List of intervals in 5-limit just intonation

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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, as well as other marked* intervals with a maximum numerator and denominator of 1,000 inclusive [unfinished], are shown in the table below.

Names Ratio Cents 12ET Cents Definition 53ET commas

(syntonic, diaschisma,

magic, etc.)

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 or X − x or

S − Λ or L − S or t − τ

1 22.64 semi-diminished octave
diesis

minor diesis
diminished second

128/125 41.06 0 D or S − x or L − X 2 45.28 augmented seventh
maximal diesis*

porcupine comma

250/243 49.17 100 x − c or t − L 2 45.28
major diesis*

diminished comma

648/625 62.57 0 D + c or L − x 3 67.92
lesser chromatic semitone
minor semitone
augmented unison
25/24 70.67 100 x or t − S or T − L or

X − c or S − D or τ − Λ

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
classic neutral second (?)* 625/576 141.34 200 x + x or t − D 6 135.85 greater neutral 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
acute major second* 729/640 225.42 200 T + c or L + X 10 226.42
classic diminished third* 144/125 244.97 200 T + D or L + S 11 249.06
semi-augmented second 125/108 253.08 300 t + x 11 249.06 semi-augmented sixth
grave minor third (?)* 729/625 266.48 200 L + L 12 271.70 acute major sixth (?)
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
classic neutral third* 625/512 345.25 400 T + x + x or T + t − D or

t + X + x or τ + X + X

15 339.62
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 above table of intervals is organized by powers of three and five such that it includes all intervals that are either in the diatonic scale or have a maximum numerator of 1,000. The black squares indicate that the interval is not in the diatonic scale and the numerator is larger than 1,000.

(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§

harmonic seventh

7/4 968.83 1000 43 973.58
9 major tone 9/8 203.91 200 9 203.77
11 undecimal semi-augmented fourth[2] 11/8 551.32 500 or 600 24 543.40
13 acute minor sixth§

tridecimal neutral sixth

13/8 840.53 800 37 837.74
15 major seventh 15/8 1088.27 1100 48 1086.79
17 limma§

large septendecimal semitone

17/16 104.96 100 5 113.21
19 Pythagorean minor third§

Otonal minor third

19/16 297.51 300 13 294.34
21 wide augmented third§

sub-fourth / narrow fourth

21/16 470.78 500 21 475.47
23 classic diminished fifth§

23-limit superaugmented fourth

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-limit large minor seventh

29/16 1029.58 1000 45 1018.87
31 augmented seventh§

31-limit ultramajor seventh

31/16 1145.04 1100 51 1154.72


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

See also[edit]

References[edit]

  1. ^ a b http://www.huygens-fokker.org/docs/intervals.html
  2. ^ "Gallery of just intervals - Xenharmonic Wiki". en.xen.wiki. Retrieved 2019-09-21.