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


== Notation ==
== Notation ==
===Sagittal notation===
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[30edo#Second-best fifth notation|30b]] and [[35edo#Sagittal notation|35]].
This notation uses the same sagittal sequence as EDOs [[30edo#Second-best fifth notation|30b]] and [[35edo#Sagittal notation|35]].


Retrieved from "https://en.xen.wiki/w/40edo"