381edo: Difference between revisions
Jump to navigation
Jump to search
Wikispaces>genewardsmith **Imported revision 241120663 - Original comment: ** |
m Text replacement - "Octave-reduced form, reduced to the first half-octave" to "Octave-reduced form, reduced to the first half-octave" |
||
| (11 intermediate revisions by 7 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | |||
381edo is [[consistent]] to the [[13-odd-limit]] and almost the [[15-odd-limit]]; the only inconsistently mapped intervals in the 15-odd-limit are [[15/11]] and its [[octave complement]]. It has a sharp tendency, with odd [[harmonic]]s 3 through 15 all tuned sharp except for 11, which is very slightly flat. | |||
As an equal temperament, it [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }}, in the 5-limit and 6144/6125 ([[porwell comma]]) and 250047/250000 ([[landscape comma]]) in the 7-limit. It provides the [[optimal patent val]] for the porwell planar temperament tempering out 6144/6125, and [[nessafof]], the {{nowrap| 99 & 282 }} temperament tempering out it and the landscape comma 250047/250000. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|381|intervals=prime}} | |||
=== Subsets and supersets === | |||
Since 381 factors into primes as {{nowrap| 3 × 127 }}, 381edo contains [[3edo]] and [[127edo]] as subsets. | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! 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| 604 -381 }} | |||
| {{Mapping| 381 604 }} | |||
| −0.1285 | |||
| 0.1284 | |||
| 4.08 | |||
|- | |||
| 2.3.5 | |||
| {{Monzo| 24 -21 4 }}, {{monzo| 25 15 -21 }} | |||
| {{Mapping| 381 604 885 }} | |||
| −0.2418 | |||
| 0.1915 | |||
| 6.08 | |||
|- | |||
| 2.3.5.7 | |||
| 6144/6125, 250047/250000, 43046721/43025920 | |||
| {{Mapping| 381 604 885 1070 }} | |||
| −0.2929 | |||
| 0.1880 | |||
| 5.97 | |||
|- | |||
| 2.3.5.7.11 | |||
| 3025/3024, 6144/6125, 19712/19683, 160083/160000 | |||
| {{Mapping| 381 604 885 1070 1318 }} | |||
| −0.2264 | |||
| 0.2144 | |||
| 6.81 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 676/675, 1001/1000, 3025/3024, 6144/6125, 10985/10976 | |||
| {{Mapping| 381 604 885 1070 1318 1410 }} | |||
| −0.2075 | |||
| 0.2002 | |||
| 6.36 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br>per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br>ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 151\381 | |||
| 475.59 | |||
| 320/243 | |||
| [[Vulture]] (5-limit) | |||
|- | |||
| 3 | |||
| 24\381 | |||
| 475.59<br>(75.59) | |||
| 320/243<br>(117/112) | |||
| [[Terture]] | |||
|- | |||
| 3 | |||
| 50\381 | |||
| 157.48 | |||
| 35/32 | |||
| [[Nessafof]] | |||
|} | |||
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct | |||
[[Category:Porwell]] | |||
[[Category:Nessafof]] | |||
Latest revision as of 13:34, 13 March 2026
| ← 380edo | 381edo | 382edo → |
381 equal divisions of the octave (abbreviated 381edo or 381ed2), also called 381-tone equal temperament (381tet) or 381 equal temperament (381et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 381 equal parts of about 3.15 ¢ each. Each step represents a frequency ratio of 21/381, or the 381st root of 2.
Theory
381edo is consistent to the 13-odd-limit and almost the 15-odd-limit; the only inconsistently mapped intervals in the 15-odd-limit are 15/11 and its octave complement. It has a sharp tendency, with odd harmonics 3 through 15 all tuned sharp except for 11, which is very slightly flat.
As an equal temperament, it tempers out the vulture comma, [24 -21 4⟩, in the 5-limit and 6144/6125 (porwell comma) and 250047/250000 (landscape comma) in the 7-limit. It provides the optimal patent val for the porwell planar temperament tempering out 6144/6125, and nessafof, the 99 & 282 temperament tempering out it and the landscape comma 250047/250000.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.41 | +1.09 | +1.25 | -0.14 | +0.42 | -1.02 | -1.45 | -1.50 | +0.34 | +1.42 |
| Relative (%) | +0.0 | +12.9 | +34.5 | +39.8 | -4.3 | +13.2 | -32.3 | -46.0 | -47.7 | +10.9 | +45.1 | |
| Steps (reduced) |
381 (0) |
604 (223) |
885 (123) |
1070 (308) |
1318 (175) |
1410 (267) |
1557 (33) |
1618 (94) |
1723 (199) |
1851 (327) |
1888 (364) | |
Subsets and supersets
Since 381 factors into primes as 3 × 127, 381edo contains 3edo and 127edo as subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [604 -381⟩ | [⟨381 604]] | −0.1285 | 0.1284 | 4.08 |
| 2.3.5 | [24 -21 4⟩, [25 15 -21⟩ | [⟨381 604 885]] | −0.2418 | 0.1915 | 6.08 |
| 2.3.5.7 | 6144/6125, 250047/250000, 43046721/43025920 | [⟨381 604 885 1070]] | −0.2929 | 0.1880 | 5.97 |
| 2.3.5.7.11 | 3025/3024, 6144/6125, 19712/19683, 160083/160000 | [⟨381 604 885 1070 1318]] | −0.2264 | 0.2144 | 6.81 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 3025/3024, 6144/6125, 10985/10976 | [⟨381 604 885 1070 1318 1410]] | −0.2075 | 0.2002 | 6.36 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 151\381 | 475.59 | 320/243 | Vulture (5-limit) |
| 3 | 24\381 | 475.59 (75.59) |
320/243 (117/112) |
Terture |
| 3 | 50\381 | 157.48 | 35/32 | Nessafof |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct