40edo
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 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
| 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 | 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 |