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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 ===
Line 16: Line 16:


=== 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
Line 68: Line 64:
|-
|-
| 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 86: Line 78:
|-
|-
| 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
Line 104: Line 92:
|-
|-
| 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
Line 122: Line 106:
|-
|-
| 10
| 10
| 266.666
| 266.7
| [[7/6]]
| [[7/6]]
| 266.871
| -0.205
| Diminished 3rd
| Diminished 3rd
| d3
| d3
Line 131: Line 113:
|-
|-
| 11
| 11
| 293.333
| 293.3
| [[32/27]]
| [[25/21]]
| 294.135
| -0.802
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 140: Line 120:
|-
|-
| 12
| 12
| 320.000
| 320.0
| [[6/5]]
| [[6/5]]
| 315.641
| 4.359
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 149: Line 127:
|-
|-
| 13
| 13
| 346.666
| 346.7
| [[11/9]]
| [[49/40]], [[60/49]]
| 347.408
| -0.741
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 158: Line 134:
|-
|-
| 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 167: Line 141:
|-
|-
| 15
| 15
| 400.000
| 400.0
| [[63/50]]
| [[63/50]]
| 400.108
| -0.108
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 176: Line 148:
|-
|-
| 16
| 16
| 426.666
| 426.7
| [[9/7]]
| [[9/7]]
| 435.084
| -8.418
| Augmented 3rd
| Augmented 3rd
| A3
| A3
Line 185: Line 155:
|-
|-
| 17
| 17
| 453.333
| 453.3
| [[13/10]]
| [[13/10]], ''[[21/16]]''
| 454.294
| -0.961
| Diminished 4th
| Diminished 4th
| d4
| d4
Line 194: Line 162:
|-
|-
| 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
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|-
|-
| 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
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|-
|-
| 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}}

Latest revision as of 11:41, 7 November 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.

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

# 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.

71edt is notable as an octave-stretched version of 45edo for having an extremely accurate 3/1, 5/1 and 11/1.

Scales

Instruments

Lumatone

See Lumatone mapping for 45edo

Music

Modern renderings

Brakence (Randall Todd Findell)

21st century

Bryan Deister
JUMBLE
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