Tartini Tones and Under/Overtones

[JohnCockburn]

Change of plan. I'll answer the points here.

David Beard:

4 hours ago, David Beard said:

??excuse me??

There is no simple relation between the pressure wave frequency and motion of individual molecules.  Only the collective presure directly has the frequency.  Individual molecules of the medium will move move in response, but not necessarily simply and directly tracking the frequency as pulses of the wave pass through.

I didn’t mention individual molecules. Each individual molecule in the air is undergoing translational motion in random directions, with a spread of velocities given by the Maxwell-Boltzmann distribution. The root mean square speed of a nitrogen molecule at room temperature is about 500 m/s. 

What’s relevant for a sound wave isn’t this individual motion. It’s the collective displacement of all the molecules in an arbitrary “slice” of gas in the direction of propagation that’s important. The molecules are, of course all still undergoing random motion, but they all have collective oscillatory motion in the direction of the pressure wave propagation. It's a bit like the drift velocity in the direction of the current experienced by the randomly-moving conduction electrons in a metal.

I don’t get what you mean by “only the collective pressure has the frequency” The acoustic pressure and the displacement oscillations are fundamentally linked. Displacement oscillations lead to local particle density fluctuations, which lead to local acoustic pressure fluctuations, as they must from the ideal gas equation of state. And all vice-versa of course, if you prefer.

 

[Bill Merkel]

^what's the formula for the periodicity shown from superimposing two random sine waves?

[David Burgess]

44 minutes ago, Bill Merkel said:

that's one way of measuring period, which is the reciprocal of frequency.  the main periodicity of a single note would usually be the fundamental or the first overtone because of their relative strength. 

Then on a violin open G string, it would probably be measuring the time between peaks of the first harmonic (octave above the open G), since the fundamental produces hardly any peak at all.

[Bill Merkel]

well, it was a guitar tuner which doesn't face that particular problem.  i don't think it was a sophisticated tuner, just a possible answer to your question about what the op was seeing on his

[JohnCockburn]

.

[David Beard]

3 hours ago, JohnCockburn said:

Change of plan. I'll answer the points here.

David Beard:

I didn’t mention individual molecules. Each individual molecule in the air is undergoing translational motion in random directions, with a spread of velocities given by the Maxwell-Boltzmann distribution. The root mean square speed of a nitrogen molecule at room temperature is about 500 m/s. 

What’s relevant for a sound wave isn’t this individual motion. It’s the collective displacement of all the molecules in an arbitrary “slice” of gas in the direction of propagation that’s important. The molecules are, of course all still undergoing random motion, but they all have collective oscillatory motion in the direction of the pressure wave propagation. It's a bit like the drift velocity in the direction of the current experienced by the randomly-moving conduction electrons in a metal.

I don’t get what you mean by “only the collective pressure has the frequency” The acoustic pressure and the displacement oscillations are fundamentally linked. Displacement oscillations lead to local particle density fluctuations, which lead to local acoustic pressure fluctuations, as they must from the ideal gas equation of state. And all vice-versa of course, if you prefer.

 

What I mean is that the wave is not medium.  While the wave causes displacements in the media, the induced displacements don't necessarily lead to oscilations of the components of the material that are simple reflections of the wave.  More importantly, the local disturbances are caused by the propagating wave, they are not it's cause.   For sound, the waves are propagating cyclic disturbance of the presure in the medium.  This closely linked to, but not necessarily the same, as vibrations of the medium.  Many such vibrations of the media give heat without giving sound.

Why does this matter?  The waves of sound are in some sense independant of the medium.  They do of course need to be expressed in a medium, but they can propagate from one medium to another.  In a larger sense, a wave is the communication of energy through a medium.   

Waves step into a connecting borderland between physical materal, energy, and information.  I'm not saying this in any mystical sense, but in a literal physics sense.  As such, mathematical effects can easily be confused with physical ones.  It isn't always easy to separate the phantom from the real.

Consider an ideal sawtooth presure wave.  Physically, the presured goes straight up linearly, then goes straight down linearly, then repeats etc. 

 Mathematically, these cyclic presure changes are same as adding together an infinite number of simple sone waves.  But physically, really, there is only one simple saw tooth cycle.  An fft's task is deliver a series of sine waves that up to the sawtooth.

A good fft does exactly that when it 'sees' a sawtooth.  It reports an endless sum of sines, but no sawtooth.

So which is the real physical wave, and which the informational illusion.

Sound is endlessly tricky in such things.

[notsodeepblue]

7 hours ago, Bill Merkel said:

well, it was a guitar tuner which doesn't face that particular problem.  i don't think it was a sophisticated tuner, just a possible answer to your question about what the op was seeing on his

If I put the same 400+600Hz file through a basic pitch-detection algorithm (autocorrelation function + thresholding + peak-selection, as Bill is likely describing for a 'simple' guitar tuner) I get the following - a detected pitch that is neither 400 or 600Hz:

What interests me most about the original question is how the same hardware/software (some black-box solution, regardless of what that might actually be) would identify the same pitch as illustrated above for one instrument, but one of the played notes for the other. I presume that the simplest explanation for this is that the Tartini pitch for one instrument generates a peak that is high enough to survive the device-specific thresholding, while the other one isn't. I am assuming that Shunyata is confident both instruments sounded equally in-tune to their ear - i.e. this specific instance of human pitch-detection has a lower threshold, which was passed for both instruments.

Other than Michael Darnton's suggestion (which seems pretty compelling, and I think might potentially account for this drop-off in the amplitude of the Tartini peak between instruments) I can only think of one basic reason for this happening - that even though the two instruments sounded equally in-tune to the player they actually weren't. In such a case, the Tartini peak would never be properly formed and so culled during thresholding leaving the lower or most in-tune of the two played notes to be identified as the pitch.

As there are clearly lots of people are thinking about sound-waves, may I ask if anyone else sees other potential reasons for the observed difference between human and machine pitch-detection for two instruments, that Shunyata is describing?

[HoGo]

11 hours ago, Bill Merkel said:

looking up some guitar tuner code just now, that particular code doesn't use an fft, but uses a comparator to square the input wave and then counts the duration of that squared wave.  if the threshold was set right, with a single note you'd usually get the fundamental or the octave above, same letter pitch.  from the same file, below you see 400hz and 600hz sine waves superimposed results in a periodicity of about 166hz.  that could easily be the frequency the tuner squares, depending on the threshold, and that may even be the pitch that you hear as a tartini tone in this case...  though there's apparently no significant component of that frequency in the spectrum, that periodicity is there...

 

This is first time I heard about this and haven't had time to read more... Just one quick question arises off the top of my head...

How would the superimposed wave differ if the oridinal waves didn't "start" at the same point. I don't know the correct nomenclature but would that be called "phase shifted" or something. My brain tells me the new period would still be there but the peaks in the amplitude will not look the same (just think if you align the first peak of frequencies then in result each 4-th peak will be at max sum of the two making latge differences between max and min, if you align max of one with zero of the other or with minimum you'll get more of a amplitude cancellind and thus lower peaks - just musing off the top of my head so I may be wrong). As a result that may be reason why on one instrument the new frequency is more "hearable" and on other not so much. Think it is mosstly the brain/ears that make it hearable and also how the SW in simple tuners work. I'd bet a stroboscope tuner won't hear it as well.

[nathan slobodkin]

This has been an interesting discussion. I had actually never heard of "Tartini tones" but I am very conscious of intonation of double stops and suspect these tones are part of what I am hearing. I am also assuming that listening for these tones is why the most accurate way to tune an instrument is by playing fifths until they just sound "right".  I find some instruments are extremely sensitive or picky about intonation while others are more forgiving and seem to "fall into" tune if my fingers are anywhere close correct position. Getting back to Shunyatas' original question: What  physical construction features are causing the difference? I was told that instruments which are hard to tune have uneven graduations or perhaps overly abrupt level changes in the graduation pattern.

As always this stuff is only useful to me if I can relate the physical construction of the instruments to the tonal properties of the finished instruments.

[Bill Merkel]

8 hours ago, HoGo said:

How would the superimposed wave differ if the oridinal waves didn't "start" at the same point. I don't know the correct nomenclature but would that be called "phase shifted" or something. My brain tells me the new period would still be there but the peaks in the amplitude will not look the same

you're right.  if you superimpose two sine waves, the result is periodic.  the actual shape would depend on where they "started" relative to each other.  it only works with sine waves. that's why i was asking above about a formula for it.  i don't know this from music, but from working as a programmer on a signal processing project a long time ago. i still remember some of the more surprising and entertaining stuff

 

[jezzupe]

6 hours ago, nathan slobodkin said:

This has been an interesting discussion. I had actually never heard of "Tartini tones" but I am very conscious of intonation of double stops and suspect these tones are part of what I am hearing. I am also assuming that listening for these tones is why the most accurate way to tune an instrument is by playing fifths until they just sound "right".  I find some instruments are extremely sensitive or picky about intonation while others are more forgiving and seem to "fall into" tune if my fingers are anywhere close correct position. Getting back to Shunyatas' original question: What  physical construction features are causing the difference? I was told that instruments which are hard to tune have uneven graduations or perhaps overly abrupt level changes in the graduation pattern.

As always this stuff is only useful to me if I can relate the physical construction of the instruments to the tonal properties of the finished instruments.

This discussion is the first you've ever heard of Tartini tones? 

[Bill Yacey]

6 hours ago, nathan slobodkin said:

 I find some instruments are extremely sensitive or picky about intonation while others are more forgiving and seem to "fall into" tune if my fingers are anywhere close correct position.

Off topic, but I have noticed a similar phenomena, where it's hard to play an off pitch note on some instruments.

[jezzupe]

I personally feel that this "anomaly" has something to do with an individual persons inner ear when the Cochlear is hyper excited in that I have been in informal listening "experiments" where some people are not hearing them while others are, and even have had people who have them come and go somewhat dependent on amplitude,location and what not.

and in all this, knowing what I know about Korg tuners, I don't find it funny that they can act funny, I have seen them grab the wrong pitch only to self correct, get very unstable when the battery is low and do other funny stuff that makes me think "well that's not very "scientific" from an observational standpoint. No offence to OP, I guess I'm just not 100% your tuner is "showing us" the "truth" or reality, on the other not saying it's not.

I just think Tartini tones are formed in our ears and that its not the violin making them, it's just the main source of the vibrations, the actual tone is created by our processing the incoming frequency information and creating an anomaly that when the vibration to electrical brain input happens a certain "crossing or wires" or additional input "messes" with the error correcting code that makes us hear the tones.

here's wiki take fwiw

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

there is only "evidence" against my theory, not proof

 

[David Beard]

7 hours ago, nathan slobodkin said:

This has been an interesting discussion. I had actually never heard of "Tartini tones" but I am very conscious of intonation of double stops and suspect these tones are part of what I am hearing. I am also assuming that listening for these tones is why the most accurate way to tune an instrument is by playing fifths until they just sound "right".  I find some instruments are extremely sensitive or picky about intonation while others are more forgiving and seem to "fall into" tune if my fingers are anywhere close correct position. Getting back to Shunyatas' original question: What  physical construction features are causing the difference? I was told that instruments which are hard to tune have uneven graduations or perhaps overly abrupt level changes in the graduation pattern.

As always this stuff is only useful to me if I can relate the physical construction of the instruments to the tonal properties of the finished instruments.

Yes.  For me, I rarely focus on hearing the Tartini tones distinctly or separately.  Getting them that clear and prominent takes a special effort.  But, I was taught that when you play an interval strongly and cleanly and you hear the intonation just sort of lock in, the Tartini tones are part of that difference you hear in the locked in interval.  

 

As for physical differences in instruments, I suspect that low Qs of the various instrument resonances helps. 

