Generator

A string mode swept through a body resonance at 460 Hz

What the string plays against what comes out. Away from the resonance the two are the same and the line is the diagonal. Near it the mode splits into a pair, and the note warbles at the difference between them — 12.9 Hz at the centre, which is slow enough to be counted and far too fast to be a tremolo. The splitting is a coupled oscillator and has nothing to do with the wolf fifth of a tuning system, which is a twenty-three cent arithmetic residue and shares only the word.

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. 10 of its 10 placements carry sound, built from the same numbers as the drawing, offering these buttons: A tone away from it, On the wolf — 12.9 Hz apart, On the wolf — 3.7 Hz apart, On the wolf — 4.4 Hz apart, On the wolf — 5.1 Hz apart, The note it is meant to be.

Called by 5 essays

the blast radius of changing it

What is struck, and where its partials land. Partial ratios for an ideal string, an ideal membrane, a kettledrum, drawn on a logarithmic axis so that a whole-number ratio is a fixed distance from the last. An ideal string: 1, 2, 3, 4, 5, 6. An ideal membrane: 1, 1.59, 2.14, 2.29, 2.65, 2.92. A kettledrum: 1, 1.50, 1.99, 2.44, 2.89. The membrane's are Bessel zeros — a derivation, and they land nowhere near the whole numbers. The kettledrum's have been pulled toward 2 : 3 : 4 : 5 by the enclosed air, on a fundamental an octave below the lowest partial present.

What a drum is doing instead

An ideal membrane's modes are zeros of Bessel functions — 1, 1.59, 2.14, 2.30 — which support no common fundamental, so a drum rings without a note. A timpani is that problem solved — the kettle's air and the radiation load drag four modes onto 1, 1.5, 2, 2.5, and the pitch a timpanist tunes is the fundamental those four imply and none of them is.

instruments · Missing fundamental

A string mode swept through a body resonance at 460 Hz. What the string plays against what comes out. Away from the resonance the two are the same and the line is the diagonal. Near it the mode splits into a pair, and the note warbles at the difference between them — 12.9 Hz at the centre, which is slow enough to be counted and far too fast to be a tremolo. The splitting is a coupled oscillator and has nothing to do with the wolf fifth of a tuning system, which is a twenty-three cent arithmetic residue and shares only the word.

The other wolf

A cellist's wolf note is a string mode landing on a body resonance, at which point the two stop being separable and start exchanging energy — the mode splits in two and the note warbles at the difference. It is a coupled oscillator. The tuning system's wolf is twelve fifths failing to close by 23.5 cents. They share a word and nothing else.

timbre · Beating

A perfectly tuned unison is the shortest note on the piano. How long a struck unison takes to fall forty decibels, against how far apart its two strings are tuned, for a note whose single string would ring for 20 seconds and whose bridge takes the in-phase mode down in 1.5. At a perfect unison the hammer excites only the mode that drives the bridge, so the note dies in 1.00 seconds. The longest note is at 2.30 cents, at 3.05 seconds — three times as long. Past 4.48 cents the two modes stop sharing a frequency and start sharing a decay, the note beats instead of ringing, and the sustain collapses.

Three strings, and the note that comes back

The usual account of a beat adds two independent sources together. A piano's strings are joined at the bridge and are not independent: the pair has normal modes, one of which drives the bridge and dies in a second and a half while the other cancels at the bridge and rings for twenty. A hammer striking a perfect unison excites only the first — so a perfectly tuned unison is the shortest note on the instrument, dying in one second, and the longest is at a deliberate detuning of about two and a half cents, at three. Past four and a half cents the two modes swap what they share and the sustain collapses.

instruments · Beating

The bow's window across a compass, with the bridge in it. Schelleng's window — the ratio of the largest usable bow force to the smallest — at a bow position of 0.09 of the length, across 196 to 1568 hertz, with the minimum scaled by the body's own admittance at each note. The window is widest between resonances and narrowest on them: it falls from 25 to 1.7 at B♭4. The notes a player finds hard to start are the local minima, and they sit on the body's resonances by construction — C♯4, B♭4, C♯5 for this body. At this bow position it never closes; move the bow toward the bridge and it does.

The note the body will not let start

Every figure until now treats the string as though it ended at a rigid point, and an earlier essay admitted it: the body feeds back on the string hard enough to make some notes difficult on one instrument and easy on another. Put the body's own admittance into Schelleng's minimum bow force and the window narrows by fifteen to one at the corpus resonances — and near the bridge it closes.

instruments · Bowed string

Where a bar and its pipe stop being two things. The two normal modes of a bar and a resonator tuned to it, against how strongly they are coupled, at 262 hertz. Below a threshold the pair has one frequency and two different decay rates — the pale curves, which are the damping splitting rather than the pitch — and above it the frequencies separate. The threshold is exact and it is not a matter of degree: it is where the coupling rate equals half the difference between the two damping rates, which for a bar of Q 197 against a tube of Q 80 is a coupling of 0.37 per cent. A marimba's own coupling is 0.62 per cent — 1.67 times the threshold, and not free: it is fixed by how much louder the tube makes the note, since the coupling that splits the pair is the coupling that carries the energy out. So the resonator model's assumption that the tube is a filter downstream of the bar is wrong at middle C, and it is wrong by less than a factor of two.

A bar and its pipe are one object

Three earlier essays treat a marimba's resonator as a filter the bar's output passes through, and both of them said in their own caveats that the coupling was not modelled. It is here, and the debt was right: the coupling is 1.67 times the threshold at which the pair acquires two frequencies instead of two decay rates, so the tube is not downstream of anything. Every consequence of that is smaller than the peaks it would have to be seen between.

instruments · Struck bar

All figures · What can be heard