46edo

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46edo

47edo →

Prime factorization

2 × 23

Step size

26.087 ¢

Fifth

27\46 (704.348 ¢)

Semitones (A1:m2)

5:3 (130.4 ¢ : 78.26 ¢)

Consistency limit

13

Distinct consistency limit

9

Special properties

Zeta gap

46 equal divisions of the octave (abbreviated 46edo or 46ed2), also called 46-tone equal temperament (46tet) or 46 equal temperament (46et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 46 equal parts of about 26.1 ¢ each. Each step represents a frequency ratio of 2^1/46, or the 46th root of 2.

Theory

In the opinion of some, 46edo is the first equal division to deal adequately with the 13-limit, though others award that distinction to 41edo or 53edo. 46edo's fifth is slightly sharp of just, which some people (e.g. Margo Schulter) prefer, sometimes strongly, over both the just fifth and fifths of temperaments with flat fifths, such as meantone. Many say that sharp fifths give a characteristic bright sound to 5-limit triads, and consider the sound of meantone triads to be more mellow in comparison. 46edo is not a zeta peak edo, but it is a zeta gap edo. It is distinctly consistent in the 9-odd-limit, and it is consistent to the 13-odd-limit or the no-15 no-19 23-odd-limit.

46edo is also notable for being the smallest equal division to approximate harmonics 3, 5, 7, 11, and 13 with less than 25% relative interval error.

Rank-2 temperaments it supports include sensi, valentine, shrutar, rodan, leapday, and unidec. The 11-odd-limit minimax tuning for valentine, (11/7)^1/10, is only 0.01 cents flat of 3\46 octaves.

Shrutar22 as srutis describes a possible use of 46edo for Indian music.

Prime harmonics

Approximation of prime harmonics in 46edo
Harmonic		2	3	5	7	11	13	17	19	23
Error	Absolute (¢)	+0.0	+2.4	+5.0	-3.6	-3.5	-5.7	-0.6	-10.6	-2.2
Error	Relative (%)	+0.0	+9.2	+19.1	-13.8	-13.4	-22.0	-2.3	-40.5	-8.4
Steps (reduced)		46 (0)	73 (27)	107 (15)	129 (37)	159 (21)	170 (32)	188 (4)	195 (11)	208 (24)

Approximation of prime harmonics in 46edo (continued)
Harmonic		29	31	37	41	43	47	53	59	61
Error	Absolute (¢)	-12.2	+2.8	+9.5	-11.7	+10.2	+12.8	-12.6	+10.4	+4.9
Error	Relative (%)	-46.7	+10.7	+36.5	-44.7	+39.2	+48.9	-48.4	+39.8	+18.6
Steps (reduced)		223 (39)	228 (44)	240 (10)	246 (16)	250 (20)	256 (26)	263 (33)	271 (41)	273 (43)

Subsets and supersets

46edo can be treated as two circles of 23edo separated by an interval of 26.087 cents.

Intervals

#	Cents	Approximate ratios^{[note 1]}	Ups and downs notation (EUs: v⁵A1 and ^^d2)			SKULO notation (K or S = 1, U = 2)			Solfege
0	0.0	1/1	perfect unison	P1	D	Perfect unison	P1	D	da	do
1	26.1	81/80, 64/63, 49/48	up unison	^1	^D	comma-wide unison, super unison	K1, S1	KD, SD	du	di
2	52.2	28/27, 36/35, 33/32	downminor 2nd	vm2	vEb	subminor 2nd, uber unison	sm2, U1	sEb, UD	fro	ro
3	78.3	25/24, 21/20, 22/21, 24/23, 23/22	minor 2nd	m2	Eb	minor 2nd, classic augmented unison	m2, kkA1	Eb, kkD#	fra	rih
4	104.3	16/15, 17/16, 18/17	upminor 2nd	^m2	^Eb	classic minor 2nd, comma-narrow aug unison	Km2, kA1	KEb, kD#	fru	ra
5	130.4	13/12, 14/13, 15/14	dupminor 2nd	^^m2	^^Eb	lesser neutral second, augmented unison	n2, A1	UEb, D#	fri	ru^{[note 2]}
6	156.5	12/11, 11/10, 23/21	dudmajor 2nd	vvM2	vvE	greater neutral second, super aug unison	N2, sA1	uE, sD#	ri	ruh^{[note 3]}
7	182.6	10/9	downmajor 2nd	vM2	vE	classic/comma-narrow major 2nd	kM2	kE	ro	reh
8	208.7	9/8	major 2nd	M2	E	major 2nd	M2	E	ra	re
9	234.8	8/7, 23/20	upmajor 2nd	^M2	^E	supermajor 2nd	SM2	SE	ru	ri
10	260.9	7/6	downminor 3rd	vm3	vF	subminor 3rd	sm3	sF	no	ma
11	287.0	13/11, 20/17	minor 3rd	m3	F	minor 3rd	m3	F	na	meh
12	313.0	6/5	upminor 3rd	^m3	^F	classic minor 3rd	Km3	KF	nu	me
13	339.1	11/9, 17/14, 28/23	dupminor 3rd	^^m3	^^F	lesser neutral 3rd	n3	UF	ni	mu^{[note 2]}
14	365.2	16/13, 26/21, 21/17	dudmajor 3rd	vvM3	vvF#	greater neutral 3rd	N3	uF#	mi	muh^{[note 3]}
15	391.3	5/4	downmajor 3rd	vM3	vF#	classic major 3rd	kM3	kF#	mo	mi
16	417.4	14/11, 23/18	major 3rd	M3	F#	major 3rd	M3	F#	ma	maa
17	443.5	9/7, 13/10, 22/17	upmajor 3rd	^M3	^F#	supermajor 3rd	SM3	SF#	mu	mo
18	469.6	21/16, 17/13	down 4th	v4	vG	sub 4th	s4	sG	fo	fe
19	495.7	4/3	perfect 4th	P4	G	perfect 4th	P4	G	fa	fa
20	521.7	27/20, 23/17	up 4th	^4	^G	comma-wide 4th	K4	KG	fu	fih
21	547.8	11/8	dup 4th	^^4	^^G	uber 4th, sub diminished 5th	U4, sd5	UG, sAb	fi/sho	fu
22	573.9	7/5, 18/13, 32/23	dudaug 4th, dim 5th	vvA4, d5	vvG#, Ab	classic augmented 4th, diminished 5th	kkA4, d5	kkG#, Ab	pi/sha	fi
23	600.0	17/12, 24/17	downaug 4th, updim 5th	vA4, ^d5	vG#, ^Ab	comma-narrow aug 4th, comma-wide dim 5th	kA4, Kd5	kG#, KAb	po/shu	seh
24	626.1	10/7, 13/9, 23/16	aug 4th, dupdim 5th	A4, ^^d5	G#, ^^Ab	augmented 4th, classic diminished 5th	A4, KKd5	G#, KKAb	pa/shi	se
25	652.2	16/11	dud 5th	vv5	vvA	super augmented 4th, unter 5th	SA4, u5	SG#, uA	pu/si	su
26	678.3	40/27, 34/23	down 5th	v5	vA	comma-narrow 5th	k5	kA	so	sih
27	704.3	3/2	perfect 5th	P5	A	perfect 5th	P5	A	sa	sol
28	730.4	32/21, 26/17	up 5th	^5	^A	super 5th	S5	SA	su	si
29	756.5	14/9, 20/13, 17/11	downminor 6th	vm6	vBb	subminor 6th	sm6	sBb	flo	lo
30	782.6	11/7	minor 6th	m6	Bb	minor 6th	m6	Bb	fla	leh
31	808.7	8/5	upminor 6th	^m6	^Bb	classic minor 6th	Km6	KBb	flu	le
32	834.8	13/8, 21/13, 34/21	dupminor 6th	^^m6	^^Bb	lesser neutral 6th	n6	UBb	fli	lu^{[note 2]}
33	860.9	18/11, 28/17, 23/14	dudmajor 6th	vvM6	vvB	greater neutral 6th	N6	uB	li	luh^{[note 3]}
34	887.0	5/3	downmajor 6th	vM6	vB	classic major 6th	kM6	kB	lo	la
35	913.0	22/13, 17/10	major 6th	M6	B	major 6th	M6	B	la	laa
36	939.1	12/7	upmajor 6th	^M6	^B	supermajor 6th	SM6	SB	lu	li
37	965.2	7/4, 40/23	downminor 7th	vm7	vC	subminor 7th	sm7	sC	tho	ta
38	991.3	16/9, 23/13	minor 7th	m7	C	minor 7th	m7	C	tha	teh
39	1017.4	9/5	upminor 7th	^m7	^C	classic/comma-wide minor 7th	Km7	KC	thu	te
40	1043.5	11/6, 20/11, 42/23	dupminor 7th	^^m7	^^C	lesser neutral 7th, sub diminished 8ve	n7, sd8	UC, sDb	thi	tu^{[note 2]}
41	1069.6	24/13, 13/7, 28/15	dudmajor 7th	vvM7	vvC#	greater neutral 7th, diminished 8ve	N7, d8	uC#, Db	ti	tuh^{[note 3]}
42	1095.7	15/8, 32/17, 17/9	downmajor 7th	vM7	vC#	classic major 7th, comma-wide dim 8ve	kM7, Kd8	kC#, KDb	to	ti
43	1121.7	48/25, 40/21, 21/11, 23/12, 44/23	major 7th	M7	C#	major 7th, classic diminished 8ve	M7, KKd8	C#, KKDb	ta	taa
44	1147.8	27/14, 35/18, 64/33	upmajor 7th	^M7	^C#	supermajor 7th, unter 8ve	SM7, u8	SC#, uD	tu	to
45	1173.9	160/81, 63/32, 96/49	down 8ve	v8	vD	comma-narrow 8ve, sub 8ve	k8/s8	kD, sD	do	da
46	1200.0	2/1	perfect 8ve	P8	D	perfect 8ve	P8	D	da	do

Interval quality and chord names in color notation

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

Quality	Color	Monzo Format	Examples
downminor	zo	[a, b, 0, 1>	7/6, 7/4
minor	fourthward wa	[a, b>, b < −1	32/27, 16/9
upminor	gu	[a, b, −1>	6/5, 9/5
dupminor	ilo	[a, b, 0, 0, 1>	11/9, 11/6
dudmajor	lu	[a, b, 0, 0, −1>	12/11, 18/11
downmajor	yo	[a, b, 1>	5/4, 5/3
major	fifthward wa	[a, b>, b > 1	9/8, 27/16
upmajor	ru	[a, b, 0, −1>	9/7, 12/7

All 46edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. Ups or downs immediately after the chord root affect the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, lu, yo and ru triads:

Color of the 3rd	JI Chord	Notes as Edosteps	Notes of C Chord	Written Name	Spoken Name
zo	6:7:9	0–10–27	C vEb G	Cvm	C downminor
gu	10:12:15	0–12–27	C ^Eb G	C^m	C upminor
ilo	18:22:27	0–13–27	C ^^Eb G	C^^m	C dupminor
lu	22:27:33	0–14–27	C vvE G	Cvv	C dudmajor or C dud
yo	4:5:6	0–15–27	C vE G	Cv	C downmajor or C down
ru	14:18:21	0–17–27	C ^E G	C^	C upmajor or C up

For a more complete list, see Ups and downs notation #Chords and chord progressions.

Notation

Ups and downs notation

46edo can be notated with ups and downs, spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

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

Another notation uses alternative ups and downs. Here, this can be done using sharps and flats with arrows, borrowed from extended Helmholtz–Ellis notation:

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

Sagittal notation

This notation uses the same sagittal sequence as 39edo.

Evo flavor

Revo flavor

Approximation to JI

alt : Your browser has no SVG support. — Selected 15-limit intervals approximated in 46edo

17-odd-limit interval mappings

The following tables show how 17-odd-limit intervals are represented in 46edo. Prime harmonics are in bold; inconsistent intervals are in italics.

17-odd-limit intervals in 46edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/7, 14/11	0.117	0.4
9/5, 10/9	0.205	0.8
17/16, 32/17	0.608	2.3
13/7, 14/13	2.137	8.2
13/11, 22/13	2.253	8.6
3/2, 4/3	2.393	9.2
5/3, 6/5	2.598	10.0
17/11, 22/17	2.884	11.1
17/12, 24/17	3.000	11.5
17/14, 28/17	3.001	11.5
11/8, 16/11	3.492	13.4
7/4, 8/7	3.609	13.8
9/8, 16/9	4.786	18.3
5/4, 8/5	4.991	19.1
17/13, 26/17	5.137	19.7
17/9, 18/17	5.393	20.7
17/10, 20/17	5.598	21.5
13/8, 16/13	5.745	22.0
11/6, 12/11	5.885	22.6
7/6, 12/7	6.001	23.0
15/8, 16/15	7.383	28.3
17/15, 30/17	7.991	30.6
13/12, 24/13	8.138	31.2
11/9, 18/11	8.278	31.7
9/7, 14/9	8.394	32.2
11/10, 20/11	8.482	32.5
7/5, 10/7	8.599	33.0
13/9, 18/13	10.531	40.4
13/10, 20/13	10.736	41.2
15/11, 22/15	10.875	41.7
15/14, 28/15	10.992	42.1
15/13, 26/15	12.958	49.7

17-odd-limit intervals in 46edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/7, 14/11	0.117	0.4
9/5, 10/9	0.205	0.8
17/16, 32/17	0.608	2.3
13/7, 14/13	2.137	8.2
13/11, 22/13	2.253	8.6
3/2, 4/3	2.393	9.2
5/3, 6/5	2.598	10.0
17/11, 22/17	2.884	11.1
17/12, 24/17	3.000	11.5
17/14, 28/17	3.001	11.5
11/8, 16/11	3.492	13.4
7/4, 8/7	3.609	13.8
9/8, 16/9	4.786	18.3
5/4, 8/5	4.991	19.1
17/13, 26/17	5.137	19.7
17/9, 18/17	5.393	20.7
17/10, 20/17	5.598	21.5
13/8, 16/13	5.745	22.0
11/6, 12/11	5.885	22.6
7/6, 12/7	6.001	23.0
15/8, 16/15	7.383	28.3
17/15, 30/17	7.991	30.6
13/12, 24/13	8.138	31.2
11/9, 18/11	8.278	31.7
9/7, 14/9	8.394	32.2
11/10, 20/11	8.482	32.5
7/5, 10/7	8.599	33.0
13/9, 18/13	10.531	40.4
13/10, 20/13	10.736	41.2
15/11, 22/15	10.875	41.7
15/14, 28/15	10.992	42.1
15/13, 26/15	13.129	50.3

Consistent circles

46edo is home to a number of consistent circles, both ones closing after generating all 46 notes and ones closing after generating 23edo.

46-note circles by gen. with related temperaments organized by period
Interval	Closing Error	Consistency	1\1	1\2
68/65	25.9%	Normal	Valentine	Semivalentine
10/9	36.1%	Normal	Mitonic	Unidec, hendec
31/24	70.2%	Weak	Sensible, sensi add-31	Bison add-31, bisensi add-31

23-note circles by gen. with related half-octave temperaments
Interval	Closing Error	Consistency	Temperaments
17/16	53.5%	Weak	Diaschismic
23/21	85.7%	Weak	Bison
44/39	12.3%	Super-strong	Abigail
21/17	53.6%	Weak	?
14/11	10.2%	Super-strong	?
21/16	107%	Sub-weak	?

For the 23rd-octave temperament that 46edo supports which combines all above 23-note circles, see icositritonic.

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
214zpi	46.008975	26.081868	7.495674	7.451949	1.356067	17.747832	1199.76592	−0.23408	14	11

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[73 -46⟩	[⟨46 73]]	−0.755	0.75	2.89
2.3.5	2048/2025, 78732/78125	[⟨46 73 107]]	−1.219	0.90	3.45
2.3.5.7	126/125, 245/243, 1029/1024	[⟨46 73 107 129]]	−0.595	1.34	5.12
2.3.5.7.11	121/120, 126/125, 176/175, 245/243	[⟨46 73 107 129 159]]	−0.274	1.36	5.20
2.3.5.7.11.13	91/90, 121/120, 169/168, 176/175, 245/243	[⟨46 73 107 129 159 170]]	+0.030	1.41	5.42
2.3.5.7.11.13.17	91/90, 121/120, 154/153, 169/168, 176/175, 245/243	[⟨46 73 107 129 159 170 188]]	+0.047	1.31	5.02
2.3.5.7.11.13.17.23	91/90, 121/120, 154/153, 169/168, 176/175, 208/207, 231/230	[⟨46 73 107 129 159 170 188 208]]	+0.101	1.23	4.72

46et is lower in relative error than any previous equal temperaments in the 17-, 19-, 23-limit, and others. The next equal temperaments doing better in the aforementioned subgroups are 72, 72, 94, respectively. 46et is even more prominent in the no-19 23-limit, and the next equal temperament doing better in this subgroup is 140.

Commas

This is a partial list of the commas that 46et tempers out with its patent val, ⟨24 38 56 67 83 89].

Prime limit	Ratio^{[note 4]}	Monzo	Cents	Color name	Name(s)
5	(16 digits)	[24 1 -11⟩	52.50	Salegu	Magus comma
5	(14 digits)	[13 5 -9⟩	32.95	Satritrigu	Valentine comma
5	2048/2025	[11 -4 -2⟩	19.55	Sagugu	Diaschisma
5	78732/78125	[2 9 -7⟩	13.40	Sepgu	Sensipent comma
5	(14 digits)	[9 -13 5⟩	6.15	Saquinyo	Amity comma
7	686/675	[1 -3 -2 3⟩	27.99	Trizo-agugu	Senga
7	245/243	[0 -5 1 2⟩	14.19	Zozoyo	Sensamagic comma
7	126/125	[1 2 -3 1⟩	13.80	Zotrigu	Starling comma
7	1029/1024	[-10 1 0 3⟩	8.43	Latrizo	Gamelisma
7	5120/5103	[10 -6 1 -1⟩	5.76	Saruyo	Hemifamity comma, aberschisma
7	(20 digits)	[31 -6 -2 -6⟩	2.69	Sasa-tribiru-agugu	Pessoalisma
7	4375/4374	[-1 -7 4 1⟩	0.40	Zoquadyo	Ragisma
11	121/120	[-3 -1 -1 0 2⟩	14.37	Lologu	Biyatisma
11	176/175	[4 0 -2 -1 1⟩	9.86	Lorugugu	Valinorsma
11	896/891	[7 -4 0 1 -1⟩	9.69	Saluzo	Pentacircle comma
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
11	441/440	[-3 2 -1 2 -1⟩	3.93	Luzozogu	Werckisma
13	91/90	[-1 -2 -1 1 0 1⟩	19.13	Thozogu	Superleap comma, biome comma
13	169/168	[-3 -1 0 -1 0 2⟩	10.27	Thothoru	Buzurgisma, dhanvantarisma
13	196/195	[2 -1 -1 2 0 -1⟩	8.86	Thuzozogu	Mynucuma
13	507/500	[-2 1 -3 0 0 2⟩	24.07	Thothotrigu
13	351/350	[-1 3 -2 -1 0 1⟩	4.94	Thorugugu	Ratwolfsma
13	352/351	[5 -3 0 0 1 -1⟩	4.93	Thulo	Minor minthma
17	256/255	[8 -1 -1 0 0 0 -1⟩	6.78	Sugu	Charisma, septendecimal kleisma
17	289/288	[-5 -2 0 0 0 0 2⟩	6.00	Soso	Semitonisma

Rank-2 temperaments

Periods per 8ve	Generator	Cents	Temperaments	MOS scales	L:s
1	1\46	26.087	Sfourth
	3\46	78.261	Valentine	1L 14s (15-tone) 15L 1s (16-tone) 16L 15s (31-tone)	4:3 ~ quasi-equal 3:1 2:1 ~ QE
	5\46	130.435	Twothirdtonic	1L 8s (9-tone) 9L 1s (10-tone) 9L 10s (19-tone) 9L 19s (28-tone) 9L 28s (37-tone)	6:5 ~ QE 5:1 4:1 3:1 2:1 ~ QE
	7\46	182.609	Minortone / mitonic	1L 5s (6-tone) 6L 1s (7-tone) 7L 6s (13-tone) 13L 7s (20-tone) 13L 20s (33-tone)	11:7 7:4 4:3 ~ QE 3:1 2:1 ~ QE
	9\46	234.783	Rodan	1L 4s (5-tone) 1L 5s (6-tone) 5L 6s (11-tone) 5L 11s (16-tone) 5L 16s (21-tone) 5L 21s (26-tone) 5L 26s (31-tone) 5L 31s (36-tone) 5L 36s (41-tone)	10:9 ~QE 9:1 8:1 7:1 6:1 5:1 4:1 3:1 2:1 ~ QE
	11\46	286.957	Gamity	4L 1s (5-tone) 4L 5s (9-tone) 4L 9s (13-tone) 4L 13s (17-tone) 4L 17s (21-tone) 21L 4s (25-tone)	11:2 9:2 7:2 5:2 3:2 ~ QE, Golden 2:1 ~ QE
	13\46	339.130	Amity / hitchcock	4L 3s (7-tone) 7L 4s (11-tone) 7L 11s (18-tone) 7L 18s (25-tone) 7L 25s (32-tone) 7L 32s (39-tone)	7:6 ~ QE 6:1 5:1 4:1 3:1 2:1 ~ QE
	15\46	391.304	Magus / amigo	1L 2s (3-tone) 3L 1s (4-tone) 3L 4s (7-tone) 3L 7s (10-tone) 3L 10s (13-tone) 3L 13s (16-tone) 3L 16s (19-tone) 3L 19s (21-tone) 3L 21s (24-tone) 3L 24s (27-tone) 3L 27s (30-tone) 3L 30s (33-tone) 3L 33s (36-tone) 3L 36s (39-tone) 3L 39s (42-tone)	16:15 ~ QE 15:1 14:1 13:1 12:1 11:1 10:1 9:1 8:1 7:1 6:1 5:1 4:1 3:1 2:1 ~ QE
	17\46	443.478	Sensi	3L 2s (5-tone) 3L 5s (8-tone) 8L 3s (11-tone) 8L 11s (19-tone) 19L 8s (27-tone)	12:5 7:5 5:2 3:2 ~ QE, Golden 2:1
	19\46	495.652	Leapday	2L 3s (5-tone) 5L 2s (7-tone) 5L 7s (12-tone) 12L 5s (17-tone) 17L 12s (29-tone)	11:8 8:3 5:3 ~ Golden 3:2 ~ QE, Golden 2:1 ~ QE
	21\46	547.826	Heinz	2L 3s (5-tone) 2L 5s (7-tone) 2L 7s (9-tone) 2L 9s (11-tone) 11L 2s (13-tone) 11L 13s (24-tone) 11L 24s (35-tone)	17:4 13:4 9:4 5:4 ~ QE 4:1 3:1 2:1 ~ QE
2	1\46	26.087	Ketchup
	2\46	52.174	Shrutar	2L 2s (4-tone) 2L 4s (6-tone) 2L 6s (8-tone) 2L 8s (10-tone) 2L 10s (12-tone) 2L 12s (14-tone) 2L 14s (16-tone) 2L 16s (18-tone) 2L 18s (20-tone) 2L 20s (22-tone) 22L 2s (24-tone)	21:2 19:2 17:2 15:2 13:2 11:2 9:2 7:2 5:2 3:2 ~ QE, Golden 2:1 ~ QE
	3\46	78.261	Semivalentine	2L 2s (4-tone) 2L 4s (6-tone) 2L 6s (8-tone) 2L 8s (10-tone) 2L 10s (12-tone) 2L 12s (14-tone) 14L 2s (16-tone) 16L 14s (30-tone)	20:3 17:3 14:3 11:3 8:3 5:3 ~ Golden 3:2 ~ QE, Golden 2:1 ~ QE
	4\46	104.348	Srutal / diaschismic	2L 2s (4-tone) 2L 4s (6-tone) 2L 6s (8-tone) 2L 8s (10-tone) 10L 2s (12-tone) 12L 10s (22-tone) 12L 22s (34-tone)	19:4 15:4 11:4 7:4 4:3 ~ QE 3:1 2:1 ~ QE
	5\46	130.435		2L 2s (4-tone) 2L 4s (6-tone) 2L 6s (8-tone) 8L 2s (10-tone) 8L 10s (18-tone) 18L 10s (28-tone)	18:5 13:5 8:5 ~ Golden 5:3 ~ Golden 3:2 ~ QE, Golden 2:1 ~ QE
	6\46	156.522	Bison	2L 2s (4-tone) 2L 4s (6-tone) 6L 2s (8-tone) 8L 6s (14-tone) 8L 14s (22-tone) 8L 22s (30-tone) 8L 30s (38-tone	17:6 11:6 6:5 ~ QE 5:1 4:1 3:1 2:1 ~ QE
	7\46	182.609	Unidec / hendec	2L 2s (4-tone) 2L 4s (6-tone) 6L 2s (8-tone) 6L 8s (14-tone) 6L 14s (20-tone) 20L 6s (26-tone)	16:7 9:7 7:2 5:2 3:2 ~ QE, Golden 2:1 ~ QE
	8\46	208.696	Abigail	2L 2s (4-tone) 4L 2s (6-tone) 6L 2s (8-tone) 6L 8s (14-tone) 6L 14s (20-tone) 6L 20s (26-tone) 6L 26s (32-tone) 6L 32s (38-tone) 6L 38s (44-tone)	15:8 8:7 ~ QE 8:1 7:1 6:1 5:1 4:1 3:1 2:1 ~ QE
	9\46	234.783	Echidnic	2L 2s (4-tone) 4L 2s (6-tone) 6L 4s (10-tone) 10L 6s (16-tone) 10L 16s (26-tone) 10L 26s (36-tone)	14:9 9:5 5:4 ~ QE 4:1 3:1 2:1 ~ QE
	10\46	260.87	Bamity	2L 2s (4-tone) 4L 2s (6-tone) 4L 6s (10-tone) 4L 10s (14-tone) 14L 4s (18-tone) 14L 18s (32-tone)	13:10 10:3 7:3 4:3 ~ QE 3:1 2:1 ~ QE
	11\46	286.957	Vines	2L 2s (4-tone) 4L 2s (6-tone) 4L 6s (10-tone) 4L 10s (14-tone) 4L 14s (18-tone) 4L 18s (22-tone) 4L 22s (26-tone) 4L 26s (30-tone) 4L 30s (34-tone) 4L 34s (38-tone) 4L 38s (42-tone)	12:11 ~ QE 11:1 10:1 9:1 8:1 7:1 6:1 5:1 4:1 3:1 2:1 ~ QE
23	1\46	26.087	Icositritonic

Scales

Main article: List of MOS scales in 46edo

Sensi

Elfleapday

Elfsensus

Harmonic scales

46edo represents overtones 8 through 16 (written as JI ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. This corresponds to scale steps of 8, 7, 6, 6, 5, 5, 5, 4.

8\46 (208.696 ¢) stands in for frequency ratio 9/8 (203.910 ¢).
7\46 (182.609 ¢) stands in for 10/9 (182.404 ¢).
6\46 (156.522 ¢) stands in for 11/10 (165.004 ¢) and 12/11 (150.637 ¢).
5\46 (130.435 ¢) stands in for 13/12 (138.573 ¢), 14/13 (128.298 ¢) and 15/14 (119.443 ¢).
4\46 (104.348 ¢) stands in for 16/15 (111.731 ¢).

Harmonic	Note (starting from C)
1	C
3	G
5	E⁠ ⁠
7	G𝄪, B⁠ ⁠
9	D
11	E♯, F⁠ ⁠
13	G♯, A⁠ ⁠
15	B⁠ ⁠

Instruments

Lumatone

Lumatone mapping for 46edo

Skip fretting

Skip fretting system 46 2 11 is a skip fretting system for playing 46-edo on a 23-edo stringed instrument.

Skip fretting system 46 7 11 is another skip fretting system for 46edo. The examples on this page are for 7-string guitar.

Harmonics

1/1: string 2 open

2/1: string 3 fret 5

3/2: not easily accessible

5/4: string 5 fret 4

Music

Modern renditions

Johann Sebastian Bach

"Ricercar a 6" from The Musical Offering, BWV 1079 (1747) – with syntonic-comma adjustment, rendered by Claudi Meneghin (2025)
"Contrapunctus 11" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)

Nicolaus Bruhns

Prelude in E Minor "The Great" – rendered by Claudi Meneghin (2023)

Scott Joplin

Maple Leaf Rag (1899) – with syntonic comma adjustment, arranged for harpsichord and rendered by Claudi Meneghin (2024)

21st century

Bryan Deister

music box 46edo (2025)

Jake Freivald (site)

A Seed Planted - (Yet another version: 46 EDO) play^{[dead link]}

Andrew Heathwaite

Rats play^{[dead link]} (2012)
Tumbledown Stew play^{[dead link]} (2012)
Hypnocloudsmack 1 play^{[dead link]} (2012)
Hypnocloudsmack 2 play^{[dead link]} (2012)
Hypnocloudsmack 3 play^{[dead link]} (2012)

Aaron Krister Johnson

Satiesque (2014)

Claudi Meneghin

Bach BWV 1029 in 46 equal Claudi Meneghin version
Bach Contrapunctus 4 Claudi Meneghin version
Chaconne et Fugue à 5 "Les Regrets"
El Rossinyol
Arietta with 5 Variations, for Organ
Fugue in 46edo (as Sensi), for two organs (2024)

Herman Miller

Light at the End (2020)

Joseph Monzo

I Love the Blackness (unfinished) (2008)

Ray Perlner

Fugue in 46EDO Bohlen-Pierce-Stearns 9 (Sensi Extension) LssLsLsLs "Lambda"

Gene Ward Smith

Chromosounds play
Music For Your Ears play – The central portion is in 27edo; the rest is in 46edo.

Tristan Bay

Catalyst (2025)

vivi mouse

valentine's day (2025)

Notes

↑ Based on treating 46edo as a 2.3.5.7.11.13.17.23 subgroup temperament; other approaches are also possible. However, ratios of 15 are not included here, as except for 15/8 and 16/15 themselves 46edo has intervals involving the 15th harmonic poorly approximated in general. This is because, while the 3rd and 5th harmonics are sharp and their deviations from just intonation add up, 7, 11, and 13 are all tuned flat, making the difference even larger. This prevents it from being consistent in the 15-odd-limit, as there is a discrepancy approximating 15/13 and 26/15—9\46 is closer to 15/13 by a hair, but 10\46 represents the difference between 46edo's 15/8 and 13/8 and is more likely to appear in chords actually functioning as 15/13.
↑ ^2.0 ^2.1 ^2.2 ^2.3 /u/ as in supraminor
↑ ^3.0 ^3.1 ^3.2 ^3.3 /ʌ/ as in submajor
↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

[interval_ratios-1] Based on treating 46edo as a 2.3.5.7.11.13.17.23 subgroup temperament; other approaches are also possible. However, ratios of 15 are not included here, as except for 15/8 and 16/15 themselves 46edo has intervals involving the 15th harmonic poorly approximated in general. This is because, while the 3rd and 5th harmonics are sharp and their deviations from just intonation add up, 7, 11, and 13 are all tuned flat, making the difference even larger. This prevents it from being consistent in the 15-odd-limit, as there is a discrepancy approximating 15/13 and 26/15—9\46 is closer to 15/13 by a hair, but 10\46 represents the difference between 46edo's 15/8 and 13/8 and is more likely to appear in chords actually functioning as 15/13.

[intervals2-2] 2.0 ^2.1 ^2.2 ^2.3 /u/ as in supraminor

[intervals3-3] 3.0 ^3.1 ^3.2 ^3.3 /ʌ/ as in submajor

[4] Ratios longer than 10 digits are presented by placeholders with informative hints.

[note 1]

[note 2]

[note 3]

[note 4]

46edo

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