What a drum is doing instead

Every resonator in this field so far has had modes in whole-number ratios, or nearly. A string’s are 1, 2, 3, 4; an open pipe’s are the same; a stopped pipe’s are the odd members of that set. A circular membrane’s are not remotely.

Why the numbers are what they are #

A string is a one-dimensional object, and its modes are sines with a whole number of half-waves along it. That is why the ratios are whole numbers: it is the only way to fit a sine between two fixed ends.

A membrane is two-dimensional and circular, so its modes are described in polar coordinates and its radial part is a Bessel function rather than a sine. The condition is that the displacement vanish at the rim, which means the rim must sit at a zero of the Bessel function — and the zeros of Bessel functions are not evenly spaced and are not in whole-number ratios to each other.

The first few come out at 1, 1.593, 2.136, 2.295, 2.653, 2.917. The mode names are two numbers, $(m,n)$: how many diameters and how many circles of nodes the pattern has. The (0,1) mode is the whole head moving together; (1,1) has a single nodal diameter, so one half of the head goes up while the other goes down.

There is nothing wrong with any of this. It is a consequence of the object being a disc rather than a line, and it is why a drum is the instrument it is.

It is worth noticing how general the point is. Whole-number mode ratios are not the normal case for a vibrating object; they are the special case that happens in one dimension. A bar, clamped at one end, has modes near 1, 6.27, 17.5 — further from whole numbers than a membrane’s. A plate is worse. A bell, before a founder gets to it, is worse still.

Almost everything that vibrates is inharmonic, and the instruments that carry melody are the exceptions. Strings and air columns are one-dimensional, which is why they are the exceptions, and a piano string’s stiffness is the ordinary two- and three-dimensional world leaking back into one of them.

Why that means no pitch #

Pitch is inferred from a pattern, and the pattern the ear looks for is a harmonic series. A set of partials at 1, 1.59, 2.14, 2.30 fits no harmonic series: there is no fundamental for which those are consecutive whole multiples, and no fundamental for which they are even approximately so.

So a struck membrane produces a sound with a definite spectral centre of gravity — it can be described as high or low — and no pitch a listener could sing back or name. Which is exactly what an untuned drum is.

The demonstration is available in the buttons above, and it is worth doing before reading on: the harmonic set has an obvious note, the membrane set has none, and the two are the same fundamental frequency with different partials over it.

What a kettle does #

A timpani is a membrane over a large sealed bowl, and the bowl changes the modes.

Two effects do the work. The enclosed air loads the head, and it loads different modes differently: a mode that moves the whole head in and out has to compress the air in the kettle and is stiffened and shifted, while a mode with a nodal diameter moves air from one side of the kettle to the other and is barely affected. The radiation load — air being pushed around outside the head — adds mass, most to the modes that radiate best.

The combination pulls the (1,1), (2,1), (3,1) and (4,1) modes toward the ratios 1 : 1.5 : 2 : 2.44. Multiply through by two: 2 : 3 : 4 : 4.9. That is a harmonic series, missing its fundamental, on a note an octave below the lowest partial present.

The fourth of those is the one worth watching. Two, three and four are exact; 4.9 is two per cent flat of five, which is a third of a semitone. On the residue machinery this collection uses that is a partial well inside the tolerance for membership, so it votes for the fundamental rather than standing out — but it is the mode a maker has least control over and the one whose departure grows fastest if the tuning drifts. A timpani’s harmonicity is three modes exact and one nearly so, which is a more precarious arrangement than the ratio list suggests and is why the instrument needs retuning between movements. The three exact ones are held there by the enclosed air, which is a fixed volume; the fourth is held by the radiation load, which is a function of how the head is sitting that day.

The pitch a timpanist tunes is a residue pitch, and it is an octave below the lowest partial the drum produces. That is the same structure as a bell’s strike note and the same structure as the laboratory demonstration with a deleted fundamental, on a third kind of object.

What “no pitch” actually means, since it is a strong claim #

It is worth being careful here, because a drum obviously has something pitch-like — a bass drum is low and a bongo is high, and a drummer tunes both.

