Are Tartini tones physically real?

April 25, 2017 at 08:46 PM · I just found out about difference tones (Tartini tones) from a book by Simon Fisher. I am interested in understanding the physics. I have done some experiments in Audacity that only confuse me. Here are some of my results:

1) The Tartini tone is audible when the two tones are played through the same speaker but not when the two tones are played through different speakers; however, I could hear the Tartini tone on a recording of the two tones played through different speakers! How can that be explained?

2) I can't remove the Tartini tone by equalization that would remove a real tone at that frequency, nor will any sine wave at the Tartini tone frequency cancel the Tartini tone.

3) I can't find the Tartini tone on a spectral analysis.

4) When I play two open strings together on my violin, I can't hear the Tartini tone, but if I wear an over-the-ears hearing protector and bring the protector into contact with the violin, I immediately hear a prominent Tartini tone.

These observations lead me to think that the Tartini tone is purely perceptual, with no physical reality, in some way akin to the phenomenon of the missing fundamental. I'm also very curious why the hearing protector brings it out.

Replies (75)

April 25, 2017 at 08:50 PM · I was always under the impression that they were some sort of auditory illusion and not really present, but I've heard other say the opposite.

April 25, 2017 at 08:55 PM · I can hear an additional tone when playing a G sharp and B natural double stop on the A and E strings respectively. Is this the type of tone you are referring to?

April 25, 2017 at 10:05 PM · They are not present in the air in the room, unless they are coming from a less than perfect amplifier from a less than perfect microphone or recording. They are present in our eardrums, though.

The optional "bass boost" on my computer monitors use this effect to recreate a low(ish) bass from a 4-inch speaker which is physically incapable of delivering real bass at an audible level. It's a horrid mess.

April 25, 2017 at 10:20 PM · If you play double stops on the violin with eardrums on, the tartini notes will become much more apparent.

April 26, 2017 at 12:07 AM · They exist, and are the ears completing the bottom note that resukts from two notes that are perfectly intune.

It is some of this phenomenon that allows the ears to synthesis the violin notes below the first C # on the G string, since the violin body is too small to support them (which is why they sound 'hollow').

Makes me wonder if Carleen Hutchins had the right idea with the alto violin, which is slighly bigger, and on recording seems to project better and have a more singing tone and low end (an overall rounder and smoother, velvety sound) than a regular violin.

April 26, 2017 at 12:40 AM · Erik, what do you mean to say here: "If you play double stops on the violin with eardrums on" ?

April 26, 2017 at 12:40 AM ·

April 26, 2017 at 01:09 AM · earplugs on?

With no eardrums, you would hear nada.

April 26, 2017 at 02:04 AM · What I have read is that two sine waves will also generate a difference tone and an additive tone. I hear the difference tone when playing a double-stop third high on the E and A string as a rather unpleasant buzz. The other time I hear it is when tuning perfect fifths. If it is slightly out of tune you hear a difference tone as a slow beat frequency. Piano tuners use the beat frequency to get the fifths 2 % short. In electronics two sine waves fed into an oscilloscope will give a visual combination wave-form.

April 26, 2017 at 08:33 AM · Doppler sonography works exactly like this.

April 26, 2017 at 10:14 AM · No, Tartini notes aren't "real", in that there isn't an actual sound wave in the air at that frequency. They're a trick of your mind and ears, analogous to an optical illusion.

April 26, 2017 at 10:46 AM · You've asked a tough question, Jeffrey, and the answer will partly depend on how you want to define "real". LOL

My current thinking is that they are real (meaning that they actually exist as pressure pulses in the air which conveys the sound from the source to the ear), because these pressure pulses at the Tartini frequency are picked up by a microphone, and can be seen by zooming in far enough on a recorded sound file.

Why don't they show up on the visual display of a spectrum analyzer? Perhaps they can, but they haven't on any of the spectrum analysis programs I have used. My guess is that something about the processing used fails to pass them through to the visual display, or filters them out, or breaks the sound down into the frequencies used to produce the Tarnini tone, rather than displaying the combination tone. If I were a mathematician, perhaps I'd be able to go through the calculations used and figure out what's going on.

Joel, I've never tried to observe these combination tones on a tube-type oscilloscope (which may be a more direct display of sound pressure and involve a lot less processing), so I can't say whether they show up there or not.

April 26, 2017 at 10:54 AM · Of course they exist, but not in a way of beeing a own frequency. During doppler sonography a sound wave gets reflected by particles in the blood. Because of the velocity the reflected sound wave has a different frequanzy (doppler effect).

Now there is a interferency between the two sound waves, the original one and the reflected one.

This causes the buzzing or tartini tone which is the easiest way to indicate the flowing velocity.

Every physicist calculates this somewhere at the beginning of university.

Of course a FT (frequency analysis) will not show this frequency, because it is analyzing down to cos/sin signals.

What you see is a frequency that changes its amplitude within time. The amplitude frequency is what you see.

Something like A cos (w1 t) sin (w2 t) with w1 beeing the tartini frequency which is way lower than w2.

April 26, 2017 at 11:16 AM · Mathematicaly it is hidden in the euler trigonometrie

Sin (a) + sin (b) = 2 cos (.5a-.5b) sin (.5a+.5b)

A and b are the original playing frequencys, abs (1/2*(a-b)) is the tartini frequency.

Frequency analysis of 2 cos (.5a-.5b) sin (.5a+.5b) will result in a,b. TThe tartini f will not be analyst,still be there in the signal,just not as "pure" frequeny

April 26, 2017 at 11:37 AM · So the processing used wants to break the Tartini tone down into its component frequencies, rather than recognizing the pressure pulses produced (which we hear and which can also be measured) when the two notes are combined? The absence of the Tartini tones on an FFT display is just an artifact of the way the processing is done?

