# 500edo

 ← 499edo 500edo 501edo →
Prime factorization 22 × 53
Step size 2.4¢
Fifth 292\500 (700.8¢) (→73\125)
Semitones (A1:m2) 44:40 (105.6¢ : 96¢)
Dual sharp fifth 293\500 (703.2¢)
Dual flat fifth 292\500 (700.8¢) (→73\125)
Dual major 2nd 85\500 (204¢) (→17\100)
Consistency limit 3
Distinct consistency limit 3

500 equal divisions of the octave (abbreviated 500edo or 500ed2), also called 500-tone equal temperament (500tet) or 500 equal temperament (500et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 500 equal parts of exactly 2.4 ¢ each. Each step represents a frequency ratio of 21/500, or the 500th root of 2.

## Theory

500edo is inconsistent to the 5-odd-limit and the error of the harmonic 3 is very large. It can be used in the 2.9.5.7.11.13.17.19.23.29.31.37.41.43 subgroup, tempering out 936/935, 4096/4095, 12376/12375, 2025/2024, 3328/3325, 9801/9800, 35750/35721, 46592/46575, 5104/5103, 12555/12544, 1296/1295, 1376/1375 and 6273/6272.

### Odd harmonics

Approximation of odd harmonics in 500edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.16 +0.09 +0.77 +0.09 +0.68 -0.53 -1.07 +0.64 +0.09 -0.38 +0.53
Relative (%) -48.1 +3.6 +32.3 +3.7 +28.4 -22.0 -44.5 +26.9 +3.6 -15.9 +21.9
Steps
(reduced)
792
(292)
1161
(161)
1404
(404)
1585
(85)
1730
(230)
1850
(350)
1953
(453)
2044
(44)
2124
(124)
2196
(196)
2262
(262)

### Subsets and supersets

Since 500 factors into 22 × 53, 500edo has subset edos 2, 4, 5, 10, 20, 25, 50, 100, 125, and 250. 1000edo, which doubles it, gives a good correction to the harmonic 3.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [317 -100 [500 1585]] -0.0142 0.0142 0.59
2.9.5 [38 -1 -15, [55 -32 20 [500 1585 1161]] -0.0219 0.0159 0.66
2.9.5.7 589824/588245, 703125/702464, 31381059609/31360000000 [500 1585 1161 1404]] -0.0853 0.1108 4.62
2.9.5.7.11 9801/9800, 46656/46585, 2252800/2250423, 703125/702464 [500 1585 1161 1404 1730]] -0.1077 0.1087 4.53
2.9.5.7.11.13 4096/4095, 9801/9800, 35750/35721, 41600/41503, 91125/91091 [500 1585 1161 1404 1730 1850]] -0.0660 0.1362 5.68
2.9.5.7.11.13.17 936/935, 4096/4095, 12376/12375, 9801/9800, 35750/35721, 11016/11011 [500 1585 1161 1404 1730 1850 2044]] -0.0791 0.1301 5.42

Francium