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)
Color name		ratio	cents			Color name		ratio	cents	Color	ratio	cents
w1	wa unison	1/1	0.000	unison	C	g1	gu comma	81/80	21.506	Lgg1	6561/6400	43.013
Lw1	large wa 1sn	2187/2048	113.685	aug. unison	C#	Ly1	large yo 1sn	135/128	92.179	yy1	25/24	70.672
sw2	small wa 2nd	256/243	90.225	minor 2nd	Db	g2	gu 2nd	16/15	111.731	gg2	27/25	133.238
w2	wa 2nd	9/8	203.910	major 2nd	D	y2	yo 2nd	10/9	182.404	syy2	800/729	160.897
Lw2	large wa 2nd	19683/16384	317.595	aug. 2nd	D#	Ly2	large yo 2nd	1215/1024	296.089	yy2	75/64	274.582
w3	wa 3rd	32/27	294.135	minor 3rd	Eb	g3	gu 3rd	6/5	315.641	gg3	243/200	337.148
Lw3	large wa 3rd	81/64	407.820	major 3rd	E	y3	yo 3rd	5/4	386.314	yy3	100/81	364.807
sw4	small wa 4th	8192/6561	384.360	dim. fourth	Fb	sg4	small gu 4th	512/405	405.866	gg4	32/25	427.373
w4	wa 4th	4/3	498.045	fourth	F	g4	gu 4th	27/20	519.551	Lgg4	2187/1600	541.058
Lw4	large wa 4th	729/512	611.730	aug. fourth	F#	y4	yo 4th	45/32	590.224	yy4	25/18	568.717
sw5	small wa 5th	1024/729	588.270	dim. fifth	Gb	g5	gu 5th	64/45	609.776	gg5	36/25	631.283
w5	wa 5th	3/2	701.955	fifth	G	y5	yo 5th	40/27	680.449	syy5	3200/2187	658.942
Lw5	large wa 5th	6561/4096	815.640	aug. fifth	G#	Ly5	large yo 5th	405/256	794.134	yy5	25/16	772.627
sw6	small wa 6th	128/81	792.180	minor 6th	Ab	g6	gu 6th	8/5	813.686	gg6	81/50	835.193
w6	wa 6th	27/16	905.865	major 6th	A	y6	yo 6th	5/3	884.359	yy6	400/243	862.852
sw7	small wa 7th	32768/19683	882.405	dim. 7th	Bbb	sg7	small gu 7th	2048/1215	903.911	gg7	128/75	925.418
w7	wa 7th	16/9	996.090	minor 7th	Bb	g7	gu 7th	9/5	1017.596	Lgg7	729/400	1039.103
Lw7	large wa 7th	243/128	1109.775	major 7th	B	y7	yo 7th	15/8	1088.269	yy7	50/27	1066.762
sw8	small wa 8ve	4096/2187	1086.315	dim. octave	Cb	sg8	small gu 8ve	256/135	1107.821	gg8	48/25	1129.328
w8	wa 8ve	2/1	1200.000	octave	C	y8	yo 8ve	160/81	1178.494	syy8	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

Music

• Duodene2 by Chris Vaisvil
• Ariel's 12-tone JI by Chris Vaisvil
• The Ballad of Jed Clampett by Paul Henning
• Do Wah Diddy Diddy by Jeff Barry and Ellie Greenwich
• Symphony 4, first movement by William Copper
• Magnificat by William Copper
• Catch for Woodwin Quintet by William Copper