9478

Semitones in Pythagorean Tuning and 12 Tone Equal Temperament

This Demonstration focuses on the aural and numerical differences between semitones in 12-ET (12-tone equal temperament) and Pythagorean tuning. 12-ET is the system of musical tuning that is used almost universally today, dividing an octave up into twelve equal semitones. Pythagorean tuning relies on the ratio of 3:2, representing the frequencies of two notes of a perfect fifth; however, the twelfth note generated by this system does not match up with the first, as a consequence of the different methods of computing tone frequencies. As the default settings show, 12 semitones in 12-ET produces a perfect octave; 12 semitones in Pythagorean tuning does not.
If the buttons are red, the frequency to be played is either unplayable or inaudible, and playing one note while another is still sounding can result in the notes not being played correctly.

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The ratio of frequencies used for a semitone in Pythagorean tuning is 256/243. This is the ratio of the frequencies between notes 3 and 4 and notes 7 and 8 of a major scale using Pythagorean tuning. It is obtained by going down five perfect fifths (so the ratio is ) and then multiplying by to get back to the original octave.
As the ratio of frequencies of two notes an octave apart is 2/1, and 12-ET splits the octave into 12 even semitones, the ratio used for a semitone is .
The formula used to calculate interval width in cents is as follows: the difference in cents is 1200 × , where and are the two frequencies.
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