Re-interpreting an Arithmetical Error in Boethius's De institutione musica

In an influential 1981 article, “Interpreting an Arithmetical Error in Boethius's De institutione musica (iii.14–16),” André Barbera drew attention to a problematic set of arithmetical proofs. At Ins. mus. 3.14ff. Boethius purports to prove that (1) the minor semitone is larger than three commas but small than four, (2) the major semitone is larger than four commas but smaller than five, and (3) the tone is larger than seven commas but smaller than eight. All of these conclusions are correct, but the mathematical procedure seems inherently flawed, for Boethius manipulates the numerical difference between the terms of a ratio as if it were an accurate quantification of the resultant interval. Barbera maintained that the rationale motivating the erroneous mathematics “lies at the heart of Pythagorean cosmogony.” Boethius (Barbera argued) set out to find the numerical truth underlying acoustic phenomena and “seems to have been satisfied by the apparent numerical verification of what he could hear.” The origin of this “arithmetical error” can be specified with greater precision than the vague invocation of “Pythagorean cosmogony” allows. First, I demonstrate that Boethius's arithmetical error is not nearly as erroneous as Barbera and others would have us believe; rather, it represents an approximate method of calculation used to prove relationships otherwise incalculable. Secondly, I argue that this approximate method was not developed by Boethius but was faithfully translated from his immediate Greek source, Nicomachus of Gerasa's (now lost) Eisagoge musike. Thirdly, I suggest that this method, or at least its basic principle, was not independently developed by Nicomachus either, for similar arithmetical methods arose within the early stages of the Greek commentary tradition on Plato's Timaeus.