to two basic divisions of quantity, multitude and magnitude. Arithmetica formed the
science of multitude per se, while musica was the discipline treating multitude in ratios;
geometrica was the subject considering fixed (immobile) magnitudes, and astronomia
formed the science of magnitude in motion. This fourfold division of mathematical
knowledge, the Quadrivium, was transmitted into the Middle Ages by Boethius’s De
institutione arithmetica and De institutione musica, and through this foundation a science
of rational, systematic reflection on the art of music was established. Cassiodorus’s
Institutiones and Isidore of Seville’s Etymologiae are two further works of late antiquity
that aided in establishing this tradition of musical reflection for the Middle Ages.
As a subject preparatory to philosophy, musica stressed “knowing,” immutable
knowledge in the philosophical sense. That knowing arose from the science of
mathematical ratios and from the physical reality that the most basic musical intervals,
the consonances of the fourth, fifth, and octave, have their scientific foundation in these
ratios. The senses were mistrusted as determinants of musical structures or as the basis of
value judgments. The philosophical hero of mythic status was the ancient Greek
Pythagoras, who by divine guidance discovered the musical ratios in a smithy. The
technique of manipulating the ratios to derive musical systems, or pitch collections, was
developed by disciples of Pythagoras, and the application of the ratios to a scientific
instrument, the monochord, became the means of determining for the senses an empirical
reality based on rational reflection. But the efficacy of rational musical reality was not
only revealed in sensual consonance; it was also present in the structure of the universe
(musica mundana) and in the physical and psychological structure of the human being
(musica humana).
Boethius’s mathematical treatises began to be studied in France during the early 9th
century, in monastic communities devoted to the realization of a liturgy that was largely
sung. No necessary affinity existed between the rational musical system of antiquity and
the practical reality of daily singing the Divine Office and Mass. Nevertheless, the
scholar and cantor—often the same person—began to demand a degree of congruence
between knowing and doing. During this period, two musical traditions are evident:
cantus and musica, or the art of singing and the discipline of musical theory. Tension
between music as a performing art and music as a theoretical discipline is evident in the
earliest theoretical and historical sources, yet each played a crucial role in shaping the
other during the Middle Ages.
Two traditions are found in the earliest truly medieval theory. On the one hand, a
collection of texts associated with the anonymous Musica enchiriadis (late 9th c.), while
aware of the rediscovery of Boethius and his mathematical basis for musical thought,
presents a theory of music largely independent of the ancient tradition. The language of
these texts tends toward the qualitative rather than the quantitative, and the myth
associated with the collection is that of Orpheus and his inability to bring the mysteries of
music into the clarity of day, rather than Boethius’s story of Pythagoras and the rational
numbers founded on hammers. Basic terminology of these texts is firmly rooted in
liturgical practice rather than ancient theoretical tradition, and the linguistic root of much
terminology is clearly Byzantine Greek. The tonal and notational system found in these
texts is independent of the system transmitted by Boethius. On the other hand, such
theorists as Aurelian of Réôme and Hucbald of Saint-Amand had read and somewhat
mastered the tradition discovered in Boethius, Cassiodorus, and Isidore and had begun
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