The voice

Why do our voices sound so silly after we have inhaled helium from a balloon?

At a given gas pressure, only two factors affect the speed of sound in the gas: its adiabatic index ${\displaystyle \gamma }$, a constant summarising the thermodynamic properties of the gas, and its density. In particular, all else being equal, the speed of sound is inversely proportional to the square root of the density of the gas.

${\displaystyle v\propto \frac{1}{\sqrt{\rho }}}$

Helium has a density of about one seventh that of air and a somewhat higher adiabatic index (1.67 against 1.4). This means that the speed of sound in helium is about three times higher than in air (about 1000 m/s against 343 m/s).

When we produce a sound, the base frequency is determined by the vibration frequency of the vocal cords, and this does not change in air or in helium. However, the timbre of the voice is filtered by the cavities of the vocal apparatus (larynx, pharynx, nasal cavities and sinuses). These cavities have resonance frequencies determined by their size and by the speed of sound in the gas inhaled at the moment the voice is produced.

Consequently, inhaling helium raises the resonance frequencies of the cavities, and so does not make the voice higher in pitch, but makes its timbre thinner.

Of course, the opposite effect can be observed when a gas heavier than air is inhaled. You will find audio examples, videos and the discussion in the section on voice and the speed of sound.

Is it possible to sing two notes simultaneously?

Yes, double sounds can be obtained by means of various vocal techniques, some of which are not yet fully understood from the acoustic point of view. See the section devoted to this subject on the page on the human voice

Why is it that when a soprano sings in a high register, it is very difficult to distinguish the vowel sounds being sung?

The difference between vowels is ultimately a difference of timbre. We distinguish the various vowels from one another because their acoustic spectrum is different, that is, because they contain different amounts of energy at different frequencies. The way the energy produced by the vibration of the vocal cords is shared among the various frequencies is determined by the resonances of the cavities of the vocal apparatus, which are modified during phonation. These resonances are determined essentially by the volume of the various cavities, and lie in frequency ranges below 6000 Hz in adults. When a soprano sings high notes, however, the higher harmonics are more widely spaced, because the frequency of the fundamental is particularly high. Consequently, fewer harmonics fall within the range of the resonances of the vocal tract. Moreover, singing technique leads the singer to concentrate as much energy as possible on the fundamental, because this makes the sound easier to hear even at a distance. All this means that the information about the vowel being sung often becomes insufficient to distinguish it "by ear". Fortunately we do not listen only with our ears, but with the whole brain, which is perfectly able to reconstruct the missing part of the information if the context allows it (for example if the singer is singing in a language we know).

To listen to sound examples of the phenomenon and see their spectral analysis, see the section on vowels in singing.

Why are we able to distinguish vowel sounds if we pronounce them all with the same intonation?

The perceptual feature of sound that allows us to distinguish one vowel from another is ultimately its timbre, but we can also recognise the same vowel when it is pronounced by different voices, with different timbres. This is why phonologists and phonolinguists use the concept of formants. The formants of a sound do not coincide with its spectrum, but are a processing of it designed precisely to identify the phoneme pronounced, independently of the timbre and intonation of the voice that pronounced it.

To see audio examples and diagrams illustrating this point, go to the section devoted to vowels.

Wind instruments

Is it true that sound also reflects at the open end of a tube and not just at the closed end?

Yes, it is true. In fact, this reflection mechanism is the basis for the formation of notes of definite pitch in wind instruments. To understand how this happens, it is useful to compare it with the oscillations of a string, which are reflected along the string both when the end is fixed and when it is free. They are reflected because at the end of the string (whether fixed or free) there is an abrupt change in the properties of the medium, which forces the string at that point to an unchanging condition called a boundary condition.

When the string is ideal and fixed, the condition imposed is a node at the end, while if it is free the condition is an antinode. At first sight it seems that the medium in a sound tube does not change at the ends: there is air outside and air inside the tube. However, we must remember that a sound wave is not the transport of air, but the transport of a variation of pressure in the air. As far as pressure is concerned, being inside or outside a tube makes a big difference.

For example, at the open end of an ideal tube there can be no change in pressure, since at the point where the tube opens to the outside the pressure must always equal the external pressure. So the open end always behaves as a pressure node. Conversely, at the closed end the pressure variation will be at a maximum, because the air presses against an ideally unyielding wall. The closed end therefore behaves as a pressure antinode.

