Giovanni Cecchi Study of Combination Tones

November 1, 2022, 1:29 PM · Has anyone else seen this yet?

If the existence of combination tones can indeed be proven, that ought to be significant for luthiers and researchers alike. I would, however, be interested to see a follow up experiment that included some old violins that didn’t sound so good and some examples of the best-sounding or most highly regarded violins by current makers in addition to good old violins and bad new ones.

Replies (40)

November 2, 2022, 7:52 AM · If you ADD together the waveforms of two tones, this COMBINED waveform can be represented by two other tones that MULTIPLY each other, essentially modifying each other's amplitude.

The frequencies of these combined tones are proportional the SUM and the DIFFERENCE of the frequencies of the original tones.

If these combined frequencies are in a certain frequency range, typically 50hz +/-, the ear can discern the amplitude changes each is causing the other. The frequency difference is the one that usually predominates.

Let's call this effect the Tartini Tones after the famous violinist who seems to have been the first to talk about it.

The effect can sometimes be heard when loudly double-stopping fifths as a soft, pulsing tone with a frequency proportional to the difference in the frequencies of the original tones.

It is important to consider the following before one gets distracted with thinking the report has discovered something magical...

When playing a single note on the violin, the only tones being sounded are the fundamental and the overtones. With the possible exception of wolfy notes, all these tones are locked in-phase and separated from each other by at least the frequency of the fundamental. With the possible exception of the lowest notes on the G string, the combined tones would have frequencies beyond what would trigger the ear to perceive the pulsing.

November 2, 2022, 10:09 AM · "IF the existance..." I can assume that the difference and sum tones, the combination tones of double stops are real, and not a mental illusion, because I can hear them, and, if you run the two signals through an oscilloscope you can see them. In a broad philosophical sense, the universe is more likely to be real, and not an illusion or a computer simulation, because through our five senses, physical phenomenon obey laws, are predictable and consistent.
November 2, 2022, 10:21 AM · I saw the article, but it leads to a question: ok, so a third tone is made when playing two notes together—a double stop. But we don’t play double stops exclusively. How does it explain good tone from single notes?
Edited: November 2, 2022, 10:37 AM · @Joel - Our senses only give a us a partial and biased account of reality. In vision we're accustomed to accepting certain well-known phenomena as illusory. In audition I believe there's a greater tendency to assume that everything we hear is "out there" but in fact all the perceptual analysis and re-synthesis of sound objects is done "in here". Our hearing does a lot better job of deciphering sound mixtures than eyeballing an oscilloscope, but it's still vulnerable to deception.
November 3, 2022, 2:07 AM · I think the author’s theory is that the increased presence of Tartini Tones in a violin correlates to greater air resonance in the body, so a more resonant violin will sound better on individual notes as well, not just when double stops are played.

From the way it’s presented, it comes across as though acoustic researchers have discounted the idea of phenomena like this as too minuscule to have any real bearing on the overall tone. If it can be demonstrated that the impact is actually quite clear, that would give more weight to claims players have been making since Tartini discovered the phenomenon.

Edited: November 3, 2022, 5:51 AM · The sum and difference frequencies are produced in non-linear mixers, NOT in linear mixers. Your ears are non-linear mixers and "tartini tones" are produced in your ears.
But I suppose it raises the question - is a violin a linear mixer or a non-linear mixer? I guess non-linear. Semiconductors are non-linear - e.g. ring modulators use diodes which are semiconductors. An oscilloscope's electronics will be non-linear to an extent and create tartini tones but at different levels from the ones created by your ears, since the ideal oscilloscope would be linear, in the same way that the ideal amplifier would be distortion-free.
If you play a sine wave at 110Hz (A) and another at 440 Hz (concert A), you will hear both and also the sum, 550Hz and the difference, 330Hz, whatever those notes are. When you play those two As on a violin or cello, the waveforms contain many audible overtones of greatly varying amplitudes, and every fundamental and overtone will create sum and difference frequencies (at greatly varying amplitudes) with every other fundamental and overtone.
Edited: November 3, 2022, 12:28 PM · I might have said this on this forum before, but our senses are very interesting. While studying electronics 40+ years ago, we were shown the maths of it - from the dimmest stars in the sky we can prove that our eyes are capable of detecting light at a rate of 8 photons per second, which is the same as persistence of vision, so our eyes are just able to detect one photon of light.
Also we can prove that our ears (when we are young and haven't been to too many Slipknot gigs, or haven't sat in front of too many brass sections or piccolo players) can just about hear air molecules jostling each other. In other words, our senses are able to detect the lowest possible information level. If audio is quieter than jostling air molecules, it is masked by them, so there would be no point in being able to detect it.

