Percussion: Membranes, Bars, and Plates

Learning Objectives¶

By the end of this chapter, you should be able to:

Explain why the modes of a two-dimensional vibrator are generally inharmonic, and why this makes most percussion instruments indefinite in pitch.
Describe the transverse modes of a free-free bar, and state the frequencies of the first few relative to the fundamental.
Explain how undercutting a marimba bar brings its second mode into a harmonic relation with its first, and what that does to the perceived pitch.
Describe the mode shapes of a circular membrane, identify nodal diameters and circles, and explain why an ideal drumhead has no definite pitch.
Explain how the air loading of a kettle turns the timpani’s inharmonic modes into a near-harmonic set, and identify which mode sounds the pitch.
Describe the modes of a flat plate, interpret a Chladni pattern, and explain what the sand reveals.
Explain why cymbals and gongs produce a dense, noise-like spectrum, and describe the nonlinear energy cascade responsible.
Describe the modes of a church bell and the tuning practice that makes the hum, prime, tierce, quint, and nominal fall in fixed relations.
Explain what it means to tune an inharmonic instrument, and identify the strategies used for bars, kettles, and bells.

Introduction¶

A pair of ornate silver kettle drums with their bowls, membranes, and tensioning screws visible. — Figure 12.1:Percussion instruments make mode shapes physical: a stretched membrane, a bowl, and an adjustable boundary all contribute to the sound. The Metropolitan Museum of Art, CC0.

Every instrument so far has had a harmonic spectrum. Strings and air columns are one-dimensional, and their boundary conditions force their modes into exact whole-number ratios; so they have clear pitches, their tones blend, and Chapter 9’s tuning systems are built out of small integers.

Percussion is different, and the difference is not a matter of degree.

A drumhead, a metal bar, a plate, a bell: these are two-dimensional or three-dimensional vibrators, and their modes fall in ratios like $1: 1.59: 2.14: 2.30: 2.65$ . Nothing forces those into whole numbers, and they are not. The ear, presented with such a set, cannot find a fundamental that explains it (Chapter 8), and reports a sound with a rough pitch region rather than a note.

Percussion therefore divides so sharply into instruments of definite and indefinite pitch, and the definite ones, timpani, marimba, tuned bells, are the result of considerable ingenuity. Each is an object that has been made to have a pitch, by moving its modes into place. This chapter is largely about how.

It is also where the book’s model starts to strain. Chapter 5’s spectrum assumed periodicity; a cymbal is not periodic. Chapter 8’s consonance assumed harmonic partials; a gamelan’s are not. The physics still works, it is just that the answers stop looking like music theory.

12.1Why Percussion Is Different¶

One Dimension Versus Two¶

A string is a one-dimensional system, and its modes are indexed by a single number $n$ . The boundary conditions, zero at both ends, force $\lambda_n = 2L/n$ and hence $f_n = nf_1$ . The whole-number ratios come from counting half-wavelengths along one direction.

A membrane is two-dimensional, and a mode must satisfy a boundary condition around a whole curve rather than at two points. Its modes are indexed by two numbers, and the frequencies are set by the zeros of Bessel functions rather than by the zeros of a sine. Bessel zeros are irrational numbers with no arithmetic relationship to each other.

Figure 12.2:The first six modes of an ideal circular membrane. Each is labeled $(m, n)$ with $m$ nodal diameters and $n{-}1$ nodal circles inside the rim. The frequency ratios, 1.00, 1.59, 2.14, 2.30, 2.65, 2.92, are not whole numbers, are not close to whole numbers, and are not multiples of anything.

A bar is one-dimensional but obeys a different equation. A string’s restoring force comes from tension; a bar’s comes from bending stiffness, and the resulting equation has a fourth derivative rather than a second. Its modes come out in the ratios

1: 2.756: 5.404: 8.933: \ldots

(12.1)

which are the squares of the roots of $\cos x \cosh x = 1$ , again, no whole numbers anywhere.

Screenshot of the Normal Modes simulation — Figure 12.3:Switch to the two-dimensional screen. The modes of a rectangular array are indexed by two numbers, and their frequencies are emphatically not whole-number multiples of the lowest, which is this chapter’s opening claim, in the simplest two-dimensional system there is.
Interactive simulation: Normal Modes

Inharmonic Modes and Indefinite Pitch¶

Chapter 8 explained what the ear does with a set of partials: it looks for a fundamental of which all are harmonics. Given $200, 300, 400, 500$ Hz it finds 100 Hz and reports it, even if nothing is there.

Given $200, 318, 428, 460, 530$ Hz it finds nothing. There is no fundamental that explains this set, and the pattern-matching mechanism has no answer to return. The listener hears a sound with a center of gravity, they can say it is high or low, and can tell one drum from another, but not a note. They cannot sing it back, and it does not participate in harmony.

Waveform and spectrum of Ideal membrane, Modeled timpano — Figure 12.4:An ideal membrane and a modeled kettledrum, struck the same way. The difference is entirely in where the partials sit, and §12.3 explains how the kettle moves them.
*What you hear:* Two struck drum sounds. The first is a dull thud with no identifiable note in it. The second is recognizably a timpano and has a clear pitch that could be sung back.
Listen: Ideal membrane, Modeled timpano

Waveform and spectrum of Real Timpani — Figure 12.5:A recorded timpani. The synthesized comparison above isolates the shifted mode ratios; this CC0 recording restores everything the model leaves out: mallet noise, room sound, small mistunings, and interacting decays.
*What you hear:* A real timpani strike begins with a short mallet impact, settles into a clearly pitched low ring, and decays for several seconds.
Listen: Real Timpani

Strike Tone and Decay¶

Two more features distinguish percussion.

The strike transient is a large part of the sound. Chapter 5 showed that attack carries an instrument’s identity; for percussion it may carry most of it. The initial contact is broadband noise, a click, and removing it makes a struck bar sound like an organ.

Different partials decay at very different rates. High modes lose energy faster, so the spectrum thins as the sound dies. A struck bell begins as a clangorous mixture and ends as a nearly pure hum tone, and the change is one of the most recognizable things about it.

12.2Bars¶

Modes of a Free-Free Bar¶

A xylophone or marimba bar is supported at two points and otherwise free. Its transverse modes are in the ratios $1: 2.756: 5.404: \ldots$ , and the second is the problem: 2.756 is close to neither 2 (an octave) nor 3 (a twelfth), and it clashes with the fundamental.

The supports are placed at the nodes of the fundamental mode, about $22\%$ of the length from each end, so that holding the bar does not damp the note it is supposed to produce. This is the same principle as touching a string at a node (Chapter 3).

Figure 12.6:Left: a plain bar’s modes against an undercut bar’s. Right: where the wood comes off. Thinning the middle is not decoration, it is what makes a marimba a pitched instrument.

Undercutting: Xylophone, Marimba, Vibraphone¶

The fix is to remove material from the underside of the bar, and the reasoning is elegant.

The fundamental has its maximum displacement, its antinode, at the center of the bar. The second mode has a node there. Removing material from the center therefore weakens the bar where the fundamental is bending hardest and where the second mode is barely bending at all, so it lowers the fundamental far more than it lowers the second mode.

Remove enough and the ratio 2.756 can be driven up to 4.0, two octaves, an exact harmonic. Continue and the third mode can be brought to about 10.

The amount of undercutting distinguishes the instruments:

Instrument	Second mode tuned to	Character
Xylophone	$3f_1$ (a twelfth)	Bright, sharp, penetrating
Marimba	$4f_1$ (two octaves)	Warm, full, mellow
Vibraphone	$4f_1$	Mellow, sustained (aluminum, long decay)

Waveform and spectrum of Plain bar, Undercut bar — Figure 12.7:The same fundamental, with the second mode at 2.76 and at 4.0 times it. Moving one partial is enough to turn a clang into a note.
*What you hear:* Two struck bars at the same fundamental. The first is clangorous and hard to assign a note to; the second has a clear, warm pitch.
Listen: Plain bar, Undercut bar

Struck Bars That Are Not Undercut¶

A glockenspiel’s bars are not undercut, and its steel bars keep the 2.756 ratio.

It works anyway, for a reason worth noticing: the glockenspiel plays high. Its fundamentals are above 2 kHz, so the inharmonic second mode lands above 5 kHz: a region where Chapter 8 showed pitch perception is poor and where the ear largely stops trying to match patterns. The partial is audible as brightness rather than as a clashing pitch.

Tubular bells exploit the same loophole from the other direction: their modes are inharmonic, but modes 4, 5, and 6 happen to fall close to the ratios $2: 3: 4$ , and the ear constructs a missing fundamental from them. A tubular bell’s perceived pitch is an octave below its lowest strong partial and corresponds to no mode of the tube at all.

12.3Membranes¶

Modes of a Circular Membrane¶

The modes of an ideal circular membrane, fixed at the rim, are

f_{mn} = \frac{\alpha_{mn}}{2\pi a}\sqrt{\frac{T}{\sigma}},

(12.2)

where $a$ is the radius, $T$ the tension per unit length, $\sigma$ the mass per unit area, and $\alpha_{mn}$ the $n$ th zero of the Bessel function $J_m$ .

