402edo: 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 36: | Line 45: | ||
| 0.2182 | | 0.2182 | ||
| 7.31 | | 7.31 | ||
|} | |||
=== 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 64: | Line 80: | ||
| 4/3<br />(200/189) | | 4/3<br />(200/189) | ||
| [[Bischismic]] (402c, 7-limit) | | [[Bischismic]] (402c, 7-limit) | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
Revision as of 13:09, 16 November 2024
| ← 401edo | 402edo | 403edo → |
Theory
402edo is only consistent to the 5-odd-limit. There are three possible mappings in the 7-limit:
- ⟨402 637 933 1129] (patent val)
- ⟨402 637 933 1128] (402d)
- ⟨402 637 934 1129] (402c)
Using the patent val, it tempers out the semicomma in the 5-limit; 4375/4374, 7381125/7340032 and 3200000/3176523 in the 7-limit. It supports abigail.
Using the 402d val, it tempers out 250047/250000, 1500625/1492992 and 2460375/2458624 in the 7-limit.
Using the 402c val, it tempers out the schisma in the 5-limit; and 3136/3125, 321489/320000 and 13060694016/12867859375 in the 7-limit. It supports bischismic.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.46 | -1.24 | +1.32 | -0.92 | +0.92 | +1.26 | +1.28 | -0.48 | +0.99 | +0.86 | -1.41 |
| Relative (%) | -15.5 | -41.5 | +44.3 | -31.0 | +30.8 | +42.3 | +43.0 | -16.0 | +33.3 | +28.8 | -47.2 | |
| Steps (reduced) |
637 (235) |
933 (129) |
1129 (325) |
1274 (68) |
1391 (185) |
1488 (282) |
1571 (365) |
1643 (35) |
1708 (100) |
1766 (158) |
1818 (210) | |
Subsets and supersets
Since 402 factors into 2 × 3 × 67, 402edo has subset edos 2, 3, 6, 67, 134, and 201. 804edo, which doubles it, gives a good correction to the harmonics 5 and 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-637 402⟩ | [⟨402 637]] | 0.1459 | 0.1459 | 4.89 |
| 2.3.5 | 2109375/2097152, [25 -48 22⟩ | [⟨402 637 933]] | 0.2752 | 0.2182 | 7.31 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 91\402 | 271.64 | 75/64 | Orson (402) |
| 1 | 115\402 | 343.28 | 8000/6561 | Raider (402) |
| 2 | 70\402 | 208.96 | 44/39 | Abigail (402) |
| 2 | 167\402 (34\402) |
498.51 (101.49) |
4/3 (200/189) |
Bischismic (402c, 7-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct