Saturday, June 19, 2010

The ‘Sound’ of Music

The ‘Sound’ of Music

Arthur L. Finkle

Any sound, whatever it might be, is caused by something vibrating. Without vibration there can be no sound. The vibrating body causes the air particles next to it to vibrate; those air particles, in turn, cause the particles next to them to vibrate. In this way a disturbance of the air moves out from the source of the sound and may eventually reach the ears of a listener. When we hear a sound, air vibrates against our eardrums causing them to vibrate also. These vibrations are detected and analyzed by our brains. Although it is usually air that acts as the transmitting medium, sound can be transmitted by other media, e.g. water, building structures.

Sound does not consist of air moving towards us in bulk; it travels through the air as a sound wave. A sound wave consists of a disturbance moving out from a source to surrounding places with the result that energy is transferred from one place to another.

As the wave passes, the disturbance of particles is in the direction of the wave travel.

The displacement of particles of the medium results in alternate regions of high particle density and low particle density. Regions of high particle density are called compressions. Regions of low particle density are called rarefactions.

Rarefactions and compressions both move in the direction of the wave travel. The particles of the medium do not move bodily in the direction of the wave movement; they vibrate about their normal positions. Each complete vibration of a particle is called a cycle ( i.e. from its starting position, to a maximum distance in one direction, back through the starting position, then to a maximum displacement in the opposite direction and back to the starting place).


The number of cycles completed in one second is called the frequency of the vibration. One of the most noticeable differences between two sounds is a difference in pitch (high to low). It is the frequency of a sound that mostly determines its pitch.

Frequency is measured in hertz, one hertz (Hz) being one cycle per second.

(One thousand hertz = 1 kilohertz = 1 kHz.) A high frequency vibration produces a high pitched note; a low frequency vibration gives a low pitched note.

The human hearing range (audible range) is about 16Hz to 16kHz. The frequencies of notes that can be played on a piano range from 27.5 Hz to just over 4kHz.

Any note played on a piano will sound different to a note of the same pitch produced by another type of instrument, e.g. a tuning fork.

The musical note produced by a tuning fork is called a pure tone because it consists of a tone of one frequency. A note played on a piano, or most other instruments, consists of several such tones all sounding together at different frequencies. These frequencies are related to the frequency (usually the lowest one) which gives the note its characteristic pitch.

In other words, when we hear sound from an instrument, we actually hear several sounds

The tone with the lowest frequency is called the fundamental. The other tones are called overtones If the overtones have frequencies that are whole number multiples (x2, x3...up to x14) of the fundamental frequency they are called harmonics. It is the difference in the harmonic content of notes that gives each musical instrument its characteristic sound or timbre ("tam-brah"). Therefore although the highest note of a piano has a fundamental frequency of just over 4kHz, equipment used to record music must be able to handle much higher frequencies to preserve the harmonics associated with each note. (1.9 Meg) To set the mood, listen to Anthony Heinrichs playing part of the cadenza from the trumpet concerto by Joe Wolfe.
• The player provides air at a pressure above that of the atmosphere (technically, from a few kPa to perhaps tens of kPa: from a few percent to a few tenths of an atmosphere). This is the source of power input to the instrument, but it is a source of continuous power. In a useful analogy with electricity, it is like DC electrical power. Sound is produced by an oscillating motion or air flow (like AC electricity).
• Once the air in the instrument is vibrating, some of the energy is radiated as sound out of the bell. A much greater amount of energy is lost as a sort of friction (viscous and thermal loss) with the wall. In a sustained note, this energy is replaced by energy put in by the player.
• The column of air in the instrument vibrates much more easily at some frequencies than at others (i.e. it resonates at certain frequencies). These resonances largely determine the playing frequency and thus the pitch, and the player in effect changes the length of the instrument---and thereby the frequencies of the resonances---by suitable combinations of inserting extra pieces of pipe via the valves, or by changing the length of the slide in the case of the trombone.
Let us now look at these components in turn and in detail.
First something about sound. If you put your finger gently on a loudspeaker you will feel it vibrate---if it is playing a low note loudly you can see it moving. (More about loudspeakers.) When it moves forwards, it compresses the air next to it, which raises its pressure. Some of this air flows outwards, compressing the next layer of air. The disturbance in the air spreads out as a travelling sound wave. Ultimately this sound wave causes a very tiny vibration in your eardrum---but that's another story.

At any point in the air near the source of sound, the molecules are moving backwards and forwards, and the air pressure varies up and down by very small amounts. The number of vibrations per second is called the frequency (f). It is measured in cycles per second or Hertz (Hz). The pitch of a note is almost entirely determined by the frequency: high frequency for high pitch and low for low. 440 vibrations per second (440 Hz) is heard as the note A in the treble clef, a vibration of 220 Hz is heard as the A one octave below, 110 Hz as the A one octave below that and so on. We can hear sounds from about 15 Hz to 20 kHz (1 kHz = 1000 Hz). A contrabassoon can play Bb0 at 29 Hz. When this note is played loudly, you may be able to hear the individual pulses of high pressure emitted as the reed opens and closes 29 times per second. Human ears are most sensitive to sounds between 1 and 4 kHz - about two to four octaves above middle C (See hearing curves). That is why piccolo players don't have to work as hard as tuba players in order to be heard. To convert from notes to frequencies and back again, see notes.
The Lips Control the Air Flow
Brass players can make musical sounds with just their lips, as you'll hear in the sound files below. This is one of the first things a brass player learns: you close your mouth, pull your lips back in a strange smile, and blow. The result may be anywhere between a low pitched 'raspberry' or a high pitched musical note, depending on the tension in your lips (how hard you pull them backwards in that smile, and other parameters).

