The intervals of 5limit just intonation (prime limit, not odd limit) are ratios involving only the powers of 2, 3, and 5. The fundamental intervals are the superparticular ratios 2/1 (the octave), 3/2 (the perfect fifth) and 5/4 (the major third). That is, the notes of the major triad are in the ratio 1:5/4:3/2 or 4:5:6.
In all tunings, the major third is equivalent to two major seconds. However, because just intonation does not allow the irrational ratio of √5/2, two different frequency ratios are used: the major tone (9/8) and the minor tone (10/9).
The intervals within the diatonic scale are shown in the table below.
Names

Ratio

Cents

12ET Cents

Definition

53ET commas

53ET cents

Representation (Makam)

Complement

unison

1/1

0.00

0


0

0


octave

syntonic comma

81/80

21.51

0

c or T − t

1

22.64


semidiminished octave

diesis
diminished second

128/125

41.06

0

D or S − x

2

45.28


augmented seventh

lesser chromatic semitone
minor semitone
augmented unison

25/24

70.67

100

x or t − S or T − L

3

67.92


diminished octave

Pythagorean minor second
Pythagorean limma

256/243

90.22

100

Λ

4

90.57


Pythagorean major seventh

greater chromatic semitone
wide augmented unison

135/128

92.18

100

X or T − S

4

90.57


narrow diminished octave

major semitone
limma
minor second

16/15

111.73

100

S

5

113.21


major seventh

large limma
acute minor second

27/25

133.24

100

L or T − x

6

135.85


grave major seventh

grave tone
grave major second

800/729

160.90

200

τ or Λ + x or t − c

7

158.49


acute minor seventh

minor tone
lesser major second

10/9

182.40

200

t

8

181.13


minor seventh

major tone
Pythagorean major second
greater major second

9/8

203.91

200

T or t + c

9

203.77


Pythagorean minor seventh

diminished third

256/225

223.46

200

S + S

10

226.42


augmented sixth

semiaugmented second

125/108

253.08

300

t + x

11

249.06



augmented second

75/64

274.58

300

T + x

12

271.70


diminished seventh

Pythagorean minor third

32/27

294.13

300

T + Λ

13

294.34


Pythagorean major sixth

minor third

6/5

315.64

300

T + S

14

316.98


major sixth

acute minor third

243/200

333.18

300

T + L

15

339.62


grave major sixth

grave major third

100/81

364.81

400

T + τ

16

362.26


acute minor sixth

major third

5/4

386.31

400

T + t

17

384.91


minor sixth

Pythagorean major third

81/64

407.82

400

T + T

18

407.55


Pythagorean minor sixth

classic diminished fourth

32/25

427.37

400

T + S + S

19

430.19


classic augmented fifth

classic augmented third

125/96

456.99

500

T + t + x

20

452.83


classic diminished sixth

wide augmented third

675/512

478.49

500

T + t + X

21

475.47


narrow diminished sixth

perfect fourth

4/3

498.04

500

T + t + S

22

498.11


perfect fifth

acute fourth^{[1]}

27/20

519.55

500

T + t + L

23

520.75


grave fifth

classic augmented fourth

25/18

568.72

600

T + t + t

25

566.04


classic diminished fifth

augmented fourth

45/32

590.22

600

T + t + T

26

588.68


diminished fifth

diminished fifth

64/45

609.78

600

T + t + S + S

27

611.32


augmented fourth

classic diminished fifth

36/25

631.29

600

T + t + S + L

28

633.96


classic augmented fourth

grave fifth^{[1]}

40/27

680.45

700

T + t + S + t

30

679.25


acute fourth

perfect fifth

3/2

701.96

700

T + t + S + T

31

701.89


perfect fourth

narrow diminished sixth

1024/675

721.51

700

T + t + S + S + S

32

724.53


wide augmented third

classic diminished sixth

192/125

743.01

700

T + t + S + L + S

33

747.17


classic augmented third

classic augmented fifth

25/16

772.63

800

T + t + S + T + x

34

769.81


classic diminished fourth

Pythagorean minor sixth

128/81

792.18

800

T + t + S + T + Λ

35

792.45


Pythagorean major third

minor sixth

8/5

813.69

800

(T + t + S + T) + S

36

815.09


major third

acute minor sixth

81/50

835.19

800

(T + t + S + T) + L

37

837.74


grave major third

major sixth

5/3

884.36

900

(T + t + S + T) + t

39

883.02


minor third

Pythagorean major sixth

27/16

905.87

900

(T + t + S + T) + T

40

905.66


Pythagorean minor third

diminished seventh

128/75

925.42

900

(T + t + S + T) + S + S

41

928.30


augmented second

augmented sixth

225/128

976.54

1000

(T + t + S + T) + T + x

43

973.58


diminished third

Pythagorean minor seventh

16/9

996.09

1000

(T + t + S + T) + T + Λ

44

996.23


Pythagorean major second

minor seventh

9/5

1017.60

1000

(T + t + S + T) + T + S

45

1018.87


lesser major second

acute minor seventh

729/400

1039.10

1000

(T + t + S + T) + T + L

46

1041.51


grave major second

grave major seventh

50/27

1066.76

1100

(T + t + S + T) + T + τ

47

1064.15


acute minor second

major seventh

15/8

1088.27

1100

(T + t + S + T) + T + t

48

1086.79


minor second

narrow diminished octave

256/135

1107.82

1100

(T + t + S + T) + t + S + S

49

1109.43


wide augmented unison

Pythagorean major seventh

243/128

1109.78

1100

(T + t + S + T) + T + T

49

1109.43


Pythagorean minor second

diminished octave

48/25

1129.33

1100

(T + t + S + T) + T + S + S

50

1132.08


augmented unison

augmented seventh

125/64

1158.94

1200

(T + t + S + T) + T + t + x

51

1154.72


diminished second

semidiminished octave

160/81

1178.49

1200

(T + t + S + T) + T + t + x + c

52

1177.36


syntonic comma

octave

2/1

1200.00

1200

(T + t + S + T) + (T + t + S)

53

1200.00


unison

(The Pythagorean minor second is found by adding 5 perfect fourths.)
The table below shows how these steps map to the first 31 scientific harmonics, transposed into a single octave.
Harmonic

Musical Name

Ratio

Cents

12ET Cents

53ET Commas

53ET Cents

1

unison

1/1

0.00

0

0

0.00

2

octave

2/1

1200.00

1200

53

1200.00

3

perfect fifth

3/2

701.96

700

31

701.89

5

major third

5/4

386.31

400

17

384.91

7

augmented sixth§

7/4

968.83

1000

43

973.58

9

major tone

9/8

203.91

200

9

203.77

11


11/8

551.32

500 or 600

24

543.40

13

acute minor sixth§

13/8

840.53

800

37

837.74

15

major seventh

15/8

1088.27

1100

48

1086.79

17

limma§

17/16

104.96

100

5

113.21

19

Pythagorean minor third§

19/16

297.51

300

13

294.34

21

wide augmented third§

21/16

470.78

500

21

475.47

23

classic diminished fifth§

23/16

628.27

600

28

633.96

25

classic augmented fifth

25/16

772.63

800

34

769.81

27

Pythagorean major sixth

27/16

905.87

900

40

905.66

29

minor seventh§

29/16

1029.58

1000

45

1018.87

31

augmented seventh§

31/16

1145.04

1100

51

1154.72

§ These intervals also appear in the upper table, although with different ratios.
See also
References

^ ^{a} ^{b} http://www.huygensfokker.org/docs/intervals.html


Numbers in brackets are the number of semitones in the interval.
Fractional semitones are approximate.


Twelve
semitone
(Western)



Other
systems



Other
intervals

Groups






Measurement



Others




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