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== Theory ==
== 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 ===
=== 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 ==
{| class="wikitable mw-collapsible mw-collapsed right-all center-3 left-6 center-7"
{| class="wikitable center-1 right-2 center-5 center-6"
|-
! rowspan="2" | Step #
! ET
! colspan="2" | Just (JI)
! rowspan="2" | Error<br>(ET−JI)
! colspan="4" rowspan="2" | [[Ups and downs notation]]
|-
|-
! #
! Cents
! Cents
! Interval
! Approximate ratios*
! Cents
! colspan="4" | [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
| 0.000
| 0.000
| Perfect Unison
| Perfect Unison
| P1
| P1
Line 41: Line 43:
|-
|-
| 1
| 1
| 26.666
| 26.7
| [[65/64]]
| [[49/48]], [[50/49]]
| 26.841
| -0.174
| Up unison
| Up unison
| ^1
| ^1
Line 50: Line 50:
|-
|-
| 2
| 2
| 53.333
| 53.3
| [[33/32]]
| [[36/35]], ''[[25/24]]'', ''[[64/63]]''
| 53.273
| 0.060
| Augmented Unison
| Augmented Unison
| A1
| A1
Line 59: Line 57:
|-
|-
| 3
| 3
| 80.000
| 80.0
| [[22/21]]
| [[21/20]]
| 80.537
| -0.537
| Diminished 2nd
| Diminished 2nd
| d2
| d2
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|-
|-
| 4
| 4
| 106.666
| 106.7
| [[17/16]]
| [[15/14]]
| 104.955
| 1.711
| Downminor 2nd
| Downminor 2nd
|vm2
| vm2
| vEb
| vEb
|-
|-
| 5
| 5
| 133.333
| 133.3
| [[27/25]]
| [[13/12]], [[14/13]], [[27/25]], ''[[16/15]]''
| 133.238
| 0.095
| Minor 2nd
| Minor 2nd
| m2
| m2
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|-
|-
| 6
| 6
| 160.000
| 160.0
| [[11/10]]
| [[54/49]]
| 165.004
| -5.004
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 95: Line 85:
|-
|-
| 7
| 7
| 186.666
| 186.7
| [[10/9]]
| [[10/9]], ''[[9/8]]''
| 182.404
| 4.262
| Major 2nd
| Major 2nd
| M2
| M2
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|-
|-
| 8
| 8
| 213.333
| 213.3
| [[9/8]]
|  
| 203.910
| 9.423
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 113: Line 99:
|-
|-
| 9
| 9
| 240.000
| 240.0
| [[8/7]]
| [[8/7]], [[15/13]]
| 231.174
| 8.826
| Augmented 2nd
| Augmented 2nd
| A2
| A2
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|-
|-
| 10
| 10
| 266.666
| 266.7
| [[7/6]]
| [[7/6]]
| 266.871
| -0.205
| Diminished 3rd
| Diminished 3rd
| d3
| d3
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|-
|-
| 11
| 11
| 293.333
| 293.3
| [[32/27]]
| [[25/21]]
| 294.135
| -0.802
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
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|-
|-
| 12
| 12
| 320.000
| 320.0
| [[6/5]]
| [[6/5]]
| 315.641
| 4.359
| Minor 3rd
| Minor 3rd
| m3
| m3
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|-
|-
| 13
| 13
| 346.666
| 346.7
| [[11/9]]
| [[49/40]], [[60/49]]
| 347.408
| -0.741
| Mid 3rd
| Mid 3rd
| ~3
| ~3
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|-
|-
| 14
| 14
| 373.333
| 373.3
| [[5/4]]
| [[5/4]], [[26/21]], ''[[16/13]]''
| 386.314
| -12.980
| Major 3rd
| Major 3rd
| M3
| M3
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|-
|-
| 15
| 15
| 400.000
| 400.0
| [[63/50]]
| [[63/50]]
| 400.108
| -0.108
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
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|-
|-
| 16
| 16
| 426.666
| 426.7
| [[9/7]]
| [[9/7]]
| 435.084
| -8.418
| Augmented 3rd
| Augmented 3rd
| A3
| A3
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|-
|-
| 17
| 17
| 453.333
| 453.3
| [[13/10]]
| [[13/10]], ''[[21/16]]''
| 454.294
| -0.961
| Diminished 4th
| Diminished 4th
| d4
| d4
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|-
|-
| 18
| 18
| 480.000
| 480.0
| [[21/16]]
| ''[[64/49]]''
| 470.781
| 9.219
| Down 4th
| Down 4th
| v4
| v4
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|-
|-
| 19
| 19
| 506.666
| 506.7
| [[4/3]]
| [[4/3]]
| 498.045
| 8.622
| Perfect 4th
| Perfect 4th
| P4
| P4
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|-
|-
| 20
| 20
| 533.333
| 533.3
| [[49/36]]
| [[49/36]]
| 533.742
| -0.409
| Up 4th or Mid 4th
| Up 4th or Mid 4th
| ^4, ~4
| ^4, ~4
Line 221: Line 183:
|-
|-
| 21
| 21
| 560.000
| 560.0
| [[18/13]]
| [[18/13]]
| 563.382
| -3.382
| Augmented 4th
| Augmented 4th
| A4
| A4
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|-
|-
| 22
| 22
| 586.666
| 586.7
| [[7/5]]
| [[7/5]]
| 582.512
| 4.155
| Upaugmented 4th
| Upaugmented 4th
| ^A4
| ^A4
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|-
|-
| 23
| 23
| 613.333
| 613.3
| [[10/7]]
| [[10/7]]
| 617.488
| -4.155
| Downdiminshed 5th
| Downdiminshed 5th
| vd5
| vd5
Line 248: Line 204:
|-
|-
| 24
| 24
| 640.000
| 640.0
| [[13/9]]
| [[13/9]]
| 636.618
| 3.382
| Diminished 5th
| Diminished 5th
| d5
| d5
Line 257: Line 211:
|-
|-
| 25
| 25
| 666.666
| 666.7
| [[72/49]]
| [[72/49]]
| 666.258
| 0.409
| Down 5th or Mid 5th
| Down 5th or Mid 5th
| v5, ~5
| v5, ~5
Line 266: Line 218:
|-
|-
| 26
| 26
| 693.333
| 693.3
| [[3/2]]
| [[3/2]]
| 701.955
| -8.622
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 275: Line 225:
|-
|-
| 27
| 27
| 720.000
| 720.0
| [[32/21]]
| ''[[49/32]]''
| 729.219
| -9.219
| Up 5th
| Up 5th
| ^5
| ^5
Line 284: Line 232:
|-
|-
| 28
| 28
| 746.666
| 746.7
| [[20/13]]
| [[20/13]], ''[[32/21]]''
| 745.786
| 0.961
| Augmented 5th
| Augmented 5th
| A5
| A5
Line 293: Line 239:
|-
|-
| 29
| 29
| 773.333
| 773.3
| [[14/9]]
| [[14/9]]
| 764.916
| 8.418
| Diminished 6th
| Diminished 6th
| d6
| d6
Line 302: Line 246:
|-
|-
| 30
| 30
| 800.000
| 800.0
| [[100/63]]
| [[100/63]]
| 799.892
| 0.108
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 311: Line 253:
|-
|-
| 31
| 31
| 826.666
| 826.7
| [[8/5]]
| [[8/5]], [[21/13]], ''[[13/8]]''
| 813.686
| 12.980
| Minor 6th
| Minor 6th
| m6
| m6
Line 320: Line 260:
|-
|-
| 32
| 32
| 853.333
| 853.3
| [[18/11]]
| [[49/30]], [[80/49]]
| 852.592
| 0.741
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 329: Line 267:
|-
|-
| 33
| 33
| 880.000
| 880.0
| [[5/3]]
| [[5/3]]
| 884.359
| -4.359
| Major 6th
| Major 6th
| M6
| M6
Line 338: Line 274:
|-
|-
| 34
| 34
| 906.666
| 906.7
| [[27/16]]
| [[42/25]]
| 905.865
| 0.802
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 347: Line 281:
|-
|-
| 35
| 35
| 933.333
| 933.3
| [[12/7]]
| [[12/7]]
| 933.129
| 0.205
| Augmented 6th
| Augmented 6th
| A6
| A6
Line 356: Line 288:
|-
|-
| 36
| 36
| 960.000
| 960.0
| [[7/4]]
| [[7/4]], [[26/15]]
| 968.826
| -8.826
| Diminished 7th
| Diminished 7th
| d7
| d7
Line 365: Line 295:
|-
|-
| 37
| 37
| 986.666
| 986.7
| [[16/9]]
|  
| 996.089
| -9.423
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 374: Line 302:
|-
|-
| 38
| 38
| 1013.333
| 1013.3
| [[9/5]]
| [[9/5]], ''[[16/9]]''
| 1017.596
| -4.262
| Minor 7th
| Minor 7th
| m7
| m7
Line 383: Line 309:
|-
|-
| 39
| 39
| 1040.000
| 1040.0
| [[20/11]]
| [[49/27]]
| 1034.996
| 5.004
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 392: Line 316:
|-
|-
| 40
| 40
| 1066.666
| 1066.7
| [[50/27]]
| [[13/7]], [[24/13]], [[50/27]], ''[[15/8]]''
| 1066.762
| -0.095
| Major 7th
| Major 7th
| M7
| M7
Line 401: Line 323:
|-
|-
| 41
| 41
| 1093.333
| 1093.3
| [[32/17]]
| [[28/15]]
| 1095.044
| -1.711
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
Line 410: Line 330:
|-
|-
| 42
| 42
| 1120.000
| 1120.0
| [[21/11]]
| [[40/21]]
| 1119.463
| 0.537
| Augmented 7th
| Augmented 7th
| A7
| A7
Line 419: Line 337:
|-
|-
| 43
| 43
| 1146.666
| 1146.7
| [[64/33]]
| [[35/18]], ''[[48/25]]'', ''[[63/32]]''
| 1146.727
| -0.060
| Diminished 8ve
| Diminished 8ve
| d8
| d8
Line 428: Line 344:
|-
|-
| 44
| 44
| 1173.333
| 1173.3
| [[128/65]]
| [[49/25]], [[96/49]]
| 1173.158
| 0.174
| Down 8ve
| Down 8ve
| v8
| v8
Line 437: Line 351:
|-
|-
| 45
| 45
| 1200.000
| 1200.0
| [[2/1]]
| [[2/1]]
| 1200.000
| 0.000
| Perfect Octave
| Perfect Octave
| P8
| P8
| D
| D
|}
|}
<nowiki/>* As a 2.3.5.7.13-subgroup temperament, using the 45f val


== Notation ==
== Notation ==
Line 495: 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.
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 ==
Line 710: Line 627:
| [[Pentadecal]]
| [[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 ==
== 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
Line 728: Line 648:


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


; [[JUMBLE]]
; [[JUMBLE]]
Line 738: Line 664:
* [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]]
[[Category:Todo:add rank 2 temperaments table]]
{{todo|add rank 2 temperaments table|add scales list}}
Retrieved from "https://en.xen.wiki/w/45edo"