Fuzzy logic helps to integrate music theory and practice

Abstract

Among the many practices of composers, instrumentalists and singers which clearly correspond with fuzzy logic our focus here is on tuning. Different criteria have been used to select the sounds that music uses. A set containing these sounds (musical notes) is called a tuning system. Several tuning systems coexist in a classical orchestra. The pitches of the notes are different and very precisely defined for each system; however the consequences of small deviations from these theoretical frequencies are not serious. Actually, the orchestra members are aware of the necessity of reaching a consensus and adjust their instruments to tune well. Because of this, many musicians feel that the mathematical arguments that justify tuning systems are impractical. Modeling the notes as fuzzy sets provides a flexible theoretical framework which helps to integrate tuning theory and practice. We illustrate our approach comparing some tuning systems and analyzing a excerpt by Béla Bartók.