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 |
---|---|---|---|

C | 1.000 | 1.000 | 1.000 |

D | 9/8=1.125 | 9/8=1.125 | 1.12246 |

E | 81/64=1.2656 | 5/4=1.2500 | 1.25992 |

F | 4/3=1.3333 | 4/3=1.3333 | 1.33483 |

G | 3/2=1.500 | 3/2=1.500 | 1.49831 |

A | 27/16=1.6875 | 5/3=1.6667 | 1.68179 |

B | 243/128=1.8984 | 15/8=1.875 | 1.88775 |

C | 2.000 | 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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