# List of intervals in 5-limit just intonation

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
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 |
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 | 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 | |

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 | 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 | |

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§
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.

## References

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