THE PHYSICS OF MUSIC

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1. Noise and sound

It is not easy to make an absolute distinction between noise and sound. After all, what may be taken as noise for one person (for example, a racing car's engine), may be a pleasing sound to another person! In general, a noise consists of random sound frequencies and is associated with an unpleasant feeling for the listener, while sound is found to consist of a limited range of frequencies, which often are mathematically related to each other. Such sounds are commonly pleasing to a listener, and when forming some sort of pattern, is recognised as "music".

The PITCH of a sound is a term used by musicians to describe the frequency of a musical note - high pitch means high frequency, while low pitch means a low frequency. The LOUDNESS of a note refers to its amplitude. Loud sounds have relatively high amplitudes, that is, the sound wave has a relatively high energy.

2. Vibrating strings

A number of musical instruments use vibrating strings in order to generate sounds. Among these may be mentioned violins, guitars, the piano and the harp. They operate on the principle that a tight string is forced to vibrate from its equilibrium position under some stimulus (rubbing, plucking or striking), creating a standing wave along the string. In turn, the vibrating string causes vibrations in the air, which are heard as musical sounds.

2.1 The sonometer:

The sonometer consist of a wooden sound box, on which a suitable string,such as piano wire, if fixed at one end, and which passes over a pulley and is connected to a mass, m. Two movable bridges, A and B, define the vibrating length, L, of the string.

If the string is energized, sound is emitted. The frequency of the sound DECREASES as the length, L, and INCREASES as the tension in the string. The frequency of a note is referred to as its PITCH. A note with a high pitch has a high frequency, while a low pitch note refers to one which has a low frequency. (Note for educators).

2.2 Harmonics:

For a given string, the method of energizing (striking, plucking, stroking) which causes the vibration will produce different modes of vibration, called HARMONICS.

There are several modes of vibration of the string, corresponding to the number of nodes in the string. If the fundamental mode of vibration has a frequency f1, the so-called second harmonic will have a frequency f2 = 2 f1, the third harmonic will have a frequency f3 = 3 f1, the fourth harmonic f4 = 4 f1, and so on.


The presence of harmonics when a string is set to vibrate gives rise to "richer", more interesting sounds. The QUALITY OF THE SOUND produced by different musical instruments is due, in part, to the harmonics generated by the instrument. Musicians decribe the quality of a sound with terms such as TONE, TONE COLOUR or TIMBRE

The figures above are simulations of oscilloscope traces of (from left to right): a 440 Hz fundamental, a 440 Hz fundamental with its 2nd harmonic (880 Hz), the 400 Hz fundamental with its 2nd (880 Hz) and 3rd (1320 Hz) harmonics. You may listen to the corresponding sounds.

Click here to find out about the violin.
Click here to find out about the piano.

3. Vibrating air columns

Many musical instruments use the principle of vibrating air columns in order to generate sound. Such instruments consist of a tube which may have various lengths and shapes, fitted with a mouthpiece. When the player blows into the mouthpiece in certain ways, the air column in the tube is caused to vibrate. Such instruments are generally called "wind instruments" (subdivided further into "woodwind" instruments, such as the flute, and "brass" instruments, such as the trumpet).

Experiments show that (1) for a straight tube, (such as an organ pipe), the frequency of the fundamental note is independent of the cross sectional shape of the tube, or the material from which the tube is made, (2), the frequency of the fundamental increases as the length of the tube decreases, and (3), a tube which is closed at the end farthest away from the mouthpiece will emit the same fundamental note as an open tube which has twice the length.

When air is blown into the tube, standing waves are set up inside the tube, as shown in the diagrams below. The effective length of the air column is varied in various ways: in brass instruments, an intricate system of valves short-circuit parts of the tube. In most woodwind instruments, keys activate pads which in turn cover and uncover holes in the tube. This alters the frequencies of the fundamental notes.

4. Additional questions


Fundamental frequency of a vibrating string

The fundamental frequency (first harmonic) f1 of a vibrating string is given by:

where L is the length of the string, T the tension in the string, and μ the mass per unit length of the string. For the sonometer described above, T = mg, where m is the mass causing the tension in the string, and g is the acceleration due to gravity.









The violin

The violin consists of a wooden sound box, with four strings of different thicknesses, made of dried animal gut. These strings are caused to vibrate by stroking them with a "bow" made with horsehair. The tension of the strings is adjusted by means of screws, thus "tuning" the strings to the fundamental frequencies of 294 Hz (D4), 330 Hz (E4), 392 Hz (G4), and 440 Hz (A4). The length of the strings are varied by the player by pressing them with the fingers as required, thus producing different fundamental notes.










The piano

The sound of a piano note is caused by the striking of a wire string by a hammer, as shown in the picture on the right

The pianos notes are arranged in OCTAVES, of 13 notes, designated C, C#, D, D#, E, F, F#, G, G#, A, A#, B, and C. The interval between the notes are called "half steps", or "chromatic steps", and the ratio of the frequencies between two adjacent notes is 21/12 (the twelfth root of 2) = 1.059463. So, for any note whose frequency, f1, is known, the frequency of any other note is fn = f1 x 2(± n/12). (For notes of lower pitch than the known note, use the negative exponent, for higher pitch, use the positive exponent).