Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum Simultaneous partials in a string: reality or myth?

In the P12 thread there was a side discussion on partials.

BDB stated the notion that a string would have to "disassemble itself to vibrate at a variety of different sine waves".

Kees agreed [Edit: in principle], stating that "decomposing a sound into partials is a (often useful) approximation invented by humans, no more, no less".

Pyropaul, on the other hand, wrote that "The string has many different modes of vibration and they all occur simultaneously - it doesn't have to "disassemble" itself."

Essentially, BDB says that observing partials is only an artefact of a stroboscope: "You are describing an effect of the strobe, not of the vibration of the string. It is statistical, rather than physical, sampling the position of the string at specific moments."

Jeff, OP of the P12 thread, asked for a new topic to be started, which is what I'd like to do here, in response (rebuttal, really) to BDB and Kees.



As did we.

If your demonstration delved into partials (which it should have, if it was any good), you should have seen something like this (again, these are taken without any stroboscopic effects, they are blurred long-time exposures):
(taken from metasynthesis.com)
(taken from physicsthebook.com)
Partials are no approximation. They are real, and they can be excited simultaneously. The resultant string movement is simply the sum of the partials at each point in space and time. For example:
(taken from projects.kmi.open.ac.uk)
And an animation showing the sum of partials 1, 2 and 3:
(taken from scratch.mit.edu)

What pyropaul wrote, is therefore true: any periodic oscillation can be expressed/reconstructed as a sum of sine and cosine waves (with individual frequencies and amplitudes). This is the gist of Fourier transforms, and the frequency spectra that Kees himself has been posting here lately.

This page also describes it rather well:
http://www.acs.psu.edu/drussell/Demos/string/Fixed.html

Partials are no "invented approximation", and there is no need for strings to "disassemble" themselves in order to vibrate simultaneously at different frequencies. What the strobe captures, is no statistical artefact, but a snapshot of a real vibration at the strobe's frequency.

Mark,

thank you very much for posting this. I don't understand why some people were so vehemently opposed to the idea. Piano tuners, of all people, are the most aware of partials in a vibrating string.

Don't forget, also, the discussions previously about longitudinal and transverse waves. These are also just as real as the partials we hear.

As you state, the position of a vibrating string in time is described by the superposition of multiple modes of oscillation with different amplitudes, frequencies and phases. And, as I mentioned earlier, inharmonicity is due to the stiffness of the string making for an effectively shorter string for the higher modes of vibration.

Paul.

To look at it another way, "If a tree falls in a forest and there is no one around to hear it, does it make an noise?"

Those little "hairs" in you inner ear is the interface between what is happening with the air that is being moved by the string and what you percieve. It could be argued that those little hairs is what breaks the string into little pieces that vibrate at different frequencies. Each little "hair" has it's own resonate frequency.

Piano Tuning 101.

"A vibrating wire subdivides itself into many
simultaneously vibrating fractions. Each fraction
produces an audible pitch of its own, called a partial."

- A. Reblitz
(See his illustration, 1993 ed., 6-2, p204)

I would have thought this is an absolute no-brainer. It staggers me that BDB would claim what he did. Not only are they measurable, but the human ear can selectively focus on many different partials as separate tones - provided they have good listening skills and know what they are listening for. I mean, you tuners are talking about hearing these things every darn day - surely you can't claim them to be mythical when you are using and hearing them in your tunings and inharmonicity judgements? As a guitarist, I am intensely aware of these partials and how they can be exploited. They are certainly real. Moreover, it's the relative strengths of different partials that help us to distinguish one instrument from another (coupled with attack/sustain/decay envelope).

It's not as simple as that. The hairs in your ear don't cover the 10 octaves we can hear - there's no way you have a hair long enough to resonate at C0, for example. The string's motion is described by the superposition of all of its harmonics, same for the motion of the air molecules that convey the energy from the string to your ear. Microphones don't have resonant hairs but they also faithfully reproduce the harmonics when the sound is converted into an electrical signal. The change of voltage is also perfectly described by superposition of sine waves too.

Paul.

It's probably more simple than QP. The little hairs detect, though I may be splitting hairs.

If the guy that wrote This Thesis was still around, he might be able to explain it. (Yeah, that's my Dad! )

It's not the hairs that resonate. It's the basilar membrane between the two cochlear channels. Pitch is not recognised by hair length, but by the point of maximum vibration of the basilar membrane. The highest frequencies are sensed right at the point of entry (oval window), where the basilar membrane is stiffest, while progressively lower frequencies cause the basilar membrane to vibrate progressively deeper into the cochlea, where it becomes progressively more supple.

Stiff bodies have high resonant frequencies, supple ones have low resonant frequencies.

http://en.wikipedia.org/wiki/Cochlea#Detailed_events
http://bio1152.nicerweb.com/Locked/media/ch50/pitch.html

So, the cochlea does cover the 10 octaves we can hear, but without the need for 10 octaves (2^10 = 1024) of hair lengths. It's the stiffness gradient of the basilar membrane that differentiates pitch.

OK, Mark, but the question remains, and I think it is the point that BDB is making in an argumentum ad absurdum, do the partials exist outside of observing them? Like does the sun rise or does the earth turn?

Yes of course they do, unless we want to go down a rat-hole of talking about quantum reality. But otherwise, partials are real, they are not an artifact of our hearing system. I find it quite surprising that so many seem to doubt this.

Would anyone argue there is such a thing as white light? At one time, yes, but Newton showed the more interesting reality. It's essentially the same argument with sound - all sound is composed of sine waves of various frequencies and intensities. There is no "piano wave" or "trumpet wave" anymore than there is a "white ray".

Paul.

Piano-sound vibrations are not periodic.

Kees

Are there square waves, triangle waves and saw tooth waves, or just superimposed sine waves?

Just the latter - it is the superimposition that creates the waveshapes we see as square, triangle or sawtooth on an oscilloscope. Try it with an analog synthesiser and see what you get when you add sine waves of the right frequencies and amplitudes.

Paul.

So? What is it your harmonic analysis of beat rates is showing, if not the partials that make up the sound? Of course the amplitudes and frequencies may vary in time, but the resultant waveform can still be perfectly described by the appropriate sum of sines at the right frequencies, amplitudes and phases. Everything goes back to the basic physics of simple harmonic motion - motion that cannot be decomposed any further. This may also be the reason that we hear sinewaves as "pure".

Paul.

How about bells? You even have a nice java applet that generates the sound of bells.

Kees Bell Generator

The write up even states: "The bell has 50 modal frequencies"

So if a bell can be composed of 50 frequencies, why do you think a piano note cannot be similarly composed?

Paul.

Jeff,

I don't think BDB's comment was a re-phrasing of the (in)famous philosophical question, "does the act of observation make it any more or less real". Read my opening post again. BDB was questioning the very presence of actual partials in a vibrating string, and the fact that a strobe makes them visible, he called a statistical artefact.

If you first strike a note on the piano and then touch the speaking length exactly halfway, the second (and fourth, sixth etc.) partial will continue to ring for a while. The same goes for the other partials. Now the act of touching cannot generate any partials, as it doesn't impart any energy to the string. This means that all those partials were already physically present in the string before you touched it.

The movement of the string is the sum, no more, no less, of all the partials physically present, each with its own frequency, amplitude and phase. Granted, as soon as more than two or three partials are combined in one string, the resulting shape of the string at any given moment will be incredibly complex. Nevertheless, the partials are all there, each one of them very much physical and real.

Whether the string vibration is observed
a) in the time domain (overall string motion vs. time), or
b) in the frequency domain (individual partials resolved by cochlea or by stroboscope), or
c) both, or
d) neither,
doesn't change the vibration [Edit: being a sum of sines[/i] one bit.

In other words, the information that's carried in both domains is exactly the same, and a transformation from time domain to frequency domain or vice versa doesn't add or remove any information from the signal.

A "sine wave" with a time dependent amplitude is not a sine wave. It does not even have a well-defined frequency.

In fact my "partials" represent the best approximation by a sum of non-sine wavelets, where each wavelets is the product of an exponential and a sinusoid. That is also a matter of choice, there are arguments the exponential envelope is better modeled by the product of a power function and an exponential. For the case of hand, it makes little difference for the approximately harmonic partials that are of interest here.

I assure you if you reassemble all the partials and resynthesize the sound you do not get the original sound back: just the part that is well approximated as a sum of partials.

This is really all old well-known stuff, investigated in depth in the literature. For example there is a sound synthesis method from the '80-ies by Rodet where you explicitly split the sound into something that is modeled by partials + a residual that can't be modeled like that.

No offense, but a little knowledge is a dangerous thing.

Kees

Kees,

I'm well aware of the approximations introduced by considering band-limited signals. Do you believe that a vibrating string simultaneously contains multiple modes of vibration? I think we're getting to the point where we're effectively arguing about the number of angels on the head of a pin. Harmonics are a real phenomenon and are used daily by piano tuners.

Paul.

So, Kees, if there are no real partials present, as BDB maintains, what gets the soundboard vibrating at those frequencies - as you've kept showing us in your spectrograms? What creates those (periodic!) beats when tuning an interval?

By the way, I would have thought it rather obvious that we're not considering the decay envelopes at this stage. We're considering constituent frequencies, and for such purposes, a slowly damped sine wave can still be considered a sine wave, I would submit.

[Edit: but perhaps, nothing is obvious. I'm not a physicist or signal processing expert... Just battled through two years of university physics.]