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{{ED intro}}
{{ED intro}}
== Theory ==
== Theory ==
Up to this point, all the multiples of 5 have had the 720{{c}} 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 the use of 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 5/4 and 6/5 will result in the blackwood 5th instead. So some may not consider it a valid perfect fifth.
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 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 [[16/13|tridecimal neutral 3rd]] and it takes a full 11 to reach the 5th harmonic.  
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).


81/80 is only tempered out in the 40c alternative val where the aforementioned high neutral 3rd 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. It 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; and [[66/65]] in the 13-limit.
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.  


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


Line 459: Line 379:


== Music ==
== 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]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/x5cnT4Bw1ZQ ''Balance Beam''] (2026)
* [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]]
; [[Claudi Meneghin]]
Line 468: Line 395:
; [[Stephen Weigel]]
; [[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=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:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/40edo"