The hammer that is heavier than its string

This ladder has varied three properties of one event. Where the exciter lands decides which partials have a node under it and are silent. How long it stays low-passes the excitation, and because felt stiffens as it compresses, a harder blow shortens the contact and brightens the note. How wide it is turns out to change almost nothing, because a mode is odd about its own node and a symmetric contact centred on one cancels exactly however wide it is.

The third rung ended by naming the fourth: the exciter’s own mass, which none of the three had touched, and which is a different shape of argument because it makes the contact time depend on the string rather than on the felt.

The ratio inverts inside one instrument #

Start with the masses, because they are the surprise.

A piano string’s mass is its linear density times its speaking length. This site already carries a string design — a stated one rather than a measured one, and the same one every inharmonicity figure here is built on — so the number is available at every pitch: about twelve grams at the bottom of the compass, two and a half at A4, a quarter of a gram at the top.

A hammer’s mass runs the other way, and much less steeply: heavy in the bass, light in the treble, by a factor of about six across the keyboard rather than fifty.

Below D3 the hammer is the lighter object and above it the string is. A bass hammer arrives at a wire that outweighs it and is stopped by it; a treble hammer arrives at a wire a thirteenth of its own mass and barely notices.

Nothing about the mechanism changes across that line. What changes is which of the two objects is doing the pushing, and that decides how the contact ends.

Two clocks, and they cross in the same octave #

A contact ends when the force between hammer and string reaches zero, and there are two ways for that to happen.

The felt lets go. Piano felt is a nonlinear spring; the hammer compresses it, the spring pushes back, and the hammer leaves. That is the mechanism the second rung of this ladder is built on, and its time is set by the hammer’s mass against the felt’s stiffness. The site’s model gives it as a fixed 1.6 milliseconds at a stated force, shortening to about a millisecond when the force is quadrupled.

The string takes the momentum. A transverse force applied at a point on a long string meets a resistance of 2Z, where Z is the string’s characteristic impedance and the two is because two waves leave, one in each direction. A mass m pushing on that resistance loses its momentum with a time constant of m/2Z. That is the longest the string will take to absorb the whole blow.

The second time is computable from the wire: Z is the linear density times the wave speed, and the wave speed is twice the frequency times the length. Both are in this site’s string design.

They cross at G3, at 196 hertz. Below it the string’s time is the longer of the two — three milliseconds at A1, twelve at A0 — so the felt gets there first and the contact is the felt’s. Above it the string’s time is shorter, falling to three quarters of a millisecond at the top, so the string gets there first.

And over six of the piano’s eight octaves the two are within a factor of two of each other. That is the finding, and it is not the tidy one. The contact time is neither the felt’s property nor the string’s; it is a coupled quantity, and a model that attributes it to one of them is describing a limiting case that the middle of the keyboard is not in.

Which makes a third crossover, and they are not the same crossover #

There is a third time in the picture and it belongs to the geometry rather than to either object.

The corner a hammer makes travels away at the wave speed, reflects at the near end, and comes back. For a strike at a fraction p of the length, that round trip takes p/f₀ — a quarter of a millisecond at A4 and four and a half at A0. If the round trip is shorter than the contact, the wave the hammer made returns while the hammer is still there.

That crossover is at E2, 82 hertz, and above it the returning corner arrives during the contact — five times over at A4, forty-five times at the top of the keyboard.

So there are three lines in the low-middle register and they are not in the same place:

• E2, where the corner starts coming back during the contact;
• D3, where the hammer stops being lighter than the wire;
• G3, where the string can absorb the blow faster than the felt can let go.

An octave and a half holds all three, which is a coincidence worth naming rather than a structure: the three quantities scale differently with pitch — the round trip as 1/f, the mass ratio as roughly f, the momentum time as 1/f to a small power — and there is no reason for their crossings to be near each other. That they are means the piano’s transition from one regime to the other is a region rather than a line, and it sits at the bottom of the instrument’s melodic range.

Whether that complication matters is a question about register rather than about the model, because the number of round trips inside the contact falls with the pitch. At the bottom of the compass it reaches zero, and there the simple picture is not an approximation to anything — it is what happens.

