500edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 13: | Line 13: | ||
== 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" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
| Line 25: | Line 26: | ||
| {{monzo| 317 -100 }} | | {{monzo| 317 -100 }} | ||
| {{mapping| 500 1585 }} | | {{mapping| 500 1585 }} | ||
| | | −0.0142 | ||
| 0.0142 | | 0.0142 | ||
| 0.59 | | 0.59 | ||
| Line 32: | Line 33: | ||
| {{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 | ||
| 0.0159 | | 0.0159 | ||
| 0.66 | | 0.66 | ||
| Line 39: | Line 40: | ||
| 589824/588245, 703125/702464, 31381059609/31360000000 | | 589824/588245, 703125/702464, 31381059609/31360000000 | ||
| {{mapping| 500 1585 1161 1404 }} | | {{mapping| 500 1585 1161 1404 }} | ||
| | | −0.0853 | ||
| 0.1108 | | 0.1108 | ||
| 4.62 | | 4.62 | ||
| Line 46: | Line 47: | ||
| 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 | ||
| 0.1087 | | 0.1087 | ||
| 4.53 | | 4.53 | ||
| Line 53: | Line 54: | ||
| 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 | ||
| 0.1362 | | 0.1362 | ||
| 5.68 | | 5.68 | ||
| Line 60: | Line 61: | ||
| 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 | ||
| 0.1301 | | 0.1301 | ||
| 5.42 | | 5.42 | ||
Latest revision as of 12:15, 21 February 2025
| ← 499edo | 500edo | 501edo → |
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
| 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 |