April 26, 2017 at 11:48 AM · Basically yes. Frequency analysis is usually some form of fourier transformation. Very unscientific speaking it is rebuilding the signal by a sum like

A1*cos (phi1+tau1)+....+An*cos(phin+taun)

Now lookig at phi and a gives the freauencys and its impotants.

This is a bit of an oversimplification though, but what you see is we get sums of simple cos (tau can make it to sin) and no products of different trigonometric functions.

April 26, 2017 at 12:02 PM · Can an FFT be written to include this, or does something like this already exist?

It seems like without this included, we might be missing an important element when we try to visualize the sound a violin is actually producing.

April 26, 2017 at 12:23 PM · Well, that would not be a FFT anymore. I can think of an extra analysis that can visualize this. The information is also hidden in the FFT, but it is not directly visible.

If you are really intetested I can put some thinking into it, before writing any analysis I think we should put some extra thinking into what exactly the goal is. A lot comes into my mind right now.

Actually this would be a nice addition to my current projects.

April 26, 2017 at 01:33 PM · Thanks, I guess I'd need to talk to some of the people more heavily involved in violin acoustic research to know if I (or we) would be really interested, or interested enough to put time and money into it. I think most of them are currently using Spectra Plus. It would be cool if an existing program could be modified to add that feature...

Is such a thing possible?

April 26, 2017 at 02:13 PM · If the software is open source or has some plugin packag it should not be to hard. I got to say I am not used to write software with front ends I usually write software for big clusters.

I dont know the existing software but will have a look on it.

This would be part of researches, time is involved but not money.

April 26, 2017 at 02:36 PM · Ok, this is propriatary software, there is no way to add functions to the core.

There might already be tools to find something in it, but I will not purchase it.

What would be easy for the beginning:

I could do some one time analysis to an example recording and you or some others take a look if this is interesting enough to find a solution you can work with or if it is just useless information.

I am absolutly willing to contribute some time into it (altough I would also agree to get one of your violins ;) I played a I think 2001 modell once and it was just beautifull. Wonderfull piece of art!)

April 26, 2017 at 03:23 PM · This is making more sense to me now. Good discussion! If I hear beats when two pure tones are played together, there is a periodic amplitude peak and trough due to interference, which is visible on the waveform. But when I run the spectral analysis, I get only the two pure frequencies. So the spectral analysis does not capture the interference. Yet the interference is physically real. David, would you let us know if you learn something further?

April 26, 2017 at 03:29 PM · Yep, this way it is right.

April 26, 2017 at 03:31 PM · Tartini tones appear to be a psycho-acoustic property of the ear and sound processing of the brain. The tone itself cannot be observed in the pressure pulses that reach the ear except in very special cases where the tone differences are carefully selected to also represent the frequency of the Tartini tone.

Here is the general case: suppose two pure tones, f1 and f2, are separated by a difference, d. (f2-f1 = d)

A spectral response analysis of the combined tones will show two prominent tones, f1 and f2, and a bunch of weak pressure pulses occurring at multiple intervals of +/- d from f1, or f2 since they are separated by the same interval.

To give a specific example, if I choose f1=440 (A4) and f2=636 (~D#5), then the Tartini tone I hear is f2-f1=196 (G3).

But no pressure pulse occurs with this frequency. You would see weak periodic pulses at 244 (440-196), 832 (440+2x196), 1028 (440+3x196), etc.

In special cases, like two tones a perfect 5th apart, the Tartini tone will coincidently be a perfect octave below the lowest tone which can help in tuning via double stops.

I find the effect to be very subtle and takes some real concentration for my conscious mind to recognize the Tartini tone being generated by my ears.

April 26, 2017 at 04:34 PM · Carmen, I disagree. A microphone is basically an instrument which picks up variations in air pressure, loosely like the human ear, and recorded sound files clearly exhibit the pressure variations at the frequency of Tartini tones.

I can't say at this point whether the "phantom" lowest G fundamental on a violin is a product of brain processing, or if it might actually exist in the sound emanating from the violin. Plenty of acoustics people have claimed that it only exists due to brain processing (and I have been a purveyor of that explanation too), but right now, with Marc's explanation of how FFT's work, and what they fail to display, I'm thinking that the violin open G fundamental could be a real waveform, which most of us so far just haven't applied the time and the tools to observe in the most acoustically meaningful way.

It's always tempting to try to stand on the shoulders of prior research, rather than starting from scratch. But it can also leave some things out that prior researchers didn't look for.

Marc and Jeffrey, thanks for your contributions.

April 26, 2017 at 04:48 PM · If we say Tartini tones are physically real as shown by the pressure wave, then what is the physical processing in the ear, or computational processing in the brain, that allows them to be perceived? If the ear/brain can do FFT (as the ability to hear the overtone series suggests), and FFT can't account for perception of the Tartini tone, then what additional mechanism can allow its perception? When I look at the waveform, I see peaks at the frequency of the Tartini tone. Does this suggest something like an amplitude filter is used and an additional FFT is done by the ear/brain?

April 26, 2017 at 05:03 PM · I often don't find hugely meaningful (relating to violin sound) the things that an FFT shows (except in a crude and general way), and I also hear things that an FFT doesn't show, and I think most hardcore musicians will too. Most musicians won't care very much about some kind of sound analysis. They like the sound and playing qualities, or they don't. So it's up to the people on my end, to present something better than the usual BS.

We all have had experiences with a soundpost adjustment making huge differences, but it's looking like its pretty hard to confirm or interpret the differences from an FFT. We're still at the point where player and live listener perception are mostly what matters.

The "double blind" player and listener testing folks have accumulated a lot of data. Maybe this will eventually be processed in such a way that they can make recommendations to contemporary makers?

So far, it's looking like select contemporary instruments can compare quite favorably, with select multi-million dollar instruments.

April 26, 2017 at 05:37 PM · Carmen, it is in the measurment, you can see it of you look at it in wave form. You will not see it in the FFT. In fact, in the cases you see it in the FFT (or DFT) it is a numerical error that can happen if it is in some dimensions compared to other parameters.

Your ear is not doing any FFT, also the brain does not. It is more like a "reverse" FFT.

This is how I understood the functioning of the ear (i am not a medic or biologist): The eardrum is very inhomogenious. This causes that it has at different positions small areas with very different eigen frequencys. So it can analyse which frequencys are inside a wave (there is also a general area that will just try to resonate exactly like the presure wave, that is what a microphone or inverse a speaker does). How ever, that is the data the brain gets.

Now our brain is doing the analysis the other way around, out of the single frequencys it forms a sound, the exact opposite way of the FFT.

Also the information is still inside the FFT, it is not lost. It is just not visible as some would expect it to be, which is obviosly if you look at

sin(a)+sin(b)=2*cos(1/2(a-b))*sin(1/2(a+b))

This is fully mathematically correct and a completly bikjective.

You can always form it back to the original data (at FT without loss, FFT and DFT with numerical and resolution errors).

April 26, 2017 at 06:10 PM · Marc I'm basically just a woodworker, who blew off most of college in order to accept a position at the Weisshaar shop, while I was still a teenager. Could you explain that in simpler language?

I'm confident that you can do it, since you've pulled it off previously in this thread.

April 26, 2017 at 06:15 PM · David, consider my "special case" of double stopping strings tuned to perfect 5ths. This is an example where the basic Tartini Tone is in step with the overtones of the strings.

Double check the example where you saw pressure measurements occurring with the frequency of the Tartini tone and see if its was a case of tones separated by perfect 5ths.

In the "general case" I mentioned, the tones are selected so that the difference in frequencies do not correspond to the overtones. In this case, the Tartini tone can be heard at the frequency difference, but there is no pressure variation measured which matches the Tartini frequency.

I can post a wave sample where I start with a strong G (196Hz), then switch to 440 and 636 played together. One can pick up a faint G sounding with the two tones, but no amplitude spikes with a frequency of 196 will appear in the sound signal.

As I mentioned, one of the reasons I think the phenomenon is useful for tuning violins is that the psycho-acoustic effect reinforces the fundamentals when you get two strings a perfect 5th apart, and detracts from quality of the sound when they are not a perfect 5th apart.

April 26, 2017 at 06:25 PM · First of all:

I did 3 plots showing the frequencies mentioned by Carmen:

www.marschalls.net/stuff/tartini.pdf

There you can see where the information is hidden in the signal. The tartini frquency is clearly in the signal.

David, I wish I was a woodworker in your sense, it is kind of my dream job and I got all the respect in the world for it.

What I mean is, that the ear is already at the eardrum breaking the signal into differenct frequencies. Different parts of the ear resonate the best with different frequencies hitting it. So it just measures the force comming from the oscillation at different parts. This is somehow equvivalent to a FFT.

But the brain is now doing the opposite, it is adding the signals up again to think of the sound.

As I stated, the information about the tartine tone is not visible in the FFT but still there. Out of a perfect FFT you can always reform the original signal. So thats why our brain can rethink it.

However, the interpretations by our brain do make "false measurements".

I dont know if this is true for the g string but the effect you talked about clearly exists. If you hear the frequenies 200Hz, 300Hz, 400Hz, 500Hz, 600Hz, ... the brain interpolates the 100Hz inside the tone as this is the typical overtone series for it.

This is for example used at telephones. The typical voice of men is lower than the lowest frequency in the signal but still we somehow hear it at the low pitch.

Edit: Those special frequencys you talked about are simply those if the following is true:

n*(w1-w2)=w1+w2

So the minima of the amplitudes hit also the minima of the inner oscillation and the FFT will recognize it.

April 26, 2017 at 07:01 PM · Carmen wrote:

"David, consider my "special case" of double stopping strings tuned to perfect 5ths. This is an example where the basic Tartini Tone is in step with the overtones of the strings.

Double check the example where you saw pressure measurements occurring with the frequency of the Tartini tone and see if its was a case of tones separated by perfect 5ths."

______________________

Carmen, I've already been there and done that many times. Perfect fifths can generate some strong "beat" tones, but other intervals can generate stronger beat tones, or tones which seem to have more significance when it comes to player and listener evaluations of violin sound.

Heterodyne tones can either add or subtract from the perception of "good" violin tone, so I think most of what matters is the frequency and amplitude at which they are generated.

April 26, 2017 at 07:18 PM · I loved your graphics at www.marschalls.net/stuff/tartini.pdf, Marc. It's crystal clear now. So, apparently, the eardrum detects these Tartini frequencies in addition to the FFT-derived frequencies. This reminds me of the concept of emergence: The combined tone has properties due to but not described by the sum of the parts.

April 27, 2017 at 03:00 AM · I think we are getting into a technical discussion that is beyond what most members would find interesting. But let me make two points on Tartini Tones specifically.

First, using Marc's mathematics, he states that the frequency of the Tartini Tone is HALF the difference of the two tones that are sounding together. But the effect that people have been studying for centuries generates tone that is exactly the difference of the two frequencies, not half the difference. So whatever effect Marc is referring to is not what is commonly regarded as a Tartini Tone.

For a brief history, technical description and a demonstration try the following link, but you can find many technical references to this effect.

https://www.youtube.com/watch?v=gPGA2pkrabU&ab_channel=AmyTurner

Personally, I have setup tests of a variety of tone differences and every time I have noticed two things: 1. The tones have to be played loudly for me to hear the difference tone, and 2. the difference tone is always the difference of the frequencies, not half the difference.

Second, the plot Marc made shows a series of physical wave forms enveloped by a sine wave generated with a frequency of 98 Hz. If you compare the wave forms in each envelope, you will see that each wave is dramatically different than any other envelope.

In fact, if you continue this plot out for several cycles, you will see that there is no physical pattern that is repeating at 98 Hz. I can envelope the plot with sine curves of many different frequencies and by the same logic make a claim that these are the Tartini Tones, but just like the 98Hz frequency, these are not what is heard.

