Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum Quadra(tic) effect (it is not about the age of the captain)

I find the original explanation on the quadractic effect (I am unsure of the name in English)

Back and forth motion of the bridge product frequencies based on the square of the frequency of all partials, the first frequency is then the octave ( a pure octave, no iH)
The taller the bridge the more preeminent the quadratic effect
It is non linear depending of force not as L modes

Same orientation (transverse) than the L Modes frequencies

Presence can be even stronger than the partials of the string

Here is the original definition by Philippe guillaume, Pianoteq designer/developper. (in French, sorry)

Talk of a new parameter in tone, not really known...

c'est l'effet de tirage produit par les cordes sur le chevalet (qui se met donc a "tanguer"). La variation de force exercee dans cette direction par la corde est le carre de celle exercee transversalement, ce qui introduit de nouvelles frequences de la forme f_i +f_j, les f_i etant les frequences des partiels de la vibration transversales. Ca agit dans la meme direction que les ondes longitudinales, mais ce ne sont pas les memes. La premiere frequence aisni obtenue est a l'octave *exacte" du fondamental (donc legerement inferieur au second partiel - à cause de l'inharmonicite)

Hi Isaac,

Thank you very much for the information above. I wonder if what you posted in French is only a part of a paper by Philippe Guillaume, in which case it would be nice to have the whole paper.

You say: ..."Presence can be even stronger than the partials of the string"...

Do you have any numerical data? Or perhaps more material on this subject?

Best regards,

Alfredo

Hello ALfredo, nice to read you.

I did not find much, the exerp is just from a conversation, and Philippe told me the power is variable and sometime it can be stronger than the real partials.

It is very mysterious, in the end. I wish to have an idea to the level of tone where the effect begin to be perceived.

As I have heard for instance M3 that had 2 sets of beats at the same level. One due to the 4th partial and 5th, the other with a different speed.
Was on STeiwnay D very lively, rich, progressive
Tested them with and ETD : the frequency was a mistery, did not match with the theoretical frequancy.

Double beating on M3, does it ring to you ? (ther is always one due to the beat at the octave above, but I believe it was something else, or a mix of partials used to regulate the speed. (at those times I was believning we have to listen only one partial beat

ALl the best


Isaac

Hi Isaac,

As you say, it remains a bit mysterious...

How does that 2:1 frequency affect the whole final tone? How relevant is it when it comes to unisons? How is it related to the phase of two or three vibrating strings?

On the other hand, perhaps we get one more clue about two facts:

firstly, that we end up having to deal with combined beats (you wrote: "Double beating on M3, does it ring to you ?"… My answer is yes, sure… also that "objective" phenomenon (like a double pulsation-rate) made me think that we can enhance/reinforce beats and get one more feedback on individual beat curves, in this case M3 beat curve);

secondly, that very small variations in string "tension" (read loads onto the bridge and beats geometry) can make a big difference, here referring to the overall resonance of any piano. Perhaps we all agree on this?

Regards, a.c.
.

Hello, I finally find one global writing about the "transitory grequencies at attack time" and where the quadratic effect is mentioned (did not read it yet, it may be complicated)

One of the things they state also is that there is a "precursory tone " due to coupling between L waves and transverse waves.
I seems that the quadratic effect is due to that coupling.

I ahet to tell you but I only find a French version of the paper.

It stresses on the importance of that coupling for the attack transitory.
Here is the doc : I hope it can be useful to some

http://hal.archives-ouvertes.fr/docs/00/53/97/71/PDF/000225.pdf

It seem to be that there is a combination of frequencies between Lmodes and transverse modes that creates a non linear behavior of the resultant frequencies.

If someone have some simplified version I will appreciate it !

To simplify even more you can measure quadratic partials, but also cubic ones as audible frequencies.

Non linear hammer & non linear bridge : non linear frequencies in all directions.

Find that in the sources :

That one may be of some interest as the same authors :
http://hal.archives-ouvertes.fr/docs/00/44/44/70/PDF/RR-7168.pdf

Then :

http://home.mit.bme.hu/~bank/publist/jasa05.pdf

http://home.mit.bme.hu/~bank/publist/smac03.pdf


P.S Interesting is the fact that the scientific description for a piano tone is 3 strings having a slightly different tension, for the formulas they dont' consider the 3 strings to have an exactly similar tension (at last in the first document)


PPS I wonder if the correct term for that effect is not :

"nonlinear hamiltonian systems of wave equations"

Which seems to be relatively recent

Nakamura and Naganuma found
in 1993 a second series of partials in piano sound spectra having one-fourth of inharmonicity compared to the main partial series

Isao Nakamura and D Naganuma. Characteristics of piano sound spectrum. 1993.