To visualise the phenomenon better, see the many animations on the page reflection in strings.

Why do the flute, clarinet and oboe play in completely different registers even though they are almost the exact same length (about 60 cm)?

The answer lies in the relations between the properties of conical and cylindrical pipes. For a detailed discussion see the page on the conical pipe. In short, remember that

• a conical pipe, whether half-open or open at both ends, has the same resonance frequencies as a cylindrical pipe open at both ends of the same effective length
• a half-open cylindrical pipe has a fundamental of half the frequency of a cylindrical pipe of the same length open at both ends and, moreover, lacks the even-order higher harmonics.

So the oboe has a conical body, and therefore produces a fundamental whose wavelength is twice the length of the instrument, the same as that of the flute, which is a cylinder open at both ends. The clarinet, instead, has a cylindrical bore, but the beating reed forces a pressure antinode at the mouthpiece, which makes it acoustically equivalent in every respect to a half-closed pipe. The clarinet therefore has a fundamental whose wavelength is about four times the length of the instrument, while the oboe and flute have a fundamental whose wavelength is twice the length of the instrument. In terms of frequency, therefore, the fundamental of the clarinet is about an octave lower than the fundamental of the flute and oboe.

It is not actually an exact octave, because the clarinet is not purely cylindrical, as it ends with a small conical bell.

This reasoning explains the difference in range between the instruments, while a description of the whole emitted spectrum is needed to account for the differences in timbre. One particularly evident feature of the clarinet's sound, however, can already be deduced from the observations above: the absence (or very strong attenuation) of the even-order harmonics. This means that the first higher harmonic of the fundamental is not the octave above, but the twelfth (an octave plus a perfect fifth).

How important is the material with which a wind instrument is made?

As far as sound production is concerned, the material of the body of a wind instrument is generally not as important as is commonly thought.

In wind instruments, in fact, the element set into vibration is directly the air, not the material of the bore. The vibration modes of a wooden or metal tube, moreover, have natural frequencies much higher than those of the air column they contain (see the discussion of the role of elasticity and density of the medium on the page on the speed of sound). The material plays a relatively important role only in the kind of friction it exerts on the air, and in any heat exchanges taking place in the first thin layer of air in contact with the walls.

The reason why, for example, a flute and a clarinet have very different ranges and timbres does not depend so much on the fact that the first has a metal body and the second a wooden one, but rather on the conditions at the ends (the first cylindrical and open at both ends, the second cylindrical but closed by the reed at one end), and on the different way the oscillations are induced in the bore (by a direct air jet in the first, by a reed in the second). See also the previous question, and the page on the conical pipe.

In string instruments, on the contrary, the vibrating element is a string, which in general does not itself couple well acoustically with the surrounding air, as shown by the fact that a plucked string produces a very weak sound unless it is supported by a suitable wooden resonating cavity. In this type of instrument the surrounding air is actually set into vibration by the body of the instrument, which is in turn excited by the string. So, in this class of instruments, the response characteristics of the body (and therefore also the material it is made of) are essential in determining the qualities of the resulting sound.

Are the vibrations one feels on the body of a trumpet wasted energy?

Partly yes, but they are not the main loss. The sound of the trumpet is produced by the vibration of the air column, which is well coupled to the outside air (otherwise no sound would come out), but very poorly to the metal of the tube, because there is a huge difference in acoustic impedance between air and metal.

The vibrations felt with the hands come mainly from two sources:

• from the lips, which vibrate directly against the mouthpiece: part of this energy is inevitably lost, but the mouthpiece serves precisely to match the lips to the air column. Different mouthpieces give different results;
• from parts that vibrate because of a defect (loose screws, imperfect solder joints or contacts): these should be fixed through maintenance.

Wall vibrations, especially in the bell, nevertheless have a small but measurable effect on the sound.

Thickening the whole tube to reduce them would make the instrument much heavier and more tiring, for a modest gain. The most important energy losses occur instead inside, in the thin layer of air in contact with the walls, where viscous friction and heat exchange damp the wave. They depend mainly on the diameter and length of the tube. The inner finish matters only if the surface is really rough or dirty.

Is it possible to play two notes simultaneously with one wind instrument?

Every instrument (wind or not) produces many sounds simultaneously even when we hear only one. Indeed, every vibrating system, when excited, can oscillate in a large number of independent fundamental modes called normal modes. The illusion of hearing a single sound is ensured by our perceptual apparatus.

That said, it is true that there are particular conditions in which a player manages to excite two modes of the instrument particularly strongly. If the frequencies of these modes are not integer multiples, the sound produced is not easily "resolved" by our perceptual system, and so two distinct sounds appear to us.

To hear the examples and see their spectral resolution, see the sections

• double sounds on the clarinet,
• double sounds on the oboe

If a free reed, like that of an accordion, is fitted in place of the head joint of a recorder, why does the note not change when the holes are opened and closed?

In the recorder the air column inside the body is excited directly by the air jet striking the labium. The instrument always plays one of the natural frequencies of the column, which depend on its effective length, that is on the open holes. The note obtained from the head joint alone plays no special role: it is just the natural frequency of the shorter air column contained in the head joint.

In reed instruments, instead, the note arises from the coupling between the vibration of the reed and that of the air column. Which one "wins" depends on how stiff the reed is compared with the air column.

• The reeds of the clarinet, saxophone and oboe are beating reeds. They are light and damped by the lips. At every cycle they close completely, interrupting the airflow and producing strong pressure variations. Their natural frequency is higher than the notes played, so they easily follow the pressure oscillations of the air column: it is the air column that decides the note (see the next question).
• The free reeds of the accordion, harmonica and harmonium are stiff, lightly damped metal tongues, precisely tuned to a fixed frequency, which swing through a slot without closing it. A cavity or a tube connected to the reed can change its frequency only slightly (harmonica players use the cavities of the mouth to "bend" some notes by a semitone or more), but cannot impose a different note on it. The reed goes on sounding at its own frequency and the tube at most reinforces it or changes its timbre.

In the organ, in fact, free-reed pipes each have their own resonator, tuned to the note of its reed: one pipe for each note. There are free-reed instruments with holes, such as the Chinese bawu, but the reed is specially built and the range of the instrument is limited.

To obtain a reed instrument with the body of a recorder a beating reed must be used, for example a clarinet mouthpiece. In this case the pipe behaves as a pipe closed at one end, so the instrument sounds lower and needs different fingerings (see the question on the registers of flute, clarinet and oboe).

When the oboe gives the A to the orchestra, is it the reed that vibrates at 440 Hz?

Yes, but it is not the reed that chooses the frequency. When the oboe plays an A at 440 Hz the reed also vibrates at 440 Hz, but this frequency is imposed by the air column inside the body of the instrument, that is by the fingering. The reed alone produces a sound of more or less fixed pitch, which for the oboe is usually around a C and which the player can raise by tightening the lips (it can be heard in the section the sound of the reed).

The mechanism is a continuous interaction between reed and air column. The reed works as a valve that opens and closes the passage of the breath. Each pressure pulse travels along the pipe, is reflected at the first open hole (or at the bell) and returns to the reed, pushing it to open or close. In the first cycles, that is within a few milliseconds, the oscillation of the reed "locks" onto a natural frequency of the air column. During this short attack transient other components are also present, which contribute to the character of the attack but do not form a second stable sound.

It should be added that the sound is not the air coming out of the instrument: it is the vibration of the air inside the instrument, which is transmitted to the outside air. Sound waves are like the waves of the sea, not like the current of a river.

The oboe gives the A to the orchestra because its intonation is stable and its timbre, rich in harmonics, is easily heard above the other instruments, not because of a property of its reed. In some wind ensembles it gives a B flat instead.

Can a saxophonist change the timbre with the mouth and throat?

Yes, to some extent. One sometimes hears that a cavity located "before" the source of the sound cannot resonate, because the sound wave would not pass through it. For a reed instrument this is not so. The reed is a valve that moves under the effect of the difference in pressure between the mouth and the mouthpiece: pressure oscillations are produced on both sides. The mouth cavity and the throat (the vocal tract) therefore also receive a sound wave. The vocal tract is thus acoustically "in series" with the bore of the instrument, on the opposite side of the reed.

Usually the bore dominates, because its resonances are much stronger than those of the vocal tract. When, however, the player shapes the mouth and throat so that a resonance of the vocal tract is strong and close to the note being played, this resonance affects the motion of the reed and hence also the intonation and the timbre. Measurements have shown that expert saxophonists tune their vocal tract to the desired note above all in the very high (altissimo) register and in glissandos, where without this technique the notes would not sound. In the normal register the effect on timbre is subtler, but real.

The sensations of vibration in the chest of singers and players are a different matter: they are vibrations transmitted through bones and tissues, which do not radiate sound significantly and should not be confused with an acoustic resonance (see the human voice).

String instruments

Why do all string instruments have a hollow body?

In short: strings on their own can vibrate at precise frequencies, suitable for producing musical sound. However, their vibration is not easily transmitted to the surrounding air, so much so that a string on its own cannot be heard even a short distance from the source. The main function of the hollow body of string instruments is therefore to amplify the vibration of the string, turning it into radiated sound energy (sound) by exploiting the phenomenon of resonance. Put in another, more technical way, the sound box acts as an impedance matching device between the strings and the surrounding air.

• You will find an extended discussion of this point on the page sound and resonance
• See also the question How does the body of a string instrument make the sound produced by its strings louder?

How important is the material with which a bowed string instrument is made?

In string instruments, the vibrating and resonating element that selects the fundamental frequency emitted by the instrument (the note played) is a thin string of gut, steel or synthetic material. However, the sound necessarily reaches our ears through the air.

Yet it is easy to check that a string on its own, played without a hollow support, emits a very weak sound. The reason is that some physical characteristics of the string, such as its density and elasticity, are very different from those of the surrounding air, and this prevents the vibrations of the string from passing easily into the air (the page refraction in strings contains animations illustrating this point). In technical jargon we say that the string has an acoustic impedance very different from that of the air. If we want the oscillation of the string to be transmitted effectively to the air, we must match the two impedances by means of suitable connecting elements.

In bowed string instruments these connecting elements make up the body of the instrument. For example, in the violin family the strings rest on a bridge, which rests on a plate, which in turn forms the top of a hollow box. This whole body essentially serves to turn the oscillation of the strings into oscillation of the air as efficiently as possible and, therefore, to radiate the sound into the surrounding space.

Impedance matching in bowed strings is a rather complex process, and the elastic properties of the wood the box is made of clearly play a fundamental role. The quality of the material is therefore of primary importance for the good sound quality of these instruments.

Why is it that when we bow, pluck or strike the same string, we get different sounds even if it is the same note?

The note perceived from the vibration of a string corresponds in all three cases to the free vibration frequency of the string (called the fundamental frequency). However, the precise shape of the string during vibration is quite different in the three cases, because the initial conditions of the motion are different. Consequently, the sound produced in the three cases is not a pure sound, but the combination (the superposition) of many pure harmonic partials. What distinguishes the three sounds is the different intensity of each of these partials.

• For a more analytical description, with an animation of the motion, see the page on string instruments.
• For more information on the physics of strings see normal modes of a string, natural frequencies of a string

Why does plucking a string at different points give different timbres? Is it true that some harmonics disappear?

Yes. When a string of length L is plucked at a distance a from one end, displacing it by h, the string starts from a triangular shape. Decomposing this shape into the normal modes one finds that the amplitude of the n-th harmonic is

${\displaystyle A_n=\frac{2h{L}^2}{{\pi }^2{n}^{2}a(L-a)}\sin \left(\frac{n\pi a}{L}\right)}$

• The factor 1/n² makes the amplitude of the harmonics decrease as n increases.
• The sine vanishes when the plucking point coincides with a node of the n-th harmonic, that is when a is a multiple of L/n. Plucking at the middle of the string removes all the even harmonics; plucking at one fifth of the length removes the fifth, tenth, fifteenth and so on.

The physical reason is simple: a mode that has a node at the plucking point does not move at that point, so displacing the string exactly there gives it no energy.

As a result, plucking near the centre gives a soft, "round" sound with few harmonics, while plucking near the bridge gives many harmonics of similar amplitude and a bright, "nasal" sound. In reality the finger or the plectrum does not touch a single point, so the harmonics with a node at the plucking point do not disappear completely, but are strongly attenuated. The same principle applies to bowed and struck strings (see the table on the page on string instruments).

The bow excites the string with many frequencies: why do we hear only the harmonics of the string?

