Acoustics & Music Theory
why it sounds like that
Pitch and tuning

It was never the strings that stopped the climb

Every figure so far holds the instrument still and moves the standard. History did the opposite — instruments were rebuilt to suit the pitch, string by string. Hold the tension instead and each string's gauge is forced: the diameter goes as one over the frequency and the stress as its square. Gut breaks at A = 525 hertz and steel at 604, and the climb stopped at 466. The ceiling was somewhere else entirely.
15 min read 6 figures Somebody measured itThe design was forcedThe instrument is not the model

Assumes: What the climb was a search for · A standard is a specification

The fifth rung of this ladder asked what the four-hundred-year climb in pitch changed besides the pitch, and found the answer was almost nothing through tension and a great deal through the filters the notes move against. Every figure in it holds the instrument fixed and moves the standard.

That is the wrong way round, historically. Nobody kept an instrument and retuned it up a semitone; instruments were rebuilt. The rung’s own last paragraph said so:

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.

Re-gauging at a fixed tension: how close sheep gut comes to breakingHolding the tension at 700 newtons and re-gauging every string to suit the standard, the diameter each note needs goes as one over its frequency — so the stress goes as the frequency squared, and the margin against breaking falls the same way. At A = 392 the worst note has 1.79 times the stress sheep gut will take; at A = 466 it has 1.27. The margin reaches one at A = 525 hertz, which is far above anything the four hundred years of climb reached.the string breaks1.791.601.491.421.351.2739040041042043044045046047000.511.522.5A above middle C, hertztimes the breaking stress the top note has sparesheep gutbreaks at A = 525 Hztension held at700 N a string
Fig. 1 Re-gauging every string to suit the standard at a fixed tension of 700 newtons, in gut. The diameter each note needs goes as one over its frequency, so the stress goes as the frequency squared and the margin against breaking falls the same way. From A = 392 to A = 466 the worst note’s headroom drops from 1.79 to 1.27, and the margin reaches one at A = 525.

What re-gauging is, in one line

The string equation is

f = (1/2L)·√(T/μ),  μ = ρπd²/4

so at a fixed tension and length,

d = (1/Lf)·√(T/ρπ)

The diameter goes as one over the frequency. Raise the standard and every string needs to be thinner to sound its note at the same tension — which is a completely available thing for a maker to do, because wire comes in gauges and a maker chooses one per note anyway.

What it costs is stress. The tension is spread over an area proportional to the square of the diameter, so

σ = T / (πd²/4) ∝ f²

and a nineteen per cent rise in the standard is a forty-one per cent rise in stress on every string, exactly. The hero figure’s margins bear that out to three figures: 1.79 at 392 and 1.27 at 466, whose ratio is 1.409 against (466/392)² = 1.413.

How near the breaking point each string already is. Frequency times length, as a fraction of what the material allows. The ceiling is half the square root of specific strength — sheep gut 240, music wire 276, nylon 114, brass 127 hertz metres — and it depends on nothing a maker can change: not the gauge, not the tension, not the workmanship. The guitar top E runs at 187 per cent of its own ceiling, which is why it is the string that breaks and why every complaint about rising pitch in the historical record is about that one string.
Fig. 2 The ceiling the material sets, from a standard is a specification: the product of frequency and length that each material can carry before it breaks, with several instruments plotted under it. Gut’s limit is about 240 hertz-metres and steel’s about 276. Every design decision about a string instrument is a point under one of those lines.

The answer, which is a negative

Push the standard up until the top string breaks and gut gives out at A = 525 hertz and steel at 604.

The historical climb ran from about 392 to about 466 before the 1939 conference settled on 440, and it never came close to either. So the string was not the constraint.

That is a genuinely useful negative, because it rules out the most intuitive explanation for why the climb stopped where it did. The usual story — that pitch rose until instruments could not take it and then a standard was imposed — has a mechanism in it, and the mechanism is not this one. At A = 466 a gut string at the top of a keyboard compass has twenty-seven per cent of its breaking stress in hand, which is not comfortable and is not failure.

A pitch standard is a number of tonnes. The total string tension of one piano, tuned to each of six pitch standards. The instrument is built once — the lengths and gauges are the same in every row — and only the frequency changes, so the tension goes as its square: 392 hertz gives 11.5 tonnes and 466 gives 16.2, a difference of 41 per cent across a range of pitch that music actually used. The model's absolute figure is low — a modern grand carries eighteen to twenty tonnes against the 14.5 here — because the scaling is this site's own and has one plain wire per note where a real bass string is wound. The ratio between rows does not depend on any of that.
Fig. 3 The constraint that does bind: the total tension a piano’s whole string band puts on its frame, at each standard, for a fixed scaling. It runs from 11.5 tonnes at A = 392 to 16.2 at 466. The rise is the same f² and it is applied to the sum over 88 notes and 230 strings — so what is a fraction of a string’s headroom at one string is nearly five tonnes across the instrument.

Where the ceiling actually was

Five tonnes is the number. The difference in total frame load between the bottom and the top of the historical range of standards is 4.7 tonnes, on the same instrument, from the pitch alone.

A wooden-framed piano — a Viennese fortepiano of 1800, say — carries in the region of two to three tonnes in total. So the increment from raising the standard across the range this climb covered is larger than the entire load a wooden case ever bore.

Which means the constraint was structural and it was at the instrument level rather than at the string level. A maker could always find a gauge; what they could not do was find a case. And the thing that removed the constraint is dateable: Babcock’s cast-iron plate is patented in 1825 and full iron frames are standard by the 1850s, which is exactly the half-century in which the standard climbs fastest and in which the disagreements this ladder’s first rung catalogues become most extreme.

So the sequence is not pitch rose until strings broke. It is: iron frames made high tension affordable, high tension made a louder instrument, the louder instrument made a higher standard attractive, and nothing stopped it until a committee did.

One thing about that argument needs stating exactly, because the two halves of this essay compute two different routes and the four tonnes belongs to the other one. The frame-load figure holds the scaling fixed and raises the pitch, which is the retune route — the same strings, tighter. Re-gauging at fixed tension, which is what a maker building at a new standard does, leaves the frame load unchanged by construction: thinner strings at the same tension pull exactly as hard. So a maker building at A = 466 needed no more case than one building at A = 392, and the 4.7 tonnes is what an existing instrument would have to bear if it were pulled up to the new standard.

Both are real and they bear on different people. The maker’s constraint is the margin above, which the re-gauge route pays in full. The frame load is what the surviving instrument pays, which is why an old case at a modern standard is the case the historical-performance argument is about. And the iron frame’s part in the story is the third route again — makers who chose more tension for more sound rather than the same tension at a higher pitch, which is a decision about loudness that this essay’s arithmetic does not contain.

Re-gauging at a fixed tension: how close music wire comes to breakingHolding the tension at 700 newtons and re-gauging every string to suit the standard, the diameter each note needs goes as one over its frequency — so the stress goes as the frequency squared, and the margin against breaking falls the same way. At A = 392 the worst note has 2.38 times the stress music wire will take; at A = 466 it has 1.68. The margin reaches one at A = 604 hertz, which is far above anything the four hundred years of climb reached.the string breaks2.382.121.981.891.791.6839040041042043044045046047000.511.522.5A above middle C, hertztimes the breaking stress the top note has sparemusic wirebreaks at A = 604 Hztension held at700 N a string
Fig. 4 The same computation in steel rather than gut. Every margin is larger and the ceiling moves from 525 hertz to 604 — eighty hertz, about three semitones. That is a real gain and it is not where the climb stopped either, which is the point: the material improved the wrong constraint. Steel arrives in quantity in the same decades as the iron frame, so the two improvements are contemporaneous and only one of them mattered for this.

Two ways to raise a pitch, and they are different instruments

The distinction this rung turns on is worth stating on its own, because a great deal of writing about historical pitch runs the two together.

Keep the strings and retune upward. The tension rises as the square of the frequency, the string gets tighter, the instrument gets louder and harder to play, and eventually something gives. That is what the fifth rung computed and it is what happens to an instrument that is left alone.

Keep the tension and re-gauge. The strings get thinner, the instrument feels the same under the hand, the tension is unchanged and the frame load is unchanged. That is what a maker does when building at a new standard, and it costs nothing at all until the wire runs out of strength.

Those two produce instruments that differ in almost every measurable respect at the same nominal pitch, and both are “an instrument at A = 440”. Which one a surviving instrument is depends entirely on whether it was built at that pitch or has been retuned to it, and that is a question about its history rather than about its dimensions.

