What the climb was a search for

A standard is a specification found a material constant under a convention: a string can be tuned no higher than its material’s specific strength allows, and the ceiling that constant sets is the reason nobody has ever proposed A = 600. It ended by naming the question the ceiling raises. If pressing against it was worth doing, something must have been on the other side — and the usual answer is brilliance.

The question is whether that answer is even coherent. Does raising a string’s tension change its spectrum as well as its pitch?

The tension effect is real and it is not brilliance. Twenty-nine per cent less inharmonicity is a measurable change and it makes the instrument less interesting rather than more — the piano’s stretched octaves exist because of inharmonicity, and reducing it shrinks the Railsback curve rather than brightening anything.

The question the rest of this rung is about is therefore a narrow one. The tension arithmetic is settled: more tension, less inharmonicity, more load. What is not settled is whether any of that is what anybody could hear, and the answer turns out to be no — which leaves the observed timbral change needing a different mechanism.

The thing tension does not do #

The obvious mechanical story is that a tighter string is a brighter one, and it does not survive contact with the model.

The claim has an appealing physical ring to it and it does not survive being written down. A plucked or bowed string’s spectrum is a property of where and how it is excited, not of how tight it is. Where the hammer lands puts a node under the excitation point and silences a partial; the amplitude of the n-th partial of an ideally flexible string plucked at a fraction p of its length depends on p and n and on nothing else. Raising the tension raises every partial’s frequency by the same ratio and leaves every amplitude where it was.

What tension does change is two things at the margins. The characteristic impedance rises as the square root of the tension, so a string at 440 delivers about six per cent more force to the bridge than the same string at 415 — half a decibel, which nobody has ever heard. And the inharmonicity falls, which moves the partials but does not change their strengths.

So a violin string at a higher standard is not a brighter string. If the brilliance story is right, the mechanism is somewhere else.

Where it actually is #

It is in the filters, and it applies to every instrument whose resonator is a fixed box rather than a tube that changes length — which includes the ones with no strings, and excludes rather more of the orchestra than it first appears. The violin and the voice are the cases; a section below is the count that says so.

A body’s resonances are fixed in absolute frequency and the notes are not. The body is the filter, and its peaks sit where the wood and the enclosed air put them; raising the standard by a hundred cents slides every note’s whole partial series up against a curve that has not moved. How far that generalises past the violin is a question with an answer, and a section below runs it over six radiators rather than asserting it.

The consequence is a different timbre for every written note, and the changes have no common sign. Some notes move a strong partial onto a resonance and get louder and brighter; some move one off and get duller.

The voice is the case where this has always been argued about, and it is argued about in exactly these terms without the arithmetic. Singers have objected to rising pitch since the nineteenth century; the objection is usually reported as being about strain, and strain is real. What the figure adds is that the timbre of particular written notes changes too, and that a singer who has learned where the difficult notes are in one standard finds them somewhere else in another.

The size is worth holding onto. A change of a few decibels on a written note is a large perceptual quantity — larger than the loudness a whole extra desk of violins buys, and comfortably above anything a listener would call a subtlety. And unlike a change in pitch, it is not something the ear normalises away: a listener with no absolute pitch cannot tell 415 from 440 at all, and can certainly tell a note that has moved onto a body resonance from one that has moved off it.

That is the shape of the answer to the fourth rung’s question. Raising the standard does change the timbre; it does not do it through the string; and the change is per-note rather than global, which is why it has never been possible to say what it sounds like in one sentence.

The one instrument where the tension story is true #

There is an exception and it is worth naming because it is the instrument the brilliance story was probably told about.

A struck string is excited by a hammer whose contact time is a property of the hammer, not of the string it hits. A hammer is not an impulse: the contact low-passes the excitation with a corner at roughly one over the contact time, and that corner is a fixed number of hertz.

Raise the standard and every note’s fundamental rises by six per cent while the corner stays put — so the corner falls at a lower partial number, and each note has fractionally fewer partials below it. A piano at a higher standard is, note for note, very slightly darker, which is the opposite of the story.

