45edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{ED intro}}
{{ED intro}}
== Theory ==
== Theory ==
''It is recommended to read the page [[regular temperament]] first to understand this section.''
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.
 
45edo effectively has two approximate 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 a flat-tending system in the [[7-limit]], with 3, 5, and 7 all flat, but the 11 is sharp.
 
It provides the [[optimal patent val]] for [[flattone]] temperament, 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 tempers out 81/80, 3125/3087, 525/512, 875/864 and 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 2.17.25.27.33.63.65 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.


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|45}}
{{Harmonics in equal|45}}


=== Octave stretch ===
=== As a tuning of other temperaments ===
45edo's approximations of 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 are all improved by [[Gallery of arithmetic pitch sequences #APS of farabs|APS3.21farab]], a [[Octave stretch|stretched-octave]] version of 45edo. The trade-off is a slightly worse 2/1.
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]].


The tuning [[126ed7]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1 and 13/1, at the cost of a slightly worse 2/1.
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]].


There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used for this same purpose: 207zpi, 208zpi and 209zpi. The main Zeta peak index page details all three tunings.
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 ==
== 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 45: Line 34:
|-
|-
| 1
| 1
| 26.666
| 26.7
| [[65/64]]
| [[49/48]], [[50/49]]
| 26.841
| -0.174
| Up unison
| Up unison
| ^1
| ^1
Line 54: Line 41:
|-
|-
| 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 63: Line 48:
|-
|-
| 3
| 3
| 80.000
| 80.0
| [[22/21]]
| [[21/20]]
| 80.537
| -0.537
| Diminished 2nd
| Diminished 2nd
| d2
| d2
Line 72: Line 55:
|-
|-
| 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
Line 90: Line 69:
|-
|-
| 6
| 6
| 160.000
| 160.0
| [[11/10]]
| [[54/49]]
| 165.004
| -5.004
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 99: Line 76:
|-
|-
| 7
| 7
| 186.666
| 186.7
| [[10/9]]
| [[10/9]], ''[[9/8]]''
| 182.404
| 4.262
| Major 2nd
| Major 2nd
| M2
| M2
Line 108: Line 83:
|-
|-
| 8
| 8
| 213.333
| 213.3
| [[9/8]]
|  
| 203.910
| 9.423
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 117: Line 90:
|-
|-
| 9
| 9
| 240.000
| 240.0
| [[8/7]]
| [[8/7]], [[15/13]]
| 231.174
| 8.826
| Augmented 2nd
| Augmented 2nd
| A2
| A2
Line 126: Line 97:
|-
|-
| 10
| 10
| 266.666
| 266.7
| [[7/6]]
| [[7/6]]
| 266.871
| -0.205
| Diminished 3rd
| Diminished 3rd
| d3
| d3
Line 135: Line 104:
|-
|-
| 11
| 11
| 293.333
| 293.3
| [[32/27]]
| [[25/21]]
| 294.135
| -0.802
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 144: Line 111:
|-
|-
| 12
| 12
| 320.000
| 320.0
| [[6/5]]
| [[6/5]]
| 315.641
| 4.359
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 153: Line 118:
|-
|-
| 13
| 13
| 346.666
| 346.7
| [[11/9]]
| [[49/40]], [[60/49]]
| 347.408
| -0.741
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 162: Line 125:
|-
|-
| 14
| 14
| 373.333
| 373.3
| [[5/4]]
| [[5/4]], [[26/21]], ''[[16/13]]''
| 386.314
| -12.980
| Major 3rd
| Major 3rd
| M3
| M3
Line 171: Line 132:
|-
|-
| 15
| 15
| 400.000
| 400.0
| [[63/50]]
| [[63/50]]
| 400.108
| -0.108
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 180: Line 139:
|-
|-
| 16
| 16
| 426.666
| 426.7
| [[9/7]]
| [[9/7]]
| 435.084
| -8.418
| Augmented 3rd
| Augmented 3rd
| A3
| A3
Line 189: Line 146:
|-
|-
| 17
| 17
| 453.333
| 453.3
| [[13/10]]
| [[13/10]], ''[[21/16]]''
| 454.294
| -0.961
| Diminished 4th
| Diminished 4th
| d4
| d4
Line 198: Line 153:
|-
|-
| 18
| 18
| 480.000
| 480.0
| [[21/16]]
| ''[[64/49]]''
| 470.781
| 9.219
| Down 4th
| Down 4th
| v4
| v4
Line 207: Line 160:
|-
|-
| 19
| 19
| 506.666
| 506.7
| [[4/3]]
| [[4/3]]
| 498.045
| 8.622
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 216: Line 167:
|-
|-
| 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 225: Line 174:
|-
|-
| 21
| 21
| 560.000
| 560.0
| [[18/13]]
| [[18/13]]
| 563.382
| -3.382
| Augmented 4th
| Augmented 4th
| A4
| A4
Line 234: Line 181:
|-
|-
| 22
| 22
| 586.666
| 586.7
| [[7/5]]
| [[7/5]]
| 582.512
| 4.155
| Upaugmented 4th
| Upaugmented 4th
| ^A4
| ^A4
Line 243: Line 188:
|-
|-
| 23
| 23
| 613.333
| 613.3
| [[10/7]]
| [[10/7]]
| 617.488
| -4.155
| Downdiminshed 5th
| Downdiminshed 5th
| vd5
| vd5
Line 252: Line 195:
|-
|-
| 24
| 24
| 640.000
| 640.0
| [[13/9]]
| [[13/9]]
| 636.618
| 3.382
| Diminished 5th
| Diminished 5th
| d5
| d5
Line 261: Line 202:
|-
|-
| 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 270: Line 209:
|-
|-
| 26
| 26
| 693.333
| 693.3
| [[3/2]]
| [[3/2]]
| 701.955
| -8.622
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 279: Line 216:
|-
|-
| 27
| 27
| 720.000
| 720.0
| [[32/21]]
| ''[[49/32]]''
| 729.219
| -9.219
| Up 5th
| Up 5th
| ^5
| ^5
Line 288: Line 223:
|-
|-
| 28
| 28
| 746.666
| 746.7
| [[20/13]]
| [[20/13]], ''[[32/21]]''
| 745.786
| 0.961
| Augmented 5th
| Augmented 5th
| A5
| A5
Line 297: Line 230:
|-
|-
| 29
| 29
| 773.333
| 773.3
| [[14/9]]
| [[14/9]]
| 764.916
| 8.418
| Diminished 6th
| Diminished 6th
| d6
| d6
Line 306: Line 237:
|-
|-
| 30
| 30
| 800.000
| 800.0
| [[100/63]]
| [[100/63]]
| 799.892
| 0.108
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 315: Line 244:
|-
|-
| 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 324: Line 251:
|-
|-
| 32
| 32
| 853.333
| 853.3
| [[18/11]]
| [[49/30]], [[80/49]]
| 852.592
| 0.741
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 333: Line 258:
|-
|-
| 33
| 33
| 880.000
| 880.0
| [[5/3]]
| [[5/3]]
| 884.359
| -4.359
| Major 6th
| Major 6th
| M6
| M6
Line 342: Line 265:
|-
|-
| 34
| 34
| 906.666
| 906.7
| [[27/16]]
| [[42/25]]
| 905.865
| 0.802
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 351: Line 272:
|-
|-
| 35
| 35
| 933.333
| 933.3
| [[12/7]]
| [[12/7]]
| 933.129
| 0.205
| Augmented 6th
| Augmented 6th
| A6
| A6
Line 360: Line 279:
|-
|-
| 36
| 36
| 960.000
| 960.0
| [[7/4]]
| [[7/4]], [[26/15]]
| 968.826
| -8.826
| Diminished 7th
| Diminished 7th
| d7
| d7
Line 369: Line 286:
|-
|-
| 37
| 37
| 986.666
| 986.7
| [[16/9]]
|  
| 996.089
| -9.423
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 378: Line 293:
|-
|-
| 38
| 38
| 1013.333
| 1013.3
| [[9/5]]
| [[9/5]], ''[[16/9]]''
| 1017.596
| -4.262
| Minor 7th
| Minor 7th
| m7
| m7
Line 387: Line 300:
|-
|-
| 39
| 39
| 1040.000
| 1040.0
| [[20/11]]
| [[49/27]]
| 1034.996
| 5.004
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 396: Line 307:
|-
|-
| 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 405: Line 314:
|-
|-
| 41
| 41
| 1093.333
| 1093.3
| [[32/17]]
| [[28/15]]
| 1095.044
| -1.711
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
Line 414: Line 321:
|-
|-
| 42
| 42
| 1120.000
| 1120.0
| [[21/11]]
| [[40/21]]
| 1119.463
| 0.537
| Augmented 7th
| Augmented 7th
| A7
| A7
Line 423: Line 328:
|-
|-
| 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 432: Line 335:
|-
|-
| 44
| 44
| 1173.333
| 1173.3
| [[128/65]]
| [[49/25]], [[96/49]]
| 1173.158
| 0.174
| Down 8ve
| Down 8ve
| v8
| v8
Line 441: Line 342:
|-
|-
| 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 ==


