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 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