1578edo

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1578edo

1579edo →

Prime factorization	2 × 3 × 263
Step size	0.760456 ¢
Fifth	923\1578 (701.901 ¢)
Semitones (A1:m2)	149:119 (113.3 ¢ : 90.49 ¢)
Consistency limit	29
Distinct consistency limit	29

1578edo is a very strong higher limit system, and is a zeta peak, peak integer, integral and gap edo. It is distinctly consistent through the 29-odd-limit, and is the first edo past 311edo with a lower 29-limit relative error. It is also the lowest past 311edo in the 31-limit, the lowest past 581edo in the 23-limit, and the lowest past 1178edo in the 19-limit. It is also quite strong taken just as an 11-limit system; the only smaller edo with a lower 11-limit relative error is 342edo.

It is also the most accurate edo below 10000 to approximate quarter-comma meantone, with its approximation of [0 0 1/4⟩ with a relative error of only 0.06%.

Some 31-limit or lower superparticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/26999… .

Prime harmonics

Approximation of prime harmonics in 1578edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.054	-0.002	-0.005	+0.013	-0.223	-0.012	-0.175	-0.137	+0.081	+0.212
	Relative (%)	+0.0	-7.1	-0.3	-0.6	+1.7	-29.4	-1.6	-23.0	-18.1	+10.6	+27.8
Steps (reduced)		1578 (0)	2501 (923)	3664 (508)	4430 (1274)	5459 (725)	5839 (1105)	6450 (138)	6703 (391)	7138 (826)	7666 (1354)	7818 (1506)

Divisors

Since 1578 factors into 2 × 3 × 263, 1578edo has subset edos 2, 3, 6, 263, 526, and 789.