NOTE: all the percussion sound samples were performed by Matteo Manzini on 04/12/2007. Sound engineer: Valerio Carboni.

Timpani are a set of large membranophones widely used in the symphony orchestras of Western music since the Baroque age.

Their popularity with composers has an obvious reason: they are among the very few membranophones able to produce sounds of definite pitch. These instruments can combine their rhythmic nature with the harmonic function of underlining the fundamental of the chords over which they are played.

In modern orchestras the timpanist plays up to four timpani, each tuned in advance to a certain note. In some modern compositions, however, the possibility of changing the pitch of a single instrument "on the fly" is also exploited, to obtain effects such as the glissando (see the section on effects).

The exceptional acoustic features of the timpani deserve careful analysis. The first to study them, and to identify them correctly, was Lord Rayleigh in 1877.^[1] The same scientist gave his name to several physical phenomena (Rayleigh waves, Rayleigh scattering, to mention just two of direct interest for this site), won the Nobel prize in 1904 for the discovery of a new noble gas, argon, and was succeeded as head of the Cavendish Laboratory in Cambridge by J.J. Thomson (Nobel prize in 1906 for the discovery of the electron), father of another Nobel laureate, G.P. Thomson (in 1937 for the discovery of the wave properties of the electron).

Structure of the instrument

Timpano	Timpano: membrane	Timpano: various mallets

The instrument consists of a copper or brass bowl (the kettle), over which a membrane is stretched; once made of animal skin, it is now typically a sheet of mylar (a synthetic polymer more resistant to environmental changes in temperature and humidity) about 0.2 mm thick. The diameter of the instrument varies from 55÷60 cm for the high timpano (in A) to 75÷80 cm for the low timpano (in D).

The kettle is suspended above a heavy cast-iron base or, in less valuable instruments, fixed to it. Of course the best sound is obtained in the first case, which allows the kettle to take a more active part in producing the sound.

The membrane is stretched over the kettle by means of six or eight screws, which must be carefully balanced before playing so that the tension is uniform.

Inside the kettle, at the upper rim, there is a metal ring whose height can be adjusted over a range of a couple of centimetres by means of a pedal under the instrument. When the ring rises it pushes the membrane upwards, increasing its tension up to roughly doubling it.

Acoustic characteristics

The variables at play

The timpano can be struck with mallets of different softness, from the hardest (with a small disc-shaped wooden head) to the softest (with a large, roughly spherical head covered with felt or other soft material). Besides the type of mallet, the player has total control over the point of impact, its intensity, the time of contact with the membrane, and the damping time of the membrane. All these elements (not always independent of each other) affect the sound produced by the instrument. Of course we do not mention here the great rhythmic variety, which is instead the dominant factor from a more strictly compositional point of view.

Many membranes in one

three normal modes of the circular membrane
Mode (0,1). It has no nodal diameters, and only one nodal circle (at the edge of the membrane). Frequency ${\displaystyle f_0}$.	Mode (1,1). Besides the nodal circle at the edge, it has one nodal diameter. The point of impact of the mallet obviously cannot fall on the diameter. Frequency ${\displaystyle 1.59\cdot f_0}$.	Mode (2,1). Besides the nodal circle at the edge, it has two perpendicular nodal diameters. Frequency ${\displaystyle 2.14\cdot f_0}$.

The sound of membranes is always a complex sound, made up of many partials. Each partial corresponds to a normal mode of vibration of the membrane, and each mode, independent of the others, is determined by the number and position of the nodal diameters and circles, which are the loci of the points that stay at rest during the oscillation of that mode. Each mode can therefore be identified by a pair of integers, as illustrated in the following table.

For an introduction to normal modes see the page normal modes. An analysis of the modes of vibration in the ideal case is on the page normal modes of a circular membrane.

The fundamental frequency of the membrane is proportional to the square root of the tension of the membrane, inversely proportional to the square root of its density and inversely proportional to its diameter (this is why the smallest timpano is the highest)

${\displaystyle f\propto \sqrt{\frac{T}{\rho }}}$.

This is a property common to all vibrating mechanical systems (see the page on the speed of mechanical waves). So, once tension and density are fixed - that is, given a certain type of membrane, and once tuning is complete - the frequency of the fundamental produced by the instrument (the lowest-frequency harmonic) is fixed.

The sound in theory

The study of the ideal membrane shows that the natural frequencies of the normal modes are not in simple ratios to each other, like those of vibrating strings, but in ratios unsuitable for forming a harmonic progression. For the circular membrane the following series is obtained

Now, from the study of the perception of pitch we know that, while a harmonic series is perceived as a sound of definite pitch, our perceptual system is confused by non-harmonic series, and is unable to attribute a definite pitch to them, perceiving instead sometimes a "noise", sometimes the individual harmonics, if some are particularly prominent.

The real sound

waveform	sonogram	Audio
		Timpano_Suono_base.mp3 A single stroke, mezzo forte, repeated

As soon as one listens to the sound example, it is immediately clear that the scenario predicted by theory does not occur at all in the case of the timpano.

• Almost all listeners easily distinguish a note (together also with the percussive noise and, possibly, individual harmonics).
• Some hear a D₂ at a frequency of 73 Hz; others swear that it is instead a D₃ at a frequency of 146 Hz.

In fact, in this case the resonance on A₃ at about 216 Hz is also clearly perceptible, above all because of its conspicuous beating. This artefact indicates that the timpano used is not of the highest quality. The beating is also evident in the waveform: the first part of the sound (up to about t = 4000 ms in the figure, that is, for a little less than a second after the strike) decays with a purely exponential law, while in the following tail one can distinguish the envelope of a slow beat, with an interval between minima of about 62 hundredths of a second. The beating occurs because, as can be seen in the picture alongside, generated at very high resolution, what looks like a single peak at about 216 Hz is in fact made up of two very close peaks, at about 215.9 Hz and at about 217.5 Hz. The beat frequency (1.6 Hz, that is, a minimum about every 0.63 s) is indeed the difference between these two. See the page on beats on this subject.

To let you form your own opinion, we have produced two pure sounds corresponding to the notes most frequently perceived by listeners: D₂ at a frequency of 73 Hz, and D₃ at a frequency of 146 Hz. Listen carefully to the sample, and try to decide which of these two notes seems to you to be the one actually played by the timpano.

sonogram		Audio
		Timpano_nota_percepita.mp3 Two pure sounds: one at 73 Hz, corresponding to the approximate pitch of the note perceived in the timpano sample above. Then the same note an octave higher, to make it easier to hear.

Note: the first sound has a very low frequency, and cannot be heard if played through computer loudspeakers. Sometimes it is not reproduced realistically even by headphones. In the two pictures, which show the same sonogram, we have drawn two grids for ease of reading. The first highlights the integer multiples of 73 Hz; the second those of 146 Hz. One can see that not all the partials fall on the grids, but certainly the first three have frequencies that are, to a very good approximation, multiples of 73.

Back to acoustics, the mystery deepens in both cases: for those who hear a sound at 73 Hz, it is enough to look carefully at the spectrogram above to notice that the most regular harmonics really seem to indicate a fundamental at 73 Hz, but there is no peak at this frequency.

On the other hand, if we take the note at 146 Hz as the fundamental, the higher harmonics are no longer in a harmonic ratio with this fundamental.

We must therefore explain the two mysteries we have discovered:

1. Why do we perceive a sound of definite pitch instead of a noise?
2. If one perceives D₂ at 73 Hz:

• "how is it possible to perceive a frequency that is not even produced by the timpano?"
• Or, if one perceives D₃ at 146 Hz: "why does the perceived fundamental not match the harmonic series?"

The "mysteries" solved

1. The problem of the harmonicity of the sound is explained if, instead of studying the modes of the ideal membrane, we introduce some more realistic parameters into our model. In particular we must consider the effects of air resistance, of the inharmonicity of the membrane, and of the coupling of the modes of the membrane with the modes of the kettle. In fact, the results given on the page on the normal modes of a circular membrane refer to an ideal membrane, that is, infinitely thin, in which the only force acting is the tension, and under the assumption that the tension is constant at any wavelength. Of all the corrections, however, the most important one seems to be due to the effect of the air.
   • The membrane of the timpano, in fact, with its diameter of about 70 cm, moves a considerable mass of air. The effect of all this air is to add inertia to the vibrating system, which behaves, for the same tension, as if it had a greater density than that of the membrane alone. Consequently the frequencies of the normal modes are slightly lowered.
   • The inharmonicity of the membrane consists in the fact that, for very short wavelengths, that is, for normal modes with high numbers, the shear elasticity of the membrane (absent in an ideal membrane) begins to become important, because the radius of curvature of the deformation becomes very small. The added stiffness raises the frequency of the highest harmonics. Paradoxically, this pushes them towards "more harmonic" ratios.
   • Finally, the kettle is not an inert element, but takes part in the vibration both through its copper walls and through the enclosed air, which behaves as an oscillator coupled to the membrane. We know (see the pages on coupled oscillators and on vibration absorption) that the coupling of two oscillating systems produces new resonance frequencies for the whole system, different from the frequencies of the individual component systems. Needless to say, it was the experience of generations of skilled makers that optimized the volume of the kettle so as to favour modes with almost integer ratios between the higher partials of the whole system.