The violin presents various masses that are driven to vibrate following the signal from the strings.  But all these masses and modes of vibration also have their own natural resonances. 

 

 

(Q is a measure of how much more the resonance responds to energy coming in at a frequency near its natural frequency versus an input energy that differs in frequency.  A high Q object like a good string very strongly favors its own natural frequency.  A much lower Q object like a good soundboard readily responds to a much broader range of driving frequencies.)

 

[David Burgess]

10 hours ago, HoGo said:

This is first time I heard about this and haven't had time to read more... Just one quick question arises off the top of my head...

How would the superimposed wave differ if the oridinal waves didn't "start" at the same point. I don't know the correct nomenclature but would that be called "phase shifted" or something. My brain tells me the new period would still be there but the peaks in the amplitude will not look the same (just think if you align the first peak of frequencies then in result each 4-th peak will be at max sum of the two making latge differences between max and min, if you align max of one with zero of the other or with minimum you'll get more of a amplitude cancellind and thus lower peaks - just musing off the top of my head so I may be wrong).

Interesting question. I don't have a solid answer. Some things on violins tend to "phase-lock", like the slip-stick action of the bow, with the note being played on the string. It wouldn't be a stretch to consider that many more such things could be going on.

[ctanzio]

This has been covered previously.

 

[nathan slobodkin]

2 hours ago, jezzupe said:

This discussion is the first you've ever heard of Tartini tones? 

I don't recall talking about these perhaps because I never had a problem playing in tune. Other than  being  the sound of properly played double tones is there some use for this?

[jezzupe]

5 minutes ago, nathan slobodkin said:

I don't recall talking about these perhaps because I never had a problem playing in tune. Other than  being  the sound of properly played double tones is there some use for this?

I think it''s considered more of a psycho-acoustic phenomenon, not something you would use really, perhaps in acoustic study. I guess it just seems surprising to me that someone who has as much violin/musical experience as you would have heard of this before now. I don't really consider it necessary information, just one of those things that's "violiny" that comes up every once in awhile.

I think the first time I heard the term was in highschool chorus where our instructor used to give us a look and say "you're so off pitch your making Tartini tones"  

[David Beard]

43 minutes ago, nathan slobodkin said:

I don't recall talking about these perhaps because I never had a problem playing in tune. Other than  being  the sound of properly played double tones is there some use for this?

Not really a use except as a component of understanding intonation.

However, easy presence of harmonics and difference tones is probably a good sign in a fiddle.

[JohnCockburn]

3 hours ago, ctanzio said:

This has been covered previously.

 

Nice spectrum in this thread. Unambiguous proof that Tartini tones aren't high frequency beats.

[JohnCockburn]

On 5/31/2020 at 7:12 PM, David Burgess said:

Amplitude fluctuation IS the signal power which is relevant to hearing.  Amplitude fluctuation is what wobbles the eardrum. Amplitude without modulation produces no sensation of sound, only something ranging between a weird feeling, and pain.

"Amplitude" doesn't mean what you seem to think it means. 

 

[David Burgess]

51 minutes ago, JohnCockburn said:

"Amplitude" doesn't mean what you seem to think it means. 

 

There's more than one definition. Are you thinking of "peak-to-peak"? "Peak-to-trough"? Displacement from equilibrium or static?

[JohnCockburn]

18 minutes ago, David Burgess said:

There's more than one definition. Are you thinking of "peak-to-peak"? "Peak-to-trough"? Displacement from equilibrium or static?

Doesn't matter. Apart from "displacement from equilibrium" which in general, isn't the definition of amplitude. Amplitude is the maximum displacement from equilibrium. 

So constant amplitude, ie with no modulation or fluctuation of the amplitude, for a sound wave, for example, means that the sound has constant intensity, or perceived loudness, if you will. Changing the amplitude changes the loudness.

"Amplitude without modulation produces no sensation of sound, only something ranging between a weird feeling, and pain"

No. Wrong.

 

[JohnCockburn]

Anyway, to get back to the discussion of “Are Tartini Tones just high frequency beats?"

 

If we want to demonstrate the phenomenon of beats, we take a signal that’s a linear superposition of 2 sinusoidal waveforms of frequencies f1 and f2.

 

ie:  S = sin (2πf­1t) + sin (2πf­2t)

 

We arrange things so that we can vary f1 and f2. When these frequencies get within a few Hz of each other, we can clearly hear “beats” ie periodic increases and decreases in the loudness of the signal, at a frequency of f2-f1. The important thing to realize is that these beats aren’t in any sense an “additional signal”. Its just that when we add waveforms together, we will encounter situations when they will interfere constructively and destructively. Sometimes they’ll boost each other, sometimes they’ll partially cancel out. This is what gives the beats. Periodic phase cancellation. So whether we can hear beats or not, the signal is always:

 

S = sin (2πf­1t) + sin (2πf­2t)

 

NOT

 

S = sin (2πf­1t) + sin (2πf­2t) + woowoo

 

Because the woowoo is already in the addition of the sine terms.

 

Fourier transforming this signal is trivial. just delta functions (spikes) at the frequencies f1 and f2. So that’s all we see in the FFT. Nothing, ever, at the difference frequency f2-f1.

 

CTstanzio’s  frequency spectrum, and ones I’ve taken myself, many times, clearly shows the Tartini tone, as well as the sum and frequency doubled signals predicted by theory. If the Tartini tone can be seen in the FFT, it can’t have its origin in beats. No way.

 

So, how can we generate a difference frequency signal, that will show on an FFT, from 2 waveforms of frequency f1 and f2?

 

We have to arrange it so that we multiply the waveforms together, rather than adding them. For waveforms on paper, we can of course, just calculate the product and display the results on our computer screen. To do it for physical waveforms we have to find a physical way to multiply or “mix” the waveforms together. This is done by passing the waves through a nonlinear system. (see the attached appendix file- I'm not going to spend ages typing out all those trig functions here )

 

One of the terms produced in the nonlinear mixing of our  S = sin (2πf­1t) + sin (2πf­2t) signal is a multiplicative one. You’ll need to look at the attached file to see exactly how that arises. The upshot is, that after application of trig identities,  we can write the multiplicative term as:

2Cos (2πf­1t)* Cos (2πf­2t) = Cos (2π(f­1+f2)t) + Cos (2π(f­1-f2)t)

(i've changed from sin's to cos's for consistency with the appendix. As I'm sure you know, this doesn't matter. A cos is a sin with a 90 degree phase shift)

 

The second term on the RHS is the tartini tone, the other the sum frequency tone. 

Fourier transforming this gives you peaks at the sum and difference frequencies in the FFT spectrum.

 

So hopefully, we can see how nonlinearity leads to the tartini tone. This nonlinearity can be in the auditory system, the instrument , the microphone/electronice you use to record the signal, etc etc. 

 

So there you go.

JC

 

 

JC_TARTINI_APPENDIX.pdf

[Shunyata]

I started this posting with the comment that I didn't want to get into a debate of whether Tartini Tones are real... but I knew we would wind up there. 

I would point out that a tone is a standing wave of high and low pressures, with amplitudes that oscillate at a fixed frequency.  An interference tone (resulting from two fundamental tones) is identical to a fundamental tone is this regard... a high amplitude that occurs with lower frequency.

Mathematically an interference tone isn't anything extra - it happens automatically when you have two fundamental tones.  FFT picks out the fundamental tones and ignores the automatic interference tone - unless the amplitude is different than you would expect from adding two fundamental tones. 

If C + E automatically produces a lower G (what my tuner says) then FFT would not show the G, unless the G was quieter or louder than you should mathematically get by adding C + E.

Ears are nonlinear in the sense that they do not perform an orthonormal decomposition of the sound signal (e.g. FFT) and discard the automatic interference pattern.  Ears hear the G!  Viva la difference!

I suspect Tartini Tones are just audible interference tones.  For some reason the interference tones are damped on some violins.