Anne Weddigen (Sorbonne Université)
This paper aims at considering the shape and function of musical diagrams transmitted in ancient musical treatises. Despite their number and variety, diagrams have only been taken marginally into account in critical editions of musical treatises (a critical assessment of the diagrams in Ptolemy’s Harmonica is still to be undertaken (CREESE), as well as for Boethius’, BOWER). Recent changes come from the field of mathematics (NETZ, ACERBI). Extant musical treatises mainly deal with harmonic theory and were written for didactical purposes. To them one might add commentaries to Plato’s Timaeus. It strikes the reader that the more abstract the theory gets and the more mathematical the proofs, the more diagrams are to be found, as if theoretical conceptualization increased the need for visualization. The main challenge is: how to depict what underlies music, but cannot, by definition, be heard?
I would like to examine two main types of musical diagrams. The first of them is the well-known lambda-shaped diagram, ascribed to Crantor, and described by Proclos, in a slightly different shape, as used by Adrastus. No Greek manuscript handed down this diagram, but it is to be found in Calcidius’ commentary to the Timaeus, and came down to us in a Latin tradition, well known to Renaissance scholars (HUGLO, MEYER, REYDAMS-SCHILS). This diagram became the symbol for the work itself and the Pythagorean-Platonic theory on the World Soul. On this example, I would like to examine the process of mapping conceptual and musical thinking, and the role played by language and metaphor as a support for visualization of concepts. Basing on the manifold meanings of the word harmonia (MATHIESEN), this efficient image acquired the status of a quasi-magical object that displays and reveals true intrinsic qualities, in a way an astrolabe would do.
The second object under investigation would be the kanon-diagram, that was developed by byzantine scholars. Using the precise study of CREESE on the kanon in Greek harmonic theory up to Ptolemy, we can show a progressive shift, from a heuristic tool bridging the gap between theory and hearing senses towards a mathematical tool of abstract visualization. In a way, byzantine scholars like Pachymeres and Bryennios rediscovered the mathematical origins of the tool and linked them to the geometrical proofs of euclidean tradition. As can be seen in Bryennios’ chapters on the division of the kanon (II, 6-7), they did not ignore the practical purposes of the tool itself, but used a realistic schema of it as a first step towards geometrical abstraction representing arithmetical proofs and calculations.
These two examples show the strong and interweaved link between theoretical thinking and diagrammatic visualization as a support for abstraction. As music is one of the quadrivial arts, its borrowing from different mathematical and astronomical tools is not surprising; but their application to music shapes music-theory as a whole and left a long-lasting mark on the perception of Ancient Greek Music.