UNREASONABLE EFFECTIVENESS OF NUMBER THEORY

5

which yields the close approximation m = 12, n = 19. In other words, if we

want to make a good fifth with an equal-tempered (equal frequency ratio)

scale, the basic interval

1:21/12,

the semitone, recommends itself. In fact, the

semitone interval has come to dominate much of Western music. The equal-

temperedfifthcomes out as

27/i2

= l 4 9 S

. . .

( 2 )

Another fortunate number-theoretic coincidence is the fact that 7, the

numerator in the exponent in (2), is coprime to 12. As a consequence we can

reach all 12 notes of the octave interval by repeating the fifth (modulo the

octave). This is the famous Circle of Fifths.

Of course, (2) is not an exact equality and some compromises have to be

made in the construction of scales. Pythagoras used the perfect fifth (3:2) and

the perfect fourth (4:3), but fudged on the minor and major thirds, which

come out as 1:1.185 and 1:1.265, rather that 6:5 and 5:4, respectively. (How

Pythagoras must have wished the fundamental theorem out of existence!)

More recently J. R. Pierce has tried to go Pythagoras one better by

constructing a musical scale for the "tritave" (the frequency interval 3:1)

based on the integer ratio 5:3. Pierce used trial and error, but we simply

expand log3 5 into a continued fraction. This yields the close approximation

5 = 319/13, the tritave should be subdivided into 13 equal intervals. (As a

bonus - a number-theoretic fluke -

31/13

also allows a very close

approximation of the next prime 7. To wit: 7 =

323/13.)

3. Concert halls and quadratic residues

There is another connection between music and numbers: concert hall

acoustics. Extensive physical tests and psychophysical evaluation of the

acoustic qualities of concert halls around the world have established the

importance of laterally traveling sound waves. (Such waves produce

dissimilar signals of a listener's two ears, a kind of stereophonic condition

that is widely preferred for music listening.)

In order to convert sound waves traveling longitudinally (from the stage

via the ceiling to the main listening areas) into lateral waves, the author has

recommended ceiling structures that scatter sound waves, without absorption,

into broad lateral patterns. In the physicist's language, concert hall ceilings

(and perhaps other surfaces too) should be reflection phase-gratings with

equal energies going into the different diffraction orders (the different lateral