Physics of Music - Notes

Pythagorean Scale

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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