We arrive at the strings.
The hammer-head movement can be separated in two independent motions which are acting together :
— The first is the hammer-head motion without shank vibration.
— The second is the additional shank vibration.
The shank can move either freely or with the roller leaning on the jack. An intermediate case is the free vibration upwards and a leaning vibration downwards.
In free motion the amplitude is smaller, but the equivalent spring contact at the head side of the shank is greater, and the flip-flop frequency is lower. The contrary is characteristic for the leaning vibration.
The shank has five different effects on the impact spectrum :
1) The impact velocity pulse will be modified by an additional vibrational frequency depending on the shank mass that we will call impact swinging, in the low frequency range of the spectrum.
2) The spectrum of the shank vibration, produced by the pianist, is added to the low frequency range of the spectrum.
3) The string contact duration can be modified by the additional small vibrations. A slight change of the contact duration can be decisive for the number of reflected agraffe pulses which are parts of the spectrum. In some critical cases the contact time can decide wether the reflected pulse from the soundboard side, whose presence is rather disastrous, will also enter or not into the spectrum composition.
4) The rubbing of the head creates longitudinal vibrations on the strings.
5) Related to the rubbing and flip-flop motion, the effect of lack of parallelism between the hammer-head and the strings.
Our study concerns essentially the first three points, the two last points will not be detailed. A partial discussion of the Una Corda acoustics and a Conclusion will close the Piano Acoustics III. chapter.
The point n°1 : in the first step we create a simplified model of the impact process where only the hammer mass (head + shank), the approximative felt spring constants and the mechanical real resistance of the strings play a role. This model will lead us to the determination of a forte/fortissimo and a pianissimo/piano approximative spectrum (basic spectrum), necessary for the second part of this chapter.
In the second step we will discuss the felt spring constants in more detail and the influence of the non-linearity on the calculations and on the physical results. The shank mass with the shank elasticity will have a separate role influenced by the presence or absence of free path before the impact. The model forms a fourth order system.
The third step : It will be question of the superposition of the shank vibration to the impact pulse. The influence of the vibrating shank is a function of its phase at the impact. The crucial point is the possibility for the pianist to control the shank vibration, its amplitude and phase..
The fourth step : We examine the role of the reflected agraffe pulses. Their spectrum will be added to the basic spectrum.
Simplified model of the impact process
The hammer velocity of the impact produces a velocity pulse on the strings. The amplitude and form of this velocity pulse determine the timbre and depend chiefly on the head mass “M”, on the felt spring constant “C” and on the wave impedance (resistance) of the strings “R” (= sixfold resistance of one single string in case of the triple C4 strings with the wave propagating in both directions).
In the following calculations “R” is considered as real, and “C” as linear but with appropriate values for forte and piano levels. The three components, M, C and R are in a mechanical series configuration, that means a parallel circuit in electrical analogic model fed by a velocity source “V”.
We use Laplace transforms of the string velocity “vR”, across the wave impedance “R” of the strings, which yield immediately the time functions :
The following diagram shows the simplified impact form at fortissimo (v = 5 m/s) and at piano level (0,5 m/s), (the piano level is multiplied by 10 to facilitate the comparison) :
We have seen the upper and lower curves with two different but mean spring constants. It is evident that the hammer at the beginning of the mezzoforte/forte impact will transmit much less velocity to the string than later. That means that the starting force is smaller for a relatively large felt compression. This corresponds to the lower curve which was drawn with a mean 0,00005 m/N spring constant.
As a working hypothesis we admit that the first portion of the upper curve, till the tenth of its final force or velocity value and till the tenth of the time needed to reach the top, moves in the same way as the lower curve and after it continues to compress the felt with a much higher force which signifies a much smaller mean spring constant.
The partition of the upper curve in two sections permits to approach the real impact form with much higher likelihood.
The vertical line at 0,03-0,04 msec shows the partition line where the difference between the two curves is about 0,45 N. Consequently the starting point of the upper curve must be diminished by 0,45 N as we have seen it in Figure 18.
Evidently, the transition between the two layers is progressive and the more detailed hammer felt models divide the felt in several short sections. Our scope is only to illustrate the non-linear proprerties of the felt without any precise calculations. The articles on piano acoustics deal mostly with this non-linear characteristic of the hammer felt, which proves the importance of this problem.
We do not want to examine this question and we limit our model only to two sections with two values of a mean but invariable spring constant at mezzoforte/forte level. We take as a basis the results published in the scientific literature(8) and shown in the Figure 18.
The method to calculate the impact velocity pulse on the string was elaborated recently by D.E.Hall(3) although the principle was known already at the end of the XIX. century /Kaufmann (4)/. The theory is based on the simple idea that the hammer on impact cannot “know” how long the string is till the moment of the returning reflected waves. For this reason the strings, supposed infinite, represent for the hammer a frequency independent wave impedance.
The form of the impact pulse on the strings is of primary importance :
Curiously, this simple form calculated by Hall was not known by the illustrious scientists of the XIX century. Helmholtz, for example, a great speculative acoustician, supposed a sinusoidal half-wave because this was the only form he could treat with the old mathematics.
Observing the two curves we can remark that in the whole dynamic range from piano/pianissimo to forte/fortissimo the string velocity is quasi-proportional to the hammer velocity and the jack force. Differences are in the limits of ± 5 %. It seems that the felt non-linearity is less significant as it is supposed to be. “Does a power low make sense ?” is the subtitle of a recent publication (12).
Evidently the diagrams are only symbolic. Perhaps the partition in three or four sections would have given a more realistic representation, and probably, the peak values would have been more closer. But, as we have mentioned before, our task is the understanding mysteries of the piano and not the accurate determination of numerical values.
The basic spectrum of the impact pulse
The curves above come from relatively simple mathematical equations, so the development in Fourier series is in the domain of our possibilities.
We will proceed to do this, but before we must insist on the feature of the impact pulse. In scientific literature such pulses are mixed with reflected agraffe pulses. As we consider the impact as a roughly linear process, the basic impact pulse must be present invariably behind the numerous agraffe pulses. For this reason we call it the basic spectrum.
As basic spectrum it does not include the additional spectral lines such as those of the shank vibration and of the impact swinging that we will point out in the following discussions.
In Figure 19 we see three spectra. The first corresponds to the pianissimo/piano level, with a felt spring constant of 0,00005 m/N. The second is a spectrum of mezzoforte/forte level with an unique 0,00001 m/N spring constant.
The third spectrum represents the case of two spring constants. To get it, we subtract from the second spectrum, the spectrum of a right-angled triangular wave with the Y-side of 3 x 0,15 m/s (-15 %) and the large X-side equal to 30% of the sound period.
Figure 19, even without the agraffe pulses, shows clearly that the forte sound is by no means an amplified piano sound.
The spectra have been calculated by means of Fourier series development shown below :
In this simplied model there is nothing which could allow the pianist to influence the basic curves and the basic spectrum. The shank vibration which depends on the touch, will be additioned to the basic spectrum. The role of this basic spectrum is to fix a reference point to appreciate the shank vibration effect, which enables the evaluation of the factors depending on the pianist’s play.
The discussion of the piano playing begins with the influence of the shank bent by the impact force. This bending is a function of the hammer velocity but also of the different ways to control this velocity by the touch. In short, it depends on the existing or lacking free path shank vibration before impact.
1) Impact process with shank bending
To examine this question we must construct a more detailed model involving the shank elasticity and a separate concentrated shank mass. Calculations with distributed shank mass will be too complicated and superfluous because the effect of bending the shank by the impact is only secondary but not negligible, so an approximate estimation is largely sufficient.
Independently of exisiting shank vibrations, and produced by the jack-roller contact, the moving shank upward can be represented by a concentrated shank mass pushing the hammer-head through a spring which is the bended shank.
In the discussion of the simplified collision we have considered the shank as rigid, and the reduced mass of the shank was added to the hammer-head mass. In reality the shank is subject to a bending, and so the shank mass and shank elasticity form a separate oscillating circuit which represents with the hammer-head mass and the felt elasticity a fourth order system.
This shank motion we will call impact swinging of the shank. The origin and importance of the impact swinging is radically different from the shank vibration of our former examinations. This one, namely, is produced by the pianist, as for the impact swinging, it is the consequence of the hammer hitting on the strings.
However as the impact swinging depends chiefly on the existing or not existing free path of the hammer, the pianist is able so to influence it indirectly.
The impact swinging results from the hammer velocity which is directly connected to the sound level. But the same sound level can be obtained with or without free path before the impact and so the pianist’s will can prevail.
With concentrated mass calculation the equivalent total shank mass acts as if it were concentrated in the 2/3 part of the total shank length. We have interest to displace this point to be at the position of maximal deflexion point. The maximal deflection point is on the hinge side, that means that the equivalent mass must be augmented proportionnally by the square of the distance between the 2/3 point and the maximal deflexion point. At the same time the mass acting on the head diminishes.
The mass correction may represent some ±2 %, and is negligible compared to the error resulting from the concentrated mass substitution for the distributed mass discussion.
The shank bent by the inertia of his mass can be considered as a hinged bar with a maximal flexion near the center, somewhere between the center and the concentrated mass location.
But there is an alternative simple calculation manner of the equivalent spring constant :
The head mass with the flip-flop frequency, which is the same also for the vertical motion of the head, must yield the spring constant for the shank :
We propose to establish the mechanical model and the analog network :
Figure 20.
The equivalent analog network is shown in the Figure 21.
.
For the moment we separate the impact swinging from the shank vibration, that means that we suppose the hammer reaching the strings whithout shank bending and shank vibration.
In the fourth order model the shank mass and the shank elasticity have a distinct role.
The calculation below leads to the following result :
The Laplace transform inversion can be found in the Laplace transform catalogues or done it himself. We have done it but the result is rather long and we put it directly in the tabulator to get the following diagram :
The spectra of the velocity impact pulse on the strings with impact swinging at mezzoforte/forte level are as follows :
Figure 23
First column : the first column is the basic spectrum on mezzoforte/forte level with two felt spring constants and with new time constants resulting from the separation of the shank and head masses.
Second column : basic spectrum of the impact when the hammer arrives after a free path motion. The spectrum is the same as before but with the superposed impact swinging of the shank. This is the normal piano spectrum independent from the will of the pianist, but without the reflected agraffe pulses.
Third column : the touch is such that the free path is missing (for example a delayed mezzoforte/forte touch).
Fourth column : the modification of the harmonics by the touch. The 3th harmonic is stronger by 10 %. The 5th 6th and 7th harmonics are weeker by 20 - 10 - 20 %. The sound approaches the piano timbre but the sound intensity is not changed. (A new example of the change of timbre without change of intensity).
The same impact spectra as above at piano level :
As the change of spectrum at mezzoforte/forte level depends on the existing or missing free path before the impact, at piano level, where the free path supposedly always exists, the impact swinging cannot be modified by the pianist’s touch.
Piano Acoustics III (suite)
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