328edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
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.
{{EDO intro|328}}


== 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 [[support]]s [[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]]. 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]].  


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


=== Prime harmonics ===
=== Divisors ===
{{Primes in edo|328}}
Since 328 factors into 2<sup>3</sup> × 41, it 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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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=== 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
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-
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| 179.27<br>(3.66)
| 179.27<br>(3.66)
| 567/512<br>(352/351)
| 567/512<br>(352/351)
| [[Hemicounterpyth]]
| [[Hemicountercomp]]
|}
|}


Revision as of 13:37, 22 December 2022

← 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

Template:EDO intro

Theory

328edo is 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.

Prime harmonics

Approximation of odd harmonics in 328edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.48 +1.49 +0.69 +0.97 +1.12 +0.94 -1.68 +1.14 -1.17 +1.17 +0.99
Relative (%) +13.2 +40.8 +18.8 +26.5 +30.6 +25.6 -46.0 +31.2 -32.0 +32.0 +27.2
Steps
(reduced) 		520
(192) 		762
(106) 		921
(265) 		1040
(56) 		1135
(151) 		1214
(230) 		1281
(297) 		1341
(29) 		1393
(81) 		1441
(129) 		1484
(172)

Divisors

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
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
(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
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