Audio: third and sixth (0:08)
third and sixth plays the interval of a third and its inversion, the sixth, on a piano. The third and sixth figure shows the score consists of the following:
The fourth pair of intervals in Pythagorean tuning are constructed in the same manner as all the rest of the intervals by using the frequency ratio of a fifth, 3:2.
The Pythagorean minor third is constructed by multiplying 3:2 three times to obtain the ratio 27:8 and halving it to obtain the octave below, 27:16, which is a minor third.
The major sixth is obtained by inverting the major third, 27:16, to obtain 16:27, then doubling it to get the octave above, 32:27, which is a major sixth.
Pythagorean tuning has now provided four pairs of intervals: unison and octave, fourth and fifth, second and seventh, and third and sixth. Together, they produce a total of seven notes plus the octave.
Put the seven notes in alphabetical order and something marvellous happens, they form a scale. The scale of D Dorian is DEFGABC. The scale of C major is CDEFGAB. These seven notes are the basis of modern scales. It is no coincidence either, Pythagorean tuning is the reason that there are seven notes in a scale.