Pythagorean Music Theory

Pythagoras

The most incomprehensible thing about the universe is that it is comprehensible."
- Albert Einstein, "Physics and Reality" (1936)

Pythagoras was a Greek mathematician who lived from 570 - 495 BC. His first musical discovery was the audible effect of a whole integer ratio series. If we take a vibrating harp string of length n and a string of length 2n (giving us a 1:2 ratio), our ears perceive the two pitches as the same note in different registers: a musician would say the two pitches were one octave apart. Continuing the series, a string of length 3n (forming a 2:3 ratio with the second string, and a 1:3 ratio with the fundamental tone) will produce a new tone, distinct from the first and second but resonating with them in a particularly consonant way. A string of length 4n (which forms a 1:4 ratio with the fundamental tone, but a 1:2 ratio with the second tone) is perceived once again as the same note as the first two tones. This whole integer series is called a "harmonic series" and can continue up ad infinitum. The example here uses "A" as the fundamental tone, because its standard frequency of 440 Hz allows us to work with mostly whole numbers.

Pythagoras was particularly interested in the 2:3 ratio between the 2n string (the second tone of the series) and the 3n string (the third tone of the series.) By applying the same ratio to the 3n string, sounding it against a string of length 4.5n, Pythagoras discovered that it produced the same consonance at a higher pitch. A new series based on the 2:3 ratio can now be created, beginning with A in the middle and moving up three pitches and down three pitches at a 2:3 frequency ratio.

By changing the octaves (i.e. manipulating the pitches according to a 1:2 ratio), we have discovered the seven-note diatonic scale that forms the basis of all Western music. Including the octave of the fundamental tone produces an eight-note scale (hence the term "octave.")