3-limit

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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, ...

3-limit intervals up to odd-limit 19683:

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

See also