The miracle here, IMO, is that there is a mechanism in the ear and brain that can detect the temporal difference between two tones and alert us to that difference. In this case in the form of a weak but distinct third tone whose frequency represents the difference in the tones, even if there is no physical pressure wave that is actually undergoing repeating variations at that frequency.

The ear and auditory recognition logic of the brain seems especially adept at finding differences in sounds along temporal variations. Anyone who as tried to make a sound loop can attest that even a minor discontinuity between the end of the loop and its start, even if it is just a few samples at 44,100 samples a second, will be detected by the ear.

April 27, 2017 at 03:15 AM · I want to give one more example for people who are having a hard time hearing a Tartini Tone. In this case, two tones that are 200hz apart are played glissando. If you listen to the sample with head phones and turn up the volume, the 200hz difference tone becomes very prominent.

http://www.earslap.com/article/combination-tones-and-the-nonlinearities-of-the-human-ear.html

You have to go part way down the page to find the glissando sample.

April 27, 2017 at 06:22 AM · Of course the 98hz pattern continues, the sound wave is not changing in any way, the 98Hz are its periodicity. I can show you another plot or you just try yourself at wolframalpha, this is true for all t € R, so for every place. Without meaning to be mean, you do not seem to understand simple interferences, this is within the first semester of every physics studies and by far no woodo. Do you not believe the equation? I can easily show you the calculation steps, altough this is a wellknown result.

Concerning the (w1-w2)/2 : this is true for every frequency. So now it can happen this is below your hearing, so you hear the overtones, starting an octave higher.

Also they are way easier to hear when

n*(w1-w2)=w1+w2 (n beeing a natural number >0)

Speakin in pictures this is when the inner frequencys match the outer one. N=2 ist obviously the most dominant and in this case you get another 0 exactly in the midle between the f_tartini 0s, making a f=f1-f2 audible.

April 27, 2017 at 07:25 AM · I now also tried the frequencys and can clearly hear the 98Hz as very deep buzzing.

April 27, 2017 at 08:11 AM · This is how it looks at n=2, it will be much easier to hear:

This btw is the original dataset with f1=440Hz and f2=636Hz in a bigger plot range. As you can see it still fits the 98Hz at other t. But you can also see a way more complex structure than at n=2. The inner oscillation is slightly shifting inside with every oscillatin whereas at n=2 it stays synced all the time.

April 27, 2017 at 08:31 AM · And a very last thought: You can hear it from two different speakers if they are in the exact same distance. If not (which is true for every real life setup) they get out of sync and the tone will not appear.

It will be sth like sin (w1 t)+ sin ( w2 (t+ tau)) which is a totally diffetent result

April 27, 2017 at 09:59 AM · Speaking of Tartini notes: I heard an additional tone yesterday when I was tuning my E string. Even when it was in tune (perfect 5th, adjusted by listening to the audible beats until they cease). That shouldn't be happening - right??

I think the additional tone was the B below the E string. Maybe there is some additional resonance in my instrument on that B? Or maybe my ears are just hearing a good 5th and helpfully filling in the next one up?

Has anyone else experienced this?

April 27, 2017 at 10:42 AM · You mean the b very close to the a you also played? That sounds nasty.

April 27, 2017 at 12:01 PM · David wrote:

I often don't find hugely meaningful (relating to violin sound) the things that an FFT shows (except in a crude and general way), and I also hear things that an FFT doesn't show, and I think most hardcore musicians will too. Most musicians won't care very much about some kind of sound analysis. They like the sound and playing qualities, or they don't. So it's up to the people on my end, to present something better than the usual BS.

I identify with that, but this is a very interesting discussion. Just to be sure that "Tartini tones" are or are not what I'm thinking of, when I play certain double-stops that are well in tune - especially certain major 6ths - I hear a 3rd tone that completes the triad. Is this a "Tartini tone" and if not, what is it? Whether it pulses out into the air or whether my brain is making the connection and I perceive a 3rd tone, I don't know. But I think it makes a difference in the instrument's resonance - how the sound rings out. Playing certain 6ths well in tune on the G and D can generate a 3rd tone that is at a pitch lower than the open G - which I think is cool!

April 27, 2017 at 12:11 PM · I have just done some recordings and put them in iZotope RX to view as a spectrogram. They are indeed there. What I hear I also see. It's more obvious on high notes.

I was interested in this topic because I also play theremin. Theremin works by heterodyning which is where two very high (beyond hearing range) frequencies are produced - one fixed and one moving according to the player. The note you hear produced is a much lower "difference tone". I knew theremin notes were real enough so I figured Tartini tones had to be real also. They are!

April 27, 2017 at 05:35 PM · Marc,

Thanks for posting the extended plot as it makes it easier to discuss what is happening.

What your plot is showing is the "beat" pattern, which is a regular variation in volume, as opposed to a regular variation in wave pattern or tone. At 196Hz, the beating is too rapid to be clearly recognized as such.

Look up references on beat patterns and you should get a clear understanding of the effect. If you take the higher 636 tone and lower it towards the 440 tone, the beating (volume variation) becomes more noticeable.

Beating can be heard clearly even when two notes are played softly. The Tartini Tone requires sufficient volume to trigger a non-linear response in the ear.

Raphael,

If you double stop a 6th, say a D and a B, then the Tartini Tone would be about a G to G# below the D. So this would sound a root position major triad, G-B-D where the 3rd, the B, is above the 5th, the D.

So does the 3rd tone sound like the root of such a major chord? Can you only hear it when you play the double stop loudly?

April 27, 2017 at 05:59 PM · Ok, lets do it this way:

You give me an high quality, esp uncompressed example of a "beating" and a tartini.

I will show you both in the measurement, predict the tartini out of the FFT and compare it.

