402653184edo

This page presents a primarily software-related tuning. This tuning is more notable for use in software rather than for its musical properties.

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402653184edo

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Prime factorization	227 × 3
Step size	2.98023e-06 ¢
Fifth	235537013\402653184 (701.955 ¢)
Semitones (A1:m2)	38146355:30274487 (113.7 ¢ : 90.23 ¢)
Dual sharp fifth	235537014\402653184 (701.955 ¢) (→ 39256169\67108864)
Dual flat fifth	235537013\402653184 (701.955 ¢)
Dual major 2nd	68420843\402653184 (203.91 ¢)
Consistency limit	7
Distinct consistency limit	7

402653184 equal divisions of the octave (abbreviated 402653184edo or 402653184ed2), also called 402653184-tone equal temperament (402653184tet) or 402653184 equal temperament (402653184et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 402653184 equal parts of about 2.98e-06 ¢ each. Each step represents a frequency ratio of 2^1/402653184, or the 402653184th root of 2.

Theory

402,653,184edo is an equal-step tuning system that allows for finer subdivisions within the standard 128-note MIDI scale based on 12edo, using 32-bit data.

Since 128 notes equal 2⁷, this leaves 25 additional bits for further subdivision of the 12-tone equal temperament. By calculating 12 × 2²⁵, we arrive at a system with 402,653,184 equally spaced divisions of the octave.

It could be applied through the 7.25 pitch attribute introduced in MIDI 2.0.

The step size is about 2.98 × 10^-6 cents, meaning that intervals rounded to this system have a precision of 1.49 × 10^-6 cents. At 20000 Hz, the phase precision is approximately 16 hours.

Prime harmonics

Approximation of prime harmonics in 402653184edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00000000	-0.00000130	-0.00000127	+0.00000005	-0.00000022	+0.00000087	+0.00000146	+0.00000093	+0.00000111	-0.00000149	+0.00000138
	Relative (%)	+0.0	-43.6	-42.5	+1.5	-7.5	+29.0	+48.9	+31.1	+37.2	-50.0	+46.2
Steps (reduced)		402653184 (0)	638190197 (235537013)	934931740 (129625372)	1130390398 (325084030)	1392951156 (184991604)	1489993835 (282034283)	1645829928 (35217192)	1710441539 (99828803)	1821426625 (210813889)	1956081515 (345468779)	1994822919 (384210183)

Approximation of prime harmonics in 402653184edo (continued)

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	+0.00000128	-0.00000031	+0.00000087	+0.00000018	+0.00000081	-0.00000048	-0.00000130	+0.00000010	-0.00000138	-0.00000010	-0.00000146
	Relative (%)	+43.0	-10.5	+29.2	+6.1	+27.1	-16.1	-43.7	+3.4	-46.3	-3.4	-49.1
Steps (reduced)		2097602985 (84337065)	2157235373 (143969453)	2184902781 (171636861)	2236572887 (223306967)	2306365409 (293099489)	2368664954 (355399034)	2388030272 (374764352)	2442530127 (26611023)	2476215258 (60296154)	2492352567 (76433463)	2538237389 (122318285)

Subsets and supersets

402653184 factorizes into 2²⁷ × 3.