# 40edo

← 39edo | 40edo | 41edo → |

^{3}× 5**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 2^{1/40}, 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

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 |