**Euler Harmonies**

Leonhard Euler (1707–1783) was certainly living in the period of tonal music, but his curiosity led him rather far away from what other eighteenth-century musicians were doing. He was a great mathematician, not a musician, but he loved to play with the numbers of the harmonic series as much as with calculus, logarithms, trigonometry, and all the numbers he found in the world. He was sufficiently passionate about music that several hundred pages are devoted to this subject in Euler’s complete works (Opera Omnia, 1926, *Tentamen novae theoriae musicae: ex certissimus harmoniae principiis dilucide expositae*). Some of these pages concern acoustics and rhythm, but most of them have to do with calculating harmonies mathematically.

Euler’s basic approach was to construct chords containing all the harmonics that can be obtained by combining small prime numbers, including the number one, as factors. For example (...)