45edo: Difference between revisions

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=== 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.

=== 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 ==

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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 ==

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== 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]]

* [https://www.youtube.com/shorts/33tKBiWZvXM ''(short clip) Fantasy in 45edo''] (2025)

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* [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|##]]

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[[Category:Quartismic]]

[[Category:Listen]]

~~[[Category:Todo:~~add rank 2 temperaments table]]

@@ Line 17: / Line 17: @@
 === 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.
+=== Comparison with [[27edo]] ===
+edo 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 ==
@@ Line 624: / Line 633: @@
 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 ==
@@ Line 637: / Line 648: @@
 == 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]]
 * [https://www.youtube.com/shorts/33tKBiWZvXM ''(short clip) Fantasy in 45edo''] (2025)
@@ Line 652: / Line 668: @@
 * [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 -->
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 [[Category:Quartismic]]
 [[Category:Listen]]
-[[Category:Todo:add rank 2 temperaments table]]
+{{todo|add rank 2 temperaments table|add scales list}}

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