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 or quinquimal 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 (harmonics 3 and 5) are 2, 3, 7, 9, 10, 12, 19, 22, 31, 34, 53, 118, 289, 323, 441, 494, 559, 612, 1171, 1783, 2513, 3684, 4296, ...

Another approach is to find EDOs which have better approximations for 5-odd-limit intervals than all smaller EDOs. This results in 1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, ...

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

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

See also

• 5-odd-limit