Mathematically, starting with a pitch f

f

f

Of course, this process can be repeated indefinately and one will stop after a while to keep the number of notes in the scale reasonable.

Here is a table which results from that procedure. I have included more notes than we usually use for the sake of illustration. Here f

Freq (Hz) | Ratio to Fundamental | Closest Ratio in Just Scale | Freq. difference of Just Scale (Hz) | Closest Ratio in Equal Tempered |
Freq. Difference of Equal Tempered |
---|---|---|---|---|---|

261.63 | 1 = 1.0000 | 1.0000 | 0 | 1.0000 | 0 |

265.20 | 531441/524288 = 1.013643 | ||||

279.39 | 2187/2048 = 1.067871 | 1.0417 | -6.9 | 1.0595 | -2.2 |

294.33 | 9/8 = 1.125000 | 1.1250 | 0 | 1.1225 | -0.7 |

298.35 | 4782969/4194304 = 1.140349 | ||||

314.31 | 19683/16384 = 1.201355 | 1.2000 | -0.4 | 1.1892 | -3.2 |

331.13 | 81/64 = 1.265625 | 1.2500 | -4.1 | 1.2599 | -1.5 |

353.60 | 177147/131072 = 1.351524 | 1.3333 | -4.8 | 1.3348 | -4.4 |

372.52 | 729/512 = 1.423828 | 1.4063 | -4.6 | 1.4142 | -2.5 |

392.45 | 3/2 = 1.500000 | 1.5000 | 0 | 1.4983 | -0.4 |

397.80 | 1594323/1048576 = 1.520465 | ||||

419.08 | 6561/4096 = 1.601807 | 1.6000 | -0.47 | 1.5874 | -3.8 |

441.50 | 27/16 = 1.687500 | 1.6667 | 5.4 | 1.6818 | -1.5 |

447.52 | 14348907/8388608 = 1.710523 | ||||

471.47 | 59049/32768 = 1.802032 | 1.8000 | -0.5 | 1.7818 | -5.3 |

496.69 | 243/128 = 1.898438 | 1.8750 | -6.1 | 1.8878 | -2.8 |

Note that the "octave" for this scale, the eighth note of the scale, should be a fifth above one of these notes, and not the usual octave. The closest would be a frequency ratio of 2.027286, slightly larger than our normal octave. Various schemes have been introduced to try to "fix" the octave for such a scale.

Scales based at least in part on this procedure were introduced by Pythagoras (the Pythagorean Scale) and can also be found in Chinese history. The ratio 81/64 is known as the Pythagorean third, for example, which is quite high compared to many other tuning schemes.

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