Beats, ratios, and commas: Teaching tuning and temperament to music students

Abstract

Undergraduate music students have many preconceived notions about tuning and temperament. For most of them, a piano tuned in equal temperament is perfectly “in tune” and A4 = 440 Hz is the only pitch standard that has ever existed. We find these naïve conceptions about tuning and temperament to be common among our introductory Physics of Music students--music majors, students pursuing majors in the sciences, and other non-science majors. In this paper we describe how we address some of these issues. We teach the mathematical underpinnings of equal, Pythagorean, just, and meantone temperaments, as well as their musical advantages and disadvantages, through the use of live demonstrations and recorded examples. We also describe our future plans to use recorded examples and available local resources—including reproductions of Baroque woodwinds tuned to A4 = 415 Hz and a 1799 organ at A4 = 409 Hz—to give students a broader understanding of historical and modern pitch standards.