433edo: Difference between revisions
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443edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. To start with, the [[patent val]] {{val| 433 686 1005 '''1216''' }} as well as the 433d [[val]] {{val| 433 686 1005 '''1215''' }} are worth considering. | 443edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. To start with, the [[patent val]] {{val| 433 686 1005 '''1216''' }} as well as the 433d [[val]] {{val| 433 686 1005 '''1215''' }} are worth considering. | ||
Using the patent val, it [[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. | Using the patent val, it [[tempering out|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 === | === Odd harmonics === | ||
Revision as of 16:16, 17 January 2025
| ← 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, it 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 |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Music
- sleeping as we don't know (2023)