
| 41
| 41
| 49\328<br />(1\328)
| 1\328
| 179.27<br />(3.66)
| 3.66
| 567/512<br />(352/351)
| 352/351
| [[Hemicountercomp]]
| [[Hemicountercomp]]
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


[[Category:Hemiwürschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Semiporwell]]
[[Category:Semiporwell]]

Latest revision as of 10:35, 12 July 2026

← 327edo 328edo 329edo →
Prime factorization 23 × 41
Step size 3.65854 ¢ 
Fifth 192\328 (702.439 ¢) (→ 24\41)
Semitones (A1:m2) 32:24 (117.1 ¢ : 87.8 ¢)
Consistency limit 13
Distinct consistency limit 13

328 equal divisions of the octave (abbreviated 328edo or 328ed2), also called 328-tone equal temperament (328tet) or 328 equal temperament (328et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 328 equal parts of about 3.66 ¢ each. Each step represents a frequency ratio of 21/328, or the 328th root of 2.

Theory

328edo is enfactored in the 5-limit, with the same tuning as 164edo, but the approximation of higher harmonics are much improved. Like 164edo, it inherits the perfect fifth from 41edo. 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.

Prime harmonics

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
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)

Subsets and supersets

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
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

Rank-2 temperaments

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 47\328 171.95 243/220 Semiosiris
2 53\328 193.90 28/25 Semihemiwürschmidt
8 13\328 47.56 36/35 Twilight
41 1\328 3.66 352/351 Hemicountercomp

* In minimal-generator form

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