5-limit

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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

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