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Theory | Linear temperaments | Just Approximation | Intervals | Compositions Other languages: Deutsch


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
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^^,
Ebvv
Quartonic
3
ro
67.92
27/26, 26/25, 25/24, 22/21
vm2
downminor 2nd
Ebv

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
Evv
Hemikleismic
8
reh
181.13
10/9
vM2
downmajor 2nd
Ev

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^^,
Fvv
Hemischis
12
ma
271.70
7/6, 75/64
vm3
downminor 3rd
Fv
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
F#vv

17
mi
384.91
5/4
vM3
downmajor 3rd
F#v

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#^^,
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

The distance from C to C# is 5 keys or frets or EDOsteps, and one up equals one fifth of a sharp. Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and Downs Notation - Chord names in other EDOs.

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