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viscount
User Manual
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. 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
. 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 between F# and C#.
A series of three natural major thirds makes a frequency of (5/4)
3
. 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
.
