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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" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
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| {{monzo| -572 361 }} | | {{monzo| -572 361 }} | ||
| {{mapping| 361 572 }} | | {{mapping| 361 572 }} | ||
| 0.1798 | | +0.1798 | ||
| 0.1798 | | 0.1798 | ||
| 5.41 | | 5.41 | ||
| Line 32: | Line 33: | ||
| {{monzo| -36 11 8 }}, {{monzo| -14 -19 19 }} | | {{monzo| -36 11 8 }}, {{monzo| -14 -19 19 }} | ||
| {{mapping| 361 572 838 }} | | {{mapping| 361 572 838 }} | ||
| 0.2230 | | +0.2230 | ||
| 0.1590 | | 0.1590 | ||
| 4.78 | | 4.78 | ||
| Line 39: | Line 40: | ||
| 4375/4374, 823543/819200, 2460375/2458624 | | 4375/4374, 823543/819200, 2460375/2458624 | ||
| {{mapping| 361 572 838 1013 }} | | {{mapping| 361 572 838 1013 }} | ||
| 0.3020 | | +0.3020 | ||
| 0.1941 | | 0.1941 | ||
| 5.84 | | 5.84 | ||
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=== 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 | ||
|- | |- | ||
| Line 60: | Line 62: | ||
|- | |- | ||
| 19 | | 19 | ||
| 150\361<br>(2\361) | | 150\361<br />(2\361) | ||
| 498.61<br>(6.65) | | 498.61<br />(6.65) | ||
| 4/3<br>(225/224) | | 4/3<br />(225/224) | ||
| [[Enneadecal]] | | [[Enneadecal]] | ||
|} | |} | ||
<nowiki>* | <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 13:33, 13 March 2026
| ← 360edo | 361edo | 362edo → |
361 equal divisions of the octave (abbreviated 361edo or 361ed2), also called 361-tone equal temperament (361tet) or 361 equal temperament (361et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 361 equal parts of about 3.32 ¢ each. Each step represents a frequency ratio of 21/361, or the 361st root of 2.
Theory
361et is consistent to the 9-odd-limit with flat tunings of harmonics 3, 5, and 7. The equal temperament tempers out 4375/4374, 703125/702464, 2460375/2458624, 43046721/43025920, and 48828125/48771072 in the 7-limit. It supports the 5-limit submajor temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.57 | -0.72 | -1.51 | -1.14 | +0.48 | +0.47 | -1.29 | +1.42 | +1.66 | +1.24 | -0.02 |
| Relative (%) | -17.1 | -21.6 | -45.5 | -34.3 | +14.5 | +14.1 | -38.8 | +42.6 | +49.8 | +37.3 | -0.6 | |
| Steps (reduced) |
572 (211) |
838 (116) |
1013 (291) |
1144 (61) |
1249 (166) |
1336 (253) |
1410 (327) |
1476 (32) |
1534 (90) |
1586 (142) |
1633 (189) | |
Subsets and supersets
361 factors into 192, with 19edo as its only edo subset.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-572 361⟩ | [⟨361 572]] | +0.1798 | 0.1798 | 5.41 |
| 2.3.5 | [-36 11 8⟩, [-14 -19 19⟩ | [⟨361 572 838]] | +0.2230 | 0.1590 | 4.78 |
| 2.3.5.7 | 4375/4374, 823543/819200, 2460375/2458624 | [⟨361 572 838 1013]] | +0.3020 | 0.1941 | 5.84 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 166\361 | 551.80 | 48/35 | Emka |
| 19 | 150\361 (2\361) |
498.61 (6.65) |
4/3 (225/224) |
Enneadecal |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct