Tempering the clavier: a corpus-based examination of Bach’s cognition of intonation in the Well-Tempered Clavier

Abstract

While it is well known today that J.S. Bach’s The Well-Tempered Clavier (WTC) was not composed with equal temperament in mind, it is unknown exactly which, if any, well temperament Bach had intended for the work. While equal temperament produces 12 uniform intervals of each type, other systems of well temperament slightly stretch and compress the size of various fifths, resulting in alterations to other intervals. A number of competing hypotheses have been published as to which temperament Bach intended, or whether he intended a specific temperament at all. While we do not have a definitive historical answer as to Bach’s preference of temperament in this work, this paper applies corpus methodologies to compare these systems. This paper develops and utilises Detuning-Emphasis Correlation (DEC): the relationship between scale-degree pairs' deviation from just intonation and their frequency of use. Frequency of use is compared to a ‘baseline’ frequency, found by averaging a scale degree pair’s prevalence across the corpus. This paper analyses the 48 fugues of Bach’s WTC by computing DECs for selected well temperaments (Werckmeister III, Bach-Lehman, Niedhardt, Vallotti, and Young), aiming to find a significant negative DEC for some temperament. While no temperament was found to fit the WTC in this way, strong correlations between these well temperaments indicated that the work would highlight similar affects for the various keys regardless of which (non-equal) well temperament is used to realise this work.