In
the 6th century BC, Greek philosopher Pythagoras proposed harmony is best
achieved between two frequencies when their ratio can be expressed as
the ratio of two small whole numbers. Newtonian physics shows that physical
systems, such as vibrating air columns and vibrating strings, naturally
produce such frequency relationships. Western music, on the other hand,
is based on the strange irrational number the-twelfth-root-of-two. Remarkably,
the tempered scale based on this number is able to produce frequency intervals
that, although not exactly equal to whole number ratios, result in notes
nearly audibly indistinguishable. Not only can the tempered scale be used
to closely approximate natural Pythagorean harmonies, it allows drastically
more flexibility in music composition. The tempered scale is also a near
perfect fit to the logarithmic frequency response characteristics of the
human ear. |