   Let us move on to the second problem: the perception of the pitch of the timpano.

2. The problem in this case does not lie in the physics of the timpano: the timpano does not emit any partial at the frequency of 73 Hz, simply because this frequency does not correspond to any normal mode of the system. However, the complex signal-processing mechanism of the human perceptual system comes into play. In a few words (for more details see the sections on the perception of sound and, in particular, on the perception of pitch):
   • In the case of the missing fundamental, this is a well-known perceptual illusion: if the fundamental is removed by filtering from a sound made up of a series of harmonic partials, the human ear continues to attribute to the sound the same pitch as the missing fundamental. In fact, the ear "anchors" itself to the fundamental of the series even when several harmonics are removed, especially if the sound has already been heard before (see the page on acoustic effects and illusions to experience the phenomenon).
   • In the second case, which incidentally occurs more often with good-quality timpani, what simply happens is that the mode (1,1), to which the peak at 146 Hz corresponds, is the one with the slowest decay and the greatest strength (second only to the "spurious" harmonic, the one subject to beating). Apparently, if the higher partials, although in harmonic ratios, decay quickly enough, the ear does not have time to form an idea of the series (essentially, it cannot compute the greatest common divisor of the frequencies), and attributes the pitch to the sound that lasts longest.

Sound samples

Now that we have clarified the mechanism of sound production and its normal characteristics, let us see what varieties of sound can be obtained from the instrument

Harmonic sounds

As for all instruments, in timpani too it is possible, if not to isolate, at least to highlight some normal modes of vibration, obtaining a sound dominated by frequencies close to that of the excited normal mode. In timpani, this selection is obtained by lightly touching the membrane with the hand while striking it at another point. Where it is touched, a node is forcibly produced, which determines the formation of a certain normal mode. In the case of membranes, however, this isolation is not as simple as in string instruments, and so the effect does not produce a single pure frequency, but rather shifts the spectral region where the maximum sound energy is concentrated towards higher frequencies. This phenomenon is evident from the sonogram, and also to the ear.

waveform	sonogram	Audio
		PC_Armonici.mp3 PC_Armonici

Different mallets

Using different mallets makes it possible to modulate the "colour" and "dryness" of the sound of the timpano. The hardest mallets are simply wooden sticks ending in a wooden disc, possibly wrapped in a leather strip. Wrapping the head of the mallet in soft material has the effect of increasing the area of impact between mallet and membrane, and hence of decreasing its curvature. Consequently the softer mallet excites the shorter-wavelength normal modes to a lesser extent. Even in piano, the greater number of harmonics obtained with the harder mallets is quite evident.

waveform	sonogram	Audio
		PC_Bacchette.mp3 PC_Bacchette

waveform	sonogram	Audio
		PC_BacchettePiano.mp3 PC_BacchettePiano

Stroke at the centre

Here is another surprise of the instrument: timpani struck exactly at the centre of the membrane "do not sound". Essentially, only by standing close to the timpano does one hear a dull thud that is quickly damped. This fact highlights the great difference between vibration of the instrument and radiation of sound. Clearly the timpano vibrates even when it is struck at the centre. Indeed, the main normal mode excited in this way is the fundamental mode (0,1) of the membrane (see above). However, this mode is also the one most affected by damping due to the air. Indeed, the membrane moves either all upwards or all downwards, trying to displace a large mass of air, and being slowed down by the air inside the kettle. In other words, the mode (0,1) gives up all its energy to the air too soon, and does not have time to complete enough oscillations to form a real sound.

In timpani the main radiating mode is the mode (1,1). In this mode, in fact, the two halves of the membrane oscillate in antiphase: when one goes up the other goes down, and it is the elasticity of the air itself that favours the oscillation, with relatively little damping. From the point of view of sound radiation, timpani therefore behave as a dipole in the mode (1,1). For the radiation diagrams of the dipole and of other configurations see the page antennas and interference.

