42edo

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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 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 double-down double-up-seven chord, written either as v3G#vv,^^7 or as ^3Abvv,^^7.

Intervals of 42edo

Degree Cents ups and downs notation
0 0.000 P1 perfect unison D
1 29.571 ^1, m2 up unison, minor 2nd ^D, Eb
2 57.143 ^^1, ^m2 double-up 1sn, upminor 2nd ^^D, ^Eb
3 85.714 ^^m2 double-up minor 2nd ^^Eb
4 114.286 v~2 downmid 2nd ^3Eb
5 143.857 ^~2 upmid 2nd v3E
6 171.429 vvM2 double-down major 2nd vvE
7 200 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 double-up minor 3rd ^^F
12 342.857 v~3 downmid 3rd ^3F
13 371.429 ^~3 upmid 3rd v3F#
14 400 vvM3 double-down major 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 double-up 4th ^^G
20 571.429 v~4 downmid 4th ^3G
21 600 ^~4, v~5 upmid 4th, downmid 5th v3G#, ^3Ab
22 628.571 ^~5 upmid 5th v3A
23 657.143 vv5 double-down 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 ^^m6 double-up minor 6th ^^Bb
29 829.571 v~6 downmid 6th ^3Bb
30 857.143 ^~6 upmid 6th v3B
31 885.714 vvM6 double-down major 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 ^m7 upminor 7th ^C
36 1028.571 ^^m7 double-up minor 7th ^^C
37 1057.143 v~7 downmid 7th ^3C
38 1085.714 ^~7 upmid 7th v3C#
39 1114.286 vvM7 double-down major 7th vvC#
40 1142.857 vM7 downmajor 7th vC#
41 1171.429 M7, v8 major 7th, down 8ve C#, vD
42 1200 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.