3159811edo

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3159811edo

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Prime factorization	29 × 108959
Step size	0.00037977 ¢
Fifth	1848371\3159811 (701.955 ¢)
Semitones (A1:m2)	299353:237578 (113.7 ¢ : 90.22 ¢)
Consistency limit	65
Distinct consistency limit	65

3159811 equal divisions of the octave (abbreviated 3159811edo or 3159811ed2), also called 3159811-tone equal temperament (3159811tet) or 3159811 equal temperament (3159811et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3159811 equal parts of about 0.00038 ¢ each. Each step represents a frequency ratio of 2^1/3159811, or the 3159811th root of 2.

3159811edo is consistent in the 65-odd-limit with a lower relative error than any previous equal temperaments in the 61-limit. It is the smallest EDO which is purely consistent^{[idiosyncratic term]} in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the harmonic series).

Prime harmonics

Approximation of odd harmonics in 3159811edo (3–15)

Harmonic		3	5	7	9	11	13	15
Error	Absolute (¢)	+0.00002113	+0.00002452	+0.00001382	+0.00004226	-0.00003126	-0.00004793	+0.00004565
	Relative (%)	+5.6	+6.5	+3.6	+11.1	-8.2	-12.6	+12.0
Steps (reduced)		5008182 (1848371)	7336854 (1017232)	8870711 (2551089)	10016364 (536931)	10931150 (1451717)	11692690 (2213257)	12345036 (2865603)

Approximation of odd harmonics in 3159811edo (17–31)

Harmonic		17	19	21	23	25	27	29	31
Error	Absolute (¢)	-0.00001818	-0.00003197	+0.00003494	+0.00006535	+0.00004904	+0.00006338	+0.00008120	+0.00000087
	Relative (%)	-4.8	-8.4	+9.2	+17.2	+12.9	+16.7	+21.4	+0.2
Steps (reduced)		12915610 (276366)	13422648 (783404)	13878893 (1239649)	14293601 (1654357)	14673708 (2034464)	15024546 (2385302)	15350302 (2711058)	15654324 (3015080)

Approximation of odd harmonics in 3159811edo (33–47)

Harmonic		33	35	37	39	41	43	45	47
Error	Absolute (¢)	-0.00001013	+0.00003834	-0.00001850	-0.00002681	+0.00009218	-0.00002310	+0.00006678	+0.00001747
	Relative (%)	-2.7	+10.1	-4.9	-7.1	+24.3	-6.1	+17.6	+4.6
Steps (reduced)		15939332 (140277)	16207565 (408510)	16460888 (661833)	16700872 (901817)	16928852 (1129797)	17145971 (1346916)	17353218 (1554163)	17551451 (1752396)

Approximation of odd harmonics in 3159811edo (49–63)

Harmonic		49	51	53	55	57	59	61	63
Error	Absolute (¢)	+0.00002763	+0.00000295	-0.00002258	-0.00000674	-0.00001084	-0.00008220	-0.00002937	+0.00005607
	Relative (%)	+7.3	+0.8	-5.9	-1.8	-2.9	-21.6	-7.7	+14.8
Steps (reduced)		17741422 (1942367)	17923792 (2124737)	18099146 (2300091)	18268004 (2468949)	18430830 (2631775)	18588040 (2788985)	18740009 (2940954)	18887075 (3088020)