Music and the Making of Modern Science

60 Chapter 4

rule ” shows how to find what he calls the “ closest approximation ” to solving an equation

through finding the integers, “ greater and less, which most nearly satisfy the equation, ”

then generating a series of differences between the values those integers generate when

substituted in the equation, from which a further refined estimate can be made, leading to

what he takes to be a converging series of approximations. Through this procedure, “ you

will undoubtedly arrive at an insensible difference ” compared with the true value: “ This

is universal reasoning and needs no other rule. ”^19 His procedures here seem to reflect the

practical sense that comes to the fore in his musical writings: “ true ” geometric (irrational)

and “ true ” arithmetic (rational) quantities sound the same; their differences are “ insensi-

ble. ” Music is the sole example he has of this kind of mediation between “ perceptual ” and

“ true ” quantities; he lacks any other kind of mathematical physics (as it would later come

to be called) that could have confronted mathematical idealizations with physical reality.

But music was sufficient for him. Musical judgments intermixed rational and irrational

quantities, supporting and paralleling Cardano ’ s working equivalence of the two in his

algebraic art and providing it with crucial examples.

The grounds on which Nicola Vicentino treated irrational quantities as numbers had

more to do with issues of melodic style and musical practice, as for Cardano, than with

the more purely theoretical questions that concerned Stifel. Boethius had enumerated

three ancient “ genera ” of melody, each genus designating a separate set of basic

intervals on which music could be constructed. In the last chapter, we considered the

diatonic genus, which Boethius considered “ somewhat more severe and natural ” ( ♪ sound

example 4.2a). The other two genera are more unfamiliar (see box 4.2 ). According to

Boethius, the chromatic genus “ departs from natural inflection and becomes more sensual

[ mollius ]. ”^20 The name “ chromatic ” persists even today to describe music that makes

Box 4.1

Pythagorean tuning, just intonation, and equal temperament

Pythagorean tuning relies on the simple intervals of the smithy story (chapter 1): octave (2:1),

perfect fifth (3:2), perfect fourth (4:3), whole tone (9:8). From these are derived more complex

(hence less consonant) intervals, such as the major third (two whole tones, 81:64), major

semitone (interval between a major third and a perfect fourth, 256:243), minor third (a whole

tone and a major semitone, 32:27) ( ♪ sound example 4.1a).

Just intonation is a system of related tunings that all begin with the Pythagorean octave,

fifth, and fourth, and then redefine other intervals to be simple whole number ratios, such as

the major third (5:4) and minor third (6:5), reflecting the growing use of those intervals as

consonances after the fifteenth century ( ♪ sound example 4.1b).

Equal temperament divides the octave into twelve mathematically equal semitones, each

defined by the irrational expression^1221 :. Except for the octave, all other intervals are slightly

“ out of tune ” with respect to just intonation or Pythagorean tuning ( ♪ sound example 4.1c).