When we do the circle of fifths calculations using rational
numbers instead of floating point numbers, we find a number of
simple-looking fractions like 3/2, 4/3, 9/8, 16/9 in our results. These
fractions lead to a geometrical interpretation of the musical intervals.
These fractions correspond with some early writings on music by the
mathematician Pythagoras.

We'll provide one set of commonly-used list of fractions for
Pythagorean tuning. These can be compared with other results to make the
whole question of scale tuning even more complex.

Name |
Ratio |

A |
1:1 |

A# |
256:243 |

B |
9:8 |

C |
32:27 |

C# |
81:64 |

D |
4:3 |

D# |
729:512 |

E |
3:2 |

F |
128:81 |

F# |
27:16 |

G |
16:9 |

G# |
243:128 |

**Pythagorean Pitches. **Develop a simple representation for the above ratios. A list of
tuples works well, for example. Use the ratio to compute the
frequencies for the various pitches, using 27.5 Hz for the base
frequency of the low "A". Compare these values with equal temperament,
overtones and circle of fifths tuning.

**Check Your Results. **The value for "G" is 27.5 * 16 / 9 = 48.88Hz.