659edo: 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" |[[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]] (%) | |||
|- | |- | ||
| 2.9 | | 2.9 | ||
| {{monzo|2089 -659}} | | {{monzo|2089 -659}} | ||
| {{mapping|659 2089}} | | {{mapping|659 2089}} | ||
| | | −0.0056 | ||
| 0.0056 | | 0.0056 | ||
| 1.02 | | 1.02 | ||
| Line 53: | Line 54: | ||
| 1575/1573, 4096/4095, 86625/86528, 31250/31213, 650000/649539 | | 1575/1573, 4096/4095, 86625/86528, 31250/31213, 650000/649539 | ||
| {{mapping|659 2089 1530 1850 2280 2439}} | | {{mapping|659 2089 1530 1850 2280 2439}} | ||
| | | −0.0313 | ||
| 0.1042 | | 0.1042 | ||
| 18.97 | | 18.97 | ||
|} | |} | ||
== Music == | |||
; [[Francium]] | |||
* "In My Wildest Dreams" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/1RQsBF3qi9n13jVfCUgbN7 Spotify] | [https://francium223.bandcamp.com/track/in-my-wildest-dreams Bandcamp] | [https://www.youtube.com/watch?v=NDciaAdI44g YouTube] | |||
Latest revision as of 13:02, 21 February 2025
| ← 658edo | 659edo | 660edo → |
659 equal divisions of the octave (abbreviated 659edo or 659ed2), also called 659-tone equal temperament (659tet) or 659 equal temperament (659et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 659 equal parts of about 1.82 ¢ each. Each step represents a frequency ratio of 21/659, or the 659th root of 2.
Theory
659edo is consistent to the 7-odd-limit and its harmonic 3 is about halfway its steps. Using the 2.9.5.7.11.17.23.31 subgroup, it tempers out 1225/1224, 2025/2024, 5832/5831, 3520/3519, 3969/3968, 790625/790272 and 1740800/1740123. It supports counterultrakleismic.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.893 | -0.274 | -0.085 | +0.035 | +0.427 | +0.747 | +0.654 | +0.659 | -0.700 | +0.843 | -0.050 |
| Relative (%) | -49.0 | -15.1 | -4.7 | +1.9 | +23.5 | +41.0 | +35.9 | +36.2 | -38.4 | +46.3 | -2.7 | |
| Steps (reduced) |
1044 (385) |
1530 (212) |
1850 (532) |
2089 (112) |
2280 (303) |
2439 (462) |
2575 (598) |
2694 (58) |
2799 (163) |
2895 (259) |
2981 (345) | |
Subsets and supersets
659edo is the 120th prime EDO. 1318edo, 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 | [2089 -659⟩ | [⟨659 2089]] | −0.0056 | 0.0056 | 1.02 |
| 2.9.5 | [10 -20 23⟩, [-83 13 18⟩ | [⟨659 2089 1530]] | +0.0357 | 0.0585 | 10.65 |
| 2.9.5.7 | 420175/419904, 703125/702464, [44 -14 5 -4⟩ | [⟨659 2089 1530 1850]] | +0.0343 | 0.0507 | 9.23 |
| 2.9.5.7.11 | 496125/495616, 420175/419904, 151263/151250, 2097152/2096325 | [⟨659 2089 1530 1850 2280]] | +0.0028 | 0.0777 | 14.15 |
| 2.9.5.7.11.13 | 1575/1573, 4096/4095, 86625/86528, 31250/31213, 650000/649539 | [⟨659 2089 1530 1850 2280 2439]] | −0.0313 | 0.1042 | 18.97 |