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Viscount Physis OrgansReference Manual
This is the Equal Temperament System (with the comma split in twelve parts).
pure: pure 5th
ETS12: Equal Tempered 5th
sc: synthonic comma
pc: pythagorean comma
pure: pure 5th
ETS12:
Equal Tempered 5th
sc:
synthonic comma
pc:
pythagorean comma
The picture is not in scale (otherwise the comma would be almost invisible) but it gives a graphical understanding
of the problem.
Now, dividing the Pythagorean comma in 12 equal parts, and subtracting this value to each pure fifth, the
result is a twelve-fifth chain, ending with the same value as seven octaves.
The Pythagorean comma can be split in larger parts and divided among a few (less than 12) fifths. This leads
to other temperaments, such as the Werckmeister III, which spreads the comma in four, equal parts, between
C-G, G-D, D-A, and B-F#.
A very important interval in the history of tuning is the major third. A chain of four pure fifths makes (3/
2)^4. A major third interval is represented by a 5/4 ratio. Therefore, rounding off, the frequency of a major
third, generated by Pythagorean fifths is 1,2656, while a natural major third is 1,2500. This surplus is called
Syntonic comma.
A series of four fifths, each one reduced by ¼ of Syntonic comma, makes a perfect major third. These fifths
are called “meantone”. If a temperament recovers only one syntonic comma, it still needs to compensate the
small difference between Pythagorean and syntonic comma. This difference is called skisma.
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