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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|40}}
{{ED intro}}
== Theory ==
== Theory ==
Up to this point, all the multiples of 5 have had the 720 cent blackwood 5th as their best approximation of 3/2. 35edo combined the small circles of blackwood and whitewood 5ths, almost equally far from just, requiring you to use both to reach all keys. 40edo adds a diatonic 5th that's closer to just. However, it is still the second flattest diatonic 5th, only exceeded by 47edo in error, which results in it being inconsistent in the 5-limit - combining the best major and minor third will result in the blackwood 5th instead. As such, calling it a perfect 5th seems very much a misnomer. Despite all keys being reachable by stacking this 5th, it does not qualify as meantone either, tempering out [[177147/163840]] and [[1053/1024]] in the patent val instead of [[81/80]], which means 4 5ths make a near perfect tridecimal neutral 3rd and it takes a full 11 to reach the 5th harmonic. 81/80 is only tempered out in the 40c alternative val where the 360 cent third is equated with 5/4 instead of the 390 cent third. The resulting [[5L 2s]] scale has large steps of 6 intervals and small ones of 5, putting sharps and flats right next to letters without any ups or downs in between and requiring a lot of them to notate more distant keys. It tempers out [[648/625]] in the 5-limit; [[225/224]] and in the 7-limit; [[99/98]], [[121/120]] and [[176/175]] in the 11-limit; and [[66/65]] in the 13-limit.
Up to this point, all the multiples of 5 have had the 720{{c}} [[blackwood]] fifth as their best approximation of [[3/2]] apart from [[35edo]], which combines the small circles of blackwood and whitewood fifths, almost equally far from just, requiring the use of both to reach all keys. 40edo adds a diatonic fifth that's closer to just. However, it is still the second flattest diatonic fifth found in the [[Val|patent val]] of an edo, only exceeded by [[47edo]] in error, which results in it being inconsistent in the [[5-limit]] - combining the best 5/4 (390{{c}}) and the best 6/5 (330{{c}}) will result in the blackwood fifth instead. So some may not consider it a valid perfect fifth.
 
Despite all keys being reachable by stacking this fifth, it does not qualify as meantone either. Instead, it supports [[deeptone]], which tempers out [[177147/163840]] and [[1053/1024]] in the patent val instead of [[81/80]], meaning that four fifths make a near perfect [[16/13|tridecimal neutral third (16/13)]] and it takes a full 11 fifths (i.e. at the augmented third) to reach the 5th harmonic. A simple 5-limit temperament 40edo does support is [[diminished]], equating 6/5 to 300 cents. Diminished temperament is familiar due to also being supported [[12edo]], but 40edo's tuning is quite different from 12edo's (at least with the 12-note or greater MOSes).
 
40edo tempers out [[648/625]] in the 5-limit; [[225/224]] and [[16807/16384]] in the [[7-limit]]; [[99/98]], [[121/120]] and [[176/175]] in the [[11-limit]] - tuning [[orwell]] though highly suboptimally; and [[66/65]] in the 13-limit.
 
81/80 is only tempered out in the 40c alternative [[val]] where the aforementioned high neutral third is equated with 5/4 instead of the much more accurate 390-cent interval. The resulting [[5L 2s]] scale has large steps of 6 intervals and small ones of 5, putting sharps and flats right next to letters without any ups or downs in between and requiring a lot of them to notate more distant keys.  


40edo is more accurate on the 2.9.5.21.33.13.51.19 [[k*N_subgroups| 2*40 subgroup]], where it offers the same tuning as [[80edo|80edo]], and tempers out the same commas. It is also the first equal temperament to approximate both the 23rd and 19th harmonic, by tempering out the 9 cent comma to 4-edo, with 10 divisions therein.
=== Odd harmonics ===
=== Odd harmonics ===
40edo is most accurate on the 2.9.5.21.33.13.51.19.23 [[k*N_subgroups| 2*40 subgroup]], where it offers the same tuning as [[80edo|80edo]], and tempers out the same commas. It is also the first equal temperament to approximate both the 23rd and 19th harmonic, by tempering out the 9 cent comma to 4-edo, with 10 divisions therein.
40edo can be treated as a [[dual-fifth system]] in the 2.3+.3-.5.7.11 subgroup, or the 2.3+.3-.5.7.11.13.19.23 subgroup for those who aren’t intimidated by lots of [[basis element]]s. Both of its fifths can sound [[consonant]] to many listeners.
{{harmonics in equal|40}}
{{harmonics in equal|40}}


