In this talk, I analyze Rameau’s recourse to the sense of touch, specifically the theorist’s notion of a “mechanics of the fingers” (“mécanique des doigts”). Specifically, I track how Rameau filters the theoretical abstraction of the fundamental bass through the physical action of the fingers in two of the theorist’s publications, the Dissertation sur les différentes méthodes d'accompagnement (1732) and the Code de musique pratique (1760). Briefly stated, Rameau defines his mécanique des doigts as a system for realizing thoroughbass expediently, based on the distribution of the fingers within chords on the keyboard, and without the need for memorizing figures or reading music.

In the Dissertation, Rameau emphasizes the mécanique des doigts as a physical response to the acoustical properties of sound despite its ostensible relationship to touch. As Hayes (1974) asserts, Rameau’s ideal keyboardist continually modulates between sensory impressions propagated to the ear and the fingers. Feeling and bon goût notwithstanding, the fingers are automated by a natural progression of the fundamental bass. In the Code, as Christensen (1993) observes, Rameau adopts a more empiricist outlook. This time, Rameau’s keyboardist senses the relationship between the fundamental bass and the motion of the fingers based on repeated experience. In both treatises, Rameau’s mécanique des doigts is secondary to hearing. Why, then, invoke the physical attributes of keyboard practice at all?

I argue that Rameau deploys the mécanique des doigts as a flexible means of subtending the universalizing properties of the fundamental bass and assuring its legibility across the shifting philosophical terrain of the French Enlightenment. In the Dissertation, Rameau’s mécanique des doigts is linked to sound’s “occult,” acoustical properties, emblematized by the fundamental bass. More crucially, it joins touching and hearing in a mechanistic, Cartesian paradigm. The fundamental bass is thus counterpoised between eighteenth-century speculative music theory and the physical demands of a vibrant tradition of accompaniment. By the publication of the Code in 1760, with Descartes’s theories readily adapted to new epistemologies of sensation and experience, Rameau recapitulates his mechanical method of touch as uniquely responsive to matters of sense and taste.

This paper explains how the five-course guitar functioned as an “instrument of music theory” (Rehding 2016) in seventeenth-century Iberia. During this period, Iberian music theory was characterized by a great “theoretical rift” separating the writings produced by church musicians from those produced by secular musicians who were predominantly guitar players (Gallardo 2012). While the former group’s treatises emphasize “conservative” topics such as plainchant and modal theory, the latter group produced “progressive” works including some of the earliest theoretical conceptions of the triad as an independent entity, and some rules for the accompaniment of melodies anticipating some of the principles of bajo continuo.

Through the study of pedagogical texts by Amat (1596), Velasco (1640), Sanz (1674), and De Huete (1702) this paper explores several of their most original explicit and implicit theoretical ideas. These include the tenets of alfabeto notation, the theoretical emancipation of the triad through the punto concept, a rule-of-thumb system for the harmonic realization of a bassline, and highly refined understandings of tonal space and chordal inversion condensed in Velasco’s musical circles and Sanz’s laberinto.

This heterogenous group of composers/performers/theorists—possibly influenced by Cartesian rationalism—demonstrate a profound interest in sophisticated topographical representations of tonal space using figures of circles. Some of these figures are practical solutions to some of the necessities of their emerging theoretical systems such as early attempts at a generalized theory of transposition. Others are original examples of speculative theoretical constructs such as intervallic cycles, and prototypical scale harmonizations. Ultimately, these new geometries of tonal space anticipate other more celebrated circular representations (such as those by Johann David Heinichen and Johann Mattheson) by several decades and arguably demonstrate a rarely discussed form of transnational intellectual dialogue between the Iberian Peninsula and other European nations in matters of harmonic syntax at the turn of the eighteenth century.

