53edo

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← 52edo 53edo 54edo →
Prime factorization 53 (prime)
Step size 22.6415¢ 
Fifth 31\53 (701.887¢)
(convergent)
Semitones (A1:m2) 5:4 (113.2¢ : 90.57¢)
Consistency limit 9
Distinct consistency limit 9
Special properties
English Wikipedia has an article on:

53 equal divisions of the octave (abbreviated 53edo or 53ed2), also called 53-tone equal temperament (53tet) or 53 equal temperament (53et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 53 equal parts of about 22.6 ¢ each. Each step represents a frequency ratio of 21/53, or the 53rd root of 2.

Theory

53edo is notable as an excellent 5-limit system, a fact apparently first noted by Isaac Newton[1]. It is the seventh strict zeta edo. In the opinion of some, 53edo is the first equal division to deal adequately with the 13-limit, while others award that distintion to 41edo or 46edo. Like 41 and 46, 53 is distinctly consistent in the 9-odd-limit, but among them, 53 is the first that finds the interseptimal intervals 15/13 and 13/10 distinctly from adjacent septimal intervals 8/7 and 7/6, and 9/7 and 21/16, respectively, which is essential to its 13-limit credibility. It also avoids equating 11/9 with 16/13, so that the former is tuned very flat to equate it with a slightly flat ~39/32 – a feature shared by 46edo. Besides, it is almost consistent to the entire 15-odd-limit, with the only inconsistency occurring at 14/11 (and its octave complement), which is mapped inconsistently sharp and equated with 9/7, but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the 27-odd-limit.

As an equal temperament, it notably tempers out Mercator's comma (353/284), the schisma (32805/32768), kleisma (15625/15552), and amity comma (1600000/1594323). In the 7-limit it tempers out the marvel comma (225/224), orwellisma (1728/1715), gariboh comma (3125/3087), and ragisma (4375/4374). In the 11-limit, it tempers out 99/98 and 121/120 (in addition to their difference, 540/539), and is the optimal patent val for big brother temperament, which tempers out both, as well as 11-limit orwell temperament, which also tempers out the 11-limit commas 176/175 and 385/384. In the 13-limit, it tempers out 169/168, 275/273, 325/324, 625/624, 676/675, 1001/1000, 2080/2079, and 4096/4095, and gives the optimal patent val for athene temperament.

53edo has also found a certain dissemination as an edo tuning for Arabic, Turkish, and Persian music. It can also be used as an extended Pythagorean tuning, since its fifths are almost indistinguishable from just.

53edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the step's role as a "comma" comes from it being an approximation of the Pythagorean comma and syntonic comma.

Prime harmonics

Approximation of prime harmonics in 53edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.00 -0.07 -1.41 +4.76 -7.92 -2.79 +8.25 -3.17 +5.69 -10.71 +9.68 -2.29
Relative (%) +0.0 -0.3 -6.2 +21.0 -35.0 -12.3 +36.4 -14.0 +25.1 -47.3 +42.8 -10.1
Steps
(reduced) 		53
(0) 		84
(31) 		123
(17) 		149
(43) 		183
(24) 		196
(37) 		217
(5) 		225
(13) 		240
(28) 		257
(45) 		263
(51) 		276
(11)
Approximation of prime harmonics in 53edo (continued)
Harmonic 41 43 47 53 59 61 67 71 73 79 83 89
Error Absolute (¢) +1.13 +9.24 -8.90 +9.51 +4.98 -7.45 +11.26 +1.44 -1.37 -2.27 +2.78 -4.84
Relative (%) +5.0 +40.8 -39.3 +42.0 +22.0 -32.9 +49.7 +6.3 -6.1 -10.0 +12.3 -21.4
Steps
(reduced) 		284
(19) 		288
(23) 		294
(29) 		304
(39) 		312
(47) 		314
(49) 		322
(4) 		326
(8) 		328
(10) 		334
(16) 		338
(20) 		343
(25)

See #Approximation to JI for details and a more in-depth discussion.

Subsets and supersets

53edo is the 16th prime edo, following 47edo and coming before 59edo.

Many of its multiples such as 159edo, 212edo, 742edo, 901edo and 954edo have good consistency limits and are each of their own interest. The mercator family comprises rank-2 temperaments with 1/53-octave periods.

