Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum Do lower partials always stay at the same amplitude ratio?

Does anyone know of experiments or formal studies about the relative amplitude of low and middle partials when a piano string is struck with varying amounts of force?

Obviously, higher partials become much more prominent with hard strikes. My question is, instead: Does the ratio of the strengths of lower partials remain exactly the same, on a given note, as force increases or decreases? More plainly, if a soft strike creates, among many partials, a fundamental and a 4th partial that are at a ratio of, say, 6:1, does that ratio persist regardless of the amount of force delivered by the hammer?

My common sense says yes, the ratio probably stays the same, but my common sense has sometimes been wrong. One thing that occurs to me is that, although the string may react linearly, the bridge and soundboard may not.

Try this paper.


http://www.acs.psu.edu/drussell/publications/pianohammer.pdf

In a similar vein to the 1998 paper Mwn mentions are the results on Steinway hammers in Henry Scarton and others' 1996 US patent# 5,537,862.

An easier way to see what is happening is to use n-Track Tuner on i-Phone. This free app shows a dynamic graph frequency analysis up to 20 kHz in decibels.

When I tried it quickly this evening my impression was that ratio of all other partials to the fundamental tends to increase with force, more rapidly for the higher ones.

Would a piano have much of a tonal range if this were not so?

Well, I'm not worried about the high partials so much. I understand that they leap in amplitude with hard strikes. But just to be sure that I understand:

You seem to be seeing that, in my example of the fundamental and the 4th partial, that if a soft blow gives an amplitude ratio of 6:1, with a harder blow, the ratio shifts--the result might look more like 5:1? In other words, the 4th partial gains in amplitude more than the fundamental on the harder strike? (You wrote that all of the partials seemed to gain on the fundamental, but I'm focusing on the lower partials.)

About the timbre range. My impression is that, yes, even if the amplitude ratio in the lower partials remains the same, a piano will still have a wide tonal range. The increasing inharmonicity of the upper partials means that as they become more prominent with harder strikes, the tone changes greatly, since the added pitches are increasingly sharp.

Thanks for this link. Isn't the article concerned with the frequency range however--how harder strikes create a non-linear increase in the upper frequencies? In this sense, yes, the ratio of the lower partials to the upper partials does change. But I meant to ask if the lower partials retained the same amplitude ratio in relation to each other, regardless of the increase in the freq range and the amplitude of the upper partials.

I focused on the lower partials, especially the 2nd, 3rd and 4th at pp to mf, for notes in the middle of the keyboard. At pp those appeared to be reduced in amplitude and at ppp their peaks were lost within the background noise patterns, at least at first glance.

At ff the partial peaks soared above the noise up to 5 - 10 kHz.

Of course the ear may be able to detect what the iPhone doesn't and fish the partials out of the the background.

Ian I amafraid that a simple recording will show much mor ethan the ear is able to catch.

Noise seem to be fairly low, then there are alsi some frequencies that appear and go, possibly depending of the force the note is played.

(plus of course all the phantom frequencies that are probably produced by the recording, the sampling and analysis processes, thye converisons from a standard to another, etc)

(while I dont know if this can really create partials.

The question asked by the OP doe snot seem to take in considertaion that we cannot listen to a string in its non transduced condition, so everything is filtered and the rendering, acoustically is modified, in both diretcions, the strings tension certainly modify the equilibrium state of the soundboard assembly as the assembly mass influences the string.

Then all became a whole complex system where separated elements are really not easy to analyse.

Jake, this is basically what voicing is in the end, playting with teh damping of returning waves, with the filtering of the impact tone by the hammer/shank and plate, (and certainly other components I am not aware of).

Changing the contact time between hammer and string modify the spectra as the enveloppe. more fundamental, or more partials ? it depends of the instrument.

I will make your recordings next week probably.

Isaac, I agree with all of that.

The app I mentioned displays the frequency analysis in real time so you can watch the partials rise up, jump around and decay. The pattern over time is quite complex and, as you say, it depends on the piano and the tuning.

What the eye sees on the screen may not be what the ear hears. That is another complex subject which I do not understand in detail. I do know the ear can continue to hear a decaying partial tone as louder than it is. That depends on how loud it seemed when the key was struck.

Ian it is also interesting to filter and focus only on some partials or a range of frequencies. (f1*4 etc)
You slow the speed, or also lower the pitch and you discover the partials that are not usually noticed

How all the partials contribute to the tone is an interesting question. Maybe we should have a topic about that.

Coming back to the original question about the the relative amplitudes of the lower partials, the Professor of Physics at Manchester University, Brian Cox, gave an interesting televised talk on fundamental principles to an invited audience recently. He was talking about the wave like nature of electrons and invited two comedians to come on to the stage.

He gave them a rope and asked each to hold one end and demonstrate how waves travelled along it. They soon had it vibrating with a peak in the middle (fundamental, first partial). Brian Cox then asked them to try a bit harder and we saw two out of phase peaks (second partial). Finally the two comedians made a frantic effort and, to Cox's surprise, produced three peaks and troughs in the rope (third partial).

The point was you need to expend more and more energy to excite more and more harmonics. There are many complicating factors in a piano but the basic principle appears to be that the ratio of amplitudes will depend on the energy of the hammer blow.

Yes, more energy is needed, but my question is: do the lower partials vary 1:1 with an increase in energy?

Oleg: It occurs to me, now, that the program I have mentioned, Spear, is what I need to use to test this out. I have more than one sample library, so I can do some work there. I predict that I will find that, as you say, there are many variables, including ghost partials and resonances, that will make it hard to say, simply, yes.

Ys Jake I confirm that Spear programm is really excellent to analyse and try to reproduce tones,

(while I dont know how much of the functions provided are necessary to build a piano tone from scratch)

It may be possible to record a few samples yourself, and test the iH level with Tunelab for instance, then verify if the 2 measures are similar.

That said, the Spear software works in Hz (?) , the precision is possibly not enough then.

The tone on a typical piano string sort of regenerate by itself after an initial drop, it is possible that iH raise a little with a stronger impact .

Anyway the level of raise in pitch during the attack is large it was surprizing to notice a pitch appearing so soon, I was expecting noise not something with a definite pitch.

(while noise is a mix of a lot of pitches together I suppose)

Jake, what I meant by "the basic principle appears to be that the ratio of amplitudes will depend on the energy of the hammer blow" is that none of the partials appear to vary 1:1 with an increase in energy.

From what I saw, and from what I understand, the amplitude of a partial relative to the fundamental increases with energy.

I suspect the extent of this depends on the piano, tuning and hammer voicing.

Why would you like the ratio to be 1:1?

This picture of C2 played at pp and then at ff seems to that the lower few partials preserve a relative intensity, but that the upper partials increase, not remain relative.

http://www.acs.psu.edu/drussell/Piano/Figures/C2-ff-pp.gif

Are there graphs for other keys and how does one get to them?

Just C2, C4, and C6.

Here's the link:


http://www.acs.psu.edu/drussell/Piano/Dynamics.html

I find it interesting that, while the soundboard is not a good radiator of frequencies below it's fundamental resonance (in the case of a large grand, around 60Hz) our ears and brain fill in the information and allow us to enjoy the sound right to the bottom of the instrument.

Also, I find it amazing, though logical, that the non-linearlity of a well voiced hammer can produce such a variation in the partial structure and the resulting gorgeous shift in richness of the tone.

Well, some of the lower partials change, too. The 4th and 5th partials, among others, show a large change in their ratio to the strength of the fundamental.

It isn't that I want the ratio to be 1:1 (an equal response to force). The question arose, in part, because I often see statements about the relative amplitudes of the partials that, to me, imply or assume that their relation to the fundamental remains constant, and that all that happens with harder strikes is that there is an increase in the upper partials and in transients. In fact, that is the assumption made in many discussions of Fourier analysis: a diagram is shown that demonstrates the partial series, with the usual descending amplitudes. I've suspected that there is more deviation in even the lower partials, caused by a varying force. Now I want to see what fairly gradual changes take place: Is there a predictable sequence? Does the 5th partial, for example, usually tend to increase more than others, and by what factor with what amount of force. The cause is, of course, the background concern. Unisons will probably be the major cause. But I'm more interested in just getting the data, first.