The constant this essay has been assuming can be checked against the coupled calculation directly.

The computed contact is longer everywhere, and the round-trip count is what says why it matters: where the corner completes several trips inside the contact, the hammer is still there when the string comes back and the two are a coupled system rather than a strike. The half-sine is not a small approximation at the bottom of the compass; it is the wrong object.

What survives, and what does not #

Two earlier claims on this site are downstream of the contact time, and the two fare differently.

The dynamics claim survives, at one pitch. A dynamic marking is not a level: on a struck string a harder blow shortens the contact from 2.26 milliseconds to 0.95, which moves the first null of the excitation from the third partial to the sixth, so a fortissimo is a different sound and not a louder one. Nothing above touches that. The felt’s nonlinearity is a fact about force at fixed pitch, and the string’s impedance does not depend on how hard the hammer is thrown.

The claim that the contact time is a property of the felt does not survive across the compass. The site’s model gives τ the same value at every pitch, because it has no pitch in it. Published measurements of real pianos give contact times falling by roughly a factor of four from bass to treble. The momentum time falls by a factor of sixteen. The felt’s model has no compass dependence at all and the measurement has one, so whatever produces it is not the felt — and the string’s impedance is the only candidate in the mechanism with the right sign.

That is a division of labour rather than a refutation: the compass dependence of a piano’s contact time is the string’s, and the dynamic dependence at a given pitch is the felt’s. Both are real, both are computable, and the ladder had only the second.

The bass note that is a different instrument #

The regime difference has an audible consequence, and it is one piano players describe without a mechanism.

At the bottom of the compass the hammer is lighter than the wire, the felt lets go before the string has taken the blow, and the corner does not come back until the hammer has gone. That is a clean impulse into a heavy string: the excitation is the felt’s pulse, the comb of missing partials is exactly where the strike point puts it, and the note that comes out is the one the inharmonicity ladder computes.

At the top the hammer outweighs the string by thirteen to one, the string sheds the blow in three quarters of a millisecond, and the corner returns forty-five times while the hammer is still down. That is not an impulse into anything; it is a heavy object resting briefly on a light one that is trying to vibrate underneath it.

A piano is two instruments with a region between them, and the region is the bottom of the melodic range rather than the middle of the keyboard. Which is where pianists say the instrument changes character, and where every scale exercise crosses.

One other quantity changes across the compass from the same string design, and the two are computed from the same numbers: the stretch a tuner puts into the octaves is a property of the wire’s stiffness, and the contact regime is a property of its impedance. A piano’s bass is a different instrument in both senses at once, and neither fact was put in — both fall out of the scaling.

Which computation produced the numbers #

Four ingredients, three of them already here.

The string’s mass and length come from pianoString, which is this site’s stated piano — a small upright with a case limit of 1.15 metres, a speaking length falling as f^−0.93 and a mild gauge taper. Every Railsback figure on this site is built on it, so the numbers here are consistent with the inharmonicity ones by construction, and with the load the whole frame carries.

The impedance is Z = μc with c = 2f₀L, which is the definition and not a model.

The felt’s contact time is hammerContactMs, unchanged: a fixed time at a stated force, falling as force to the power −1/4, which is inside Hall and Askenfelt’s measured range for felt.

The hammer masses are the new number and they are a design rather than a measurement. The scaling used is 11.5 grams at C2 falling as a power −0.36 of the pitch, which gives about 12 grams in the bass and 3 in the treble and is the right order for an upright. It is stated in the machinery as a design. Every crossover above moves if it is wrong: a uniformly heavier hammer moves the momentum-time crossing up the keyboard and the mass crossing down it.

How much of the answer is the hammers #

That last paragraph confesses a design and then leaves it there, which is the wrong place to leave it when three crossovers depend on it. Sweeping it is one loop and it sorts the three findings into one that is safe and two that are not.

The corner crossing does not move at all. It compares the felt’s recoil against a round trip along the string, and neither quantity contains the hammer’s mass, so E2 is a fact about a piano’s geometry and its felt and is immune to everything the hammer maker decides. That is the one number in this essay that would survive being wrong about the masses.

