3072edo
| ← 3071edo | 3072edo | 3073edo → |
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.
Although consistent to the 11-limit, it makes more sense to actually see 3072edo as a 2.3.5.7.13 subgroup tuning, due to lower relative error. There it tempers out 140625/140608 and 1990656/1990625. Overall in the 13-limit, the patent val still has smaller errors than any other val despite incosistency. In higher limits, it is not as impressive, with only 53/32 being 17% off and 2.3.5.67.71 subgroup having less than 4% error.
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) | |
Subsets and supersets
3072 factors as 210 x 3, with subset edos 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal
8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [37 25 -33⟩, [161 -84 -12⟩ | [⟨3072 4869 7133]] | -0.002 | 0.003 | |
| 2.3.5.7 | 250047/250000, [-2 -25 1 14⟩, [-53 -1 9 12⟩ | [⟨3072 4869 7133 8624]] | 0.006 | 0.013 | |
| 2.3.5.7.11 | 9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840 | [⟨3072 4869 7133 8624 10627]] | 0.013 | 0.019 | |
| 2.3.5.7.11.13 | 9801/9800, 140625/140608, 151263/151250, 196625/196608, 3327500/3326427 | [⟨3072 4869 7133 8624 10627 11638]] | 0.006 | 0.022 | |
See also
- Equal multiplications of MIDI-resolution units