did not find that one

suite :
They attributed these to the horizontal polarization of the
string. Giordano and Korty measured in 1996 that the longitudinal motion amplitude is a nonlinear function of the transversal amplitude ([13]), which confirms that the longitudinal vibration is
generated by the transversal motion and not by the misalignment of the hammer, as some theories
had put forward. Conklin measured again the partials found by Nakamura and Naganuma, and
named them “phantom partials”. He observed that the vibration relation between transverse and
longitudinal modes greatly influences the tone quality ([8]), but also that the phantom partials
are generated by a nonlinear coupling of these modes, noticing that the measured frequencies are
sums or differences of the transversal model frequencies ([9]). Bank and Sujbert explained in 2005
that this result can be predicted by an approximated nonlinear model, coupling transversal and
longitudinal modes ([2]). The model used in their paper is an approximation of the “geometrically
exact” model introduced in [30]. The “geometrically exact” model comes from a geometric description of the string, a stress-strain relation and Newton’s law. It is often exploited in the literature
in an approximated form. The model and its approximations are presented in section (1).

This presentation has some good diagrams showing quadratic effects (page 33) and cubic effects (page 34). Also the effects of non-linearities and longitudinal modes on the tone of C2 on a Steinway D (pages 55-58).

This website explains things in reasonably plain English.

Isaac, would you ask Juliette Chabassier if any audio samples are available? See PM.

THanks, yes I will

On the same subject I find that (listening tests) :

You could ask on your side to the concerned persons , may be (samples should certainly be interesting ; using Pianoteq version>3.0? The quadratic effect can be regulated (I did not notice it clearly however , but never tetsted Pianoteq with ideal equipent)

http://lib.tkk.fi/Diss/2010/isbn9789526034133/article6.pdf

exerp :
6. Listeners’ opinion
Besides the tests, the listeners were interviewed about their musical background, previous experience in listening tests, how hard the test was, and, most importantly, how they describe the
difference between the tones with and without longitudinal components. The test was easier for
low tones and quite hard for the high ones, as the test results also suggest. The tones with
longitudinal components are considered more realistic by all listeners and the ones without are
often described as more synthetic. According to the subjects, the tones with longitudinal components sound “broader,” are more lively, and have a more powerful attack, especially in the low
range. Some subjects also indicated that the beginning of the tones with the longitudinal components is “out of tune,” harsher, or sound distorted, and the tones without longitudinal components are softer, duller, and even boring. Subjects reported that the difference is subtle for the
high notes, in agreement with the results of the tests.

I was surprised to see that the partials concerned are in high ranges. It was not what I wrote, nor I've been said. I wait for an answer...

Withindale,
Thank you for the link.
Olek,
It does seem to me that Quadratic Effect and Phantom Partials refer to the same phenomena. Do you agree? Thank you for your posting of this subject.

They probably are , while I suppose that "phantom partials" are created by the addition of the quadratic partials themselves due to a mix between normal partials and L mode . ?

Not exactly as Philippe Guillaumme explained me, as he was talking of low level partials, the second 3, 4th etc.

On the readings above, the partials considered are at f8 minimally which is very high (if I read well)

SO it is not totally clear to me yet.

I have read also that those tones are "quasi harmonic" (?)

They certainly appear above a certain strength, also..

Was not so conscious of the importance of Lmode for supporting the attack transient (?)

I am not able to read more than resumed explanations unfortunately, I could understand a process at large, not in its details.

I missed the recordings of simulated and actual sounds of D#1, C2, F3, C#5 and G6 on a Steinway shown on this page: http://modelisation.piano.free.fr/P...Entries/2012/3/27_Voyages_en_Orient.html

All in all, I think this shows that the "phantom partials" due the quadratic and cubic effects illustrated in Patrick Joly's presentation are a fundamental part of the story but not the whole of it.

The actual sounds sustain longer and seem more complex than the simulated ones. I suspect the behaviour of the soundboard and the rest of the piano is not yet fully understood. For example the influence of voicing and the plate and rim on tone.

I guess I should have understood that with the word quadra, the minimum is F8. WHen rubbing a string with a finger covered with rosin, the L mode can be clearly exited (as when rubbing a cristal glass)

a partial tone in the high range, that I will try to record.

Comparing that pitch using the lenght and the diameter of the string as reference, allow to compute/ascertain MOE/e-modulus) with some level of accuracy (15% ?)

whatever the tension, the L-mode will be always the same pitch so only the original lenght and string diameter are necessary.

I am impressed !

They did not even hire a decent piano tuner prior to their tests, I cannot understand how scientist could be as dumb

WHat a deception , this begin to go above my brain and my nerves !

