section_path: Contributions to mathematics and physics/Music
section_name: Music
paragraph_text: Music
  
  One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae (Attempt at a New Theory of Music), hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians. Even when dealing with music, Euler's approach is mainly mathematical, for instance, his introduction of binary logarithms as a way of numerically describing the subdivision of octaves into fractional parts. His writings on music are not particularly numerous (a few hundred pages, in his total production of about thirty thousand pages), but they reflect an early preoccupation and one that remained with him throughout his life.
  
  A first point of Euler's musical theory is the definition of "genres", i.e. of possible divisions of the octave using the prime numbers 3 and 5. Euler describes 18 such genres, with the general definition 2 A, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2 (where "m is an indefinite number, small or large, so long as the sounds are perceptible"), expresses that the relation holds independently of the number of octaves concerned. The first genre, with A = 1, is the octave itself (or its duplicates); the second genre, 2.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2.5, major third + minor sixth (C–E–C); the fourth is 2.3, two-fourths and a tone (C–F–B ♭ –C); the fifth is 2.3.5 (C–E–G–B–C); etc. Genres 12 (2.3.5), 13 (2.3.5) and 14 (2.3.5) are corrected versions of the diatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2.3.5) is the "diatonico-chromatic", "used generally in all compositions", and which turns out to be identical with the system described by Johann Mattheson. Euler later envisaged the possibility of describing genres including the prime number 7.
  
  Euler devised a specific graph, the Speculum musicum, to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (see above). The device drew renewed interest as the Tonnetz in Neo-Riemannian theory (see also Lattice (music)).
  
  Euler further used the principle of the "exponent" to propose a derivation of the gradus suavitatis (degree of suavity, of agreeableness) of intervals and chords from their prime factors – one must keep in mind that he considered just intonation, i.e. 1 and the prime numbers 3 and 5 only. Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form d s = ∑ i (k i p i − k i) + 1, {\displaystyle ds=\sum _{i}(k_{i}p_{i}-k_{i})+1,} where p are prime numbers and k their exponents.