465edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|465}} == Theory == 465et is consistent to the 5-odd-limit. It can be considered for the 2.3.5.11.13.17 subgroup, tempering out 936/935,..." |
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== Theory == | == Theory == | ||
465edo is only [[consistent]] to the [[5-odd-limit]], and the errors of [[harmonic]]s beyond [[3/1|3]] tend to be quite large. It can be considered for the 2.3.5.11.13.17 [[subgroup]], [[tempering out]] [[936/935]], 1377/1375, 71874/71825, 131648/131625 and 225000/224939. It [[support]]s [[counterschismic]] in the 5-limit, and [[birds]] and [[belobog]] in the 7-limit using the [[patent val]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
465 factors into 3 × 5 × 31, | Since 465 factors into 3 × 5 × 31, 465edo has subset edos {{EDOs| 3, 5, 15, 31, 93, and 155 }}. [[930edo]], which doubles it, gives a good correction to the harmonic 7. | ||
== 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|-737 465}} | | {{monzo| -737 465 }} | ||
|{{mapping|465 737}} | | {{mapping| 465 737 }} | ||
| +0.0062 | | +0.0062 | ||
| 0.0062 | | 0.0062 | ||
| 0.24 | | 0.24 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|25 15 -21}}, {{monzo|-22 30 -11}} | | {{monzo| 25 15 -21 }}, {{monzo| -22 30 -11 }} | ||
|{{mapping|465 737 1080}} | | {{mapping| 465 737 1080 }} | ||
| -0.1083 | | -0.1083 | ||
| 0.1619 | | 0.1619 | ||
| Line 46: | Line 46: | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|193\465 | | 193\465 | ||
|498.06 | | 498.06 | ||
|4/3 | | 4/3 | ||
|[[Counterschismic]] | | [[Counterschismic]] | ||
|- | |- | ||
|5 | | 5 | ||
|322\465<br>(43\465) | | 322\465<br>(43\465) | ||
|830.97<br>(110.97) | | 830.97<br>(110.97) | ||
|80/49<br>(15/14) | | 80/49<br>(15/14) | ||
|[[Qintosec]] | | [[Qintosec]] (465) | ||
|} | |} | ||
<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 13:36, 18 February 2024
| ← 464edo | 465edo | 466edo → |
Theory
465edo is only consistent to the 5-odd-limit, and the errors of harmonics beyond 3 tend to be quite large. It can be considered for the 2.3.5.11.13.17 subgroup, tempering out 936/935, 1377/1375, 71874/71825, 131648/131625 and 225000/224939. It supports counterschismic in the 5-limit, and birds and belobog in the 7-limit using the patent val.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.02 | +0.78 | -1.08 | +0.94 | +0.76 | +0.85 | -0.74 | -1.18 | +0.10 | +0.77 |
| Relative (%) | +0.0 | -0.8 | +30.3 | -42.0 | +36.4 | +29.6 | +33.0 | -28.6 | -45.6 | +3.9 | +29.9 | |
| Steps (reduced) |
465 (0) |
737 (272) |
1080 (150) |
1305 (375) |
1609 (214) |
1721 (326) |
1901 (41) |
1975 (115) |
2103 (243) |
2259 (399) |
2304 (444) | |
Subsets and supersets
Since 465 factors into 3 × 5 × 31, 465edo has subset edos 3, 5, 15, 31, 93, and 155. 930edo, 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 | [-737 465⟩ | [⟨465 737]] | +0.0062 | 0.0062 | 0.24 |
| 2.3.5 | [25 15 -21⟩, [-22 30 -11⟩ | [⟨465 737 1080]] | -0.1083 | 0.1619 | 6.27 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 193\465 | 498.06 | 4/3 | Counterschismic |
| 5 | 322\465 (43\465) |
830.97 (110.97) |
80/49 (15/14) |
Qintosec (465) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct