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'''45edo''' divides the octave into 45 equal parts of 26.667 [[cent]]s. It has two major thirds, each almost equally far from Just, but as the flat one is slightly closer, it qualifies as a meantone temperament, forming a good approximation to 2/5 comma meantone. It is the [[optimal patent val]] for [[flattone]] temperament, the [[7-limit]] 525/512 planar [[Avicennmic_temperaments|avicennmic]] temperament, the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank four temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is a flat-tending system in the 7-limit, with 3, 5 and 7 all flat, but the 11 is sharp. Also supports messed-up [[ennealimmal]], if you want. Since 45 is a multiple of 5 and 9, it can be used to model [[Indonesian]] music in both [[wikipedia:Slendro|Slendro]] (~ 5edo) and [[wikipedia:Pelog|Pelog]] (~ modes of 9edo) tunings.
{{Infobox ET}}
{{ED intro}}


[[Category:Edo]]
== Theory ==
[[Category: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 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.


{| class="wikitable center-all"
=== Odd harmonics ===
{{Harmonics in equal|45}}
 
=== As a tuning of other temperaments ===
It tempers out [[81/80]], [[525/512]], [[875/864]], and [[3125/3087]] in the 7-limit, and [[45/44]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7- and 11-limit flattone temperament, and the 45f val is an excellent tuning for [[13-limit]] flattone. It also provides the optimal patent val for the 7-limit rank-3 [[avicennmic]] temperament, [[tempering out]] [[525/512]], the 11-limit [[calliope]] temperament, tempering out [[45/44]] and [[81/80]], and the rank-4 temperament tempering out 45/44. It is also the unique equal temperament tuning whose [[patent val]] tempers out both the syntonic comma and the [[ennealimma]].
 
45edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.7/3.33-subgroup [[The Quartercache #Direct quartismic|direct quartismic]] temperament, in which it approximates the [[33/32]] quartertone with 2 steps and [[7/6]] with 10 steps. A bit more broadly, it maps the 2.27.25.63.33.65.17 subgroup to great precision; this is the part of the [[17-limit]] shared with [[270edo]].
 
Otherwise, it can be treated as a 2.5/3.7/3-subgroup system (borrowing 5/3 from [[15edo]] and 7/3 from [[9edo]]) and is a good tuning for [[gariberttet]], defined by tempering out [[3125/3087]] in this subgroup, approximating 2/5-comma gariberttet.
 
=== 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.
 
== Intervals ==
{| class="wikitable center-1 right-2 center-5 center-6"
|-
|-
| rowspan="2" | Step #
! #
| style="text-align:center;" | ET
! Cents
| colspan="2" | Just
! Approximate ratios*
| Difference <br> (ET minus Just)
! colspan="4" | [[Ups and downs notation]]
| colspan="3" | [[Ups and Downs Notation]]
|-
| Cents
| Interval
| Cents
|
|
|
|
|-
|-
| 0
| 0
| 0¢
| 0.0
|1:1
| [[1/1]]
| 0
| Perfect Unison
|0
| P1
| unison
| 1
| D
| D
|-
|-
| 1
| 1
| 26.666
| 26.7
| 65:64
| [[49/48]], [[50/49]]
|26.841
| Up unison
| -0.174
| ^1
|  
| ^D
|  
| D^
|-
|-
| 2
| 2
| 53.333
| 53.3
|33:32
| [[36/35]], ''[[25/24]]'', ''[[64/63]]''
|53.273
| Augmented Unison
|0.060
| A1
|
| D#
|
|D#
|-
|-
| 3
| 3
|80.0
| 80.0
|22:21
| [[21/20]]
|80.537
| Diminished 2nd
| -0.537
| d2
|
| Ebb
|
|D#^
|-
|-
| 4
| 4
|106.666
| 106.7
|17:16
| [[15/14]]
|104.955
| Downminor 2nd
|1.7112
| vm2
|
| vEb
|
|Ebv
|-
|-
| 5
| 5
| 133.333
| 133.3
|27:25
| [[13/12]], [[14/13]], [[27/25]], ''[[16/15]]''
|133.238
| Minor 2nd
|0.095
| m2
|
| Eb
|
|Eb
|-
|-
| 6
| 6
|160
| 160.0
|11:10
| [[54/49]]
|165.004
| Mid 2nd
|<nowiki>-5.004</nowiki>
| ~2
|
| vE
|
|Ev
|-
|-
| 7
| 7
|186.666
| 186.7
|10:9
| [[10/9]], ''[[9/8]]''
|182.404
| Major 2nd
|4.262
| M2
|
| E
|
|E
|-
|-
| 8
| 8
|213.333
| 213.3
|9:8
|  
|203.910
| Upmajor 2nd
|9.423
| ^M2
|
| ^E
|
|E^
|-
|-
| 9
| 9
|240
| 240.0
|8:7
| [[8/7]], [[15/13]]
|231.174
| Augmented 2nd
|8.826
| A2
|
| E#
|
|E#
|-
|-
| 10
| 10
|266.666
| 266.7
|7:6
| [[7/6]]
|266.871
| Diminished 3rd
|<nowiki>-0.205</nowiki>
| d3
|
| Fb
|
|Fb
|-
|-
| 11
| 11
|293.333
| 293.3
|32:27
| [[25/21]]
|294.135
| Downminor 3rd
| -0.802
| vm3
|
| vF
|
|Fv
|-
|-
| 12
| 12
|320
| 320.0
|6:5
| [[6/5]]
|315.641
| Minor 3rd
|4.359
| m3
|
| F
|  
|F
|-
|-
| 13
| 13
|346.666
| 346.7
|11:9
| [[49/40]], [[60/49]]
|347.408
| Mid 3rd
| -0.741
| ~3
|
| ^F
|
|F^
|-
|-
| 14
| 14
|373.333
| 373.3
|5:4-
| [[5/4]], [[26/21]], ''[[16/13]]''
|386.314
| Major 3rd
|<nowiki>-12.980</nowiki>
| M3
|
| F#
|
|F#
|-
|-
| 15
| 15
|400
| 400.0
|5:4+
| [[63/50]]
|386.314
| Upmajor 3rd
|13.686
| ^M3
|
| ^F#
|
|F#^
|-
|-
| 16
| 16
|426.666
| 426.7
|9:7
| [[9/7]]
|435.084
| Augmented 3rd
| -8.418
| A3
|
| Fx
|
|Gbv
|-
|-
| 17
| 17
|453.333
| 453.3
|13:10
| [[13/10]], ''[[21/16]]''
|454.294
| Diminished 4th
| -0.961
| d4
|
| Gb
|
|Gb
|-
|-
| 18
| 18
|480
| 480.0
|21:16
| ''[[64/49]]''
|470.781
| Down 4th
|9.219
| v4
|
| vG
|
|Gv
|-
|-
| 19
| 19
|506.666
| 506.7
|4:3
| [[4/3]]
|498.045
| Perfect 4th
|8.622
| P4
|
| G
|
|G
|-
|-
| 20
| 20
|533.333
| 533.3
|49:36
| [[49/36]]
|533.742
| Up 4th or Mid 4th
| -0.409
| ^4, ~4
|
| ^G
|
|G^
|-
|-
| 21
| 21
|560
| 560.0
|18:13
| [[18/13]]
|
| Augmented 4th
|<nowiki>-3.382</nowiki>
| A4
|
| G#
|
|G#
|-
|-
| 22
| 22
|586.666
| 586.7
|7:5
| [[7/5]]
|582.512
| Upaugmented 4th
|4.155
| ^A4
|
| ^G#
|
|G#^
|-
|-
| 23
| 23
|613.333
| 613.3
|10:7
| [[10/7]]
|617.488
| Downdiminshed 5th
|<nowiki>-4.155</nowiki>
| vd5
|
| vAb
|
|Abv
|-
|-
| 24
| 24
|640
| 640.0
|13:9
| [[13/9]]
|
| Diminished 5th
|3.382
| d5
|
| Ab
|
|Ab
|-
|-
| 25
| 25
|666.666
| 666.7
|72:49
| [[72/49]]
|66.258
| Down 5th or Mid 5th
|0.409
| v5, ~5
|
| vA
|
|Av
|-
|-
| 26
| 26
|693.333
| 693.3
|3:2
| [[3/2]]
|701.955
| Perfect 5th
| -8.622
| P5
|
| A
|
|A
|-
|-
| 27
| 27
| 720
| 720.0
| style="text-align:center;" |32:21
| ''[[49/32]]''
|
| Up 5th
|<nowiki>-9.219</nowiki>
| ^5
|
| ^A
|
|A^
|-
|-
| 28
| 28
|746.666
| 746.7
| 20:13
| [[20/13]], ''[[32/21]]''
|  
| Augmented 5th
| 0.961
| A5
|
|
| A#
| A#
|-
|-
| 29
| 29
|773.333
| 773.3
| style="text-align:center;" |14:9
| [[14/9]]
|
| Diminished 6th
|8.418
| d6
|
| Bbb
|
|A#^
|-
|-
| 30
| 30
|800
| 800.0
| style="text-align:center;" |8:5-
| [[100/63]]
|
| Downminor 6th
|<nowiki>-13.686</nowiki>
| vm6
|
| vBb
|
|Bbv
|-
|-
| 31
| 31
|826.666
| 826.7
| style="text-align:center;" |8:5+
| [[8/5]], [[21/13]], ''[[13/8]]''
|
| Minor 6th
|12.980
| m6
|
| Bb
|
|Bb
|-
|-
| 32
| 32
|853.333
| 853.3
| style="text-align:center;" |18:11
| [[49/30]], [[80/49]]
|
| Mid 6th
|0.741
| ~6
|
| vB
|
|Bv
|-
|-
| 33
| 33
|880
| 880.0
| style="text-align:center;" |5:3
| [[5/3]]
|
| Major 6th
|<nowiki>-4.359</nowiki>
| M6
|
| B
|
|B
|-
|-
| 34
| 34
|906.666
| 906.7
| style="text-align:center;" |27:16
| [[42/25]]
|
| Upmajor 6th
|0.802
| ^M6
|
| ^B
|
|B^
|-
|-
| 35
| 35
|933.333
| 933.3
| style="text-align:center;" |12:7
| [[12/7]]
|
| Augmented 6th
|0.205
| A6
|
| B#
|
|B#
|-
|-
| 36
| 36
|960
| 960.0
| style="text-align:center;" |7:4
| [[7/4]], [[26/15]]
|
| Diminished 7th
|<nowiki>-8.826</nowiki>
| d7
|
| Cb
|
|Cb
|-
|-
| 37
| 37
|986.666
| 986.7
| style="text-align:center;" |16:9
|  
|  
|<nowiki>-9.423</nowiki>
| Downminor 7th
|
| vm7
|
| vC
|Cv
|-
|-
| 38
| 38
|1013.333
| 1013.3
| style="text-align:center;" |9:5
| [[9/5]], ''[[16/9]]''
|
| Minor 7th
|<nowiki>-4.262</nowiki>
| m7
|
| C
|
|C
|-
|-
| 39
| 39
|1040
| 1040.0
| style="text-align:center;" |20:11
| [[49/27]]
|
| Mid 7th
|5.004
| ~7
|
| ^C
|
|C^
|-
|-
| 40
| 40
|1066.666
| 1066.7
| style="text-align:center;" |50:27
| [[13/7]], [[24/13]], [[50/27]], ''[[15/8]]''
|
| Major 7th
|<nowiki>-0.095</nowiki>
| M7
|
| C#
|
|C#
|-
|-
| 41
| 41
|1093.333
| 1093.3
| style="text-align:center;" |32:17
| [[28/15]]
|
| Upmajor 7th
| -1.7112
| ^M7
|
| ^C#
|
|C#^
|-
|-
| 42
| 42
|1120
| 1120.0
| style="text-align:center;" |21:11
| [[40/21]]
|
| Augmented 7th
|0.537
| A7
|
| Cx
|
|Dbv
|-
|-
| 43
| 43
|1146.666
| 1146.7
| style="text-align:center;" |64:33
| [[35/18]], ''[[48/25]]'', ''[[63/32]]''
|
| Diminished 8ve
|<nowiki>-0.060</nowiki>
| d8
|
| Db
|
|Db
|-
|-
| 44
| 44
|1173.333
| 1173.3
| style="text-align:center;" |128:65
| [[49/25]], [[96/49]]
|
| Down 8ve
|0.174
| v8
|
| vD
|
|Dv
|-
|-
| 45
| 45
|1200
| 1200.0
| style="text-align:center;" |2:1
| [[2/1]]
|1200
| Perfect Octave
|0
| P8
|Octave
| D
|
|}
|D
<nowiki/>* As a 2.3.5.7.13-subgroup temperament, using the 45f val
 
