465edo: Difference between revisions
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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" | ||
|- | |||
! 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 | | 2.3 | ||
| Line 24: | Line 33: | ||
| {{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 | ||
| 0.1619 | | 0.1619 | ||
| 6.27 | | 6.27 | ||
|} | |||
=== Rank-2 temperaments === | === 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 | | 1 | ||
| Line 43: | Line 59: | ||
| 80/49<br />(15/14) | | 80/49<br />(15/14) | ||
| [[Qintosec]] (465) | | [[Qintosec]] (465) | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
Latest revision as of 06:20, 21 February 2025
| ← 464edo | 465edo | 466edo → |
465 equal divisions of the octave (abbreviated 465edo or 465ed2), also called 465-tone equal temperament (465tet) or 465 equal temperament (465et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 465 equal parts of about 2.58 ¢ each. Each step represents a frequency ratio of 21/465, or the 465th root of 2.
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 distinct