The Physics of Music and Musical Instruments

THE PHYSICS OF MUSIC ANDMUSICAL INSTRUMENTS

DAVID R. LAPP, FELLOWWRIGHT CENTER FOR INNOVATIVE SCIENCE EDUCATION

TUFTS UNIVERSITYMEDFORD, MASSACHUSETTS

f1 f3 f5 f7

http://www.tufts.edu/as/wright_center/

http://www.tufts.edu

http://www.tamhigh.org/lapp

TABLE OF CONTENTSIntroduction 1

Chapter 1: Waves and Sound 5Wave Nomenclature 7Sound Waves 8ACTIVITY: Orchestral Sound 15Wave Interference 18ACTIVITY: Wave Interference 19

Chapter 2: Resonance 20Introduction to Musical Instruments 25Wave Impedance 26

Chapter 3: Modes, overtones, and harmonics 27ACTIVITY: Interpreting Musical Instrument Power Spectra 34Beginning to Think About Musical Scales 37Beats 38

Chapter 4: Musical Scales 40ACTIVITY: Consonance 44The Pythagorean Scale 45The Just Intonation Scale 47The Equal Temperament Scale 50A Critical Comparison of Scales 52ACTIVITY: Create a Musical Scale 55ACTIVITY: Evaluating Important Musical Scales 57

Chapter 5: Stringed Instruments 61Sound Production in Stringed Instruments 65INVESTIGATION: The Guitar 66PROJECT: Building a Three Stringed Guitar 70

Chapter 6: Wind Instruments 72The Mechanical Reed 73Lip and Air Reeds 74Open Pipes 75Closed Pipes 76The End Effect 78Changing Pitch 79More About Brass Instruments 79More about Woodwind instruments 81INVESTIGATION: The Nose flute 83INVESTIGATION: The Sound Pipe 86INVESTIGATION: The Toy Flute 89INVESTIGATION: The Trumpet 91PROJECT: Building a Set of PVC Panpipes 96

Chapter 7: Percussion Instruments 97Bars or Pipes With Both Ends Free 97Bars or Pipes With One End Free 99Toward a “Harmonic” Idiophone 100INVESTIGATION: The Harmonica 102INVESTIGATION: The Music Box Action 107PROJECT: Building a Copper Pipe Xylophone 110

References 111

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“Everything is determined … by forces over which we have nocontrol. It is determined for the insects as well as for the star.Human beings, vegetables, or cosmic dust – we all dance to amysterious tune, intoned in the distance by an invisible piper.”– Albert Einstein

INTRODUCTIONHIS MANUAL COVERS the physics of waves, sound, music, andmusical instruments at a level designed for high school physics. However,it is also a resource for those teaching and learning waves and sound frommiddle school through college, at a mathematical or conceptual level. Themathematics required for full access to the material is algebra (to include

logarithms), although each concept presented has a full conceptual foundation thatwill be useful to those with even a very weak background in math.

Solomon proclaimed that there is nothing newunder the Sun and of the writing of books there is noend. Conscious of this, I have tried to producesomething that is not simply a rehash of what hasalready been done elsewhere. In the list of references Ihave indicated a number of very good sources, someclassics that all other writers of musical acousticbooks refer to and some newer and more accessibleworks. From these, I have synthesized what I believeto be the most useful and appropriate material for thehigh school aged student who has neither abackground in waves nor in music, but who desires afirm foundation in both. Most books written on thetopic of musical acoustics tend to be either verytheoretical or very cookbook style. The theoreticalones provide for little student interaction other thansome end of the chapter questions and problems. Theones I term “cookbook” style provide instructions forbuilding musical instruments with little or noexplanation of the physics behind the construction.This curriculum attempts to not only marry the bestideas from both types of books, but to includepedagogical aids not found in other books.

This manual is available as both a paper hardcopy as well as an e-book on CD-ROM. The CD-ROM version contains hyperlinks to interestingwebsites related to music and musical instruments. Italso contains hyperlinks throughout the text to soundfiles that demonstrate many concepts being developed.

MODES OF PRESENTATIONAs the student reads through the text, he or she

will encounter a number of different presentationmodes. Some are color-coded. The following is a keyto the colors used throughout the text:

Pale green boxes cover tables and figuresthat are important reference material.

Notes Frequencyinterval (cents)

Ci 0D 204E 408F 498G 702A 906B 1110Cf 1200

Table 2.8: Pythagoreanscale interval ratios

Light yellow boxes highlight derivedequations in their final form, which will be used forfuture calculations.

†

f1 =

Tm

2L

T

IIINNNTTTRRROOODDDUUUCCCTTTIIIOOONNN

2

Tan boxes show step-by-step examples formaking calculations or reasoning through questions.