The other two are violently sensitive. Halving the hammer set moves the mass crossing nineteen semitones up the keyboard and the governance crossing seventeen down — they swap sides. Increasing it by half removes the mass crossing from the compass entirely and pushes the governance crossing to E♭6.

So “an octave and a half holds all three” is not a robust feature of pianos. It is a feature of this hammer scaling, and a design thirty per cent either side of it disperses the three across four octaves or off the instrument. The section above was right to call it a coincidence rather than a structure and understated how narrow the coincidence is.

The same sweep prices the claim the next section makes about why the scaling exists, and it comes out stronger than stated. With a uniform 11.5-gram hammer the momentum time at the top of the keyboard is 3.4 milliseconds, which is fourteen periods of C8 — a contact lasting fourteen cycles of the note it is trying to excite, which cannot excite it. The graded hammer gives 0.76 milliseconds there, which is 3.2 periods, and the two designs cross one period at A5 and at A4 respectively.

Three periods is still not an impulse, and that is the finding hiding inside the confirmation: the shipped scaling does not rescue the treble, it only stops it being hopeless. The top octave of a piano is excited by a contact spanning three cycles whatever the hammer weighs, which is a constraint on the instrument rather than on its hammers.

Whose music, and when #

The piano is the instrument, and the design decisions are nineteenth-century and were arrived at empirically. Hammer scaling — heavier in the bass, lighter in the treble, with a graded felt hardness on top of it — is in every hammer maker’s tables and in no textbook’s derivation.

What the arithmetic above suggests is that the scaling is not a free choice. A uniform hammer mass across the compass would put the momentum-time crossing at the wrong end: a heavy treble hammer would take several milliseconds to shed its momentum into a light string, which is several periods of a treble note, and the excitation would be low-passed below its own fundamental — the same failure the tone-hole lattice’s cutoff produces at the top of a woodwind, arrived at from a completely different direction. A treble hammer has to be light because a treble string is light, and the graded scaling is what keeps the contact short enough to excite the note at all. That is the same shape of argument as the hammer’s position being a manufacturing tolerance rather than a design flourish.

That is a prediction about why the tables say what they say, and it is the kind that could be checked against a real hammer set in an afternoon by anybody with a balance.

What the picture cannot show #

The bass strings are wound and the model’s are not. pianoString reports the core diameter for the stiffness and refuses to pretend it knows the winding, so the bass string masses above are underestimates — the real ones carry copper. The mass crossing therefore sits higher up the keyboard than D3, and how much higher is not computable from anything here.

The two times are compared and not combined. A real contact is one interaction in which the felt’s stiffness and the string’s impedance act at once; taking the smaller of two independent estimates is a bound rather than a solution. The proper treatment integrates the hammer’s motion against a nonlinear felt force and a string reflection history, which is a numerical problem and not a formula. A bowed string’s start-up needed the same kind of treatment and got a bound instead, for the same reason.

Nothing here computes the spectrum the mass argument implies. If the returning corner arrives forty-five times during a treble contact, the excitation is not a half-sine force pulse and the rolloff every figure in this ladder uses is the wrong shape. What that does to the partial amplitudes is the obvious next calculation and it is not this one.

And no hammer was weighed. The masses are a stated scaling, the felt time is a published fit, and the string is a design. Three of the four ingredients are honest models and one of them — the impedance — is arithmetic.

Where this ladder goes next #

Four rungs and four variables of one event: where, how long, how wide, and how heavy. The fourth is the one that reaches outside the exciter, because a mass is only heavy or light compared with something, and the something is the string.

The rung after it is the one this makes computable and this essay declined to attempt: step the hammer’s motion forward against the felt’s nonlinear force and the string’s reflected wave, and read the resulting force pulse rather than assuming a half-sine. The pulse that was assumed does exactly that, and the part of the prediction above that it kept is the one about where — the bass, where the corner never returns during contact, differs hardly at all, and the treble differs completely. The part it refused is the attribution, which went the other way round from what this rung expected.

What is still owed here is a hammer on a balance. Every crossing in the table above is a consequence of a stated scaling that nobody has checked against a real set, and the sweep says which measurement would be worth making: not the absolute masses, but the taper, because the exponent moves the two sensitive crossings further than the overall weight does.