The note you hear when you press a piano key lasts for seconds. The physics that determines which harmonics are present in that note lasts for less than four milliseconds.
That physics happens at the piano hammer, and it is more interesting than most audio engineers expect.
Not a Spring
Most objects that are struck or compressed behave approximately like linear springs at small amplitudes: double the compression, double the restoring force (Hooke's law). Piano hammer felt does not work this way. The force-compression relationship for wool felt follows a power law:
F(u) = E * u^pwhere F is the force exerted on the string, u is the compression of the felt in millimeters, E is a stiffness constant measured in newtons per meter, and p is the nonlinearity exponent. For a linear spring, p equals 1. For piano hammer felt, p typically falls between 1.5 and 2.8 for new hammers, and rises to between 2.5 and 3.9 for hammers that have been repeatedly packed down by use.
Anatoli Stulov at the Institute of Cybernetics at Tallinn University of Technology formalized this model and published a detailed hysteretic version in the Journal of the Acoustical Society of America in 1995. The hysteresis matters: piano felt has memory. The stiffness during compression is not the same as during rebound. The loading curve and unloading curve do not overlap, and the gap between them represents mechanical energy absorbed rather than returned to the string. This energy goes into heat and internal deformation of the felt structure.
The practical consequence: the felt is effectively harder at higher impact velocities. A harder hammer means a shorter contact duration. And contact duration is the primary variable that determines what the string hears.
Contact Time Controls the Spectrum
When the hammer contacts the string, it drives a force pulse into the string for the duration of the contact. The string then vibrates freely once the hammer rebounds. The spectral content of the driven pulse, what frequencies were present and at what amplitudes, determines the initial energy distribution across the string's harmonics.
A longer contact time means the pulse is wider in the time domain, which means its Fourier transform is narrower in the frequency domain. Fewer high harmonics receive significant energy. A shorter contact time means a narrower pulse, with energy extending further into the upper partials.
Askenfelt and Jansson at the Royal Institute of Technology in Stockholm measured these contact durations carefully. Contact time falls from roughly 4 ms in the lowest bass register down to under 1 ms in the highest treble, driven primarily by the decrease in hammer mass across the keyboard. Bass hammers can weigh around 11 grams; treble hammers can be as light as 3.5 grams. Within any given register, hitting a key harder shortens the contact time further, because the higher impact velocity compresses the felt faster and the nonlinear stiffness increases more sharply.
The result: louder piano notes are not just louder. They have a different harmonic distribution. The spectrum tilts toward higher frequencies as velocity increases. This is a property of the hammer mechanics, not of the string itself.
A useful way to characterize a hammer is its cutoff frequency, the point at which the hammer's force spectrum begins to fall off. Measurements from voiced piano hammers show this falls around 1.5 kHz for bass hammers and around 3.5 kHz for treble hammers. Harmonics in the string that sit below the cutoff receive full excitation. Harmonics above it receive progressively less energy from the hammer. This is why a grand piano bass note has a different texture from the same pitch played on a bass guitar even when both are struck hard: the piano hammer acts as a velocity-dependent low-pass filter whose cutoff rises as the player hits harder.
Voicing: Adjusting the Exponent by Hand
Piano technicians can physically adjust the nonlinearity of each hammer through a process called voicing.
The primary softening technique is needling: piercing the felt with steel needles to disrupt the compressed fiber structure. This reduces the density of the outer felt layers, lowers the stiffness constant E, and reduces the effective exponent p. Softer felt means longer contact time at any given velocity, which means less high-frequency content in the driven pulse, warmer tone.
Hardening works in reverse. Technicians apply a lacquer or glue-based compound to the felt surface, increasing density and raising the exponent. Harder felt, shorter contact time, brighter tone.
This is why pianos need regular voicing. Each hammer strike compresses the felt and gradually packs the fiber structure. The exponent p drifts upward over years of playing, and the piano slowly brightens without anyone intentionally changing anything. A well-voiced concert grand receiving daily voicing maintenance holds its character far better than one left to its own trajectory.
Voicing is not applied uniformly. Technicians working on concert instruments will adjust individual hammers across the range to smooth tonal unevenness, targeting specific partials on specific keys by varying the depth and angle of the needle.
Implications for Sampling and Physical Modeling
This physics explains something that anyone working with piano sample libraries has probably noticed without being able to articulate: velocity layers are not just about loudness. They capture fundamentally different harmonic content.
A sample library recording a piano at pianissimo, mezzo-forte, and fortissimo is capturing three different filter states applied to the same string. Crossfading between velocity layers without accounting for the spectral shift introduces a characteristic smoothing quality that identifies underpowered piano libraries. The fundamental is roughly stable across velocities, but the upper harmonics change substantially.
Physical modeling approaches to piano synthesis handle this directly by computing the hammer force at each sample using the nonlinear equation. Vesa Välimäki at Aalto University and colleagues have built piano models that feed a hammer-string interaction computed from the nonlinear stiffness model into a digital waveguide string model. The felt parameters E and p are tuned per register to match measured data from real instruments. This means the timbre shifts correctly under velocity changes, not because additional samples were recorded, but because the underlying mechanics are right.
The hammer mass matters too. The transition from a 4 ms contact time in the bass to under 1 ms in the treble explains much of why a well-sampled bass note sounds physically different from a well-sampled treble note, not just in pitch, but in the density and ring of the attack. These are not incidental features of the instrument. They are the direct output of the power-law relationship between felt compression and string force.
One Piece of Felt
Modern digital pianos and synthesizers can model piano timbre convincingly at a static velocity, but the variation across dynamic levels is where linear approximations fail. The Hertz-form power law with a nonlinearity exponent of roughly 2 is computationally cheap to implement, and the physical insight it provides, that the hammer becomes harder as it moves faster, and that harder means spectrally brighter, is not intuitive until you have seen the numbers.
A piano hammer is a piece of compressed wool. Its behavior under impact is governed by a single exponent. That exponent, and the few milliseconds in which it operates, is the entire mechanism by which a player's touch connects to the character of the sound that follows.