David Goldman in First Things:
“It is consoling to think that the emotions that music arouses in us have something to do with the makeup of the universe. The eternal relation of math and music has been a perennial question since Plato, from Boethius and Cassiodorus in late antiquity, through Dante’s celestial harmony in Paradiso and Shakespeare’s discussion in The Merchant of Venice. The deeper affinity between mathematics and music, though, is less consoling and more challenging: The modern concept of a higher-order number begins with St. Augustine’s fifth-century treatise on music, and a red thread links it to Leibniz’ invention of the calculus in the seventeenth.
Music employs number both in its harmonic foundation and its metrical presentation in time. But what sort of number is it? In the sixth book of his De Musica, Augustine asserted the existence of a higher order of number that in some way stands above the senses, the numeri iudiciales or “numbers of judgment” which “come from God” and enable the mind to judge what it perceives and remembers, as well as what it expects. Augustine’s assertion is arresting in all three of its parts: first, that neither our sense perception nor even our memory explains how we hear music; second, that the faculty by which we judge the numbers (rhythms or harmonies) of music is also a kind of number; and third, that this higher-order number comes from God.”