Chapter 3: Pitched Sound

Introduction

What is special about pitched sounds?

• Listen to the sound on the web site of a cellist playing the open strings of the instrument.
• We can open the cello recording in Audacity and zoom in on the waveform.
• These vibrations travel through the air and are interpreted by our ears.

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• What we see is a graph, with time in seconds on the horizontal axis and displacement on the vertical axis. The computer reads the displacement measurements and tells the speakers how to vibrate so you can hear the sound.

3.1 Periodicity

Definitions.

• A pattern occurring in time is periodic if it repeats the same pattern at regular time intervals.
• Its period is the duration, in seconds, of the smallest portion of the overall pattern that can be repeated to form the overall pattern.
• A cycle is a portion of the overall pattern whose length equals one period.

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One cycle

Periodic pattern

Period = 3.770-3.760 = 0.01

Exercises

Compute the period of this “sawtooth” wave below

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Is this the wave on the right periodic? �Explain. It represents static.

3.2 Frequency

Pitched sound is sound produced by rapid, regular vibration. Frequency measures the speed of vibration.

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Definitions. The frequency of a periodic vibration is its number of cycles per second. The units of frequency are Hertz, written Hz.

Humans can hear pitched sounds that range from about 20 Hz to 20,000 Hz. Other animals can hear sounds with lower or higher frequencies.

Period-frequency conversion. The frequency and period of any periodic waveform are reciprocals.

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That is, period = 1/frequency and frequency = 1/period.

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The frequency is measured in Hertz (cycles per second) and the period is measured in seconds per cycle. Note that the units of measurement are also reciprocals.

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Example

Compute the frequency of the cello sound. �Use Audacity to generate a sound with the �same frequency and play them at the same �time. Is your answer correct?

Compute the frequency of the sawtooth wave. Zoom in on the waveform to see if your picture looks like this one.

Pitched sound

Definition. A pitched sound is an audible sound produced by a periodic vibration. That is, it has a periodic waveform and its frequency is in the range 20 Hz - 20,000 Hz.

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A few observations...

• If two pitched sounds have the same frequency, they are said to have the same pitch.
• If one sound has a higher frequency than another, it is said to have higher pitch, or to be a higher sound.
• If one sound has a lower frequency than another, it is said to have lower pitch, or to be a lower sound.
• Because frequency and period are reciprocals, high pitches correspond to short periods, while low pitches correspond to long periods.

3.3 Amplitude and Envelope

The amplitude, or strength of the displacement, is the maximum displacement of the waveform over a small unit of time. The envelope is a graph that shows how the amplitude changes. You can draw it by connecting the peaks of the waveform. Sounds with larger amplitudes have larger volumes, though volume is a complex psychological phenomenon that can’t purely be described by amplitude.

Exercises

Describe the following sound in as much detail as possible:

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What does the Envelope tool in Audacity do?

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If a song fades out at the end, what do you expect its envelope to look like?

3.4 Sinusoids

A tuning fork vibrates�in a wavy pattern called a�sinusoid. ��Here’s are pictures of the waveform of a tuning fork’s vibration. The picture at the top was made by attaching a pen to the tuning fork, while the bottom picture was made by Audacity.

Sinusoids and “sine”

You may remember studying the sine function in a geometry class. As a point travels around a circle that has radius = 1, the sine of the angle it makes with the positive t-axis is the same as its y-coordinate. As you move around the circle repeatedly, the point’s y-coordinates trace a wavy pattern.

Graphing sinusoids

Here is the graph of y = sin (t), where t is measured in radians. It crosses the t-axis at 0, pi, 2*pi, 3*pi, … and also -2*pi, -3*pi, etc.

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The general formula for a sine wave is

y = A sin (2*pi*f*t)

A = amplitude f = frequency in Hz t = time in seconds

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Of course, pi is the number 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349

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Example

Here is the graph of a sine wave with

Amplitude = 0.8 period = 0.01 seconds

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To find its formula, you must calculate its frequency = 1/period = 1/0.01 = 100 Hz. So its formula is y = 0.8 sin ( 2*pi*100*t) = 0.8 sin (200*pi*t).