Because on a string of finite length any disturbance, once left free, moves back and forth between the two ends and returns to the same position after the time 2L/v (L is the length of the string, v the speed of the waves). The motion is therefore periodic with the period of the fundamental. A periodic motion contains only the fundamental and its integer multiples, that is the harmonics (see Fourier's theorem and the point of view of travelling waves). On an ideal string there are no waves "of other frequencies" that can keep travelling along it: every shape of the string is a sum of its normal modes.

Components at other frequencies appear only while an external force acts that does not follow this rhythm, that is during the attack. In a bowed string the bow "grips" and "releases" the string irregularly for a short time, until the motion settles into a regular oscillation in which each release is synchronised with the return of the wave (the so-called Helmholtz motion). From then on the sound contains practically only harmonics. The irregular components of the attack are part of the character of the sound (see the attack of the sound in the violin).

In real strings the partials are not exactly harmonic because the string has some stiffness. In a bowed string, however, the Helmholtz motion still imposes an exactly periodic, and hence harmonic, oscillation.

If we use harmonics to tune an instrument, will we always get good tuning?

• Not always. It depends on which harmonics are used, and on the range of the instrument.
• The only harmonic that guarantees perfect tuning is the octave, because the octave is consonant in all scales. However, if harmonics are used to tune different strings (as is often done on the guitar), the instrument will be tuned in perfect fourths and fifths, as in the Pythagorean scale. This scale provides optimal intonation only for a particular key and, consequently, the instrument will be in tune for Western music only in a limited number of chords, and may sound out of tune when modulating.
• While on the guitar this mistuning may not be particularly relevant for certain kinds of music, it is almost always intolerable on the piano. Because of the wider range of its notes, the mistuning can become progressively larger the further one moves from the central octave of the instrument. Natural harmonics can be used as a first approximation, but corrections that bring the scale back to some form of temperament are absolutely necessary in this case.
• This theoretical and practical problem has troubled the whole history of music. In the individual sections (e.g. natural scale, Pythagorean scale, etc.) you will find descriptions and information on the individual scales, while in the section comparisons between the different scales you will find listening examples to learn to hear the differences between the various tunings.

We can create natural and artificial harmonic sounds on a violin and a guitar. How are they different?

A sound that is harmonic with respect to a reference sound called the fundamental is a sound whose frequency is an integer multiple of the fundamental. On string instruments, the fundamental of a string corresponds to the sound obtained from its free oscillation, while harmonic sounds are obtained for string lengths equal to 1/2, 1/3, 1/4, 1/5, etc. of its length.

The difference between natural and artificial harmonics is the following:

• natural harmonics are obtained only with respect to the fundamentals of the open strings, by lightly touching the string at the positions indicated above. In this way a node is always imposed at the point of contact (the string cannot move there), but the excitation of the part of the string on either side of the node is not prevented. The waves travelling along the string can in fact propagate, giving a standing wave with a frequency of 2, 3, 4, 5, etc. times the fundamental.
• artificial harmonics are obtained with respect to any note the instrument can produce. On the violin they are obtained by selecting the desired note with the index finger of the left hand, and lightly touching the same string with the little finger at a fixed distance equal to 1/4 of the length of the vibrating part of the string. The first finger selects the fundamental, while the second imposes a node by light touch, corresponding to a harmonic two octaves above the fundamental. Moving the hand along the fingerboard, harmonics two octaves above all the other notes can be obtained.

For an illustration of the physics of the phenomenon see

• standing waves,
• reflection in strings,
• interference between reflected waves

To listen to sound examples see

• The dedicated section on the page on the violin

Why do we speak of a trumpet (or a tenor sax) in B flat, but not, for example, of a "violin in B flat"?

• The violin, like almost all other chordophones, has a set of several strings, each tuned to a different frequency, that is, corresponding to a different note. Starting from this set of fixed notes, all the other notes can be obtained by using the fingers to shorten the vibrating portion of the strings. Since a violin is not restricted to playing only a few fixed notes, there is no need to specify a note as the "origin of the reference system".
• The case of the trumpet (and of all the other brass instruments) is different: originally the trumpet had no valves. The fundamental note produced therefore corresponded to the first usable harmonic given the length of the tube, and the instrument could not produce sounds other than this one and its higher harmonics. So, for the trumpet there is a "privileged reference system", fixed by the length of its tube. Changing the size of the instrument, for example by lengthening the tube, meant obtaining an instrument completely different from the first, since it could produce a different set of notes. This also influenced musical notation, giving rise to transposing notation, and both the nomenclature and the notation have survived to the present day, even after the trumpet, with the invention of valves (in the early 19th century), acquired a piston mechanism that allows it to produce all the notes of the modern chromatic scale.
• For further information and examples see the page on the trumpet.

