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
the semitone, recommends itself. In fact, the
semitone interval has come to dominate much of Western music. The equal-
temperedfifthcomes out as
= 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 -
also allows a very close
approximation of the next prime 7. To wit: 7 =
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
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