BUT there is not inconsiderable distortion created by our eyes and ears, and it is compensated for by our brains.

November 3, 2022, 10:54 AM · Can the 'Tartini-metric, hey, "Tat-met"' be far behind? A measurable quantity to boost the provenance - and value of violins? Will there be a stampede (OK, a legato walk) to make violins with bigger and better Tartini-metrics???

'Your Strad is only mediocre with a 3.7 Tat-met, whereas mine is all of 3.82 !'

November 3, 2022, 11:13 AM · Shhhh, Elise. Auction houses might be reading this!

If luthiers often build aiming at certain air resonances, is this actually anything that would change what they do, or is this merely descriptive?

Edited: November 3, 2022, 11:39 AM · Don't worry, the last thing they want the buyers to learn is that the violin is lemon. You can sell lemonade by marketing its fruit, not using a scale of its acid ;) [Auctions work almost entirely by provenance - they are mostly scared of sound .... ]
November 3, 2022, 12:52 PM · Non-linearity?
Not just the ears themselves, but microphones, amplifiers and speakers, which is why these tone can be seen on a screen.
From a musical angle, they seem to treat the simultaneous notes as harmonics of missing fundamentals, which is why we seem to hear low sounds on a speaker too small to effectively radiate them.
November 3, 2022, 1:20 PM · Gordon,

Your description of the non-linear mixer is the explanation I’ve always heard for combination tones. The “extra” tones we hear are considered subjective, the result of processing in the inner ear.

What intrigued me about this study was the claim that there are “objective combination tones generated by some instruments.” That claim is the basis for Cecchi’s measurement practices, which involve recording with microphones or other equipment to potentially make objective assessments.

It remains to be seen if this holds up to scrutiny. I hope there will be more information forthcoming.

Edited: November 3, 2022, 2:01 PM · I have long "believed" that combination tones have a lot to do with appreciation of one violin over another, and have spent tons of time studying it.
While some will claim that these tones exist only in the ear, or in the brain, sufficiently expanding the wave form of a recording and comparing it against the timeline will show that these patterns of air pressure differences (sound) quite clearly exist, before they ever get to the ear.

Carmen Tanzio, when playing a single note on the violin, the only tones being sounded are NOT just the fundamental and the overtones of the played note. There are also "drone" frequencies produced by the resonances of the violin parts and air spaces, similar to those produced by an impact hammer. One could think of the string vibration input into the violin body as a series of tiny impacts.

November 3, 2022, 9:53 PM · David, a quick glance at a frequency vs sound intensity (dB) plot of a single note will show considerable "noise" and small peaks MANY decibels below the intensity of the overtone and its fundamentals.

Most of this stuff is related to the effects of friction like the slip-stick and scratching of the bow hairs across a string, certain movements of the tailpiece parts, and non-transverse movement of the strings (compression along the string and twisting). This can sometimes excite resonances that are not closely coupled to the main movement of the string-bridge-body sound train.

But what all these secondary effects have in common is that they are virtually inaudible beyond a short distance from the player. And any combination tones that might be generated in the ear are well beyond what the ear would notice from any research I have seen.

In terms of the classic Tartini Tones, yes, there are actual air pressure patterns occurring with the frequency of a combined tone. I have also seen some research that seemed to prove that hearing these combined tones is more than just a creation of the ear and brain. But beyond a certain narrow frequency range, these patterns become unnoticeable.

And noticing the effect still requires that the two interacting tones both be of significant amplitude to each other. So frictional hiss and any droning it might incite seem unlikely to trigger the effect with the fundamental or overtones of a played note.

The ear does appear to be sensitive to rapid changes in air pressure over short time periods. This seems to be the main driver to quality of tone judgments people make.

For the complex pressure wave generated by a violin, I think this is more a function of the relative amplitudes of the fundamental and overtones to each other, and phase shifts of the various overtones due to them not aligning to the frequencies of the natural modes of vibration of the violin.

It is possible to come close to synthesizing an actual violin tone and wave shape by extracting the relative amplitudes and phases of the overtones, combining them mathematically using sine functions, then feeding the digital results into a digital synthesizer.

November 4, 2022, 9:41 AM · David Burgess,

I’m intrigued by your comment! If you’re willing to share anything about your experiments, I’d love to hear it.

The question to which I think this (and other acoustic research as well) leads, is: if this system allows us to measure existing instruments to objectively identify important characteristics, how can new makers use this in their own making to reproduce the results?

Edited: November 4, 2022, 5:04 PM · Carmen, thank you for your further comments.

Even if the "tartini tones" or combination tones due to underlying resonance combinations within the instrument itself may not be strong enough to be perceived as separate notes by many or most people, I don't think that is what matters the most in their contribution to violin tone quality assessment. I remain conviced that they are strong enough to color the sound in various ways.

To most people, even a synthesized violin sound resembles the sound of a real violin decently enough. Then you've also got the snobs (highly experienced people) who can identify the difference in a heartbeat. (Well, maybe five heartbeats) LOL
Even when every possible note on a particular violin has been sampled, copied and reproduced, the bow pressure and speed variations as well as all the various attack and decay rates, as well as the vibrato variations, have not yet been reproduced to the extent that they can resemble a real human player, to the best of my knowledge and experience.

November 4, 2022, 6:08 PM · I haven't studied this, but this is the first I've heard of an objective difference tone out of the violin (something that can be measured by precise linear equipment). Something must be nonlinear somewhere.

As for this being a measure of goodness of a violin, I seriously doubt it. Unless I missed something, the difference tone is primarily due to the fundamentals of the notes played, and mostly in the lower register. Old violins tend to be thinner and more compliant, producing a strong fundamental tone. The referenced paper only used 5 violins, which is not much. “It wouldn’t be difficult to find high-quality contemporary violins that would show this effect just as strongly."... Jim Woodhouse.

David... the "series of tiny impacts" is right, and it's true that an impact will excite all of the violin's body resonances. However, each tiny impact is indeed tiny, and only those resonances that are multiples of the impact periodicity will build up any appreciable amplitude. The rest will be tiny exitations that are out of phase with each other on each "impact". Wolf notes are something else entirely.

Edited: November 4, 2022, 9:09 PM · I have now read a whole thread about difference tones and nobody mentioned why Tartini "talked about them" as Carmen notes.

He pointed them out as intonation help for double stops: If you play a third (the interval at which I can hear them best) and you don't hear a Tartini tone your third is lout of tune. If the Tartini tone is not consonant with the two notes of your third you are still not quite in tune.

It helps a great deal to have your ear out for those tones, especially for thirds and sixths.

The physics behind combination tones is explained here:

I can imagine that the ability of an instrument to resonate with combination tones is somehow related to its quality but I am almost certain it would not be an easily applicable quality measure, useful for dealers and buyers.

Edited: November 5, 2022, 11:18 AM · In other fields, these are called heterodyne frequencies. When I was flying remote control aircraft years ago, there were certain transmitter frequencies which we were not supposed to use at the same time, because the combination of the two would produce a third frequency, risking loss of control of another aircraft, which not only could result in crashing the aircraft, but also risk the safety of operators and bystanders.

Nowadays, most of the better transmitter/receiver combinations for remote-control aircraft use additional signal coding, making each transmitter/receiver combination more unique, and much less likely to interfere with each other.

In some other applications, heterodyning is used deliberately to combine frequencies which would otherwise be outside of the human perception range, to produce "tarnini" frequencies which are within the human aural or visual perception range.

My examples are over-simplified, so here's more on that:

November 5, 2022, 12:11 PM · If we define secondary tones as anything other than the fundamental or overtones of the note being played, my personal experience is that, yes, these can have a dramatic effect on the player's appreciation of the violin.

Anecdotally, I can think of two examples.

When I first started playing, I developed a rash on my eyelids. It turns out I have a skin allergy to most rosins. I eventually found a hypoallergenic rosin that I could use but it had a very annoying effect: the scratching sound of the bow hairs on the string became very noticeable.

Other players noticed the same when they played my violin, but we could not notice the scratching beyond a couple of feet from the violin. So it was definitely an under-the-ear effect only.

After a while, I realized I could control the timbre of the scratching with bow pressure and speed and could use it to tell when I hit that sweet spot between the glassy sound of the bow skating over the strings, and the crushing sound of too much pressure.

At times I still find it annoying, but I have learned to live with it.

The second example occurs when I switch violin strings. Some brands will emit a shrill, squeek-like sound during a bow change with a quick attack. This is also noticeable by listeners more than a couple of feet away and I can control it by easing into a bow change.

