385edo: Difference between revisions
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== Theory == | == Theory == | ||
385edo has a reasonable approximation to the 11-limit, and perhaps beyond. The equal temperament [[tempering out|tempers out]] [[19683/19600]], [[589824/588245]], and [[703125/702464]] in the 7-limit; [[540/539]], [[8019/8000]], 43923/43904, 151263/151250, 160083/160000, 166698/166375, and 172032/171875 in the 11-limit. It [[support]]s [[hemipental]] and provides the [[optimal patent val]] for the 7-limit version thereof. Using the [[patent val]], it tempers out [[1575/1573]], [[1716/1715]], [[2200/2197]], [[4096/4095]], [[6656/6655]] and [[10648/10647]] in the 13-limit; and [[936/935]], [[1275/1274]], 1377/1375, and [[2601/2600]] in the 17-limit. | 385edo has a reasonable approximation to the 11-limit, and perhaps beyond. The equal temperament [[tempering out|tempers out]] [[19683/19600]], [[589824/588245]], and [[703125/702464]] in the 7-limit; [[540/539]], [[8019/8000]], 43923/43904, 151263/151250, 160083/160000, 166698/166375, and 172032/171875 in the 11-limit. It [[support]]s [[hemipental]] and provides the [[optimal patent val]] for the 7-limit version thereof. Using the [[patent val]], it tempers out [[1575/1573]], [[1716/1715]], [[2200/2197]], [[4096/4095]], [[6656/6655]], and [[10648/10647]] in the 13-limit; and [[936/935]], [[1275/1274]], 1377/1375, and [[2601/2600]] in the 17-limit. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 54: | Line 54: | ||
| 540/539, 1575/1573, 2200/2197, 4096/4095, 8019/8000 | | 540/539, 1575/1573, 2200/2197, 4096/4095, 8019/8000 | ||
| {{mapping| 385 610 894 1081 1332 1425 }} | | {{mapping| 385 610 894 1081 1332 1425 }} | ||
| | | −0.0394 | ||
| 0.2207 | | 0.2207 | ||
| 7.08 | | 7.08 | ||
| Line 61: | Line 61: | ||
| 540/539, 936/935, 1377/1375, 1575/1573, 2200/2197, 4096/4095 | | 540/539, 936/935, 1377/1375, 1575/1573, 2200/2197, 4096/4095 | ||
| {{mapping| 385 610 894 1081 1332 1425 1574 }} | | {{mapping| 385 610 894 1081 1332 1425 1574 }} | ||
| | | −0.0693 | ||
| 0.2171 | | 0.2171 | ||
| 6.97 | | 6.97 | ||
| Line 100: | Line 100: | ||
| [[Pental (temperament)|Pental]] (5-limit) | | [[Pental (temperament)|Pental]] (5-limit) | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
[[Category:Hemipental]] | [[Category:Hemipental]] | ||
Revision as of 19:45, 15 January 2025
| ← 384edo | 385edo | 386edo → |
Theory
385edo has a reasonable approximation to the 11-limit, and perhaps beyond. The equal temperament tempers out 19683/19600, 589824/588245, and 703125/702464 in the 7-limit; 540/539, 8019/8000, 43923/43904, 151263/151250, 160083/160000, 166698/166375, and 172032/171875 in the 11-limit. It supports hemipental and provides the optimal patent val for the 7-limit version thereof. Using the patent val, it tempers out 1575/1573, 1716/1715, 2200/2197, 4096/4095, 6656/6655, and 10648/10647 in the 13-limit; and 936/935, 1275/1274, 1377/1375, and 2601/2600 in the 17-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.66 | +0.18 | +0.52 | +0.37 | +1.03 | +1.02 | -1.41 | +1.34 | -1.01 | -1.14 |
| Relative (%) | +0.0 | -21.1 | +5.8 | +16.8 | +11.9 | +33.1 | +32.7 | -45.2 | +42.9 | -32.3 | -36.6 | |
| Steps (reduced) |
385 (0) |
610 (225) |
894 (124) |
1081 (311) |
1332 (177) |
1425 (270) |
1574 (34) |
1635 (95) |
1742 (202) |
1870 (330) |
1907 (367) | |
Subsets and supersets
Since 385 factors into 5 × 7 × 11, 385edo has subset edos 5, 7, 11, 35, 55, and 77.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-122 77⟩ | [⟨385 610]] | +0.2070 | 0.2071 | 6.64 |
| 2.3.5 | [-28 25 -5⟩, [38 -2 -15⟩ | [⟨385 610 894]] | +0.1122 | 0.2158 | 6.92 |
| 2.3.5.7 | 19683/19600, 589824/588245, 703125/702464 | [⟨385 610 894 1081]] | +0.0374 | 0.2274 | 7.30 |
| 2.3.5.7.11 | 540/539, 8019/8000, 151263/151250, 172032/171875 | [⟨385 610 894 1081 1332]] | +0.0085 | 0.2114 | 6.78 |
| 2.3.5.7.11.13 | 540/539, 1575/1573, 2200/2197, 4096/4095, 8019/8000 | [⟨385 610 894 1081 1332 1425]] | −0.0394 | 0.2207 | 7.08 |
| 2.3.5.7.11.13.17 | 540/539, 936/935, 1377/1375, 1575/1573, 2200/2197, 4096/4095 | [⟨385 610 894 1081 1332 1425 1574]] | −0.0693 | 0.2171 | 6.97 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 62\385 | 193.25 | 262144/234375 | Luna |
| 1 | 162/385 | 504.94 | 4/3 | Countermeantone |
| 5 | 80\385 (3\385) |
249.35 (9.35) |
81/70 (176/175) |
Hemipental |
| 5 | 160\385 (6\385) |
498.70 (18.70) |
4/3 (81/80) |
Pental (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct