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'''45edo''' divides the octave into 45 equal parts of 26.667 [[cent]]s. 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.
 
=== 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"
|-
! #
! Cents
! Approximate ratios*
! colspan="4" | [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| Perfect Unison
| P1
| D
|-
| 1
| 26.7
| [[49/48]], [[50/49]]
| Up unison
| ^1
| ^D
|-
| 2
| 53.3
| [[36/35]], ''[[25/24]]'', ''[[64/63]]''
| Augmented Unison
| A1
| D#
|-
| 3
| 80.0
| [[21/20]]
| Diminished 2nd
| d2
| Ebb
|-
| 4
| 106.7
| [[15/14]]
| Downminor 2nd
| vm2
| vEb
|-
| 5
| 133.3
| [[13/12]], [[14/13]], [[27/25]], ''[[16/15]]''
| Minor 2nd
| m2
| Eb
|-
| 6
| 160.0
| [[54/49]]
| Mid 2nd
| ~2
| vE
|-
| 7
| 186.7
| [[10/9]], ''[[9/8]]''
| Major 2nd
| M2
| E
|-
| 8
| 213.3
|
| Upmajor 2nd
| ^M2
| ^E
|-
| 9
| 240.0
| [[8/7]], [[15/13]]
| Augmented 2nd
| A2
| E#
|-
| 10
| 266.7
| [[7/6]]
| Diminished 3rd
| d3
| Fb
|-
| 11
| 293.3
| [[25/21]]
| Downminor 3rd
| vm3
| vF
|-
| 12
| 320.0
| [[6/5]]
| Minor 3rd
| m3
| F
|-
| 13
| 346.7
| [[49/40]], [[60/49]]
| Mid 3rd
| ~3
| ^F
|-
| 14
| 373.3
| [[5/4]], [[26/21]], ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
| 15
| 400.0
| [[63/50]]
| Upmajor 3rd
| ^M3
| ^F#
|-
| 16
| 426.7
| [[9/7]]
| Augmented 3rd
| A3
| Fx
|-
| 17
| 453.3
| [[13/10]], ''[[21/16]]''
| Diminished 4th
| d4
| Gb
|-
| 18
| 480.0
| ''[[64/49]]''
| Down 4th
| v4
| vG
|-
| 19
| 506.7
| [[4/3]]
| Perfect 4th
| P4
| G
|-
| 20
| 533.3
| [[49/36]]
| Up 4th or Mid 4th
| ^4, ~4
| ^G
|-
| 21
| 560.0
| [[18/13]]
| Augmented 4th
| A4
| G#
|-
| 22
| 586.7
| [[7/5]]
| Upaugmented 4th
| ^A4
| ^G#
|-
| 23
| 613.3
| [[10/7]]
| Downdiminshed 5th
| vd5
| vAb
|-
| 24
| 640.0
| [[13/9]]
| Diminished 5th
| d5
| Ab
|-
| 25
| 666.7
| [[72/49]]
| Down 5th or Mid 5th
| v5, ~5
| vA
|-
| 26
| 693.3
| [[3/2]]
| Perfect 5th
| P5
| A
|-
| 27
| 720.0
| ''[[49/32]]''
| Up 5th
| ^5
| ^A
|-
| 28
| 746.7
| [[20/13]], ''[[32/21]]''
| Augmented 5th
| A5
| A#
|-
| 29
| 773.3
| [[14/9]]
| Diminished 6th
| d6
| Bbb
|-
| 30
| 800.0
| [[100/63]]
| Downminor 6th
| vm6
| vBb
|-
| 31
| 826.7
| [[8/5]], [[21/13]], ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
| 32
| 853.3
| [[49/30]], [[80/49]]
| Mid 6th
| ~6
| vB
|-
| 33
| 880.0
| [[5/3]]
| Major 6th
| M6
| B
|-
| 34
| 906.7
| [[42/25]]
| Upmajor 6th
| ^M6
| ^B
|-
| 35
| 933.3
| [[12/7]]
| Augmented 6th
| A6
| B#
|-
| 36
| 960.0
| [[7/4]], [[26/15]]
| Diminished 7th
| d7
| Cb
|-
| 37
| 986.7
|
| Downminor 7th
| vm7
| vC
|-
| 38
| 1013.3
| [[9/5]], ''[[16/9]]''
| Minor 7th
| m7
| C
|-
| 39
| 1040.0
| [[49/27]]
| Mid 7th
| ~7
| ^C
|-
| 40
| 1066.7
| [[13/7]], [[24/13]], [[50/27]], ''[[15/8]]''
| Major 7th
| M7
| C#
|-
| 41
| 1093.3
| [[28/15]]
| Upmajor 7th
| ^M7
| ^C#
|-
| 42
| 1120.0
| [[40/21]]
| Augmented 7th
| A7
| Cx
|-
| 43
| 1146.7
| [[35/18]], ''[[48/25]]'', ''[[63/32]]''
| Diminished 8ve
| d8
| Db
|-
| 44
| 1173.3
| [[49/25]], [[96/49]]
| Down 8ve
| v8
| vD
|-
| 45
| 1200.0
| [[2/1]]
| Perfect Octave
| P8
| 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:Ennealimmal]]
[[Category:Quartismic]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/45edo"