40edo

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40edo is the equal division of the octave into 40 parts of exactly 30 cents each. 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 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.

Step # ET Just Difference

(ET minus Just)

Notation
Cents Interval Cents
0 1:1 0 0 Unison 1 D
1 30 59:58 29.5944 0.40553 Up Unison D#
2 60 29:28 60.7512 -0.75128 Downminor 2nd D##
3 90 20:19 88.8006 1.19930 Minor 2nd D###/Ebbb
4 120 15:14 119.4428 0.55719 Upminor 2nd Ebb
5 150 12:11 150.6370 -0.63705 Downmajor 2nd Eb
6 180 10:9 182.4037 -2.40371 Major 2nd E
7 210 9:8 203.9100 6.08999 Upmajor 2nd E#
8 240 8:7 231.1741 8.82590 Augmented 2nd E##
9 270 7:6 266.8709 3.12909 Diminished 3rd Fbb
10 300 19:16 297.5130 2.48698 Downminor 3rd Fb
11 330 6:5 315.6412 14.3587 Upminor 3rd F
12 360 16:13 359.4723 0.52766 Neutral 3rd F#
13 390 5:4 386.3137 3.68628 Major 3rd F##
14 420 14:11 417.5079 2.49203 Augmented 3rd F###/Gbbb
15 450 22:17 446.3625 3.63746 Diminished 4th Gbb
16 480 21:16 470.781 9.219 Blackwood 4th Gb
17 510 4:3 498.0449 11.9550 Diatonic 4th G
18 540 11:8 551.3179 -11.3179 Augmented 4th G#
19 570 25:18 568.7174 1.2825 Minor Tritone G##
20 600 7:5 582.5121 17.4878 Perfect Tritone G###/Abbb
21 630 23:16 628.2743 1.72565 Major Tritone Abb
22 660 16:11 648.6820 11.3179 Diminished 5th Ab
23 690 3:2 701.9550 -11.9550 Diatonic 5th A
24 720 32:21 729.2191 -9.219 Blackwood 5th A#
25 750 17:11 753.6374 -3.63746 Augmented 5th A##
26 780 11:7 782.4920 -2.49203 Diminished 6th A###/Bbbb
27 810 8:5 813.6862 -3.68628 Minor 6th Bbb
28 840 13:8 840.5276 -0.52766 Neutral 6th Bb
29 870 5:3 884.3587 -14.3587 Downmajor 6th B
30 900 32:19 902.4869 -2.48698 Upmajor 6th B#
31 930 12:7 933.1291 -3.12909 Augmented 6th B##
32 960 7:4 968.8259 -8.82590 Harmonic 7th Cbb
33 990 16:9 996.0899 -6.08999 Downminor 7th Cb
34 1020 9:5 1017.5962 2.40371 Minor 7th C
35 1050 11:6 1049.3629 0.63705 Upminor 7th C#
36 1080 28:15 1080.5571 -0.55719 Downmajor 7th C##
37 1110 19:10 1111.1993 -1.19930 Major 7th C###/Dbbb
38 1140 56:29 1139.2487 0.75128 Upmajor 7th Dbb
39 1170 116:59 1170.4055 -0.40553 Down Octave Db
40 1200 2:1 1200 0 Octave D