Around 500 BC Pythagoras studied the musical scale and the ratios between the lengths of vibrating strings needed to produce them. Since the string length (for equal tension) depends on 1/frequency, those ratios also provide a relationship between the frequencies of the notes. He developed what may be the first completely mathematically based scale which resulted by considering intervals of the octave (a factor of 2 in frequency) and intervals of fifths (a factor of 3/2 in frequency). The procedure is described in the book by Jeans. The resulting scale divides the octave with intervals of "Tones" (a ratio of 9/8) and "Hemitones" (a ratio of 256/243). Here is a table for a C scale based on this scheme.
|Note||Ratio to Fundamental||Closest Ratio
in Just Scale
in Equal Tempered
The intervals between all the adjacent notes are "Tones" except between
E and F, and between B and C which are "Hemitones."
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