Musical Acoustics and Continued Fractions - Abstract

Richard Krantz and Jack Douthett
Bridges: Mathematical Connections in Art, Music, and Science (2001)
Pages 323–323


Equal-tempered systems (ETS) "close the octave" by forcing all frequency ratios to be the same for adjacent notes. In our familiar 12-tone equal-tempered system, the frequency ratio of adjacent notes is 2^1/12. Therefore, the octave is reproduced every twelve notes. ETS make transposition and modulation easy but compromise the consonant/harmonious ('just") intervals. The generation of equal-tempered musical scales that best approximate just intervals has a continued fraction basis.