53edo

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Theory

The famous 53 equal division divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a 5-limit system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. In the 7-limit it tempers out 225/224, 1728/1715 and 3125/3087, the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out 99/98 and 121/120, 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 comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for athene temperament. It is the eighth zeta integral edo and the 16th prime edo, following 47edo and coming before 59edo.

53EDO has also found a certain dissemination as an EDO tuning for Arabic/Turkish/Persian music.

It can also be treated as a no-elevens, no-seventeens tuning, on which it is consistent all the way up to the 21-limit.

Wikipeda article about 53edo

Linear temperaments

List of edo-distinct 53et rank two temperaments

Just Approximation

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 tone 9/8 9 −0.14 cents
minor tone 10/9 8 −1.27 cents
diat. semitone 16/15 5 +1.48 cents

One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.

The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the septimal kleisma, 225/224.

Intervals

degree solfege cents pions 7mus approximate ratios ups and downs notation generator for
0 do 0.00 1/1 P1 unison D
1 di 22.64 24 28.98 (1C.FB16) 81/80, 64/63, 50/49 ^1 up unison ^D
2 daw 45.28 48 57.96 (39.F616) 49/48, 36/35, 33/32, 128/125 ^^1,

vvm2

double-up unison,

double-down minor 2nd

^^D,

vvEb

Quartonic
3 ro 67.925 72 86.94 (56.F1816) 27/26, 26/25, 25/24, 22/21 vm2 downminor 2nd vEb
4 rih 90.57 96 115.925 (73.EC16) 21/20, 256/243 m2 minor 2nd Eb
5 ra 113.21 120 144.91 (90.E816) 16/15, 15/14 ^m2 upminor 2nd ^Eb
6 ru 135.85 144 173.89 (AD.E316) 14/13, 13/12, 27/25 v~2 downmid 2nd ^^Eb
7 ruh 158.49 168 202.87 (CA.DE16) 12/11, 11/10, 800/729 ^~2 upmid 2nd vvE Hemikleismic
8 reh 181.13 192 231.85 (E7.D916) 10/9 vM2 downmajor 2nd vE
9 re 203.77 216 260.83 (104.D4816) 9/8 M2 major 2nd E
10 ri 226.42 240 289.81 (121.D16) 8/7, 256/225 ^M2 upmajor 2nd ^E
11 raw 249.06 264 318.79 (13E.CB16) 15/13, 144/125 ^^M2,

vvm3

double-up major 2nd,

double-down minor 3rd

^^E,

vvF

Hemischis
12 ma 271.70 288 347.77 (15B.C616) 7/6, 75/64 vm3 downminor 3rd vF Orwell
13 meh 294.34 312 376.755 (178.C116) 13/11, 32/27 m3 minor 3rd F
14 me 316.98 336 405.74 (195.BC16) 6/5 ^m3 upminor 3rd ^F Hanson/Catakleismic
15 mu 339.62 360 434.72 (1B2.B816) 11/9, 243/200 v~3 downmid 3rd ^^F Amity/Hitchcock
16 muh 362.26 384 463.7 (1CF.B316) 16/13, 100/81 ^~3 upmid 3rd vvF#
17 mi 384.91 408 492.68 (1EC.AE16) 5/4 vM3 downmajor 3rd vF#
18 maa 407.55 432 521.66 (209.A916) 81/64 M3 major 3rd F#
19 mo 430.19 456 550.64 (226.A416) 9/7, 14/11 ^M3 upmajor 3rd ^F# Hamity
20 maw 452.83 480 579.62 (243.9F16) 13/10, 125/96 ^^M3,

vv4

double-up major 3rd,

double-down 4th

^^F#,

vvG

21 fe 475.47 504 608.6 (260.9B16) 21/16, 675/512, 320/243 v4 down 4th vG Vulture/Buzzard
22 fa 498.11 528 637.585 (27D.9616) 4/3 P4 perfect 4th G
23 fih 520.75 552 666.57 (29A.9116) 27/20 ^4 up 4th ^G
24 fu 543.40 576 695.55 (2B7.8C16) 11/8, 15/11 v~4 downmid 4th ^^G
25 fuh 566.04 600 724.53 (2D4.8716) 18/13 ^~4,

vd5

upmid 4th,

downdim 5th

vvG#,

vAb

Tricot
26 fi 588.68 624 753.51 (2F1.8216) 7/5, 45/32 vA4,

d5

downaug 4th,

dim 5th

vG#,

Ab

27 se 611.32 648 782.49 (30E.7E16) 10/7, 64/45 A4,

^d5

aug 4th,

updim 5th

G#,

^Ab

28 suh 633.96 672 811.47 (32B.7916) 13/9 ^A4,

v~5

upaug 4th,

downmid 5th

^G#,

^^Ab

29 su 656.60 696 840.45 (348.7416) 16/11, 22/15 ^~5 upmid 5th vvA
30 sih 679.25 720 869.43 (365.6F16) 40/27 v5 down 5th vA
31 sol 701.89 744 898.415 (382.6A16) 3/2 P5 perfect 5th A Helmholtz/Garibaldi
32 si 724.53 768 927.4 (39F.6516) 32/21, 243/160, 1024/675 ^5 up 5th ^A
33 saw 747.17 792 956.38 (3BC.6116) 20/13, 192/125 ^^5,

