53edo

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The famous 53 equal division divides the octave into 53 equal comma-sized parts of 22.642 cents each.

Theory

53edo 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 275/273, 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 23-limit.

See also: 53 Equal Temperament - Wikipedia

Intervals

# Solfege Cents Approximate ratios Ups and downs notation Generator for
0 do 0.00 1/1 P1 unison D
1 di 22.64 81/80, 64/63, 50/49 ^1 up unison ^D
2 daw 45.28 49/48, 36/35, 33/32, 128/125 ^^1,
vvm2
double-up unison,
double-down minor 2nd
^^D,
vvEb
Quartonic
3 ro 67.92 25/24, 28/27, 22/21, 27/26, 26/25 vm2 downminor 2nd vEb
4 rih 90.57 21/20, 256/243 m2 minor 2nd Eb
5 ra 113.21 16/15, 15/14 ^m2 upminor 2nd ^Eb
6 ru 135.85 14/13, 13/12, 27/25 v~2 downmid 2nd ^^Eb
7 ruh 158.49 12/11, 11/10, 800/729 ^~2 upmid 2nd vvE Hemikleismic
8 reh 181.13 10/9 vM2 downmajor 2nd vE
9 re 203.77 9/8 M2 major 2nd E
10 ri 226.42 8/7, 256/225 ^M2 upmajor 2nd ^E
11 raw 249.06 15/13, 144/125 ^^M2,
vvm3
double-up major 2nd,
double-down minor 3rd
^^E,
vvF
Hemischis
12 ma 271.70 7/6, 75/64 vm3 downminor 3rd vF Orwell
13 meh 294.34 13/11, 32/27 m3 minor 3rd F
14 me 316.98 6/5 ^m3 upminor 3rd ^F Hanson/Catakleismic
15 mu 339.62 11/9, 243/200 v~3 downmid 3rd ^^F Amity/Hitchcock
16 muh 362.26 16/13, 100/81 ^~3 upmid 3rd vvF#
17 mi 384.91 5/4 vM3 downmajor 3rd vF#
18 maa 407.55 81/64 M3 major 3rd F#
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#,
vvG
21 fe 475.47 21/16, 675/512, 320/243 v4 down 4th vG 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 v~4 downmid 4th ^^G
25 fuh 566.04 18/13 ^~4,
vd5
upmid 4th,
downdim 5th
vvG#,
vAb
Tricot
26 fi 588.68 7/5, 45/32 vA4,
d5
downaug 4th,
dim 5th
vG#,
Ab
27 se 611.32 10/7, 64/45 A4,
^d5
aug 4th,
updim 5th
G#,
^Ab
28 suh 633.96 13/9 ^A4,
v~5
upaug 4th,
downmid 5th
^G#,
^^Ab
29 su 656.60 16/11, 22/15 ^~5 upmid 5th vvA
30 sih 679.25 40/27 v5 down 5th vA
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,
vvBb
34 lo 769.81 14/9, 25/16, 11/7 vm6 downminor 6th vBb
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 vvB
39 la 883.02 5/3 vM6 downmajor 6th vB
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,
vvC
43 ta 973.58 7/4 vm7 downminor 7th vC
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 vvC#
48 ti 1086.79 15/8 vM7 downmajor 7th vC#
49 tih 1109.43 40/21, 243/128 M7 major 7th C#
50 to 1132.08 48/25, 27/14, 21/11, 52/27, 25/13 ^M7 upmajor 7th ^C#
51 taw 1154.72 96/49, 35/18, 64/33, 125/64 ^^M7,
vv8
double-up major 7th,
double-down 8ve
^^C#,
vvD
52 da 1177.36 160/81, 63/32, 49/25 v8 down 8ve vD
53 do 1200.00 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. 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 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 14:18:21 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.

Relationship to 12-edo

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

53-edo spiral.png

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.

Selected just intervals by error

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23
Error absolute (¢) 0.00 -0.07 -1.41 +4.76 -7.92 -2.79 +8.26 -3.17 +5.69
relative (%) 0.0 -0.3 -6.2 +21.0 -35.0 -12.3 +36.4 -14.0 +25.1
fifthspan 0 +1 -8 -14 +23 +20 +7 -3 +18

The following table shows how 15-odd-limit intervals are represented in 53edo. Octave-reduced prime harmonics are bolded; inconsistent intervals are in italic.

Interval, complement Error (abs, ¢)
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

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 53et.

3-limit 5-limit 7-limit 11-limit 13-limit 2.3.5.7.13.19 2.3.5.7.13.19.23
Octave stretch (¢) +0.022 +0.216 -0.262 +0.248 +0.332 +0.075 -0.115
Error absolute (¢) 0.022 0.276 0.861 1.279 1.183 0.850 0.915
relative (%) 0.10 1.22 3.81 5.64 5.22 3.75 4.04
  • 53et has a lower relative error than any previous ETs in the 3-, 5-, and 13-limit. The next ET that does better in these subgroups is 306, 118, and 58, respectively.
  • 53et is prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups. The next ET that does better in either case is 130.

Linear temperaments

Music

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