User:Francium/3389edo
| ← 3388edo | 3389edo | 3390edo → |
3389 equal divisions of the octave (abbreviated 3389edo or 3389ed2), also called 3389-tone equal temperament (3389tet) or 3389 equal temperament (3389et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3389 equal parts of about 0.354 ¢ each. Each step represents a frequency ratio of 21/3389, or the 3389th root of 2.
Theory
3389edo is consistent to the 7-limit, and its harmonic 3 is about halfway its steps. As an equal temperament, it tempers out 2460375/2458624, 1640558367/1638400000 and [63 29 -30 -14⟩ in the 7-limit. 3389edo is strong in the 2.5.7.11 subgroup, tempering out 214375000/214358881, [49 -17 4 -6⟩ and [-30 -8 21 -3⟩.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.155 | -0.005 | -0.045 | +0.044 | -0.005 | +0.074 | -0.160 | -0.146 | -0.080 | +0.154 | -0.124 |
| Relative (%) | -43.8 | -1.4 | -12.6 | +12.4 | -1.4 | +21.0 | -45.2 | -41.2 | -22.6 | +43.6 | -35.1 | |
| Steps (reduced) |
5371 (1982) |
7869 (1091) |
9514 (2736) |
10743 (576) |
11724 (1557) |
12541 (2374) |
13240 (3073) |
13852 (296) |
14396 (840) |
14886 (1330) |
15330 (1774) | |
Subsets and supersets
3389edo is the 477th prime edo. 6778edo, which doubles it, gives a good correction to its harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [10743 -3389⟩ | [⟨3389 10743]] | −0.0069 | 0.0069 | 1.95 |
| 2.9.5 | [144 -11 -47⟩, [57 -59 56⟩ | [⟨3389 10743 7869]] | −0.0039 | 0.0071 | 2.01 |
| 2.9.5.7 | 24414062500/24407490807, 33232930569601/33215062500000, [47 -9 -14 5⟩ | [⟨3389 10743 7869 9514]] | +0.0010 | 0.0105 | 2.97 |
| 2.9.5.7.11 | 151263/151250, 184549376/184528125, 1722499009/1721868840, 46894531250/46877879169 | [⟨3389 10743 7869 9514 11724]] | +0.0011 | 0.0094 | 2.65 |
| 2.9.5.7.11.13 | 10648/10647, 1146880/1146717, 105644/105625, 140625/140608, 1377751375/1377495072 | [⟨3389 10743 7869 9514 11724 12541]] | −0.0024 | 0.0117 | 3.30 |