The ratios were listed in §12.1, and none of them is anything.

Worked example: the fundamental of a drumhead

A snare drum’s batter head has radius $a = 0.178$ m (a 14-inch drum), areal mass density $\sigma = 0.20$ kg/m², and is tensioned to $T = 3200$ N/m. Using $\alpha_{01} = 2.405$ , find its $(0,1)$ fundamental.

f_{01} = \frac{\alpha_{01}}{2\pi a}\sqrt{\frac{T}{\sigma}} = \frac{2.405}{2\pi(0.178\ \text{m})}\sqrt{\frac{3200\ \text{N/m}}{0.20\ \text{kg/m}^2}} = (2.151\ \text{m}^{-1})(126.5\ \text{m/s}) = 272\ \text{Hz}.

(12.3)

The next mode up, $(1,1)$ , sits at 1.59 times this, about 432 Hz, an interval of

1200\log_2(1.59) = 804\ \text{cents}.

(12.4)

The nearest ratio of small whole numbers is $8{:}5$ (813.7 cents), itself an unremarkable interval, not a number that organizes a scale, and even that near-miss does not extend to the modes above it. A snare drum, unlike a timpano, has no enclosed kettle of air to pull this family of ratios toward consonance, so its head produces this pattern undisguised: a perfectly well-defined $f_{01}$ , and no definite pitch at all.

Nodal Diameters and Nodal Circles¶

The index $m$ counts nodal diameters, straight lines across the head that do not move, and $n-1$ counts nodal circles.

This has a practical consequence percussionists use constantly. Striking at the center excites only the modes with no nodal diameter, $(0,1)$ , $(0,2)$ , and so on, because every mode with $m \ge 1$ has a node there. Striking near the rim excites the $(m,1)$ family strongly.

Timpanists strike about a quarter of the way in from the rim, and §12.3 explains why that is exactly right.

The Timpani: How the Kettle Fixes the Pitch¶

A timpano is a membrane with a pitch, and it gets one from three things at once.

The air in the kettle loads the head. The enclosed air resists being compressed, which adds stiffness, and it has mass, which the head must drag along. Crucially the loading is different for different modes: the $(0,1)$ mode moves the whole head in the same direction and compresses the kettle air strongly, so it is stiffened a great deal. The $(1,1)$ mode moves one half up while the other goes down, so the air merely sloshes sideways and is barely compressed at all.

The result is that the modes move by different amounts, and they move into a near-harmonic set.

A bar chart comparing the mode ratios of an ideal membrane, 1, 1.59, 2.14, 2.65, 3.16, with measured timpani values of 1, 1.50, 2.00, 2.44, 2.90, against dotted lines at 1, 1.5, 2, 2.5 and 3. — Figure 12.8:Ideal against real. The air-loaded timpani modes sit very close to the ratios $1: 1.5: 2: 2.5: 3$ , which is $2: 3: 4: 5: 6$ of a fundamental one octave below the lowest mode. The ear finds that fundamental and reports it.

The pitch heard is a missing fundamental. Multiply the measured set by two and it reads $2: 3: 4: 5: 6$ , harmonics 2 through 6 of a note an octave below the $(1,1)$ mode. Exactly as with the brass pedal tone of Chapter 11, the ear supplies what the instrument does not radiate.

The $(0,1)$ mode, which moves the whole head as a piston, is the one mode that does not fit this series. It is also the loudest radiator, because a piston pushes far more air than a mode with nodal diameters. So a timpanist strikes about a quarter of the way in, near a node of $(0,1)$ , suppressing the one mode that would spoil the pitch, and favoring the ones that make it.

12.4Plates, Shells, and Bells¶

Flat Plates and Chladni Patterns¶

A flat plate has modes indexed by two numbers, like a membrane, but stiffness-controlled, like a bar. Their ratios are inharmonic and depend on the plate’s shape and boundary conditions.

The classic way to see them is Chladni’s method, published in 1787: scatter sand on the plate and excite a mode, and the sand bounces away from the moving regions and collects along the nodal lines.

