Raphi Giangiulio's Homemade Pipe Organ

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Temperament

The need for temperament can be explained by the fact that you can't fit twelve perfect fifths into seven octaves. The span of seven octaves is 2^7 = 128 but the span of twelve perfect fifths is 1.5^12 = 129.75. This extra amount is called the ditonic comma. Some or all of the fifths must be slightly detuned or tempered some fraction of the comma. For any temperament, all the fractions of comma must add up to negative one. A temperament is typically described by listing the fraction of comma added to or taken away from each fifth. An interval with zero comma would be a perfect fifth. The following is the equation to calculate the frequency ratio for the interval of a fifth given its amount of comma:

f (x) = (128 / 1.5^(12 + 1 / x))^(-1 * x)

For example, to calculate the interval of a fifth minus 1/4 comma, us -0.25 for the value of x. This gives a frequency ratio of 1.49493. Here's a table of frequency ratios for the different fractions of comma. Using this table, you can easily calculate the frequencies for any temperament.
Here is an example of how to calculate the frequencies for the Valotti temperament. The Valotti temperament has six perfect fifth intervals and six intervals with 1/6 comma. The perfect fifths are B-F#, F#-C#, C#-G#, G#-Eb, Eb-Bb and Bb-F. The 1/6 comma fifths are C-G, G-D, D-A, A-E, E-B and F-C. The frequency ratio for a perfect fifth is 1.5 and for a 1/6 comma is 1.49662. The following table shows the series of calculations. Several times there will be division by two to keep the frequencies in the same octave. Start on A = 440 Hz and use the frequency ratios to go around the circle of fifths.

Here is a table of some other temperaments:

Here are the corresponding frequencies all based on A = 440 Hz: