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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]] fifth as their best approximation of [[3/2]]. 35edo combined 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, 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.
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.  
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.
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.
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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
|}
|}


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