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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|45}}
{{ED intro}}
 
== Theory ==
== Theory ==
45edo effectively 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.
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]].


45edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.33/32.7/6 subgroup [[The Quartercache#Direct quartismic|direct quartismic]] temperament, in which it approximates 33/32 quartertone with 2 steps and 7/6 with 10 steps. It is also the unique equal temperament tuning that tempers out both the syntonic comma and the [[ennealimma]].
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.


=== Odd harmonics ===
=== Subsets and supersets ===
{{harmonics in equal|45}}
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 ==
== 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
| D
| D
|-
|-
| 1
| 1
| 26.666
| 26.7
| [[65/64]]
| [[49/48]], [[50/49]]
| 26.841
| -0.174
| Up unison
| Up unison
|^1
| ^1
| ^D
| ^D
|-
|-
| 2
| 2
| 53.333
| 53.3
| [[33/32]]
| [[36/35]], ''[[25/24]]'', ''[[64/63]]''
| 53.273
| 0.060
| Augmented Unison
| Augmented Unison
|A1
| A1
| D#
| D#
|-
|-
| 3
| 3
| 80.000
| 80.0
| [[22/21]]
| [[21/20]]
| 80.537
| -0.537
| Diminished 2nd
| Diminished 2nd
|d2
| d2
| Ebb
| Ebb
|-
|-
| 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
| Eb
| Eb
|-
|-
| 6
| 6
| 160.000
| 160.0
| [[11/10]]
| [[54/49]]
| 165.004
| -5.004
| Mid 2nd
| Mid 2nd
|~2
| ~2
| vE
| vE
|-
|-
| 7
| 7
| 186.666
| 186.7
| [[10/9]]
| [[10/9]], ''[[9/8]]''
| 182.404
| 4.262
| Major 2nd
| Major 2nd
|M2
| M2
| E
| E
|-
|-
| 8
| 8
| 213.333
| 213.3
| [[9/8]]
|  
| 203.910
| 9.423
| Upmajor 2nd
| Upmajor 2nd
|^M2
| ^M2
| ^E
| ^E
|-
|-
| 9
| 9
| 240.000
| 240.0
| [[8/7]]
| [[8/7]], [[15/13]]
| 231.174
| 8.826
| Augmented 2nd
| Augmented 2nd
|A2
| A2
| E#
| E#
|-
|-
| 10
| 10
| 266.666
| 266.7
| [[7/6]]
| [[7/6]]
| 266.871
| -0.205
| Diminished 3rd
| Diminished 3rd
|d3
| d3
| Fb
| Fb
|-
|-
| 11
| 11
| 293.333
| 293.3
| [[32/27]]
| [[25/21]]
| 294.135
| -0.802
| Downminor 3rd
| Downminor 3rd
|vm3
| vm3
| vF
| vF
|-
|-
| 12
| 12
| 320.000
| 320.0
| [[6/5]]
| [[6/5]]
| 315.641
| 4.359
| Minor 3rd
| Minor 3rd
|m3
| m3
| F
| F
|-
|-
| 13
| 13
| 346.666
| 346.7
| [[11/9]]
| [[49/40]], [[60/49]]
| 347.408
| -0.741
| Mid 3rd
| Mid 3rd
|~3
| ~3
| ^F
| ^F
|-
|-
| 14
| 14
| 373.333
| 373.3
| [[5/4]]
| [[5/4]], [[26/21]], ''[[16/13]]''
| 386.314
| -12.980
| Major 3rd
| Major 3rd
|M3
| M3
| F#
| F#
|-
|-
| 15
| 15
| 400.000
| 400.0
| [[63/50]]
| [[63/50]]
| 400.108
| -0.108
| Upmajor 3rd
| Upmajor 3rd
|^M3
| ^M3
| ^F#
| ^F#
|-
|-
| 16
| 16
| 426.666
| 426.7
| [[9/7]]
| [[9/7]]
| 435.084
| -8.418
| Augmented 3rd
| Augmented 3rd
|A3
| A3
| Fx
| Fx
|-
|-
| 17
| 17
| 453.333
| 453.3
| [[13/10]]
| [[13/10]], ''[[21/16]]''
| 454.294
| -0.961
| Diminished 4th
| Diminished 4th
|d4
| d4
| Gb
| Gb
|-
|-
| 18
