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 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