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#2038562 - 02/24/13 04:59 PM Do lower partials always stay at the same amplitude ratio?
Jake Jackson Offline
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Registered: 02/17/09
Posts: 580
Loc: Atlanta, GA
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.

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#2038576 - 02/24/13 05:18 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Mwm Offline
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Registered: 02/20/13
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#2038641 - 02/24/13 07:01 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Withindale Offline
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Registered: 02/09/11
Posts: 1940
Loc: Suffolk, England
Originally Posted By: Jake Jackson

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?

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?


Edited by Withindale (02/24/13 07:13 PM)
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Schiedmayer & Soehne, 1925 Model 14, 55" upright
Ibach, 1922 49" upright (project piano)

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#2038656 - 02/24/13 07:45 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Withindale]
Jake Jackson Offline
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Loc: Atlanta, GA
Originally Posted By: Withindale

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.

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#2038687 - 02/24/13 09:00 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Mwm]
Jake Jackson Offline
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Loc: Atlanta, GA
Originally Posted By: Mwm


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.

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#2038764 - 02/25/13 12:58 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Withindale Offline
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Registered: 02/09/11
Posts: 1940
Loc: Suffolk, England
Originally Posted By: Jake Jackson
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.)

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.
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Schiedmayer & Soehne, 1925 Model 14, 55" upright
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#2038789 - 02/25/13 02:43 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Olek Online   content
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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.
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#2038791 - 02/25/13 02:48 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Olek Online   content
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Registered: 03/14/08
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Loc: France
Originally Posted By: Jake Jackson
Originally Posted By: Mwm


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.


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.



Edited by Olek (02/25/13 02:48 AM)
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#2038881 - 02/25/13 07:23 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Olek]
Withindale Offline
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Registered: 02/09/11
Posts: 1940
Loc: Suffolk, England
Originally Posted By: Olek
Ian I am afraid that a simple recording will show much more than 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.

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.
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Ibach, 1922 49" upright (project piano)

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#2038901 - 02/25/13 08:07 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Olek Online   content
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Loc: France
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


Edited by Olek (02/25/13 08:08 AM)
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#2038921 - 02/25/13 08:57 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Olek]
Withindale Offline
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Registered: 02/09/11
Posts: 1940
Loc: Suffolk, England
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.
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#2038989 - 02/25/13 12:12 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Jake Jackson Offline
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Loc: Atlanta, GA
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.

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#2038995 - 02/25/13 12:27 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Olek Online   content
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Registered: 03/14/08
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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)
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#2039001 - 02/25/13 12:44 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Withindale Offline
1000 Post Club Member

Registered: 02/09/11
Posts: 1940
Loc: Suffolk, England
Originally Posted By: Jake Jackson
Yes, more energy is needed, but my question is: do the lower partials vary 1:1 with an increase in energy?


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?
_________________________
Ian Russell
Schiedmayer & Soehne, 1925 Model 14, 55" upright
Ibach, 1922 49" upright (project piano)

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#2039064 - 02/25/13 02:40 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Mwm Offline
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Registered: 02/20/13
Posts: 752
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

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#2039089 - 02/25/13 03:06 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Mwm]
Withindale Offline
1000 Post Club Member

Registered: 02/09/11
Posts: 1940
Loc: Suffolk, England
Originally Posted By: Mwm
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?


Edited by Withindale (02/25/13 03:06 PM)
_________________________
Ian Russell
Schiedmayer & Soehne, 1925 Model 14, 55" upright
Ibach, 1922 49" upright (project piano)

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#2039093 - 02/25/13 03:08 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Mwm Offline
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Registered: 02/20/13
Posts: 752
Just C2, C4, and C6.

Here's the link:


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

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#2039099 - 02/25/13 03:14 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Mwm Offline
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Registered: 02/20/13
Posts: 752
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.

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#2039124 - 02/25/13 03:48 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Mwm]
Jake Jackson Offline
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Registered: 02/17/09
Posts: 580
Loc: Atlanta, GA
Originally Posted By: Mwm
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


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.

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#2039134 - 02/25/13 03:59 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Withindale]
Jake Jackson Offline
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Registered: 02/17/09
Posts: 580
Loc: Atlanta, GA
Originally Posted By: Withindale
Originally Posted By: Jake Jackson
Yes, more energy is needed, but my question is: do the lower partials vary 1:1 with an increase in energy?


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?


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.



Edited by Jake Jackson (02/25/13 04:03 PM)

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#2039137 - 02/25/13 04:04 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Olek Online   content
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Registered: 03/14/08
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There is something you may take in account and it is the tuning, even dampened notes are modifying the spectra of the upper notes.

ANd it is noticed (more in the 2 top octaves) any note played get reactions from a lot of others, open or not . I just made a recording where notes out of line have a less clear spectra, some false beats, that get cleared once thy line well at the octave and double octave level
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#2039204 - 02/25/13 06:21 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Withindale Offline
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Registered: 02/09/11
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Loc: Suffolk, England
Jake, I've had a closer look this evening and it does appear that
as a general rule the power of the partials that are excited have for a note have a reasonably constant relationship to each other from pp to ff. There are some fluctuations and there are wide variations across the keyboard. I wouldn't be surprised if some of those are due to soundboard characteristics.


Edited by Withindale (02/25/13 07:09 PM)
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Schiedmayer & Soehne, 1925 Model 14, 55" upright
Ibach, 1922 49" upright (project piano)

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#2039361 - 02/25/13 11:48 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
Del Offline
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As has been suggested, there are many variables affecting the relative strengths of the fundamental and various harmonics in the tone envelope produced by the piano.

With this particular question one of the most important of these variables is the resilience characteristic of the hammer. Assuming the hammer is acting as a non-linear spring the relative amplitudes of the all of the vibrating partials in the struck string will vary depending on the velocity of the hammer at impact. Depending on the scale—the note in question, the speaking length of the string, its diameter and, hence, its tension—this relationship may vary more or less (I’ve not actually tested to determine how much) but the relationship between the energy at the fundamental and that of all of the partials does change depending on hammer impact velocity.

A very hard hammer—i.e., one not acting like a non-linear spring—will produce less variation in the respective amplitudes of the vibrating partials no matter its velocity at impact. This is why pianos with hard, dense hammers tend to sound linear—i.e., loud and less-loud with little or no timbral change—no matter how it is played.

ddf
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#2039454 - 02/26/13 07:45 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Withindale]
Roy123 Offline
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Originally Posted By: Withindale


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.


I don't think that's correct. Say you impart some energy into a stretched string by striking it as in a piano. That energy can be distributed across a number of harmonics of the string. The distribution of energy across the harmonics will be influenced the placement of the strike, i.e., where along the string the strike occurs, as well as the temporal characteristics of the strike, i.e., how much time it takes for the force of the strike to build up and then die down.

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

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#2039591 - 02/26/13 12:11 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Roy123]
BDB Online   content
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Originally Posted By: Roy123
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.
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#2039613 - 02/26/13 12:54 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: BDB]
Roy123 Offline
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Originally Posted By: BDB
Originally Posted By: Roy123
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.


You misunderstood my point. What I tried to explain is this; imagine a string is vibrating at its lowest mode, with a certain amount of energy. Now, let's say you cause the same string to vibrate at one of its higher nodes at the same energy. To be clear, we are not superimposing modes--in the first case, only the lowest mode was active, and in the second case, only a higher node is active. One will observe that the string displacement is higher for the lower mode.

Nothing about which node is vibrating implies anything about energy--the energy of any mode is determined by how that mode is excited, be it by a piano-hammer strike or by any other means.

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#2039823 - 02/26/13 07:33 PM Re: Do lower partials always stay at the same amplitude ratio? [Re: Mwm]
Withindale Offline
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Originally Posted By: Mwm
Just C2, C4, and C6.
Here's the link: http://www.acs.psu.edu/drussell/Piano/Dynamics.html

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

Originally Posted By: Roy123
Originally Posted By: Withindale
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.

I don't think that's correct.

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.
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#2039959 - 02/27/13 02:40 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Jake Jackson]
BDB Online   content
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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.
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#2040018 - 02/27/13 08:16 AM Re: Do lower partials always stay at the same amplitude ratio? [Re: Withindale]
Roy123 Offline
1000 Post Club Member

Registered: 09/20/04
Posts: 1710
Loc: Massachusetts
Originally Posted By: Withindale
Originally Posted By: Mwm
Just C2, C4, and C6.
Here's the link: http://www.acs.psu.edu/drussell/Piano/Dynamics.html

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

Originally Posted By: Roy123
Originally Posted By: Withindale
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.

I don't think that's correct.

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 think the most likely cause of the higher partials at fff is the nonlinear compression of the hammer, which causes the hammer strike to be quicker, i.e., the hammer bounces off the string more quickly. Another possibility is some small amount of nonlinear behavior of the string or soundboard at a very forceful hammer strike.

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#2040020 - 02/27/13 08:22 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
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.


I think your have made a good start. You will find that the decay is much more complicated, however. The decay tends to be exponential, not linear, and pianos always show two regions of decay, an initial region in which the decay is more rapid, and a secondary region in which the decay is quite a bit slower.

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