Pitch and tuning

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.

Assumes: The pitch nobody agreed on, for four hundred years · The note that moved by a minor third

This ladder began by observing that nobody agreed on where A should be for four centuries, and treated the disagreement as a fact about institutions: church organs, opera houses, and eventually a conference. Its third rung found that raising the standard is not a uniform operation, because a performer’s own resonances do not transpose.

Both of those treat the choice as a decision somebody made. This rung is about the part that was not a decision.

Tension goes as the square

A string’s frequency is fixed by three things: how long it is, how heavy it is per unit length, and how hard it is pulled. Rearranged, the tension needed for a given pitch on a given string is proportional to the square of that pitch.

That squaring is the whole of the argument. A ten per cent rise in pitch is a twenty-one per cent rise in load, and pitch standards have varied by rather more than ten per cent.

Fig. 1 Where A has been put. From the French church organs at 392 to north German Chorton at 465 is a range of nineteen per cent — very nearly a minor third — and every instrument that crossed between them had to be rebuilt, restrung or replaced.

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. 2 One piano’s total string tension at six pitch standards. The instrument is built once: every length and gauge in the calculation is the same in all six rows, and only the frequency moves. From 392 to 466 the load rises by forty-one per cent — three and a half tonnes in this model — and the frame has to carry all of it.

The model’s absolute figure is low. It gives fourteen and a half tonnes at A440 where a modern concert grand is usually quoted at eighteen to twenty, because this site’s own string scaling puts one plain wire on every note where a real bass string is wound and there are three strings to a note only in the treble. The ratio between rows does not depend on any of that: it is exactly the square of the frequency ratio in every case.

A cast-iron frame is designed for a load. Tuning an instrument twenty-six hertz above what it was scaled for is not a musical decision made about it; it is a change to its structural specification, made by whoever wrote the pitch on the programme.

The ceiling nobody can engineer round

The interesting constraint is not tension, because tension can be reduced by using a thinner string. It is something a maker cannot escape at all.

Write the frequency of a string in terms of its length, its density and the stress in it — the tension divided by the cross-sectional area — and the cross-section cancels. What is left is that frequency times length is at most half the square root of the material’s strength over its density, and nothing else appears.

f · L ≤ ½√(σ/ρ)

A thinner string does not help: it takes less tension and it has less area to take it with, and the stress is unchanged. A better maker does not help. The only things that move the ceiling are the material and the length.

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. 3 Where each string of several instruments sits against its own material’s ceiling. Gut allows 240 hertz-metres and music wire 277; nylon allows 114 and brass 127. The violin E is the string closest to its limit by a wide margin, and it is the only one on the chart above eighty per cent.

A violin’s speaking length is 32.5 centimetres and its top string is E. At A440 that is 659 hertz, so the product is 214 hertz-metres, which is 89 per cent of what gut allows.

At A415 it is 84 per cent. At the Chorton pitch of 465 it is 94.5. And the pitch at which it reaches a hundred is A = 493.

That is one whole tone above the modern standard. A gut violin E string cannot be tuned there, at any gauge, by anybody.

What that number is worth, and how to get a better one

A493 is the most strength-sensitive figure in this essay and gut is the least well-specified material in it, so the two should be put together before the number is used for anything.

gut strength	ceiling, Hz·m	violin E at A440	standard at which the E reaches 100%
200 MPa	196	109%	A = 403
250	219	98%	A = 450
300 (used here)	240	89%	A = 493
350	259	83%	A = 533
400	277	77%	A = 570

The headline runs from A403 to A570 across the range a reasonable person would quote for gut, which is not a margin of error so much as an absence of one. At the bottom of it the model says a gut E is over its own breaking stress at modern pitch and cannot exist.

So run it backwards

That last row is the useful one, because it is a falsifiable consequence and history has already falsified it. Gut violin E strings existed, at A440 and well above it, in quantity, for centuries. The model therefore cannot be run from the material constant to the pitch; it can be run from the pitch to the material constant, and it is much better determined in that direction.

a standard gut E strings demonstrably survived	strength that requires
A415, baroque	≥ 212 MPa, with no margin at all
A440, modern	≥ 239 MPa
A465, north German Chorton	≥ 267 MPa

Those are the stresses at which the string breaks. A string strung at its breaking stress breaks, so a real one needs a safety factor, and at a modest one and a half the same three rows require 319, 358 and 400 megapascals — the last of which is above the top of the range anybody quotes for gut.

That is the essay’s finding turned into something sharper than it was. Either gut is considerably stronger than the 300 megapascals used here, or violin E strings at high baroque pitch were being run at essentially zero margin. The second is not a reductio; it is a description of the historical record, in which E strings broke constantly, were sold in bulk, and were the string players complained about. The arithmetic and the complaints agree, and where they agree they pin the constant: gut’s working strength, inferred from the fact that the strings existed at Chorton, is around 270 megapascals, and the margin at that pitch was nearly nothing.

