Thomas’s Mathematical Music

Metaphysics of the Music, part 15

The previous post noted the comparatively spare use Aquinas, unlike earlier theologians such as Augustine or Boethius, made of musical imagery as a metaphor for cosmic harmony. As we shall see, rather, the relevance of Thomas’s views on music for understanding Tolkien, ironically, have more to do with his view of music as exhibiting an exceedingly abstract, almost mathematical kind of existence. In his commentary on Boethius’s De Trinitate, Thomas closely associates music with mathematics on account of the way music derives its first principles from arithmetic and applies these principles to natural things: “In another way, one science is contained under another as subalternated to it. This occurs when in a higher science there is given the reason for what a lower science knows only as a fact. This is how music is contained under arithmetic.” (Commentary on Boethius’s De Trinitate 5.1 ad 5, trans. Maurer). For Aquinas, music represents an “intermediate” between mathematics and natural science, yet he says it bears “a closer affinity to mathematics” since music is more “formal” and thus more separated from matter and motion than is the case in natural science: “music considers sounds, not inasmuch as they are sounds, but inasmuch as they are proportionable according to numbers” (5.3). Behind Thomas’s argument here is his teaching that, although concepts of both mathematics and natural objects involve an act of mental abstraction separating their intelligible principles from the physical, sensible substances in which these principles are actually experienced, mathematics and natural science nevertheless differ in their respective degrees of abstraction (5.1-2). In the case of a mathematical object such as a circle, there is no reference in the concept of a circle to the kind of matter that real (i.e., non-mental) circles are actually made of, since circles can be made out of virtually anything. The case is otherwise with concepts of natural substances such as man, for which the kind of matter the thing is made out of comprises an integral part of the substance’s essence or form. Thus, while the concept of man, like the concept of a circle, is produced by the mind’s abstracting it from the determinate or “signate” matter out of which individual men or circles are actually made, the concept of man nevertheless retains a notional reference to the kind of matter out of which real men are made, namely flesh and bones. To return to the question of music, then, for Thomas, while music as we experience it is of course an inherently physical, sensible, and sensuous phenomenon, in terms of the formal qualities which constitute its sounds as musical sounds, the comparative indifference of music to the particular, material environment, circumstances, or conditions under which it is played makes it similar, in Thomas’s mind, to the heightened degree of mental abstraction involved in mathematics. For Thomas, in short, music is a highly abstract reality that is ultimately concerned with sound, not as sound (i.e., an inherently physical phenomenon), but as a peculiarly mathematical and proportionate kind of sound.

Boethius, Music of the Spheres, and the Music of the Ainur

Metaphysics of the Music, part 6

The next influential Christian thinker after Augustine to turn his attention to the significance of music, in terms of both its own principles as a science and as a metaphor for cosmic harmony, is the sixth-century Boethius, whose views on music Bradford Lee Eden has compared at some length with Tolkien’s narrative (Eden, “The ‘Music of the Spheres’: Relationships between Tolkien’s The Silmarillion and Medieval Cosmological and Religious Theory,” 183-193). Possibly the most influential treatise ever written on the subject of music, it was through Boethius’s De institutione musica that classical musical theory was primarily transmitted to the Middle Ages and Renaissance (Godwin, The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music, 86). Toward the beginning of his treatise, Boethius distinguishes three kinds of music: cosmic (musica mundana, or “music of the spheres”), human (musica humana, the music of the human body and soul), and instrumental (musica instrumentalis) (Boethius, Fundamentals of Music, 1.2, trans. Bower. See also Godwin, The Harmony of the Spheres, 86). The three primary examples of the cosmic music distinguished by Boethius include the movement of the heavenly bodies, the combination of the physical elements, and the changing of the seasons. Of the heavenly bodies, for example, Boethius thinks it impossible that “so swift a heavenly machine moves on a mute and silent course” and “that such extremely fast motion of such large bodies should produce absolutely no sound…,” and in a later chapter Boethius even correlates each of the planetary spheres with the various standard musical strings (“the hypate meson is assigned to Saturn, whereas the parhypate is like the orbit of Jupitor,” etc.) (Boethius, Fundamentals of Music, 1.27).

Although it is with the classical idea of the “music of the spheres” that commentators have most often compared the Music of the Ainur, it is worth noting that the Ainulindalë itself does not in fact ever refer to the heavenly bodies, nor are they elsewhere in Tolkien’s mythology ever described as producing any kind of sound or music. Of greater relevance to the Ainulindalë, therefore, it would seem, are Boethius’s second and third examples of cosmic music, namely the harmony of the elements and seasons. Of the former, for example, Boethius asks: “If a certain harmony did not join the diversities and opposing forces of the four elements, how would it be possible that they could unite in one mass and contrivance?” (Boethius, Fundamentals of Music, 1.2). Similarly, in the Ainulindalë it is in a state of Boethian harmony that the four elements first appear to the Ainur in the Vision:

And they observed the winds and the air, and the matters of which Arda was made, of iron and stone and silver and gold and many substance: but of all these water they most greatly praised. And it is said by the Eldar that in water there lives yet the echo of the Music of the Ainur more than in any substance else that is in this Earth; and many of the Children of Ilúvatar hearken still unsated to the voices of the Sea, and yet know not for what they listen. (Silmarillion 19)

As for his third category of cosmic music, Boethius compares the “consonance” of the four seasons with the attunement of lower and higher strings of an instrument, so that “the whole corpus of pitches is coherent and harmonious with itself”: “For what winter confines, spring releases, summer heats, and autumn ripens, and the seasons in turn either bring forth their own fruit or give aid to others in bringing forth their own” (Ibid.1.2). In a comparable expression found in the Ainulindalë of the accord between seasons and weather patterns, upset only by the disruptions of Melkor, Ilúvatar informs the Valar Ulmo:

‘Behold the snow, and the cunning work of frost! Melkor hath devised heats and fire without restraint, and hath not dried up thy desire nor utterly quelled the music of the sea. Behold rather the height and glory of the clouds, and the everchanging mists; and listen to the fall of rain upon the Earth! And in these clouds thou art drawn nearer to Manwë, they friend, whom thou lovest.’

            Then Ulmo answered: ‘Truly, Water is become now fairer than my heart imagined, neither had my secret thought conceived the snowflake, nor in all my music was contained the falling of the rain. I will seek Manwë, that he and I may make melodies for ever to they delight!’ (Silmarillion 19)