1536edo
Template:EDO intro Its adjacent step is known as heptamu (seventh MIDI-resolution unit, 7mu, 27 = 128 equal divisions of the 12edo semitone). The internal data structure of the 7mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and 6 other bits which are not used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.
| ← 1535edo | 1536edo | 1537edo → |
Theory
1536edo is inconsistent to the 5-odd-limit and both harmonics 3 and 5 are about halfway between its steps. Otherwise it is excellent in approximating harmonics 7, 9, 13, and 15, making it suitable for a 2.9.15.7.13 subgroup interpretation, with an optional addition of either 11 or 17.
If we do use it for the 5-limit, then ⟨1536 2434 3566] (1536b val) and ⟨1536 2435 3567] (1536c val) are worth considering, but 1536b is enfactored through the 17-limit (see 768edo), leaving us with 1536c alone. It tempers out 6115295232/6103515625 (vishnuzma) and [169 -111 3⟩ in the 5-limit; 250047/250000, 134217728/133984375, and 12111126300875/12050326889856 in the 7-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.389 | -0.376 | -0.076 | -0.004 | +0.245 | +0.097 | +0.013 | -0.268 | +0.143 | +0.313 | -0.149 |
| Relative (%) | +49.8 | -48.2 | -9.7 | -0.5 | +31.3 | +12.5 | +1.6 | -34.3 | +18.3 | +40.0 | -19.1 | |
| Steps (reduced) |
2435 (899) |
3566 (494) |
4312 (1240) |
4869 (261) |
5314 (706) |
5684 (1076) |
6001 (1393) |
6278 (134) |
6525 (381) |
6747 (603) |
6948 (804) | |
Subsets and supersets
Since 1536 factors into 29 × 3, 1536edo has subset edos 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, and 768.
See also
- Equal multiplications of MIDI-resolution units