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Studio B • Thursday evening, 7:30–10:30

History of Theory

Thomas Christensen (University of Chicago), Chair

Caleb Mutch (Indiana University)

The Triad in Dispute: Genre and Audience in the Writings of Johannes Lippius

André Redwood (University of Massachusetts Amherst)

Combinatorics, Composition, Copia: Mersenne's Mathematics and the Erasmian Impulse

Derek Remeš (Eastman School of Music)

J. S. Bach's Chorales: Reconstructing Eighteenth-Century German Figured-Bass Pedagogy in Light of a New Source

Deborah Burton (Boston University)

From Rameau to Riemann: Antoniotto's L'arte armonica as a Missing Link from Fundamental Bass to the Tonnetz

Abstracts

The Triad in Dispute: Genre and Audience in the Writings of Johannes Lippius

Caleb Mutch (Indiana University)

Johannes Lippius is best known for his path-breaking articulation of the concept of the triad. The reception of this innovation chiefly relies upon his treatise Synopsis novae musicae, despite the fact that Lippius had already published many of this book’s notable points in earlier texts, which he called “disputations on music.” Yet the genre-specific peculiarities of the disputatio, which was a university-based event featuring a lecturer and a respondent who attempted to rebut the lecturer's ideas, have been overlooked in previous studies of Lippius's music-theoretical content. By situating his writings within the university system of the day and the formalized genres in which he wrote, this paper reveals an important instance of how music theory has been shaped by the demands of audience and genre.

Through comparison of how Lippius presented material first in the form of disputations and then reworked it for publication in his treatise, this paper reveals finds that some textual features are clearly attributable to generic demands, such as the disputations’ frequent appeals to authorities and the treatise’s adoption of an organizational framework which was heavily indebted to the pedagogical theories of Johann Sturm. It also casts light on unexpected aspects of those genres, like Lippius’s expectation that readers of his introductory musical treatise could comprehend both Latin and Greek. Further discrepancies between the texts offer tantalizing glimpses of how the oral disputation may have encouraged Lippius to clarify his ideas, particularly in his recasting of the analogy between the triad and the trinity.

Combinatorics, Composition, Copia: Mersenne's Mathematics and the Erasmian Impulse

André Redwood (University of Massachusetts Amherst)

Marin Mersenne, author of the Harmonie universelle (1636-37), has long been recognized for his obsession with mathematical permutation, combination, and calculation. His passion for combinatorics seems to account for one of his most infamous passages, in which he systematically writes out all 720 permutations of the Guidonian hexachord—twice. Historians of mathematics and science, and more recently of music theory, have tended to situate these exercises within narratives about the early history of combinatorics and explorations of its applications, to which Mersenne was an important contributor. Yet his decision, on certain occasions, exhaustively to notate each permutation leaves unanswered the question of why he chose to do so when simple arithmetic calculation might have sufficed.

This paper argues that Mersenne’s hexachord permutations, and other exercises like it, are part of a deliberate rhetorical strategy of abundant display—a strategy that draws on the rhetorical principle of copia, most famously theorized by Erasmus in his popular De Copia (1512). Although copia has received thoughtful attention in musicological studies centering on variation in seventeenth- and eighteenth-century composition, its role as an animating principle of musico-rhetorical theorization remains unexplored. In Mersenne’s case, the display of hexachord permutation invites comparison to, and is clarified by, analogous ‘accumulative’ tactics found in his other writings. Furthermore, copia may have broader historiographical relevance, for it offers a means by which rhetoric can be understood to have a meaningful ‘life’ long after its supposed music-theoretical ‘death’ at the turn of the nineteenth century.

J. S. Bach's Chorales: Reconstructing Eighteenth-Century German Figured-Bass Pedagogy in Light of a New Source

Derek Remeš (Eastman School of Music)

According to Emanuel Bach, his father used figured-bass chorales to teach harmony: first, students added alto and tenor voices to a given outer-voice framework; next, students composed their own basslines. Robin Leaver recently identified a new source which likely originates from J. S. Bach’s students in Dresden, c. 1740. The anonymous manuscript, dubbed the Sibley Choralbuch, contains 236 chorales with outer voices and figures. Therefore, this source likely documents the first stage of Bach’s teaching method. I will use a variety of eighteenth-century German sources to reconstruct Bach’s pedagogy in detail.

We might think to use Bach’s published four-voice chorales as models, but these are vocal, not keyboard works—Emanuel Bach reduced them from four to two staves for ease of reading at the keyboard. This format has led to confusion between Bach’s vocal and keyboard styles of harmonization. The Sibley Choralbuch implies that there was a simpler, keyboard-based Bach chorale tradition, which formed the starting point in Bach’s pedagogy. I argue that the goal of Bach’s method was to bridge from the simpler, keyboard style to the more complex, vocal style. As evidence, I cite among other things the multiple-bass chorale tradition of Bach’s students, Kittel and Kirnberger, where each bassline becomes increasingly ornate.

After chorales, Bach introduced fugue. This ordering suggests he believed contrapuntal thinking depends on a foundation of figured-bass harmony. This harmony-centered perspective informs our understanding of Bach’s music and encourages us to reconsider our own teaching methods.

From Rameau to Riemann: Antoniotto's L'arte armonica as a Missing Link from Fundamental Bass to the Tonnetz

Deborah Burton (Boston University)

Giorgio Antoniotto’s 1760 treatise L’arte armonica, with subscribers including Burney, Arne, Hawkins and Dr. Johnson, is presented here as a link between Rameau and Riemann. Using Rameau’s fundamental bass, Antoniotto generates scales from sequential perfect fifths. He posits two systems: Natural (diatonic) and General (chromatic). In the latter, he explores the complete chromatic, whole-tone scales, and major- and minor-third cycles. His grid of the General moves by perfect fifths horizontally and vertically, with one diagonal a whole-tone scale, and the other unisons. Another example shows whole-tone lines in the soprano, tenor and bass parts, with a chromatic line in the alto. He demonstrates a major-third cycle (C-E-G#-C) in yet another example, and in a fourth, a minor-third cycle passes through the major keys of C, A, F# and Eb, before returning to C major. Euler’s 1739 “genus diatonicum chromaticum” and 1774 “Speculum Musicum” are considered forerunners of the Neo-Riemannian Tonnetz, and Antoniotto has no grid precisely equivalent to Euler’s discoveries. However, he does create one in which the horizontal and vertical axes consist of Natural (diatonic) thirds, with one diagonal perfect fifths, and the other unisons. In addition to exploring Antoniotto’s concepts, I place them in the context of contemporary geometric representations of musical structures, by Euler, Hartung (1749), Mace (1676), Butler (1636) and Smith (1759).