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. 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

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.

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
  • 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)

Music

Eliora

• Etude for Celtic Harp in Whoosh (Op. 3, No. 5) (2023)