3-limit

A 3-limit interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as 2^a 3^b, where a and b can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are 3/2, 4/3, 9/8. Confining intervals to the 3-limit is known as Pythagorean tuning, and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music.

EDOs which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the continued fraction for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, ...

Another approach is to find EDOs which have more accurate 3 than all smaller EDOs. This results in 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, ...

3-limit intervals up to odd-limit 19683:

Ratio	Monzo	Size (¢)	Color name		Interval category
1/1	[0⟩	0.000	w1	wa unison	unison	C
2187/2048	[-11 7⟩	113.685	Lw1	lawa 1sn	aug. unison	C#
256/243	[8 -5⟩	90.225	sw2	sawa 2nd	minor 2nd	Db
9/8	[-3 2⟩	203.910	w2	wa 2nd	major 2nd	D
19683/16384	[-14 9⟩	317.595	Lw2	lawa 2nd	aug. 2nd	D#
32/27	[5 -3⟩	294.135	w3	wa 3rd	minor 3rd	Eb
81/64	[-6 4⟩	407.820	Lw3	lawa 3rd	major 3rd	E
8192/6561	[13 -8⟩	384.360	sw4	sawa 4th	dim. fourth	Fb
4/3	[2 1⟩	498.045	w4	wa 4th	fourth	F
729/512	[-9 6⟩	611.730	Lw4	lawa 4th	aug. fourth	F#
1024/729	[10 -6⟩	588.270	sw5	sawa 5th	dim. fifth	Gb
3/2	[-1 1⟩	701.955	w5	wa 5th	fifth	G
6561/4096	[-12 8⟩	815.640	Lw5	lawa 5th	aug. fifth	G#
128/81	[7 -4⟩	792.180	sw6	sawa 6th	minor 6th	Ab
27/16	[-4 3⟩	905.865	w6	wa 6th	major 6th	A
32768/19683	[15 -9⟩	882.405	sw7	sawa 7th	dim. 7th	Bbb
16/9	[4 -2⟩	996.090	w7	wa 7th	minor 7th	Bb
243/128	[-7 5⟩	1109.775	Lw7	lawa 7th	major 7th	B
4096/2187	[12 -7⟩	1086.315	sw8	sawa 8ve	dim. octave	Cb
2/1	[1⟩	1200.000	w8	wa 8ve	octave	C

See also

• Harmonic limit
• 3-odd-limit
• Gallery of just intervals
• Pythagorean tuning - Wikipedia