3072edo
| ← 3071edo | 3072edo | 3073edo → |
3072 equal divisions of the octave (abbreviated 3072edo or 3072ed2), also called 3072-tone equal temperament (3072tet) or 3072 equal temperament (3072et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3072 equal parts of about 0.391 ¢ each. Each step represents a frequency ratio of 21/3072, or the 3072nd root of 2.
Theory
3072edo is consistent to the 11-odd-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 (landscape comma), [-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-odd-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.
As a tuning standard
A step of 3072edo is known as an 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 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 × 3, with subset edos 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, and 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 | |
Music
See also
- Equal multiplications of MIDI-resolution units