769edo: 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 distinct | <nowiki />* [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Music == | |||
; [[Francium]] | |||
* "Is Madness Sexy?" from ''Questions, Vol. 2'' (2025) − [https://open.spotify.com/track/744opccKPUbk721EsD6Hoa Spotify] | [https://francium223.bandcamp.com/track/is-madness-sexy Bandcamp] | [https://www.youtube.com/watch?v=8EZbxtTIukU YouTube] − in Newt, 769edo tuning | |||
Latest revision as of 11:31, 12 April 2025
| ← 768edo | 769edo | 770edo → |
769 equal divisions of the octave (abbreviated 769edo or 769ed2), also called 769-tone equal temperament (769tet) or 769 equal temperament (769et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 769 equal parts of about 1.56 ¢ each. Each step represents a frequency ratio of 21/769, or the 769th root of 2.
Theory
769edo is consistent to the 9-odd-limit, despite of a large error in its harmonic 5. As an equal temperament, it tempers out 2401/2400, 359661568/358722675 and [8 14 -13 0⟩ in the 7-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.256 | +0.682 | +0.225 | +0.511 | -0.473 | +0.565 | -0.622 | -0.404 | +0.536 | +0.480 | +0.594 |
| Relative (%) | +16.4 | +43.7 | +14.4 | +32.8 | -30.3 | +36.2 | -39.9 | -25.9 | +34.4 | +30.8 | +38.1 | |
| Steps (reduced) |
1219 (450) |
1786 (248) |
2159 (621) |
2438 (131) |
2660 (353) |
2846 (539) |
3004 (697) |
3143 (67) |
3267 (191) |
3378 (302) |
3479 (403) | |
Subsets and supersets
769edo is the 136th prime edo. 1538edo, which doubles it, gives a good correction to the harmonic 5.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1219 -769⟩ | [⟨769 1219]] | −0.0807 | 0.0806 | 5.17 |
| 2.3.5 | [8 14 -13⟩, [103 -43 -15⟩ | [⟨769 1219 1786]] | −0.1517 | 0.1202 | 7.70 |
| 2.3.5.7 | 2401/2400, 359661568/358722675, 1224440064/1220703125 | [⟨769 1219 1786 2159]] | −0.1338 | 0.1086 | 6.96 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 202\769 | 315.215 | 6/5 | Parakleismic |
| 1 | 220\769 | 343.303 | 8000/6561 | Raider |
| 1 | 225\769 | 351.105 | 49/40 | Newt |
| 1 | 334\769 | 521.196 | 875/648 | Maviloid |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct