## Musical scale based on fifths

One can create a musical scale based solely on the "fifth" and the octave.
First, pick a starting pitch, now go up a fifth (multiply the frequency
by 3/2), then go up another fifth and convert this back down an octave,
go up a fifth from that - if the result is beyond the octave, go back down
an octave.

Mathematically, starting with a pitch f_{0}, the next pitch is
f_{1} = 3f_{0}/2, and f_{2} = (3/2)f_{1}/2.
More generally, given the pitch f_{i}, then

f_{i+1} = (3/2) f_{i} if that result is less than 2 f_{0}

f_{i+1} = (3/4) f_{i} if the previous result was not less.

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_{0} = 261.63 Hz was used as an example and corresponds
to "middle C." Frequency differences (in Hz) are based on this f_{0}.

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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