April 27, 2017 at 07:15 PM · I think Carmen is right actually... Beating is ubiquitous, you have it with any two sine waves (which start together) whatever their frequencies. But the Tartini tone only happens for specific pairs of frequencies. Or, Marc, can you prove that the phenomenon you are trying to explain also only happens for specific pairs of frequencies? Anyway, I am thrilled to see this type of discussion on v.com!

April 27, 2017 at 07:23 PM · I posted multiple times the condition for it beeing more tartini like than buzzing like. Its just a special case imo. This is due to false detection imo of the same phenomenon.

I think other than dooing a academic discussion we should look at actual data. I am sure I can identify the tartini inside the data. If not, well than you are right and I will agree on your point.

April 27, 2017 at 08:24 PM · Actually, I just did some experimenting on my violin and I now understand that Marc is indeed right :-) Tartini tones *are* beating patterns! Indeed, contrary to what Carmen was claiming I believe, the Tartini tone is by no means limited to playing specific double stops such as a minor or a major third. They occur for any pair of frequencies, and they are indeed generated by what Carmen calls a beating pattern and so nicely illustrated in Marc's graphics. It's the same thing by another name.

Just try the following: play the major third double-stop C-E on your violin, with open E, and C on the A-string. You clearly hear a Tartini tone. Now play sustained strokes on both strings, keep the E open, and slowly slide the C higher and higher. You hear the Tartini tone gradually lowering. Equivalently, the beating pattern decreases in frequency. When you have almost reached E on the A-string, so when you have almost reached a unison, the beating frequency has become so slow that you hear the individual beats that are well known when we are approaching a unison (and also when approaching a fifth, as when tuning the violin). So they are one and the same thing! It's just that the frequency of the beating is originally so high that you hear it as a tone. That's what Marc has been saying all along (sorry Marc).

Sorry if I am just restating the obvious here.

April 27, 2017 at 08:30 PM · and, when that beat/interference/difference pattern is faster then 16 c p s, it begins to be perceived as a pitch.

April 27, 2017 at 09:07 PM · Carmen I hear a G as the root of the G major triad. I don't have to play especially loudly but I'll experiment tomorrow with different volumes. I also noticed such tones with some 3rds today.

April 28, 2017 at 02:39 AM · Jean,

The beating phenomenon is not a Tartini Tone. The Tartini Tones sound like actual notes.

There are a great many examples of both phenomena on the internet. Plug in your earphones and do some searches. The link I posted about a glissando experiment is a great place to start for a clear illustration of Tartini Tones.

An even more dramatic experiment in that link is where a tone is generated with a random frequency with the second frequency always 200Hz above it. If you look at the wave form, it is a complete and utter mess. Yet the Tartini tone comes through clear and unwavering.

The beat phenomenon when generated at a higher frequency than about 10Hz does not become a note. It adds a subtle quality to the combined tone but it does not generate a 3rd tone.

Raphael,

What you are hearing is certainly consistent with the physics of a Tartini tone. Since the tone is generated by the ear, you might try muting the G string with a finger while double stopping an open D and the B on the A string and see if you can still hear it. If you can, it would eliminate a sympathetic vibration of the G string as the source of the tone.

I've encountered people who are much more sensitive to Tartini Tones than I, so it is possible you can hear the tones when playing at modest volume levels.

Can you give me some examples of double stopped 3rds that seem to generate a third tone for you? Do not tell me what you think the third tone is. I will do a Tartini Tone computation for the 3rd tone and tell you what it would be.

April 28, 2017 at 05:45 AM · Again, give me data, I will show you tartini in them.

n*(w1-w2)=w1+w2 will surely be mistaken as a sound instead of a beating if the frequency is high enough. You now can argue that it is still caused by our hearing, but I claim physical (also linear) effects as root.

Also would it be a very big incident when the tone you claim is exactly an octave higher than the beating pattern but has nothing to do with it.

April 28, 2017 at 09:04 AM · Carmen wrote:

"The beating phenomenon is not a Tartini Tone. The Tartini Tones sound like actual notes...

...The beat phenomenon when generated at a higher frequency than about 10Hz does not become a note. It adds a subtle quality to the combined tone but it does not generate a 3rd tone."

___________________________________

I disagree. It certainly can, and the only reason I won't assert that it DOES is that conditions and individual listener perception thresholds can vary. The interval chosen can also effect the strength of the beats, and some sets of conditions can "mask" the phenomenon to a listener.

I've spent quite a bit of time over the years generating sound files like Marc has illustrated (experimenting with a wide variety of intervals) because I've suspected that these might be very important to our perception of violin sound, and that taking these into account might be able to take us much farther than we've been able to get by observing conventional FFTs (which don't clearly display this phenomenon).

I also strongly suspect that even a single note on a violin will generate some of these "combination tones", since a single note played on a violin will actually produce numerous pitches, with quite a variety of interval relationships.

April 28, 2017 at 09:46 AM · Do you have wav or similar files of it you can share with me? I could try to write a detection filter, two different would be enough for me.

The higher octaves might also be caused by higher pitches on the violin interfering.

April 28, 2017 at 12:01 PM · Carmen - in fact I did think to put a finger down on the open G and I still heard it! But why don't you want me to tell you what I hear with a 3rd? To me it's more important what I do hear than what mathematics or physics says I ought to hear.

April 28, 2017 at 01:39 PM · Raphael - I didn't want you to tell me what the note of the 3rd tone was in order to make the "test" a bit blind on my part. ;-)

But since the Tartini tone is computed mathematically, I guess there is no way I can skew the results if you tell me the note you hear.

It just so happens I have been practicing etudes that have double stopped 6ths in them (open D, stopped B). I seem to hear something extra happening but I am sure my sense of pitch is not as sensitive or as refined as yours. Assuming I have the D and A strings tuned to within 1c of a perfect 5th, would I need to play the B slightly sharp or flat of its Equal Tempered value to get a 3rd tone that completes the triad?

