45edo: Difference between revisions

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== Theory ==

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 [[3/1|3]], [[5/1|5]], and [[7/1|7]] all flat, ~~but the~~ [[11/1|11]] is sharp.

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 ===

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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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| 5

| 133.3

| [[13/12]], [[14/13]], ''[[16/15]]''

| [[13/12]], [[14/13]], [[27/25]], ''[[16/15]]''

| Minor 2nd

| m2

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| 40

| 1066.7

| [[13/7]], [[24/13]], ''[[15/8]]''

| [[13/7]], [[24/13]], [[50/27]], ''[[15/8]]''

| Major 7th

| M7

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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.

== 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 ==

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| [[Pentadecal]]

|}

<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== 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.

[[71edt]] is notable as an octave-stretched version of 45edo for having an extremely accurate 3/1, 5/1 and 11/1.

== Scales ==

* [[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

* 12-tone 5&9edo scale{{idio}}: 5 4 1 5 3 2 5 2 3 5 5 5

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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)

* [https://www.youtube.com/watch?v=Xblr-4aGBtM ''<nowiki>Twin Arrows [45edo]</nowiki>''] (2026)

; [[JUMBLE]]

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* [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=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|##]]

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[[Category:Quartismic]]

[[Category:Listen]]

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

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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 [[3/1|3]], [[5/1|5]], and [[7/1|7]] all flat, but the [[11/1|11]] is sharp.
+edo 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 ===
@@ Line 16: / Line 16: @@
 === 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 63: / Line 72: @@
 | 5
 | 133.3
-| [[13/12]], [[14/13]], ''[[16/15]]''
+| [[13/12]], [[14/13]], [[27/25]], ''[[16/15]]''
 | Minor 2nd
 | m2
@@ Line 308: / Line 317: @@
 | 40
 | 1066.7
-| [[13/7]], [[24/13]], ''[[15/8]]''
+| [[13/7]], [[24/13]], [[50/27]], ''[[15/8]]''
 | Major 7th
 | M7
@@ Line 399: / Line 408: @@
 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 ==
@@ Line 614: / Line 627: @@
 | [[Pentadecal]]
 |}
-<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Octave stretch and compression ==
-edo'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.
+edo'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.
+[[71edt]] is notable as an octave-stretched version of 45edo for having an extremely accurate 3/1, 5/1 and 11/1.
 == Scales ==
 * [[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
 * 12-tone 5&9edo scale{{idio}}: 5 4 1 5 3 2 5 2 3 5 5 5
@@ Line 632: / 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)
+* [https://www.youtube.com/watch?v=Xblr-4aGBtM ''<nowiki>Twin Arrows [45edo]</nowiki>''] (2026)
 ; [[JUMBLE]]
@@ Line 642: / Line 664: @@
 * [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=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 -->
@@ Line 648: / Line 677: @@
 [[Category:Quartismic]]
 [[Category:Listen]]
-[[Category:Todo:add rank 2 temperaments table]]
+{{todo|add rank 2 temperaments table|add scales list}}

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