461edo: 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" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | |- | ||
! rowspan="2" |[[Comma list | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |Optimal<br>8ve | ! rowspan="2" | [[Mapping]] | ||
! colspan="2" |Tuning | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
|- | ! colspan="2" | Tuning error | ||
![[TE error|Absolute]] (¢) | |- | ||
![[TE simple badness|Relative]] (%) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
|{{monzo|731 -461}} | | {{monzo|731 -461}} | ||
|{{val|461 731}} | | {{val|461 731}} | ||
| | | −0.2729 | ||
| 0.2728 | | 0.2728 | ||
| 10.48 | | 10.48 | ||
|} | |} | ||
Latest revision as of 17:46, 20 February 2025
| ← 460edo | 461edo | 462edo → |
461 equal divisions of the octave (abbreviated 461edo or 461ed2), also called 461-tone equal temperament (461tet) or 461 equal temperament (461et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 461 equal parts of about 2.6 ¢ each. Each step represents a frequency ratio of 21/461, or the 461st root of 2.
Theory
461edo is only consistent to the 3-odd-limit. It can be considered for the 2.3.11.13.29.31.37.41.43 subgroup, tempering out 1189/1188, 1333/1332, 3224/3219, 4433/4428, 39904/39897, 19778/19773, 17303/17298 and 2564692/2558061. It supports quartemka in the 5-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.86 | -1.06 | -0.50 | -0.87 | +0.53 | +0.25 | -0.20 | -0.83 | -0.77 | +0.37 | -0.94 |
| Relative (%) | +33.2 | -40.9 | -19.1 | -33.5 | +20.2 | +9.7 | -7.7 | -32.0 | -29.5 | +14.2 | -36.2 | |
| Steps (reduced) |
731 (270) |
1070 (148) |
1294 (372) |
1461 (78) |
1595 (212) |
1706 (323) |
1801 (418) |
1884 (40) |
1958 (114) |
2025 (181) |
2085 (241) | |
Subsets and supersets
461edo is the 89th prime edo. 1383edo, which triples it, gives a good correction to the harmonic 5.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [731 -461⟩ | ⟨461 731] | −0.2729 | 0.2728 | 10.48 |