Intervals

# Cents Approximate Ratios[note 1] Ups and Downs Notation Solfeges
0 0.00 1/1 P1 unison D Da Do
1 22.64 81/80, 64/63, 50/49 ^1 up unison ^D Du Di
2 45.28 49/48, 36/35, 33/32, 128/125 ^^1, vvm2 dup unison, dudminor 2nd ^^D, vvEb Di / Fre Daw
3 67.92 25/24, 28/27, 22/21, 27/26, 26/25 vvA1, vm2 dudaug 1sn, downminor 2nd vvD#, vEb Fro Ro
4 90.57 19/18, 20/19, 21/20, 256/243 vA1, m2 downaug 1sn, minor 2nd vD#, Eb Fra Rih
5 113.21 16/15, 15/14 A1, ^m2 aug 1sn, upminor 2nd D#, ^Eb Fru Ra
6 135.85 14/13, 13/12, 27/25 ^^m2 dupminor 2nd ^^Eb Fri Ru
7 158.49 35/32, 12/11, 11/10, 57/52, 800/729 vvM2 dudmajor 2nd vvE Re Ruh
8 181.13 10/9 vM2 downmajor 2nd vE Ro Reh
9 203.77 9/8 M2 major 2nd E Ra Re
10 226.42 8/7, 256/225 ^M2 upmajor 2nd ^E Ru Ri
11 249.06 15/13, 144/125, 125/108 ^^M2, vvm3 dupmajor 2nd, dudminor 3rd ^^E, vvF Ri / Ne Raw
12 271.70 7/6, 75/64 vm3 downminor 3rd vF No Ma
13 294.34 13/11, 19/16, 32/27 m3 minor 3rd F Na Meh
14 316.98 6/5 ^m3 upminor 3rd ^F Nu Me
15 339.62 11/9, 243/200 ^^m3 dupminor 3rd ^^F Ni Mu
16 362.26 16/13, 100/81 vvM3 dudmajor 3rd vvF# Me Muh
17 384.91 5/4 vM3 downmajor 3rd vF# Mo Mi
18 407.55 19/15, 24/19, 81/64 M3 major 3rd F# Ma Maa
19 430.19 9/7, 14/11 ^M3 upmajor 3rd ^F# Mu Mo
20 452.83 13/10, 125/96, 162/125 ^^M3, vv4 dupmajor 3rd, dud 4th ^^F#, vvG Mi / Fe Maw
21 475.47 21/16, 25/19, 675/512, 320/243 v4 down 4th vG Fo Fe
22 498.11 4/3 P4 perfect 4th G Fa Fa
23 520.75 27/20 ^4 up 4th ^G Fu Fih
24 543.40 11/8, 15/11, 26/19 ^^4 dup 4th ^^G Fi / She Fu
25 566.04 18/13 vvA4, vd5 dudaug 4th, downdim 5th vvG#, vAb Pe / Sho Fuh
26 588.68 7/5, 45/32 vA4, d5 downaug 4th, dim 5th vG#, Ab Po / Sha Fi
27 611.32 10/7, 64/45 A4, ^d5 aug 4th, updim 5th G#, ^Ab Pa / Shu Se
28 633.96 13/9 ^A4, ^^d5 upaug 4th, dupdim 5th ^G#, ^^Ab Pu / Shi Suh
29 656.60 16/11, 19/13, 22/15 vv5 dud 5th vvA Pi / Se Su
30 679.25 40/27 v5 down 5th vA So Sih
31 701.89 3/2 P5 perfect 5th A Sa Sol
32 724.53 32/21, 38/25, 243/160, 1024/675 ^5 up 5th ^A Su Si
33 747.17 20/13, 192/125, 125/81 ^^5, vvm6 dup 5th, dudminor 6th ^^A, vvBb Si / Fle Saw
34 769.81 14/9, 25/16, 11/7 vm6 downminor 6th vBb Flo Lo
35 792.45 19/12, 30/19, 128/81 m6 minor 6th Bb Fla Leh
36 815.09 8/5 ^m6 upminor 6th ^Bb Flu Le
37 837.74 13/8, 81/50 ^^m6 dupminor 6th ^^Bb Fli Lu
38 860.38 18/11, 400/243 vvM6 dudmajor 6th vvB Le Luh
39 883.02 5/3 vM6 downmajor 6th vB Lo La
40 905.66 22/13, 27/16, 32/19 M6 major 6th B La Laa
41 928.30 12/7 ^M6 upmajor 6th ^B Lu Li
42 950.94 26/15, 125/72, 216/125 ^^M6, vvm7 dupmajor 6th, dudminor 7th ^^B, vvC Li / The Law
43 973.58 7/4 vm7 downminor 7th vC Tho Ta
44 996.23 16/9 m7 minor 7th C Tha Teh
45 1018.87 9/5 ^m7 upminor 7th ^C Thu Te
46 1041.51 64/35, 11/6, 20/11, 729/400 ^^m7 dupminor 7th ^^C Thi Tu
47 1064.15 13/7, 24/13, 50/27 vvM7 dudmajor 7th vvC# Te Tuh
48 1086.79 15/8 vM7 downmajor 7th vC# To Ti
49 1109.43 19/10, 36/19, 40/21, 243/128 M7 major 7th C# Ta Tih
50 1132.08 48/25, 27/14, 21/11, 52/27, 25/13 ^M7 upmajor 7th ^C# Tu To
51 1154.72 96/49, 35/18, 64/33, 125/64 ^^M7, vv8 dupmajor 7th, dud 8ve ^^C#, vvD Ti / De Taw
52 1177.36 160/81, 63/32, 49/25 v8 down 8ve vD Do Da
53 1200.00 2/1 P8 perfect 8ve 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) with 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) with b > 1 9/8, 27/16
upmajor ru (a, b, 0, -1) 9/7, 12/7

All 53edo chords can be named using ups and downs. An up, down or mid 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). Alterations are always enclosed in parentheses, additions never are.

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 0-12-31 C vEb G Cvm C downminor
gu 10:12:15 0-14-31 C ^Eb G C^m C upminor
ilo 18:22:27 0-15-31 C ^^Eb G C^^m C dupminor
lu 22:27:33 0-16-31 C vvE G Cvv C dudmajor or C dud
yo 4:5:6 0-17-31 C vE G Cv C downmajor or C down
ru 14:18:21 0-19-31 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

Using Helmholtz–Ellis notation accidentals, 53edo can be notated using ups and downs notation:

Step Offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp Symbol
Heji18.svg
Heji19.svg
Heji20.svg
Heji23.svg
Heji24.svg
Heji25.svg
Heji26.svg
Heji27.svg
Heji30.svg
Heji31.svg
Heji32.svg
Heji33.svg
Heji34.svg
Flat Symbol
Heji17.svg
Heji16.svg
Heji13.svg
Heji12.svg
Heji11.svg
Heji10.svg
Heji9.svg
Heji6.svg
Heji5.svg
Heji4.svg
Heji3.svg
Heji2.svg

Here, a sharp raises by five steps (commas), 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, then in some cases certain notes may be best spelled with triple arrows.

Sagittal

The following table shows sagittal notation accidentals in one apotome for 53edo.

Steps 0 1 2 3 4 5
Symbol Sagittal natural.png Sagittal pai.png Sagittal phai.png Sagittal sharp phao.png Sagittal sharp pao.png Sagittal sharp.png

Relationship to 12edo

Whereas 12edo has a circle of twelve 5ths, 53edo has a spiral of twelve 5ths (since 31\53 is on the 7\12 kite in the scale tree). This shows 53edo in a 12edo-friendly format. Excellent for introducing 53edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.

53-edo spiral.png

Approximation to JI

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Selected 7-limit intervals approximated in 53edo

53edo provides excellent approximations for the classic 5-limit just chords and scales, such as the Ptolemy-Zarlino "just major" scale.

Interval Ratio Size Difference
Perfect fifth 3/2 31 −0.07 cents
Major third 5/4 17 −1.40 cents
Minor third 6/5 14 +1.34 cents
Major second 9/8 9 −0.14 cents
10/9 8 −1.27 cents
Minor second 16/15 5 +1.48 cents

Because the 5th is so accurate, 53edo also offers good approximations for Pythagorean thirds. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the septimal kleisma, 225/224.

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 53edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 53edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.068 0.3
9/8, 16/9 0.136 0.6
9/5, 10/9 1.272 5.6
15/13, 26/15 1.316 5.8
5/3, 6/5 1.340 5.9
13/10, 20/13 1.384 6.1
5/4, 8/5 1.408 6.2
15/8, 16/15 1.476 6.5
13/9, 18/13 2.655 11.7
13/12, 24/13 2.724 12.0
13/8, 16/13 2.792 12.3
7/4, 8/7 4.759 21.0
7/6, 12/7 4.827 21.3
9/7, 14/9 4.895 21.6
13/11, 22/13 5.130 22.7
7/5, 10/7 6.167 27.2
15/14, 28/15 6.235 27.5
15/11, 22/15 6.445 28.5
11/10, 20/11 6.514 28.8
13/7, 14/13 7.551 33.3
11/9, 18/11 7.785 34.4
11/6, 12/11 7.854 34.7
11/8, 16/11 7.922 35.0
11/7, 14/11 9.961 44.0
15-odd-limit intervals in 53edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.068 0.3
9/8, 16/9 0.136 0.6
9/5, 10/9 1.272 5.6
15/13, 26/15 1.316 5.8
5/3, 6/5 1.340 5.9
13/10, 20/13 1.384 6.1
5/4, 8/5 1.408 6.2
15/8, 16/15 1.476 6.5
13/9, 18/13 2.655 11.7
13/12, 24/13 2.724 12.0
13/8, 16/13 2.792 12.3
7/4, 8/7 4.759 21.0
7/6, 12/7 4.827 21.3
9/7, 14/9 4.895 21.6
13/11, 22/13 5.130 22.7
7/5, 10/7 6.167 27.2
15/14, 28/15 6.235 27.5
15/11, 22/15 6.445 28.5
11/10, 20/11 6.514 28.8
13/7, 14/13 7.551 33.3
11/9, 18/11 7.785 34.4
11/6, 12/11 7.854 34.7
11/8, 16/11 7.922 35.0
11/7, 14/11 12.681 56.0

