# 40edo

 ← 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 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, 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 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 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.

### Odd harmonics

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 Approximate ratios Difference Notation
0 1:1 0 0 perfect unison P1 D
1 30 59:58 29.5944 0.40553 augmented 1sn A1 D#
2 60 29:28 60.7512 -0.75128 double-aug 1sn AA1 Dx
3 90 20:19 88.8006 1.19930 double-dim 2nd dd2 D#x, Ebbb
4 120 15:14 119.4428 0.55719 diminished 2nd d2 Ebb
5 150 12:11 150.6370 -0.63705 minor 2nd m2 Eb
6 180 10:9 182.4037 -2.40371 major 2nd M2 E
7 210 9:8 203.9100 6.08999 augmented 2nd A2 E#
8 240 8:7 231.1741 8.82590 double-aug 2nd AA2 Ex
9 270 7:6 266.8709 3.12909 double-dim 3rd dd3 Fbb
10 300 19:16 297.5130 2.48698 diminished 3rd d3 Fb
11 330 6:5 315.6412 14.3587 minor 3rd m3 F
12 360 16:13 359.4723 0.52766 major 3rd M3 F#
13 390 5:4 386.3137 3.68628 augmented 3rd A3 Fx
14 420 14:11 417.5079 2.49203 double-aug 3rd AA3 F#x, Gbbb
15 450 22:17 446.3625 3.63746 double-dim 4th dd4 Gbb
16 480 21:16 470.781 9.219 diminished 4th d4 Gb
17 510 4:3 498.0449 11.9550 perfect 4th P4 G
18 540 11:8 551.3179 -11.3179 augmented 4th A4 G#
19 570 25:18 568.7174 1.2825 double-aug 4th AA4 G##
20 600 7:5 582.5121 17.4878 triple-aug 4th,

triple-dim 5th

AAA4,

ddd5

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

Claudi Meneghin