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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. This fifth qualifies for [[flattone]], a variant of meantone with flat fifths, although 40edo's fifth is a bit extreme even for flattone. 40edo's fifth is flat enough that the meantone major third falls into submajor or even sharp neutral third territory at 360 cents, while the minor third is supraminor although not quite high enough to be considered neutral at 330 cents. 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 up to 3 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" |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"