I can see where a player trying a new violin might think it is an unwanted feature of the violin and not realize it is a string issue. I add that string brand to my list of strings not to buy again. But it I wonder if it could also be related to setup, rosin or bow tension issues.

November 5, 2022, 6:35 PM · Carmen, your examples are of high frequency noise that has nothing to do with the notes being (or attempted to be) played, so I wouldn't call them a "tone" at all, and not secondary. Additional sounds, perhaps.

The "difference tones" under discussion are at a frequency lower than the notes being played. Personally, I think this is mostly academic, and not very relevant to the violin, violin quality or music played on it. Overtones are infinitely more important.

November 6, 2022, 12:17 PM · Don, aren't the overtone structures highly dependent on the underlying resonating properties of an instrument (or its parts)?
November 6, 2022, 12:29 PM · Sure, overtone charcteristics are extremely dependent on how the wooden box vibrates, and that is very much how "good" and "bad" are determined.

The difference tones, as I understand them, are only perceptable as a combination of the fundamental note(s), and I don't see much connection to quality of the instrument directly. How often are these precise double-stops used, anyway?

November 6, 2022, 2:21 PM · Indeed getting thirds and sixths in tune with combination tones means pure, un-tempered intervals, which are lovely, but tricky to integrate with tempered accompaniments. For example a Pythagorean third can be shrunk to a pure third from the top, from the bottom, or symmetrically..
November 6, 2022, 5:47 PM · Don, I'm not talking about fingered double stops. I'm talking about double stops built into the resonances of the instrument itself.
November 6, 2022, 6:43 PM · David, presumably you mean body resonances that are tuned in some way to each other. I think your mental model differs from how it really works. We can talk about this at VSA.
November 7, 2022, 9:51 AM · David, I am trying not to get too technical since this is probably not the forum for detailed discussion about the physics of vibration. But I might be able to briefly summarize what is happening.

Pick a mode, any natural mode of a violin. Say A0, the air mode. For some hypothetical violin it might have a frequency of 280Hz.

Now play an open G, say 196Hz (using my Even Temperament piano frequency reference).

The string will oscillate at 196Hz, 2x196=392Hz, 3x196=588Hz etc. Moreover, all these oscillations are locked in phase with each other because of the restraints imposed by the nut and bridge.

When these vibrations reach the sound cavity, something dramatically different occurs. First, the cavity vibrates AT THE STRING FREQUENCIES, not its natural frequency. The intensity of the vibration decreases the further the string's frequencies are from the cavity's natural frequency.

For the fundamental of the open G and its first overtone, the air cavity will reproduce the sound at roughly the same intensity because they are about the same "distance" apart from the natural frequency of A0. All the other overtones of the open G will also drive the air cavity at the overtone frequencies, but with decreasing intensity.

But there is another effect that changes the phase of each overtone that is reproduced by the air cavity. Where the fundamental and the overtones of the string are locked in-phase, the fundamental and the overtones reproduced by the air cavity will be phase shifted by an amount dependent on the difference between the natural frequency of the air cavity and the frequency of each overtone.

The resulting pressure wave produced by the air cavity becomes a complex sound because of the changes in intensity and phase of the fundamental and overtones from each other.

Now redo this calculation for each natural mode of the violin. They are all vibrating at the frequencies of the fundamental and the overtones, but with wildly different intensities and phases.

November 7, 2022, 10:09 AM · Carmen, refer back to what I was saying about the string impulses acting somewhat like tiny impact hammer pulses. In that case, they will excite more resonances than just those of the string fundamental and harmonics.
Edited: November 7, 2022, 10:50 AM · It might help to think of someone sitting on a swing, and you give them a short push every time a metronome clicks. If the metronome and the swing natural frequency are close, you can get them moving well after a while. If the frequencies are far apart, you will be pushing at the wrong time to build up any amplitude... and after things stabilze, the swinger will only be getting a small swing, and it will be at the frequency you are pushing... not the natural frequency of the swing.

If you want to carry this thought out to more complexity, imagine that there are LOTS of swings, all with different natural frequencies. If you drive them all with the same little push at the same metronome rate, the swings closest to 1X, 2X, 3X (etc) the metronome rate will get the highest amplitudes, and the swings without that alignment won't get much. But even the unaligned swings will be moving at some multiple of the push frequency, not their own frequency.

