393edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|393}} == Theory == 393et is only consistent to the 5-limit, with three mappings possible for the 7-limit: * {{val|393 623 913 1103}} (patent val),..." |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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== 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|623 -393}} | ! rowspan="2" | [[Comma list]] | ||
|{{mapping|393 623}} | ! rowspan="2" | [[Mapping]] | ||
| | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo|623 -393}} | |||
| {{mapping|393 623}} | |||
| −0.1057 | |||
| 0.1057 | | 0.1057 | ||
| 3.43 | | 3.43 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|393216/390625, {{monzo|-46 51 -15}} | | 393216/390625, {{monzo|-46 51 -15}} | ||
| | | −0.2819 | ||
| 0.2636 | | 0.2636 | ||
| 8.63 | | 8.63 | ||
| Line 47: | Line 48: | ||
=== 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 | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|127\393 | | 127\393 | ||
|387.79 | | 387.79 | ||
|5/4 | | 5/4 | ||
|[[Würschmidt]] | | [[Würschmidt]] | ||
|} | |} | ||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
Latest revision as of 13:31, 13 March 2026
| ← 392edo | 393edo | 394edo → |
393 equal divisions of the octave (abbreviated 393edo or 393ed2), also called 393-tone equal temperament (393tet) or 393 equal temperament (393et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 393 equal parts of about 3.05 ¢ each. Each step represents a frequency ratio of 21/393, or the 393rd root of 2.
Theory
393et is only consistent to the 5-limit, with three mappings possible for the 7-limit:
- ⟨393 623 913 1103] (patent val),
- ⟨393 623 912 1103] (393c),
- ⟨393 623 913 1104] (393d).
Using the patent val, it tempers out 393216/390625 and [-46 51 -15⟩ in the 5-limit; 10976/10935, 393216/390625 and 5250987/5242880 in the 7-limit.
Using the 393c val, it tempers out 2109375/2097152 and [32 -48 19⟩ in the 5-limit; 2401/2400, 1071875/1062882 and 2109375/2097152 in the 7-limit.
Using the 393d val, it tempers out 393216/390625 and [-46 51 -15⟩ in the 5-limit; 250047/250000, 2460375/2458624 and 2097152/2083725 in the 7-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.34 | +1.47 | -0.89 | +0.67 | +1.35 | -0.83 | -1.25 | -1.14 | -1.33 | -0.55 | +0.73 |
| Relative (%) | +11.0 | +48.2 | -29.0 | +21.9 | +44.3 | -27.3 | -40.8 | -37.3 | -43.6 | -18.1 | +24.0 | |
| Steps (reduced) |
623 (230) |
913 (127) |
1103 (317) |
1246 (67) |
1360 (181) |
1454 (275) |
1535 (356) |
1606 (34) |
1669 (97) |
1726 (154) |
1778 (206) | |
Subsets and supersets
393 factors into 3 × 131, with 3edo and 131edo as its subset edos. 786edo, which doubles it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [623 -393⟩ | [⟨393 623]] | −0.1057 | 0.1057 | 3.43 |
| 2.3.5 | 393216/390625, [-46 51 -15⟩ | −0.2819 | 0.2636 | 8.63 | |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 127\393 | 387.79 | 5/4 | Würschmidt |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct