323edo: Difference between revisions
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→Regular temperament properties: +semigamera |
Update templates; link to deuteromere |
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{{EDO intro|323}} | |||
== Theory == | == Theory == | ||
323et [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], 589824/588245 and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]]. In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, 5632/5625, [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as the 2.3.5.11 subgroup temperament tempering out 14641/14580 | 323et [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], 589824/588245 and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]]. In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, 5632/5625, [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580. | ||
323 = 17 × 19, and shares the excellent approximations of [[25/24]] in [[17edo]] and of the [[28/27]] and the [[6/5]] in [[19edo]]. | 323 = 17 × 19, and shares the excellent approximations of [[25/24]] in [[17edo]] and of the [[28/27]] and the [[6/5]] in [[19edo]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{ | {{Harmonics in equal|323}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
Revision as of 14:33, 21 May 2022
| ← 322edo | 323edo | 324edo → |
Theory
323et tempers out the vulture comma, [24 -21 4⟩ and the luna comma, [38 -2 -15⟩, in the 5-limit; 4375/4374, 589824/588245 and 703125/702464 in the 7-limit, supporting 7-limit vulture, lunatic, enneadecal, and gamera. In the 11-limit, the 323e val and the patent val are comparable in errors. 1375/1372, 5632/5625, 14641/14580, and 19712/19683 are tempered out in the patent val; 540/539, 6250/6237, 12005/11979, and 16384/16335 are tempered out in the 323e val. It provides the optimal patent val for the rank-5 temperament tempering out 1573/1568, the lambeth comma, as well as deuteromere, the 2.3.5.11 subgroup temperament tempering out 14641/14580.
323 = 17 × 19, and shares the excellent approximations of 25/24 in 17edo and of the 28/27 and the 6/5 in 19edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.21 | +0.06 | +0.83 | -1.47 | -0.90 | -0.93 | -0.30 | -0.41 | -0.48 | -0.76 |
| Relative (%) | +0.0 | +5.7 | +1.7 | +22.4 | -39.6 | -24.2 | -25.0 | -8.1 | -11.1 | -12.8 | -20.5 | |
| Steps (reduced) |
323 (0) |
512 (189) |
750 (104) |
907 (261) |
1117 (148) |
1195 (226) |
1320 (28) |
1372 (80) |
1461 (169) |
1569 (277) |
1600 (308) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [512 -323⟩ | [⟨323 512]] | -0.0669 | 0.0669 | 1.80 |
| 2.3.5 | [24 -21 4⟩, [38 -2 -15⟩ | [⟨323 512 750]] | -0.0538 | 0.0577 | 1.55 |
| 2.3.5.7 | 4375/4374, 589824/588245, 703125/702464 | [⟨323 512 750 907]] | -0.1146 | 0.1165 | 3.14 |
| 2.3.5.7.11 | 540/539, 4375/4374, 12005/11979, 16384/16335 | [⟨323 512 750 907 1118]] (323e) | -0.2213 | 0.2375 | 6.39 |
| 2.3.5.7.11 | 1375/1372, 4375/4374, 5632/5625, 14641/14580 | [⟨323 512 750 907 1117]] (323) | -0.0066 | 0.2399 | 6.46 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 26\323 | 96.59 | 200/189 | Hemiluna (323) |
| 1 | 30\323 | 111.46 | 16/15 | Stockhausenic (323) |
| 1 | 31\323 | 115.17 | 77/72 | Semigamera (323) |
| 1 | 52\323 | 193.19 | 352/315 | Luna / lunatic (323e) |
| 1 | 62\323 | 230.34 | 8/7 | Gamera |
| 1 | 128\323 | 475.54 | 320/243 | Vulture |
| 17 | 134\323 (9\323) |
248.92 (33.44) |
[-23 5 9 -2⟩ (100352/98415) |
Chlorine |
| 19 | 134\323 (2\323) |
497.83 (7.43) |
4/3 (225/224) |
Enneadecal |