433edo: Difference between revisions
Jump to navigation
Jump to search
→Theory: put to the right subsection; -6701edo (addressed in the relevant edo page) |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| Line 15: | Line 15: | ||
== 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 | ||
|- | |- | ||
| Line 69: | Line 70: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br>Ratio* | ! Associated<br />Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 82: | Line 84: | ||
| [[Orson]] | | [[Orson]] | ||
|} | |} | ||
{{orf}} | |||
== Music == | == Music == | ||
Revision as of 01:06, 16 November 2024
| ← 432edo | 433edo | 434edo → |
Theory
443edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. To start with, the patent val ⟨433 686 1005 1216] as well as the 433d val ⟨433 686 1005 1215] are worth considering.
Using the patent val, the equal temperament tempers out 19683/19600 and 4096000/4084101 in the 7-limit; 3025/3024, 4000/3993, 6250/6237, 161280/161051, and 180224/180075 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.80 | -1.09 | +1.15 | +1.17 | +0.18 | -0.80 | +0.88 | +0.36 | -0.98 | +0.35 | +0.82 |
| Relative (%) | -28.9 | -39.5 | +41.5 | +42.2 | +6.6 | -29.0 | +31.6 | +12.9 | -35.3 | +12.7 | +29.8 | |
| Steps (reduced) |
686 (253) |
1005 (139) |
1216 (350) |
1373 (74) |
1498 (199) |
1602 (303) |
1692 (393) |
1770 (38) |
1839 (107) |
1902 (170) |
1959 (227) | |
Subsets and supersets
433edo is the 84th prime edo. It might be interesting due to being the smallest subset edo of the nanotemperament 2901533edo, an extremely high-precision/complexity microtemperament.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-686 433⟩ | [⟨433 686]] | 0.2525 | 0.2525 | 9.11 |
| 2.3.5 | 2109375/2097152, [-29 52 -23⟩ | [⟨433 686 1005]] | 0.3254 | 0.2306 | 8.32 |
| 2.3.5.7 | 19683/19600, 4096000/4084101, 2109375/2097152 | [⟨433 686 1005 1216]] | 0.1414 | 0.3759 | 13.56 |
| 2.3.5.7.11 | 3025/3024, 6250/6237, 30375/30184, 180224/180075 | [⟨433 686 1005 1216 1498]] | 0.1026 | 0.3451 | 12.45 |
| 2.3.5.7.11.13 | 2080/2079, 625/624, 3025/3024, 18954/18865, 41472/41405 | [⟨433 686 1005 1216 1498 1602]] | 0.1217 | 0.3179 | 11.47 |
| 2.3.5.7.11.13.17 | 2080/2079, 375/374, 715/714, 936/935, 1377/1372, 76032/75803 | [⟨433 686 1005 1216 1498 1602 1770]] | 0.0919 | 0.3033 | 10.94 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 98\433 | 271.594 | 75/64 | Orson |
Music
- sleeping as we don't know (2023)