ExampleIf the sound intensity of a screaming baby were

†

1¥10-2 Wm 2 at 2.5 m away, what would it be at

6.0 m away?The distance from the source of sound is greater by afactor of

†

6.02.5 = 2.4 . So the sound intensity is decreased

by

†

1(2.4)2 = 0.174 . The new sound intensity is:

†

(1¥10-2 Wm 2 )(0.174) =

†

1.74 ¥10-2 Wm 2

Gray boxes throughout the text indicatestopping places in the reading where students areasked, “Do you get it?” The boxes are meant toreinforce student understanding with basic recallquestions about the immediately preceding text. Thesecan be used to begin a discussion of the reading witha class of students.

Do you get it? (4)A solo trumpet in an orchestra produces a soundintensity level of 84 dB. Fifteen more trumpets jointhe first. How many decibels are produced?

In addition to the “Do you get it?” boxes,which are meant to be fairly easy questions doneindividually by students as they read through the text,there are three additional interactions students willencounter: Activities , Investigations, andProjects. Activities more difficult than the “Do youget it?” boxes and are designed to be done eitherindividually or with a partner. They either require ahigher level of conceptual understanding or draw onmore than one idea. Investigations are harder still anddraw on more than an entire section within the text.Designed for two or more students, each onephotographically exposes the students to a particularmusical instrument that they must thoroughly

consider. Investigations are labs really, often requiringstudents to make measurements directly on thephotographs. Solutions to the “Do you get it?”boxes, Activities, and Investigations are provided inan appendix on the CD-ROM. Finally, projectsprovide students with some background for buildingmusical instruments, but they leave the type ofmusical scale to be used as well as the key theinstrument will be based on largely up to the student.

PHYSICS AND … MUSIC?

“Without music life would be amistake.”– Friedrich Nietzsche

With even a quick look around most schoolcampuses, it is easy to see that students enjoy music.Ears are sometimes hard to find, covered byheadphones connected to radios or portable CDplayers. And the music flowing from them has thepower to inspire, to entertain, and to even mentallytransport the listener to a different place. A closerlook reveals that much of the life of a student eitherrevolves around or is at least strongly influenced bymusic. The radio is the first thing to go on in themorning and the last to go off at night (if it goes offat all). T-shirts with logos and tour schedules of


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popular bands are the artifacts of many teens’ mostcoveted event … the concert. But the school bellringing for the first class of the day always bringswith it a stiff dose of reality.

H. L. Mencken writes, “School days, I believe,are the unhappiest in the whole span of humanexistence. They are full of dull, unintelligible tasks,new and unpleasant ordinances, brutal violations ofcommon sense and common decency.” This maypaint too bleak a picture of the typical student’sexperience, but it’s a reminder that what is taughtoften lacks meaning and relevance. When I think backto my own high school experience in science, I findthat there are some classes for which I have nomemory. I’m a bit shocked, but I realize that it wouldbe possible to spend 180 hours in a science classroomand have little or no memory of the experience if theclassroom experience werelifeless or disconnectedfrom the reality of my life.Middle school and highschool students are a toughaudience. They want to beentertained … but theydon’t have to be. What theyreally need is relevance.They want to see directconnections and immediateapplications. This is thereason for organizing anintroduction to the physicsof waves and sound aroundthe theme of music andmusical instruments.

It’s not a stretch either.Both music and musicalinstruments are intimatelyconnected to the physics ofwaves and sound. To fullyappreciate what occurs in amusical instrument when itmakes music or to

understand the rationale for the development of themusical scales one needs a broad foundation in mostelements of wave and sound theory. With that said,the approach here will be to understand music andmusical instruments first, and to study the physics ofwaves and sound as needed to push the understandingof the music concepts. The goal however is a deeperunderstanding of the physics of waves and sound thanwhat would be achieved with a more traditionalapproach.

SOUND, MUSIC, AND NOISEDo you like music? No, I guess a better question

is, what kind of music do you like? I don’t thinkanyone dislikes music. However, some parentsconsider their children’s “music” to be just noise.Likewise, if the kids had only their parent’s music tolisten to many would avoid it in the same way theyavoid noise. Perhaps that’s a good place to start then– the contrast between music and noise. Is there anobjective, physical difference between music andnoise, or is it simply an arbitrary judgment?

After I saw the movie 8 Mile, the semi-autobiographical story of the famous rapper Eminem,I recommended it to many people … but not to mymother. She would have hated it. To her, his music isjust noise. However, if she hears an old Buddy Hollysong, her toes start tapping and she’s ready to dance.But the music of both of these men would beconsidered unpleasant by my late grandmother whoseemed to live for the music she heard on theLawrence Welk Show. I can appreciate all three“artists” at some level, but if you ask me, nothing

beats a little Bob Dylan. It’sobviously not easy to definethe difference between noiseand music. Certainly there isthe presence of rhythm in thesounds we call music. At amore sophisticated level thereis the presence of tones thatcombine with other tones inan orderly and ... “pleasing”way. Noise is often associatedwith very loud and gratingsounds – chaotic sounds whichdon’t sound good together orare somehow “unpleasant” tolisten to. Most would agreethat the jackhammer tearingup a portion of the street isnoise and the sound comingfrom the local marching bandis music. But the distinctionis much more subtle than that.If music consists of soundswith rhythmic tones of certainfrequencies then thejackhammer might be


