5-limit
The 5-limit consists of all just intonation intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called regular numbers. Some examples of 5-limit intervals are 5/4, 6/5, 10/9 and 81/80. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.
The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a hexagonal lattice or as a square lattice; this can be done automatically by Scala. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a hexagonal tiling.
EDOs which do relatively well in approximating the 5-limit are 2edo, 3edo, 7edo, 9edo, 10edo, 12edo, 19edo, 22edo, 31edo, 34edo, 53edo, 118edo and 289edo.
Syntonic Comma Pairs
A significant interval in 5-limit JI is 81/80, the syntonic comma or Didymus' comma, which measures about 21.5¢. Although it rarely appears as an interval in a scale, it represents the difference between many 5-limit intervals and a nearby 3-limit (Pythagorean) interval. 81/80 is tempered out in 12edo, meantone, and many other related systems, meaning that those 5- and 3-limit distinctions are obliterated and one interval stands in for each. Living in a largely 12edo musical culture from birth, we are not accustomed to distinguishing two different major thirds, two different minor seconds, etc. Below is a list of some common intervals involving 3 and 5 which are distinguished by 81/80. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). Bold fractions are simplest for this interval category.
wa (3-limit) interval | interval category | yo or gu (5-limit) interval (81/80) | yoyo or gugu interval (6561/6400) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ratio | cents | Color name | ratio | cents | Color name | ratio | cents | Color | ||||
1/1 | 0.000 | w1 | wa unison | unison | C | 81/80 | 21.506 | g1 | gu comma | 6561/6400 | 43.013 | Lgg1 |
2187/2048 | 113.685 | Lw1 | large wa 1sn | aug. unison | C# | 135/128 | 92.179 | Ly1 | large yo 1sn | 25/24 | 70.672 | yy1 |
256/243 | 90.225 | sw2 | small wa 2nd | minor 2nd | Db | 16/15 | 111.731 | g2 | gu 2nd | 27/25 | 133.238 | gg2 |
9/8 | 203.910 | w2 | wa 2nd | major 2nd | D | 10/9 | 182.404 | y2 | yo 2nd | 800/729 | 160.897 | syy2 |
19683/16384 | 317.595 | Lw2 | large wa 2nd | aug. 2nd | D# | 1215/1024 | 296.089 | Ly2 | large yo 2nd | 75/64 | 274.582 | yy2 |
32/27 | 294.135 | w3 | wa 3rd | minor 3rd | Eb | 6/5 | 315.641 | g3 | gu 3rd | 243/200 | 337.148 | gg3 |
81/64 | 407.820 | Lw3 | large wa 3rd | major 3rd | E | 5/4 | 386.314 | y3 | yo 3rd | 100/81 | 364.807 | yy3 |
8192/6561 | 384.360 | sw4 | small wa 4th | dim. fourth | Fb | 512/405 | 405.866 | sg4 | small gu 4th | 32/25 | 427.373 | gg4 |
4/3 | 498.045 | w4 | wa 4th | fourth | F | 27/20 | 519.551 | g4 | gu 4th | 2187/1600 | 541.058 | Lgg4 |
729/512 | 611.730 | Lw4 | large wa 4th | aug. fourth | F# | 45/32 | 590.224 | y4 | yo 4th | 25/18 | 568.717 | yy4 |
1024/729 | 588.270 | sw5 | small wa 5th | dim. fifth | Gb | 64/45 | 609.776 | g5 | gu 5th | 36/25 | 631.283 | gg5 |
3/2 | 701.955 | w5 | wa 5th | fifth | G | 40/27 | 680.449 | y5 | yo 5th | 3200/2187 | 658.942 | syy5 |
6561/4096 | 815.640 | Lw5 | large wa 5th | aug. fifth | G# | 405/256 | 794.134 | Ly5 | large yo 5th | 25/16 | 772.627 | yy5 |
128/81 | 792.180 | sw6 | small wa 6th | minor 6th | Ab | 8/5 | 813.686 | g6 | gu 6th | 81/50 | 835.193 | gg6 |
27/16 | 905.865 | w6 | wa 6th | major 6th | A | 5/3 | 884.359 | y6 | yo 6th | 400/243 | 862.852 | yy6 |
32768/19683 | 882.405 | sw7 | small wa 7th | dim. 7th | Bbb | 2048/1215 | 903.911 | sg7 | small gu 7th | 128/75 | 925.418 | gg7 |
16/9 | 996.090 | w7 | wa 7th | minor 7th | Bb | 9/5 | 1017.596 | g7 | gu 7th | 729/400 | 1039.103 | Lgg7 |
243/128 | 1109.775 | Lw7 | large wa 7th | major 7th | B | 15/8 | 1088.269 | y7 | yo 7th | 50/27 | 1066.762 | yy7 |
4096/2187 | 1086.315 | sw8 | small wa 8ve | dim. octave | Cb | 256/135 | 1107.821 | sg8 | small gu 8ve | 48/25 | 1129.328 | gg8 |
2/1 | 1200.000 | w8 | wa 8ve | octave | C | 160/81 | 1178.494 | y8 | yo 8ve | 12800/6561 | 1156.987 | syy8 |
It is important to note that 5-limit music does not mean favoring intervals of 5 over intervals of 3. It means allowing for both 3's and 5's in generating harmonic material, and so it is an interplay between both. The 5-limit includes the 3-limit -- a work in 5-limit JI will utilize intervals from both sides of the chart above.
See Harmonic Limit