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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, with slightly sharper fifths like those of [[26edo]] or [[45edo]] being considered optimal 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]]

Latest revision as of 13:41, 5 May 2025

← 39edo 40edo 41edo →
Prime factorization 23 × 5
Step size 30 ¢ 
Fifth 23\40 (690 ¢)
Semitones (A1:m2) 1:5 (30 ¢ : 150 ¢)
Dual sharp fifth 24\40 (720 ¢) (→ 3\5)
Dual flat fifth 23\40 (690 ¢)
Dual major 2nd 7\40 (210 ¢)
Consistency limit 3
Distinct consistency limit 3

40 equal divisions of the octave (abbreviated 40edo or 40ed2), also called 40-tone equal temperament (40tet) or 40 equal temperament (40et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 40 equal parts of exactly 30 ¢ each. Each step represents a frequency ratio of 21/40, or the 40th root of 2.

Theory

Up to this point, all the multiples of 5 have had the 720 ¢ 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 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.

Odd harmonics

40edo is most accurate on the 2.9.5.21.33.13.51.19.23 2*40 subgroup, where it offers the same tuning as 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 elements. 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.

Approximation of odd harmonics in 40edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -12.0 +3.7 -8.8 +6.1 -11.3 -0.5 -8.3 -15.0 +2.5 +9.2 +1.7
Relative (%) -39.9 +12.3 -29.4 +20.3 -37.7 -1.8 -27.6 -49.9 +8.3 +30.7 +5.8
Steps
(reduced) 		63
(23) 		93
(13) 		112
(32) 		127
(7) 		138
(18) 		148
(28) 		156
(36) 		163
(3) 		170
(10) 		176
(16) 		181
(21)

Intervals

# Cents Notation Approximate ratios Difference
0 0¢ perfect unison P1 D 1:1 0 0
1 30 augmented 1sn A1 D# 59:58 29.5944 0.40553
2 60 double-aug 1sn AA1 Dx 29:28 60.7512 -0.75128
3 90 double-dim 2nd dd2 D#x, Ebbb 20:19 88.8006 1.19930
4 120 diminished 2nd d2 Ebb 15:14 119.4428 0.55719
5 150 minor 2nd m2 Eb 12:11 150.6370 -0.63705
6 180 major 2nd M2 E 10:9 182.4037 -2.40371
7 210 augmented 2nd A2 E# 9:8 203.9100 6.08999
8 240 double-aug 2nd AA2 Ex 8:7 231.1741 8.82590
9 270 double-dim 3rd dd3 Fbb 7:6 266.8709 3.12909
10 300 diminished 3rd d3 Fb 19:16 297.5130 2.48698
11 330 minor 3rd m3 F 6:5 315.6412 14.3587
12 360 major 3rd M3 F# 16:13 359.4723 0.52766
13 390 augmented 3rd A3 Fx 5:4 386.3137 3.68628
14 420 double-aug 3rd AA3 F#x, Gbbb 14:11 417.5079 2.49203
15 450 double-dim 4th dd4 Gbb 22:17 446.3625 3.63746
16 480 diminished 4th d4 Gb 21:16 470.781 9.219
17 510 perfect 4th P4 G 4:3 498.0449 11.9550
18 540 augmented 4th A4 G# 11:8 551.3179 -11.3179
19 570 double-aug 4th AA4 G## 25:18 568.7174 1.2825
20 600 triple-aug 4th,

triple-dim 5th

AAA4,

ddd5

Gx#, Abbb 7:5 582.5121 17.4878
21 630 double-dim 5th dd5 Abb 23:16 628.2743 1.72565
22 660 diminished 5th d5 Ab 16:11 648.6820 11.3179
23 690 perfect 5th P5 A 3:2 701.9550 -11.9550
24 720 augmented 5th A5 A# 32:21 729.2191 -9.219
25 750 double-aug 5th AA5 Ax 17:11 753.6374 -3.63746
26 780 double-dim 6th dd6 A#x, Bbbb 11:7 782.4920 -2.49203
27 810 diminished 6th d6 Bbb 8:5 813.6862 -3.68628
28 840 minor 6th m6 Bb 13:8 840.5276 -0.52766
29 870 major 6th M6 B 5:3 884.3587 -14.3587
30 900 augmented 6th A6 B# 32:19 902.4869 -2.48698
31 930 double-aug 6th AA6 Bx 12:7 933.1291 -3.12909
32 960 double-dim 7th dd7 Cbb 7:4 968.8259 -8.82590
33 990 diminished 7th d7 Cb 16:9 996.0899 -6.08999
34 1020 minor 7th m7 C 9:5 1017.5962 2.40371
35 1050 major 7th M7 C# 11:6 1049.3629 0.63705
36 1080 augmented 7th A7 Cx 28:15 1080.5571 -0.55719
37 1110 double-aug 7th AA7 C#x, Dbbb 19:10 1111.1993 -1.19930
38 1140 double-dim 8ve dd8 Dbb 56:29 1139.2487 0.75128
39 1170 diminished 8ve d8 Db 116:59 1170.4055 -0.40553
40 1200 perfect octave P8 D 2:1 1200 0

Notation

Sagittal notation

This notation uses the same sagittal sequence as EDOs 30b and 35.

Sagittal notationPeriodic table of EDOs with sagittal notationlimma-fraction notation

Scales

Music

Claudi Meneghin

Instruments

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