42edo

← 41edo

42edo

43edo →

Prime factorization	2 × 3 × 7
Step size	28.5714¢
Fifth	25\42 (714.286¢)
Semitones (A1:m2)	7:1 (200¢ : 28.57¢)
Dual sharp fifth	25\42 (714.286¢)
Dual flat fifth	24\42 (685.714¢) (→4\7)
Dual major 2nd	7\42 (200¢) (→1\6)
Consistency limit	7
Distinct consistency limit	7

42 equal divisions of the octave (abbreviated 42edo or 42ed2), also called 42-tone equal temperament (42tet) or 42 equal temperament (42et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 42 equal parts of about 28.6 ¢ each. Each step represents a frequency ratio of 2^1/42, or the 42nd root of 2.

Theory

42edo has a patent val fifth (the step of which is not from 7edo, this being a first for EDOs of the form 7n) and a third both over 12 cents sharp, using the same 400 cent interval to represent 5/4 as does 12edo, 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 dud dup-seven chord, written either as v³G#vv,^^7 or as ^³Abvv,^^7.

Odd harmonics

Approximation of odd harmonics in 42edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+12.3	+13.7	+2.6	-3.9	-8.5	-12.0	-2.6	+9.3	-11.8	-13.6	+0.3
	relative (%)	+43	+48	+9	-14	-30	-42	-9	+33	-41	-48	+1
Steps (reduced)		67 (25)	98 (14)	118 (34)	133 (7)	145 (19)	155 (29)	164 (38)	172 (4)	178 (10)	184 (16)	190 (22)

Intervals

#	Cents	Ups and Downs Notation
0	0.000	P1	perfect unison	D
1	28.571	^1, m2	up unison, minor 2nd	^D, Eb
2	57.143	^^1, ^m2	dup 1sn, upminor 2nd	^^D, ^Eb
3	85.714	^^m2	dupminor 2nd	^^Eb
4	114.286	v~2	downmid 2nd	^3Eb
5	143.857	^~2	upmid 2nd	v3E
6	171.429	vvM2	dudmajor 2nd	vvE
7	200.000	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	dupminor 3rd	^^F
12	342.857	v~3	downmid 3rd	^3F
13	371.429	^~3	upmid 3rd	v3F#
14	400.000	vvM3	dudmajor 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	dup 4th	^^G
20	571.429	v~4	downmid 4th	^3G
21	600.000	^~4, v~5	upmid 4th, downmid 5th	v3G#, ^3Ab
22	628.571	^~5	upmid 5th	v3A
23	657.143	vv5	dud 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	800.000	^^m6	dupminor 6th	^^Bb
29	828.571	v~6	downmid 6th	^3Bb
30	857.143	^~6	upmid 6th	v3B
31	885.714	vvM6	dudmajor 6th	vvB
32	914.286	vM6	downmajor 6th	vB
33	942.857	M6	major 6th	B
34	971.429	m7	minor 7th	C
35	1000.000	^m7	upminor 7th	^C
36	1028.571	^^m7	dupminor 7th	^^C
37	1057.143	v~7	downmid 7th	^3C
38	1085.714	^~7	upmid 7th	v3C#
39	1114.286	vvM7	dudmajor 7th	vvC#
40	1142.857	vM7	downmajor 7th	vC#
41	1171.429	M7, v8	major 7th, down 8ve	C#, vD
42	1200.000	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.

Instruments

Lumatone

See Lumatone mapping for 42edo