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Diesis - major thirds versus octave

Diesis is the name of a comma, a very small interval. It is a measure of the difference between three just major thirds, each of the ratio 5:4, and an octave, the proportion 2:1. Mathematically, the three successive major thirds equals (5/4) to the power of 3 = 125/64 = 1.953, which falls somewhat short of 2, as it only gives us 97.66% of the octave value.
In musical terms, this corresponds to 41.06 cents, where an octave is 1200 cents, and an 12-TET semitone 100 cents. It is thus somewhat less than a quartertone.
Illustrated in the frame where the octave represents one rotation of a circle or spiral, and each 12-TET semitone equal 30 °, the just major third is 386.31 cents translates to 115.89°. Three of these interval angles is 347.68 °, so the difference between an octave and three just major thirds equals 360-347.68° = 12.32°.

Lesser Diesis appears directly as 125:128 in the octave spiral illustrating distribution of the partials in the harmonic series. In the octave 64: 128 (= 1:2), 64:80 (= 4:5) is the first just major third; 80:100 (= 4:5) is the next, and 100:125 (= 4:5) the third. 125:128 is the missing fraction, the comma between just thirds and the octave.

In ancient times until the Renaissance musical tuning theory was understood as a matter of proportions between integers – and was clearly related to geometry and arithmetics. Mathematically speaking, the Delian problem – doubling the cube, whose solution requires that one can derive the cube root of a value – is inextricably linked to Diesis, and there is no doubt whatsoever that in the time of Plato scholars were aware that the geometric problem had a musical counterpart in respect of the discrepancy between the just thirds and the perfect octave.

On modern keyboard the major thirds are tempered, so their relative frequency increase is not the value from the harmonic series 5/4 (= 1.25), but 1.2599 (cube root of 2).

The difference between the just major third and the modern equal temperament major third is almost 14 cents, or about 1/7 semitone. In the Renaissance musicians rejected the Pythagorean tuning because its major thirds differed 21 cents from the just intonation, so the harmonies based on thirds, sounded false. Our relationship to the experience of such an important interval as the major third is thus still problematic. We have become accustomed to impurity. Often musicians playing Renaissance music will tend to tune tothe so-called mean-tone temperament where the major thirds are just.

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