Sand on a metal plate forms distinct geometric patterns at successive resonant frequencies. — Figure 12.9:Physics Girl excites successive resonances of a Chladni plate. Sand leaves the moving regions and gathers on nodal lines, making each two-dimensional mode visible as a different geometric pattern.
Watch on YouTube: Singing Plates: Standing Waves on Chladni Plates

Four square plates showing nodal-line patterns of increasing complexity, from a simple diagonal cross to an intricate lattice with a star at the center. — Figure 12.10:Select different resonant frequencies and watch the particles collect along the lines that do not move. Each pattern belongs to one mode, and the number of lines generally rises with the mode number. Violin makers still use the method to check the symmetry of a top or back plate before assembly; an asymmetric pattern means the plate is thicker on one side.
Interactive simulation: Resonance — Chladni patterns

Sand rearranges into nodal patterns as a plate is driven at different frequencies. — Figure 12.11:BBC Earth Science gives a slower explanatory view of the same method, showing why grains migrate away from vibrating regions and settle along the lines that do not move.
Watch on YouTube: Seeing Sound With Sand

Cymbals, Gongs, and the Nonlinear Cascade¶

A cymbal’s modes are so numerous and so closely spaced that by a few hundred hertz the ear cannot resolve them. Its spectrum is effectively continuous, and it is heard as noise.

Something further happens when a cymbal is struck hard, and it is genuinely different physics. The plate’s deflection becomes large enough that the restoring force is no longer proportional to displacement, the small-amplitude assumption of Chapter 1 fails, and the modes stop being independent. Energy fed into low modes cascades into higher ones, spreading across the spectrum over a few tenths of a second.

A crash cymbal therefore blooms: it does not simply start loud and decay, it starts with a limited spectrum and then fills out, brightening as it goes. It is one of the very few places in this book where superposition genuinely fails, and it is audible.

Waveform and spectrum of Cymbal — Figure 12.12:A cymbal-like sound built from hundreds of randomly spaced partials. There are no lines to count in the spectrum, and no fundamental for the ear to find, which is the definition of indefinite pitch.
*What you hear:* A metallic crash that swells slightly after the initial strike and then decays over several seconds, with no pitch in it at all.
Listen: Cymbal

Waveform and spectrum of Real Cymbal — Figure 12.13:The real orchestral instrument. Its continuously filled spectrum makes clear why a list of a few ideal modes cannot capture a cymbal crash. CC0 recording from VSCO Community Edition.
*What you hear:* A real pair of orchestral clash cymbals produces a sharp metallic impact followed by a dense, shimmering decay with no stable pitch.
Listen: Real Cymbal

The Church Bell and Its Five Tuned Partials¶

A bell is the most remarkable object in this chapter: a three-dimensional shell whose modes have been forced, by centuries of trial and refinement, into deliberate musical relationships.

A tuned bell has five named partials:

Five labeled vertical bars at frequency ratios 0.5, 1.0, 1.2, 1.5 and 2.0, named hum, prime, tierce, quint and nominal, and annotated as octave below, the strike note, minor third, fifth and octave above. — Figure 12.14:The five tuned partials of a church bell. Four are placed on consonant intervals; the **tierce** is deliberately a *minor* third, so bells sound solemn rather than bright, and a major-third bell, which founders can now make, sounds subtly wrong to most listeners.

Two things are worth drawing out.

The strike note is not a partial. The pitch a listener assigns to a bell corresponds to the prime, but it is largely constructed by the ear from the nominal, the quint, and higher partials, another missing fundamental. A bell’s perceived pitch can be measured, and it does not correspond to any single measured mode.

The minor third is a choice. Bell founders tune by turning the bell on a lathe and removing metal from the inside at specific heights, each of which affects different modes by different amounts. The traditional tierce is minor, and Dutch founders in the 1980s worked out how to produce a major-third bell. It sounds, to ears raised on the traditional kind, slightly wrong, which is a nice demonstration that expectation is doing work alongside acoustics.

Waveform and spectrum of Bell — Figure 12.15:A bell with the five tuned partials and several above them. Listen to the *change* rather than the sound: the high partials die first, so the clangorous strike settles into the hum tone. That evolution is what makes a bell sound like a bell.
*What you hear:* A bell strike: an initial clang containing many partials, resolving over several seconds into a steady low hum tone as the higher partials fade.
Listen: Bell

12.5Tuning the Untunable¶

Three Strategies¶

Percussion instruments with definite pitch are made so by one of three strategies. They are named together here because the whole chapter is instances of them.

Move the modes. Undercut a bar, turn a bell on a lathe, load a membrane with air. This is direct and it is what most tuned percussion does.

Suppress the modes that do not fit. Strike a timpano off-center to avoid the $(0,1)$ piston mode. Support a bar at the fundamental’s nodes.

Exploit the missing fundamental. Arrange for several partials to be harmonics of a note that is not there, and let the ear construct it. Timpani, tubular bells, and church bells all do this.

Pitch Where the Spectrum Is Inharmonic¶

“Pitch” for these instruments is not quite what it means for a violin.

