Mathematical Foundations of Complex Tonality
For centuries, musicians and mathematicians have struggled with the challenges of temperament and tonality. The problem is that there is a paradox at the heart of Western music. The paradox is that no tuning system exists for keyboard instruments such that all the harmonious intervals are perfectly tuned. Here we propose a radical solution based on the notion of complex tonality. We begin with an overview of the just system of intonation, which originated in antiquity and was much studied in our era during the 16th, 17th and 18th centuries, reaching a high point in the work of Euler (1739, 1774).
