The Physics of Musical Sound

1287 words | 5 page(s)

A sound is a wave that moves through air. The disturbance itself is what moves, not the individual particles. The individual particles oscillate back and forth in place. Sound produces compressed air. The collision of air molecules against one another is what allows the sound wave to travel from one place to another. The speed at which sound waves propagate depends on the properties of the air, such as the density and temperature. Sound moves more quickly in less dense air. The temperature of air affects the speed of sound because that affects the density of the air and, in turn, how easily one molecule can run into the next.

For musical sound, we are interested in periodic, repetitive waves, i.e., repetitive compressions that move at the same speed. Wavelength is the distance between identical points on a repetitive wave, and frequency is the number of cycles a wave makes each second (Hz).

Wavelength and frequency are related to each other. High frequencies always make short wavelengths, and low frequencies always make long wavelengths. The speed of a wave is equal to its wavelength times its frequency (v = wavelength x f).

The range of human hearing is between 20 Hz and 20,000 Hz, and the wavelengths of sounds that we can hear are between 17cm (associated with the higher frequencies) and 17m (associated with the lower frequencies). The speed of sound in air is 343 m/s.

The pitch of a note or sound is related to frequency. The higher the pitch the higher the frequency, and the lower the pitch the lower the frequency. A musical octave corresponds to a doubling of the frequency. The range of frequencies for a piano is 27.5 Hz to about 4000 Hz and for the human voice from about 100 Hz to about 1000 Hz. The musical fifth is the interval between a note and the higher note with a frequency 1.5 times the given note. For example, a musical fifth of C (262 Hz) is G (392 Hz).

The musical pitch that we hear is determined by the frequency of the sound wave, and the frequency of the sound wave is determined by how often the air is shaken by the instrument. So, the question becomes, what determines the frequency at which the instrument shakes? All objects have natural frequencies at which they shake or vibrate. These are called resonances. Resonant frequencies are the natural vibration frequencies of any object. For example, if you blow into a wind instrument, bow a stringed instrument, or hit a drum, there are resonant frequencies which occur.

There are resonant frequencies at which the waves collide together and add up to create bigger waves at certain places and places where the waves cancel each other out and don’t move. The places where the waves add up to create bigger waves are called antinodes and the places where the waves cancel each other out are called nodes. For example, the string in the demonstration had resonant frequencies that were multiples of 40 Hz. Multiple nodes and antinodes could be seen depending on the resonant frequency at which the string was vibrated. Because the string is tied down at the ends and could not move, the ends of the string would be nodes. The number of nodes in the string are determined by the resonant frequency applied to the string and are distributed evenly along the string, depending on the frequency.

The fundamental resonance is called the first resonance. Since the wavelength is equal to two times the length of the string (Wavelength = 2 x L) and the frequency is equal to the velocity divided by the wavelength (f = v/wavelength), then the frequency is also equal to the velocity divided by two times the length of the string (f = v/2L). To increase the pitch that a string plays, the frequency of the fundamental resonance must be increased. There are two ways to increase the fundamental resonance as indicated in the formula f = v/2L. One is by decreasing the length of the string by fingering, and the other is by increasing the velocity by increasing the tension or decreasing the mass.

The physical properties of a string (length, mass, and tension) determine its resonant frequencies. The strings of a stringed instrument are shorter for high notes and longer for low notes. In addition, thinner (lower mass) strings are used for the higher notes and thicker (higher mass) strings for the lower notes. This is because mass makes a difference in how fast a wave propagates. Therefore, if mass is increased, this causes the velocity of the wave to decrease, which will cause frequency to decrease. Pulling on the string increases the tension on the string. This will increase the velocity and thus the frequency.

Each of the resonances of a string is associated with a particular shape of the motion of the string. These can be described in terms of the wavelength of the wave created. The fundamental resonance has the longest wavelength (2xL). The second resonance (second harmonic) has a wavelength equal to the length of the string (L). The third resonance (third harmonic) has a wavelength equal to 2/3rds the length of the string (2L/3). The fourth resonance (fourth harmonic) has a wavelength equal to 1/2 the length of the string (L/2), and the fifth resonance (fifth harmonic) has a wavelength equal to 2/5ths the length of the string (2L/5).

Using the wavelengths, the frequencies or pitches of the notes that can be played on a string can be determined. The frequencies of the vibrations for these particular resonances are related to each other. All of the resonant frequencies are just multiples of the lowest one. When you know the fundamental frequency, you can determine the frequencies of all the harmonics.

Different instruments playing the same note (pitch) sound differently. This is because the timbre or tone quality is different from instrument to instrument. The frequencies are all the same, but the shape of the sound waves are different. Real sounds are complex waves. Instruments just aren’t vibrating in one resonant motion, but in some combination of multiple resonant motions. For example, if a piano, soprano, and whistle all play the same note, they each sound differently. For a piano, most of the resonant motion is coming from the fundamental frequency. With a soprano, there is a lot of the fundamental frequency, but there are significant contributions from the 2nd and 3rd harmonics. Even though a whistle has most of the resonant motion coming from the 6th harmonic, the same note is heard because there is still a fair amount of fundamental frequency present.

Several harmonics have to be added in before the note sounds like an instrument that is recognizable. The timbre of the sound distinguishes one instrument from another.

The second harmonic is an octave which has a ratio of 2:1 from the fundamental frequency. The third harmonic is a fifth which has a ratio of 3:2 from the second harmonic. The fourth harmonic is a fourth which has a ratio of 4:3 from the third harmonic. The fifth harmonic is a Major 3rd which has a ratio of 5:4 from the fourth harmonic. The 6th harmonic is a Minor 3rd which has a ratio of 6:5 from the fifth harmonic.

The harmonics of the string and the notes of the scale go together. Not every instrument has resonances that are simple integer multiples. Those that don’t are called anharmonic instruments. The drum is an anharmonic instrument. Its resonances are much more complicated than the resonances of a string (f, 1.59f, 2.14f, 2.30f, and 2.91f). There is more of a thud or noise with less pitch in anharmonic instruments. There is a whole continuum between harmonic instruments and anharmonic instruments. The bell falls somewhere in between.