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 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, ...

See Harmonic Limit.

3-limit intervals up to odd-limit 19683:

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