Tartini’s third tones

  • During the Baroque era ,theoreticians tried to link up music to  acoustics.  Nevertheless ,natural origin of the minor chord were difficult to explain. Rameau suggested the lower harmonics (the  Fm chord resulting  from the mirror of C Maj) that would be in sympathy with upper harmonics according to the  theory of resonance but he failed since in rejected the  7th division and mixed up the origin of  the  string division :

« The major third appears between the fourth and  the fifth  division of a string. Since the minor third appears between the fifth and sixth division of the string, the minor third is of the same kind of the Major third. »  

The issue is  the fundamental sound which is different:

Fourth division yields C

Fifth division     «  »    E

Sixth  division «  »         G

so  CE =major third  root=C

EG=minor third  root=E
Tartini evolved another  theory from the differential or resultant tones:Two notes played in pure intonation yield a faint note lower .


Those terzo suoni (third tones) or ghost tones are useful for intonation in double stops but unfortunately failed to explain the natural origin of minor chord since the minor third produces a major third (17 th) below.
Remark: interval from the bottom note of the double stops are similar to harmonics


Beware: for both Rameau and Tartini. »Natural » sometimes means  « physical »  based on resonance  and sometimes   purely mathematical .



I am a bit doubtful about the   Tartini’ s third tone, because nobody  seems talking about the same thing.

The physical phenomena used  for lower sound in organ or in otoacoustic emission study requires pure sounds which don’t fit with a violin (futhermore a Stradivarius )

The issue is that some bearings of the resultant tone,additive tone for example, are   posterior to Tartini,so we don’t know any longer what Tartini’s third tone is exactly.

I have read some excerpt of his treatise on  (differential) resultant tones that was obviously based upon  a mathematics view. I think this topics was only  supposed to prove the origin of the minor third  while his description of intonation technique,although related to resultant tone, was  from another  point of view.

Two simultaneous near frequencies yield beats which slow down then disappear  when frequencies fuse but speed up and swell when they stray until another pure tone appears (the third tone) when both sounds frequencies tally  with the  harmonics of a fundamental sound.

Later on  otoacoustic emission  was discovered: inner ear cells produce a rather loud sound in respond to two pure sounds with a similar intensity. This compound  sound consists of manifold sounds called « Distortion products » which frequencies are multiple of both injected sounds frequencies. D.P intensity is changeable  accord to their frequency the loudest intensity appears  when the ratio  F2/F1=1,22 (near the fifth value)

Neglecting his mathematical approach, we have a better understanding of intonation  Tartini’s  technique: a tuning with a inferior limit through beats disappearance  and a superior limit through a sound reinforcement .