Marc and David - Sound perception is certainly a highly subjective (and occasionally charged!) subject. My assertions about Tartini Tones vs Beat Patterns are clearly demonstrated by simple experiments with sine waves with the results published all over the internet. Of course, as we all know, if it is on the 'Net, it must be true! >grin<

I especially appreciate David weighing in with his considerable experience. It certainly suggests that focusing on a single phenomenon whose PHYSICS is well-understood might mask other effects that can be occurring simultaneously.

The only real way to resolve the difference of opinions on this matter is to devise a specific experiment that we all reproduce independently and then compare notes. But for now, I think the forum members have more than enough information to explore the topic on their own if they so desire.

April 28, 2017 at 02:11 PM · Again, one single audio file and I will proove its physical and in the data ;)

Actually what I found on the internet is no series science, it will be somewhere for sure, but I did not find it.

Its not in the FFT and many seam to think that this means its not in the data. Although opening a lot of possibilities the FFT is a very simple numerical procedure with a hell lot of problems in a lot of cases.

I think we can agree that we dont agree and without data I have nothing to add. I am very close to buying a good mic for recording right now.

If I find data I will analyse them and tell you the result, of course also if I cannot find the note. I just disagree on one more point, if you hear it and I can find it in the data I think nobody needs to double the experiment. Of course we should do more than one measurement though. Sadly I dont think its relevant enough to publish a whole paper, otherwise I could even do it during work. Maybe it can be combined with some other violin analysis, but I had no valid idea yet. (The not measurable changing with sound post adjustment are maybe something to put a finger on).

So long and thank you for the discussion shortening my daily train travel.

April 29, 2017 at 03:15 AM · Marc Marschall,

If it's any consolation, Tartini tones were thought to be a high frequency beat phenomenon for about 100 years after their discovery.

However, it's been known that this is not the case since Helmholtz's "On the Sensations of Tone" more than 150 years ago. In that work, he realized that Tartini tones are non-linear and therefore cannot be the same as beats. Helmholtz, however, erroneously thought that Tartini tones originated in the middle ear. We now understand that non-linear response happens in the cochlea.

There is a significant amount of both experimental and theoretical scientific literature on the subject (more than 150 years worth). While I can appreciate your curiosity and desire for evidence, your steadfast argument with Carmen Tanzio has only confused others on this board.

Since you seem to appreciate physics, the current model for Tartini tones that agrees with all the experimental data is that of a damped harmonic oscillator (i.e. the inner ear) with the two waves as forcing terms. The sum and difference tones arise as quadratic non-linearities. There are also additional tones that arise from higher order non-linearities. You can both derive these pitches mathematically as well as observe them in the inner ear.

You said you don't know where to find serious science for this. If you don't know where to find authoritative information on a topic, you should not be speaking authoritatively about it. Technical discussions can be found in non-linear acoustics books, books on the physics of hearing, medical texts on hearing tests, books on the physics of music, etc.

April 29, 2017 at 03:17 AM · Jeffrey Freed,

There is a disappointing amount of misinformation on this thread. I'll try to answer your original questions as clearly as I can--

We have to first be careful about what we mean by Tartini tone. It has a very specific meaning, and many people confuse them with beats even though they are completely different phenomena.

Humans perceive beats when pitches are very close in frequency (usually within about 15 hertz of each other). This is how we tune our violin strings. This is only possible because violin strings do not produce pure tones and have reasonably strong overtones. For example, when we tune the A and E strings against each other, the 2nd overtone of the A string is a very high E and should be the same frequency as the 1st E overtone. We are actually tuning the beats of the overtones against each other rather than the beats of the fundamentals (i.e. the open A and open E). This confuses a lot of people. People think the you can hear the beats between the open A and open E themselves and this is not true.

Tartini tones, on the other hand, require two loud, continuous pitches played against each other. As counterintuitive as it is, the loudness is key to perceiving Tartini tones. The actual loudness threshold to hear them depends on the individual person and the unique structure of their inner ears. Judging by your description, I think you are hearing Tartini tones.

So with all of that said, let me answer your questions:

1) Without being able to examine the recordings, I would guess that this is a result of the requisite loudness. On the same speaker, the two pitches will be equally loud hitting your ears and if your playback is loud enough, you will hear the Tartini tone.

Again, I don't know your speaker configuration, but you need both tones hitting a single ear with enough loudness to trigger the effect. When the tones are coming out different speakers, your ears might not be pointed at both speakers and so the volume requirement may not be met. You can try pointing both speakers are the same ear and turning up the volume and seeing if you hear the Tartini tone then.

When you heard the recording of the two tones being played through different speakers, were both pitches coming out of both speakers? This case is then the same as the first.

Note that Marc's incorrect beat explanation cannot explain any aspect of this question.

2 and 3) These are both correct and expected.

4) This is pretty interesting. I'm not 100% sure what's happening, but I have an educated guess. Tartini tones work best with pure pitches. As we know, the violin does not produce pure pitches and instead is very rich in overtones. So Tartini tones might not always occur or be obvious when just playing two open strings, say. However, with the over-ear protectors, you dampen the sound. Then, when you bring it into contact with the violin, the low frequencies (i.e. fundamentals) get transmitted better than high frequencies. This is the same phenomenon whereby you can hear the bass from someone's car radio well before you can hear the treble. High frequencies get absorbed much more easily. So under my conjecture, you are filtering out high frequencies while transmitting the fundamentals. As long as the fundamentals are loud enough, you should clearly hear a Tartini tone.

Despite these answers, Tartini tones are not purely perceptual. By this I mean, this is not something that is happening in the brain. There are very funky vibrations going on in the fluid of the inner ear, so in that sense, they are very physical.

It's worth noting you can also produce Tartini tones not in your ear, but electronically if the circuit has nonlinear distortion. In this situation, you absolutely do see the Tartini tones in a spectrogram.

April 29, 2017 at 03:20 AM · On a general note, when some random person on the internet actively disagrees with a science article on Wikipedia (in this case, https://en.wikipedia.org/wiki/Combination_tone), you should be pretty skeptical.

