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Created page with "{{Infobox ET}} {{EDO intro|361}} == Theory == 361et is consistent to the 9-odd-limit, tempering out 43046721/43025920, 2460375/2458624, 703125/702464, 48828125/48..." |
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== Theory == | == Theory == | ||
361et is consistent to the [[9-odd-limit]], tempering out [[ | 361et is [[consistent]] to the [[9-odd-limit]] with flat tunings of [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]]. The equal temperament [[tempering out|tempers out]] [[4375/4374]], [[703125/702464]], 2460375/2458624, [[43046721/43025920]], and 48828125/48771072 in the 7-limit. It [[support]]s the 5-limit [[submajor]] temperament. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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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|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|-572 361}} | | {{monzo| -572 361 }} | ||
|{{mapping|361 572}} | | {{mapping| 361 572 }} | ||
| 0.1798 | | 0.1798 | ||
| 0.1798 | | 0.1798 | ||
| 5.41 | | 5.41 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{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 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|4375/4374, 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 | ||
| Line 53: | Line 53: | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|166\361 | | 166\361 | ||
|551.80 | | 551.80 | ||
|48/35 | | 48/35 | ||
|[[Emka]] | | [[Emka]] | ||
|- | |- | ||
|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 lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | <nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | ||
Revision as of 19:57, 11 January 2024
| ← 360edo | 361edo | 362edo → |
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 it is distinct