Arithmetic, geometric and harmonic means in music theory

Some milestones in the development of music theory can be represented as a dialectical contrast between means. In the Pythagorean harmony, the arithmetic and harmonic means are generators of musical scales, while a specific theorem by Archytas precludes the geometric mean from the same opportunity. The developments in the Renaissance music theory are described by its major protagonist, Zarlino, in terms of means. He identifies the overcoming of the Pythagorean harmony with the intervention of the contrharmonic mean, the superiority of the major mode over the minor mode with the superiority of the harmonic mean over the arithmetic one, the contrast between natural and tempered tunings with the opposition between the arithmetic mean and the geometric one. Also Rameau, the major theorist of the Baroque era, describes his “fundamental bass” using the three means (for the first time all together): the arithmetic and harmonic means applied to the vertical dimension of the harmony, the geometric one to the horizontal dimension of the melody.