Viscount Cantorum Duo Mode D'emploi page 96

User Manual
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.
This is the Equal Temperament System (with the comma split in twelve parts).
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.
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A chain of four pure fifths makes (3/2) . 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. For example, the Kirnberger II temperament is
based on the syntonic comma. It spreads the comma equally between D-A and A-E and the skisma
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between F# and C#. A series of three natural major thirds makes a frequency of (5/4) . The difference
between that and the octave is called enharmonic comma. When a temperament makes up for more than
the Pythagorean comma, usually one fifth is much wider than the rest and becomes unusable. It is called
the wolf fifth.
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