And the ordering is exact rather than robust

The caveat below says the ordering — that the E is the tightest constraint on the instrument by a factor of nearly two — is robust to the spread in gut’s strength. It is better than robust. It is invariant, and trivially so: the strength enters every string’s percentage through the same denominator, so it cancels in any ratio.

On a violin the cancellation goes further, because all four strings share a speaking length. The E’s fraction of the ceiling divided by the A’s is then just 659.3 over 440, which is 1.498 — a perfect fifth. The “factor of nearly two” between the two most-stressed strings on the instrument is the interval between them, and it would be that number on any material, at any pitch standard, for any maker. Nothing about it is measured.

Fig. 4 The other thing a maker can change, and how little room it leaves. Holding the tension fixed 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 shrinks as the square of the pitch. Gut, brass and steel have different margins and the same slope. A standard is therefore not only a decision about A: it is a decision about how close every string on the instrument runs to its own material limit, and the top of the compass is where it is decided.

Which is why the complaints are always about the E string

The historical record on rising pitch is remarkably consistent about where the pain was, and the arithmetic above says why.

Every other string on the instrument has margin. The A at 440 is at 60 per cent of the ceiling, the cello A at 220 on a 69-centimetre string is at 63. Only the E is near the edge, and its margin is a few per cent — small enough that the difference between one town’s organ and the next town’s decides whether the string survives a rehearsal.

So the ceiling on historical pitch was set by one string on one instrument, and it was set by a ratio of two material constants that nobody at the time could have written down.

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 other half of the same constraint, at the bottom of the range. A plain wire low enough for the bottom of a piano would have to be metres long, so the bass strings are wound — mass added without stiffness. Winding raises the mass per unit length without touching the strength, so it lowers the pitch a given length can reach and does nothing at all for the ceiling. The constraint above is on the *top* of every instrument and there is no engineering answer to it.

It is worth putting that beside the two other things this collection has found sitting at the top of an instrument’s range. The top of a piano is where the strings are shortest and the inharmonicity worst, and the top of a woodwind is where the tone holes stop working. Three instruments, three completely different mechanisms, and in all three the constraint that decides the design is at the top of the compass. That is not a coincidence: every one of them is a consequence of some quantity scaling with frequency, and frequency is largest there.

Steel changes the picture and it arrived late. Music wire allows 277 hertz-metres, so a steel E is at 77 per cent at A440 rather than 89 — which is why steel E strings displaced gut ones almost completely in the twentieth century, and why the change happened at the top of the instrument first and worked downward.

Put through the inversion above, the size of that change is larger than eleven percentage points suggests. The pitch at which a steel E reaches its own ceiling is A570, a fourth above the modern standard, against gut’s A493 — so steel did not improve the margin, it removed the constraint from the argument entirely. No pitch standard anybody has ever proposed comes near A570, which is why the E string stopped being the string people complained about and why nothing in the twentieth century’s own pitch disputes is about breakage.

There is a second consequence and it runs the other way. A ceiling on f · L is also a floor on length. An instrument’s top string cannot be shortened indefinitely, so the compass of a string instrument at the top is set by the same constant, and a maker wanting a higher top note must either use a stronger material or accept a shorter string with less sound in it. The violin’s length is what it is partly because of where the E has to reach.

That makes the constraint a design parameter rather than an accident, and it is the reason the family has the shape it has: each member’s length is set by its lowest string’s need for mass and its highest string’s need for margin, and the two pull in opposite directions.

The instruments that cannot be retuned at all

A string player meeting a new pitch standard buys new strings. A wind player cannot.

The sounding length of a tube is set by where the holes are, and the frequency it produces is inversely proportional to that length. Raising A from 415 to 440 requires the tube to be 5.7 per cent shorter — which is not a tuning adjustment, it is a different instrument.

And an end correction makes a tube acoustically longer than it is, by a fixed few millimetres — which is why a wind instrument cannot simply be scaled to a new standard either: the correction does not scale with the length.

The practical response was the one the arithmetic forces: instruments were built in families for particular pitches, and a player moving between a church and a theatre carried two. The surviving stock of baroque woodwind is a stock of instruments at several different pitches, and a modern maker copying one has to decide which pitch to copy it at before cutting anything.

That is worth setting beside the whole essay. The standards argued over for four hundred years are separated by amounts that a change in the weather moves a wind section by — and the difference between two of them is comfortably above the smallest pitch change anybody can hear while being well below the point at which a piece would be unrecognisable. The whole dispute lives in a band a few per cent wide, bounded below by audibility and above by the strength of sheep gut.

A tube’s modes are set by its length and the speed of sound in it, and the speed of sound depends on temperature — so a wind instrument’s contribution to the standard moves with the room while a string’s moves the other way, which is the argument the thermometer essay makes in full.

And the voice does not transpose either

The third rung of this ladder found that a singer’s register break sits at a fixed frequency, so raising the standard moves the break onto a different note of the tune.

And a singer’s two laryngeal mechanisms overlap by about eight semitones, so a rise in the standard moves the seam between them relative to the written notes — which is the vocal version of the same specification problem, and the reason singers were the loudest objectors to nineteenth-century pitch inflation.

So there are three separate things going on at once and none of them is a matter of taste. A string has a breaking point, a tube has a length, and a larynx has a seam. A fourth could be added: a fretted instrument’s frets are cut for a scale length and moving the pitch does not move them, so its compensation is wrong at the new standard by an amount nobody notices only because it was already wrong. All three are fixed in absolute frequency; all three move relative to the music when the standard moves; and none of them is what “concert pitch” sounds like it is about.

Read on the ruler at 415 hertz the intervals are exactly the intervals at 440, because a ratio is a ratio — which is precisely why the argument has to be about tensions, lengths, materials and temperatures rather than about tuning.

That is the last thing worth extracting from the whole ladder. A pitch standard is a number written down to three significant figures about a system that moves by tens of cents when the room changes temperature. Its precision is not a claim about how accurately anybody plays; it is a coordination device, and what it coordinates is which instruments can be in the room together.

Which computation produced the numbers

The frequency-length ceiling is ½√(σ/ρ) with σ the tensile strength and ρ the density. The values used are 300 megapascals and 1300 kilograms per cubic metre for gut, 2.4 gigapascals and 7850 for music wire, 60 megapascals and 1150 for nylon, 550 megapascals and 8500 for brass. All four are quoted mid-range figures for the material and every percentage in the essay scales inversely with the square root of the strength assumed; gut in particular is a natural material with a wide spread, and taking 250 megapascals instead of 300 puts the violin E at A440 at 98 per cent rather than 89.

Those are breaking stresses, and this file’s other string figure uses 800 megapascals for steel, which is a working stress — what a string is actually strung at. The ratio between the two is the safety factor a maker leaves, and it is about three. The two numbers are not in conflict and it is worth saying so, because they appear in the same collection describing the same material.

The frame load sums the tension of every string of an 88-note instrument, with lengths and gauges from this site’s own scaling evaluated at A440 and held fixed while the frequency changes. One string per note in the bass, two in the tenor, three in the treble. The tension of each is 4L²f²μ with μ the mass per unit length of a plain steel wire of the modelled diameter.

The violin figures use a speaking length of 32.5 centimetres and a cello length of 69, both standard. The wind length change is exactly the inverse frequency ratio.

The strength sweep is the same formula evaluated at seven values of σ, and the inverted table is that formula solved the other way: σ = ρ(2·f·L)², with f the violin E’s frequency at each historical standard. Nothing in the inversion is new machinery — it is the same identity read right to left, which is what makes the comparison between the two directions worth anything. The safety factor of one and a half is stated rather than derived; it is at the low end for a component that is expected to fail occasionally and is deliberately generous to the essay’s own figure.

What the picture cannot show

Gut is not one material. Its strength depends on the animal, the twist, the treatment and the humidity on the day, and the spread is wide enough that the 89 per cent figure could reasonably be anywhere from 77 to over 100 — the section above tabulates it, and concludes that the sensible direction to run the argument is from the surviving instruments to the constant rather than the other way. The ordering is not merely robust to the spread but algebraically independent of it.

Nothing here is a historical claim. That players complained about rising pitch and that E strings broke are things the literature reports; this essay supplies an arithmetic that is consistent with them and does not establish them. In particular the causal story could run the other way, with pitch settling where it did for institutional reasons that happen to sit under the ceiling.

The frame-load model is crude in the ways stated above, and it also ignores the downbearing on the bridge, the plate’s own prestress and the fact that a real piano is scaled with a break between wound and plain strings that this model does not have.

The organ is left out entirely, and it is the instrument that decided most of the historical standards. An organ pipe’s pitch is a length, like a wind instrument’s, and an organ cannot be retuned at all without being rebuilt — which is precisely why a town’s organ fixed the pitch of everything played with it and why the standards cluster where organs were built.

And it says nothing about why anybody wanted the pitch higher. The usual explanation is brilliance — a violin at a higher tension is louder and brighter — and that is a claim about spectra which this collection has the machinery to test and has not tested.

The ladder from here

This rung took a convention and found a material constant under it. What the ladder still owes is the brilliance question above, which is the reason the ceiling was pressed against in the first place: if raising a string’s tension changes its spectrum as well as its pitch, then the four-hundred-year climb was a search for a timbre and not for a pitch at all, and the same argument the dynamic-marking rung makes about a hammer would apply to a whole orchestra.