From the waveform one sees that the intensity in the mode (0,1) halves (going from 90 to 87 dB) in a few periods, that is, in about 5 hundredths of a second, while in the basic stroke (the membrane is struck at a distance from the edge between 1/3 and 1/4 of the radius) this time is at least ten times longer.

waveform	sonogram	Audio
		PC_ColpoAlCentro.mp3 PC_ColpoAlCentro

Tuning

In this example the timpanist progressively adjusts the tension of the membrane using the pedal so as to form a scale. It is a way to better appreciate the definite pitch of the instrument and its possibilities of accurate intonation.

waveform	sonogram	Audio
		PC_Scala.mp3 PC_Scala

Dynamic range

Timpani can radiate large amounts of sound energy, and are therefore suited to communication over long distances. Indeed, examples of drums dating back to 4000 BC are known. In this case we have artificially reduced the dynamic range, so as not to make listening unpleasant. Note that the pitch of the two strokes, forte and piano, changes. This change is more evident in the crescendo sample (see among the effects), and is illustrated there. The two sounds nevertheless give a different perceptual impression. The stroke in forte excites many more modes of vibration, and alters the ratios between the intensities of the harmonics with respect to the second.

waveform	sonogram	Audio
		PC_PianoForte.mp3 PC_PianoForte

Effects

As one soon realizes, also by looking at the many waveforms on this page, timpani, like all other percussion instruments, after a brief initial impact that excites the instrument, vibrate freely once the mallet has lost contact with the membrane. This explains the exponential decay of the sound, but is also a limitation for the musician who, on some occasions, wants the sound to last longer. The only solution is to strike the membrane several times in rapid succession. This technique, called a roll, actually produces a whole series of new timbres and "atmospheres". Here is an example of three rolls played at the same rate of about 11 strokes per second, but with different dynamics.

waveform	sonogram	Audio
		PC_PianoMedioForte.mp3 PC_PianoMedioForte

The roll is entirely controlled in intensity and rate by the neuromotor skill of the player, and its different varieties can be blended continuously to obtain crescendo and decrescendo effects.

In this sample the change of pitch between piano (when the roll begins), forte (at the climax of the roll), and also in the final damped part after the forte, is particularly evident. Spectral analysis shows that this is not a psychoacoustic effect, but an objective physical difference between the pitches of the partials produced in piano and in forte. From a structural point of view, the different pitch is a demonstration of the behaviour of the real membrane, as opposed to the ideal case. In forte, that is, when the amplitude of vibration is large, the tension of the membrane is no longer independent of the amplitude. Larger amplitudes correspond to larger tensions, and hence to a higher natural frequency. High-resolution examination of the fundamental of the spectrum shows a shift of about 15 Hz. Once the maximum intensity is reached, the timpanist stops the roll and damps the membrane. The amplitude decreases very rapidly (from maximum to zero in about 2 tenths of a second), and the frequency therefore shifts rapidly downwards.

waveform	sonogram	Audio
		PC_Crescendo.mp3 PC_Crescendo

waveform	sonogram	Audio
		PC_Decrescendo.mp3 PC_Decrescendo

There is a technique called the double stroke that makes extremely fast rolls possible. The timpanist strikes the membrane twice: the first time using the wrist, and the second using the fingers and exploiting the natural rebound of the mallet. In this example the difference between the two effects can be appreciated. In bands, which cannot include timpani because of their size, the roll effect is entrusted to a special drum (called, precisely, the snare drum). In that case, however, the instrument has an additional device that makes the effect easier (see the page on the snare drum on this subject). It is not easy to appreciate the effect by looking at the waveform, because the strokes are so close together that they are hard to count. So, alongside the original recording, we have extracted two short portions of the sample and played them back at half speed. It can be estimated that the ratio between the stroke rates of the two rolls is between 3:2 and 8:5.

waveform	sonogram	Audio
Audio at normal speed	Audio at half speed
PC_RulloTimpano.mp3 Roll and double roll at normal speed	PC_RulloTimpano_mezza.mp3 Roll and double roll played back at half speed

Glissandos

The glissando is obtained by striking the membrane and changing the tension with the pedal during its free oscillation. This effect is used particularly by B. Bartók. One can actually speak of two distinct effects, since the upward and the downward glissando have completely different damping times, as clearly seen in the pictures here.

waveform	sonogram	Audio
		PC_GlissandoUp.mp3 PC_GlissandoUp

waveform	sonogram	Audio
		PC_GlissandoDown.mp3 PC_GlissandoDown

Check your understanding

Links and bibliography

• We invite you to read the essential page Sound and resonance, where you will find the general principles of how a musical instrument works. Reading it will help you better appreciate the peculiarities of the instrument described on this page.
• Related questions and answers:
  • Why do timpani produce a sound of definite pitch?