What is "Tartini's third sound" and how is it obtained?

Tartini's third sound is a lower sound heard when two sufficiently intense sounds of suitable frequencies are played together. It does not correspond to any of the notes played. The conditions are easy to obtain for a violinist (as Tartini was): playing loudly and together a C₅ on the second string and an E₅ on the first string, tuned to a just major third, one hears a C₃. Its frequency is the difference between the fundamentals of the two notes (about 654 − 523 ≈ 131 Hz).

Where does it arise? In air, at normal levels, sounds add linearly (see superposition principle): when they overlap they create no new frequencies. A difference frequency appears only where the system the sound passes through behaves non-linearly (see non-linear effects and intermodulation distortion). This can happen in two places.

• In the ear. This is the most common case. The cochlea behaves non-linearly and itself generates combination frequencies. They are not an illusion: they are real vibrations inside the ear, which can even be measured with a small microphone inserted in the ear canal. The component 2f₁ − f₂ appears already at moderate volumes, while the simple difference f₂ − f₁ requires higher volumes: this is why the third sound is heard well only when playing loudly.
• Before the ear. If the instrument itself, or something in contact with it, behaves non-linearly (for example a part that vibrates by striking against the body of the instrument, or the coupling of the two strings through the bow and the bridge), the third sound can already be present in the radiated sound. The same can happen in a recording or amplification system pushed to its limits.

How can the two cases be told apart? Just record the sound with a good microphone, at a moderate volume, and look at its spectrum. If a line appears at the difference frequency, the third sound is present in the sound itself. An even clearer sign is a line at the sum frequency f₁ + f₂, which would have no other explanation. If instead the spectrum contains only the harmonics of the two notes, the third sound arises in the listener's ear.

With two harmonic sounds the difference frequency often coincides with a multiple of their common fundamental: in the example the C₃ is exactly the common fundamental of C₅ and E₅. In these cases the mechanism of the missing fundamental can also help make a low note heard. For this reason hearing a lower note is not enough to establish which mechanism is at work.

For audio examples and diagrams see the section devoted to the third sound.

Why are some guitar strings made from nylon while others are covered with metal?

The guitar (like bowed string instruments) has a set of strings all of the same length, but each vibrating at a different frequency (tuning).

For a given string length, two elements determine its vibration frequency: its mass and its tension. For details see the page on the natural frequencies of a string. It is clear, therefore, that to obtain different notes the various strings of a guitar must differ in these parameters. Winding steel spirals around a nylon core serves precisely to increase the mass of the strings that must produce the lowest notes, so that their tension does not have to be increased excessively, which would subject the wooden structure to overly asymmetric stresses.

Percussion instruments

Why don't all percussion instruments produce a precise note the way string instruments do?

• No musical instrument, when excited, emits sound at a single frequency: each emits a whole "spectrum". This happens even when we perceive "a single precise note", and it characterises the timbre of each instrument. The perceived note is almost always the fundamental, that is, the note corresponding to the sound emitted at the lowest frequency. All the other sounds are called harmonics.
• Now, there is a fundamental difference between the spectrum of harmonic sounds emitted by a string or a sound tube, as in bowed strings and wind instruments, and those emitted by a plate or a membrane, as in many percussion instruments. The difference is that the harmonics emitted by a string or a tube all have frequencies that are integer multiples of the fundamental, while those emitted by a plate or a membrane do not.
• The physical explanation of this phenomenon is at the same time very important and fairly simple, and has to do only with the fact that a string or a sound tube are essentially one-dimensional systems, while a plate or a membrane are essentially two-dimensional.

• On the pages normal modes of a string, normal modes of a circular membrane and normal modes of a rectangular membrane you can find an illustration of the phenomenon, with animations and sounds to see and hear the various harmonics.
• Sound examples of instruments of indefinite pitch on the pages on the snare drum and other percussion instruments

Why are some percussion instruments, such as the timpani, tuned while others, such as the cymbals, are not?

