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Article ID: CBB139505675

Arithmetic, Geometric and Harmonic Means in Music Theory (2014)

Bollettino di Storia delle Scienze Matematiche
Volume: 34
Issue: 2
Pages: 201-244
Publication date: 2014
Language: English

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Mathematics and art Geometry Music Baroque Pythagoreanism Arithmetic Harmony (music theory) Music theory Renaissance Europe

Some milestones in the development of music theory can be represented as a dialectical contrast between means. In Pythagorean harmony, arithmetic and harmonic means are generators of musical scales, while a specific theorem by Archytas precludes the geometric mean from this opportunity. The developments in Renaissance music theory are described by its protagonist, Zarlino, in terms of means. He identifies the overcoming of Pythagorean harmony with the intervention of the contrharmonic mean, the superiority of the major 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. Rameau, the major musical theorist of the Baroque era, also 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 harmony, the geometric one to the horizontal dimension of melody.

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