328edo: Difference between revisions

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Cleanup; clarify the title row of the rank-2 temp table; -redundant categories
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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! 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
| 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
| &minus;0.298
| 0.229
| 0.229
| 6.27
| 6.27
Line 32: Line 24:
| 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
| &minus;0.303
| 0.205
| 0.205
| 5.61
| 5.61
Line 39: Line 31:
| 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
| &minus;0.295
| 0.188
| 0.188
| 5.15
| 5.15
Line 46: Line 38:
| 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
| &minus;0.293
| 0.174
| 0.174
| 4.77
| 4.77
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by 164et are not listed.  
Note: 5-limit temperaments supported by 164et are not listed.  


{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 81: Line 67:
|-
|-
| 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)
| [[Hemicountercomp]]
| [[Hemicountercomp]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


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

Revision as of 04:15, 16 November 2024

← 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, 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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

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

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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