5-limit

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Five-limit tuning

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, … (OEIS: A060525)

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, … (OEIS: A054540)

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.

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.

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
• Harmonic limit