| 18
| 480.000
| 480.0
| [[21/16]]
| ''[[64/49]]''
| 470.781
| 9.219
| Down 4th
| Down 4th
|v4
| v4
| vG
| vG
|-
|-
| 19
| 19
| 506.666
| 506.7
| [[4/3]]
| [[4/3]]
| 498.045
| 8.622
| Perfect 4th
| Perfect 4th
|P4
| P4
| G
| G
|-
|-
| 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
| ^G
| ^G
|-
|-
| 21
| 21
| 560.000
| 560.0
| [[18/13]]
| [[18/13]]
| 563.382
| -3.382
| Augmented 4th
| Augmented 4th
|A4
| A4
| G#
| G#
|-
|-
| 22
| 22
| 586.666
| 586.7
| [[7/5]]
| [[7/5]]
| 582.512
| 4.155
| Upaugmented 4th
| Upaugmented 4th
|^A4
| ^A4
| ^G#
| ^G#
|-
|-
| 23
| 23
| 613.333
| 613.3
| [[10/7]]
| [[10/7]]
| 617.488
| -4.155
| Downdiminshed 5th
| Downdiminshed 5th
|vd5
| vd5
| vAb
| vAb
|-
|-
| 24
| 24
| 640.000
| 640.0
| [[13/9]]
| [[13/9]]
| 636.618
| 3.382
| Diminished 5th
| Diminished 5th
|d5
| d5
| Ab
| Ab
|-
|-
| 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
| vA
| vA
|-
|-
| 26
| 26
| 693.333
| 693.3
| [[3/2]]
| [[3/2]]
| 701.955
| -8.622
| Perfect 5th
| Perfect 5th
|P5
| P5
| A
| A
|-
|-
| 27
| 27
| 720.000
| 720.0
| [[32/21]]
| ''[[49/32]]''
| 729.219
| -9.219
| Up 5th
| Up 5th
|^5
| ^5
| ^A
| ^A
|-
|-
| 28
| 28
| 746.666
| 746.7
| [[20/13]]
| [[20/13]], ''[[32/21]]''
| 745.786
| 0.961
| Augmented 5th
| Augmented 5th
|A5
| A5
| A#
| A#
|-
|-
| 29
| 29
| 773.333
| 773.3
| [[14/9]]
| [[14/9]]
| 764.916
| 8.418
| Diminished 6th
| Diminished 6th
|d6
| d6
| Bbb
| Bbb
|-
|-
| 30
| 30
| 800.000
| 800.0
| [[100/63]]
| [[100/63]]
| 799.892
| 0.108
| Downminor 6th
| Downminor 6th
|vm6
| vm6
| vBb
| vBb
|-
|-
| 31
| 31
| 826.666
| 826.7
| [[8/5]]
| [[8/5]], [[21/13]], ''[[13/8]]''
| 813.686
| 12.980
| Minor 6th
| Minor 6th
|m6
| m6
| Bb
| Bb
|-
|-
| 32
| 32
| 853.333
| 853.3
| [[18/11]]
| [[49/30]], [[80/49]]
| 852.592
| 0.741
| Mid 6th
| Mid 6th
|~6
| ~6
| vB
| vB
|-
|-
| 33
| 33
| 880.000
| 880.0
| [[5/3]]
| [[5/3]]
| 884.359
| -4.359
| Major 6th
| Major 6th
|M6
| M6
| B
| B
|-
|-
| 34
| 34
| 906.666
| 906.7
| [[27/16]]
| [[42/25]]
| 905.865
| 0.802
| Upmajor 6th
| Upmajor 6th
|^M6
| ^M6
| ^B
| ^B
|-
|-
| 35
| 35
| 933.333
| 933.3
| [[12/7]]
| [[12/7]]
| 933.129
| 0.205
| Augmented 6th
| Augmented 6th
|A6
| A6
| B#
| B#
|-
|-
| 36
| 36
| 960.000
| 960.0
| [[7/4]]
| [[7/4]], [[26/15]]
| 968.826
| -8.826
| Diminished 7th
| Diminished 7th
|d7
| d7
| Cb
| Cb
|-
|-
| 37
| 37
| 986.666
| 986.7
| [[16/9]]
|  
| 996.089
| -9.423
| Downminor 7th
| Downminor 7th
|vm7
| vm7
| vC
| vC
|-
|-
| 38
| 38
| 1013.333
| 1013.3
| [[9/5]]
| [[9/5]], ''[[16/9]]''
| 1017.596
| -4.262
| Minor 7th
| Minor 7th
|m7
| m7
| C
| C
|-
|-
| 39
| 39
| 1040.000
| 1040.0
| [[20/11]]
| [[49/27]]
| 1034.996
| 5.004
| Mid 7th
| Mid 7th
|~7
| ~7
| ^C
| ^C
|-
|-
| 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
| C#
| C#
|-
|-
| 41
| 41
| 1093.333
| 1093.3
| [[32/17]]
| [[28/15]]
| 1095.044
| -1.711
| Upmajor 7th
| Upmajor 7th
|^M7
| ^M7
| ^C#
| ^C#
|-
|-
| 42
| 42
