mysterious mathematics relating natural Pythagorean and tempered western music.
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
intervals can be heard
in bugle tunes. Such tunes result from a fixed vibrating air column stimulated
at different frequencies. The ratios of the four bugle tones to the tonic are
3/4, 1, 5/4 and 3/2. To here an MP3 of the bugle melody Taps, click HERE.
To hear Revelry, click HERE.|
Mathematics of Classic Western Harmony - Pythagoras to Bach to Fourier
to Today" is a Power Point introduction to the tempored
HERE to view it.
Well Tempered Pythagorean: The Remarkable Relation Between Western
Harmonic Music" contains a comparative analysis between Pythagorean
and western music. The remarkable relationships between the two systems are explored
using a number of examples. The basis for the Pythagorean system is derived using
basic Newtonian physics applied to a vibrating string. Click HERE to
view a pdf copy of the paper.