What a drum has is a spectral centroid: a centre of gravity in the spectrum, which moves up when the head is tightened and down when it is loosened. That is a perceptual dimension, it is ordered, and it is what “tuning a drum kit” adjusts. It is not a pitch in the sense the rest of this site uses, because a listener cannot sing it back, cannot name its interval to another drum, and cannot say whether two drums are an octave apart.

The distinction is the same one the category essay draws from the other direction. A pitch is something that falls into a category and can be an interval; a brightness is a continuum with no categories on it. A membrane gives the second and not the first.

Where a timpanist strikes, and why #

A timpani is struck about a third of the way in from the rim, not at the centre. That is the rule from the excitation-point essay applied to a disc.

The centre is a point where every mode with a nodal diameter has a node — the (1,1), (2,1), (3,1) and (4,1) modes all have zero displacement there. Striking the centre therefore excites none of the modes the whole design exists to tune, and excites instead the (0,1) and (0,2) modes, which are the ones that move the whole head and which the kettle’s air has stiffened into something high and dead. A timpani struck in the centre is a thud with no pitch, which is a legitimate effect and is not the instrument’s normal sound.

Struck a third of the way in, all four tuned modes have substantial displacement and the residue is strong. The exact position is a compromise between them, and the compromise can be solved rather than described: each mode’s excitation at radius r is the Bessel function of its own order evaluated at r/a times that order’s first zero, so the four curves can be drawn and read together.

The position that maximises the weakest of the four is r/a = 0.70 and the position that maximises their product is 0.63; the traditional beating spot at 0.67 sits between them and is within one per cent of either optimum. So the spot a player learns by ear is the solution to a two-criterion optimisation over four Bessel functions, and the two criteria bracket it.

The more useful reading is the row rather than the maximum. At the beating spot the four excitations are 0.480, 0.466, 0.434 and 0.395 — within twenty per cent of each other, where at 0.40 they span a factor of four and at 0.10 a factor of two hundred. That evenness is what the position is really buying, and it is what a residue needs: a fundamental is implied most strongly when the partials implying it are all present at comparable strength, and one weak member of a four-partial set weakens the inference more than a uniform reduction of all four would.

The centre confirms the same arithmetic from the other end. At a tenth of the radius the four excitations are 0.188, 0.032, 0.005 and 0.001, so a strike near the middle gives the first tuned mode a fifth of its available amplitude and the fourth a thousandth of it. That is not a weak version of the drum’s sound; it is a different partial set with no common fundamental in it, which is the untuned membrane the kettle was built to escape.

The head is also tuned, in six places #

The other control is tension, and a timpani’s head is tensioned by six or eight tuning rods around the rim plus a pedal that changes all of them together.

The pedal sets the pitch. The individual rods set something else: the evenness of the tension, which decides whether the modes are where the design expects. An unevenly tensioned head has modes split into pairs — a nodal diameter in one direction sees a different tension from one at right angles to it — and a split mode produces two frequencies a few hertz apart where there should be one.

Two other drums that solve it differently #

The timpani is not the only membrane that has been made to produce a pitch, and the other solutions are instructive because they attack different terms.

A tabla carries a black patch of loaded paste on its head, off-centre on the bayan and central on the dayan. The added mass changes the modes’ frequencies differentially, and the result is a set of partials in near-harmonic ratios — a definite pitch, from a membrane, achieved by loading the head rather than by enclosing air. It predates the acoustics by centuries and it works.

A mridangam does the same with a similar patch, and C. V. Raman studied both in the 1920s specifically because a drum with a pitch was a theoretical problem.

Three traditions, three mechanisms — an air cavity, a mass patch, a shaped shell — and one target. That convergence is the strongest available evidence that what is being aimed at is the listener’s inference rather than any property of the object: nothing about a kettle resembles anything about a lump of paste, and both produce partials in whole-number ratios because that is what makes a pitch inferable.

An inharmonic set is also a rough one #

There is a second consequence of Bessel-zero ratios and it belongs to a different field of this site.

Roughness is computed from pairs of partials falling close together without coinciding, and an inharmonic set is one whose partials fall wherever they fall. Two harmonic notes at a simple ratio share partials, and shared partials do not beat. Two inharmonic sounds share nothing, so almost every pair of partials contributes some roughness.

