40edo
| ← 39edo | 40edo | 41edo → |
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 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.
| 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.
Scales
- Amulet [idiosyncratic term ], (approximated from magic in 25edo): 3 2 3 3 2 3 5 3 3 2 3 5 3
- Equipentatonic: 8 8 8 8 8
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Todo: complete section, add examples |
Music
- Happy Birthday Canon, 6-in-1 Canon in 40edo
- Canon on Twinkle Twinkle Little Star, for Baroque Oboe & Viola (2023) – (for Organ)