The effect is small — six per cent of a partial number is a fraction of a decibel on most notes — and the direction is unambiguous and wrong for the received account.

Every rung of this ladder has computed strings, because a string’s tension and gauge give a closed-form scaling law. Half the orchestra does not have one.

A woodwind cannot be pulled to a new standard, only lengthened at a joint, and lengthening at one place puts the tone holes in the wrong places relative to the new length. So the four-hundred-year climb was not a thing the whole orchestra could follow at the same cost: the strings retuned, and the winds were rebuilt or replaced.

Which computation produced the numbers #

Four quantities, all of them the site’s own.

The frame load is frameLoad, which is the fourth rung’s own function: every note’s tension from its scaling and its pitch, summed over eighty-eight notes and their unisons. It goes as the square of the standard because tension goes as the square of frequency at fixed length and mass.

The inharmonicity ratio is B ∝ 1/T at fixed geometry, which follows directly from the site’s inharmonicityB — the coefficient carries the tension in its denominator and nothing else in it moves when a string is tuned up.

The violin’s filter is radiatedPartials on the measured body resonance list, and the vowel’s is formantGain on a published formant triple. In both cases the note’s whole partial series is evaluated through the fixed filter at both standards and the total power compared, which is the honest measure because a filter can move energy between partials without changing the total and this asks whether it does.

The hammer’s corner is hammerRolloff at the contact time this site publishes, which is the ladder’s own.

Where the model stops #

Nobody retunes a violin without changing its strings. The figures above move the standard and leave the instrument alone, which is what happens over an afternoon and not what happens over a century. A maker working at a higher standard uses lighter gut, moves the bass bar, adjusts the bridge — and the “fixed” body curve is fixed only until somebody rebuilds the instrument around it. The historical instruments that survive have mostly been rebuilt at least once.

The body list is one violin. Every number in the violin figure is computed on a single measured resonance set, and two violins differ far more than a hundred cents of standard does. The mechanism is general; the specific notes that gain and lose are not.

And the total power is a crude measure of timbre. Two spectra with the same total and different shapes sound different, and the figure would score them equal. A centroid would catch that and would introduce its own arbitrary weighting; both were computed and the ordering of which notes gain and lose is the same, so the finding does not turn on the choice.

Whose music, and what the argument has actually been about #

The claim is about instruments. The argument is about performance practice and it has been running for two hundred years.

The rise itself is documented and is not in dispute: the note moved by a minor third across the surviving organ evidence, and it moved unevenly, by place and by function. The explanations offered at the time were about brilliance, about ensembles wanting to be heard, and about instrument makers competing on impressiveness.

What this rung says is that the brilliance explanation is true and its mechanism is not the one anybody gave. It is not that strings get brighter when tightened. It is that a whole orchestra of fixed filters — bodies, tracts, bores, hammer contacts, and the room, whose modes are fixed too — sits under a set of notes that is being slid upward, and the result is a different sound for every written note.

That reframes the historical-performance argument in a useful way. Playing baroque repertoire at 415 is usually defended as restoring the pitch the composer heard; the arithmetic here says the pitch is the smaller part of it, because the pitch is a hundred cents and a hundred cents is a category boundary a listener barely notices in isolation. What is restored is the relationship between a fixed filter and a moving note, and that is worth several decibels on particular notes — which is a much larger perceptual quantity than a semitone of absolute pitch.

The exception, again, is the singer, and it is the one place the received account and the arithmetic agree completely. A voice at a higher standard is both differently coloured and working harder, and the second is not in any figure here.

The instrument that could not follow #

There is one more consequence of the tension arithmetic and it is the reason the climb stopped where it did.

A frame carrying sixteen tonnes rather than eleven and a half is a different object from one carrying eleven and a half. Wooden-framed pianos of the early nineteenth century were built for standards near 430 and they do not survive being tuned much above it; the cast-iron frame, patented in the 1820s, is what made the rest of the climb possible at all. The pitch standard and the frame material are the same history, and the tension curve above is why.

The same arithmetic bounds it from above, which is the fourth rung’s own result: a string can be tuned no higher than its material’s specific strength allows, whatever the frame is made of. The climb from 392 to 465 spends a third of the headroom between an ordinary working tension and the breaking one, and the reason nobody proposed 600 is not taste.

What the picture cannot show #

Which direction anybody preferred. The figures say the notes change and do not share a sign; nothing says that the resulting sound was judged better, and the claim that it was is the part of the historical account this arithmetic cannot reach.

How much survives an ensemble is now run, and it survives — for a reason that narrows the claim. A section below does the averaging.

And the room is not in it. A hall’s modes are fixed in hertz too, and the low modes are countable — so the same reshuffling happens once more at the bottom of the range, in a filter that belongs to the building rather than to the instrument.

Averaged over an ensemble, and which instruments it is actually about #

The obvious objection is that an orchestra is dozens of fixed filters at once, and a reshuffling that raises one instrument’s written G may lower another’s — so the effect might average away. Running the same 415-to-440 comparison through six of this collection’s radiators, over eighteen written notes, says it does not, and says something more useful on the way.

Averaging six instruments reduces the per-note spread by a factor of 1.7, not to nothing. So the effect survives an ensemble — but not because the reshufflings combine. It survives because it is concentrated in two of the six, and averaging a large effect with four small ones divides it rather than cancelling it.

That is the refinement the section above needs. “Every instrument in the orchestra including the ones with no strings” is too broad, and the arithmetic says which ones and why. A filter is only fixed if the resonator does not change size with the note. A violin’s body and a singer’s tract are boxes of constant dimensions with sharp resonances, and the notes slide across them. A clarinet’s principal resonator is the sounding length of its own bore, which shortens with every fingering — so most of its filtering moves with the note and has nothing fixed to slide against. What is left fixed on a wind instrument is the bell’s radiation curve and the tone-hole lattice’s cutoff, and both are gentle.

An organ flue pipe is the limiting case and is exactly zero, for the reason the dynamics rung gives: its spectrum is set at the voicing bench and a stop is one shape at one level, so there is nothing for a standard to slide.

So the historical argument belongs to the strings and the voices, which is where it has always been conducted. Singers objected and violinists were rebuilt; wind players’ complaints about a rising standard are about fingering, breath and the fact that a bore cannot be pulled far — a different mechanism, and a rung of its own.

The awkward corollary #

If the mechanism is a fixed filter under a moving note, then it does not need a change of standard to bite. It bites whenever an instrument’s fixed resonances and its playing range are in the wrong relationship, which is a permanent condition of every instrument in the orchestra.

That makes this rung’s finding a special case of one the timbre ladder already has. An instrument is not one timbre established that a fixed filter under a moving fundamental gives a different spectrum at every pitch — which is a statement about playing up a scale. Raising a pitch standard is the same operation applied to the whole scale at once, by a hundred cents, and produces the same kind of change in a smaller amount.

So the historical argument and the register argument are one argument. A violin’s wolf note, the weak spot in a soprano’s passaggio and the effect of moving from 415 to 440 are three readings of one fact about fixed resonances, and only the third of them is usually described as being about tuning at all.

Where this ladder goes next #

Five rungs. Four hundred years without agreement; a memory for the note itself; the note moved by a minor third; a standard is a specification with a material constant under it; and now, what the climb changed besides the pitch — which is almost nothing through tension and a great deal through the filters the notes move against.

The rung after it is the one the retuning limitation names. Every figure here holds the instrument still and moves the standard, and history did the opposite: instruments were rebuilt to suit the standard, string by string and bar by bar. A model of that would take the site’s own string scaling, hold the tension fixed rather than the string, and ask what gauge each note needs at each standard — which would say whether the climb was affordable at all without the metal frame, and would put a date on the point at which it stopped being.