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There are two major thirds used in Intonalism, and in most great music.

One of the major thirds is tunable, the pure major third of the harmonic series.    If you play the open C string on a viola, and listen carefully, you can hear that third (two octaves and a third higher).  That is the fifth overtone or fifth harmonic in the harmonic series.   If you sing the third of a major triad in pure intonation it should be the same pitch as that harmonic.

The other major third, often called the Pythagorean third, is not directly tunable.  It is much sharper than the pure, tunable major third.  However, it is used in tonal modulations along the circle of fifths.   It is the result of a stack of pure fifths. From the same open C string on a viola, each successive string is tuned a pure fifth higher:  C – G – D – A on the viola.   A fifth above the viola’s A string is the violin E string, another pure fifth higher.  So the violin E string is two octaves and a Pythagorean major third above the viola C string.

If a portion of music were to modulate from C to G and on to D and A, and finally to E, it would be properly tuned to that violin E string, a Pythagorean third away.   If a different portion of music were to modulate from C by means of a pivot on the third scale step, it would be properly tuned to the harmonic third, and the violin open E string would sound badly out of tune.

The third on the organ or piano, in equal temperament, must fulfill both purposes, so as a reasonable compromise it is somewhere in between in tuning, rather closer to the Pythagorean third than to the Harmonic third.

The audio example gives all three over a viola open C string.

In relation to the equal tempered third, the Pythagorean third is approximately 8 cents sharper  and the Harmonic third is 14 cents flatter.   (A cent is a rather artificial measurement, meaning 1 / 100 of an equal tempered semitone).

The first, the harmonic third, is played lightly by a flute, and blends almost undetectably into the viola sound.