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

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 the augene temperament.

42edo is on the optimal ET sequence of the eugene, joan, lemba, neutron, qeema, seville, sevond, skateboard, tritikleismic and vines temperaments.

While not an accurate tuning on the full 7-limit, 42edo 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).

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 (%)	+43.2	+47.9	+9.1	-13.7	-29.6	-41.8	-8.9	+32.7	-41.3	-47.7	+1.0
Steps (reduced)		67 (25)	98 (14)	118 (34)	133 (7)	145 (19)	155 (29)	164 (38)	172 (4)	178 (10)	184 (16)	190 (22)

Subsets and supersets

Since 42 factors into 2 × 3 × 7, 42edo contains subset edos 2, 3, 6, 7, 14, and 21.

Octave stretch

42edo’s inaccurate 3rd and 5th harmonics can be greatly improved through octave stretching or octave shrinking. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with.

The following is a table comparing stretched and shrunk tunings of 42edo:

Key

• “Great” = <20% relative error
• “Good” = 20 to 30% relative error
• “Poor” = >30% relative error

Tuning	42ed257/128 (~41.77edo)	189zpi (~41.83edo)	42edo	191zpi (~42.19edo)	42ed255/128 (~42.24edo)
Summary	great: 3, 5 good: 2, 7 poor: 11, 13	great: 2, 5 good: 3, 11, 13 poor: 7	great: 2, 7 good: 11 poor: 3, 5, 13	great: 2, 3, 5, 11, 13 good: - poor: 7	great: 3, 5, 11 good: 2 poor: 7, 13
2/1 relative error	23.5%	17.2%	0.0%	18.8%	23.9%
3/1 rel. err.	19.6%	29.6%	43.2%	13.4%	5.4%
5/1 rel. err.	2.4%	12.2%	47.9%	4.3%	7.5%
7/1 rel. err.	24.9%	42.6%	9.1%	43.6%	42.2%
11/1 rel. err.	48.3%	29.8%	29.6%	5.5%	12.1%
13/1 rel. err.	45.1%	21.7%	41.8%	11.3%	30.1%

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	^3m	trupminor 2nd	^3Eb
5	143.857	v3M	trudmajor 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	^3m3	trupminor 3rd	^3F
13	371.429	v3M3	trudmajor 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	^34, ^^d5	trup 4th, dupdim 5th	^3G, ^^Ab
21	600.000	v3A4, ^3d5	trudaug 4th, trupdim 5th	v3G#, ^3Ab
22	628.571	vvA4, v35	dudaug 4th, trud 5th	vvG#, 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	^3m6	trupminor 6th	^3Bb
30	857.143	v3M6	trudmajor 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	^3m7	trupminor 7th	^3C
38	1085.714	v3M7	trudmajor 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.

Notation

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.

Instruments

Lumatone

See Lumatone mapping for 42edo

Music

• Improvisation in 42edo - composed and played by Bryan Deister (May 2023), transcribed by Stephen Weigel (Sept 2024)
• Glory of Them - Mundoworld (July 2024)