647edo
| ← 646edo | 647edo | 648edo → |
647 equal divisions of the octave (abbreviated 647edo or 647ed2), also called 647-tone equal temperament (647tet) or 647 equal temperament (647et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 647 equal parts of about 1.85 ¢ each. Each step represents a frequency ratio of 21/647, or the 647th root of 2.
Theory
647edo is consistent to the 7-odd-limit and its harmonic 3 is about halfway its steps. It tends heavily flat in the first few harmonics. Using the patent val it tempers out 2401/2400, 7381125/7340032 and 1224440064/1220703125 in the 7-limit; 8019/8000, 2401/2400, 46656/46585 and 1265625/1261568 in the 11-limit. Alternatively, the 2.9.5.7.11.13 subgroup might be worth considering.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.873 | -0.533 | -0.665 | +0.109 | -0.468 | -0.342 | +0.448 | +0.763 | -0.759 | +0.316 | +0.474 |
| Relative (%) | -47.1 | -28.7 | -35.9 | +5.9 | -25.2 | -18.4 | +24.2 | +41.2 | -40.9 | +17.1 | +25.5 | |
| Steps (reduced) |
1025 (378) |
1502 (208) |
1816 (522) |
2051 (110) |
2238 (297) |
2394 (453) |
2528 (587) |
2645 (57) |
2748 (160) |
2842 (254) |
2927 (339) | |
Subsets and supersets
647edo is the 118th prime EDO. 1294edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [2051 -647⟩ | [⟨647 2051]] | -0.0171 | 0.0171 | 0.92 |
| 2.9.5 | [-50 -4 27⟩, [63 -25 7⟩ | [⟨647 2051 1502]] | +0.0651 | 0.1172 | 6.32 |
| 2.9.5.7 | 703125/702464, 3500000000/3486784401, 4202539929/4194304000 | [⟨647 2051 1502 1816]] | +0.1081 | 0.1258 | 6.78 |
| 2.9.5.7.11 | 496125/495616, 1684375/1679616, 151263/151250, 40960000/40920957 | [⟨647 2051 1502 1816 2238]] | +0.1135 | 0.1131 | 6.10 |
| 2.9.5.7.11.13 | 4459/4455, 35750/35721, 47432/47385, 60025/59904, 1146880/1146717 | [⟨647 2051 1502 1816 2238 2394]] | +0.1100 | 0.1035 | 5.58 |