# Semitones in Pythagorean Tuning and 12 Tone Equal Temperament

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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.

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Contributed by: David Harris (August 2012)

Open content licensed under CC BY-NC-SA

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## Details

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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