vvm6

double-up 5th,

double-down minor 6th

^^A,

vvBb

34 lo 769.81 816 985.36 (3D9.5C16) 14/9, 25/16, 11/7 vm6 downminor 6th vBb
35 leh 792.45 840 1014.34 (3F6.5716) 128/81 m6 minor 6th Bb
36 le 815.09 864 1043.32 (413.5216) 8/5 ^m6 upminor 6th ^Bb
37 lu 837.74 888 1072.3 (430.4D16) 13/8, 81/50 v~6 downmid 6th ^^Bb
38 luh 860.38 912 1101.28 (44D.4816) 18/11, 400/243 ^~6 upmid 6th vvB
39 la 883.02 936 1130.26 (46A.4416) 5/3 vM6 downmajor 6th vB
40 laa 905.66 960 1159.245 (487.3F16) 22/13, 27/16 M6 major 6th B
41 lo 928.30 984 1188.23 (4A4.3A16) 12/7 ^M6 upmajor 6th ^B
42 law 950.94 1008 1217.21 (4C1.3516) 26/15, 125/72 ^^M6

vvm7

double-up major 6th,

double-down minor 7th

^^B,

vvC

43 ta 973.58 1032 1246.19 (4DE.316) 7/4 vm7 downminor 7th vC
44 teh 996.23 1056 1275.17 (4FB.2B816) 16/9 m7 minor 7th C
45 te 1018.87 1080 1304.15 (518.2716) 9/5 ^m7 upminor 7th ^C
46 tu 1041.51 1104 1333.13 (535.2216) 11/6, 20/11, 729/400 v~7 downmid 7th ^^C
47 tuh 1064.15 1128 1362.11 (552.1D16) 13/7, 24/13, 50/27 ^~7 upmid 7th vvC#
48 ti 1086.79 1152 1391.09 (56F.1816) 15/8 vM7 downmajor 7th vC#
49 tih 1109.43 1176 1420.075 (58C.1316) 40/21, 243/128 M7 major 7th C#
50 to 1132.08 1200 1449.06 (5A9.0E816) 48/25, 27/14 ^M7 upmajor 7th ^C#
51 taw 1154.72 1224 1478.04 (5C6.0A16) 125/64 ^^M7,

vv8

double-up major 7th,

double-down 8ve

^^C#,

vvD

52 da 1177.36 1248 1507.02 (5E3.0516) 160/81 v8 down 8ve vD
53 do 1200 1272 1536 (60016) 2/1 P8 perfect 8ve D

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
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) 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. 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 Cv~ C downmid
lu 22:27:33 0-16-31 C vvE G C^~ C upmid
yo 4:5:6 0-17-31 C vE G Cv C downmajor or C down
ru 14:18:27 0-19-31 C ^E G C^ C upmajor or C up

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

This chart shows 53-edo in a 12-edo-friendly format.

53-edo spiral.png

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 53edo (ordered by absolute error).

Interval, complement Error (abs., in cents)
4/3, 3/2 0.068
9/8, 16/9 0.136
10/9, 9/5 1.272
15/13, 26/15 1.316
6/5, 5/3 1.340
13/10, 20/13 1.384
5/4, 8/5 1.408
16/15, 15/8 1.476
18/13, 13/9 2.655
13/12, 24/13 2.724
16/13, 13/8 2.792
8/7, 7/4 4.759
7/6, 12/7 4.827
9/7, 14/9 4.895
13/11, 22/13 5.130
7/5, 10/7 6.167
15/14, 28/15 6.235
15/11, 22/15 6.445
11/10, 20/11 6.514
14/13, 13/7 7.551
11/9, 18/11 7.785
12/11, 11/6 7.854
11/8, 16/11 7.922
14/11, 11/7 9.961

Compositions

Bach WTC1 Prelude 1 in 53 by Bach and Mykhaylo Khramov

Bach WTC1 Fugue 1 in 53 by Bach and Mykhaylo Khramov

Whisper Song in 53EDO play by Prent Rodgers

Trio in Orwell play by Gene Ward Smith

Desert Prayer by Aaron Krister Johnson

Whisper Song in 53 EDO by Prent Rodgers

Elf Dine on Ho Ho play and Spun play by Andrew Heathwaite

The Fallen of Kleismic15play by Chris Vaisvil

mothers by Cam Taylor