53edo

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Theory[edit]

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.

See also Wikipeda article about 53edo

Linear temperaments[edit]

See List of edo-distinct 53et rank two temperaments

Just Approximation[edit]

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[edit]

degree solfege cents approximate ratios ups and downs notation generator for
0 do 0.00 1/1 P1 unison C
1 di 22.64 81/80, 64/63, 50/49 ^1 upmajor unison C^
2 daw 45.28 49/48, 36/35, 33/32, 128/125 ^^1,
vvm2 		upupmajor unison C^^ Quartonic
3 ro 67.92 27/26, 26/25, 25/24, 22/21 vm2 downdownaugmented unison C#vv
4 rih 90.57 21/20, 256/243 m2 downaugmented unison C#v
5 ra 113.21 16/15, 15/14 ^m2 augmented unison C#
6 ru 135.85 14/13, 13/12, 27/25 v~2 upupminor second Db^^
7 ruh 158.49 12/11, 11/10, 800/729 ^~2 downdownmajor second Dvv Hemikleismic
8 reh 181.13 10/9 vM2 downmajor second Dv
9 re 203.77 9/8 M2 major second D
10 ri 226.42 8/7, 256/225 ^M2 upmajor second D^
11 raw 249.06 15/13, 144/125 ^^M2,
vvm3 		upupmajor second Hemischis
12 ma 271.70 7/6, 75/64 vm3 downminor third Ebv Orwell
13 meh 294.34 13/11, 32/27 m3 minor third Eb
14 me 316.98 6/5 ^m3 upminor third Eb^ Hanson/Catakleismic
15 mu 339.62 11/9, 243/200 v~3 upupminor third Eb^^ Amity/Hitchcock
16 muh 362.26 16/13, 100/81 ^~3 downdownmajor third Evv
17 mi 384.91 5/4 vM3 downmajor third Ev
18 maa 407.55 81/64 M3 major third E
19 mo 430.19 9/7, 14/11 ^M3 upmajor 3rd F#^ Hamity
20 maw 452.83 13/10, 125/96 ^^M3,
vv4 		double-up major 3rd,
double-down 4th 		F#^^,
Gvv
21 fe 475.47 21/16, 675/512, 320/243 v4 down 4th Gv Vulture/Buzzard
22 fa 498.11 4/3 P4 perfect 4th G
23 fih 520.75 27/20 ^4 up 4th G^
24 fu 543.40 11/8, 15/11 ^^4 double-up 4th G^^
25 fuh 566.04 18/13 vvA4,
vd5 		double-down aug 4th,
downdim 5th 		G#vv,
Abv 		Tricot
26 fi 588.68 7/5, 45/32 vA4,
d5 		downaug 4th,
dim 5th 		G#v,
Ab
27 se 611.32 10/7, 64/45 A4,
^d5 		aug 4th,
updim 5th 		G#,
Ab^
28 suh 633.96 13/9 ^A4,
^^d5 		upaug 4th,
double-up dim 5th 		G#^,
Ab^^
29 su 656.60 16/11, 22/15 vv5 double-down 5th Avv
30 sih 679.25 40/27 v5 down 5th Av
31 sol 701.89 3/2 P5 perfect 5th A Helmholtz/Garibaldi
32 si 724.53 32/21, 243/160, 1024/675 ^5 up 5th A^
33 saw 747.17 20/13, 192/125 ^^5,
vvm6 		double-up 5th,
double-down minor 6th 		A^^,
Bbvv
34 lo 769.81 14/9, 25/16, 11/7 vm6 downminor 6th Bbv
35 leh 792.45 128/81 m6 minor 6th Bb
36 le 815.09 8/5 ^m6 upminor 6th Bb^
37 lu 837.74 13/8, 81/50 v~6 downmid 6th Bb^^
38 luh 860.38 18/11, 400/243 ^~6 upmid 6th Bvv
39 la 883.02 5/3 vM6 downmajor 6th Bv
40 laa 905.66 22/13, 27/16 M6 major 6th B
41 lo 928.30 12/7 ^M6 upmajor 6th B^
42 law 950.94 26/15, 125/72 ^^M6,
vvm7 		double-up major 6th,
double-down minor 7th 		B^^,
Cvv
43 ta 973.58 7/4 vm7 downminor 7th Cv
44 teh 996.23 16/9 m7 minor 7th C
45 te 1018.87 9/5 ^m7 upminor 7th C^
46 tu 1041.51 11/6, 20/11, 729/400 v~7 downmid 7th C^^
47 tuh 1064.15 13/7, 24/13, 50/27 ^~7 upmid 7th C#vv
48 ti 1086.79 15/8 vM7 downmajor 7th C#v
49 tih 1109.43 40/21, 243/128 M7 major 7th C#
50 to 1132.08 48/25, 27/14 ^M7 upmajor 7th C#^
51 taw 1154.72 125/64 ^^M7,
vv8 		double-up major 7th,
double-down 8ve 		C#^^,
Dvv
52 da 1177.36 160/81 v8 down 8ve Dv
53 do 1200 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}, b < -1 32/27, 16/9
upminor gu {a, b, -1} 6/5, 9/5
downmid lova {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 53edo chords can be named using ups and downs. Here are the zo, gu, lova, 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 Ebv G C.vm C downminor
gu 10:12:15 0-14-31 C Eb^ G C.^m C upminor
lova 18:22:27 0-15-31 C Eb^^ G C.v~ C downmid
yo 4:5:6 0-17-31 C Ev G C.v C downmajor or C dot down
ru 14:18:27 0-19-31 C E^ G C.^ C upmajor or C dot up

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

Selected just intervals by error[edit]

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 12.681

Compositions[edit]

de:53edo

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