== Intervals ==
== Intervals ==
{{Todo|cleanup|inline=1}}
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! #
! #
! style="text-align:center;" |Cents
! style="text-align: center;" | Cents
! colspan="2" |Approximate ratios
! colspan="3" | Notation
!Difference
! Approximate ratios
!colspan="3" |Notation
|-
|-
|0
! 0
|0¢
| 0¢
|1:1
| perfect unison
|0
| P1
|0
| D
|perfect unison
| [[1/1]]
|P1
|D
|-
|-
|1
! 1
|30
| 30
|59:58
| augmented 1sn
|29.5944
| A1
|0.40553
| D#
|augmented 1sn
| [[33/32]], ''[[36/35]]'', [[49/48]], [[81/80|''80/81'']]
|A1
|D#
|-
|-
|2
! 2
|60
| 60
|29:28
| double-aug 1sn
|60.7512
| AA1
| -0.75128
| Dx
|double-aug 1sn
| ''[[21/20]]'', ''[[64/63]]''
|AA1
|Dx
|-
|-
|3
! 3
|90
| 90
|20:19
| triple-aug 1sn, double-dim 2nd
|88.8006
| dd2
|1.19930
| D#x, Ebbb
|double-dim 2nd
| ''[[25/24]]''
|dd2
|D#x, Ebbb
|-
|-
|4
! 4
|120
| 120
|15:14
| diminished 2nd
|119.4428
| d2
|0.55719
| Ebb
|diminished 2nd
|[[14/13]], [[15/14]], [[16/15]]
|d2
|Ebb
|-
|-
|5
! 5
|150
| 150
|12:11
| minor 2nd
|150.6370
| m2
| -0.63705
| Eb
|minor 2nd
| [[11/10]], [[12/11]], [[13/12]]
|m2
|Eb
|-
|-
|6
! 6
|180
| 180
|10:9
| major 2nd
|182.4037
| M2
| -2.40371
| E
|major 2nd
| ''[[9/8]]''
|M2
|E
|-
|-
|7
! 7
|210
| 210
|9:8
| augmented 2nd
|203.9100
| A2
|6.08999
| E#
|augmented 2nd
| ''[[10/9]]''
|A2
|E#
|-
|-
|8
! 8
|240
| 240
|8:7
| double-aug 2nd
|231.1741
| AA2
|8.82590
| Ex
|double-aug 2nd
| [[8/7]], [[15/13]]
|AA2
|Ex
|-
|-
|9
! 9
|270
| 270
|7:6
| double-dim 3rd
|266.8709
| dd3
|3.12909
| Fbb
|double-dim 3rd
| [[7/6]]
|dd3
|Fbb
|-
|-
|10
! 10
|300
| 300
|19:16
| diminished 3rd
|297.5130
| d3
|2.48698
| Fb
|diminished 3rd
| ''[[6/5]]''
|d3
|Fb
|-
|-
|11
! 11
|330
| 330
|6:5
| minor 3rd
|315.6412
| m3
|14.3587
| F
|minor 3rd
|  ''[[27/22]]'', [[39/32]]
|m3
|F
|-
|-
|12
! 12
|360
| 360
|16:13
| major 3rd
|359.4723
| M3
|0.52766
| F#
|major 3rd
| ''[[11/9]]'', [[16/13]]
|M3
|F#
|-
|-
|13
! 13
|390
| 390
|5:4
| augmented 3rd
|386.3137
| A3
|3.68628
| Fx
|augmented 3rd
| [[5/4]]
|A3
|Fx
|-
|-
|14
! 14
|420
| 420
|14:11
| double-aug 3rd, triple-dim 4th
|417.5079
| AA3
|2.49203
| F#x, Gbbb
|double-aug 3rd
| ''[[9/7]]''
|AA3
|F#x, Gbbb
|-
|-
|15
! 15
|450
| 450
|22:17
| double-dim 4th
|446.3625
| dd4
|3.63746
| Gbb
|double-dim 4th
| [[13/10]]
|dd4
|Gbb
|-
|-
|16
! 16
|480
| 480
|21:16
| diminished 4th
|470.781
| d4
|9.219
| Gb
|diminished 4th
| ''[[27/20]]''
|d4
|Gb
|-
|-
|17
! 17
|510
| 510
|4:3
| perfect 4th
|498.0449
| P4
|11.9550
| G
|perfect 4th
| [[4/3]]
|P4
|G
|-
|-
|18
! 18
|540
| 540
|11:8
| augmented 4th
|551.3179
| A4
| -11.3179
| G#
|augmented 4th
| [[11/8]], [[15/11]], ''[[18/13]]''
|A4
|G#
|-
|-
|19
! 19
|570
| 570
|25:18
| double-aug 4th
|568.7174
| AA4
|1.2825
| G##
|double-aug 4th
| [[7/5]]
|AA4
|G##
|-
|-
|20
! 20
|600
| 600
|7:5
| triple-aug 4th,
|582.5121
|17.4878
|triple-aug 4th,
triple-dim 5th
triple-dim 5th
|AAA4,
| AAA4,
ddd5
ddd5
|Gx#, Abbb
| Gx#, Abbb
| ''[[25/18]]'', ''[[36/25]]''
|-
|-
|21
! 21
|630
| 630
|23:16
| double-dim 5th
|628.2743
| dd5
|1.72565
| Abb
|double-dim 5th
|
|dd5
|Abb
|-
|-
|22
! 22
|660
| 660
|16:11
| diminished 5th
|648.6820
| d5
|11.3179
| Ab
|diminished 5th
|
|d5
|Ab
|-
|-
|23
! 23
|690
| 690
|3:2
| perfect 5th
|701.9550
| P5
| -11.9550
| A
|perfect 5th
|
|P5
|A
|-
|-
|24
! 24
|720
| 720
|32:21
| augmented 5th
|729.2191
| A5
| -9.219
| A#
|augmented 5th
|
|A5
|A#
|-
|-
|25
! 25
|750
| 750
|17:11
| double-aug 5th
|753.6374
| AA5
| -3.63746
| Ax
|double-aug 5th
|
|AA5
|Ax
|-
|-
|26
! 26
|780
| 780
|11:7
| double-dim 6th
|782.4920
| dd6
| -2.49203
| A#x, Bbbb
|double-dim 6th
|
|dd6
|A#x, Bbbb
|-
|-
|27
! 27
|810
| 810
| style="text-align:center;" |8:5
| diminished 6th
|813.6862
| d6
| -3.68628
| Bbb
|diminished 6th
| style="text-align: center;" |
|d6
|Bbb
|-
|-
|28
! 28
|840
| 840
|13:8
| minor 6th
|840.5276
| m6
| -0.52766
| Bb
|minor 6th
|
|m6
|Bb
|-
|-
|29
! 29
|870
| 870
| style="text-align:center;" |5:3
| major 6th
|884.3587
| M6
| -14.3587
| B
|major 6th
| style="text-align: center;" |
|M6
|B
|-
|-
|30
! 30
|900
| 900
| style="text-align:center;" |32:19
| augmented 6th
|902.4869
| A6
| -2.48698
| B#
|augmented 6th
| style="text-align: center;" |
|A6
|B#
|-
|-
|31
! 31
|930
| 930
| style="text-align:center;" |12:7
| double-aug 6th
|933.1291
| AA6
| -3.12909
| Bx
|double-aug 6th
| style="text-align: center;" |
|AA6
|Bx
|-
|-
|32
! 32
|960
| 960
| style="text-align:center;" |7:4
| double-dim 7th
|968.8259
| dd7
| -8.82590
| Cbb
|double-dim 7th
| style="text-align: center;" |
|dd7
|Cbb
|-
|-
|33
! 33
|990
| 990
| style="text-align:center;" |16:9
| diminished 7th
|996.0899
| d7
| -6.08999
| Cb
|diminished 7th
| style="text-align: center;" |
|d7
|Cb
|-
|-
|34
! 34
|1020
| 1020
| style="text-align:center;" |9:5
| minor 7th
|1017.5962
| m7
|2.40371
| C
|minor 7th
| style="text-align: center;" |
|m7
|C
|-
|-
|35
! 35
|1050
| 1050
| style="text-align:center;" |11:6
| major 7th
|1049.3629
| M7
|0.63705
| C#
|major 7th
| style="text-align: center;" |
|M7
|C#
|-
|-
|36
! 36
|1080
| 1080
| style="text-align:center;" |28:15
| augmented 7th
|1080.5571
| A7
| -0.55719
| Cx
|augmented 7th
| style="text-align: center;" |
|A7
|Cx
|-
|-
|37
! 37
|1110
| 1110
| style="text-align:center;" |19:10
| double-aug 7th, triple-dim 8ve
|1111.1993
| AA7
| -1.19930
| C#x, Dbbb
|double-aug 7th
| style="text-align: center;" |
|AA7
|C#x, Dbbb
|-
|-
|38
! 38
|1140
| 1140
| style="text-align:center;" |56:29
| double-dim 8ve
|1139.2487
| dd8
|0.75128
| Dbb
|double-dim 8ve
| style="text-align: center;" |
|dd8
|Dbb
|-
|-
|39
! 39
|1170
| 1170
| style="text-align:center;" |116:59
| diminished 8ve
|1170.4055
| d8
| -0.40553
| Db
|diminished 8ve
| style="text-align: center;" |
|d8
|Db
|-
|-
|40
! 40
|1200
| 1200
| style="text-align:center;" |2:1
| perfect octave
|1200
| P8
|0
| D
|perfect octave
| style="text-align: center;" |
|P8
|D
|}
|}


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
== Notation ==
[[Category:Subgroup]]
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[30edo#Second-best fifth notation|30b]] and [[35edo#Sagittal notation|35]].
 
<imagemap>
File:40-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 647 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:40-EDO_Sagittal.svg]]
</imagemap>
 
== Octave stretch or compression ==
'''127ed9''' optimises 40edo for [[dual-fifth]] usage by distributing error evenly between its two fifths. 127ed9 is just 40edo with [[octave shrinking|octaves compressed]] by 1.9{{c}}.
{{harmonics in equal|127|9|1|intervals=integer|collapsed=yes}}
{{harmonics in equal|40|2|1|intervals=integer|collapsed=yes}}
 
== Scales ==
* Amulet{{idiosyncratic}}, (approximated from [[magic]] in [[25edo]]): 3 2 3 3 2 3 5 3 3 2 3 5 3
* [[Equipentatonic]]: 8 8 8 8 8
* Evacuated planet{{idiosyncratic}} (approximated from [[66afdo|66]][[afdo]]): 4 13 6 12 5
* Approximations of [[gamelan]] scales:
** 5-tone pelog: 4 5 14 3 14
** 7-tone pelog: 4 5 8 6 3 10 4
** 5-tone slendro: 8 8 8 8 8
* 12-tone 4&10edo scale: 4 4 2 2 4 4 4 4 2 2 4 4
* 12-tone 5&8edo scale: 5 3 2 5 1 4 4 1 5 2 3 5
 
{| class="wikitable mw-collapsible mw-collapsed"
|+Approximated from [[96edo]]
|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
* Flattened major: 7 6 4 6 7 7 3
* Sharpened minor: 7 3 7 6 4 7 6
* Sharpened harmonic minor: 7 3 7 7 3 11 2
* Flattened major pentatonic: 6 7 10 7 10
* Sharpened minor pentatonic: 10 7 6 11 6
* Evened minor hexatonic: 6 4 7 6 10 7
* Roughened augmented: 11 3 10 3 10 3
* Evened dominant pentatonic: 7 7 9 10 7
* Sharpened Dorian: 7 3 7 7 7 3 6
* Flattened Ionian pentatonic: 13 3 7 13 4
* Sharpened Dorian harmonic: 7 3 10 4 7 3 6
* Evened Mixolydian pentatonic: 13 4 6 10 7
* Evened Phrygian dominant: 4 9 4 6 4 6 7
* Evened Phrygian dominant hexatonic: 3 10 4 6 10 7
* Sharpened Phrygian pentatonic: 4 7 13 3 13
* Sharpened minor harmonic pentatonic I: 7 3 14 13 3
* Evened hirajoshi: 7 4 12 4 13
* Sharpened hirajoshi: 7 4 13 4 12
* Extra-sharp hirajoshi: 8 3 13 4 12
* Evened akebono I: 6 5 12 6 11
* Sharpened akebono I: 7 3 14 6 10
* Extra-sharp akebono I: 7 4 13 7 9
* Evened Javanese pentachordal: 4 7 9 4 16
* Moonbeam: 7 4 12 14 3
* Palace (type of [[equiheptatonic]]): 5 6 6 6 6 6 5
* Underpass: 11 12 8 3 6
|}
 
== Instruments ==
* [[Lumatone mapping for 40edo]]
 
== Music ==
=== Modern Renderings ===
; [https://www.nts.live/artists/83437-hi-posi Hi-Posi]
* [https://www.youtube.com/shorts/YY8rZ6ZcoXc ''Boku De Artitai''] (1994) – microtonal cover in 40edo by [[Bryan Deister]] (2026)
 
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/x5cnT4Bw1ZQ ''Balance Beam''] (2026)
* [https://www.youtube.com/shorts/fREuVtq875U ''40edo improv''] (2026)
* [https://www.youtube.com/shorts/JriWAth4vBY ''40edo idea''] (2026)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=EMZu6ZE6A3g ''Happy Birthday Canon'', 6-in-1 Canon in 40edo]
* [https://www.youtube.com/watch?v=eu854Ld_uLE ''Canon on Twinkle Twinkle Little Star'', for Baroque Oboe & Viola] (2023) – ([https://www.youtube.com/watch?v=l7vDHwsboLE for Organ])
 
; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=tLmaQK10aYM ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!''] (2025; mostly in 42edo, but also some in 40edo)
** [https://www.youtube.com/watch?v=NE77rwCsGHw live performance of the above in Munich, Germany] (2026)
 
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/40edo"