Conference Agenda

Beginning in the early ninth century, the copying and glossing of Boethius’s Musica marks what is arguably the greatest contribution of the Carolingian Renaissance to the history of music theory. A virtual compendium of ancient Greek harmonics, firmly rooted in Pythagorean mathematics, Boethius’s work was not only read on its own terms, but also inspired singers in the Latin West to apply its mathematical rigor to the repertoire of plainchant, writing new theoretical tracts in the late ninth century and beyond. Foremost among these was a multi-layered treatise that carried the designation Alia musica, “another treatise on music.”

The present session devotes itself to studies of the manuscript tradition of the Musica of Boethius and to the first two layers of the Alia musica, those composed by authors given the designations “First Quidam” and “Expositor” respectively by the 1964 editor of the Alia musica, Jacques Chailley. It begins with a paper on Boethius’s Musica, showing that in certain locales the way in which the treatise was read, understood, and copied changed significantly during the course of the ninth and tenth centuries. Indeed, the new orientation in the diagrams of several tenth-century manuscripts points the way toward a new theoretical foundation for Western music itself.

The second paper likewise suggests a change, not in the medieval, but in the modern understanding of the Alia musica. Chailley said of its principal author, the Expositor, that he was “totally ignorant of Greek music in general and the modes of transposition in particular.” This paper takes the opposite view--that the Expositor did understand Boethius, and that he consciously employed Boethian theory in order to situate the toni of his own tradition within a rational and venerable music-theoretical context.

The final paper presents a new explication of the mathematical proportions lying at the heart of the Alia musica’s source treatise, that written by Chailley’s “First Quidam.” This paper offers--for the first time ever--a logical explanation of the numerical proportions of the First Quidam. Its title says it all: “The Number System of the First Quidam of the Alia musica: A Mystery Solved.”

The first three books of Boethius’s De institutione musica develop the theory of mathematical ratios possessing musical verity to a degree witnessed by no other treatise in Latin before the ninth century. Indeed, by the time the reader arrives at Book III and plows through the proofs of the relative sizes of the major and minor semitone and the negative arithmetical arguments against Aristoxenus, the patience of even the most avid reader becomes severely tested. Yet in chapters 9 and 10, in the midst of Book III and its demanding arithmetic, Boethius introduces a linear approach to musical space, leaving the world of complicated ratios as he projects consonances onto a geometric line and assigns points with letters on the line to represent pitches. With a single shift in emphasis, the eye--visual perception--aided by the rationale of consonances established through multiple and superparticular ratios, becomes the means of navigating the ethereal realm of musical pitch, enabling one to grasp even the most complex of diagrams, such as that in chapter 10, one of the more intricate drawings found in medieval theory.

The idea of explicating pitch relationships visually seems to have attracted the attention of students in several Benedictine monasteries in the tenth century, who saw its potential for rationalizing their own musical practice. But there was a problem: whereas for Boethius the placement of higher or lower pitch seems to have been entirely arbitrary--the diagrams in the earliest manuscripts show ascending or descending pitch unfolding in various directions--the monks apparently wanted consistency with their own conception of musical space, in which higher pitch was to the right, lower pitch to the left. Consequently, certain of the diagrams had to be revised. But how and when did ‘right’ become the placement of higher pitch?

This paper will examine revised diagrams in three tenth-century manuscripts from Fleury and Einsiedeln (Paris, BnF, lat. 7200 and Einsiedeln, Stiftsbibliothek 298 and 358) and reflect on the developing nexus between theory and practice exemplified in an approach to visual representation of pitch that was consistent with the monks´ own spatial conception.

The Principal Author of the Alia musica, the Expositor, has been referred to as the “Wrong-Way Corrigan” of medieval music history. As opposed to Boethius, the Expositor segmented the Greater Perfect System from the lowest pitch to the highest, thereby yielding octave species that were upside-down in relation to those of his 5th/6th-c. predecessor. Moreover, rather than using the octave species to identify the transpositions of the System that constituted the ancient Greek tonoi in Boethius, the Expositor took the octave species themselves to be equivalent to Boethius's modes. As a result, he charted a course that earned him the epithet applied to the 20th-century aviator who ostensibly was planning to fly to California, but landed in Dublin instead. But does the Expositor really deserve a reputation equivalent to that of “the worst navigator in aviation history?”

In this paper I shall argue that the Expositor’s course was a logical one and that it is consistent within itself. I shall demonstrate that the Expositor did understand Boethius, and that he drew upon his own understanding of the species/modes in order to situate the proportional and musical analyses of the ecclesiastical toni in his source treatise within a venerable mathematical and musical tradition. When he states that the “lichanos hypaton [D] of the Hypodorian is the proslambanomenos [D] of the Dorian” and that “the mese [d] of the Dorian, which is the paranete diezeugmenon [d] of the Hypodorian is a perfect fourth higher than the mese [a] of the same Hypodorian,” it is clear that the Expositor had Boethius’s wing diagram before him, and that he understood it. It is also clear that he understood the distinction between the modi of Boethius and the toni of plainchant, and could apply the former in explication of the latter, as for example when he states that “The fourth tonus, which we call Hypophrygian, contains 2 x 12 to 2 x 9 and 3 x 8 to 3 x 6 in one consonance of numbers.” These and other factors lead me to conclude that the Expositor was on course after all.

The Alia musica is one of the earliest treatises to describe technical aspects of the ecclesiastical modes. It consists of at least four layers written over the course of the ninth and tenth centuries. Although it is best known for its last three layers, which fuse Boethian species theory with that of the eight toni based on the octoechos, the earliest layer--written by an author designated as the “First Quidam” by Jacques Chailley--describes the eight toni by means of a complex system of numerical relationships that yield intervallic combinations characteristic of each. This layer was probably composed in the mid- to late ninth century. It appears in the four complete manuscripts of the Alia musica, in two manuscripts from Florence, and in slightly expanded form in a manuscript from Karlsruhe.

These manuscripts appear to be the unique source for the First Quidam’s numerical theory, which elaborates an observation made by Aurelianus Reomensis that each of the four authentic toni is characterized by a specific interval. In the Alia musica the proportions derived from the numbers 6, 8, 9, and 12, which define the perfect intervals, are multiplied by a series of coefficients whose derivation is not explained; the products are then compared with one another to yield the intervals that characterize each tonus. For instance, in the authentic tritus there is one twelve, four sixes (24), three eights (24), and two nines (18). The products of 12:24 and 12:18 yield a diapason and a diapente respectively (and theoretically a diatessaron,18:24, which the First Quidam does not acknowledge), the intervals typical of the tritus autentus.

Previous attempts to explain the derivation of these coefficients have been unsatisfactory. They have attempted to connect the coefficients in varying ways with the octoechos-based toni, but they discount the inconsistencies that result. This paper demonstrates that the coefficients can be explained as mathematically convenient sets intended not to discover what the appropriate intervals ought to be, but rather to filter out undesired intervals, leaving behind a set of intervals that conform to the author’s preconceptions about which intervals are characteristic of each tonus.