1578edo: Difference between revisions
m Categories |
mNo edit summary |
||
| (11 intermediate revisions by 7 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | |||
{{ED intro}} | |||
1578edo is a very strong higher limit system, and is a [[zeta edo|zeta peak, peak integer, integral and gap edo]]. It is [[consistency|distinctly consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness|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 additionally contains an extremely accurate approximation of [[quarter-comma meantone]] inherited from 789edo. | |||
[[ | 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 === | |||
{{Harmonics in equal|1578|columns=11}} | |||
=== Subsets and supersets === | |||
Since 1578 factors into {{factorization|1578}}, 1578edo has subset edos [[2edo|2]], [[3edo|3]], [[6edo|6]], [[263edo|263]], [[526edo|526]], and [[789edo|789]]. | |||
Latest revision as of 02:30, 2 April 2025
| ← 1577edo | 1578edo | 1579edo → |
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 additionally contains an extremely accurate approximation of quarter-comma meantone inherited from 789edo.
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
| 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.