Exercises

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1. Find the amplitude, frequency, and period of the waveform �y = 0.4 sin (100*pi*t).
2. Find the formula for this waveform.

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• Sketch a sine wave with amplitude = 3 and period = 4 seconds. Be sure to label both the horizontal and vertical axes of your graph. Find the formula for this wave and use a graphing program to check your answer.

3.5 The Superposition Principle

The Superposition Principle explains how multiple sounds combine.

• It is the result of a basic principle of physics stating that the impact of two forces at a point is the sum of those forces.
• Superposition predicts that it is possible for two sounds to reinforce each other, cancel each other out, or something in between.

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The Superposition Principle: the waveform of two sounds that reach the same place at the same time is the sum of the waveforms of each sound on its own.

Adding waveforms

OK, so the Superposition Principle says that waveforms “add.” What does “add” mean? In mathematics, adding two graphed functions means that you add their y-values.

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With recorded sound, the displacements are added, which means that you add y-values of the graphed waveforms.

Mixing

In Audacity, if you select two waveforms, then choose Tracks -> Mix and Render To New Track, the sum of the waveforms will appear in a new track.

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The picture shows the result of mixing a 440 Hz sine wave with amplitude 0.4 (top) and a 330 Hz square wave with amplitude 0.3 (middle) to produce a new wave (bottom), which is their sum.

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Results of the Superposition Principle

If two absolutely identical waveforms are combined, the result is a waveform with double the original amplitude.

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*It is possible to cook up weird examples when this doesn’t work--for example, where the sounds combine to produce silence.

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Normally, the combination of two sounds with the same frequency, f, also has frequency f.*

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Constructive and destructive interference

Constructive interference occurs when identical waveforms are combined. The result is a waveform with double the original amplitude.

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Destructive interference occurs when one waveform cancels another to produce silence. The second wave’s displacement is always in the opposite direction as the first wave’s displacement. Mathematically, the y-values are multiplied by -1.

Alternately, you could move the first wave horizontally (ahead in time) by ½ period to get the second wave. The two waves are “out of phase” in that they are at different points in their cycle at the same time.

3.6 Beats

What happens when you play two sounds with close, but not identical, frequency? (Musicians call this being “out of tune.”) Here are 100 and 110 Hz waves played over 0.10 sec. The 100Hz wave has 10 cycles and the 110Hz wave has 11 cycles.

100 Hz

110 Hz

Beats: micro scale (0.01 sec.)

Beats: macro scale (1 sec.)

The combination of 100 Hz and 110 Hz tones produces a tone pulsing at 10 Hz.

Beats: macro scale (1 sec.)

The combination of 100 Hz and 106 Hz tones produces a tone pulsing at 6 Hz.

100 Hz

106 Hz

6 Hz “beats”

Beats: macro scale (1 sec.)

The combination of 100 Hz and 102 Hz tones produces a tone pulsing at 2 Hz.

Conjecturing a Beats Formula

Definition. Beats are pulses that are heard when two pitched sounds with close but not equal frequency combine due to superposition.

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What is the frequency of the beats? Make a conjecture.

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Conjecture: The beats’ frequency is the difference between the frequencies of the two pitched sounds.

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• Example: Suppose a 440 Hz and a 441 Hz tone combine due to superposition. What is the frequency of the resulting beats?
• Example: Suppose you hear a 440 Hz tone in your right ear and a 441 Hz tone in your left ear. Explain why beats are not heard.
• Example: Suppose a 440 Hz and a 330 Hz tone combine due to superposition. Explain why beats are not heard.

The Beats Formula

The Beats Formula. Suppose two pitched sounds with close* but not equal frequencies, f₁ and f₂, combine due to superposition. Then

• The frequency of the beats is the difference of the two individual frequencies, |f₁ - f₂|.
• The combination sound is a pitched sound that has frequency equal to the average of the two individual frequencies, (f₁+f₂)/2.

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*There’s not a hard-and-fast limit, but I’ll say “close” means “less than 20 Hz apart.”

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Note: In order to form beats, (1) superposition must occur, and (2) the two frequencies have to be “close” (within 20 Hz of each other).

Why the Beats Formula?

The justification comes from a formula you might have seen in trigonometry:

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The formula predicts how sine waves with different frequencies will add.

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Average

½ the difference

Why an accordion sounds bad

The button accordion has four reeds for each different tone. When you play an “A”, two of them sound almost the same, one sounds high, and the other sounds low. You also hear a rough pulsing sound. Here’s a picture of the individual waveforms of the “almost the same” tones, plus a mixed waveform of both of them playing together. What happened?

Binaural beats?

Is it possible for your brain to perceive beats when different (but close) frequencies are played in each ear? These “beats” are called binaural beats.

Some people claim that binaural beats have mystical powers, though no one has proven that they can actually be heard -- or affect you at all! What do you think? Try listening with headphones.

3.7 The wave equation

The wave equation is a mathematical formula that predicts the frequencies of pitched sounds made by musical instruments. The one-dimensional wave equation is used for stringed and wind instruments.

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A string that has been stretched tight vibrates periodically when you pluck it. As long as the length and tension of the string don’t change, the frequency will stay the same when you pluck it a second time. Any other string made of the same material with the same length and tension will play the same frequency.

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Frequency of a vibrating string

The one-dimensional wave equation predicts that a plucked string’s vibration is a sum of sinusoids whose frequencies equal an/(2L), where

a is a number determined by the tension and material of the string.

L is the length of the string

n is the number of the harmonic, so n can be any of the numbers 1, 2, 3, etc. The fundamental frequency is played when n = 1.

Four ways to play the violin: (1) Changing length

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Changing the length of the part of the string that’s �vibrating by pressing it down to the neck is the �most common way to change the�string’s frequency.

Since L is in the denominator, �large lengths correspond to �small frequencies - that is, �lower pitches.

L is the length of the string

Anatomy of a violin

Body

Neck

Tuning pegs

Fine tuners

Length-frequency ratio formula

If the length of a string is changed without changing its tension, the ratio of lengths equals the reciprocal of the ratio of fundamental frequencies. That is,

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where the original string has length L₁ and fundamental frequency f₁ and the altered string has length L₂ and fundamental frequency f₂.

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Why? The wave equation says that f₁ = an/(2L₁) and f₂ = an/(2L₂). If n=1 and a doesn’t change, then an = 2f₁L₁ and an = 2f₂L₂, so f₁L₁ = f₂L₂ and

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Example: length-frequency ratio formula

The fundamental frequency of a violin's A string is 440 Hz and the length of the part of the string that vibrates is 327 mm. Where would the player need to press down to make the string vibrate at 660 Hz?

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ANSWER: In the formula, use L₁ = 327, f₁ = 440, and f₂ = 660. We want to find L₂. Plugging into the formula below, 327/L₂ = 660/440 = 3/2, L₂ = 327*2/3 = 218 mm. The player presses down 218 mm from the bridge end of the string. That is, 2/3 of the original length is vibrating to produce a sound that is 3/2 times higher.

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Four ways to play the violin: (2) Changing tension

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When the tension is greater, the frequency is higher.

a is a number determined by the tension and material of the string.

Four ways to play the violin: (3) Changing string mass

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When the string is heavier, the frequency is lower.

a is a number determined by the tension and material of the string.

The precise meaning of “a”

The relationship between a, tension, and length is

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This explains why

• increasing the tension makes a larger, and therefore the frequency, an/(2L), higher.
• Increasing the mass makes a smaller, and therefore the frequency lower.

Buying violin strings

Why do violin strings come in sets of four? That is, why can’t use use the same material for all four strings?

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ANSWER: Since having strings with very different tensions is harmful to the violin and may break the strings, fatter strings are needed for lower sounds.

Four ways to play the violin: (4) Changing harmonic

n is the number of the harmonic, so n can be any of the numbers 1, 2, 3, etc.

Standing waves and modes of vibration

The first seven modes of vibration produce the first seven harmonics

1st

2nd

3rd

4th

5th

6th

7th

Harmonics and overtones

Definition. The fundamental frequency of a string, f, is its frequency when it vibrates in the first mode of vibration. The harmonics of a string are the frequencies f, 2f, 3f, 4f, … which correspond to its modes of vibration.

Example. If a string’s fundamental frequency is 100 Hz, its harmonics will be

100 Hz 200 Hz 300 Hz 400 Hz etc.

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Example: harmonics

The points at which the string is stationary are called nodes.

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The nth harmonic can be produced by lightly touching the string at a point 1/n of the way along the string. That is

• You produce the first harmonic by not touching the string.
• You produce the second harmonic by touching the string in the middle.
• You produce the third harmonic by touching the string ⅓ of the way along its length.
• You produce the first harmonic by touching the string ¼ of the way along its length.

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Four ways to play the violin

Suppose a violin’s A string is tuned to 440 Hz (its fundamental frequency). Describe three ways you could change its pitch to 880 Hz without using a different string.

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1. Touch the string lightly in the middle to create a node and play the second harmonic.
2. Press the string down ½ of the way along, thus changing its length.
3. Increase its tension by a factor of 4. This will probably break the string--or the violin!

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If you are allowed to replace the string with another one of the same length and tension, how could you produce 880 Hz, while still playing the first harmonic.

ANS: Replace the string by one that has 1/4 times the mass.

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Four ways to play the cello

Suppose a cello’s A string is tuned to 220 Hz (its fundamental frequency). Describe three ways you could change its pitch to 660 Hz without using a different string.

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• Touch the string lightly either ⅓ or ⅔ of the distance from the nut to bridge to create a note and play the third harmonic.
• Press the string down ⅓ of the way from the bridge, thus changing its length to ⅓ of the original length.
• Increase its tension by a factor of 9. This will probably break the string--or the cello!

If you are allowed to replace the string with another one of the same length and tension, how could you produce 660 Hz, while still playing the first harmonic.

ANS: Replace the string by one that has 1/9 times the mass.

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Wind instruments

Wind instruments include the flute, saxophone, trumpet, etc.

• The one-dimensional wave equation also holds true for wind instruments.
• Instead of vibrating strings, vibration is produced by periodic differences in pressure inside the instrument’s tube.
• The modes of vibration look similar to those of stringed instruments.

First mode of vibration �(sounds the first harmonic = f )

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Second mode of vibration �(sounds the second harmonic = 2f )

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Third mode of vibration�(sounds the third harmonic = 3f )

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How to play (most) wind instruments

The frequency formula is still where

• n is the number of the harmonic, with n = 1 being the fundamental
• L is the length of the part of the air column that vibrates
• a is a number depending on air pressure

You can

• Change n by blowing harder (a “register key” helps in some instruments)
• Change L by making small adjustments in the length (used for tuning) or by uncovering some holes.

The value of a can’t be changed.

Clarinets

Unlike a flute, saxophone, or trumpet, the air column of a clarinet is open on one end but closed on the other. Therefore, its standing waves have a node at one end and an antinode at the other. The formula for the frequency of a clarinet’s modes of vibrations are where n must be an odd number.

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If the fundamental frequency of a clarinet is 100 Hz, then its harmonics are

• 100 Hz (n = 1)
• 300 Hz (n = 3)
• 500 Hz (n = 5)
• etc.

The two-dimensional wave equation

The modes of vibration for a two-dimensional (flat) object that is vibrating, like the head of a drum, are complicated, and there’s not a good formula for the frequencies produced.

The three-dimensional wave equation

Three-dimensional instruments include bells and the singing� bowl. The Bianzhong of Marquis Yi of Zeng was made �in 433 B.C.