Nearly every old instrument played today is the first kind, because the pitch rose after they were made. Which is why the historical-performance movement’s decision to play at A = 415 is not an aesthetic preference so much as a decision to stop asking three-hundred-year-old wood to carry a nineteenth-century load.

How long a plain wire would have to be. The length a plain steel wire needs to sound each note at a fixed diameter and tension, against the length a piano actually has. The bottom A at 27.5 Hz would need 5.9 metres, which is longer than most rooms. Doubling the diameter instead halves the frequency and multiplies the inharmonicity coefficient by 5.7 — so the note arrives with its partials so sharp that it stops being one note. Winding copper over a thin core adds the mass without the bending stiffness, because a helix carries almost no bending moment.
Fig. 5 The scaling every one of these figures rests on: the length each pitch wants, against the length a case allows, with the point where the two part company and the string has to be wound instead. The wound string is the maker’s answer to the bass and there is no equivalent answer at the top, which is why the top note is always the one at risk in the hero figure.

What a maker gains by re-gauging, and what they lose

Re-gauging at a fixed tension is not free even where it is affordable, and the cost is in a quantity this site has an anchor about.

A thinner string at the same length and pitch has a smaller inharmonicity coefficient — B goes as the fourth power of the diameter over the fourth power of the length — so re-gauging upward in pitch by thinning the strings makes the instrument less inharmonic, not more. The stretched octaves a tuner sets get slightly smaller and the treble gets slightly clearer.

That is a real and slightly surprising benefit of the climb, and it runs opposite to the usual account, which is that higher pitch means more tension means a harder-driven and harsher instrument. At fixed tension the opposite happens. The harshness in the usual account comes from the other route — instruments built with more tension for more power — which is a different decision that happened to be made at the same time — and which shows up in the hammer’s own spectrum rather than in the string’s gauge.

Three octaves, and none of them is 2:1. How far above an exact doubling the upper note of an octave is set, against frequency. The listener's octave is measured with pure tones, which have no partials to beat against each other, so nothing about a stiff string can account for it. The piano's stretch is a different quantity with a different cause, and the two are drawn together only so that the difference is visible.
Fig. 6 The quantity that improves: the octave stretch a tuner has to set, which follows the inharmonicity coefficient. Thinner strings at the same length and pitch mean a smaller coefficient and a flatter curve. Re-gauging up is a small gift to the tuner, which nobody appears to have noticed at the time and which is dwarfed by everything else that changed.

Which computation produced the numbers

The string lengths come from this site’s own piano scaling — the ideal length for each pitch, capped where the case runs out — evaluated at the pitch the instrument was built for, which is taken as A = 415. That distinction matters: the lengths are a property of the frame and do not change when the standard does.

For each standard, each note’s required diameter is computed from the string equation at a fixed 700 newtons, its stress from that diameter, and its margin as the material’s breaking stress over that. The worst note is always the top one, because it has the shortest string and therefore needs the smallest diameter.

The breaking stresses are this site’s own material constants — 300 megapascals for gut and 2.4 gigapascals for music wire — and they are measured properties quoted from tables rather than anything derived, which is the one place in this essay where a number comes from outside.

The frame load is the sum over all 88 notes of the tension each string needs at the standard, with the string counts a real piano uses — one per note at the bottom, two in the tenor, three above — at the site’s own scaling. It is the same computation the fifth rung ran and is reported here at more standards.

The breaking standard is found by bisection: the pitch at which the worst note’s margin reaches exactly one.

Where the model stops

A fixed tension is a modelling choice, not a historical fact. Makers did not hold tension constant across a change of standard; they made whatever decisions they made, and the surviving evidence is instruments rather than intentions. What holding tension does is isolate one variable — it answers what would it cost to keep the instrument feeling the same — and a maker who accepted a higher tension for a bigger sound was making a different trade this figure does not draw.