November 7, 2022, 4:20 PM · I know it can be difficult to wrap one's head around an impact consisting primarily of a fundamental frequency and overtones that are each an integer multiple of the fundamental.

If one looks at slow motion videos of a vibrating violin string, it does look like a sharp impact-like wave traveling along the string.

It seems miraculous that one could add up a series of continuous sine waves and end up with a shape that looks like a sharp impact pulse. But because of the way the string deforms, and the displacement conditions imposed at the nut and bridge, that is what happens.

Certainly, because the string, nut and bridge are not "perfect" in a theoretical sense, you will get some periodic excitations that are not the fundamental or integer overtones. But the intensity is so much smaller than the fundamental and overtones that is seems unlikely anyone could actually hear those effects.

Still, I am willing to keep an open mind if someone is able to demonstrate an observable effect.

November 8, 2022, 1:27 PM · Carmen wrote:
"Certainly, because the string, nut and bridge are not "perfect" in a theoretical sense, you will get some periodic excitations that are not the fundamental or integer overtones. But the intensity is so much smaller than the fundamental and overtones that is seems unlikely anyone could actually hear those effects."

Most violins sound pretty much like a violin, don't they, rather easily identifiable as a violin?
In my opinion, understanding the small differences between an OK violin and a great violin will need investigation going beyond your chosen envelope.

November 8, 2022, 5:03 PM · David wrote: "In my opinion, understanding the small differences between an OK violin and a great violin will need investigation going beyond your chosen envelope."

In my opinion, the not-so-small acoustic differences between an OK and great violin have been established for quite some time. The difficult part is creating a new violin that actually does that great acoustic thing... and it's not hiding in the difference tones.

November 9, 2022, 6:58 AM · In my experience, there is next-to-nothing on a violin that can be changed, without having SOME effect on sound and playing properties.
Edited: November 9, 2022, 7:58 AM · You can tune a Malibu forever, and it still won't behave like a supercharged Corvette.

Changing something on a violin can change the way energy gets from the string to the body, or modify the body vibrating character to some degree. But I thought we were talking about the difference between OK and Great instruments, which IMO is more than adjustments.

Edited: November 9, 2022, 9:46 AM · Did I restrict this to adjustments? If you were under the impression that I did, what about adjustments to the thicknesses or wood properties?

By the way, I have never owned a Malibu, but I once owned a Pontiac Fiero. Pontiac Fieros were not known to be very fast, but mine sure as heck was!
Lots of tweaks one can do to a Malibu too.

November 9, 2022, 10:09 AM · I have to admit my bias. In the engineering world that I came from, we were mostly trying to suppress vibrations, not enhance them.

The clear advantage I had over the violin maker is that I could quantify the performance goal. But as many discussions on MN has revealed, where are the quantifiable performance goals that make a great violin?

The only wisdom I can offer is what we found could be manipulated to influence vibrations. To the extent that I was successful in manipulating structural shapes and material properties to control vibration effects, I can only offer those things I found to be of significance.

There are also two rules we used to help focus our efforts:

1. If you cannot measure it, it does not exist.

2. If you can measure it, and it does not rise about the order of magnitude of noise, ignore it.

November 9, 2022, 10:17 AM · Carmen...duh! Volume!
November 9, 2022, 10:32 AM · Carmen, I can accept no.2, but not no.1!
If we can hear something (repeatedly) but not measure it, surely it is the measurement method which is deficient?
November 9, 2022, 11:17 AM · Christian, I stand corrected! >grin<

Adrian, if you can hear a sound, you can measure a variety of things about it.

I think the discussion is about how to measure the sound quality of a great violin. And that has to start with a consensus on what is a great sound.

If Christian is correct, then it isn't difficult to build a great violin. In fact, I bet I can redo a VSO to be quantifiably great. But it does appear that many great violins can pump out the dBs.

But returning to the car analogies, would a violin that can be played beautifully in a chamber setting not be considered great because it cannot speak over a full orchestra?

And to flip that question, I once heard a Strad played in a chamber setting that was so loud and screechy that it destroyed any harmony that was built into the music for the other instruments. But I would not be surprised if it sounded fantastic in a concert hall with a full orchestra.

November 9, 2022, 11:57 AM · Carmen, interestingly enough, one of the major researchers in violin and guitar acoustics is a now-retired Boeing aircraft vibration analyst. He has said something similar... that while the final objectives are different, and might even be opposites, vibration is vibration.

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