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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
382edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]]s [[3/1|3]] and [[7/1|7]] are about halfway between its steps. It is also bad at approximating [[11/1|11]], [[13/1|13]], [[15/1|15]], and [[17/1|17]], though its [[5/1|5]], [[9/1|9]], [[19/1|19]], [[21/1|21]], and [[23/1|23]] are good, making it suitable for a 2.9.5.21.19.23 [[subgroup]] interpretation. | |||
Using the [[patent val]] nonetheless, the equal temperament [[tempering out|tempers out]] [[65625/65536]] in the 7-limit; [[540/539]], [[4000/3993]], and [[9801/9800]] in the 11-limit. It [[support]]s [[bastille]]. | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 9: | Line 11: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
382 factors into 2 × 191 with [[2edo]] and [[191edo]] as its subset edos. [[764edo]], which doubles it, gives a good correction to the | 382 factors into 2 × 191 with [[2edo]] and [[191edo]] as its subset edos. [[764edo]], which doubles it, gives a good correction to the harmonics 3, 7, 11, 13, and 17. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | |- | ||
! rowspan="2" |[[Comma list | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |Optimal<br>8ve | ! rowspan="2" | [[Mapping]] | ||
! colspan="2" |Tuning | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.9 | | 2.9 | ||
|{{monzo|1211 -382}} | | {{monzo| 1211 -382 }} | ||
|{{mapping|382 1211}} | | {{mapping| 382 1211 }} | ||
| | | −0.0439 | ||
| 0.0439 | | 0.0439 | ||
| 1.40 | | 1.40 | ||
|- | |- | ||
|2.9.5 | | 2.9.5 | ||
|{{monzo|38 -1 -15}}, {{monzo|25 -24 22}} | | {{monzo| 38 -1 -15 }}, {{monzo| 25 -24 22 }} | ||
|{{mapping|382 1211 887}} | | {{mapping| 382 1211 887 }} | ||
| | | −0.0399 | ||
| 0.0363 | | 0.0363 | ||
| 1.16 | | 1.16 | ||
|- | |- | ||
|2.9.5. | | 2.9.5.21 | ||
| | | 4375/4374, 52734375/52706752, {{monzo| 31 0 -2 -6 }} | ||
|{{mapping|382 1211 887 | | {{mapping| 382 1211 887 1678 }} | ||
| | | −0.0552 | ||
| 0. | | 0.0412 | ||
| | | 1.31 | ||
|} | |} | ||
Latest revision as of 12:22, 21 February 2025
| ← 381edo | 382edo | 383edo → |
382 equal divisions of the octave (abbreviated 382edo or 382ed2), also called 382-tone equal temperament (382tet) or 382 equal temperament (382et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 382 equal parts of about 3.14 ¢ each. Each step represents a frequency ratio of 21/382, or the 382nd root of 2.
Theory
382edo is consistent to the 7-odd-limit, but harmonics 3 and 7 are about halfway between its steps. It is also bad at approximating 11, 13, 15, and 17, though its 5, 9, 19, 21, and 23 are good, making it suitable for a 2.9.5.21.19.23 subgroup interpretation.
Using the patent val nonetheless, the equal temperament tempers out 65625/65536 in the 7-limit; 540/539, 4000/3993, and 9801/9800 in the 11-limit. It supports bastille.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.43 | +0.07 | -1.29 | +0.28 | +1.56 | +1.36 | -1.36 | -1.29 | +0.92 | +0.42 | -0.00 |
| Relative (%) | -45.6 | +2.3 | -41.0 | +8.9 | +49.7 | +43.2 | -43.2 | -41.1 | +29.2 | +13.5 | -0.1 | |
| Steps (reduced) |
605 (223) |
887 (123) |
1072 (308) |
1211 (65) |
1322 (176) |
1414 (268) |
1492 (346) |
1561 (33) |
1623 (95) |
1678 (150) |
1728 (200) | |
Subsets and supersets
382 factors into 2 × 191 with 2edo and 191edo as its subset edos. 764edo, which doubles it, gives a good correction to the harmonics 3, 7, 11, 13, and 17.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [1211 -382⟩ | [⟨382 1211]] | −0.0439 | 0.0439 | 1.40 |
| 2.9.5 | [38 -1 -15⟩, [25 -24 22⟩ | [⟨382 1211 887]] | −0.0399 | 0.0363 | 1.16 |
| 2.9.5.21 | 4375/4374, 52734375/52706752, [31 0 -2 -6⟩ | [⟨382 1211 887 1678]] | −0.0552 | 0.0412 | 1.31 |