46edo
← 45edo | 46edo | 47edo → |
46 equal divisions of the octave (abbreviated 46edo), or 46-tone equal temperament (46tet), 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 46 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. 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
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 |
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 augmented 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 augmented 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 augmented 4th, comma-wide diminished 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 diminished 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
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.
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 |
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 | |
---|---|---|---|---|---|
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 |
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 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¢).
Instruments
- Isomorphic layout 11\46 x 2\46: A skip-fretting system for playing 46-edo on a 23-edo stringed instrument.
- Lumatone mapping for 46edo
Music
Modern renditions
- Prelude in E Minor "The Great" – rendered by Claudi Meneghin (2023)
21st century
- 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)
- Locrian Suite Gavotte (2020)
- Satiesque (2014)
- 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
- Chromosounds play
- Music For Your Ears play – The central portion is in 27edo; the rest is in 46edo.
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints