The hardness of a piano hammer has a great deal of influence on the resulting piano sound. Hard hammers are better at exciting high frequency modes of a piano string's vibration so that the resulting tone quality may be characterized as being bright, tinny, or harsh. Soft hammers, on the other hand, do not excite high frequencies very well, and the resulting tone is somewhat dull or dark. The static hardness of a piano hammer may be tested by a durometer or hardness tester. In a typical piano, treble hammers are much harder than bass hammers.
Static hardness is not the only factor in determining piano tone. Dynamic hardness also plays an important role. A piano hammer behaves somewhat as a hardening spring; for large impact forces the hammer felt appears harder than it does for low impact forces. In the piano this means that a loud note sounds much brighter (i.e. contains more high frequencies) than a quiet note. It is difficult to test the dynamic hardness of hammers except by listening to them in a finished piano. If the piano tone is not as desired then the hammer hardness must be adjusted by voicing.
However, when a steadily increasing force is applied to a wool felt pad, the resulting compression does not obey the linear relationship F = k x. Instead, one finds that the felt appears to become stiffer the more it compresses, so that a larger force must be applied to produce the same increase in compression. In this case, the wool felt used for piano hammers acts as a nonlinear hardening spring. Measurements (reported in the literature) of applied force versus felt compression for many different brands of piano hammers show that the nonlinear relationship between applied force and felt compression appears to be of the form F = K x p , where K is a generalized stiffness of the hammer, and the exponent p describes how the stiffness changes with applied force. The greater the value of p the greater the range of hammer stiffness for a given range of force. This exponent p is called the effective nonlinearity exponent
Static measurements (apply force, measure compression, change force, measure new compression . . .) typically produce values of p ranging from 2.2 to 3.5 for hammers taken from pianos, and 1.5 to 2.8 for unused hammers, with no definite trend from bass to treble. According to surveys of several performers, the preferred range of values for the nonlinearity exponent is 2< p < 3. If p = 1 the hammer is too linear and loud notes are simply amplified soft notes. If p > 3 there is too much contrast - fortissimos are too harsh and pianissimos are too bland.
|Soft linear hammer||Hard linear hammer|| Measured linear hammer
Furthermore, in contrast to a linear hammer, the contact duration of a real hammer decreases for increasing peak force as shown in the figure below (hammer A3 at impact velocities of 1 m/s and 4 m/s). For a real hammer, hard blows have a shorter duration than soft blows.
The effective contact duration for a hammer blow on a rigid surface is related to the maximum force by τe ≈ Fmax(1-p)/2p where the exponent p is the same effective nonlinearity exponent as measured from static force-compression curves. By plotting the pulse half-width versus peak force on a log-log plot (the figure below is for a D2 hammer with p=2.2), one can calculate the exponent p.
The plot below shows exponent values measured for a Steinway voiced hammer action (black) and a set of hammers removed from a Steinway because they were too hard (white).
The relatively smooth increase from p≈2 in the bass, to p≈4 in the treble seems to agree with other measurements in the literature. From this figure we might predict that treble hammers will show greater changes in stiffness than bass hammers if both are tested over the same range of velocities. The data also shows that most of the hard hammers have lower exponent values than the voiced hammers, a result that agrees with measurements made by Hall and Askenfelt. This suggests that the hard hammers should show smaller changes in stiffness than the voiced hammers.