I'd like to recommend a free book to people who are interested in these sorts of topics:

https://homepages.abdn.ac.uk/mth192/pages/html/music.pdf

It is a math and physics book, but different sections require different amounts of background to read. With that said, it is extremely well researched with plenty of references to both recent and seminal papers. It also has correct explanations of many common questions about sound and music.

Lastly, there is a very cool, practical application of Tartini tones. Outside of medicine, it is not well known that the inner ear actually *produces* sounds. The sounds are very quiet and require very fancy microphones to detect. These two phenomena are then combined to test the hearing in infants and children.

Basically, two tones are played loudly right outside the child's ear and then immediately stopped. The child's inner ear should then produce these very quiet emissions. Because the inner ear is where the Tartini tones originate, they are also emitted back out of the ear. The microphone should then pick up both of the original tones, the difference tone, and typically another kind of Tartini tone that is not often experienced outside of a controlled environment like this testing. All of these pitches will be clearly visible on a spectrogram.

If the microphone picks up the Tartini tones being emitted, everything in the inner ear is assumed to be ok. If the microphone does not pick up the Tartini tones, then the doctor immediately knows something is wrong with the structure of the inner ear or the fluids in the inner ear. You can read about this here and here:

https://en.wikipedia.org/wiki/Otoacoustic_emission

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3614374/

The distortion product otoacoustic emission is the one using Tartini tones.

April 29, 2017 at 09:04 AM · "Since you seem to appreciate physics, the current model for Tartini tones that agrees with all the experimental data is that of a damped harmonic oscillator (i.e. the inner ear) with the two waves as forcing terms. The sum and difference tones arise as quadratic non-linearities. There are also additional tones that arise from higher order non-linearities. You can both derive these pitches mathematically as well as observe them in the inner ear."

Which makes it ... a beating. The resonancy just becomes overdriven and therefore "hops" into the next order. Its like you can drive your violin by a single frequency but still hear the whole violin sound. If you overdrive it you hear the flagelot like sound.

Btw the same reason why it is exactly one octave above from what I pressumed at the first example. You cant denie the resemblency.

"You said you don't know where to find serious science for this."

I cannot find it at google fast because of all the pseudo science done on that topic and people were told to google it. Of course I know how to do reserches on topics but I ment it more as a warning.

" On a general note, when some random person on the internet actively disagrees with a science article on Wikipedia (in this case, https://en.wikipedia.org/wiki/Combination_tone), you should be pretty skeptical."

First of all, wikipedia is the last place to look for science that is at a topic of rare interest. Even very basic articles like Huygen Fresnel Diffraction contain severe mistakes (some of them got corrected by me a few years ago when some physics students tryed to use it and we realised they had a wrong source for there home paper as a lot had the same wrong preassumptions)

Also you disagree yourself!

"Despite these answers, Tartini tones are not purely perceptual. By this I mean, this is not something that is happening in the brain. There are very funky vibrations going on in the fluid of the inner ear, so in that sense, they are very physical."

You basically told us its a second order resonance from the damped oscillator in the inner ear. This resonances gets driven by the force of the tones and the beating. So basically it still is the same. Of course there happens the jump in the next order and therefore it is happening something to it in the process of measurment in the ear that changes it. There is this step of energy that forces it into the next order, thats why you need it loudly but still its the very same physics happening. This by the way is exactly why I wrote that thing about n*(w1-w2)=w1+w2 . Here you will fastly jump into the second mode.

I did not read your paper yet as it is quite long but from looking quickly it is a very general collection which might be interesting esp for its sources.

I think we dont disagree as much as it seems. I do agree now that it is not directly the beating but still it is just a kind of overdriving during measurement of the beating. If you look at the beatinfg in the air or reform it inside the oscillator again, I dont know if this is something important.

However I will, this takes some time, try to do complete calculations with the inner ear oscillator and try to show what I mean and why it is not very different from the original beating effect. Give me some time to read the papers first and get typicall parameters of the inner ear.

April 29, 2017 at 11:19 AM · Helmholtz discusses Tartini tones and other related phenomena in Chap VII of his Sensations of Tone. Incidentally, Helmholtz tells us that Tartini tones were first discovered in 1745 by a German organist Sorge, and later incorrectly associated with Tartini.

April 29, 2017 at 12:06 PM · Carmen - I see...

Anyway, I'm a bit busy these days in my practicing - with a recital coming up (a separate announcement coming in another thread) to stop and make a list. But concentrating my listening in a certain way I did notice Tartini tones in a number of 3rds. Sometimes, like with 6ths, I heard a 3rd tone that completed the triad. Sometimes the 3rd tone seemed to double one of the 2 fundamental tones I was playing, an octave down. And sometimes I could not clearly discern an exact tone but heard more of a kind of not unpleasant rumble, reminiscent of an organ - but much softer. I'm finding that listening for these tones seems to make the instrument ring out more when I get them.

April 30, 2017 at 08:03 PM · Raphael - double stop major thirds would yield a Tartini Tone somewhere in the range of an "augmented unison" relative to the lower tone being double stopped. If I did the math right, it would sound two octaves lower.

So double stopping an A# on the G with the open D would yield a Tartini Tone somewhere between an A# and B, but two octaves lower. Might be related to the rumbling or duplicate tone you hear. >shrug<

It is also possible you might be hearing something related to the overtones of the notes being double-stopped. Tones on a violin can be generated with overtones that are stronger than the fundamental tone being played. Determining if this can generate a significant Tartini Tone sounds like a major research project.

April 30, 2017 at 08:13 PM · OK - anyway, between my recital plans and life demands I will leave it here for my part. This has been a good quality discussion.

May 1, 2017 at 02:09 AM · When people are talking about "beats" they are talking about simple linear differences in frequency. My interpretation of Marc's posts is that he was implicating these linear differences which would be perceived as an overtone if at sufficiently high frequency -- and they would have to be loud enough too (which is not a given if we are talking about differences among overtones). On the other hand, what Seph is describing is a non-linear type of response, and that seems fundamentally different from what I would call "beats." Perhaps where there is some common ground is in the generation of the overtones because on the violin the overtone spectrum changes with the amplitude of the principle.