| 1120.000
| 1120.0
| [[21/11]]
| [[40/21]]
| 1119.463
| 0.537
| Augmented 7th
| Augmented 7th
|A7
| A7
| Cx
| Cx
|-
|-
| 43
| 43
| 1146.666
| 1146.7
| [[64/33]]
| [[35/18]], ''[[48/25]]'', ''[[63/32]]''
| 1146.727
| -0.060
| Diminished 8ve
| Diminished 8ve
|d8
| d8
| Db
| Db
|-
|-
| 44
| 44
| 1173.333
| 1173.3
| [[128/65]]
| [[49/25]], [[96/49]]
| 1173.158
| 0.174
| Down 8ve
| Down 8ve
|v8
| v8
| vD
| vD
|-
|-
| 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 ==
=== 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 ==
== 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 ===
=== Commas ===
This is a partial list of the [[commas]] that 45edo [[tempers out]] with its patent [[val]], {{val| 45 71 104 126 143 156 167 }}.
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"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
Line 452: Line 459:
| 5
| 5
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{Monzo| -4 4 -1 }}
| 21.51
| 21.51
| Gu
| Gu
| Syntonic comma, Didymus comma, meantone comma
| Syntonic comma, Didymus' comma, meantone comma
|-
|-
| 5
| 5
| <abbr title="7629394531250/7625597484987">(26 digits)</abbr>
| <abbr title="7629394531250/7625597484987">(26 digits)</abbr>
| {{monzo| 1 -27 18 }}
| {{Monzo| 1 -27 18 }}
| 0.86
| 0.86
| Satritribiyo
| Satritribiyo
Line 466: Line 473:
| 7
| 7
| [[16807/16384]]
| [[16807/16384]]
| {{monzo| -14 0 0 5}}
| {{Monzo| -14 0 0 5 }}
| 44.13
| 44.13
| Laquinzo
| Laquinzo
Line 473: Line 480:
| 7
| 7
| [[525/512]]
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| {{Monzo| -9 1 2 1 }}
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
Line 480: Line 487:
| 7
| 7
| [[875/864]]
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| {{Monzo| -5 -3 3 1 }}
| 21.90
| 21.90
| Zotrigu
| Zotrigu
Line 487: Line 494:
| 7
| 7
| [[3125/3087]]
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyo
Line 494: Line 501:
| 7
| 7
| <abbr title="40353607/40310784">(16 digits)</abbr>
| <abbr title="40353607/40310784">(16 digits)</abbr>
| {{monzo| -11 -9 0 9 }}
| {{Monzo| -11 -9 0 9 }}
| 1.84
| 1.84
| Tritrizo
| Tritrizo
Line 501: Line 508:
| 7
| 7
| [[4375/4374]]
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| {{Monzo| -1 -7 4 1 }}
| 0.40
| 0.40
| Zoquadyo
| Zoquadyo
Line 508: Line 515:
| 11
| 11
| [[45/44]]
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| {{Monzo| -2 2 1 0 -1 }}
| 38.91
| 38.91
| Luyo
| Luyo
Line 515: Line 522:
| 11
| 11
| [[385/384]]
| [[385/384]]
| {{monzo|-7 -1 1 1 1 }}
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
Line 522: Line 529:
| 11
| 11
| <abbr title="117440512/117406179">(18 digits)</abbr>
| <abbr title="117440512/117406179">(18 digits)</abbr>
| {{monzo| 24 -6 0 1 -5 }}
| {{Monzo| 24 -6 0 1 -5 }}
| 0.51
| 0.51
| Saquinlu-azo
| Saquinlu-azo
| [[Quartisma]]
| [[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 ==
== Instruments ==
Line 534: Line 637:


