User:Francium/1667edo
| ← 1666edo | 1667edo | 1668edo → |
1667 equal divisions of the octave (abbreviated 1667edo or 1667ed2), also called 1667-tone equal temperament (1667tet) or 1667 equal temperament (1667et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1667 equal parts of about 0.72 ¢ each. Each step represents a frequency ratio of 21/1667, or the 1667th root of 2.
Theory
1667edo is consistent to the 7-limit, tempering out 1640558367/1638400000, 184528125/184473632 and 117546246144/117236328125 in the 7-limit. It is strong in the 2.3.7.11.17 subgroup, tempering out 34992/34969, 142633953/142606336, 32019867/32000528 and 2030754159/2030043136.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.095 | +0.249 | +0.100 | +0.092 | +0.264 | +0.144 | -0.212 | +0.160 | -0.183 | +0.255 |
| Relative (%) | +0.0 | -13.2 | +34.6 | +13.9 | +12.7 | +36.7 | +19.9 | -29.5 | +22.2 | -25.4 | +35.5 | |
| Steps (reduced) |
1667 (0) |
2642 (975) |
3871 (537) |
4680 (1346) |
5767 (766) |
6169 (1168) |
6814 (146) |
7081 (413) |
7541 (873) |
8098 (1430) |
8259 (1591) | |
Subsets and supersets
1667edo is the 262nd prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-2642 1667⟩ | [⟨1667 2642]] | +0.0301 | 0.0301 | 4.18 |
| 2.3.5 | [31 20 -27⟩, [-99 61 1⟩ | [⟨1667 2642 3871]] | −0.0157 | 0.0692 | 9.61 |
| 2.3.5.7 | 1640558367/1638400000, 184528125/184473632, 117546246144/117236328125 | [⟨1667 2642 3871 4680]] | −0.0207 | 0.0606 | 8.42 |
| 2.3.5.7.11 | 160083/160000, 172032/171875, 1240029/1239040, 52612659/52521875 | [⟨1667 2642 3871 4680 5767]] | −0.0219 | 0.0542 | 7.53 |
| 2.3.5.7.11.13 | 4096/4095, 34398/34375, 40656/40625, 91125/91091, 10513152/10504375 | [⟨1667 2642 3871 4680 5767 6169]] | −0.0301 | 0.0528 | 7.33 |
| 2.3.5.7.11.13.17 | 1701/1700, 4096/4095, 12376/12375, 14400/14399, 11016/11011, 10007712/10000165 | [⟨1667 2642 3871 4680 5767 6169 6814]] | −0.0308 | 0.0490 | 6.81 |