8539edo

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8539edo

8540edo →

Prime factorization	8539 (prime)
Step size	0.140532¢
Fifth	4995\8539 (701.956¢)
Semitones (A1:m2)	809:642 (113.7¢ : 90.22¢)
Consistency limit	27
Distinct consistency limit	27
Special properties	zeta peak zeta peak integer zeta integral zeta gap

8539 equal divisions of the octave (abbreviated 8539edo or 8539ed2), also called 8539-tone equal temperament (8539tet) or 8539 equal temperament (8539et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 8539 equal parts of about 0.141 ¢ each. Each step represents a frequency ratio of 2^1/8539, or the 8539th root of 2.

While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a unit of interval measure, the tina (see Tonalsoft Encyclopedia | Tina). This is because it is a very strong higher-limit system, distinctly consistent through the 27-odd-limit, and is both a zeta peak, zeta integral, and zeta gap tuning. In the 13-limit, the only smaller systems with a lower logflat badness are 72, 270, 494, 5585 and 6079; in the 17-limit, that becomes 72, 494, 1506, 3395 and 7033. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until 20203. Some 17-limit commas it tempers out are 28561/28560, 31213/31212 and 37180/37179; in the 19-limit it tempers out 27456/27455 and 43681/43680.

Prime harmonics

Approximation of prime harmonics in 8539edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	absolute (¢)	+0.0000	+0.0007	+0.0079	-0.0005	-0.0122	-0.0077	+0.0218	-0.0075	+0.0428	-0.0421	+0.0165
	relative (%)	+0	+1	+6	-0	-9	-5	+15	-5	+30	-30	+12
Steps (reduced)		8539 (0)	13534 (4995)	19827 (2749)	23972 (6894)	29540 (3923)	31598 (5981)	34903 (747)	36273 (2117)	38627 (4471)	41482 (7326)	42304 (8148)

Subsets and supersets

8539edo is the 1065th prime edo. On that basis the tina as a unit of measure could be criticized; however, some people prefer primes for this sort of job, as they do not imply a preference for one smaller edo over another.