== Music ==
== 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]]
; [[JUMBLE]]
* [https://www.youtube.com/watch?v=tbc_OxHp-ec ''Fishbowl''] (2023)
* [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=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:Equal divisions of the octave|##]] <!-- 2-digit number -->

Latest revision as of 14:11, 29 May 2026

← 44edo 45edo 46edo →
Prime factorization 32 × 5
Step size 26.6667 ¢ 
Fifth 26\45 (693.333 ¢)
Semitones (A1:m2) 2:5 (53.33 ¢ : 133.3 ¢)
Consistency limit 7
Distinct consistency limit 7

45 equal divisions of the octave (abbreviated 45edo or 45ed2), also called 45-tone equal temperament (45tet) or 45 equal temperament (45et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 45 equal parts of about 26.7 ¢ each. Each step represents a frequency ratio of 21/45, or the 45th root of 2.

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 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, 5, and 7 all flat. However, harmonics 11 and 13 are sharp, but this can be fixed with the 45ef val.

Odd harmonics

Approximation of odd harmonics in 45edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -8.6 -13.0 -8.8 +9.4 +8.7 +12.8 +5.1 +1.7 -4.2 +9.2 +11.7
Relative (%) -32.3 -48.7 -33.1 +35.3 +32.6 +48.0 +19.0 +6.4 -15.7 +34.6 +44.0
Steps
(reduced) 		71
(26) 		104
(14) 		126
(36) 		143
(8) 		156
(21) 		167
(32) 		176
(41) 		184
(4) 		191
(11) 		198
(18) 		204
(24)

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 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 32 × 5, 45edo has subset 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

# Cents Approximate ratios* 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

* 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.

Step offset 0 1 2 3 4 5
Sharp symbol   
 
  
   
  
   
Flat symbol
 
  
   
  
   

Quarter-tone notation

Since a sharp raises by two steps, quarter-tone accidentals can also be used.

Step offset −4 −3 −2 −1 0 +1 +2 +3 +4
Symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

Sagittal notation

This notation uses the same sagittal sequence as EDOs 52 and 59b.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation36/351053/1024

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation36/351053/1024

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation36/351053/1024

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 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

The following tables show how 15-odd-limit intervals are represented in 45edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 45edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/6, 12/7 0.204 0.8
11/9, 18/11 0.741 2.8
13/10, 20/13 0.881 3.3
13/9, 18/13 3.382 12.7
15/11, 22/15 3.617 13.6
13/11, 22/13 4.124 15.5
7/5, 10/7 4.154 15.6
9/5, 10/9 4.263 16.0
5/3, 6/5 4.359 16.3
11/10, 20/11 5.004 18.8
13/7, 14/13 5.035 18.9
15/8, 16/15 5.065 19.0
13/12, 24/13 5.239 19.6
15/13, 26/15 7.741 29.0
9/7, 14/9 8.417 31.6
3/2, 4/3 8.622 32.3
11/8, 16/11 8.682 32.6
7/4, 8/7 8.826 33.1
11/7, 14/11 9.159 34.3
11/6, 12/11 9.363 35.1
9/8, 16/9 9.423 35.3
15/14, 28/15 12.776 47.9
13/8, 16/13 12.806 48.0
5/4, 8/5 12.980 48.7
15-odd-limit intervals in 45edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/6, 12/7 0.204 0.8
13/11, 22/13 4.124 15.5
7/5, 10/7 4.154 15.6
9/5, 10/9 4.263 16.0
5/3, 6/5 4.359 16.3
9/7, 14/9 8.417 31.6
3/2, 4/3 8.622 32.3
11/8, 16/11 8.682 32.6
7/4, 8/7 8.826 33.1
15/14, 28/15 12.776 47.9
13/8, 16/13 12.806 48.0
5/4, 8/5 12.980 48.7
9/8, 16/9 17.243 64.7
11/6, 12/11 17.304 64.9
11/7, 14/11 17.508 65.7
13/12, 24/13 21.427 80.4
15/8, 16/15 21.602 81.0
13/7, 14/13 21.632 81.1
11/10, 20/11 21.662 81.2
13/10, 20/13 25.786 96.7
11/9, 18/11 25.925 97.2
13/9, 18/13 30.049 112.7
15/11, 22/15 30.284 113.6
15/13, 26/15 34.408 129.0
15-odd-limit intervals by 45ef val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/6, 12/7 0.204 0.8
11/9, 18/11 0.741 2.8
13/10, 20/13 0.881 3.3
13/9, 18/13 3.382 12.7
15/11, 22/15 3.617 13.6
13/11, 22/13 4.124 15.5
7/5, 10/7 4.154 15.6
9/5, 10/9 4.263 16.0
5/3, 6/5 4.359 16.3
11/10, 20/11 5.004 18.8
13/7, 14/13 5.035 18.9
13/12, 24/13 5.239 19.6
15/13, 26/15 7.741 29.0
9/7, 14/9 8.417 31.6
3/2, 4/3 8.622 32.3
7/4, 8/7 8.826 33.1
11/7, 14/11 9.159 34.3
11/6, 12/11 9.363 35.1
15/14, 28/15 12.776 47.9
5/4, 8/5 12.980 48.7
13/8, 16/13 13.861 52.0
9/8, 16/9 17.243 64.7
11/8, 16/11 17.985 67.4
15/8, 16/15 21.602 81.0

