42edo

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	Size (Cents)	relative notation		absolute notation
0	0.000	perfect unison	P1	D
1	29.571	up 1sn, minor 2nd	^1, m2	^D, Eb
2	57.143	double-up 1sn, upminor 2nd	^^1, ^m2	^^D, ^Eb
3	85.714	double-up minor 2nd	^^m2	^^Eb
4	114.286	downmid 2nd	v~2	^3Eb
5	143.857	upmid 2nd	^~2	v3E
6	171.429	double-down major 2nd	vvM2	vvE
7	200	downmajor 2nd	vM2	vE
8	228.571	major 2nd	M2	E
9	257.143	minor 3rd	m3	F
10	285.714	upminor 3rd	^m3	^F
11	314.286	double-up minor 3rd	^^m3	^^F
12	342.857	downmid 3rd	v~3	^3F
13	371.429	upmid 3rd	^~3	v3F#
14	400	double-down major 3rd	vvM3	vvF#
15	428.571	downmajor 3rd	vM3	vF#
16	457.143	major 3rd, down 4th	M3, v4	F#, vG
17	485.714	perfect 4th	P4	G
18	514.286	up 4th	^4	^G
19	543.857	double-up 4th	^^4	^^G
20	571.429	downmid 4th	v~4	^3G
21	600	upmid 4th, downmid 5th	^~4, v~5	v3G#, ^3Ab
22	628.571	upmid 5th	^~5	v3A
23	657.143	double-down 5th	vv5	vvA
24	685.714	down 5th	v5	vA
25	714.286	perfect 5th	P5	A
26	742.857	up 5th, minor 6th	^5, m6	^A, Bb
27	771.429	upminor 6th	^m6	^Bb
28	800	double-up minor 6th	^^m6	^^Bb
29	829.571	downmid 6th	v~6	^3Bb
30	857.143	upmid 6th	^~6	v3B
31	885.714	double-down major 6th	vvM6	vvB
32	914.286	downmajor 6th	vM6	vB
33	942.857	major 6th	M6	B
34	971.429	minor 7th	m7	C
35	1000	upminor 7th	^m7	^C
36	1028.571	double-up minor 7th	^^m7	^^C
37	1057.143	downmid 7th	v~7	^3C
38	1085.714	upmid 7th	^~7	v3C#
39	1114.286	double-down major 7th	vvM7	vvC#
40	1142.857	downmajor 7th	vM7	vC#
41	1171.429	major 7th, down 8ve	M7, v8	C#, vD
42	1200	perfect 8ve	P8	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.