The most obvious relationships between mathematics
and music appear in acoustics, the science of musical
sound, and particularly in the analysis of the intervals
between pairs of pitches. With the development
of polyphonic music in the Renaissance period, the
Pythagorean conception of consonance based on the
simple ratios of the integers from 1 to 4 eventually
came into conflict with musical practice.

The acoustically pure perfect consonances of Pythagorean tuning were well-suited for medieval parallel organum, but in the fifteenth and sixteenth centuries use was increasingly made of the so-called imperfect consonances, that is, major and minor thirds and their octave inversions, minor and major sixths. In Pythagorean tuning, intervals are derived by successions of perfect fifths, so the corresponding frequency ratios are powers of 3/2 . In conventional Western music, twelve perfect fifths in succession, C–G–D–A–E–B–F*–C*–G*–D*–A*–E*–B*, are supposed to equal seven octaves (C = B*), but this does not work in Pythagorean tuning, since (3/2)^12 does not equal 2^7.

Indeed, a succession of Pythagorean perfect fifths will never result in a whole number of octaves. As it happens, twelve Pythagorean perfect fifths give an interval slightly larger than seven octaves. The difference is a small interval known as the Pythagorean comma, which corresponds to a ratio of (3/2)^12/2^7, which is about 1.013643.

The acoustically pure perfect consonances of Pythagorean tuning were well-suited for medieval parallel organum, but in the fifteenth and sixteenth centuries use was increasingly made of the so-called imperfect consonances, that is, major and minor thirds and their octave inversions, minor and major sixths. In Pythagorean tuning, intervals are derived by successions of perfect fifths, so the corresponding frequency ratios are powers of 3/2 . In conventional Western music, twelve perfect fifths in succession, C–G–D–A–E–B–F*–C*–G*–D*–A*–E*–B*, are supposed to equal seven octaves (C = B*), but this does not work in Pythagorean tuning, since (3/2)^12 does not equal 2^7.

Indeed, a succession of Pythagorean perfect fifths will never result in a whole number of octaves. As it happens, twelve Pythagorean perfect fifths give an interval slightly larger than seven octaves. The difference is a small interval known as the Pythagorean comma, which corresponds to a ratio of (3/2)^12/2^7, which is about 1.013643.