Higher-limit JI

There is also a cluster of usable higher primes starting at 71; even 89 (4.84 ¢ flat), 97 (4.63 ¢ sharp) and 101 (2.6 ¢ sharp) are usable if placed in just the right context.

Approximation of prime harmonics in 53edo
Harmonic 71 73 79 83
Error Absolute (¢) +1.44 -1.37 -2.27 +2.78
Relative (%) +6.3 -6.1 -10.0 +12.3
Steps
(reduced) 		326
(8) 		328
(10) 		334
(16) 		338
(20)

This make 53edo excellent (for its size) in the 2.3.5.7.11.13.19.23.37.41.71.73(.79.83.101) subgroup, although some higher error primes like 11 and 23 require the right context to be convincing.

Note that the high primes, in rooted (/2n) position, essentially act as alternate interpretations of LCJI intervals, if you want to force a rooted interpretation; namely: 71/64 as ~10/9, 73/64 as ~8/7, 79/64 as ~16/13, and perhaps most questionably in the context of 53edo, 83/64 as ~13/10.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-84 53⟩ [⟨53 84]] +0.022 0.022 0.10
2.3.5 15625/15552, 32805/32768 [⟨53 84 123]] +0.216 0.276 1.22
2.3.5.7 225/224, 1728/1715, 3125/3087 [⟨53 84 123 149]] −0.262 0.861 3.81
2.3.5.7.11 99/98, 121/120, 176/175, 2200/2187 [⟨53 84 123 149 183]] +0.248 1.279 5.64
2.3.5.7.11.13 99/98, 121/120, 169/168, 176/175, 275/273 [⟨53 84 123 149 183 196]] +0.332 1.183 5.22
2.3.5.7.11.13.19 99/98, 121/120, 169/168, 176/175, 209/208, 275/273 [⟨53 84 123 149 183 196 225]] +0.391 1.105 4.88
  • 53et is lower in relative error than any previous equal temperaments in the 3-, 5-, and 13-limit. The next equal temperaments doing better in these subgroups are 306, 118, and 58, respectively. It is even more prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups, and the next equal temperament doing better in either subgroup is 130.

Commas

Commas that 53edo tempers out using its patent val, ⟨53 84 123 149 183 196 225], include:

Prime
limit 		Ratio[note 2] Monzo Cents Color name Name(s)
3 (52 digits) [-84 53⟩ 3.62 Tribilawa 53-comma, Mercator's comma
5 (14 digits) [-21 3 7⟩ 10.06 Lasepyo Semicomma
5 15625/15552 [-6 -5 6⟩ 8.11 Tribiyo Kleisma
5 (14 digits) [9 -13 5⟩ 6.15 Saquinyo Amity comma
5 (22 digits) [24 -21 4⟩ 4.20 Sasaquadyo Vulture comma
5 32805/32768 [-15 8 1⟩ 1.95 Layo Schisma
7 3125/3087 [0 -2 5 -3⟩ 21.18 Triru-aquinyo Gariboh comma
7 1728/1715 [6 3 -1 -3⟩ 13.07 Triru-agu Orwellisma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Marvel comma, septimal kleisma
7 4375/4374 [-1 -7 4 1⟩ 0.40 Zoquadyo Ragisma
11 99/98 [-1 2 0 -2 1⟩ 17.58 Loruru Mothwellsma
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 (22 digits) [33 -23 0 0 1⟩ 6.35 Trisalo Pythrabian comma
11 385/384 [-7 -1 1 1 1⟩ 4.50 Lozoyo Keenanisma
11 540/539 [2 3 1 -2 -1⟩ 3.21 Lururuyo Swetisma
13 275/273 [0 -1 2 -1 1 -1⟩ 12.64 Thuloruyoyo Gassorma
13 169/168 [-3 -1 0 -1 0 2⟩ 10.27 Thothoru Buzurgisma, dhanvantarisma
13 625/624 [-4 -1 4 0 0 -1⟩ 2.77 Thuquadyo Tunbarsma
13 676/675 [2 -3 -2 0 0 2⟩ 2.56 Bithogu Island comma, parizeksma
13 1001/1000 [-3 0 -3 1 1 1⟩ 1.73 Tholozotrigu Fairytale comma, sinbadma
13 2080/2079 [5 -3 1 -1 -1 1⟩ 0.83 Tholuruyo Ibnsinma, sinaisma
13 4096/4095 [12 -2 -1 -1 0 -1⟩ 0.42 Sathurugu Schismina

