3072edo

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3072edo

3073edo →

Prime factorization	210 × 3
Step size	0.390625 ¢
Fifth	1797\3072 (701.953 ¢) (→ 599\1024)
Semitones (A1:m2)	291:231 (113.7 ¢ : 90.23 ¢)
Consistency limit	11
Distinct consistency limit	11

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. 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

Approximation of prime harmonics in 3072edo

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.

See also

• Equal multiplications of MIDI-resolution units
  • 24edo (1mu tuning)
  • 48edo (2mu tuning)
  • 96edo (3mu tuning)
  • 192edo (4mu tuning)
  • 384edo (5mu tuning)
  • 768edo (6mu tuning)
  • 1536edo (7mu tuning)
  • 6144edo (9mu tuning)
  • 12288edo (10mu tuning)
  • 24576edo (11mu tuning)
  • 49152edo (12mu tuning)
  • 98304edo (13mu tuning)
  • 196608edo (14mu tuning)