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The 42 equal division divides the octave into 42 equal parts of 28.571 cents each. It has a 3 (the size of which being coprime to its cardinality, this being a first for a composite equal division of cardinality 7n) and a 5 both over 12 cents sharp, using the same 400 cent interval to represent 5/4 as does 12, 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 regular edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. 42edo is one of the most difficult regular edos to notate, because no other regular 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 roughly 5edo-ish 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 G#v3 - B#v5 - D#v3 - F#v or as Ab^3 - C^ - Eb^3 - Gb^5. This is a dot-double-down double-up-seven chord, written either as G#v3.vv,^^7 or as Ab^3.vv,^^7.

Intervals of 42edo


Degree
Size (Cents)
relative notation
absolute notation
0
0
perfect unison
P1
D
1
29
up 1sn, minor 2nd
^1, m2
D^, Eb
2
57
double-up 1sn, upminor 2nd
^^1, ^m2
D^^, Eb^
3
86
double-up minor 2nd
^^m2
Eb^^
4
114
downmid 2nd
v~2
Eb^3
5
143
upmid 2nd
^~2
Ev3
6
171
double-down major 2nd
vvM2
Evv
7
200
downmajor 2nd
vM2
Ev
8
229
major 2nd
M2
E
9
257
minor 3rd
m3
F
10
286
upminor 3rd
^m3
F^
11
314
double-up minor 3rd
^^m3
F^^
12
343
downmid 3rd
v~3
F^3
13
371
upmid 3rd
^~3
F#v3
14
400
double-down major 3rd
vvM3
F#vv
15
429
downmajor 3rd
vM3
F#v
16
457
major 3rd, down 4th
M3, v4
F#, Gv
17
486
perfect 4th
P4
G
18
514
up 4th
^4
G^
19
543
double-up 4th
^^4
G^^
20
571
triple-up 4th
^34
G^3
21
600
triple-down aug 4th, triple-up dim 5th
v3A4, ^3d5
G#v3, Ab^3
22
629
triple-down 5th
v35
Av3
23
657
double-down 5th
vv5
Avv
24
686
down 5th
v5
Av
25
714
perfect 5th
P5
A
26
743
up 5th, minor 6th
^5, m6
A^, Bb
27
771
upminor 6th
^m6
Bb^
28
800
double-up minor 6th
^^m6
Bb^^
29
829
downmid 6th
v~6
Bb^3
30
857
upmid 6th
^~6
Bv3
31
886
double-down major 6th
vvM6
Bvv
32
914
downmajor 6th
vM6
Bv
33
943
major 6th
M6
B
34
971
minor 7th
m7
C
35
1000
upminor 7th
^m7
C^
36
1029
double-up minor 7th
^^m7
C^^
37
1057
downmid 7th
v~7
C^3
38
1086
upmid 7th
^~7
C#v3
39
1114
double-down major 7th
vvM7
C#vv
40
1143
downmajor 7th
vM7
C#v
41
1171
major 7th, down 8ve
M7, v8
C#, Dv
42
1200
perfect 8ve
P8
D