That is why percussion sections play rhythm and pitched instruments play harmony, and it is a fact about Bessel zeros rather than about musical convention. It also predicts what happens once a drum is tuned: a set of timpani can play a bass line, and does, because the residue pitches behave like pitches and can form intervals.

The prediction has a limit worth stating. Two timpani a fifth apart share the residue’s harmonics and not the drums’ actual partials, so the consonance is thinner than two strings at the same interval — there is less to coincide. Orchestrators know that timpani do not blend the way strings do, and this is a reason for it rather than a restatement of it.

What the picture cannot show #

The ratios are the ideal and a real timpani departs from them. Published measurements put the (2,1) mode nearer 1.50 and the (4,1) nearer 2.44 to 2.50 depending on the drum, the head and the tension, and the fifth tuned mode is often nowhere near where the pattern would want it.

The head’s own stiffness is ignored. A real head has bending stiffness, which raises the higher modes exactly as a piano string’s stiffness raises its partials, and the effect grows with mode number. That is one reason the fifth mode is the one that will not come into line.

Decay is not drawn. A timpani’s tuned modes decay slowly and its untuned ones fast, so the pitched part of the sound emerges after the strike rather than being present in it. The pitch a listener names is inferred from the second half of a note.

And the buttons are additive synthesis at the stated ratios. They demonstrate the point of the essay — that one partial set has an inferable fundamental and the other does not — and they are not a recording of a drum.

The kettle is not a resonator in the usual sense #

One point of vocabulary, because it is the thing most often got wrong about a timpani.

A guitar’s body, a violin’s body and a marimba’s tubes are resonators in the ordinary sense: they take what the vibrating element produces and radiate some frequencies more strongly than others. They are filters, they sit downstream, and they do not change the source’s own modes at all — which is the whole architecture the vowel essays established and the next essay in this field is about.

A timpani’s kettle is not doing that. It changes which frequencies the head vibrates at. The head with a kettle under it has different modes from the same head without one; it is not the same source heard through a different filter.

The distinction has an audible consequence. A filter can be changed without retuning the instrument — a violin’s body affects every note the same way. A kettle cannot: change the kettle and the drum’s tuning changes, because the modes move. That is why a timpani’s bowl is a fixed part of the instrument and a violin’s body, in principle, is swappable.

Both mechanisms are common in instruments and both are usually described with the word “resonator”, which is why it is worth keeping them apart: one changes what an object does, and one changes what escapes from it.

A violin body is a filter: a fixed response curve with the partials of the note sliding underneath it as the note changes. Nothing about it moves the string’s modes — the string vibrates at exactly the same frequencies whatever body it is mounted on, and the body decides only how much of each is radiated. A timpani’s kettle is not doing that. It changes which frequencies the head vibrates at: the head with a kettle under it has different modes from the same head without one, so it is not the same source heard through a different filter. The audible consequence is that a filter can be swapped without retuning the instrument and a kettle cannot. Both are called resonators, which is why it is worth keeping them apart: one changes what an object does, and one changes what escapes from it.

Whose drums, and when #

The Bessel-function treatment of a circular membrane is Euler’s and Poisson’s, from the eighteenth and early nineteenth centuries, and the mode ratios are a standard result long before anybody applied them to an instrument. The measurement of a real timpani’s modes, and the identification of the air load as the mechanism that tunes them, is twentieth-century — Rossing’s work from the 1970s and 80s is the standard reference. Raman’s tabla papers are from 1920 and 1934.

The kettledrum itself is much older than any of it and reached its modern form, with a pedal, in the nineteenth century. As with the bell, the craft arrived at an arrangement that makes a residue pitch inferable without any theory of residue pitch — and in this case without even a theory of what the kettle was for. It was there to make the drum louder.

Where this goes #

This is the fourth rung of the missing-fundamental ladder and the last one this phase writes. What it leaves open is the case the ladder has been circling: an object whose partials support two incompatible fundamentals, which is what a badly tuned bell and a split-mode drum both are, and what a listener does with the ambiguity.

The field turns next to bodies. A drum’s kettle and a bell’s shell are resonators that change an object’s own modes; a violin’s body is a filter that changes what is radiated without touching the string’s modes at all, and the distinction between those two kinds of thing is one worth having.