402edo: Difference between revisions
-rank-3 temps that are obvious from the commas ("parahemwuer" is just an alias for hemimean); note abigail and bischismic in the table of rank-2 temps |
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
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| 0.2182 | | 0.2182 | ||
| 7.31 | | 7.31 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
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|- | |- | ||
| 2 | | 2 | ||
| 167\402<br>(34\402) | | 167\402<br />(34\402) | ||
| 498.51<br>(101.49) | | 498.51<br />(101.49) | ||
| 4/3<br>(200/189) | | 4/3<br />(200/189) | ||
| [[Bischismic]] (402c, 7-limit) | | [[Bischismic]] (402c, 7-limit) | ||
{{rank-2 end}} | |||
{{orf}} | |||
Revision as of 02:14, 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
Template:Comma basis begin |- | 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 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 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)
Template:Rank-2 end
Template:Orf