# 53edo

(Redirected from 106ed4)
 ← 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:

The famous 53 equal divisions of the octave (53edo), or 53(-tone) equal temperament (53tet, 53et) when viewed from a regular temperament perspective, divides the octave into 53 equal comma-sized parts of around 22.6 cents each.

## Theory

53edo is notable as a 5-limit system, a fact apparently first noted by Isaac Newton, notably tempering out the schisma (32805/32768) and the kleisma (15625/15552). More complex 5-limit commas tempered out include the amity comma (1600000/1594323), the semicomma (2109375/2097152), and the vulture comma ([24 -21 4). 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 (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. It is the seventh strict zeta edo. It can also be treated as a no-elevens, no-seventeens tuning, on which it is consistent all the way up to the 23-odd-limit.

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

53edo's step size is sometimes called the "Holdrian comma", despite not being a rational one.

### Prime harmonics

Approximation of prime harmonics in 53edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.00 -0.07 -1.41 +4.76 -7.92 -2.79 +8.25 -3.17 +5.69 -10.71 +9.68
relative (%) +0 -0 -6 +21 -35 -12 +36 -14 +25 -47 +43
Steps
(reduced)
53
(0)
84
(31)
123
(17)
149
(43)
183
(24)
196
(37)
217
(5)
225
(13)
240
(28)
257
(45)
263
(51)

### 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 limit and are each of their own interest. The mercator family comprises rank-2 temperaments with 1/53-octave periods.

## Intervals

# Cents Approximate Ratios* 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 dud-aug 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 v~2 downmid 2nd ^^Eb Fri Ru
7 158.49 35/32, 12/11, 11/10, 57/52, 800/729 ^~2 upmid 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 v~3 downmid 3rd ^^F Ni Mu
16 362.26 16/13, 100/81 ^~3 upmid 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 v~4 downmid 4th ^^G Fi / She Fu
25 566.04 18/13 ^~4,
vd5
upmid 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,
v~5
upaug 4th,
downmid 5th
^G#,
^^Ab
Pu / Shi Suh
29 656.60 16/11, 19/13, 22/15 ^~5 upmid 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 v~6 downmid 6th ^^Bb Fli Lu
38 860.38 18/11, 400/243 ^~6 upmid 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 v~7 downmid 7th ^^C Thi Tu
47 1064.15 13/7, 24/13, 50/27 ^~7 upmid 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

* based on interpreting 53edo as a no-17's 19-limit temperament. Italics represent inconsistent intervals which are mapped by the 19-limit patent val to their second-closest (as opposed to closest) approximation in 53edo.

### 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
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.

## Notation

### Sagittal

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

 Steps Symbol 0 1 2 3 4 5

## 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.

## JI 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 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 very accurate, 53edo also offers good approximations for Pythagorean thirds. In addition, the 43\53 interval is only 4.8 cents wider than the just ratio 7/4, so 53edo can also be used for 7-limit harmony, tempering out the septimal kleisma, 225/224.

### 15-odd-limit interval mappings

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

15-odd-limit intervals by direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢) Error (rel, %)
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 by patent val mapping
Interval, 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

## 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.

### 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 Orson / 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 Helmholtz / garibaldi / pontiac
1 25\53 566.04 18/13 Tricot
1 26\53 588.68 45/32 Untriton / aufo

## Music

### Modern renderings

Johann Sebastian Bach
Nicolaus Bruhns

### 21st century

Francium
Andrew Heathwaite
Hideya