No surprise that some of the public stated that the real tones where harsh and sound out of tune.

Then they also modelized a piano tone with a bad unison.

Thanks for all the concert tuners, devoted to get the best of the instruments ;

In fact I think we can feel insulted by such demonstration

Thank you, Isaac, and thank you, Ian, for your posting and those links.

Isaac, do not get angry... perhaps in a while... ;-) I too listened to those sounds:

http://modelisation.piano.free.fr/Piano/..._en_Orient.html

and to me too it sounds a bit pathetic.

I agree, Ian, the whole story is very complex, many variables and... many ways all variables can affect each other; which - beyond the urge of modeling - is not bad, as that makes every real piano unique and every "job" a challenge.

On models and string length, perhaps this adds on:

"Reduction of longitudinal modes in musical instruments strings":

http://www.patentgenius.com/patent/5874685.html

Regards, a.c.
.

THank you ALfredo, what I wish to understand better is how those modes relate to the strength of the attack and how e tune their result be it not voluntarily.

Philippe Guillaumme, who developed the modeling software Pianoteq (not alone), was a piano tuner for 12 years.

So he have at last some credentials on that side.

Modeling is limited by the computation time necessary (yet those days)
so shortcuts are used, in a millisecond a huge number of parameters are computed, some may be asking too much for the moment, plus those interactions seem to be terribly complicated, as you say.

I also cannot see how electronic speakers can reproduce the tone of of polarization plane change, for instance ...

Despite those big mistakes they do, those scientists certainly help things to be better understood...

Best wishes

Hi Isaac,

Perhaps I should first ask you what you mean by "strength of the attack", just to make sure I'm not misunderstanding. I say this because I may understand "strength" what you (or others) would call "noise", or any other "suitable" word...

If you like we may remain in this thread, I would not mind, and deepen on what is (for you and all) a "nice" attack, or what is there to be avoided...

As for models, in general, I think we share their meaning, their goal and... limits, when it comes to practice.

We may have a chance to meet sometime in March, Isaac, cross fingers..

Best wishes,

Alfredo
.

Bonsoir ALfredo,

I will try a clear answer , it begins to sound like something , a decent construction, in my mind .

I find interesting what is written by James Ellis :


Immediately following impact by a hammer near one end of a string in a piano, or plucking by a quill or plectrum in a harpsichord, a pure fundamental tone of the note is absent. Instead, all of the component frequencies of the whole tone are contained in a complex pulse-like wave that travels from end to end along the string, reflecting from one end to the other, over and over again, until it finally disperses into the discrete transverse standing waves that make up the timbre of the note,and persist for its duration. As explained above, when the string is struck or plucked near one end, the vector sum of forces produces a longitudinal as well as a transverse wave. When the complex transverse pulse reaches the termination at the opposite end of the string, it is reflected back inverted. A virtual inverted image of the initiation of the pulse occurs during the first reflection, and each following reflection is an inverted image of the previous reflection. This process continues until the transverse pulse has dispersed into discrete standing waves. A virtual image of the force vectors that initiated the longitudinal wave is repeated each time the transverse pulse is reflected, inverting at each reflection. If each reflection of this transverse pulse happens to be in phase with a reflection of the longitudinal wave, the vector sum of forces during each reflection will cause the transverse pulse to give up some of its energy to the longitudinal wave, causing the longitudinal mode to build up and the transverse mode to decay more rapidly than it otherwise would. The relative tuning between transverse and longitudinal modes required to make this happen is critical. The very slightest relative detuning of the two modes will cause this energy transfer to disappear.

That is why "noise" is indeed an accepteable terminology, but the tuner have some influence on the way it behaves,

This is in the way we build tone that those agreements are.

I begin to think we can have a tone that comes from the second partial (first) to energize the fundamental, or another construction with a more immediate raising of fundamental , that then adsorb and rule more partials.

That energy wave that contain the impact noise and excites the longitudinal mode (while using it as a vector (?)
we are probably dealing with it by the way we use some of the early high pitches produced by the lModes to energize our partials at the moment of attack.

Reading that the Lmodes (and their results hence quadra and other "phantom partials" are modified by the way the strings are coupling, sound logical to me.

I am may be totally wrong and only coupling of the transverse waves occur with the energy deal related to it.

I feel that the tuner is expecting to tune the "noise" - as it cannot be manipulated as it is he only can use the spectra and the energy dissipation to damp some part and enhance others.

I see unison tuning as being partly an energy regulation job

If we pluck the string to tune an unison the result will be very different, energy wise, than when the note is played.

But it is late !

I hope we can meet, for sure.

I will be there normally in March, but not at the end of February where I am going for a few days

An now I am going to bed (if you want to know !)

ALl the best

P.S may be it relates with the age of the captain too, finally

I suspect that we use the Lmodes (or the phantom partials) as a support for tuning.

It happened for instance on a G2 that I could not obtain a nice tone without playing extra FFF, then only the part of the tone I was waiting fort became apparent, and I could tune with some nicely ringing partials where before I only heard a dull and carboard like tone.

But the most surprising, particularely when one is used to get a justness approach via an ETD, is to see that one can provide a global tone, beat wise, that does not agree with the single, or multi partials analysis, using a richer tone including many unstable pitches that cannot be defined precisely, but add a lot of coloration.
(tuning with the sustain pedal engaged open the door to those pitches, as it oblige the tuner to take in consideration all available tones coming from the strings, when you have clean unisons with sustain pedal engaged, chances are that other vibrations modes than the vertical are taken in account)

I have done that once to a concert grand prior a recording, and the sound engineers told me thy did not understood what I did to the piano, that begin to be "singing like birds" they say.

AT another moment, trying to do a similar job I ended with too much beats to be acceptable, so there is a listening mode that may go along that trick to use it correctly.

And this can only be done with a non discriminating ear .

That is what I call "listening to the tone of the instrument apart" sometime possible in a small room as well, taking the inner reverberation in account when tuning unisons.

A too close listening is mostly good for recording sessions.

The tone can be cleared ans straightened to a point the L modes become too prominent and rob a lot of energy from the tone.

But I have always related that to a "tune the attack" job.

I liked the recent phrase from RXD where he states that "its been a long time he did not listen the fundamental"

The most important process in unison happens at the 2nd partial level so this seem evident that we must focus our job there.

Then one can recognize the energy behavior (as done when checking unison) and get there by other perceptions, or more probably with mixed perceptions, touch, hearing, and the feel of the tinkling in the tuning lever (we are rarely aware of).

"Attack tuning" mean regulating the way the first waves of energy at the hammer impact stabilize, we feel (or imagine) the motion of the initial hammer impact in its first travel toward the bridge and the agrafe, and modulate the coupling between the string as to have some emphasis of the energy.

Then this is taking some power and cannot be done more than a certain limit.

SO in the end the best choice is most probably to only change the force we use to play , to modify the harmonic content of unison.

A tuning made playing very strong will have emphasis on the attack (it can help some pianos and some pianists)

Done more lightly the tone is more open .

We can obtain the same effect playing lighty but it is easy to be mistaken, so at last a few test at FFF are necessary.

I like your definition, Isaac, when you say "build the tone", it says well what I feel I'm doing on a piano, also when I work on voicing and "touch" regulation. As I know you, your conscientious dedication, I'm not sure I can add something to your actual knowledge. Anyway...

Perhaps I relate the "attack" more to the way the string is first solicited, and in my experience that sound-event depends on the "object", say on "what" is soliciting the string and on its dynamic, say "how" the object solicits the string. By controlling the elasticity together with the dynamic of the hammer, I try to control the first "reaction" of a three-strings body, quality wise, say color wise, also remembering that unisons can (and will) influence the final color.

As you say, this has to do with energy indeed, although in general I tend to appreciate "quantity" (of energy) only when it is strictly related to quality, as if quality is what (for me) justifies quantity. In a way, energy then becomes a relative factor, relative to the partials that actually build the tone (and that we are enabled to shape) and to the executor, to the pianist that needs to be able to convert any possible intention into (quantitative) sound and (qualitative) tone.

As you know, I use one precise type of unison as a reference, it is the unison that reinforces the fundamental and makes it long-lasting; this happens right below the third partial (in inverse order), that is where the second partial appears and when I can phase it and lock it (as a beam) with the fundamental. On the bases of that unison I evaluate the attack, the presence (read accent) of other partials at the attack, which gives me the idea of the tone's nature and character.

I cannot anticipate how "regularly" the T-modes and the L-modes will combine together (in practice), and it is difficult to transpose a theoretical "norm" onto a piano where many factors will continuously (and fortunately) determine a special case, nor about other "random" partials that, with variable intensity, come out from somewhere else (see the bridge), but I'm sure we can enhance energy and resonance by tuning, by voicing and regulating the action.

Even knowing that the L-mode is 10 or 20 times faster than the T-mode does not really help my practice, but in force of my experience I know that part of the string energy can be transferred to the pin block as well. There again I establish a "reference mode", that has to do with the pin's grip, with the pin-Vs-string relation and how the string's tension is distributed onto its three lengths. Say that I'm not dreaming, that the pin-block too can transfer energy all around, will it be due to the L-mode? To some other M-combinations?

Un caro saluto, a.
.