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-71 45⟩ [⟨45 71]] +2.72 2.73 10.2
2.3.5 81/80, [-27 1 11⟩ [⟨45 71 104]] +3.68 2.61 9.75
2.3.5.7 81/80, 525/512, 2401/2400 [⟨45 71 104 126]] +3.55 2.27 8.49
2.3.5.7.13 65/64, 81/80, 105/104, 2401/2400 [⟨45 71 104 126 166]] (45f) +3.59 2.03 7.60

Commas

This is a partial list of the commas that 45et tempers out with its patent val, ⟨45 71 104 126 143 156 167].

Prime
limit 		Ratio[note 1] Monzo Cents Color name Name(s)
5 81/80 [-4 4 -1⟩ 21.51 Gu Syntonic comma, Didymus' comma, meantone comma
5 (26 digits) [1 -27 18⟩ 0.86 Satritribiyo Ennealimma
7 16807/16384 [-14 0 0 5⟩ 44.13 Laquinzo Cloudy comma
7 525/512 [-9 1 2 1⟩ 43.41 Lazoyoyo Avicennma, Avicenna's enharmonic diesis
7 875/864 [-5 -3 3 1⟩ 21.90 Zotrigu Keema
7 3125/3087 [0 -2 5 -3⟩ 21.18 Triru-aquinyo Gariboh comma
7 (16 digits) [-11 -9 0 9⟩ 1.84 Tritrizo Septimal ennealimma
7 4375/4374 [-1 -7 4 1⟩ 0.40 Zoquadyo Ragisma
11 45/44 [-2 2 1 0 -1⟩ 38.91 Luyo Undecimal 1/5-tone
11 385/384 [-7 -1 1 1 1⟩ 4.50 Lozoyo Keenanisma
11 (18 digits) [24 -6 0 1 -5⟩ 0.51 Saquinlu-azo Quartisma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
(4\45) 		506.7
(106.7) 		4/3
(15/14) 		Lithium
5 19\45
(1\45) 		506.7
(26.7) 		4/3
(49/48) 		Cloudtone
9 12\45
(2\45) 		320.0
(53.3) 		6/5
(36/35) 		Ennealimmal
15 19\45
(1\45) 		506.7
(26.7) 		4/3
(126/125) 		Pentadecal

* Octave-reduced form, reduced to the first half-octave, and 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 stretched-octave version of 45edo, such as 161ed12 or 116ed6. The trade-off is a slightly worse 2/1. 207zpi also improves on all of those harmonics except for 17/1.

The 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

Instruments

Lumatone

See Lumatone mapping for 45edo

Music

Bryan Deister
JUMBLE
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