The Consonant Eleventh and the Expansion of the Musical Tetractys: A Study of Ancient Pythagoreanism

Abstract

The interval of the eleventh, that is, the octave plus fourth, stands at the crease between rationalism and empiricism in ancient musical treatises. For convenience, I refer to the interval as the eleventh, and to its neighbor as the twelfth, although most ancient and medieval treatises refer to these intervals, respectively, as the diapason plus diatessaron and diapason plus diapente. By extracting the eleventh, I perform a biopsy on ancient Pythagoreanism and thus hope to elucidate the Pythagorean world view as well as to indicate some avenues along which ancient Pythagoreanism may have traveled to the Middle Ages.' By tracing the Pythagorean treatment of the eleventh, I show the role of number in early musical theory to be both causal and descriptive. The causal nature of number would seem to be more primitive and therefore older than the descriptive role played by number in Pythagorean harmonics. Modern musical theory tends to use numbers as a general, abstract system or metaphor for sound. Thus, our modern disposition invites us to see a development from a time when numbers determined the consonance or dissonance of an interval to a time when numbers simply described an interval, that is, provided a quantitative metaphor without evaluating the interval according to its consonant or dissonant character. The history of mathematics reflects this development

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