• Clearly cymbals cannot be tuned, since all the physical parameters characterising them (size, shape, density of the material) are given and fixed. The only variable under the player's control is the way they are struck (point of contact, force and duration of the stroke, material of the beater, etc.). These parameters largely determine the acoustic result of the instrument, but even this is not enough to allow real tuning, simply because cymbals emit sound with a broad spectrum, and the partial components of this spectrum are not in harmonic ratios. For a mathematical treatment see natural frequencies of circular plates.

The timbral peculiarities of plates (cymbals, antique cymbals, gongs, tam-tams) are not, however, to be considered a limitation. On the contrary, the effects that can be obtained can be surprising (for an example see the sound of a tam-tam with its sonogram).

• Timpani are percussion instruments of the so-called membranophone class: the sound is obtained by striking with a special mallet a membrane (once of animal skin, now typically of mylar, a synthetic polymer) stretched over a hollow support similar to a large pot with a rounded bottom. Around the instrument there are tensioning rods that can be adjusted by means of a pedal under the base. These tensioners, controlled by the pedal, can change the tension of the membrane, up to doubling it. Elementary vibration theory (see for example the page on natural frequencies) shows that increasing the tension by a factor of 2 is equivalent to increasing the frequency of the fundamental by a factor of ${\displaystyle \sqrt{2}}$, which, in musical terms, corresponds to an interval of about an augmented fourth.

From the mathematical treatment we know that the circular membrane of the timpani shares with other membranes, with plates, and with all two-dimensional systems in general, the property of having higher harmonics at frequencies in non-integer ratios to the fundamental. The real question, then, is the following:

Why do timpani produce a sound of definite pitch?

The answer lies in the effect of the air: timpani are large instruments, and the air above the membrane outside the drum and, to a lesser extent, below the membrane inside the hollow body of the instrument, has the effect of lowering the frequencies of the modes, especially the lowest ones, until the ratios between some of them become very close to integers. Finally, the player helps to excite the harmonic modes by striking the membrane at a standard position about a quarter of the radius in from the edge.

Further details and sound examples on the page on the timpani.

In a snare drum, why does the frequency change depending on whether the two heads vibrate in phase or in antiphase?

The two heads of the snare drum are connected by the air inside the shell, which behaves like a spring. The system resembles two coupled oscillators. In such systems the coupling always produces two different frequencies, one for each way of oscillating together. To see which is higher, just recall that for any mechanical oscillation

${\displaystyle f\propto \sqrt{\frac{K}{m}}}$

where K is the elasticity of the system and m its mass. Consider the simplest mode, in which each head moves entirely in the same direction, with maximum amplitude at the centre.

• Heads in phase: the two heads move in the same direction, so the volume of air between them stays almost constant. The air is not compressed and does not act as a spring, but it must be moved together with the heads: its inertia adds to that of the heads. m increases, so the frequency is lower than (or at most equal to) that of an isolated head.
• Heads in antiphase: the heads move towards and away from each other, so the air between them is alternately compressed and rarefied. In this case the air does not move as a whole, but adds its spring effect to that of the heads. K increases, so the frequency increases too.

In the animations on the snare drum page the point that oscillates most, at the centre of the head, is an antinode of the vibration mode, while the edge, which stays still, is a nodal circle (see normal modes of a circular membrane).

This model is only a first approximation. A complete description requires solving together the equations of the two heads and the wave equation in the air of the shell, which in practice is done numerically.

Why does cutting a tubular bell in half not give the same note an octave higher?

Because a tubular bell is neither a string nor an air column, but a bar vibrating in bending. For strings and air columns the frequency is inversely proportional to the length, whereas for a free bar the fundamental frequency is

${\displaystyle {\nu }_0=3.561\frac{R}{L^2}\sqrt{\frac{Y}{\rho }}}$

that is, it is inversely proportional to the square of the length L (R depends on the cross-section of the tube, Y and ρ on the material). Halving the length makes the frequency four times higher: two octaves up, not one. To go up an octave the length must instead be multiplied by ${\displaystyle 1/\sqrt{2}\approx 0.707}$. In general, for the same tube, to go from a frequency f₁ to a frequency f₂ the length must become

${\displaystyle L_2=L_1\sqrt{\frac{f_1}{f_2}}}$

For example, to go up a semitone the tube must be shortened by about 3%.

In practice the calculated lengths do not always match those found by ear, for several reasons.