== Notation ==
 
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{Ups and downs sharpness}}
 
=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used.
{{sharpness-sharp2}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[52edo#Sagittal notation|52]] and [[59edo#Second-best fifth notation|59b]].
 
==== Evo flavor ====
<imagemap>
File:45-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[36/35]]
rect 140 80 300 106 [[1053/1024]]
default [[File:45-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:45-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[36/35]]
rect 140 80 300 106 [[1053/1024]]
default [[File:45-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:45-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 511 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[36/35]]
rect 140 80 300 106 [[1053/1024]]
default [[File:45-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.
 
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 ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -71 45 }}
| {{Mapping| 45 71 }}
| +2.72
| 2.73
| 10.2
|-
| 2.3.5
| 81/80, {{monzo| -27 1 11 }}
| {{Mapping| 45 71 104 }}
| +3.68
| 2.61
| 9.75
|-
| 2.3.5.7
| 81/80, 525/512, 2401/2400
| {{Mapping| 45 71 104 126 }}
| +3.55
| 2.27
| 8.49
|-
| 2.3.5.7.13
| 65/64, 81/80, 105/104, 2401/2400
| {{Mapping| 45 71 104 126 166 }} (45f)
| +3.59
| 2.03
| 7.60
|}
|}
=== Commas ===
This is a partial list of the [[commas]] that 45et [[tempering out|tempers out]] with its [[patent val]], {{val| 45 71 104 126 143 156 167 }}.
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| <abbr title="7629394531250/7625597484987">(26 digits)</abbr>
| {{Monzo| 1 -27 18 }}
| 0.86
| Satritribiyo
| [[Ennealimma]]
|-
| 7
| [[16807/16384]]
| {{Monzo| -14 0 0 5 }}
| 44.13
| Laquinzo
| Cloudy comma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[875/864]]
| {{Monzo| -5 -3 3 1 }}
| 21.90
| Zotrigu
| Keema
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| <abbr title="40353607/40310784">(16 digits)</abbr>
| {{Monzo| -11 -9 0 9 }}
| 1.84
| Tritrizo
| [[Septimal ennealimma]]
|-
| 7
| [[4375/4374]]
| {{Monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[45/44]]
| {{Monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal 1/5-tone
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| <abbr title="117440512/117406179">(18 digits)</abbr>
| {{Monzo| 24 -6 0 1 -5 }}
| 0.51
| Saquinlu-azo
| [[Quartisma]]
|}
<references group="note" />
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 1\45
| 26.7
| 49/48
| [[Sfourth]]
|-
| 1
| 2\45
| 53.3
| 36/35
| [[Chromo]]
|-
| 1
| 7\45
| 186.7
| 10/9
| [[Mintone]]
|-
| 1
| 11\45
| 293.3
| 25/21
| [[Quasitemp]]
|-
| 1
| 14\45
| 373.3
| 5/4
| [[Submerged]]
|-
| 1
| 16\45
| 426.7
| 9/7
| [[Squares]]
|-
| 1
| 23\45
| 453.3
| 13/10
| [[Maja]]
|-
| 1
| 19\45
| 506.7
| 4/3
| [[Flattone]]
|-
| 3
| 19\45<br>(4\45)
| 506.7<br>(106.7)
| 4/3<br>(15/14)
| [[Lithium]]
|-
| 5
| 19\45<br>(1\45)
| 506.7<br>(26.7)
| 4/3<br>(49/48)
| [[Cloudtone]]
|-
| 9
| 12\45<br>(2\45)
| 320.0<br>(53.3)
| 6/5<br>(36/35)
| [[Ennealimmal]]
|-
| 15
| 19\45<br>(1\45)
| 506.7<br>(26.7)
| 4/3<br>(126/125)
| [[Pentadecal]]
|}
<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. [[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.
== 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
== Instruments ==
'''Lumatone'''
See [[Lumatone mapping for 45edo]]
== Music ==
; [[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]]
* [https://www.youtube.com/watch?v=tbc_OxHp-ec ''Fishbowl''] (2023)
* [https://www.youtube.com/watch?v=WqEOi4cd1Og ''Archipelago Arpeggio''] (2024)
* [https://www.youtube.com/watch?v=Kd4t_iKiKMA ''Fallen Angel''] (2024)
* [https://www.youtube.com/watch?v=DPztb8W6ykY ''Solar Guardian''] (2024)
* [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)
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Ennealimmal]]
[[Category:Quartismic]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
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