500edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|500}}
{{ED intro}}


== Theory ==
== Theory ==
500et is only consistent to the [[3-odd-limit]] and the error of the [[harmonic]] [[3/1|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.
500edo is in[[consistent]] to the [[5-odd-limit]] and the error of the [[harmonic]] [[3/1|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 ===
=== Odd harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
500 factors into 2<sup>2</sup> * 5<sup>3</sup>, with subset edos {{EDOs|2, 4, 5, 10, 20, 25, 50, 100, 125, and 250}}. [[1000edo]], which doubles it, gives a good correction to the harmonic 3.
Since 500 factors into {{factorization|500}}, 500edo has subset edos {{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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
|-
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | [[Mapping]]
! colspan="2" |Tuning Error
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.9
| 2.9
|{{monzo|317 -100}}
| {{monzo| 317 -100 }}
|{{mapping|500 1585}}
| {{mapping| 500 1585 }}
| -0.0142
| −0.0142
| 0.0142
| 0.0142
| 0.59
| 0.59
|-
|-
|2.9.5
| 2.9.5
|{{monzo|38 -1 -15}}, {{monzo|55 -32 20}}
| {{monzo| 38 -1 -15 }}, {{monzo| 55 -32 20 }}
|{{mapping|500 1585 1161}}
| {{mapping| 500 1585 1161 }}
| -0.0219
| −0.0219
| 0.0159
| 0.0159
| 0.66
| 0.66
|-
|-
|2.9.5.7
| 2.9.5.7
|703125/702464, 589824/588245, 31381059609/31360000000
| 589824/588245, 703125/702464, 31381059609/31360000000
|{{mapping|500 1585 1161 1404}}
| {{mapping| 500 1585 1161 1404 }}
| -0.0853
| −0.0853
| 0.1108
| 0.1108
| 4.62
| 4.62
|-
|-
|2.9.5.7.11
| 2.9.5.7.11
|9801/9800, 46656/46585, 2252800/2250423, 703125/702464
| 9801/9800, 46656/46585, 2252800/2250423, 703125/702464
|{{mapping|500 1585 1161 1404 1730}}
| {{mapping| 500 1585 1161 1404 1730 }}
| -0.1077
| −0.1077
| 0.1087
| 0.1087
| 4.53
| 4.53
|-
|-
|2.9.5.7.11.13
| 2.9.5.7.11.13
|4096/4095, 9801/9800, 35750/35721, 41600/41503, 91125/91091
| 4096/4095, 9801/9800, 35750/35721, 41600/41503, 91125/91091
|{{mapping|500 1585 1161 1404 1730 1850}}
| {{mapping| 500 1585 1161 1404 1730 1850 }}
| -0.0660
| −0.0660
| 0.1362
| 0.1362
| 5.68
| 5.68
|-
|-
|2.9.5.7.11.13.17
| 2.9.5.7.11.13.17
|936/935, 4096/4095, 12376/12375, 9801/9800, 35750/35721, 11016/11011
| 936/935, 4096/4095, 12376/12375, 9801/9800, 35750/35721, 11016/11011
|{{mapping|500 1585 1161 1404 1730 1850 2044}}
| {{mapping| 500 1585 1161 1404 1730 1850 2044 }}
| -0.0791
| −0.0791
| 0.1301
| 0.1301
| 5.42
| 5.42
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== Music ==
== Music ==
; [[Francium]]
; [[Francium]]
* [https://www.youtube.com/watch?v=CKXtJSihX-A ''loveyouforever''] (2024)
* "loveyouforever" from ''albumwithoutspaces'' (2024) – [https://open.spotify.com/track/63WBz0dSUO6Q58WMZmckJ2 Spotify] | [https://francium223.bandcamp.com/track/loveyouforever Bandcamp] | [https://www.youtube.com/watch?v=CKXtJSihX-A YouTube]

Latest revision as of 12:15, 21 February 2025

← 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

Music

Francium
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