By Hans Jørgen Jensen

Tartini Square in Piran, Slovenia

What are Tartini Tones?

Tartini Tones are an excellent tool for training the mind’s ear to perceive the beautiful intonation of chords and double stops. When two notes are played in tune by a single instrument or by two instruments, a third lower note can be heard. This third note is often referred to as either a Tartini tone or a difference tone. The frequency of this tone is the difference between the fundamental frequencies of the two notes being played.

The history of Tartini tones

The phenomenon described above was discovered in 1714 by the famous composer and music scientist Giuseppe Tartini (1692-1770). These Tartini tones or difference tones were also advocated by the famous violin pedagogue Leopold Mozart in his Treatise on the Fundamental Principle of Violin Playing as a tool to play in tune with a good strong tone. Leopold Mozart was also a strong advocate of just intonation like all performers in the baroque period. Tartini tones fell out of favor in the 20^th century due to the romantic style of playing string instruments and the abundant use of vibrato amongst many performers. 

In the last 15 years there has been a strong resurgence of interest in just intonation and Tartini tones due to string quartets, other chamber ensembles, and individual performers becoming very interested in earlier styles of music. In addition to their utilization on the modern concert stage, Tartini tones are a wonderful tool for helping string players improve their intonation and listening.

Theoretical Background

When the interval between two notes in a double stop is perfectly tuned using the just intonation system, the Tartini tones are clearly heard. It is also easier to hear the Tartini tones when playing double stops in a higher register because Tartini tones have lower frequencies. The frequency of a Tartini tone or difference tone is the difference between the fundamental frequency of the two notes being played. You can see that clearly in the following two diagrams.

VIOLIN VERSION OF TARTINI TONES

990Hz - 660Hz=330Hz

CELLO VERSION OF TARTINI TONES

330Hz - 220Hz = 110Hz

6 samples of Tartini Tones:

Different intervals create different Tartini tones due to the fact that the distance between the intervals changes depending on the notes that make up the double stop. 

Interval                   Tartini Tone

Major third                    Two octaves below the low note

Minor third                    One octave + one fifth below the high note

Perfect fourths              Two octaves below the high note

Perfect fifths                 One octave below the low note

Major sixth                    One fifth below the low note

Minor sixth                    Major sixth below the low note

Three ways of tuning double stops

Tuning intervals with the following three tuning systems is an excellent exercise in improving your knowledge and control of intonation and a great tool for finding the Tartini tones. The Tartini tones ring out loud and clear when the double stop is in tune using just intonation. Beats emanate mostly from the lowest common overtones that the two notes share. When the double stop is in tune according to just tuning, the two common overtones vibrate and swing together in complete unison.

In other words, when two notes are perfectly in tune in a consonant double stop, you will not hear any beats between the different pitches. However, if the double stops are tuned using equal temperament or Pythagorean tuning (meaning they are not perfectly in tune), beats will emanate from the double stops. The more dissonant the double stops, the faster the beat rate will be. The following double stops played here clearly illustrate that with Pythagorean and equal temperament the double stops don’t sound as pure and in tune as with just intonation.

Cents indicates the size of the interval

Cents are used here to illustrate the size of the various thirds. Knowing how big an interval is in cents is not important, but it is helpful to be able to hear the various intervals with our mind’s ear. A half step on a piano is 100 cents and a major sixth is 9 half steps, so an equal tuned major sixth is 900 cents. A Pythagorean major sixth is 906 cents—slightly larger—and a just major sixth is 884 cents, so quite a bit smaller. Remember that when an interval is in tune using just intonation the Tartini tone is very clear, and it sounds great!

VIOLIN

 

 

CELLO

 

Pythagorean tuned major sixth

Violin sample no 1A: Pythagorean tuned major sixth- 906 cents

For this Pythagorean tuned major sixth use the D natural on the A string one octave above the D string and for the B use the B that tunes with the harmonic B on the E string. This double stop will be out of tune and the two common overtones that the two pitches have in common will clash slightly against each other and beats will emanate from the double stops.

Cello sample no 1B: Pythagorean tuned major sixth- 906 cents

For this Pythagorean tuned major sixth use the G natural on the D string one octave above the G string and for the E use the E that tunes with the harmonic E on the E string. This double stop will be out of tune and the two common overtones that the two pitches have in common will class slightly against each other and beats will emanate from the double stops.

Equal tempered tuned major sixth

Violin sample no 2A: Equal tempered tuned major sixth- 900 cents

For this equal tempered tuned major sixth use the D natural on the A string one octave above the D string and for the B use the B that is slightly lower than the Pythagorean B natural. This double stop will be slightly out of tune and the two common overtones that the two pitches have in common will class slightly against each other and beats will emanate from the double stops.

Cello sample no 2B: Equal tempered tuned major sixth- 900 cents

For this equal tempered tuned major sixth use the G natural on the D string one octave above the G string and for the E use the E that is slightly lower than the Pythagorean E natural. This double stop will be slightly out of tune and the two common overtones that the two pitches have in common will class slightly against each other and beats will emanate from the double stops.

Just tuned major sixth

Violin Sample no 3A: Just tuned major sixth- 884 cents

When a double stop is played justly in tune, the two lowest overtones that the two notes have in common will line up and swing together in complete unison and the Tartini tone will be very loud. For this double stop use the D natural on the A string that tunes beautifully with the D string and for the B natural use the B that tunes justly with the D natural. That B is the same pitch as the harmonic B on the G string except it is two octaves higher.

Cello Sample no 3B: Just tuned major sixth- 884 cents

When a double stop is played justly in tune the two lowest overtones that the two notes have in common will line up and swing together in complete unison and the Tartini tone will be very loud. For this double stop use the G natural on the D string that tunes beautifully with the G string and for the E use the E that tunes justly with the G natural. That E is the same pitch as the harmonic E on the C string except it is two octaves higher.

How do we hear the difference tones?

For a long time, it was thought that the difference tones were only created inside the ear. But we now know that difference tones can be measured outside the ear. A study from 2010 notes the following:

“For the purpose of clarity, combination tones that are generated inside the ear will be called “intra-aural“, those that occur outside and independently of the ear “extra-aural“. Intra-aural combination tones (ICT) have their origin in the human ear, therefore they cannot be detected by a microphone. Accordingly, how intense intra-aural combination tones are perceived varies depending on the individual auditory capacities of the person. In contrast to the intra-aural combination tones, extra-aural combination tones (ECT) can be measured as vibrations in the air; they follow physical principles.”[1]

Conclusion

It is very helpful to use Tartini tones for tuning double stops. Musicians with great intonation hear Tartini tones intuitively. If the Tartini tones are not audible to you, don’t spend more than one or two minutes on listening for Tartini tones each day because the ability to hear the Tartini tones will get better over time if you repeat the short exercise over several days. Take your time when adjusting and tuning the pitches and enjoy the process of searching for these special tones! Happy practicing and tuning!

 

[1] ARCHIVES OF ACOUSTICS DOI: 10.2478/v10168-011-0049-1 36, 4, 727–740 (2011)

Combination Tones in Violins Angela LOHRI(1), Sandra CARRAL(2), Vasileios CHATZIIOANNOU(2)