The 5-limit consists of all justly tuned 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[edit]

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.

3-limit interval		interval category		\|5-limit interval (81/80)		\|Another 5-limit (6561/6400)
ratio	cents value			ratio	cents value	ratio	cents value
1/1	0.000	unison	C	81/80	21.506	6561/6400	43.013
2187/2048	113.685	aug. unison	C#	135/128	92.179	25/24	70.672
256/243	90.225	minor 2nd	Db	16/15	111.731	27/25	133.238
9/8	203.910	major 2nd	D	10/9	182.404	800/729	160.897
19683/16384	317.595	aug. 2nd	D#	1215/1024	296.089	75/64	274.582
32/27	294.135	minor 3rd	Eb	6/5	315.641	243/200	337.148
81/64	407.820	major 3rd	E	5/4	386.314	100/81	364.807
8192/6561	384.360	dim. fourth	Fb	512/405	405.866	32/25	427.373
4/3	498.045	fourth	F	27/20	519.551	2187/1600	541.058
729/512	611.730	aug. fourth	F#	45/32	590.224	25/18	568.717
1024/729	588.270	dim. fifth	Gb	64/45	609.776	36/25	631.283
3/2	701.955	fifth	G	40/27	680.449	3200/2187	658.942
6561/4096	815.640	aug. fifth	G#	405/256	794.134	25/16	772.627
128/81	792.180	minor 6th	Ab	8/5	813.686	81/50	835.193
27/16	905.865	major 6th	A	5/3	884.359	400/243	862.852
32768/19683	882.405	dim. 7th	Bbb	2048/1215	903.911	128/75	925.418
16/9	996.090	minor 7th	Bb	9/5	1017.596	729/400	1039.103
243/128	1109.775	major 7th	B	15/8	1088.269	50/27	1066.762
4096/2187	1086.315	dim. octave	Cb	256/135	1107.821	48/25	1129.328
2/1	1200.000	octave	C	160/81	1178.494	12800/6561	1156.987

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