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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.


Due to their historical significance, 5-limit intervals go by various names, including classic(al)[1], pental[2], or quinquimal[citation needed].

Recently, composers Catherine Lamb and Marc Sabat have adopted quintal for the HC5[citation needed] since the corresponding Latin numerals are used to refer to higher prime limits such as septimal for the 7-limit and undecimal for the 11-limit. Pental is less consistent due to its Greek origins. However, that creates a conflict of usage as quintal has been the adjective associated with the fifth diatonic degree. (Quintal harmony does not mean 5-limit harmony, but harmony with chords stacked by fifths – cf. secundal harmony, tertian harmony, quartal harmony.)

A finite set of 5-limit intervals are labeled just, especially when the interval in question is the simplest in the category. For example, 5/4 is known as the just major third[3]. Indeed, just intonation traditionally meant specifically the 5-limit version thereof. Even so, justness is not to be generalized to all 5-limit intervals, nor can we assume all just intervals 5-limit in contemporary usage.

The term ptolemaic could also refer to the 5-limit[4]. On this wiki it is part of the Pythagorean-commatic interval naming system, and refers specifically to intervals that contain a single factor of harmonic 5. We distinguish multi-order 5-limit intervals by diptolemaic, triptolemaic, and so on.

Edo approximation

A list of edos with progressively better tunings (TE error) for 5-limit intervals: 2, 3, 4, 5, 7, 12, 19, 31, 34, 46, 53, 118, 171, 289, 323, 388, 441, 559, 612, 1171, 1783, 2513, 3684, 4296, …

Another list of edos that do relatively well in approximating harmonics 3 and 5: 2, 3, 7, 9, 10, 12, 19, 22, 31, 34, 53, 118, 289, 323, 441, 494, 559, 612, 1171, 1783, 2513, 3684, 4296, … (OEIS: A060525[5])

Another list of edos that do relatively well in approximating 5-odd-limit intervals: 1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, … (OEIS: A054540[6])

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. The syntonic comma is tempered out in 12edo, meantone, and many other related systems, meaning that those 5-limit 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, and so forth- however, perhaps a handful of people in the Xenharmonic community are at least starting to take the idea of such a distinction more seriously. 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 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 name
unison C 1/1 0.000 w1 wa unison 81/80 21.506 g1 gu comma 6561/6400 43.013 Lgg1 lagugu 1sn
aug. 1sn C# 2187/2048 113.685 Lw1 lawa 1sn 135/128 92.179 Ly1 layo 1sn 25/24 70.672 yy1 yoyo 1sn
minor 2nd Db 256/243 90.225 sw2 sawa 2nd 16/15 111.731 g2 gu 2nd 27/25 133.238 gg2 gugu 2nd
major 2nd D 9/8 203.910 w2 wa 2nd 10/9 182.404 y2 yo 2nd 800/729 160.897 syy2 sayoyo 2nd
aug. 2nd D# 19683/16384 317.595 Lw2 lawa 2nd 1215/1024 296.089 Ly2 layo 2nd 75/64 274.582 yy2 yoyo 2nd
minor 3rd Eb 32/27 294.135 w3 wa 3rd 6/5 315.641 g3 gu 3rd 243/200 337.148 gg3 gugu 3rd
major 3rd E 81/64 407.820 Lw3 lawa 3rd 5/4 386.314 y3 yo 3rd 100/81 364.807 yy3 yoyo 3rd
dim. 4th Fb 8192/6561 384.360 sw4 sawa 4th 512/405 405.866 sg4 sagu 4th 32/25 427.373 gg4 gugu 4th
4th F 4/3 498.045 w4 wa 4th 27/20 519.551 g4 gu 4th 2187/1600 541.058 Lgg4 lagugu 4th
aug. 4th F# 729/512 611.730 Lw4 lawa 4th 45/32 590.224 y4 yo 4th 25/18 568.717 yy4 yoyo 4th
dim. 5th Gb 1024/729 588.270 sw5 sawa 5th 64/45 609.776 g5 gu 5th 36/25 631.283 gg5 gugu 5th
5th G 3/2 701.955 w5 wa 5th 40/27 680.449 y5 yo 5th 3200/2187 658.942 syy5 sayoyo 5th
aug. fifth G# 6561/4096 815.640 Lw5 lawa 5th 405/256 794.134 Ly5 layo 5th 25/16 772.627 yy5 yoyo 5th
minor 6th Ab 128/81 792.180 sw6 sawa 6th 8/5 813.686 g6 gu 6th 81/50 835.193 gg6 gugu 6th
major 6th A 27/16 905.865 w6 wa 6th 5/3 884.359 y6 yo 6th 400/243 862.852 yy6 yoyo 6th
dim. 7th Bbb 32768/19683 882.405 sw7 sawa 7th 2048/1215 903.911 sg7 sagu 7th 128/75 925.418 gg7 gugu 7th
minor 7th Bb 16/9 996.090 w7 wa 7th 9/5 1017.596 g7 gu 7th 729/400 1039.103 Lgg7 lagugu 7th
major 7th B 243/128 1109.775 Lw7 lawa 7th 15/8 1088.269 y7 yo 7th 50/27 1066.762 yy7 yoyo 7th
dim. 8ve Cb 4096/2187 1086.315 sw8 sawa 8ve 256/135 1107.821 sg8 sagu 8ve 48/25 1129.328 gg8 gugu 8ve
octave C 2/1 1200.000 w8 wa 8ve 160/81 1178.494 y8 yo 8ve 12800/6561 1156.987 syy8 sayoyo 8ve

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.


Modern renderings

Jeff Barry and Ellie Greenwich
Paul Henning

20th century

Ben Johnston
performed by Kepler Quartet
  • String Quartet No. 3 (1966, 1973) – Bandcamp | YouTube – performed by Kepler Quartet

21st century

Axoid Music
William Copper (site 1 site 2[dead link])
Carlo Serafini
Chris Vaisvil

See also


  1. A note on the naming of musical intervals by Dave Keenan
  2. Gallery of Just Intervals « Music & Techniques by Chris Vaisvil
  3. The Helmholtz-Ellis JI Pitch Notation (HEJI) by Marc Sabat and Thomas Nicholson from Plainsound Music Edition
  4. Fundamental Principles of Just Intonation and Microtonal Composition by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."
  5. The description reads: a list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to four of the simple ratios of musical harmony: 5/4, 4/3, 3/2 and 8/5. It is yet to be found out what criterion is used.
  6. The description reads: a list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the six simple ratios of musical harmony: 6/5, 5/4, 4/3, 3/2, 8/5 and 5/3. It is yet to be found out what criterion is used.