367edo
| ← 366edo | 367edo | 368edo → |
367 equal divisions of the octave (abbreviated 367edo or 367ed2), also called 367-tone equal temperament (367tet) or 367 equal temperament (367et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 367 equal parts of about 3.27 ¢ each. Each step represents a frequency ratio of 21/367, or the 367th root of 2.
Theory
367et is only consistent to the 5-odd-limit, with three mappings possible for the 7-limit:
- ⟨367 582 852 1030] (patent val)
- ⟨367 582 852 1031] (367d val)
- ⟨367 582 853 1031] (367cd val)
Using the patent val, it tempers out 15625/15552 and [102 -57 -5⟩ in the 5-limit; 5120/5103 and 40353607/39858075 in the 7-limit.
Using the 367d val, it tempers out 15625/15552 and [102 -57 -5⟩ in the 5-limit; 2460375/2458624 and 2097152/2083725 in the 7-limit.
Using the 367cd val, it tempers out 268435456/263671875 and [33 -34 9⟩ in the 5-limit; 5120/5103, 7558272/7503125 and 235298/234375 in the 7-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.04 | -0.48 | -0.98 | -1.19 | +1.27 | -0.20 | +0.56 | -0.32 | +0.03 | +0.06 | -0.48 |
| Relative (%) | +31.9 | -14.8 | -29.9 | -36.2 | +38.9 | -6.1 | +17.1 | -9.9 | +1.1 | +2.0 | -14.7 | |
| Steps (reduced) |
582 (215) |
852 (118) |
1030 (296) |
1163 (62) |
1270 (169) |
1358 (257) |
1434 (333) |
1500 (32) |
1559 (91) |
1612 (144) |
1660 (192) | |
Subsets and supersets
367edo is the 73rd prime edo. 1101edo, which triples it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [582 -367⟩ | [⟨367 582]] | -0.3288 | 0.3287 | 10.05 |
| 2.3.5 | 15625/15552, [102 -57 -5⟩ | [⟨367 582 852]] | -0.1500 | 0.3688 | 11.28 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 28\367 | 91.55 | [46 -7 -15⟩ | Gross |
| 1 | 97\367 | 317.17 | 6/5 | Hanson |
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