In this paper I investigate sonorities (Fuller, 1986; Hartt, 2010)—precursors to chords—in Renaissance polyphony through the dual lenses of theory and practice. With the aid of a small corpus and close analytical readings, I outline a system for quantitatively comparing sonorities, identify which sonorities composers prefer over others, and explore divergences between sonority theory and compositional practice. Music theorists active in the sixteenth century became preoccupied with taxonomizing multi-voice sonorities, evincing a shift toward theorizing textures, or multiple voice parts simultaneously (Bergquist, 1964). In practice, such sonorities are products of the modal-contrapuntal frameworks that generate them, but their intervallic structure bears intrinsic textural information that can be quantified and compared in several ways. Pitch locus is calculated by the mean of the distances between the lowest-sounding voice and each other voice. Tightness is calculated by the mean of the intervallic distance between each pair of voices. And Evenness is calculated by the standard deviation of intervallic distances between each pair of adjacent voices. Together, these measurements generate holistic quantitative profiles of a sonority based on its spacing and the pitch space it occupies.

I use this system to analyze a corpus of polyphonic works for four, five, and six voices as a preliminary exploration of how perfect sonorities (Fuller, 1986)—those containing only thirds, fifths, and octaves (and octave transpositions) above the lowest sounding note—were used to conclude sections of works, hence, final sonorities. My preliminary corpus work suggests that only a handful of possible final sonorities are used with any frequency, regardless of the lowest-sounding pitch used and despite composers having many available voicing options. This observation underpins the broader, oft-observed divergence between music theory and practice in the sixteenth century. I use the analytical framework presented here to identify relationships between voicing structure, lowest-sounding pitch, form, and text setting. This forms the basis for my broader research on aesthetic and perceptual qualities of final sonorities, adding to general scholarship on chordal perception and affect. As such, researching sonorities in Renaissance polyphony via their voicing arrangement speaks to their structural importance and affective power in this repertoire.

Zhu Zaiyu 朱載堉 formulated twelve-tone equal temperament mathematically in 1584, independently of contemporaneous European scholars such as Simon Stevin. As Rehding (2022) demonstrates, the coincidence of the Chinese and European inventions of equal temperament affords a multicultural perspective for studying the global history of music theory. However, both Rehding (2020, pt. III; 2022, n. 10) and Martin (2022, 178) acknowledge that the lack of translations of primary sources poses a major obstacle to studying Zhu’s theories. To overcome this linguistic obstacle, I suggest that contemporary scholarship from the Sinosphere warrants further examination.

This paper reviews the secondary literature from Sinophone scholarship and provides my translations of selected passages that offer fresh insights into Zhu’s work. First, I discuss how Chinese scholars have evaluated the mathematical-theoretical basis of Zhu’s calculations (Li and Zhu 1985; Dai 1986; Xu 1994). These evaluations concern how Zhu arrived at the value of the twelfth root of 2 by combining the principle of geometric sequence and the Chinese gougu 勾股 theorem, equivalent to the Pythagorean theorem in the West. These sources elucidate Zhu’s detailed calculations, which refine the current understanding of Zhu’s mathematical thinking in Anglophone academia (Needham 1962, 212–28; Rehding 2022).

Second, I discuss practical musical issues surrounding Zhu’s invention of twelve-tone equal temperament. I consider the relevance of his invention to practical musical contexts of his time. As Rehding (2020, pt. IV) asks, “what problem exactly did Zhu Zaiyu’s temperament solve in the context of Ming-dynasty yayue [雅樂, i.e., court music]?” I answer this question by referring to discussions in the secondary literature (Huang 1986; Guo 1993; Miao 1996), which identify the problem of xuangong 旋宮—roughly translatable to “modulation”—as the primary motivation behind Zhu’s invention. Moreover, I argue that Zhu used the sheng 笙 as a “music-theoretical instrument” for his formulation of twelve-tone equal temperament (Rehding 2016; 2022). Mechanical features of the sheng allow Zhu to use it as an instrument to test the precision of his new temperament and replace traditional tuning instruments such as the qin 琴 zither and the lü 律 pitch pipes (Sun 1987; Guo 1994).