385edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== 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. | |||
=== Prime harmonics === | |||
===Prime harmonics=== | |||
{{Harmonics in equal|385}} | {{Harmonics in equal|385}} | ||
===Subsets and supersets=== | === Subsets and supersets === | ||
385 factors into | Since 385 factors into {{factorization|385}}, 385edo has subset edos {{EDOs| 5, 7, 11, 35, 55, and 77 }}. | ||
==Regular temperament properties== | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|-122 77}} | ! 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 | |||
| {{monzo| -122 77 }} | |||
| {{mapping| 385 610 }} | |||
| +0.2070 | | +0.2070 | ||
| 0.2071 | | 0.2071 | ||
| 6.64 | | 6.64 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|-28 25 -5}}, {{monzo|38 -2 -15}} | | {{monzo| -28 25 -5 }}, {{monzo| 38 -2 -15 }} | ||
|{{ | | {{mapping| 385 610 894 }} | ||
| +0.1122 | | +0.1122 | ||
| 0.2158 | | 0.2158 | ||
| 6.92 | | 6.92 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|19683/19600, 703125/702464 | | 19683/19600, 589824/588245, 703125/702464 | ||
|{{ | | {{mapping| 385 610 894 1081 }} | ||
| +0.0374 | | +0.0374 | ||
| 0.2274 | | 0.2274 | ||
| 7.30 | | 7.30 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|540/539, 8019/8000, | | 540/539, 8019/8000, 151263/151250, 172032/171875 | ||
|{{ | | {{mapping| 385 610 894 1081 1332 }} | ||
| +0.0085 | | +0.0085 | ||
| 0.2114 | | 0.2114 | ||
| 6.78 | | 6.78 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|540/539, | | 540/539, 1575/1573, 2200/2197, 4096/4095, 8019/8000 | ||
|{{ | | {{mapping| 385 610 894 1081 1332 1425 }} | ||
| | | −0.0394 | ||
| 0.2207 | | 0.2207 | ||
| 7.08 | | 7.08 | ||
|- | |- | ||
|2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
|540/539, 936/935, 1377/1375, | | 540/539, 936/935, 1377/1375, 1575/1573, 2200/2197, 4096/4095 | ||
|{{ | | {{mapping| 385 610 894 1081 1332 1425 1574 }} | ||
| | | −0.0693 | ||
| 0.2171 | | 0.2171 | ||
| 6.97 | | 6.97 | ||
Line 71: | Line 68: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|62\385 | | 62\385 | ||
|193. | | 193.25 | ||
| | | 262144/234375 | ||
|[[Luna]] | | [[Luna]] | ||
|- | |- | ||
|1 | | 1 | ||
|162/385 | | 162/385 | ||
|504. | | 504.94 | ||
|4/3 | | 4/3 | ||
|[[Countermeantone]] | | [[Countermeantone]] | ||
|- | |- | ||
|5 | | 5 | ||
|160\385<br>(6\385) | | 80\385<br />(3\385) | ||
|498. | | 249.35<br />(9.35) | ||
|4/3<br>(81/80) | | 81/70<br />(176/175) | ||
|[[Pental]] | | [[Hemipental]] | ||
|- | |||
| 5 | |||
| 160\385<br />(6\385) | |||
| 498.70<br />(18.70) | |||
| 4/3<br />(81/80) | |||
| [[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 distinct | |||
[[Category:Hemipental]] |
Latest revision as of 06:30, 21 February 2025
← 384edo | 385edo | 386edo → |
385 equal divisions of the octave (abbreviated 385edo or 385ed2), also called 385-tone equal temperament (385tet) or 385 equal temperament (385et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 385 equal parts of about 3.12 ¢ each. Each step represents a frequency ratio of 21/385, or the 385th root of 2.
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