The mysterious mathematics relating natural Pythagorean and tempered western music.

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.
Simple harmonious Pythagorean 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.
"Musimatics: Mathematics of Classic Western Harmony - Pythagoras to Bach to Fourier to Today" is a Power Point introduction to the tempored scale. Click HERE to view it.

"The Well Tempered Pythagorean: The Remarkable Relation Between Western and Natural 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.