Tonal consonance parameters link microscopic and macroscopic properties of music exposing a hidden order in melody

Consonance is related to the perception of pleasantness arising from a combination of sounds and has been approached quantitatively using mathematical relations, physics, information theory, and psychoacoustics. Tonal consonance is present in timbre, musical tuning, harmony, and melody, and it is used for conveying sensations, perceptions, and emotions in music. It involves the physical properties of sound waves and is used to study melody and harmony through musical intervals and chords. From the perspective of complexity, the macroscopic properties of a system with many parts frequently rely on the statistical properties of its constituent elements. Here we show how the tonal consonance parameters for complex tones can be used to study complexity in music. We apply this formalism to melody, showing that melodic lines in musical pieces can be described in terms of the physical properties of melodic intervals and the existence of an entropy extremalization principle subject to psychoacoustic macroscopic constraints with musical meaning. This result connects the human perception of consonance with the complexity of human creativity in music through the physical properties of the musical stimulus.