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 the augene temperament.

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

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 fifth is as sharp (see 47edo for the opposite extreme).

Odd harmonics

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.

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
Error	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)

Octave stretch

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

The following table compares three stretched tunings of 42edo:

Tuning	42ed257/128	APS720jot	189zpi	42edo
Steps / octave	~41.77	~41.81	~41.83	42.00
Approximation of harmonics	great: 5 good: 2, 3, 7 okay: bad: 11, 13	great: 5 good: 2, 3 okay: 7, 11, 13 bad:	great: 5 good: 2, 13 okay: 3, 11 bad: 7	great: 2, 7 good: okay: 11 bad: 3, 5, 13
“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%

The following table compares three compressed tunings of 42edo:

Tuning	42edo	APS715jot	191zpi	42ed255/128
Steps / octave	42.00	~42.10	~42.19	~42.24
Approximation of harmonics	great: 2, 7 good: okay: 11 bad: 3, 5, 13	great: 2 good: 5, 7, 13 okay: 3, 11 bad:	great: 5, 11, 13 good: 2, 3 okay: bad: 7	great: 3, 5, 11 good: 2 okay: 13 bad: 7
“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%

For a more detailed comparison see Table of stretched 42edo tunings.

Subsets and supersets

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

Intervals

#	Cents	Ups and downs notation
0	0.0	P1	perfect unison	D
1	28.6	^1, m2	up unison, minor 2nd	^D, Eb
2	57.1	^^1, ^m2	dup 1sn, upminor 2nd	^^D, ^Eb
3	85.7	^^m2	dupminor 2nd	^^Eb
4	114.3	^³m	trupminor 2nd	^³Eb
5	143.9	v³M	trudmajor 2nd	v³E
6	171.4	vvM2	dudmajor 2nd	vvE
7	200.0	vM2	downmajor 2nd	vE
8	228.6	M2	major 2nd	E
9	257.1	m3	minor 3rd	F
10	285.7	^m3	upminor 3rd	^F
11	314.3	^^m3	dupminor 3rd	^^F
12	342.9	^³m3	trupminor 3rd	^³F
13	371.4	v³M3	trudmajor 3rd	v³F#
14	400.0	vvM3	dudmajor 3rd	vvF#
15	428.6	vM3	downmajor 3rd	vF#
16	457.1	M3, v4	major 3rd, down 4th	F#, vG
17	485.7	P4	perfect 4th	G
18	514.3	^4	up 4th	^G
19	543.9	^^4	dup 4th	^^G
20	571.4	^³4, ^^d5	trup 4th, dupdim 5th	^³G, ^^Ab
21	600.0	v³A4, ^³d5	trudaug 4th, trupdim 5th	v³G#, ^³Ab
22	628.6	vvA4, v³5	dudaug 4th, trud 5th	vvG#, v³A
23	657.1	vv5	dud 5th	vvA
24	685.7	v5	down 5th	vA
25	714.3	P5	perfect 5th	A
26	742.9	^5, m6	up 5th, minor 6th	^A, Bb
27	771.4	^m6	upminor 6th	^Bb
28	800.0	^^m6	dupminor 6th	^^Bb
29	828.6	^³m6	trupminor 6th	^³Bb
30	857.1	v³M6	trudmajor 6th	v³B
31	885.7	vvM6	dudmajor 6th	vvB
32	914.3	vM6	downmajor 6th	vB
33	942.9	M6	major 6th	B
34	971.4	m7	minor 7th	C
35	1000.0	^m7	upminor 7th	^C
36	1028.6	^^m7	dupminor 7th	^^C
37	1057.1	^³m7	trupminor 7th	^³C
38	1085.7	v³M7	trudmajor 7th	v³C#
39	1114.3	vvM7	dudmajor 7th	vvC#
40	1142.9	vM7	downmajor 7th	vC#
41	1171.4	M7, v8	major 7th, down 8ve	C#, vD
42	1200.0	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

Ups and downs 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 ups and downs with three arrows, and if chords are to be spelled correctly four or more arrows may be required in certain cases. For example, a 1/1 – 5/4 – 3/2 – 9/5 chord with a root on the edostep midway between G and A would be written either as v³G♯–v⁵B♯ – v³D♯ – vF♯ or as ^³A♭ – ^C – ^³E♭ – ^⁵G♭. This is a dud dup-seven chord, written either as v³G♯vv,^^7 or as ^³A♭vv,^^7.

In this table, dup is equivalent to quidsharp, trup is equivalent to quudsharp, trudsharp is equivalent to quup, dudsharp is equivalent to quip, etc.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Sharp symbol
Flat symbol

Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
Sharp symbol
Flat symbol

Sagittal notation

Best fifth notation

This notation uses the same sagittal sequence as 35b.

Evo flavor

Revo flavor

Second-best fifth notation

This notation uses the same sagittal sequence as 47edo, and is a superset of the notations for edos 21, 14, and 7.

JI approximation

The following tables show how 15-odd-limit intervals are represented in 42edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 42edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
5/3, 6/5	1.356	4.7
15/8, 16/15	2.554	8.9
7/4, 8/7	2.603	9.1
13/10, 20/13	2.929	10.3
13/11, 22/13	3.495	12.2
9/8, 16/9	3.910	13.7
13/12, 24/13	4.284	15.0
11/9, 18/11	4.551	15.9
15/14, 28/15	5.157	18.0
15/11, 22/15	5.906	20.7
11/10, 20/11	6.424	22.5
9/7, 14/9	6.513	22.8
11/6, 12/11	7.780	27.2
13/9, 18/13	8.046	28.2
11/8, 16/11	8.461	29.6
15/13, 26/15	9.402	32.9
7/6, 12/7	9.728	34.0
9/5, 10/9	10.975	38.4
11/7, 14/11	11.063	38.7
7/5, 10/7	11.084	38.8
13/8, 16/13	11.956	41.8
3/2, 4/3	12.331	43.2
5/4, 8/5	13.686	47.9
13/7, 14/13	14.013	49.0

15-odd-limit intervals in 42edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
5/3, 6/5	1.356	4.7
7/4, 8/7	2.603	9.1
13/11, 22/13	3.495	12.2
11/8, 16/11	8.461	29.6
7/6, 12/7	9.728	34.0
9/5, 10/9	10.975	38.4
11/7, 14/11	11.063	38.7
7/5, 10/7	11.084	38.8
13/8, 16/13	11.956	41.8
3/2, 4/3	12.331	43.2
5/4, 8/5	13.686	47.9
13/7, 14/13	14.559	51.0
11/6, 12/11	20.792	72.8
9/7, 14/9	22.059	77.2
11/10, 20/11	22.147	77.5
15/14, 28/15	23.414	82.0
13/12, 24/13	24.287	85.0
9/8, 16/9	24.661	86.3
13/10, 20/13	25.643	89.7
15/8, 16/15	26.017	91.1
11/9, 18/11	33.122	115.9
15/11, 22/15	34.478	120.7
13/9, 18/13	36.618	128.2
15/13, 26/15	37.973	132.9

Zeta peak index

Tuning			Strength				Octave (cents)		Integer limit
ZPI	Steps per 8ve	Step size (cents)	Height		Integral	Gap	Size	Stretch	Consistent	Distinct
ZPI	Steps per 8ve	Step size (cents)	Tempered	Pure	Integral	Gap	Size	Stretch	Consistent	Distinct
190zpi	41.998867	28.572199	2.689436	2.688894	0.303886	9.541545	1200.032365	0.032365	8	8

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[67 -42⟩	[⟨42 67]]	−3.89	3.88	13.57
2.3.5	128/125, 5000000/4782969	[⟨42 67 98]]	−4.55	3.30	11.55
2.3.5.7	64/63, 126/125, 6860/6561	[⟨42 67 98 118]]	−3.65	3.26	11.42

Scales

Main article: List of MOS scales in 42edo
MOS scales

Eugene/Tritikleismic[9]: 3 8 3 3 8 3 3 8 3
Eugene/Tritikleismic[15]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3
Lemba[16]: 3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3
Qeema/Skateboard[15]: 2 5 2 2 2 5 2 2 2 5 2 2 2 5 2
Qeema/Skateboard[19]: 2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2
Seville/Sevond[14] 1st mode: 1 5 1 5 1 5 1 5 1 5 1 5 1 5
Seville/Sevond[14] 2nd mode: 5 1 5 1 5 1 5 1 5 1 5 1 5 1
Seville/Sevond[21]: 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4

Subsets of MOS scales

(Names used are idiosyncratic.)

Eugene/Tritikleismic[9]
- Groovy aeolian pentatonic: 11 6 8 3 14
- Otonal mixolydian pentatonic: 14 3 8 11 6
- Pseudo-equipentatonic: 11 6 8 6 11
- Septimal melodic minor pentatonic: 8 3 14 14 3
- Septimal Picardy pentatonic: 8 6 11 3 14
- Undecimal lydian-aeolian pentatonic: 8 14 3 11 6
- Yokai pentatonic: 3 14 8 3 14

Approximations of gamelan scales

5-tone pelog: 4 5 15 3 15
7-tone pelog: 4 5 9 6 3 10 5
5-tone slendro: 8 9 8 9 8

Instruments

Lumatone

Main article: Lumatone mapping for 42edo

Skip fretting

Skip fretting system 42 3 11: One way to play 42edo on a 14edo guitar is to tune the strings 11\42, or approximately a just 6/5, apart. All examples on this page are for 7-string guitar.

Prime intervals

1/1: string 2 open

2/1: string 5 fret 3

3/2: string 4 fret 1 and string 7 fret 4

5/4: string 3 fret 1

7/4: string 1 fret 1 and string 4 fret 4

11/8: string 7 fret 2

13/8: string 3 fret 6

17/16: string 1 fret 5

19/16: string 1 fret 7

23/16: string 4 open and string 7 fret 3

29/16: string 5 fret 1

31/16: string 1 fret 3 and string 4 fret 6

Chords

Minor 7th: 100123X

Music

Modern renderings

Johann Sebastian Bach

"Ricercar a 3" from The Musical Offering, BWV 1079 (1747) – rendered by Claudi Meneghin (2024)

Bing Crosby

White Christmas - 42edo reimagining by Todd Harrop (2024)

21st century

Bryan Deister

improv 42edo (2023)
Improvisation in 42edo (2023), transcribed by Stephen Weigel (2024)

James Kukula

Circulating and Traversing (2024) - see the composer’s notes

Budjarn Lambeth

Four Short Experiments in Octave Stretched 42edo (2024)

Herman Miller

Through the Dark (2024) - uses mostly Augene[15] with some chromaticism

Mundoworld

Glory of Them (2024)

42edo

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