Generator

What is struck, and where its partials land

Partial ratios for an ideal string, an ideal membrane, a kettledrum, a church bell, 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. A church bell: 0.50, 1, 1.20, 1.50, 2, 2.50. 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. The bell's ratios are a founder's target rather than a computation — arrived at by shaving metal off a casting, and given here as measured.

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. 13 of its 13 placements carry sound, built from the same numbers as the drawing, offering these buttons: Bar free at both ends, Church bell, Ideal membrane, Ideal string, Kettledrum, Only the three that imply the strike note, The bell, The strike note itself, 294 Hz.

Called by 3 essays

the blast radius of changing it

A bell tuned to 294 Hz, and the note it is heard at. The partials of a well-tuned church bell, as ratios to the prime, with the three that imply the strike note marked. The nominal, twelfth and double octave sit at 2, 3 and 4, which is a harmonic series on 1 — so they imply a fundamental at 294 Hz, an octave below the loudest partial the bell has. The tierce at 1.2 is a MINOR third above the prime, which is why a bell has a minor quality by construction rather than by choice.

A bell has no fundamental

The note a listener names when a church bell is struck is not any partial the bell has. Its nominal, twelfth and double octave sit at 2, 3 and 4 times the prime, which is a harmonic series on a pitch an octave below the loudest thing in the sound — and that pitch is supplied by the listener. Founders have been tuning it by ear since the fifteenth century.

instruments · Missing fundamental

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

Second order and fourth order, from the same sequence of wavenumbers. Frequency against wavenumber for a string and for a bar, with each object's own wavenumbers marked on the axis. A string's equation is second order in space, so frequency goes as wavenumber and the straight line carries its modes at 1, 2, 3, 4. A bar's is fourth order, so frequency goes as the SQUARE of wavenumber and the parabola carries the same kind of sequence to 1, 2.757, 5.404, 8.933. Both sequences of wavenumbers are arithmetic — a string's are the even multiples of π/2L and a bar's the odd ones, except that the first is 3.011 rather than 3 — and squaring an arithmetic sequence is the whole of why one object has a fundamental and the other has none. Every number is derived from the boundary condition; none is measured.

A bar's partials are the odd numbers, squared

A string's wave equation is second order in space and a bar's is fourth, so a string's frequencies go as its wavenumbers and a bar's go as their squares. Both objects fit a nearly arithmetic sequence of wavenumbers between their ends; squaring one is the whole of why a marimba key has no note in it. The ratios are 1 : 2.756 : 5.404 : 8.933, which are the odd squares over nine, uniformly 12.9 cents flat, and the 12.9 cents is one root that refused to be 3π/2.

instruments · Struck bar

All figures · What can be heard