A fuzzy framework to explain musical tuning in practice

A theoretical tuning system is a set of pitches that can be used to play music. It is a fact that the human ear perceives notes with very close frequencies as if they were the same note. Therefore, in our approach a musical note and its pitch sensation are modeled as L–R fuzzy numbers with a modal interval and a bounded support. We pay particular attention to the 12-tone equal temperament (12-TET) for being the most widely used tuning system and we define the fuzzy 12-TET composed of 12 fuzzy notes. A similarity relation between a fuzzy note and a theoretical note can be defined, and subsequently a similarity class associated to each one of the fuzzy notes in the fuzzy 12-TET arises. Finally, we study the similarity between the theoretical notes in a given score and those notes played by a musician making use of an OWA operator. Some numerical examples illustrate our approach.

► Musical notes and their pitch sensations modeled as L–R fuzzy numbers. ► Fuzzy tuning systems are analyzed, especially interested in 12-TET. ► A similarity relation between offered and theoretical notes is defined. ► Similarities between notes can be aggregated using an OWA operator. ► Three saxophonists helped us to perform our numerical experiments.