For a harmonic instrument, the pitch is the fundamental, and the partials agree unanimously about what it is. For an inharmonic one, the partials disagree, and the perceived pitch is a best compromise the auditory system arrives at. It can therefore be ambiguous, some listeners hear a bell an octave away from others, and it can depend on how loud the sound is and what came before it.

This is not a defect of hearing. It is what happens when a mechanism evolved to group harmonically related partials is presented with partials that are not harmonically related, and the fact that it produces any answer at all is impressive.

The Steelpan as a Modern Solution¶

The steelpan, developed in Trinidad in the 1940s, is the newest acoustic instrument in this book and a genuine addition to the strategies above.

A section of oil drum is hammered into a concave dish, and distinct areas of the surface are shaped into individual note regions. Each region is a local resonator, tuned to a note, and the skill of the pan tuner is in shaping each so that its own second and third modes fall on the octave and twelfth above its fundamental.

So the steelpan turns one inharmonic plate into many nearly harmonic vibrators, and it does so by geometry alone, no undercutting, no air loading, no exploiting a missing fundamental. It is the best evidence in this chapter that tuned percussion is an engineering problem rather than a natural category, and that the problem is still being solved in new ways.

12.6Summary¶

Two-dimensional and stiffness-controlled vibrators have inharmonic modes. A circular membrane’s are set by Bessel zeros ( $1: 1.59: 2.14: 2.30: \ldots$ ); a free-free bar’s by the roots of $\cos x\cosh x = 1$ ( $1: 2.76: 5.40: \ldots$ ).
Inharmonic partials give indefinite pitch, because the pattern-matching mechanism of Chapter 8 finds no fundamental that explains them.
Undercutting a bar removes material at the center, where the fundamental has an antinode and the second mode a node, lowering the first far more than the second. Xylophone bars are tuned to $3f_1$ , marimba bars to $4f_1$ .
Bars are supported at the fundamental’s nodes, about $22\%$ from each end, so that holding them does not damp the note.
A timpano’s kettle loads the modes unequally, moving them into the near-harmonic set $1: 1.5: 2: 2.5: 3$ . The perceived pitch is a missing fundamental an octave below the lowest of these.
Striking a timpano a quarter of the way in suppresses the $(0,1)$ piston mode, which is the loudest radiator and the one that does not fit the series.
Chladni figures show nodal lines directly, and are still used to check the symmetry of violin plates.
A hard-struck cymbal is nonlinear: energy cascades from low modes into high ones over a few tenths of a second, which is the audible “bloom”.
A tuned bell has five named partials, hum, prime, tierce, quint, nominal, placed deliberately, with the tierce a minor third by tradition. The strike note is a construction of the ear rather than a measured mode.
Tuned percussion uses three strategies: move the modes, suppress the ones that do not fit, or exploit the missing fundamental. The steelpan adds a fourth by making many small resonators out of one plate.

12.7Check Your Understanding¶

Five short, auto-graded questions cycle within one compact activity. One question uses the chapter’s Chladni-pattern figure as a visual prompt.

Multiple choice. Where should a free-free bar be supported to preserve its fundamental vibration: the center antinode, both ends, or the nodes about $22\%$ from each end?
Visual true or false. In a Chladni pattern, sand collects along antinodes where the plate moves most.
Drag the words. Match the initial modal-frequency ratios $1:2$ , $1:2.76$ , $1:1.59$ , and $1:1.5$ to an ideal string, free-free bar, circular membrane, and timpano.
Fill in the blanks. The perceived timpani pitch is a ___ fundamental one ___ below its lowest prominent partial.
Mark the words. Identify the five named bell partials: “The five named partials are hum, prime, tierce, quint, and nominal; the traditional tierce is a minor third.”

12.8Conceptual Questions¶

Explain why a one-dimensional string has harmonic modes and a two-dimensional membrane does not, without invoking Bessel functions by name.
A drum and a violin are struck and bowed respectively. Explain why one has a definite pitch and the other does not, in terms of what the ear is trying to do.
Explain why removing wood from the center of a marimba bar lowers the fundamental much more than the second mode.
Explain why a marimba bar is supported at about $22\%$ of its length from each end rather than at the ends themselves.
A timpanist strikes about a quarter of the way in from the rim rather than at the center. Give two reasons.
The pitch a listener assigns to a timpano corresponds to no mode of the instrument. Explain how this can be.
A glockenspiel’s bars are not undercut and yet it sounds pitched. Explain why it gets away with it.
Explain what is happening in a hard-struck cymbal that does not happen in a lightly struck one, and why this violates an assumption made throughout the book.