User:Francium/1031edo: Difference between revisions
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<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | <nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | ||
== Music == | |||
; [[Francium]] | |||
* "sweet gravy" from ''have nice day'' (2025) – [https://open.spotify.com/track/6F1HyBmCUC6t7FUaGswzK1 Spotify] | [https://francium223.bandcamp.com/track/sweet-gravy Bandcamp] | [https://www.youtube.com/watch?v=5YC--2hRyms YouTube] – in Sycamore, 1031edo tuning | |||
Latest revision as of 17:10, 6 November 2025
| ← 1030edo | 1031edo | 1032edo → |
1031 equal divisions of the octave (abbreviated 1031edo or 1031ed2), also called 1031-tone equal temperament (1031tet) or 1031 equal temperament (1031et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1031 equal parts of about 1.16 ¢ each. Each step represents a frequency ratio of 21/1031, or the 1031st root of 2.
Theory
1031edo is consistent to the 9-odd-limit, tempering out 420175/419904, 5250987/5242880 and 152946081792/152587890625 in the 7-limit. It is strong in the 2.3.5.13.17 subgroup, tempering out 2601/2600, 1601613/1600000, 33564375/33554432 and 4782969/4780672.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.112 | +0.107 | -0.446 | +0.379 | -0.178 | -0.203 | +0.450 | +0.242 | +0.491 | +0.260 |
| Relative (%) | +0.0 | -9.6 | +9.2 | -38.3 | +32.6 | -15.3 | -17.4 | +38.7 | +20.8 | +42.2 | +22.4 | |
| Steps (reduced) |
1031 (0) |
1634 (603) |
2394 (332) |
2894 (832) |
3567 (474) |
3815 (722) |
4214 (90) |
4380 (256) |
4664 (540) |
5009 (885) |
5108 (984) | |
Subsets and supersets
1031edo is the 273rd prime edo. 3093edo, which triples it, gives a good correction to its harmonic 11.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1634 1031⟩ | [⟨1031 1634]] | 0.0354 | 0.0354 | 3.04 |
| 2.3.5 | [38 -2 -15⟩, [-38 65 -28⟩ | [⟨1031 1634 2394]] | 0.0082 | 0.0481 | 4.13 |
| 2.3.5.7 | 420175/419904, 5250987/5242880, 152946081792/152587890625 | [⟨1031 1634 2394 2894]] | 0.0458 | 0.0774 | 6.65 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 166\1031 | 193.210 | 262144/234375 | Luna |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct