Among other things, the Pythagoreans quantified “harmonious” musical intervals in terms of proportions (ratios) of the rst few whole numbers: a unison is 1:1, octave is 2:1, perfect h is 3:2, perfect fourth is 4:3, and so on (Miranda 2001, p. 6). e Pythagorean scale was refined over centuries to produce well-tempered and equal-tempered scales (Livio 2002, pp. 29, 186).