Keyboard Temperament Estimation from Symbolic Data: A Case Study on Bach's Well-Tempered Clavier

Abstract

In this paper we address the task of keyboard temperament estimation from symbolic data. The aim is to find a keyboard temperament that minimizes the deviations from pure intervals, given a corpus of music. The problem of finding a suitable temperament has been studied for centuries. Many solutions have been proposed. By taking a data-driven approach, we contribute a method to this field. We define a loss function that measures the deviation from pure intervals, with a reward for exactly pure intervals. Three optimization methods are explored: Basin Hopping, Differential Evolution, and Dual Annealing. We validate our method with synthetic data, and by comparing with c. 1,500 existing temperaments, including equal temperament. Our method improves on any existing temperament. As a case study, we apply the method to Bach’s Well-Tempered Clavier. Our findings show interesting correspondence to existing proposals in musicological literature.