3395edo

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3395edo

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Prime factorization	5 × 7 × 97
Step size	0.353461¢
Fifth	1986\3395 (701.973¢)
Semitones (A1:m2)	322:255 (113.8¢ : 90.13¢)
Consistency limit	21
Distinct consistency limit	21
Special properties	Zeta peak Zeta integral Zeta gap

3395 equal divisions of the octave (abbreviated 3395edo or 3395ed2), also called 3395-tone equal temperament (3395tet) or 3395 equal temperament (3395et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3395 equal parts of about 0.353 ¢ each. Each step represents a frequency ratio of 2^1/3395, or the 3395th root of 2.

Theory

3395edo is an extremely strong 17- and 19-limit system, and a zeta peak, integral and gap edo. It has a lower 17-limit TE relative error than any edo until 7033, and a lower 19-limit relative error than any edo until 8269. Besides, it provides the optimal patent val for the 13-limit rank-5 temperament tempering out 6656/6655, the jacobin comma, and for quartismic, which also tempers out 123201/123200. A basis for the 17-limit commas is {6656/6655, 12376/12375, 28561/28560, 31213/31212, 37180/37179, 937125/937024}, and for the 19-limit, {6656/6655, 12376/12375, 12636/12635, 13377/13376, 14365/14364, 23409/23408, 28561/28560}.

Prime harmonics

Approximation of prime harmonics in 3395edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.018	+0.019	+0.011	+0.081	+0.003	+0.022	+0.101	-0.174	+0.055	-0.175
	Relative (%)	+0.0	+5.2	+5.4	+3.0	+23.0	+0.7	+6.4	+28.6	-49.3	+15.5	-49.6
Steps (reduced)		3395 (0)	5381 (1986)	7883 (1093)	9531 (2741)	11745 (1560)	12563 (2378)	13877 (297)	14422 (842)	15357 (1777)	16493 (2913)	16819 (3239)

Subsets and supersets

Since 3395 factors into 5 × 7 × 97, 3395edo has subset edos 5, 7, 35, 97, 485, and 679.

Regular temperament properties

3395edo has a lower 17-limit TE relative error than any edo until 7033, and a lower 19-limit relative error than any edo until 8269.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	2319\3395	819.676	55115776/34328125	Genojacobin
35	1409\3395 (51\3395)	498.027 (18.026)	4/3 (?)	Bromine
97	1409\3395 (9\3395)	498.027 (3.181)	4/3 (?)	Berkelium

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct