467edo: Difference between revisions
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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 38: | Line 47: | ||
| 0.1617 | | 0.1617 | ||
| 6.05 | | 6.05 | ||
|} | |||
=== 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 54: | Line 70: | ||
| 6/5 | | 6/5 | ||
| [[Counterhanson]] | | [[Counterhanson]] | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Music == | == Music == | ||
Revision as of 12:51, 16 November 2024
| ← 466edo | 467edo | 468edo → |
Theory
467edo is consistent to the 9-odd-limit with harmonics 3, 5, and 7 all tuned flat. Using the patent val, the equal temperament tempers out 4375/4374, 2100875/2097152, 5250987/5242880, and [-16 4 9 -4⟩ in the 7-limit. It supports mitonic and counterkleismic, supplying the optimal patent val for the latter.
In the 11-limit, the 467e val scores much better than the patent val. The 467e val tempers out 1375/1372, 24057/24010, 35937/35840, and 41503/41472, and in the 13-limit, 625/624, 729/728, 1716/1715, and 2200/2197. The patent val tempers out 540/539, 6250/6237, 12005/11979, and 14700/14641, and in the 13-limit, 625/624, 729/728, and 2080/2079.
In the 17-limit, it supplies the optimal patent val for the rank-6 temperament tempering out 375/374.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.46 | -0.87 | -0.09 | -0.91 | +1.14 | -0.27 | +1.24 | +0.40 | +0.56 | -0.55 | +1.28 |
| Relative (%) | -17.7 | -34.0 | -3.5 | -35.5 | +44.5 | -10.5 | +48.2 | +15.5 | +21.8 | -21.2 | +49.7 | |
| Steps (reduced) |
740 (273) |
1084 (150) |
1311 (377) |
1480 (79) |
1616 (215) |
1728 (327) |
1825 (424) |
1909 (41) |
1984 (116) |
2051 (183) |
2113 (245) | |
Subsets and supersets
467edo is the 91st prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-740 467⟩ | [⟨467 740]] | 0.1439 | 0.1439 | 5.38 |
| 2.3.5 | [-36 11 8⟩, [-16 35 -17⟩ | [⟨467 740 1084]] | 0.2215 | 0.1608 | 6.02 |
| 2.3.5.7 | 4375/4374, 2100875/2097152, [-16 4 9 -4⟩ | [⟨467 740 1084 1311]] | 0.1741 | 0.1617 | 6.05 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 71\467 | 182.441 | 10/9 | Mitonic |
| 1 | 123\467 | 316.060 | 6/5 | Counterhanson |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct