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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== Theory == | == Theory == | ||
328edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[164edo]], but the approximation of higher | 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]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|328|intervals=prime|columns=11}} | {{Harmonics in equal|328|intervals=prime|columns=11}} | ||
=== | === Subsets and supersets === | ||
Since 328 factors into 2<sup>3</sup> × 41, | Since 328 factors into 2<sup>3</sup> × 41, 328edo has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 24: | Line 24: | ||
| 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 }} | ||
| -0.298 | | -0.298 | ||
| 0.229 | | 0.229 | ||
| Line 31: | Line 31: | ||
| 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 }} | ||
| -0.303 | | -0.303 | ||
| 0.205 | | 0.205 | ||
| Line 38: | Line 38: | ||
| 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 }} | ||
| -0.295 | | -0.295 | ||
| 0.188 | | 0.188 | ||
| Line 45: | Line 45: | ||
| 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 }} | ||
| -0.293 | | -0.293 | ||
| 0.174 | | 0.174 | ||
| Line 57: | Line 57: | ||
|+Table of rank-2 temperaments by generator | |+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 | ||
|- | |- | ||
| Line 98: | Line 98: | ||
| [[Hemicountercomp]] | | [[Hemicountercomp]] | ||
|} | |} | ||
<nowiki>*</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:Hemiwürschmidt]] | ||
[[Category:Semiporwell]] | [[Category:Semiporwell]] | ||
Revision as of 08:15, 22 January 2024
| ← 327edo | 328edo | 329edo → |
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
| 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.
| 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