1578edo

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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
Special properties

1578 equal divisions of the octave (abbreviated 1578edo or 1578ed2), also called 1578-tone equal temperament (1578tet) or 1578 equal temperament (1578et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1578 equal parts of about 0.76 ¢ each. Each step represents a frequency ratio of 21/1578, or the 1578th root of 2.

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)

Subsets and supersets

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