46edo

 ← 45edo 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

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 21/46, 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 (¢) +0.00 +2.39 +4.99 -3.61 -3.49 -5.75 -0.61 -10.56 -2.19 -12.19 +2.79
Relative (%) +0.0 +9.2 +19.1 -13.8 -13.4 -22.0 -2.3 -40.5 -8.4 -46.7 +10.7
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[note 1] 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 dupminor 2nd ^^m2 ^^Eb lesser neutral second, augmented unison n2, A1 UEb, D# fri ru[note 2]
6 156.522 12/11, 11/10, 23/21 dudmajor 2nd vvM2 vvE greater neutral second,

super aug unison

N2, sA1 uE, sD# ri ruh[note 3]
7 182.609 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 dupminor 3rd ^^m3 ^^F lesser neutral 3rd n3 UF ni mu[note 2]
14 365.217 16/13, 26/21, 21/17 dudmajor 3rd vvM3 vvF# greater neutral 3rd N3 uF# mi muh[note 3]
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 dup 4th ^^4 ^^G uber 4th, sub diminished 5th U4, sd5 UG, sAb fi/sho fu
22 573.913 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
23 600.000 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, dupdim 5th A4, ^^d5 G#, ^^Ab augmented 4th,

classic diminished 5th

A4, KKd5 G#, KKAb pa/shi se
25 652.174 16/11 dud 5th vv5 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 dupminor 6th ^^m6 ^^Bb lesser neutral 6th n6 UBb fli lu[note 2]
33 860.870 18/11, 28/17, 23/14 dudmajor 6th vvM6 vvB greater neutral 6th N6 uB li luh[note 3]
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 dupminor 7th ^^m7 ^^C lesser neutral 7th, sub diminished 8ve n7, sd8 UC, sDb thi tu[note 2]
41 1069.565 24/13, 13/7, 28/15 dudmajor 7th vvM7 vvC# greater neutral 7th,

diminished 8ve

N7, d8 uC#, Db ti tuh[note 3]
42 1095.652 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

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

Sagittal

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

 Steps Symbol 0 1 2 3 4 5

Ups and downs notation

Using Helmholtz–Ellis accidentals, 46edo can also be notated using ups and downs notation:

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

Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with three arrows.

Approximation to JI

Selected 15-limit intervals approximated in 46edo

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 46edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-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
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 12.958 49.7
15-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
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

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 ? ?
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.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
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

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[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
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. 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 B
9 D
11 F
13 A
15 B

Music

Modern renditions

Johann Sebastian Bach
Nicolaus Bruhns
Scott Joplin
• Maple Leaf Rag (1899) – with syntonic comma adjustment, arranged for harpsichord and rendered by Claudi Meneghin (2024)

21st century

Jake Freivald (site)
Andrew Heathwaite
Aaron Krister Johnson
Claudi Meneghin
Joseph Monzo
Ray Perlner
Gene Ward Smith

Notes

1. 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. 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. /u/ as in supraminor
3. /ʌ/ as in submajor
4. Ratios longer than 10 digits are presented by placeholders with informative hints