The safety margin is not a design margin, and naming a margin tells the model exactly where the practical ceiling is. A margin of 1.27 means the string is at seventy-nine per cent of its breaking stress, and no maker has ever worked there. Real practice keeps a top string somewhere around half its breaking stress. Solving for the standard at which gut reaches each margin:

design margin gut gives out at steel
1.00, breaking 525 Hz 604
1.25 470 541
1.50 429 494
2.00, half breaking stress 371 427

At a fifty-per-cent design margin the ceiling is A = 371 hertz, which is below the bottom of the historical climb. Even at a margin of 1.5 it is 429, which is under A = 440. So the negative this essay is built on holds only for the breaking criterion: a gut top string was never comfortable at any standard in the range, and every step of the climb took it further past where a maker would choose to work.

That does not overturn the finding; it makes it a better one. The string was not a hard limit and it was a continuous complaint. Nothing snapped at a particular standard and forced a stop, which is what the essay’s negative claims and is right — and at the same time the top strings of gut-strung instruments were working at 60 to 80 per cent of breaking throughout, which is why the historical record is full of broken trebles and why the top note is where they go. A constraint that never binds and never relents is the shape of an irritation rather than a ceiling.

The margin, incidentally, does not depend on the tension at all. Holding tension fixed makes the diameter go as the square root of the tension, so the stress is the tension over an area proportional to it, and the two cancel — the margin is a function of frequency, length and material and nothing else. A maker cannot buy headroom by slackening the instrument, only by shortening the string.

Gut is not one material. Its breaking stress varies with the animal, the twist, the humidity and the age by considerably more than the difference between the standards being compared. A single number for it is a convenience, and it is the number this whole essay’s headline ceiling rests on.

The margin is quoted at the top note and the top note is not where a real instrument fails. A keyboard’s highest string is short, thin and cheap to replace; what actually breaks in a piano is usually a bass or tenor string at a bearing point, where a stress concentration nothing here models does the damage. The arithmetic identifies where the material is most loaded and not where the instrument is weakest, and the bearing arrangement is a separate object entirely.

And the piano is doing all the work. The frame-load argument is a piano argument, and the four-hundred-year climb is much older than the piano. What constrained a sixteenth-century organ, a harpsichord or a violin band is not five tonnes of frame load, and this essay has computed nothing about any of them.

Whose instruments, and when

The climb’s arithmetic is one thing and its social history is another, and this rung has only computed the first.

What is worth stating is that the two constraints identified here have different dates. The string’s ceiling has been at 525 hertz for as long as there have been gut strings, and nothing about it changed. The frame’s ceiling moved in 1825, and the standard’s steepest rise is after it.

That is at least consistent with the frame being the binding constraint, and it is not proof. A pitch standard is a coordination problem between organ builders, opera houses, wind makers and orchestras, and the wind instruments — which cannot be re-gauged at all, and which have to be rebuilt to change pitch — are the party with the strongest interest in stability and the loudest complaint. The first rung of this ladder catalogues four hundred years of that argument and none of it is about breaking strings. What it is about is organs, which cannot be retuned at all without being rebuilt, and which are therefore the immovable object every other instrument in a town had to agree with.

What the picture cannot show

Why anybody wanted the pitch higher. The usual answer is brilliance, and the fifth rung computed what brilliance the climb actually buys: twelve per cent more tension on a fixed string, eleven per cent less inharmonicity, and a brighter radiated spectrum. This rung’s version of the same instrument re-gauged gets some of that and not all of it, and the difference between the two is a maker’s decision that no arithmetic here reaches.

And it cannot show the wind band, which is the part of the orchestra that could not follow. A string player re-gauges; a wind player buys a new instrument, because a wind instrument is a thermometer and its pitch is set by a length it cannot change. That asymmetry is the whole reason a pitch standard was ever a matter of dispute rather than of preference, and this essay’s arithmetic applies to exactly the half of the orchestra that had the least to lose.

Where this ladder goes next

Six 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; what the climb changed besides the pitch; and now what it would have cost to follow it, which turns out not to be what stopped it.

What is owed is the other half of the orchestra. Everything in this ladder computes strings, because strings have a closed-form scaling law and wind instruments do not. But this collection now has a bore solver — the horn equation, set up two essays ago — and a wind instrument at a new standard is a bore scaled by a length ratio with tone holes that do not scale the same way. What that does to a woodwind’s intonation across its compass is computable for the first time, and it is the question the four-hundred-year argument was actually about.

