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	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.