There are two forms of minor third in intonalism, and in much great music.
The first exists in the harmonic series between the 5th and 6th partials or overtones. Above a C played on the viola open C string, you can hear the E, two octaves and a harmonic major third above, and, listening closely, the G above that. There is also a G in the harmonic series an octave lower, the 4th partial, so it is difficult to distinguish the two Gs sometimes.
Like the major third, the other minor third might be called the Pythagorean minor third. If someone knows a better name, tell me please! Pythagorean is hard to type. It may be constructed by a series of three perfect fifths: from a G, say, up to A and then up to E. (And down an octave: I’ll get to Major Sixths soon…)
Again like the major third, the two versions are approximately 22 cents apart. This distance is often called a ‘comma’ (don’t ask me why). http://en.wikipedia.org/wiki/Syntonic_comma has a discussion of the syntonic comma or comma of Didymus.
From a given note, the harmonic minor third is 16 cents sharper than an equal tempered minor third, and the pythagorean is 6 cents flatter than an equal tempered minor third. Again, the equal tempered compromise is between the two ‘natural’ thirds.
It is more difficult to hear the beats in minor third tuning, because they are very fast, but here is another audio clip with a recording of a harmonic minor third (introduced by the same viola C string, an octave and a fifth below the upper voice), then the Pythagorean minor third, and then an equal tempered minor third.
As opposed to the major third, where the Pythagorean third should never be heard in a harmonic combination of tones, the Pythagorean minor third is heard regularly, between the fifth and seventh in a dominant seventh (major-minor seventh) chord. It is a dissonant third, and thus the old rules for treatment of the seventh make perfect sense: it can’t be tuned correctly by ear, so the seventh has to be prepared, i.e., introduced earlier, in a way that its tuning can be established, before the dissonance is heard.
After the minor thirds, a natural minor triad and an equal tempered minor triad. Finally, two dominant seventh chords: to my ear, the first, tuned correctly, has the tendencies of the individual notes clearly distinguishable. The second, equal tempered, is just a blur of beats with the tendencies much more difficult to pick up.