When we do the circle of fifths calculations using rational numbers instead of floating point numbers, we find a number of simple-looking fractions like 3/2, 4/3, 9/8, 16/9 in our results. These fractions lead to a geometrical interpretation of the musical intervals. These fractions correspond with some early writings on music by the mathematician Pythagoras.

We'll provide one set of commonly-used list of fractions for Pythagorean tuning. These can be compared with other results to make the whole question of scale tuning even more complex.

Pythagorean Pitches. Develop a simple representation for the above ratios. A list of tuples works well, for example. Use the ratio to compute the frequencies for the various pitches, using 27.5 Hz for the base frequency of the low "A". Compare these values with equal temperament, overtones and circle of fifths tuning.

Check Your Results. The value for "G" is 27.5 * 16 / 9 = 48.88Hz.