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

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.

One thing that is true, is that for a given amount of energy, the lower harmonics create more string movement.

Not necessarily true. A string which vibrates in a higher mode will have more energy at that mode than at the lower modes. This may happen in the lowest notes of a piano.

This would be similar to whipping a jump rope so there is a node in the middle of it. It vibrates at the second mode, not the primary mode, so there is more movement in the second harmonic than the fundamental.

Just C2, C4, and C6.
Here's the link: http://www.acs.psu.edu/drussell/Piano/Dynamics.html

I don't think that's correct.

Thank you Mwm. This article describes and illustrates the effects of on piano sound of the non-linearity Del mentions.


Maybe I did not put my point clearly, but the frequency analyses of C6 in the article show 2 or 3 partials at pp (less energy) and 4 or 5 partials at ff (more energy).

It's quite obvious that the ratios of the second and third partials differ in the two diagrams.

On my piano the frequency analyses of C6 show only the fundamental at ppp but eleven partials at fff. That is what I meant by more and more energy exciting more and more harmonics.

I would like to toss in what I have done to understand how partials work, which is probably the opposite of what a lot of you do. I start by assuming that the vibration can be approximated by a Fourier series, or rather something close to it. The simplest version will be the summation of over n of 1/n*sin(n*t). In this case, n would be the partials and t would be time. A decent graphing calculator can show you that for n ranging from 1 to 8, which is a decent approximation for a string struck at 1/8 its speaking length. When I do that, it looks remarkably like the way that I would expect the string to look almost immediately after striking, which show it is a reasonable assumption. Then you can add (multiply, really) in a damping factor, which would be something like 1/t. This maintain the ratio, but the higher harmonics would be smaller and smaller and have less of an effect. If you think that there is more of a drop-off of the higher harmonics, it might be something like 1/(t*n) with maybe some fudge factor tossed in.

It is just that it is often easier to come up with a mathematical theory that seems reasonable, and then see if it approximates the physics, than to look at the physics and hope to come up with the appropriate mathematics.

-1 is an exponent, and a hyperbola has an area of rapid decay followed by slower decay. That has nothing to do with the question, which is whether the decay also varies according to n, the partial. That is the question that physics has to answer. It probably does, but then, it is probably negligible compared to the hyperbolic decay.

I believe that research has shown that the decay of a piano tone has two distinct regions that don't follow a smooth curve such as one exponential or a hyperbola. I also believe that there is no doubt that the higher partials decay more quickly. To put it another way, the relative amplitudes of the various partials change substantially as the note decays.

I hope to be able to check some recordings in about two days. That will be on Friday. Unfortunately, I will only be looking at recordings of one or two notes from one or two pianos, so I don't think any large questions will be answered.

About the hammer, however: remember that I'm curious about the relation between force and the relative lower partial amplitude. If the hammer is nonlinear, that will only mean that the amount of force delivered to the wire is nonlinear. So, while the overall amplitudes of the lower partials may make sudden leaps, isn't it possible, and predicted by Fourier, that the relative amplitude of the lower partials will remain constant? (Although, yes the upper partials will also enter the picture and affect the tone.)

But, yes, the recently posted pictures of soft and hard strikes appear to contradict that expectation. I hope to see if there is a predictable pattern. Do the same lower partials always tend to get louder as force increases? Is there a predictable relation between their amplitude and the amplitude of the fundamental? In predictable increments or within predictable ranges of increments? If so, of course, the question of the causes arises. Sympathetic resonance of other strings picked up by the mic (which would make sense for the 5th partial, at least)? Unisons (but shouldn't they couple, as much as they ever do, as force increases)? Difference or sum tones created by the interaction between the lower partials and the upper partials as they increase in amplitude with force? Or is it the wire itself? Assuming that there is a steady pattern of some kind, and we can't say that yet...

You cannot say much about something which is non-linear (whatever that may mean in the case of a hammer). There are more curves than lines.

Tried mind-reading to find out what publication it might be, found nothing there. In any case, it has nothing to do with the original question.