328edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>3</sup> × 41
{{ED intro}}
| Step size = 3.65854¢
| Fifth = 192\328 (702.44¢) (→ [[41edo|24\41]])
| Semitones = 32:24 (117.07¢ : 87.80¢)
| Consistency = 13
}}
The '''328 equal divisions of the octave''' ('''328edo'''), or the '''328(-tone) equal temperament''' ('''328tet''', '''328et''') when viewed from a [[regular temperament]] perspective, divides the octave into 328 [[equal]] parts of about 3.66 [[cent]]s each.


== Theory ==
== Theory ==
328edo 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 supports [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].  
328edo is [[enfactoring|enfactored]] in the [[5-limit]], with the same tuning as [[164edo]], but the approximation of higher [[harmonic]]s are much improved. It has a sharp tendency, with harmonics 3 through 17 all tuned sharp. The equal temperament [[tempering out|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]].  


328 factors into 2<sup>3</sup> × 41, with subset edos 2, 4, 8, 41, 82, and 164.
=== Prime harmonics ===
{{Harmonics in equal|328|intervals=prime|columns=11}}


=== Prime harmonics ===
=== Subsets and supersets ===
{{Primes in edo|328}}
Since 328 factors into {{Factorisation|328}}, 328edo has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! 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
|-
|-
Line 29: Line 25:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 3136/3125, 589824/588245
| 2401/2400, 3136/3125, 589824/588245
| [{{val| 328 520 762 921 }}]
| {{mapping| 328 520 762 921 }}
| -0.298
| −0.298
| 0.229
| 0.229
| 6.27
| 6.27
Line 36: Line 32:
| 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
| [{{val| 328 520 762 921 1135 }}]
| {{mapping| 328 520 762 921 1135 }}
| -0.303
| −0.303
| 0.205
| 0.205
| 5.61
| 5.61
Line 43: Line 39:
| 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
| [{{val| 328 520 762 921 1135 1214 }}]
| {{mapping| 328 520 762 921 1135 1214 }}
| -0.295
| −0.295
| 0.188
| 0.188
| 5.15
| 5.15
Line 50: Line 46:
| 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
| [{{val| 328 520 762 921 1135 1214 1341 }}]
| {{mapping| 328 520 762 921 1135 1214 1341 }}
| -0.293
| −0.293
| 0.174
| 0.174
| 4.77
| 4.77
Line 57: Line 53:


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by 164et are not listed.
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 84: Line 83:
|-
|-
| 2
| 2
| 111\328<br>(53\328)
| 111\328<br />(53\328)
| 406.10<br>(193.90)
| 406.10<br />(193.90)
| 495/392<br>(28/25)
| 495/392<br />(28/25)
| [[Semihemiwürschmidt]]
| [[Semihemiwürschmidt]]
|-
|-
| 8
| 8
| 136\328<br>(13\328)
| 136\328<br />(13\328)
| 497.56<br>(47.56)
| 497.56<br />(47.56)
| 4/3<br>(36/35)
| 4/3<br />(36/35)
| [[Twilight]]
| [[Twilight]]
|-
|-
| 41
| 41
| 49\328<br>(1\328)
| 49\328<br />(1\328)
| 179.27<br>(3.66)
| 179.27<br />(3.66)
| 567/512<br>(352/351)
| 567/512<br />(352/351)
| [[Hemicounterpyth]]
| [[Hemicountercomp]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave]]
[[Category:Hemiwürschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Semiporwell]]
[[Category:Semiporwell]]

Latest revision as of 14:41, 20 February 2025

← 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. 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
2 111\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 distinct

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