• The natural frequencies of a bar are not harmonic (see the table on the page natural frequencies of the bar). The pitch perceived in a tubular bell does not correspond to the lowest mode, but is due mainly to the fourth, fifth and sixth modes, whose frequencies are approximately in the ratio 2:3:4. As in the case of the missing fundamental, the brain hears a note about an octave below the fourth mode. With inharmonic partials the perceived pitch can be ambiguous: it is easy to get the octave or the interval wrong.
• The plug closing one end of tubular bells adds mass and changes the frequencies of the various modes differently: it serves to make the perceived note clearer, but it alters the lengths calculated for an ideal tube. The hole for suspension and irregularities of the tube also have a small effect.

It is therefore best to calculate the length with the formula, cut the tube a little longer and tune it by shortening it gradually.

How does a handpan work, and how are its notes tuned?

The handpan (the first model, invented in 2000, was called the Hang) consists of two shaped steel shells joined along the rim. Several slightly concave or convex areas are hammered into the upper shell: each one is a note. In the centre is the lowest note (the "ding"). In the lower shell there is a circular hole.

• The notes. Each area vibrates as a plate, not as a membrane: the sheet resists bending thanks to its own stiffness, whereas a membrane vibrates only because it is stretched. In a flat plate of uniform thickness the partials would not be harmonic: the sound would have no definite pitch. By hammering, the maker changes the curvature and internal stresses of the sheet and gives each area a slightly elongated (elliptical) shape, until the first partials of the area are tuned in harmonic ratios, typically fundamental, octave and fifth above the octave (1:2:3). This is the reason for the very "in tune" sound and also the hardest part of making the instrument: there are no formulas giving the right shape directly. Tuning is done by ear and with spectrum analysis.
• The cavity. The air enclosed between the two shells, together with the lower hole, forms a Helmholtz resonator, which is tuned close to the lowest note and reinforces its sound. The size of the hole and the internal volume determine this frequency, which can be lowered slightly by bringing a hand close to the hole.
• The size matters mainly for the cavity and for the space available for the notes: areas too close together influence each other, and one note can make its neighbours vibrate in sympathy.

Similar instruments are the steelpans of the Caribbean, made from steel drums, whose notes are tuned in the same way.

How are the bars of a vibraphone or a marimba retuned?

The bars are free bars vibrating in bending. The practical rule is simple:

• removing material near the centre (where the bar bends most) makes the frequency go down, because it mainly reduces the stiffness;
• removing material near the ends (where the bar moves most but bends little), or shortening the bar, makes the frequency go up, because it mainly reduces the mass.

Thinning the bar at the centre, that is deepening the arch cut underneath, therefore lowers the note, it does not raise it. Flat bars are corrected instead by removing a little material at the ends.

The arch under the bar does not only lower the fundamental: its shape also tunes the higher partials, which in a uniform bar are not harmonic (see the question on tubular bells). In the vibraphone and the marimba the second partial is tuned two octaves above the fundamental (ratio 4:1), in the xylophone usually to a twelfth (3:1). When removing material one must therefore check that the partials also stay in place, proceeding in small steps and measuring with a tuner or a spectrum analyser.

The holes for the suspension cord are near the nodes of the fundamental (about 22% of the length from each end in a uniform bar). As long as they stay near the nodes their exact position matters little: if they were far from them, the cord would damp the vibration.

Can the frequency of a drum be calculated from its dimensions?

Only in part, because the frequency depends above all on a quantity that the dimensions do not tell: the tension of the head. For an ideal circular membrane the fundamental frequency is (see natural frequencies of the circular membrane)

${\displaystyle f_0=0.382\frac{1}{R}\sqrt{\frac{T}{\mu }}}$

where R is the radius, T the tension (in N/m) and μ the mass per unit area of the head. From this formula the proportions follow:

• with the same head and tension, a drum of twice the diameter sounds an octave lower;
• doubling the tension raises the frequency by a factor √2, that is by about six semitones;
• a heavier head (thicker, or double-ply) sounds lower.

The depth of the shell does not appear in the formula: it affects mainly the air inside the drum and the coupling between the two heads, and hence the timbre and duration of the sound more than the pitch. In real drums the air lowers the frequencies appreciably compared with the ideal membrane, the two heads are coupled (see the question on the snare drum) and the partials are not harmonic. For this reason the frequency of a drum is not calculated from its dimensions alone: it is measured, for example with a microphone and a spectrum analysis program. Then it is adjusted with the tension rods.