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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 Closest Ratio in Equal Tempered C1.000 1.000 1.000 D9/8=1.125 9/8=1.125 1.12246 E81/64=1.2656 5/4=1.2500 1.25992 F4/3=1.3333 4/3=1.3333 1.33483 G3/2=1.500 3/2=1.500 1.49831 A27/16=1.6875 5/3=1.6667 1.68179 B243/128=1.8984 15/8=1.875 1.88775 C2.000 2.000 2.000 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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