Now, I'm not an expert on this kind of thing, and I'm not sure I've ever heard "Tartini Tones" (my ears ring constantly), but I think it would be pretty cool to set up some kind of laboratory-scale device that would model the same behavior, perhaps owing to the particular response characteristics of a fluid or other types of materials, which can then be studied independently using techniques such as dynamic mechanical analysis. Maybe someone has already done that. I do have the software tools and database/subscription access necessary to search that literature but unfortunately neither the time -- nor the inclination -- to do so.

May 1, 2017 at 08:05 AM · Indeed, Tartini tones are not just accelerated beats, even if the calculations seem similar at first sight.

May 1, 2017 at 08:07 PM · I guess there are many ways of looking at this. The "implied fundamental" which Seth suggests does not actually exist, but is implied by the harmonic structure (and is a result of ear and brain processing filling in the gap) will show up in the air waves, and can be picked up by a microphone.

Furthermore, beats, or pressure variations at various frequencies can be heard as tones or pitches. I've tested and experimented with this over and over again.

I think a lot of the "current knowledge" on the topic is heavily based on prior knowledge, under the assumption that the prior knowledge was accurate.

May 2, 2017 at 10:28 AM · The maths are accurate. What I call "beats" are the pulsating volume-level of a combination of very close tones or their overtones. They are not in themselves sounds. The "implied fundamental" is produced in the ear itself and cannot be picked up by a microphone, although the microphone itself, like the ear, can produce them.

My maths are too rusty to demonstrate this now, but I used to manage in my student days.

Combination tones, as opposed to beats, can also occur in an electrical circuits and loudspeakers, not to mention the extremely non-linear vibrations of a violin body: then they will become air waves.

May 2, 2017 at 12:45 PM · If Tartini falls in the woods....

May 2, 2017 at 11:32 PM · Hi David,

Thanks for the reply.

I never used the words "implied fundamental" that you are quoting, so I think you may have misunderstood something.

I think what you are calling beats is the additive interference that was illustrated nicely in the graphs above. A key problem with your hypothesis is that this interference is _always_ present whenever you have two or more pitches sounded simultaneously. So unless you hear Tartini tones 100% of the time that two or more pitches are sounded on any instrument, at any volume, at any frequency, then beats are not the cause of Tartini tones.

So while you may have been looking at the waveform and seen the envelope of amplitude changes that was described above (which, as I said, is always present with two or more tones) at the same time that you heard an extra tone, one is not caused by the other.

May 3, 2017 at 07:46 PM · Seph, apologies for the mis-attribution.

While the amplitude changes (or beats?) may exist with any two or more tones, it appears that some intervals create regular and repeating amplitude changes which are much larger, and more obvious than those which are created by other intervals. So I wouldn't expect one to be able to hear these on just any combination of two tones.

A perfect fifth is an example an interval which creates a very strong and distinct amplitude variation, occurring at a frequency which is an octave below the lower of the two tones, and coincidentally (?), where the Tartini tone happens to be. Some other intervals create almost nothing which is discernible.

May 4, 2017 at 11:07 AM · David,

When sounding two tones of different frequencies, the maximum amplitude the beat phenomenon can achieve is a complicated function of the two frequencies (or intervals).

One can both theoretically and experimentally plot the peak observed amplitude as a function of the intervals (more technically the ratio of the two frequencies). What you discover is that the loudness of the beat pattern will peak dramatically for specific intervals.

For pure tones, the first dramatic loudness peak occurs in the area of a perfect 5th (3/2 frequency ratio). A broader peak occurs around a ratio of 2.5. Then an even broader one at 3.5, etc.

Add in damping, overtones and the rather complicated cross gain between two strings being bowed simultaneously, then the intervals at which the phenomenon occurs on a violin would certainly vary.

Scientists tend to use very specific definitions for phenomena that at times can seem at odds with what people perceive. Double stopping perfect 5ths and the perception of beating versus a tone is an example.

When one double stops open strings tuned to a perfect 5th, the frequency of each string is an integer multiple of the beat frequency. This means that the temporal shape of the wave matches up perfectly with the periodic beating pattern. To my ear, this causes the double stop to blend nicely into what appears to be a continuous tone with no beating effect.

If the two strings are slightly mistuned from a perfect 5th, then the temporal shape of the wave shifts with time inside the beat pattern which the ear detects as something extra going on.

I am a little rusty on what the ear is actually latching onto here, but I think what it is hearing is the frequency at which the wave pattern is shifting inside the beat pattern. So that even though the beat pattern frequency is rather high, the rate at which the wave is shifting inside that pattern is much lower.

I think what people say is beating when tuning perfect 5ths might not be technically the physical phenomenon called beating due to two tones being nearly identical in frequency.

May 4, 2017 at 07:57 PM · When I play a d'-f#' third on the A and E strings, the D string starts to vibrate in its fundamental - visibly. Trying to explain that, I believe that it has to do with Helmholtz, or stick-slip, motion. Each time a string "slips" the bridge gets a jolt from the string. One in four pulses from the d' coincides with one in five from the f#', so that the D string is driven, through the motion of the bridge, by pulses at half its natural frequency. That's enough for it to start vibrating on its own.

May 5, 2017 at 09:56 AM · Beats. Two slightly de-tuned pure tones will pulse (beat)at the difference between the two frequencies. On an oscilloscope screen we see the combined waveform with an ampltitude variation of half the difference. What we hear as beats are not the peaks of the combined waveform, but their gradients (either side of the peaks) i.e. the variations of air pressure, upwards or downwards.

With complexe waveforms, full of harmonics, we can have beats between the fundamental of one tone and the harmonics of another.

All this has nothing whatever to do with Combination (Tartini) Tones.

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