4349edo
| ← 4348edo | 4349edo | 4350edo → |
4349 equal divisions of the octave (abbreviated 4349edo or 4349ed2), also called 4349-tone equal temperament (4349tet) or 4349 equal temperament (4349et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4349 equal parts of about 0.276 ¢ each. Each step represents a frequency ratio of 21/4349, or the 4349th root of 2.
Theory
4349edo is a fairly strong 29-limit system, consistent to the 29-odd-limit. We may note it is a counterquectismic, fortune, and euzenius system. Some of the simpler commas in tempers out in the higher limits include 151263/151250 and 1771561/1771470 in the 11-limit; 123201/123200 in the 13-limit; 12376/12375 in the 17-limit; 10241/10240, 13377/13376, 89376/89375 in the 19-limit; 12168/12167, 25025/25024, 76545/76544 in the 23-limit; 47125/47124 and 25840/25839. Since it tempers out 12168/12167, it allows vicetertismic chords in the 23-odd-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.001 | -0.018 | -0.051 | -0.019 | -0.059 | -0.104 | -0.065 | +0.008 | -0.099 | +0.055 |
| Relative (%) | +0.0 | -0.2 | -6.5 | -18.7 | -6.8 | -21.2 | -37.6 | -23.7 | +2.9 | -35.9 | +20.0 | |
| Steps (reduced) |
4349 (0) |
6893 (2544) |
10098 (1400) |
12209 (3511) |
15045 (1998) |
16093 (3046) |
17776 (380) |
18474 (1078) |
19673 (2277) |
21127 (3731) |
21546 (4150) | |
Subsets and supersets
4349edo is the 594th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-6893 4349⟩ | [⟨4349 6893]] | +0.0002 | 0.0002 | 0.07 |
| 2.3.5 | [-107 47 14⟩, [-35 -79 69⟩ | [⟨4349 6893 10098]] | +0.0027 | 0.0036 | 1.30 |
| 2.3.5.7 | 78125000/78121827, [-1 -18 -3 13⟩, [-52 17 12 -1⟩ | [⟨4349 6893 10098 12209]] | +0.0066 | 0.0074 | 2.68 |
| 2.3.5.7.11 | 151263/151250, 21437500/21434787, 246071287/246037500, 369140625/369098752 | [⟨4349 6893 10098 12209 15045]] | +0.0064 | 0.0067 | 2.43 |
| 2.3.5.7.11.13 | 123201/123200, 151263/151250, 196625/196608, 492128/492075, 5175625/5174928 | [⟨4349 6893 10098 12209 15045 16093]] | +0.0079 | 0.0070 | 2.54 |
| 2.3.5.7.11.13.17 | 12376/12375, 37180/37179, 123201/123200, 194481/194480, 221221/221184, 1328125/1328096 | [⟨4349 6893 10098 12209 15045 16093 17776]] | +0.0104 | 0.0089 | 3.23 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 803\4349 | 221.5682 | 8388608/7381125 | Fortune |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct