3072edo
Theory
3072edo is consistent to the 11-limit and it is an extremely accurate 5-limit tuning, tempering out [37 25 -33⟩ (whoosh) and [161 -84 -12⟩ (atom) in the 5-limit; 250047/250000, [-2 -25 1 14⟩, and [-53 -1 9 12⟩; in the 7-limit; 9801/9800, 151263/151250, 184549376/184528125, and 73525096183/73466403840 in the 11-limit.
Significance in digital audio software
3072edo's step is known as Octamu (eighth MIDI-resolution unit, 8mu, 28 = 256 equal divisions of the 12edo semitone). The internal data structure of the 8mu 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 5 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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.002 | +0.014 | -0.076 | -0.146 | +0.097 | +0.123 | +0.143 | -0.149 | +0.110 | -0.114 |
| Relative (%) | +0.0 | -0.5 | +3.7 | -19.4 | -37.4 | +24.9 | +31.4 | +36.7 | -38.2 | +28.2 | -29.1 | |
| Steps (reduced) |
3072 (0) |
4869 (1797) |
7133 (989) |
8624 (2480) |
10627 (1411) |
11368 (2152) |
12557 (269) |
13050 (762) |
13896 (1608) |
14924 (2636) |
15219 (2931) | |
See also
- Equal multiplications of MIDI-resolution units