Audio: second and seventh (0:08)
second and seventh plays Pythagorean seconds and sevenths on a piano. The second and seventh figure shows the score consists of the following:
The major second is an interval of a fifth above the fifth. To construct it using Pythagorean tuning, multiply the frequency ratio of a fifth, 3:2, by itself to get 9:4, then half it to get 9:8, the octave below, which is an interval of a major second. The inverse of 9:8 is 8:9, the octave above is 16:9, and that produces the interval of a major seventh.
Pythagorean tuning has now produced a third pair of intervals: a major second and a minor seventh with frequency ratios 9:8 and 16:9 respectively.