45edo: Difference between revisions
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== Theory == | == Theory == | ||
45edo effectively has two approximate [[5/4 | 45edo effectively has two approximate [[5/4]] major thirds, each almost equally far from just, but the flat one is slightly closer. Combined with a [[3/2|perfect fifth]] 8.6 cents flat of just, it can be used as a [[meantone]] tuning, forming a good approximation to [[2/5-comma meantone]] (in fact falling into the [[flattone]] range). It is a flat-tending system in the [[7-limit]], with harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] all flat. However, harmonics [[11/1|11]] and [[13/1|13]] are sharp, but this can be fixed with the 45ef val. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 45 factors into primes as {{nowrap| 3<sup>2</sup> × 5 }}, 45edo has subset edos {{EDOs| 3, 5, 9, and 15 }}. [[135edo]], which triples it, corrects its primes 3, 7, and 11 to near-just qualities, and 270edo offers even more. | Since 45 factors into primes as {{nowrap| 3<sup>2</sup> × 5 }}, 45edo has subset edos {{EDOs| 3, 5, 9, and 15 }}. [[135edo]], which triples it, corrects its primes 3, 7, and 11 to near-just qualities, and 270edo offers even more. | ||
=== Comparison with [[27edo]] === | |||
45edo can be thought of as a mirror twin of [[27edo]] in the [[7-limit]]. In 27edo, harmonics 3, 5, and 7 are all noticeably sharp; in 45edo, they are all flat, with almost the same absolute errors pointing in the opposite direction. | |||
Both tunings inherit [[9edo]]’s remarkably accurate approximation of [[7/6]]: 266.667 cents, only about 0.204 cents flat. Consequently, harmonics 3 and 7 have almost identical tuning errors within each system: nearly the same sharpness in 27edo, and nearly the same flatness in 45edo. Their shared displacement preserves an almost pure relationship between them. | |||
An especially interesting detail is that, in both tunings, the error of harmonic 5 is almost exactly 50% greater in magnitude than the error of harmonic 3. When harmonic 9 is constructed consistently as 3 × 3, its error is twice that of 3. This places the error of 5 almost exactly halfway between those of 3 and 9. In 27edo, these errors are approximately +9.16, +13.69, and +18.31 cents; in 45edo, they are −8.62, −12.98, and −17.24 cents. In both cases, the midpoint relationship is accurate to within 0.05 cents. | |||
Musically, this offers an explanation for why the 5 can blend so well into the 1:3:5:7:9 chord in both tunings: its deviation fits naturally into the pattern established by 3 and 9, while 7 closely follows 3. The chord takes on a strongly stretched character in 27edo and a strongly compressed character in 45edo. | |||
== Intervals == | == Intervals == | ||
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In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO. | In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO. | ||
== Approximation to JI== | |||
=== Interval mappings === | |||
{{Q-odd-limit intervals|45}}{{Q-odd-limit intervals|44.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 45ef val mapping}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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== Octave stretch and compression == | == Octave stretch and compression == | ||
45edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by a [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed12|161ed12]] or [[ed6|116ed6]]. The trade-off is a slightly worse 2/1. | 45edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by a [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed12|161ed12]] or [[ed6|116ed6]]. The trade-off is a slightly worse 2/1. [[207zpi]] also improves on all of those harmonics except for 17/1. | ||
The tuning [[equal tuning|183ed17]] may also be used, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1 (with different mappings for many) but at the cost of a noticeably worse 2/1 than the others. | The tuning [[equal tuning|183ed17]] may also be used, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1 (with different mappings for many) but at the cost of a noticeably worse 2/1 than the others. | ||
[[71edt]] is notable as an octave-stretched version of 45edo for having an extremely accurate 3/1, 5/1 and 11/1. | |||
== Scales == | == Scales == | ||
* [[Cloudtone]][10] - recommended by [[Maeve Gutierrez]]: 8 1 8 1 8 1 8 1 8 1 | * [[Cloudtone]][10] - recommended by [[Maeve Gutierrez]]: 8 1 8 1 8 1 8 1 8 1 | ||
* [[JUMBLE]]'s "moment of chaos scale": 3 9 6 1 4 7 2 5 8 (used in several works including [https://www.youtube.com/watch?v=WqEOi4cd1Og ''Archipelago Arpeggio''] and [https://www.youtube.com/watch?v=4iwJFVIWEII ''FERAL (45edo microtonal ambient track)'']) | |||
* 13-tone 5&9edo scale: 5 4 1 5 3 2 5 2 3 5 1 4 5 | * 13-tone 5&9edo scale: 5 4 1 5 3 2 5 2 3 5 1 4 5 | ||
* 12-tone 5&9edo scale{{idio}}: 5 4 1 5 3 2 5 2 3 5 5 5 | * 12-tone 5&9edo scale{{idio}}: 5 4 1 5 3 2 5 2 3 5 5 5 | ||
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== Music == | == Music == | ||
=== Modern renderings === | |||
; {{W|Brakence}} (Randall Todd Findell) | |||
* [https://www.youtube.com/shorts/sXIMdX5-9UA ''Crystallized''] (2026) — microtonal cover in 45edo by [[Bryan Deister]] (2026) | |||
=== 21st century === | |||
; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/shorts/33tKBiWZvXM ''(short clip) Fantasy in 45edo''] (2025) | * [https://www.youtube.com/shorts/33tKBiWZvXM ''(short clip) Fantasy in 45edo''] (2025) | ||
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* [https://www.youtube.com/watch?v=24gnhAbHtiw ''Qúchze úzeq Qávka''] (2025) | * [https://www.youtube.com/watch?v=24gnhAbHtiw ''Qúchze úzeq Qávka''] (2025) | ||
* [https://www.youtube.com/watch?v=K2p7HOI3TUE ''Sodium Light (45edo Microtonal Chillwave)''] (2026) | * [https://www.youtube.com/watch?v=K2p7HOI3TUE ''Sodium Light (45edo Microtonal Chillwave)''] (2026) | ||
* [https://www.youtube.com/watch?v=ex9WfmWVibY ''Yēú Zee Kiidhai (45edo microtonal ambient)''] (2026) | |||
* [https://www.youtube.com/watch?v=4iwJFVIWEII ''FERAL (45edo microtonal ambient track)''] (2026) | |||
* [https://www.youtube.com/watch?v=cXZ3RkTDE-I ''Chmelui-Múzeq - Haoýoze (45edo Microtonal Ambient)''] (2026) | |||
* [https://www.youtube.com/watch?v=ZK_Obi5B9bQ ''Chmelui-Múzeq - Luásoidq (45edo microtonal ambient)''] (2026) | |||
* [https://www.youtube.com/watch?v=aJjGdwolEOY ''Chmelui-Múzeq - Qidloloba (45edo microtonal dark ambient)''] (2026) | |||
* [https://www.youtube.com/watch?v=ItyazeyLfW8 ''JOYRIDE! (Funky 45edo Driving Music)''] (2026) | |||
* [https://www.youtube.com/watch?v=3ejEWLb-KMY ''may you live in boring times''] (2026) | |||
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number --> | [[Category:Equal divisions of the octave|##]] <!-- 2-digit number --> | ||
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[[Category:Quartismic]] | [[Category:Quartismic]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||
{{todo|add rank 2 temperaments table|add scales list}} | |||