2153edo
| ← 2152edo | 2153edo | 2154edo → |
2153 equal divisions of the octave (abbreviated 2153edo or 2153ed2), also called 2153-tone equal temperament (2153tet) or 2153 equal temperament (2153et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2153 equal parts of about 0.557 ¢ each. Each step represents a frequency ratio of 21/2153, or the 2153rd root of 2.
Theory
2153edo is consistent to the 7-odd-limit and the error of its harmonic 3 is quite large. It can be used in the 2.9.5.7.11.13.17.19.23.29 subgroup, tempering out 12376/12375, 10241/10240, 2737/2736, 25025/25024, 131648/131625, 104272/104247, 194481/194480, 145775/145728 and 25840/25839. Using the 2.5.7.11.13.23.37 subgroup, it tempers out 77441/77440.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.236 | -0.062 | -0.131 | +0.084 | -0.087 | -0.026 | +0.259 | -0.171 | +0.118 | +0.190 | -0.128 |
| Relative (%) | -42.4 | -11.1 | -23.5 | +15.1 | -15.6 | -4.7 | +46.5 | -30.7 | +21.2 | +34.1 | -22.9 | |
| Steps (reduced) |
3412 (1259) |
4999 (693) |
6044 (1738) |
6825 (366) |
7448 (989) |
7967 (1508) |
8412 (1953) |
8800 (188) |
9146 (534) |
9457 (845) |
9739 (1127) | |
Subsets and supersets
2153edo is the 325th prime edo. 4306edo, 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 | [6825 -2153⟩ | [⟨2153 6825]] | -0.0133 | 0.0133 | 2.39 |
| 2.9.5 | [-53 5 16⟩, [92 -103 101⟩ | [⟨2153 6825 4999]] | +0.0000 | 0.0218 | 3.91 |
| 2.9.5.7 | 420175/419904, [47 -9 -14 5⟩, [1 -8 25 -12⟩ | [⟨2153 6825 4999 6044]] | +0.0117 | 0.0276 | 4.95 |
| 2.9.5.7.11 | 420175/419904, 151263/151250, 184549376/184528125, 781258401/781250000 | [⟨2153 6825 4999 6044 7448]] | +0.0144 | 0.0253 | 4.54 |
| 2.9.5.7.11.13 | 10648/10647, 1146880/1146717, 105644/105625, 1890625/1889568, 140625/140608 | [⟨2153 6825 4999 6044 7448 7967]] | +0.0132 | 0.0233 | 4.18 |