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God the Geometer determining measurements of the cosmos, depicted as a crystallized geode containing earth, sun, moon, and stars. 13th Century, Anonymous |
Mathematics is music for the mind;
Music is mathematics for the soul.
The history of ideas rewards its readers with the pleasure of discovering soul-mates—historical figures who affirm our convictions, clarify our thinking, encourage our resolve. In the sphere of mathematics, music, and faith, the first among my soul-mates is St. Augustine (354–430). I feel confident that he would have embraced the saying printed above, had he known it. It has long delighted me.
Augustine's particular mathematical interest was "number"—what we would call arithmetic. Number and music gave Augustine a sense of immediate, contemplative, and deeply affecting contact with reality. He believed that the patterns of number and music disclose something of the structure of creation, something of the ineffable Origin and Wisdom that determine the measurements of the cosmos.
Augustine expresses these convictions in writings dating from all phases of his career, from The Confessions (397–8) to the City of God (c413–c426). This repetition tells us that number, music, and faith were more than mere passing interests for him.
On the other hand, this dispersion of passages throughout Augustine's works makes it difficult for readers to gain a comprehensive survey of his thinking, especially since some of his lesser works are difficult to come by. My aim in this posting is to bring together some of Augustine's widely dispersed passages on number, music, and faith.
Augustine's particular mathematical interest was "number"—what we would call arithmetic. Number and music gave Augustine a sense of immediate, contemplative, and deeply affecting contact with reality. He believed that the patterns of number and music disclose something of the structure of creation, something of the ineffable Origin and Wisdom that determine the measurements of the cosmos.
Augustine expresses these convictions in writings dating from all phases of his career, from The Confessions (397–8) to the City of God (c413–c426). This repetition tells us that number, music, and faith were more than mere passing interests for him.
On the other hand, this dispersion of passages throughout Augustine's works makes it difficult for readers to gain a comprehensive survey of his thinking, especially since some of his lesser works are difficult to come by. My aim in this posting is to bring together some of Augustine's widely dispersed passages on number, music, and faith.
In his commentary on the Genesis account of God's creation of the world, Augustine repeatedly quotes a text from the Wisdom of Solomon (11:20):
You [God] have arranged all things by measure and number and weight.
In one passage of his commentary he alludes to this biblical text eight times in four pages.1 "Number" is fundamental here, being the means by which the other two quantities, measure and weight, are specified.
Augustine elaborates upon eternal Wisdom's arrangement of all things by number:
Augustine elaborates upon eternal Wisdom's arrangement of all things by number:
Wherever you turn, Wisdom speaks to you through the imprint it has stamped upon its works.... Look at the sky, the earth, and the sea, and at whatever in them shines from above or crawls, flies, or swims below. These have form because they have number.2Augustine describes Wisdom and number as "somehow one and the same thing," even "identical," and asserts that "both are true, and immutably true."3
Immersed in the profound changes and uncertainties of the 5th-century Roman Empire, Augustine marvels at the universality and permanence of number:
Seven and three are ten, not only now, but forever. There has never been a time when seven and three were not ten, nor will there ever be a time when they are not ten. Therefore, I have said that the truth of number is incorruptible and common to all who think.4
He draws encouragement from his conviction that immutable number allows us trustworthy knowledge:
Whether they are considered in themselves or applied to the laws of figures, or of sound, or of some other motion, numbers have immutable rules not instituted by men but discovered.5
Augustine's language here about applying number to the laws "of sound" brings us to his appreciation for the mathematics of music. He was well acquainted with the discovery attributed to Pythagoras (6th–5th centuries BCE) that musical intervals are defined by mathematical ratios—ratios, for example, of the lengths of a musical instrument's strings. The instrument used for demonstrating these ratios was the monochord: a device consisting of one string under constant tension, with a moveable fret and a ruler to measure the fret's placement along the pluckboard.
Pythagorean experimenters with the monochord discovered that the most consonant or pleasing musical intervals are those defined by the simplest mathematical ratios: the musical octave by the ratio 1:2; the musical fifth by the ratio 2:3; the musical fourth by the ratio 3:4.
Augustine expresses particular wonderment over the consonance of the octave, which he calls "the single and the double." He recognizes that the two notes of an octave are the same note, distinguished only by being higher and lower. As an illustration, we might think of a director asking a chorus of mixed voices to sing in unison; they respond by singing in octaves, the women higher, the men lower; and the director is satisfied that the unison instruction is being followed.
