# 42edo

 ← 41edo 42edo 43edo →
Prime factorization 2 × 3 × 7
Step size 28.5714¢
Fifth 25\42 (714.286¢)
Semitones (A1:m2) 7:1 (200¢ : 28.57¢)
Dual sharp fifth 25\42 (714.286¢)
Dual flat fifth 24\42 (685.714¢) (→4\7)
Dual major 2nd 7\42 (200¢) (→1\6)
Consistency limit 7
Distinct consistency limit 7

42 equal divisions of the octave (abbreviated 42edo or 42ed2), also called 42-tone equal temperament (42tet) or 42 equal temperament (42et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 42 equal parts of about 28.6 ¢ each. Each step represents a frequency ratio of 21/42, or the 42nd root of 2.

## Theory

42edo has a patent val fifth (the step of which is not from 7edo, this being a first for EDOs of the form 7n) and a third both over 12 cents sharp, using the same 400 cent interval to represent 5/4 as does 12edo, which means it tempers out 128/125. In the 7-limit, it tempers out 64/63 and 126/125, making it a tuning supporting augene temperament.

While not an accurate tuning on the full 7-limit, it does an excellent job on the 2.9.15.7.33.39 2*42 subgroup, having the same tuning on it as does 84edo. On this subgroup 42 has the same commas as 84.

42edo is a diatonic edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. 42edo is one of the most difficult diatonic edos to notate, because no other diatonic edo's 5th is as sharp (see 47edo for the opposite extreme). Assuming the natural notes form a chain of fifths, the major 2nd is 8 edosteps and the minor 2nd is only one. The naturals create a 5edo-like scale, with two of the notes inflected by a comma-sized edostep:

D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D

D# is next to E. The notation requires triple ups and downs, even more if chords are to be spelled correctly. For example, a 1/1 - 5/4 - 3/2 - 9/5 chord with a root on the key or fret midway between G and A would be written either as v3G# - v5B# - v3D# - vF# or as ^3Ab - ^C - ^3Eb - ^5Gb. This is a dud dup-seven chord, written either as v3G#vv,^^7 or as ^3Abvv,^^7.

### Odd harmonics

Approximation of odd harmonics in 42edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +12.3 +13.7 +2.6 -3.9 -8.5 -12.0 -2.6 +9.3 -11.8 -13.6 +0.3
Relative (%) +43.2 +47.9 +9.1 -13.7 -29.6 -41.8 -8.9 +32.7 -41.3 -47.7 +1.0
Steps
(reduced)
67
(25)
98
(14)
118
(34)
133
(7)
145
(19)
155
(29)
164
(38)
172
(4)
178
(10)
184
(16)
190
(22)

## Intervals

# Cents Ups and Downs Notation
0 0.000 P1 perfect unison D
1 28.571 ^1, m2 up unison, minor 2nd ^D, Eb
2 57.143 ^^1, ^m2 dup 1sn, upminor 2nd ^^D, ^Eb
3 85.714 ^^m2 dupminor 2nd ^^Eb
4 114.286 ^3m trupminor 2nd ^3Eb
5 143.857 v3M trudmajor 2nd v3E
6 171.429 vvM2 dudmajor 2nd vvE
7 200.000 vM2 downmajor 2nd vE
8 228.571 M2 major 2nd E
9 257.143 m3 minor 3rd F
10 285.714 ^m3 upminor 3rd ^F
11 314.286 ^^m3 dupminor 3rd ^^F
12 342.857 ^3m3 trupminor 3rd ^3F
13 371.429 v3M3 trudmajor 3rd v3F#
14 400.000 vvM3 dudmajor 3rd vvF#
15 428.571 vM3 downmajor 3rd vF#
16 457.143 M3, v4 major 3rd, down 4th F#, vG
17 485.714 P4 perfect 4th G
18 514.286 ^4 up 4th ^G
19 543.857 ^^4 dup 4th ^^G
20 571.429 ^34, ^^d5 trup 4th, dupdim 5th ^3G, ^^Ab
21 600.000 v3A4, ^3d5 trudaug 4th, trupdim 5th v3G#, ^3Ab
22 628.571 vvA4, v35 dudaug 4th, trud 5th vvG#, v3A
23 657.143 vv5 dud 5th vvA
24 685.714 v5 down 5th vA
25 714.286 P5 perfect 5th A
26 742.857 ^5, m6 up 5th, minor 6th ^A, Bb
27 771.429 ^m6 upminor 6th ^Bb
28 800.000 ^^m6 dupminor 6th ^^Bb
29 828.571 ^3m6 trupminor 6th ^3Bb
30 857.143 v3M6 trudmajor 6th v3B
31 885.714 vvM6 dudmajor 6th vvB
32 914.286 vM6 downmajor 6th vB
33 942.857 M6 major 6th B
34 971.429 m7 minor 7th C
35 1000.000 ^m7 upminor 7th ^C
36 1028.571 ^^m7 dupminor 7th ^^C
37 1057.143 ^3m7 trupminor 7th ^3C
38 1085.714 v3M7 trudmajor 7th v3C#
39 1114.286 vvM7 dudmajor 7th vvC#
40 1142.857 vM7 downmajor 7th vC#
41 1171.429 M7, v8 major 7th, down 8ve C#, vD
42 1200.000 P8 perfect 8ve D

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and Downs Notation #Chords and Chord Progressions.

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