hammerhead mass

[Stephen Birkett]

I have been following the hammerweight discussion with interest. I
recently received a grant to study the interaction between hammer and
string. This project (which is just getting started) involves
developing a computer model of the dynamics of the energy transfer
from hammer to string in fortepianos. As part of the project I will
be collecting a database of action parameters for fortepianos and
these will be used in the model to investigate design principles.
E.g. what are the significant parameters, hammerhead weight, shank
compliance, hammerhead compliance etc. As fortepianos were strung
more heavily these parameters were changed to compensate, but the
precise changes were devised empirically. My objective is to quantify
the dynamics of the design, so that modern builders, who cannot
possibly make the numbers of pianos needed for experimentation, may
better understand the design principles.

The methodology of this study will of course be applicable to the
modern piano also. I would like to make a few observations on some of
the previous posts on this subject.

1)
In this discussion the effect of hammer velocity has been ignored.
The non-linear aspect of the interaction means that tonal spectra (as
well as total tonal energy) will be significantly different depending
on the velocity of the hammerhead when it strikes the string. For
higher velocities the higher spectral components get comparatively
more energy...thus a hard hit produces a brighter, as well as louder,
sound. This is independent of hammer head mass.

2) Hartman said:
> It's also true that as the hammer mass is
> increased a point will be reach when the hammer contact with the
> string will last longer than the string's fundamental effectively
> damping the string. As far as achieving a powerful tone the
> Hammer's weight has to be designed to work with the string scale,
> strike point and the soundboard response first. How the hammer
> weight effects the action is a problem to be considered second. If
> a heavier hammer is opted for then the action leverage will have
> to be less. This reduces the friction at the knuckle and requires
> fewer leads in the key for balance. If a lighter hammer is chosen
> the leverage will need to be greater providing the hammers with
> greater acceleration on it's way to the string.
>
The relationship of hammerhead mass to *string mass* is crucial to
the dynamics of the interaction. Thus it is the string diameters
which are important here, rather than the scale...(e.g. a 1780 piano
at c2 = 302mm, strung in soft iron, will have much thinner strings
than an 1860 Streicher with the same scale...the hammers on the
earlier piano will be correspondingly much lighter).

3)
Actually the strike ratio enters in here too: Typical interaction of
hammer and string involves 3 periods: (i) 0-1ms: impulsive...the
string is displaced and the hammer velocity goes to zero at about
1ms, when the hammerhead has maximum displacement; (ii) 1ms-5ms:
hammerhead begins downward travel, string displacement approx.
constant; (iii) 5ms-7ms: the first reflected wave returns from the
short end of the string, accelerating the hammerhead downward and
eventually off the string. Toward the end of phase (ii) the
hammerhead may lose contact temporarily with the string, and this is
apparently more common for bass hammers. [This summary is based on
work of H. Suzuki and D. Hall c1985...my project will proceed along
a different track, but I can't confirm nor deny the above yet since
I'm only just getting started on my project.]

The larger the hammerhead/string mass ratio the longer is the contact
time. Efficiency of energy transfer varies according to the mass
ratio...for larger ratios, which include longer phase (iii), energy
transfer is actually *less* efficient because more energy is returned
to the hammerhead to accelerate it. In all practical cases (I think
even for the Cristofori...approx. 1g bamboo hammerheads) the first
reflected wave returns before the hammerhead has finally left contact
with the string.

4)
A factor which is apparently very significant (at least for Viennese
actions, maybe in general), especially in the treble, is hammer
*shank* compliance. Modern fortepiano builders who don't use
`toothpick' thin hammer shanks get poor tone...esp. dull in the
treble. Didn't someone mention a high-speed photographic experiment
on hammer shank compliance recently? Any results yet.

5) Hartman wrote:
> The above in no way alters the importance of the relationship
> between
> hammer weight and leverage that you have been working on. At the
> bottom of this is probably some very simple and elegant relationship
> between leverage, hammer weight and friction and some sort of
> equilibrium between, key acceleration, hammers acceleration and the
> hammer's compliance. In fact I would guess that the piano action, as
> we know it, is the expression of these relationships. Our theory is
> now catching up with the evolutionary process that created the
> modern piano.
>
Equilibrium isn't really the right word to use here. From the
player's point of view the action kinematics and dynamics are
critical to achieving a controlled hammerhead velocity with an
appropriately sensitive range. The velocity must be balanced against
the dynamic parameters of the hammer/string interaction (hammerhead
and shank mass, compliance, string mass, strike ratio etc.) This kind
of non-linear multi-parameter problem makes relationships difficult to
sort out.

6)
It seems that hammerhead/string mass ratio will be more important
than the absolute mass of the hammerhead. Does anyone have data or
specs across a piano compass for hammerhead masses and string
diameters? Would be interesting to calculate the ratios from bass to
treble. Also correlate this data with strike ratios for the same
piano. Anyone have any measurements they could pass on to me?

7)
Stanwood said:
> Note: The hammer weight may be found by
> subtracting the shank strike weight from the strike
> weight.  The shank strike weight is the weight of the
> shank taken at the strike line radius.
>
This definition of shank weight is not clear to me. David please
clarify.

8) Someone (?) said:
> Now, should we throw into the equation the ability of the voicer?
> Some are able to get an incredible amount of tone out of any hammer,
> be it "light" or "heavy.
>
Seems that the art of voicing is essentially based on controlling
the non-linearities of the hammerhead compliance to achieve a desired
tonal spectrum...a process which can compensate for some, but not
all, of the other parameters in the interaction.

> This is why the heavier the hammer the harder it must be.  A harder
> hammer will rebound more quickly than a lighter hammer thereby
> cancelling out the fact that the higher weight causes it to rebound
> more slowly, (provided that resiliency is maintained).  The net
> result is that the tone of the instrument is the same but the
> heavier hammer yeilds more tonal energy.
>
According to Suzuki (1987) the total time that a hard vs soft hammer
is in contact with the string is much the same. The impulsive phase
has higher force and a shorter time interval for the hard hammer thus
the softer hammer reaults in a smoother string displacement
curve...corresponding to less tonal energy in the higher modes.

[N.B. Above comments are not based on my own work, but on published
work of Hall, Suzuki and others.]


Stephen Birkett (Fortepianos)
Authentic Reproductions of 18th and 19th Century Pianos
Waterloo, Ontario, Canada
tel: 519-885-2228
fax: 519-763-4686