Augustine calls our sense of consonance in musical intervals an occulta familiaritas, a "hidden affinity."6 He considers this affinity—founded as it is in immutable number—to be God-given. He speaks again of the octave:
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| In this hexachord the strings are multiple and of variable tensions, as determined by the attached weights. From Franchino Gaffurio, Theorica musicae (Milan, 1492) |
Augustine expresses particular wonderment over the consonance of the octave, which he calls "the single and the double." He recognizes that the two notes of an octave are the same note, distinguished only by being higher and lower. As an illustration, we might think of a director asking a chorus of mixed voices to sing in unison; they respond by singing in octaves, the women higher, the men lower; and the director is satisfied that the unison instruction is being followed.
Augustine calls our sense of consonance in musical intervals an occulta familiaritas, a "hidden affinity."6 He considers this affinity—founded as it is in immutable number—to be God-given. He speaks again of the octave:
The harmony between the single and the double...has been naturally so implanted in us (by whom indeed, if not by Him who created us?) that not even the illiterate can remain unaware of it, whether they themselves are singing, or whether they are listening to others; for by means of it the treble and the bass voices blend together, and anyone who sounds a note that does not harmonize with it commits an offense not only against the musical art of which most people are ignorant, but against our very sense of hearing. It would require a long treatise, however, to prove this, but one familiar with the subject can demonstrate it to the ear itself on a properly adjusted monochord.7Augustine's reflections on the mathematics of music are intrinsic to his life of faith. It is well known that he was profoundly moved not only by the mathematics of music but by sounding music as well—to such a degree, in fact, that he names the distraction of listening among his besetting temptations.8 He insists that our delight in music, auditory and numerical, should not be an end in itself but should transcend the sensual and practical. Music should direct our faith toward God's Wisdom:
And so with a determined retreat from every wanton movement where lies the fault of the soul's essence, and with a restored delight in reason's number, our whole life is turned to God....9Augustine's love of "reason's number" in music often echoes in his theological images. He ponders, for instance,
how the single of our Lord and Savior Jesus Christ corresponds to our double [our human nature], and in what manner it harmonizes with it for our salvation.10He speaks of musical consonance as a metaphor for a just society:
The concord of different sounds, controlled in due proportion, suggests the unity of a well-ordered city, welded together in harmonious variety.11Sometimes Augustine's love of musical consonance goes beyond metaphor to play a more essential role in his theology. After years of struggle concerning the nature of evil, vividly chronicled in The Confessions, Augustine finally resolves his theological perplexity. He expresses his resolution in an image of evil as discord—intrinsic to creation's harmonic structure, but in unresolved tension:
For you [God] evil has no being at all, and this is true not of yourself only but of everything you have created, since apart from you there is nothing that could burst in and disrupt the order you have imposed on it. In some parts of it certain things are regarded as evil because they do not suit [or "do not accord with": non conveniunt] certain others; but these things do fit in ["do accord": conveniunt] elsewhere, and they are good there, and good in themselves.12I give thanks for St. Augustine as a soul-mate. I do not regard his discussions of number and music as arguments for belief—still less as any kind of theological proofs. For me Augustine is a mentor and companion in questioning, defining, and enjoying faith in "God, the ineffably and invisibly great, the ineffably and invisibly beautiful."13
*****
1 The Literal Meaning of Genesis 4.3.7–4.5.12, tr. John Hammond Taylor, in Ancient Christian Writers: The Works of the Fathers in Translation, ed. Johannes Quasten (Paulist Press, 1982) No. 41, Vol. I, Books 1–6, pp.107–11.
2 On Free Choice of the Will 2.16.163–4, tr. Anna S. Benjamin and L. H. Hackstaff (Bobbs-Merrill, 1964), pp.73–4.
3 On Free Choice of the Will 2.11.125, p.64.
4 On Free Choice of the Will 2.8.83, p.54. See also The Confessions 6.4, tr. Maria Boulding (Vintage, 1997), p.102.
5 On Christian Doctrine 2.38.56, tr. D. W. Robertson, Jr. (Macmillan, 1958), p.73.
6 Confessiones 10.33, ed. W. H. D. Rouse (William Heinemann, 1968), Vol. 2, p.166.
7 The Trinity 4.2.4, tr. Stephen McKenna, in The Fathers of the Church, ed. Roy Joseph Deferrari (Catholic University of America Press, 1963), Vol. 45, p.134.
8 The Confessions 10.33, pp.229-30.
8 The Confessions 10.33, pp.229-30.
9 On Music, 6.11.33, tr. Robert C. Taliaferro; in Writings of Saint Augustine, Vol.2; in The Fathers of the Church, ed. Hermigild Dressler (Catholic University of America,1947), Vol.2, p.358. See also The Literal Meaning of Genesis 4.4.9–10, pp.109-10.
10 The Trinity 4.3.5, p.134.
11 Concerning the City of God 17.14, tr. Henry Bettenson (Penguin, 1984), p.744. See also 2.21, p.72.
12 The Confessions 7:12, p.136.
13 Concerning the City of God 11.4, p.432.
