List of intervals in 5-limit just intonation

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The intervals of 5-limit 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 semi-diminished 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
semi-augmented 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 Pyth minor 3.png Pythagorean major sixth
minor third 6/5 315.64 300 T + S 14 316.98 Just minor 3.png 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 Just major 3.png minor sixth
Pythagorean major third 81/64 407.82 400 T + T 18 407.55 Pyth major 3.png 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
semi-diminished 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.

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