1536edo
| ← 1535edo | 1536edo | 1537edo → |
1536 equal divisions of the octave (abbreviated 1536edo or 1536ed2), also called 1536-tone equal temperament (1536tet) or 1536 equal temperament (1536et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1536 equal parts of about 0.781 ¢ each. Each step represents a frequency ratio of 21/1536, or the 1536th root of 2.
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.
As a tuning standard
A step of 1536edo is known as a 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 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.
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