Sound and Music

A Deliciously Delightful Discussion of Resonance, Harmonics, Beats, and Periods and

Frequencies

Presented by Giuliano Godorecci, Lisa Schalk, Patrick Woodbury, and Sean Perry (respectively)

Resonance•Resonance is defined as “The reinforcement or prolongation of sound by reflection from a surface or by the synchronous vibration of a neighboring object”.•An example of resonance can easily be found in musical instruments of all kinds, ranging from woodwinds to stringed instruments.•The mass of a string, the length, and the tension of the string, all dictate what frequency the string will vibrate at and, thus, what pitch of sound will be emitted.•The strongest resonance of an instrument is located at the natural harmonics of the instrument, such as the 5th, 7th, and 12th fret of a guitar.

Resonance (Cont.)•Resonance is the movement up and down or back and forth of an object. This motion is called an oscillation.•The higher up on a guitar’s neck one plays, the higher the pitch of the notes will become because the strings length becomes shorter and shorter, causing the oscillations of the strings to increase in rate.•Some strings can start oscillating at their fundamental frequencies when other strings are played, just as an E string on a guitar will sound after an A string is plucked.

References• “The ABC’s of Resonance.” Intuitor.com. April 2, 1996. web. 2007.<http://www.intuitor.com/resonance/abcRes.html>

HarmonicsHarmonics

A harmonic is a sinusoidal component of a periodic wave or quantity having a frequency that is an integral multiple of the fundamental frequency by definition.

Harmonics are all periodic at the fundamental frequency, which means the sum of harmonics is also periodic at that frequency. In other words, the second harmonic always has exactly half the wavelength (and twice the frequency) of the fundamental; the third harmonic always has exactly a third of the wavelength (and so three times the frequency) of the fundamental, and so on. This is pictured on the right.

Guitar Harmonics Guitar Harmonics

When a guitar string is plucked, the string vibrates most notably at its fundamental frequency, but simultaneously it also vibrates at all integer multiples of that frequency. The vibration along the entire length of the string is known as the fundamental, while vibrations occurring between points along the string (known as nodes) are referred to as overtones. The fundamental and overtones, when sounded together, are perceived by the listener as a single tone.

Beats• A beat is an effect that occurs when two waves of opposite form and slightly different frequency occur simultaneously. This should not be confused with the very different concept of musical beats.

(Image sourced from http://www.aaroncake.net/

Beats (cont.)• When the waves first meet, they are exactly opposite one another. This results in complete

destructive interference – the waves cancel each other out completely, and no sound is produced.

Here, the red line represents the sound generated by the combined waves.

Beats (cont.)• As the waves become increasingly desynchronized, however, they gradually move from

destructive interference to constructive interference. When the waves cause complete destructive interference, they are said to be out of phase; when they cause complete constructive interference, they are in phase.

• As this pattern continues, the combined waves form a third wave. The listener perceives this wave as a gradual rise from softness to loudness and back.

Beats (cont.)• A beat occurs at the moment the two waves are completely in phase – that is, synchronized –

and thus are at their loudest.

• The magnitude of the difference between the waves’ frequencies determines how often beats occur.• For waves whose frequencies differ by 1 Hz, a beat will occur once every second because the combined waves have a period of 1.• If the difference in frequency is 2 Hz, then a beat will occur every 1/2 second.•A difference of 3 Hz will result in a beat every 1/3 second.•In the same vein, a difference of 4 Hz results in a beat every 1/4 second, 5 Hz results in a beat every 1/5 second, and so on.• In other words, the period of each beat is equal to the reciprocal of the difference in frequency between the waves in question.•As the difference between the waves’ frequencies increases, the speed of the transition from softness to loudness increases until eventually, the listener perceives two distinct tones.

Beats (cont.)• The distinctive effect created by a very small difference in frequency is sometimes taken

advantage of to create an unusual sound.• The supplementary video at http://titan-physics.wikispaces.com/The+Music+Group

demonstrates the sound created by beats with increasingly large differences in frequency.

References• That good ol’ physics textbook you’ve been hauling around most of the last year.• Also, http://www.aaroncake.net/ for an image of a single sine wave, which I painstakingly

copied, pasted, and manipulated to create my illustrations.

Period and Frequency

Painstakingly created by Sean Perry

Period

• The period of an object moving in a set pattern is the time it takes the object to move from its point of origin to the furthest part of it’s path, and back the point of origin.

• This is usually measured in seconds.

Frequency

• The frequency of an object moving in a set pattern is the number of complete cycles (A cycle is from origin to end back to origin.) it makes in a set amount of time.

• This is usually measured in hertz.

Switching back and forth between the two, just like

a pendulum.• To find the period of a moving

object from its frequency, you invert the frequency, dividing 1 by it.

• To find the frequency of a moving object from its period, you invert the period, dividing the number 1 by it.

How I can possibly relate this to music.

• The higher the frequency of a moving object, such as a string, the higher pitched the sound is. The lower the frequency, the lower we perceive the sound.

• Inversely, the lower the period of a moving object, the higher the sound is that we perceive, and the higher the period, the lower the sound is that we perceive.

My… numerous references

• http://www.engineeringtoolbox.com/sound-frequency-wavelength-d_56.html

And that’s it.