=== Ups and Downs notation ===
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{sharpness-sharp2a}}
{{Ups and downs sharpness}}


=== Quarter-tone notation ===
=== Quarter-tone notation ===
Line 500: Line 400:
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 ==
== Approximation to JI==
=== Zeta peak index ===
=== Interval mappings ===
{{ZPI
{{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}}
| zpi = 207
| steps = 44.8397488079316
| step size = 26.7619697233392
| tempered height = 5.252141
| pure height = 1.593330
| integral = 0.837163
| gap = 14.719415
| octave = 1204.28863755026
| consistent = 7
| distinct = 7
}}


== 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 45et [[tempering out|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"
Line 530: 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 544: 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 551: 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 558: 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 565: 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 572: 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 579: 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 586: 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 593: 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 600: 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 612: 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)
Line 617: Line 646:
* [https://www.youtube.com/watch?v=Kd4t_iKiKMA ''Fallen Angel''] (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=DPztb8W6ykY ''Solar Guardian''] (2024)
 
* [https://www.youtube.com/watch?v=24gnhAbHtiw ''Qúchze úzeq Qávka''] (2025)
== Notes ==
* [https://www.youtube.com/watch?v=K2p7HOI3TUE ''Sodium Light (45edo Microtonal Chillwave)''] (2026)
<references group="note" />
* [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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