Music and the Making of Modern Science

18 Chapter 1

Nevertheless, the Greek theorists found a way to include geometric proportion in music

by introducing a harmonic mean that is a hybrid of the arithmetic and geometric means

( box 1.1 ), positioning music between arithmetic and geometry. Like music, astronomy (the

fourth of the sister sciences) is also poised between arithmetic proportions of the ratios

between the movements of the heavenly bodies and their description using spherical

geometry. Music and astronomy bridge the invisible realm of mathematical forms and the

sensual realm of experience. Socrates argued that both music and astronomy use the expe-

rience of the senses to “ summon or arouse thought, ” whereas many other modes of percep-

tion “ do not summon thought to inquiry. ” Through beholding “ intricate traceries in the

heavens, ” astronomy “ compels a soul to look upward, ” toward “ what is and is invisible. ”^32

Music, in a different way, joins imperceptible numerical ratios with the perceptible inter-

vals. Both music and astronomy connect the purely mathematical and the sensually per-

ceptible. In what follows, both these sister sciences will play out their intermediary roles.

But whereas the formative influence of astronomy in the development of science has long

been acknowledged, the ways music has entered this story remain to be told.

These matters aroused deep and enduring controversy. The Pythagorean view of music

as mathematical ratios was opposed by Aristoxenus, the great contrarian voice in Greek

music theory, so famous that the Romans referred to him simply as “ the musician. ” Where

the Pythagoreans exalted reason over sensual judgment, Aristoxenus, like his teacher

Aristotle, emphasized the fundamental role of the senses: “ Through hearing we assess the

magnitude of intervals, and through reason we apprehend their functions. ”^33 In reasserting

the sensual, experiential character of music, Aristoxenus called into question the relation

between music and mathematics. But Boethius, adhering to Pythagorean views, treated

Aristoxenus briefly and dismissively. Because all musical study during the subsequent

millennium relied on Boethius, Aristoxenus fell into obscurity until his texts were redis-

covered in the sixteenth century with powerful effects on musical and mathematical

thought, as we shall see in chapter 4.

Then too, Ptolemy took the side of the Pythagoreans against Aristoxenus. Ptolemy ’ s

Mathematical Composition (Syntaxis) , which Arabic scholars called Almagest ( “ The

Box 1.1

The arithmetic, geometric, and harmonic means.

Arithmetic mean Geometric mean Harmonic mean

1:2:3 1: √ 2:2 1:2:4

2 is the arithmetic mean

between 1 and 3, an equal

difference from both:

2 – 1 = 3 – 2

√ 2 lies at an equal ratio

( √ 2:1) between 1 and 2

because 1: √ 2 = √ 2:2

The differences 2 – 1= 1,

4 – 2 = 2 are in the same

ratio as the terms,

namely 1:2