769edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|769}} == Theory == 769edo is consistent to the 9-odd-limit, despite of a large error in its harmonic 5. As an equal temperament..." |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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== 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 | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |Optimal<br>8ve | ! rowspan="2" | [[Mapping]] | ||
! colspan="2" |Tuning | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
|- | ! colspan="2" | Tuning error | ||
![[TE error|Absolute]] (¢) | |- | ||
![[TE simple badness|Relative]] (%) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo|1219 -769}} | | {{monzo|1219 -769}} | ||
| {{mapping|769 1219}} | | {{mapping|769 1219}} | ||
| | | −0.0807 | ||
| 0.0806 | | 0.0806 | ||
| 5.17 | | 5.17 | ||
| Line 32: | Line 33: | ||
| {{monzo|8 14 -13}}, {{monzo|103 -43 -15}} | | {{monzo|8 14 -13}}, {{monzo|103 -43 -15}} | ||
| {{mapping|769 1219 1786}} | | {{mapping|769 1219 1786}} | ||
| | | −0.1517 | ||
| 0.1202 | | 0.1202 | ||
| 7.70 | | 7.70 | ||
| Line 39: | Line 40: | ||
| 2401/2400, 359661568/358722675, 1224440064/1220703125 | | 2401/2400, 359661568/358722675, 1224440064/1220703125 | ||
| {{mapping|769 1219 1786 2159}} | | {{mapping|769 1219 1786 2159}} | ||
| | | −0.1338 | ||
| 0.1086 | | 0.1086 | ||
| 6.96 | | 6.96 | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 77: | Line 79: | ||
| [[Maviloid]] | | [[Maviloid]] | ||
|} | |} | ||
<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