'Buzzing' with the lips alone, and varying the tension.
How does that work? Lips are springy: if you pull your top lip forward with your fingers and let it go, it will spring back to its original position. Lips also have mass, and a mass and a spring together can oscillate, although lip vibrations are more complicated than the linear mass-and-spring oscillator that one finds in introductory physics books. What happens here is a cycle that converts the DC air pressure in your lungs into an oscillating air pressure and air current. The air pressure in your mouth (1) forces the lips open (2), which lets the air rush out. This lowers the pressure in the mouth (3), so the tension in the lips pulls them closed (4), and the cycle repeats. (This explanation is rather simplified: the Bernouilli effect due to the flow of air through the opening also acts to close the lips. Further, the internal motion of each lip is a little complicated.)

The more tension you apply to your lips (the harder you pull your lips backwards in a strange smile), the more quickly they spring back into position. If the whole cycle takes a time T (called the period), then there are (one second)/T cycles per second. So the frequency f, in cycles per second, is just f = 1/T. All else equal, high lip tension gives high frequency and so high pitch. In the sound files above (play them again), the lip tension is increased and decreased smoothly.
The film clip below was made using a transparent yidaki (didjeridu). On the left you see a high speed video of the lips, on the right schlieren images of the air jet from the lips. More details on our yidaki (didjeridu) site.
Adding (only) a mouthpiece makes relatively minor changes: it can reduce the amount of the lips that move, and it allows the pressure outside the lips to be a little different from atmospheric. We'll discuss mouthpieces in detail later, but for now here is a sketch of the lips and a horn mouthpiece. Many players position the mouthpiece asymmetrically, so that it covers more of the upper lip than the lower lip, as is shown here.

Playing Softly and Loudly
This simple picture already allows us to explain something about how the timbre changes when we go from playing softly to loudly. If we play softly, and especially if we play a high note softly, the lips don't move fast enough and don't have enough time to close completely. In this case we observe nearly sinusoidal vibration: the system is behaving like the linear mass-and-spring oscillator of physics texts. This means that the fundamental in the sound spectrum is strong, but that the higher harmonics are weak. This gives rise to a mellow timbre. Playing loudly, the lips do close, and may even close abruptly. This gives what physicists call clipping and nonlinear behaviour, which produces more high harmonics. As well as making the timbre brighter, adding more harmonics makes the sound louder as well, because the higher harmonics fall in the frequency range where our hearing is most sensitive (See hearing curves for details).
The Effects of the Bell.
Now there are (at least) two problems with a cylindrical pipe like this: first, the notes are too far apart to be musically useful. Second, it's not loud enough (and what's a brass instrument for, eh?). Adding a flare and a bell reduces both of these problems. The flared section of the bore in many instruments are almost conical. First let's look at what this does to the spacing of the frequencies. In the page about pipes and harmonics, we saw that closed conical pipes have resonances whose frequencies are both higher and more closely spaced than those of a closed cylindrical pipe.

In a Shofar, the bell is minimized and is usually as narrow as the bore so the bell has minmal effect.

The Effect of the Mouthpiece
The mouthpieces of brass instruments have a rounded section that fits comfortably against the lips, an enclosed volume of air, a narrow constriction, and a taper that widens out to meet the bore of the body of the instrument. The enclosed volume may be approximately conical, as in many horn mouthpieces or cup shaped, as in most other brass instruments.

The Shofar has a distinctive mouthpiece unique to the instrument because all instruments are custom made. Some mouthpieces are shallow (giving a blaring sound); some are deep (giving a mellow tone).

When placed against the player's lips, the enclosed volume is sealed at one end by the lips, and has the constricted part of the pipe at the other end. You might imagine it as a tiny bottle, with your lips at the base of the bottle, and the constriction representing the neck of the bottle. Now, just as the bottle has a resonance that you can excite by blowing over the top, the mouthpiece has a resonance that you can excite by slapping the wide end against the palm of your hand. When shopping for mouthpieces, brass players sometimes do this to compare what they call the 'pop tone'. It is much easier to hear the pitch if you compare two: say a trumpet and a trombone mouthpiece. As you might expect, the larger the volume (all else equal), the lower the pitch of the pop tone. This is an example of a Helmholtz resonator, whose frequency depends on the enclosed volume and the geometry of the constriction.

The mouthpiece does a few things. First, it allows you to connect the pipe to a comfortably large section of lips. Then there are the acoustic effects of the enclosed volume and constriction. One effect is to lower the frequency of the very highest resonances opposing the effect of the flare (bell), which tends to raise all the resonances.

Another is that to strengthen some of the resonances.

For more information about Shofar and other Holy Temple instruments, we have written extensively on the Shofar and have three websites

1) Joint Effort with Michael Chusid, an expert Shofar sounder and commentator
2) Shofar Sounders WebPage
3) Shofar WebPage

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