Charles André Barbera, Ph.D. 1980

Advisor: Calvin Bower

Dissertation Title: The Persistence of Pythagorean Mathematics in Ancient Musical Thought

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Dissertation Abstract:

Several ways of knowing music exist in Western civilization, two of which predominate: grammatical (linguistic) and mathematical. As early as the sixth and fifth centuries B.C., Pythagoras and Pythagoreans initiated and developed a mathematical way of knowing about the world in general, and in particular about music. Historians of mathematics have long recognized that the Euclidean generalization of mathematics during the fourth century B.C. rendered obsolete the qualitative, substantive mathematics of the Pythagoreans, which previously had played a participatory role in the development of Greek mathematics. This generalization, culminating in Euclid’s compilation of the Elements of Geometry (c. 300 B.C.), transformed mathematics into an abstract theory, capable of accommodating incommensurable magnitudes and generally applicable to all physical sciences.

Historians of music have long recognized that several ancient musical treatises, most of which date from well after the fourth century B.C., contain and rely upon Pythagorean mathematics. My study investigates why a mathematical way of knowing that was rendered obsolete during the fourth century B.C. by the Euclidean generalization lived on in the musical treatises, persevering for over a millennium after having been superseded. I conclude that the link and strength between Pythagorean mathematics and ancient musical theory was substantive number. Pythagorean number is as corporeal as sound, and in this way Pythagorean harmonics (musical theory) distinguishes itself from the incorporeal harmonics of Plato. In addition to mathematical changes and developments, during the fourth century B.C. Pythagorean musical theory was threatened by the geometrically conceived musical theory of Aristoxenus, but withstood this threat on its own merits. Pythagorean mathematics survived because Aristoxenus’s Elements of Harmony did not eradicate Pythagorean musical theory. The link between the Pythagorean mathematical and musical theories was of sufficient philosophical strength to withstand the turn of events during the fourth century B.C.

In this study I present a brief history of Pythagorean mathematics in order to discuss its connection to sound and to music on the bases of: classification, proportional theory, and transfer of terms. In so doing I define a central tradition for the transmission of Pythagorean mathematics in ancient musical treatises as the corpus of treatises that, in devoting themselves exclusively or largely to musical matters, exhibit Pythagorean mathematical reasoning. The major mathematical traits and issues occurring in this tradition include: the relation of reason to sensory perception; the myth of the Pythagorean hammers; the treatment of the semitone; the division of the tetrachord; the arithmetic, geometric, and harmonic means; and the assignment of numbers to notes and ratios to intervals. This tradition includes the following treatises: Sectio canonis, Nicomachus’s Manual of Harmony, Theon of Smyrna’s Expositio rerum mathematicarum ad legendum Platonem utilium, Gaudentius’s Introduction to Harmony, and Boethius’s De institutione musica. In addition to these treatises, I discuss works by the following authors: Aristides Quintilianus, Cassiodorus, Censorinus, Chalcidius, Iamblichus, Macrobius, Martianus Capella, Porphyry, and Proclus. Finally, I use Ptolemy’s Harmonica to evaluate the major mathematical traits and issues found in the musical treatises under consideration.

Dr. Barbera was a Professor at St. John’s College in Annapolis, MD.