User:Francium/1699edo: Difference between revisions
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<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | <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]] | |||
* "STOP ACTING SO SPECIAL YOU'RE A DO SI DO" from ''Things to Scream at People'' (2025) – [https://open.spotify.com/track/3FZFZ5lBnLL0jZAv3ndhwC Spotify] | [https://francium223.bandcamp.com/track/stop-acting-so-special-youre-a-do-si-do Bandcamp] | [https://www.youtube.com/watch?v=se9MXY6UHr0 YouTube] – in Tridecimal Olympic, 1699edo tuning | |||
Latest revision as of 12:18, 20 October 2025
| ← 1698edo | 1699edo | 1700edo → |
1699 equal divisions of the octave (abbreviated 1699edo or 1699ed2), also called 1699-tone equal temperament (1699tet) or 1699 equal temperament (1699et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1699 equal parts of about 0.706 ¢ each. Each step represents a frequency ratio of 21/1699, or the 1699th root of 2.
Theory
1699edo is consistent to the 17-limit, tempering out 4375/4374, 131072/130977, 820125/819896 and 3294225/3294172 in the 11-limit, 2080/2079, 4096/4095, 4375/4374, 225000/224939 and 3132421875/3132286976 in the 13-limit; and 2080/2079, 4096/4095, 4375/4374, 2500/2499, 31213/31212 and 224939/224910 in the 17-limit. As an equal temperament, it supports tridecimal olympic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.105 | +0.031 | +0.215 | +0.301 | -0.033 | +0.283 | -0.162 | +0.331 | +0.205 | -0.127 |
| Relative (%) | +0.0 | +14.9 | +4.4 | +30.4 | +42.6 | -4.7 | +40.1 | -22.9 | +46.8 | +29.0 | -18.0 | |
| Steps (reduced) |
1699 (0) |
2693 (994) |
3945 (547) |
4770 (1372) |
5878 (781) |
6287 (1190) |
6945 (149) |
7217 (421) |
7686 (890) |
8254 (1458) |
8417 (1621) | |
Subsets and supersets
1699edo is the 266th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2693 -1699⟩ | [⟨1699 2693]] | −0.0331 | 0.0331 | 4.69 |
| 2.3.5 | [77 -31 -12⟩, [-13 -46 37⟩ | [⟨1699 2693 3945]] | −0.0266 | 0.0286 | 4.05 |
| 2.3.5.7 | 4375/4374, 52734375/52706752, [72 -21 -7 -8⟩ | [⟨1699 2693 3945 4770]] | −0.0390 | 0.0329 | 4.66 |
| 2.3.5.7.11 | 4375/4374, 131072/130977, 820125/819896, 3294225/3294172 | [⟨1699 2693 3945 4770 5878]] | −0.0486 | 0.0351 | 4.97 |
| 2.3.5.7.11.13 | 2080/2079, 4096/4095, 4375/4374, 225000/224939, 3132421875/3132286976 | [⟨1699 2693 3945 4770 5878 6287]] | −0.0390 | 0.0386 | 5.47 |
| 2.3.5.7.11.13.17 | 2080/2079, 4096/4095, 4375/4374, 2500/2499, 31213/31212, 224939/224910 | [⟨1699 2693 3945 4770 5878 6287 6945]] | −0.0433 | 0.0372 | 5.27 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 263\1699 | 185.756 | [24 4 -13⟩ | Pirate |
| 1 | 366\1699 | 258.505 | [-32 13 5⟩ | Lafa |
| 1 | 616\1699 | 435.079 | 9/7 | Supermajor |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct