46edo

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. In fact, while 41 is a zeta peak and zeta integral edo but not a zeta gap edo, 46 is zeta gap but not zeta peak or zeta integral. (Their sum, 87edo, does better than both in the 13-limit at the expense of a high note count.) Like 41, 46 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'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.

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	29	31
Error	absolute (¢)	+0.0	+2.4	+5.0	-3.6	-3.5	-5.7	-0.6	-10.6	-2.2	-12.2	+2.8
Error	relative (%)	+0	+9	+19	-14	-13	-22	-2	-40	-8	-47	+11
Steps (reduced)		46 (0)	73 (27)	107 (15)	129 (37)	159 (21)	170 (32)	188 (4)	195 (11)	208 (24)	223 (39)	228 (44)

Subsets and supersets

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

Intervals

#	Cents	Approximate Ratios*	Ups and Downs Notation			SKULO notation (K or S = 1, U = 2)			Solfeges
0	0.000	1/1	perfect unison	P1	D	Perfect unison	P1	D	da	do
1	26.087	81/80, 64/63, 49/48	up unison	^1	^D	comma-wide unison, super unison	K1, S1	KD, SD	du	di
2	52.174	28/27, 36/35, 33/32	downminor 2nd	vm2	vEb	subminor 2nd, uber unison	sm2, U1	sEb, UD	fro	ro
3	78.261	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	104.348	16/15, 17/16, 18/17	upminor 2nd	^m2	^Eb	classic minor 2nd, comma-narrow aug unison	Km2, kA1	KEb, kD#	fru	ra
5	130.435	13/12, 14/13, 15/14	downmid 2nd	v~2	^^Eb	lesser neutral second, augmented unison	n2, A1	UEb, D#	fri	ru **
6	156.522	12/11, 11/10, 23/21	upmid 2nd	^~2	vvE	greater neutral second, super aug unison	N2, sA1	uE, sD#	ri	ruh ***
7	182.609	10/9	downmajor 2nd	vM2	vE	classic/comma-narrow major 2nd	kM2	kE	ro	reh
8	208.696	9/8	major 2nd	M2	E	major 2nd	M2	E	ra	re
9	234.783	8/7, 23/20	upmajor 2nd	^M2	^E	supermajor 2nd	SM2	SE	ru	ri
10	260.870	7/6	downminor 3rd	vm3	vF	subminor 3rd	sm3	sF	no	ma
11	286.957	13/11, 20/17	minor 3rd	m3	F	minor 3rd	m3	F	na	meh
12	313.043	6/5	upminor 3rd	^m3	^F	classic minor 3rd	Km3	KF	nu	me
13	339.130	11/9, 17/14, 28/23	downmid 3rd	v~3	^^F	lesser neutral 3rd	n3	UF	ni	mu **
14	365.217	16/13, 26/21, 21/17	upmid 3rd	^~3	vvF#	greater neutral 3rd	N3	uF#	mi	muh ***
15	391.304	5/4	downmajor 3rd	vM3	vF#	classic major 3rd	kM3	kF#	mo	mi
16	417.391	14/11, 23/18	major 3rd	M3	F#	major 3rd	M3	F#	ma	maa
17	443.478	9/7, 13/10, 22/17	upmajor 3rd	^M3	^F#	supermajor 3rd	SM3	SF#	mu	mo
18	469.565	21/16, 17/13	down 4th	v4	vG	sub 4th	s4	sG	fo	fe
19	495.652	4/3	perfect 4th	P4	G	perfect 4th	P4	G	fa	fa
20	521.739	27/20, 23/17	up 4th	^4	^G	comma-wide 4th	K4	KG	fu	fih
21	547.826	11/8	downmid 4th	v~4	^^G	uber 4th, sub diminished 5th	U4, sd5	UG, sAb	fi/sho	fu
22	573.913	7/5, 18/13, 32/23	upmid 4th, dim 5th	^~4, d5	vvG#, Ab	classic augmented 4th, diminished 5th	kkA4, d5	kkG#, Ab	pi/sha	fi
23	600.000	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	626.087	10/7, 13/9, 23/16	aug 4th, downmid 5th	A4, v~5	G#, ^^Ab	augmented 4th, classic diminished 5th	A4, KKd5	G#, KKAb	pa/shi	se
25	652.174	16/11	upmid 5th	^~5	vvA	super augmented 4th, unter 5th	SA4, u5	SG#, uA	pu/si	su
26	678.261	40/27, 34/23	down 5th	v5	vA	comma-narrow 5th	k5	kA	so	sih
27	704.348	3/2	perfect 5th	P5	A	perfect 5th	P5	A	sa	sol
28	730.435	32/21, 26/17	up 5th	^5	^A	super 5th	S5	SA	su	si
29	756.522	14/9, 20/13, 17/11	downminor 6th	vm6	vBb	subminor 6th	sm6	sBb	flo	lo
30	782.609	11/7	minor 6th	m6	Bb	minor 6th	m6	Bb	fla	leh
31	808.696	8/5	upminor 6th	^m6	^Bb	classic minor 6th	Km6	KBb	flu	le
32	834.783	13/8, 21/13, 34/21	downmid 6th	v~6	^^Bb	lesser neutral 6th	n6	UBb	fli	lu **
33	860.870	18/11, 28/17, 23/14	upmid 6th	^~6	vvB	greater neutral 6th	N6	uB	li	luh ***
34	886.957	5/3	downmajor 6th	vM6	vB	classic major 6th	kM6	kB	lo	la
35	913.043	22/13, 17/10	major 6th	M6	B	major 6th	M6	B	la	laa
36	939.130	12/7	upmajor 6th	^M6	^B	supermajor 6th	SM6	SB	lu	li
37	965.217	7/4, 40/23	downminor 7th	vm7	vC	subminor 7th	sm7	sC	tho	ta
38	991.304	16/9, 23/13	minor 7th	m7	C	minor 7th	m7	C	tha	teh
39	1017.391	9/5	upminor 7th	^m7	^C	classic/comma-wide minor 7th	Km7	KC	thu	te
40	1043.478	11/6, 20/11, 42/23	downmid 7th	v~7	^^C	lesser neutral 7th, sub diminished 8ve	n7, sd8	UC, sDb	thi	tu **
41	1069.565	24/13, 13/7, 28/15	upmid 7th	^~7	vvC#	greater neutral 7th, diminished 8ve	N7, d8	uC#, Db	ti	tuh ***
42	1095.652	15/8, 32/17, 17/9	downmajor 7th	vM7	vC#	classic major 7th, comma-wide dim 8ve	kM7, Kd8	kC#, KDb	to	ti
43	1121.739	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	1147.826	27/14, 35/18, 64/33	upmajor 7th	^M7	^C#	supermajor 7th, unter 8ve	SM7, u8	SC#, uD	tu	to
45	1173.913	160/81, 63/32, 96/49	down 8ve	v8	vD	comma-narrow 8ve, sub 8ve	k8/s8	kD, sD	do	da
46	1200.000	2/1	perfect 8ve	P8	D	perfect 8ve	P8	D	da	do

* Based on treating 46edo as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46edo has intervals involving the 15th harmonic poorly approximated, except for 15/8 and 16/15 themselves, 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, preventing it from being consistent in the 15-odd-limit. This can be demonstrated with the discrepancy approximating 15/13 and 26/15. 9\46 is closer to 15/13 by a hair; 10\46 represents the difference between, for instance, 46edo's 15/8 and 13/8, and is more likely to appear in chords actually functioning as 15/13.

** -u as in supraminor

*** -uh as in submajor

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
downmid	ilo	{a, b, 0, 0, 1}	11/9, 11/6
upmid	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. An up, down or mid immediately after the chord root affects 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	Cv~	C downmid
lu	22:27:33	0-14-27	C vvE G	C^~	C upmid
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

Sagittal

The following table shows sagittal notation accidentals in one apotome for 46edo.

Steps	0	1	2	3	4	5
Symbol

Approximation to JI

Selected 15-limit intervals approximated in 46edo

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 46edo. Prime harmonics are in bold; inconsistent intervals are in italic.

15-odd-limit intervals by direct mapping (even if inconsistent)
Interval, complement	Error (abs, ¢)	Error (rel, %)
11/7, 14/11	0.117	0.4
9/5, 10/9	0.205	0.8
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
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
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
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
18/13, 13/9	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

15-odd-limit intervals by patent val mapping
Interval, 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
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
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
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
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

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	[⟨41 65 95 115 142 152 174 188]]	+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	[⟨41 65 95 115 142 152 174 188 208]]	+0.101	1.23	4.72

Commas

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

Prime Limit	Ratio^[1]	Monzo	Cents	Color name	Name(s)
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
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
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
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
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

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.

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, Pathological
	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! Pathological
	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 ~ Pathological 2:1 ~ QE, Pathological
	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, Pathological
	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 ~ Pathological 2:1 ~ QE, Pathological
	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 ~ Pathological 2:1 ~ QE, Pathological
23	1\46	26.087	Icositritonic

Scales

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. In steps-in-between, that's 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¢).