The Notes Tell Us How to Tune

Abstract

Abstract:Notes have different names (such as D♭ vs C#) because they belong to different diatonic scales. From the sixteenth century forward, there were equally spaced systems of intonation holding a general principle where sharps are pitched lower than nearby flats. The naming distinctions matter because these notes were at least a comma apart from one another, differently pitched to fit into the scales. To play the correct notes within ensembles, and to play keyboard solos, keyboards with only twelve key levers presented a practical problem. It was necessary to compromise (tastefully adjust) some of their sharp or flat pitches toward one another to play acceptable approximations.Bach's keyboard parts and solos show that he required more than twelve differently named notes per composition throughout his career. For example, he frequently used both G# and A♭ within the same piece. He did this with such impunity and flexibility that he obviously had practical ways to make these notes sound acceptable in their different scale functions. The Well-Tempered Clavier Book I (1722) was not exceptional in its requirements of such enharmonic overlapping. A temperament to play Bach's keyboard music therefore must be able to play the sharps D#, A#, E#, and B#, along with the flats A♭, D♭, G♭, C♭, and F♭. These notes are primary evidence for compromised keyboard tuning. Temperaments are demonstrably wrong where they do not provide notes such as D♭ and E# that are far from the C-major scale.This article presents a close look at this evidence of the required notes in Bach's music, with more than 400 pieces beyond the Well-Tempered Clavier Book I. Measured within eighteenth-century expectations of expert practice, enharmonically flexible temperaments can have no notes out of tune by as far as a comma, and no beatless major thirds. This rules out quarter-comma temperaments. With these technical constraints on keyboard temperaments to play this repertoire, this article proposes that Bach required a sixth-comma tempérament ordinaire to play his sharps and flats. That is the same practical tuning procedure and set of principles presented by this author in 2005. The background and enharmonic measurements from 2005 are explained more thoroughly in this article.Documents from Pier Francesco Tosi, Johann Joachim Quantz, Georg Andreas Sorge, Friedrich Marpurg, et al. provide context in recognizing the scale requirements, regular systems of intonation, avoidance of Pythagorean thirds, and matters of taste. An equal-spacing illustration shows how to transpose music and observe the regularity or irregularities of temperaments. The example of Johann Philipp Kirnberger's published temperament shows why Kirnberger was unreliable as a witness for Bach, because the behavior of melodic motion is crucial in assessing a temperament's suitability.