Acoustics & Music Theory
why it sounds like that
Generator

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.
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.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 15 of its 15 placements carry sound, built from the same numbers as the drawing, offering these buttons: The highest note music wire 1.20 metres long can be tuned to before it breaks: 230 hertz, The highest note music wire 1.40 metres long can be tuned to before it breaks: 197 hertz, The highest note sheep gut 0.65 metres long can be tuned to before it breaks: 370 hertz, The highest note sheep gut 1.20 metres long can be tuned to before it breaks: 200 hertz, The same written A at 392 hertz, the lowest standard here, The same written A at 466 hertz, the highest — 299 cents above it.

Called by 4 essays

the blast radius of changing it

The worst interval in each open chord, before and after the best possible compensation. For each shape, the largest departure of any interval from the just interval its name implies, in cents. Before compensation the worst is 15.6 cents; after a search over per-string saddle compensation it is 15.6. The search has six numbers to fix eight shapes with, and on this measure it finds nothing worth changing — because what is being measured is almost entirely equal temperament's own thirds, and a saddle moves a length rather than a temperament.

A guitar cannot be in tune

Three errors land on the same instrument — equal temperament's own thirds, the sharpening that comes from pressing a string down to a fret, and the fact that the only correction available is one length per string. Searching over every setting a luthier could choose leaves a worst-case error of about fifteen cents, and most of what is left is the temperament, which no saddle can reach.

tuning · The comma 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.

tuning · Pitch standard The window is a different width on each string. The width of Schelleng's bow-force window across each string's own playable range, with both terms in it: the body's admittance, which was added earlier, and the string's own characteristic impedance, which every figure had held at one value. The four curves are the same shape displaced vertically, because the impedance is a constant per string — the G string's is 1.81 times the E string's, and the window is one over that. Where the ranges overlap, the same pitch sits on two or three curves at once.

The same note is a different width

Schelleng's two bounds both carry the string's characteristic impedance and they carry it to different powers — the maximum force as Zc and the minimum as Zc squared — so the window a player has to stay inside goes as one over Zc. A violin has four strings whose impedances differ by a factor of 1.81, the same written pitch is available on two or three of them, and the tolerance is nearly twice as wide on one as on another. The two lightest strings turn out to have almost identical impedance, which nobody chose by accident.

instruments · Bowed string Re-gauging at a fixed tension: how close sheep gut comes to breaking. Holding 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.

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.

tuning · Pitch standard

All figures · What can be heard