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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, as stacking 4 of them results in a near perfect tridecimal neutral third rather than a major one. 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 [[tempering_out|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 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.
 
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 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. As a dual-fifth system, it really shines, as both of its fifths have low enough [[harmonic entropy]] to sound [[consonant]] to many listeners, giving two consonant intervals for the price of one.
{{harmonics in equal|40}}
{{harmonics in equal|40}}


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


[[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>
 
== 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
 
== Music ==
; [[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])
 
== Instruments ==
* [[Lumatone mapping for 40edo]]
 
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/40edo"