Dialogue between humans and nature

José Rodríguez Alvira

To compare different tuning systems we will use *cents*. Any octave can be divided in 1,200 cents. In the case of equal temperament each semitone has 100 cents. In the table below we compare the size of the major third and perfect fifth of the harmonic series with discussed temperaments:

Third | Difference | Fifth | Difference | |

Harmonic series | 386 | 702 | ||

Pythagoras | 408 | +22 | 702 | 0 |

Meantone | 386 | 0 | 697 | -5 |

Well temperament | 390 to 408 | +4 to +22 | 696 or 702 | -6 or 0 |

Equal temperament | 400 | +14 | 700 | -2 |

Notes:

- Pythagorean major third tone is almost 1/4 semitone larger than the acoustic third
- The major thirds of the well temperament range from somewhat larger (390) than the acoustic third to the size of Pythagorean major third (408)
- Equal temperament third is larger than the acoustic third but not as large as the Pythagorean third
- Equal temperament has smaller fifths than the harmonic series but they are closer (2 cents) than the meantone (5) fifth or the well temperament reduced fifths (6)

The following chart compares the tuning systems discussed with equal temperament (horizontal line on the 0). The vertical scale measures are in cents:

Notes:

- The meantone temperament (green) is a mirror of the Pythagorean tuning (blue)
- Equal temperament is kind of average of the different tuning systems

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José Rodríguez Alvira.