Linear temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 2\53 45.28 36/35 Quartonic
1 5\53 113.21 16/15 Misneb
1 7\53 158.49 11/10 Hemikleismic
1 9\53 203.77 9/8 Baldy
1 10\53 226.42 8/7 Semaja
1 11\53 249.06 15/13 Hemischis / hemigari
1 12\53 271.70 7/6 Orwell
1 13\53 294.34 25/21 Kleiboh
1 14\53 316.98 6/5 Hanson / catakleismic / countercata
1 15\53 339.62 11/9 Amity / houborizic
1 16\53 362.26 16/13 Submajor
1 18\53 407.55 1225/972 Ditonic / coditone
1 19\53 430.19 9/7 Hamity
1 21\53 475.47 21/16 Vulture / buzzard
1 22\53 498.11 4/3 Garibaldi / pontiac
1 23\53 520.75 4/3 Mavila (53bbcc)
1 25\53 566.04 18/13 Tricot
1 26\53 588.68 45/32 Untriton / aufo

* Octave-reduced form, reduced to the first half-octave

Scales

MOS scales

While there is only one possible generator for the diatonic mos scale supported by this edo, there are a greater number of generators for other mosses such as the antidiatonic scale.

Scales approximated from JI

  • The eagle 53 scale described by John O'Sullivan
  • Ptolmey—Zarlino justly-intonated major: 9 8 5 9 8 9 5
  • Ptolmey—Zarlino justly-intonated minor: 9 5 8 9 5 9 8
  • Composite Cliffedge (approximated from 60afdo): 12 10 9 19 3
  • Composite Deja Vu (approximated from 101afdo): 14 17 5 9 8
  • Composite Dungeon (approximated from 30afdo): 17 5 9 4 18
  • Composite Freeway (approximated from 6afdo): 12 10 9 8 7 7
  • Composite Geode (approximated from 6afdo): 12 10 9 15 7
  • Composite Labyrinth (approximated from 30afdo): 7 7 17 5 17
  • Composite Mushroom (approximated from 30afdo): 12 10 9 3 19
  • Composite Underpass (approximated from 10afdo): 14 17 10 4 8
  • Spectral Arcade (approximated from 32afdo): 17 4 10 12 10
  • Spectral Mechanical (approximated from 16afdo): 13 4 14 12 10
  • Spectral Moonbeam (approximated from 16afdo): 9 4 18 17 5
  • Spectral Springwater (approximated from 8afdo): 9 8 14 12 10
  • Spectral Starship (approximated from 68ifdo): 4 13 4 10 12 10
  • Spectral Volcanic (approximated from 16afdo): 5 12 14 12 10

Other scales

  • Sequar5m: 6 3 6 2 3 6 2 3 6 3 2 6 3 2
  • Palace (approximated from Porky in 29edo): 7 7 8 9 7 7 8

Music

See also: Category:53edo tracks

Modern renderings

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

21st century

Francium
Andrew Heathwaite
Hideya
Nathan Ho
Aaron Krister Johnson (site⁠ ⁠[dead link])
MortisTheneRd
Prent Rodgers
Subhraag Singh
Gene Ward Smith
  • Trio in Orwell (archived 2010) – detail | play – Orwell[9] in 53edo
Nick Stephens
Cam Taylor
Chris Vaisvil
  • The Fallen of Kleismic15 (2013) – blog | play – Kleismic[15] in 53edo
Valeriana of the Night
Randy Wells
Xotla

Instruments

Notes

  1. ↑ Based on treating 53edo as a no-17's 19-limit temperament; other approaches are also possible. Italics represent inconsistent intervals which are mapped by the 19-limit patent val to their second-closest (as opposed to closest) approximation in 53edo.
  2. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

References

  1. ↑ Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". Empirical Musicology Review 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.
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