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#2040095 - 02/27/13 11:26 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
BDB Online   content
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Registered: 06/07/03
Posts: 21522
Loc: Oakland
-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.
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#2040116 - 02/27/13 12:20 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: BDB]
Roy123 Offline
1000 Post Club Member

Registered: 09/20/04
Posts: 1710
Loc: Massachusetts
Originally Posted By: BDB
-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.


Edited by Roy123 (02/27/13 12:22 PM)

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#2040411 - 02/27/13 09:49 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Roy123]
Jake Jackson Offline
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Registered: 02/17/09
Posts: 580
Loc: Atlanta, GA
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...


Edited by Jake Jackson (02/27/13 09:52 PM)

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#2040432 - 02/27/13 10:43 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Mwm Offline
500 Post Club Member

Registered: 02/20/13
Posts: 752
With a non-linear hammer, as the force increases, the hammer becomes harder, which decreases the hammer/string contact time. Shorter contact time means fewer partials are quenched by the hammer felt.

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#2040573 - 02/28/13 06:54 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Roy123]
Olek Offline
7000 Post Club Member

Registered: 03/14/08
Posts: 7540
Loc: France
Originally Posted By: Roy123
Originally Posted By: BDB
-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.


WIth just one string we can see a drop then a hop, under some circumstances, I dont know what they relates too, but the curve is all but straight then.

Sometime the curve is way less pronounced, more a straight slant, that gives the impression that less manipulations of the tone are possible.

Of course when you add another string, plus another one, the interations (modified by the tuning) can go in the direction of straightening, or inflationg the original curve.
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#2040600 - 02/28/13 07:49 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Roy123 Offline
1000 Post Club Member

Registered: 09/20/04
Posts: 1710
Loc: Massachusetts
Originally Posted By: Jake Jackson
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...


The principal effect of the nonlinear springiness of hammers is to change the force vs. time curve of the hammer as it presses on the string(s). At a light blow, the hammer's effective spring constant is lower and the hammer will stay in contact with the string longer. At a hard blow, the higher effective spring constant will cause the hammer to spend less time in contact with the string.

This is getting technical, but if you calculated the Fourier integral for the force vs. time curve, you would end up with a distribution of energy vs. frequency. The longer the hammer is in contact with the string, the less energy there is at higher frequencies.

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#2040712 - 02/28/13 11:14 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
BDB Online   content
Yikes! 10000 Post Club Member

Registered: 06/07/03
Posts: 21522
Loc: Oakland
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.
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#2040769 - 02/28/13 01:11 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: BDB]
Roy123 Offline
1000 Post Club Member

Registered: 09/20/04
Posts: 1710
Loc: Massachusetts
Originally Posted By: BDB
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.


Sure you can--if you have some data. Check out the publication on the piano hammer as a nonlinear spring--it will show just what I said. Besides, any spring that becomes stiffer as it compresses ("which is what it means in the case of a hammer") will produce a quicker bounce as its compression increases--unless there are additional esoteric effects that come into play.

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#2040925 - 02/28/13 05:26 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
BDB Online   content
Yikes! 10000 Post Club Member

Registered: 06/07/03
Posts: 21522
Loc: Oakland
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.
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#2041211 - 03/01/13 07:20 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: BDB]
Roy123 Offline
1000 Post Club Member

Registered: 09/20/04
Posts: 1710
Loc: Massachusetts
Originally Posted By: BDB
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.


Here's the link. I typed in 'piano hammer nonlinear spring' into Google and it was the 1st item in the list.

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