Uncertainty and tuning in music

Rok, strany: 1997, 113 - 129

The tuning in music is an excellent example of that the human perceptive mechanism using various many valued systems of information coding. The ambiguous intervals in the Just Intonation Set are the second and the minor seventh and the tritone (the relative frequencies $10/9, 9/8, 8/7$ and $7/4, 16/9, 18/10$ and $45/32, 64/45$, respectively). Pythagorean Tuning is also $17$-valued. In the present paper we find a $17$-valued diatonic tone system which is a consequence of the unique solution of a Diophantine equation describing the basic acoustic relations among octave, perfect fifth and major (minor) third. This system has properties of the Just Intonation Set (it involves octave, perfect fifth, perfect fourth, major third, minor third, major whole tone, minor whole tone, diatonic semitone and chromatic semitone) and also of Pythagorean Tuning. We bring applications of our theory to superparticular ratios and partial monounary algebras.

ISO 690:

Haluška, J. 1997. Uncertainty and tuning in music. In Tatra Mountains Mathematical Publications, vol. 12, no.3, pp. 113-129. 1210-3195.

APA:

Haluška, J. (1997). Uncertainty and tuning in music. Tatra Mountains Mathematical Publications, 12(3), 113-129. 1210-3195.