Dialogue between humans and nature

José Rodríguez Alvira

Equal temperament - commonly used today - divides the octave into 12 semitones of equal size. It was proposed since the sixteenth century:

- 1581: Vincenzo Galilei (1520-1591) suggests dividing the octave into 12 equal parts using the ratio 18/17 for the semitone
- ca. 1580: the Dutch mathematician Simon Stevin (1548-1630) calculates the size of the semitone
- 1584: in China Zhu Zaiyu accurately calculated the scale using the ratio 749: 500 for the fifth

However it was not until the nineteenth century that it begins to be used with the reluctance of some:

- 1906: Busoni considers that it is an
*evil of civilization* - 1907: Saint-Saëns considered it
*a devastating tyrant and heresy ...*recognizes, however*,*that*it has been too fruitful to be ignored*

(from* Musique et témperaments, Pierre-Yves Asselin*)

How do we calculate the size of the semitone that divides an octave into twelve equal parts? If we were able to split the third in two tones of equal size using the square root, then we can divide the octave into 12 equal parts by calculating the twelve root of 2 (octave):

The following table shows that we can use this number to calculate the frequency of each semitone:

C | 262 | |

C# | 277.6 | 262 x 1.0594630943593 |

D | 294.1 | 277.6 x 1.0594630943593 |

D# | 311.6 | 294.1 x 1.0594630943593 |

E | 330.1 | 311.6 x 1.0594630943593 |

F | 349.7 | 330.1 x 1.0594630943593 |

F# | 370.5 | 349.7 x 1.0594630943593 |

G | 392.6 | 370.5 x 1.0594630943593 |

G# | 415.9 | 392.6 x 1.0594630943593 |

A | 440.6 | 415.9 x 1.0594630943593 |

A# | 466.8 | 440.6 x 1.0594630943593 |

B | 494.6 | 466.8 x 1.0594630943593 |

C | 524 | 494.6 x 1.0594630943593 |

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José Rodríguez Alvira.