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Tartini Tones

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A recent thread brought up the subject of Tartini tones. Some people thought they were an "illusion" due to some feature of the ear or brain. Actually, it is a response to an actual physical effect that is occurring in the sound wave.

A Tartini tones can occur when two notes are played simultaneously. Due to periodic reinforcement and cancellation of the two sound waves with each other, additional pressure pulses may occur that the ear detects as additional tones.

If the two notes have frequencies F1 and F2, then tones may be detected that have frequencies F2-F1 and F2+F1.

If one generates two pure sine wave notes, say of 440Hz and 660Hz, then a very distinctive pulse can be seen with frequency 220Hz, the F2-F1 Tartini tone. As more overtones are added to the notes, it becomes increasingly difficult to distinguish the Tartini pulses from the note fundamentals and overtones.

To illustrate the effect on a violin, I captured the sound of double stopped open A (440Hz) and E (660Hz) strings. The wave form appears as follows:

tartiniwave.jpg.2a2c0e19ca33fc6d01f3126e40c08440.jpg

You can see two distinct wave form shapes.

The distance between similar shapes is the F2-F1 Tartini tone, or 220Hz.

The distance between successive peaks is note F1, or 440Hz.

The F2 note, 660Hz, is more difficult to pick out because of the periodic cancellation and reinforcement of the various overtones.

An FFT analysis of the wave form detects both Tartini tones.

tartiniFFT.jpg.64d0a365275cc58913d1a81bc48b56e9.jpg

The sound levels of the Tartini tones for the combination of complex sound patterns are much smaller than the note fundamentals and the first few overtones. This makes the Tartini tones difficult to detect unless the notes are being played very loudly.

A more dramatic illustration can be done by combining pure sine waves.

 

 

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Difference tones aren't an illusion. You hear them because of the nonlinear response of the auditory system. Or in your case the nonlinear response of your recording equipment (which also shows the sum tones and 2nd harmonic generation predicted by nonlinear response theory).

It's interesting that we only hear the difference tones and not the other nonlinear products predicted by theory and observed in measurements as shown above.

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In an old thread, some were arguing that they are only a psycho-acoustic phenomena and strictly a product produced in our mind, but in my experiences, these two or more frequencies mix, and can be picked up by a microphone and observed on an oscilloscope. If they didn't exist, music harmony would be a useless endeavor that we wouldn't be able to hear.

A related, if not intertwined topic is interference waves.

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People misunderstand the true meaning of "psychoacoustic". The word implies some kind of imagined or illusory phenomenon, but really it describes the response of our auditory system to external stimuli. These can, in some cases be illusory, but in other cases (like tartini tones) it's a completely measurable effect.

 

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Acoustics and Psychoacoustics by Howard and Angus defines psychoacoustics as the study of how humans perceive sound. It is a fascinating, albeit not easy, read and there is an explanation of difference tones in there.

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There is a psychoacoustic layer to all our experiences of sound.  But that doesn't mean there isn't a physical real aspect to many/most sound phenomena, and a math side also.

Some phenomena, like Tartini difference tones, require very pure, stable, and reasonable powerful tones to emerge.  For the Tartini tone to be 'in tune', the base tones should form a pure tempered interval.

 

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3 hours ago, David Beard said:

There is a psychoacoustic layer to all our experiences of sound.  But that doesn't mean there isn't a physical real aspect to many/most sound phenomena, and a math side also.

Some phenomena, like Tartini difference tones, require very pure, stable, and reasonable powerful tones to emerge.  For the Tartini tone to be 'in tune', the base tones should form a pure tempered interval.

A pure sine wave is the best because any byproduct tones stand out well.

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Yeah.  Using them on a real instrument isn't so easy.  Tartini's discovery and use of them is a testament to the purity and depth of his skill.  And the quality and setup of his Strad.

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I think the OP is referring to my post, suggesting that Tartini tones may be a psychoacoustic rather than an acoustic phenomenon, like "missing fundamental" tones. The case can be argued either way (also for the missing fundamental) and I suspect both factors are partially responsible.

However I certainly wouldn't suggest that either perception is an "illusion" of the "mind". As other commentators have said, psychoacoustics is the proper science of how air vibrations are translated into perceptions (sounds) by the neuronal processes of the inner ear and the brain. The perception of musical pitch is only to a small degree determined by the frequency spectrum: the frequency resolution provided by the cochlea is very approximate so individual hair cells may respond to sound frequencies over a frequency range of possibly an octave or more. How can this be translated into the sharply tuned pitch sense responsible for music? Regardless of the actual frequencies present, complex harmonic tones (and repeating noise patterns) have an envelope that repeats with the same regularity as the fundamental, whether or not the latter is actually present, and it is nerve impulses following this regularity with high precision that probably (I'd say "must") underly the sensation of musical pitch.

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Most of this Tartini tone effect is basically a beat frequency, where frequencies very close together give a tremolo effect.  If the frequencies are far enough apart, then the beat frequency gets up to where it is perceived as a separate note.  FFT shows two distinct frequencies, time history shows a modulated wave... but it's all the same thing, looked at (or heard) different ways.

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1 hour ago, Don Noon said:

Most of this Tartini tone effect is basically a beat frequency, where frequencies very close together give a tremolo effect.  If the frequencies are far enough apart, then the beat frequency gets up to where it is perceived as a separate note.  FFT shows two distinct frequencies, time history shows a modulated wave... but it's all the same thing, looked at (or heard) different ways.

Sorry, Don, that's not correct. The effect is due to the response of a non-linear system to 2 tones of different frequency. It's analogous to sum, difference and multiple harmonic generation in nonlinear optics.

As the OP's spectrum shows the "Tartini" tone shows up in a spectrum, a beat note doesn't.

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Additionally (I may be speculating above my pay grade here) the combined acoustic envelope of the two tones contains a third periodicity equal to the reciprocal of the frequency difference between the tones. Neuronal firing in the eighth nerve may phase-lock to any of these three periods in the envelope.

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I particularly recall hearing Menuhin play the Brahms concerto once (in 1956) with his Khevenhhuller Strad (I checked in the dressing room afterwards -- his case was open). You could hear those low, 'cello range Tartini tones even on single notes, from well back in the audience (!)

The only other fiddle I recall that ever did that was Stern's Ysaye Del Gesu.

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Ahh, Tartini tones.  Be cautious fiddling with this stuff, guys, the Devil's in the details.  [Goes back to practicing trills.]  :ph34r:;)

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On 4/20/2019 at 12:57 PM, JohnCockburn said:

Difference tones aren't an illusion. You hear them because of the nonlinear response of the auditory system. Or in your case the nonlinear response of your recording equipment

Are you saying if f1 and f2 were sine waves, frequencies other than f1 and f2 in the FFT plot would be harmonic distortion?   Actually...I remember that is right, and in fact if frequencies were in essence generating themselves through some kind of combining the plot would be useless.  Best I can remember you get the kind of periodicity seen in the first graph with any combination of sine waves, and only with sine waves.  And in the second plot, those tall peaks that aren't f1 or f2 are just harmonics, either distortion from the equipment or else harmonics that should be there.

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