Part 6 of 14

One essay in the series on pitch standard. The essays either side of this one:

What the climb costs, in tonnes and in inharmonicity. The total string tension a piano frame carries at each historical standard, computed from the string scaling used here. Tension goes as the square of the pitch, so the climb from 392 to 465 hertz is a rise from 11.5 to 16.2 tonnes on one instrument. The same ratio runs the other way through inharmonicity, which is inversely proportional to tension: the same wire at 465 hertz has 29 per cent less of it than at 392, so its partials are that much closer to a true series. What the climb was a search for Part 5 — The four-hundred-year rise in the pitch standard is usually explained as a search for brilliance, and an earlier essay ended by asking whether that was even coherent: does raising a string's tension change its spectrum as well as its pitch? Mostly it does not. What changes the timbre is something else entirely, and it is a mechanism that applies to instruments with no strings at all — every filter in the chain is fixed in absolute frequency while the notes move against it, so raising the standard is not a brightening but a reshuffling, and some notes lose. A woodwind cannot be pulled to a new standard. Every earlier figure computes strings, because a string's tension and gauge give a closed-form scaling law. A wind instrument does not have one, and this is why. To move from 440 to 415 hertz the player pulls out 11.7 millimetres at the joint, which lengthens the sounding tube of every fingering by the same absolute amount — so the interval each note drops is a fixed length against a shrinking one, exactly the shape of an end correction. The tuning note lands where it should and nothing else does: D3 is 67 cents sharp of where it belongs and G5 is 75 flat, a spread of 142 cents across the compass. A rebuilt instrument has no such problem — scale every length by one ratio and every mode moves by the same interval, and the tone-hole lattice cutoff moves with it, from 1766 hertz to 1666, which is 101 cents and therefore the same instrument transposed. That is the difference between an afternoon and a year. A woodwind cannot be pulled to a new standard Part 7 — Every earlier essay computes strings, because a string has a closed-form scaling law. A wind instrument does not. Pulling out at the joint lengthens every fingering's sounding tube by the same number of millimetres, which is a fixed length against a shrinking one — so the tuning note lands and the compass spreads by 142 cents. The strings could be regauged in an afternoon; the winds had to be rebuilt.

Where this goes next

What the climb costs, in tonnes and in inharmonicity. The total string tension a piano frame carries at each historical standard, computed from the string scaling used here. Tension goes as the square of the pitch, so the climb from 392 to 465 hertz is a rise from 11.5 to 16.2 tonnes on one instrument. The same ratio runs the other way through inharmonicity, which is inversely proportional to tension: the same wire at 465 hertz has 29 per cent less of it than at 392, so its partials are that much closer to a true series. What the climb was a search for The four-hundred-year rise in the pitch standard is usually explained as a search for brilliance, and an earlier essay ended by asking whether that was even coherent: does raising a string's tension change its spectrum as well as its pitch? Mostly it does not. What changes the timbre is something else entirely, and it is a mechanism that applies to instruments with no strings at all — every filter in the chain is fixed in absolute frequency while the notes move against it, so raising the standard is not a brightening but a reshuffling, and some notes lose. How near the breaking point each string already is. Frequency times length, as a fraction of what the material allows. The ceiling is half the square root of specific strength — sheep gut 240, music wire 276, nylon 114, brass 127 hertz metres — and it depends on nothing a maker can change: not the gauge, not the tension, not the workmanship. The guitar top E runs at 187 per cent of its own ceiling, which is why it is the string that breaks and why every complaint about rising pitch in the historical record is about that one string. A standard is a specification Choosing where to put A looks like a convention and is a mechanical decision. Tension goes as the square of frequency, so a piano built at 440 and tuned to 466 carries twelve per cent more load — a tonne and a half in this model's arithmetic. And there is a hard ceiling nobody can engineer round: frequency times length is capped by half the square root of a material's specific strength, which for gut is 240 hertz-metres. A violin E at A440 runs at 89 per cent of that. At A493 it is at a hundred, and every complaint in the historical record about rising pitch is about that one string. Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A440. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded. The pitch nobody agreed on, for four hundred years A440 is a committee decision from 1955. Before it, the same written note was played anywhere between about 392 and 465 hertz depending on the town, the building and the decade — a spread of very nearly a minor third, and every piece of theory here is untouched by it.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

Breaking stressInstrument designIntonationMaterialsPianoPitch standardScalingString tension