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 2^1/500, 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

Music

Francium

• "loveyouforever" from albumwithoutspaces (2024) – Spotify | Bandcamp | YouTube