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considered a musical instrument. After all, itpummels the street with a very regular frequency. Andif noise consists of loud sounds, which are unpleasantto listen to, then the cymbals used to punctuate theperformance of the marching band might beconsidered noise. I suppose we all define the pointwhere music becomes noise a bit differently. Perhapsit’s based on what we listen to most or on thegeneration we grow up in or … make a break from.But we need to be careful about being cavalier as Iwas just now when I talked about “pleasing” sounds.John Bertles of Bash the Trash® (a company dedicatedto the construction and performance of musicalinstruments from recycled materials:http://www.bashthetrash.com/ ) was quick to cautionme when I used the word “pleasing” to describemusical sound. Music that is pleasing to one personmay not be pleasing to others. Bertles uses thedefinition of intent rather than pleasing whendiscussing musical sound. He gives the example of anumber of cars all blaring their horns chaotically atan intersection. The sound would be considered noiseto most anyone. But the reason for the noise is not somuch the origin of the sound, but the lack of intentto organize the sounds of the horns. If someone at theintersection were to direct the car horns to beep atparticular times and for specific periods, the noisewould evolve into something more closely related tomusic. And no one can dispute that whether it’sEminem, Buddy Holly, Lawrence Welk, or BobDylan, they all create(d) their particular recordedsounds with intent.

“There are two means of refuge fromthe miseries of life: music and cats.”– Albert Schweitzer

BEGINNING TO DEFINE MUSICMusic makes us feel good, it whisks us back in

time to incidents and people from our lives; it rescuesus from monotony and stress. Its tempo and pace jivewith the natural rhythm of our psyche.

The simplest musical sound is some type ofrhythmical beating. The enormous popularity of thestage show Stomp http://www.stomponline.com/ andthe large screen Omnimax movie, Pulsehttp://www.Pulsethemovie.com/ gives evidence for

the vast appreciation of this type of music. Definingthe very earliest music and still prominent in manycultures, this musical sound stresses beat overmelody, and may in fact include no melody at all.One of the reasons for the popularity of rhythm-onlymusic is that anyone can immediately play it at somelevel, even with no training. Kids do it all the time,naturally. The fact that I often catch myselfspontaneously tapping my foot to an unknown beator lie in bed just a bit longer listening contentedly tomy heartbeat is a testament to the close connectionbetween life and rhythm.

Another aspect of music is associated with moreor less pure tones – sounds with a constant pitch.Whistle very gently and it sounds like a flute playinga single note. But that single note is hardly a song,and certainly not a melody or harmony. No, to makethe single tone of your whistle into a musical soundyou would have to vary it in some way. So you couldchange the way you hold your mouth and whistleagain, this time at a different pitch. Going back andforth between these two tones would produce acadence that others might consider musical. Youcould whistle one pitch in one second pulses for threeseconds and follow that with a one second pulse ofthe other pitch. Repeating this pattern over and overwould make your tune more interesting. And youcould add more pitches for even more sophistication.You could even have a friend whistle with you, butwith a different pitch that sounded good with the oneyou were whistling.

If you wanted to move beyond whistling tomaking music with other physical systems, youcould bang on a length of wood or pluck a taut fiberor blow across an open bamboo tube. Adding morepieces with different lengths (and tensions, in the caseof the fiber) would give additional tones. To add morecomplexity you could play your instrument alongwith other musicians. It might be nice to have thesound of your instrument combine nicely with thesound of other instruments and have the ability toplay the tunes that others come up with. But to dothis, you would have to agree on a collection ofcommon pitches. There exist several combinations ofcommon pitches. These are the musical scales.

Here we have to stop and describe what the pitchof a sound is and also discuss the variouscharacteristics of sound. Since sound is a type ofwave, it’s additionally necessary to go even furtherback and introduce the idea of a wave.

http://www.bashthetrash.com

http://www.stomponline.com/

http://www.Pulsethemovie.com/

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“It is only by introducing the young to great literature, dramaand music, and to the excitement of great science that we opento them the possibilities that lie within the human spirit –enable them to see visions and dream dreams.”– Eric Anderson

CHAPTER 1WAVES AND SOUND

SK MOST PEOPLEto define what a waveis and they get amental image of awave at the beach.

And if you really press them foran answer, they're at a loss. Theycan describe what a wave lookslike, but they can't tell you what itis. What do you think?

If you want to get energy from one placeto another you can do it by transferring itwith some chunk of matter. For example, ifyou want to break a piece of glass, you don'thave to physically make contact with ityourself. You could throw a rock and theenergy you put into the rock would travel onthe rock until it gets to the glass. Or if apolice officer wants to subdue a criminal, he doesn'thave to go up and hit him. He can send the energy tothe criminal via a bullet. But there's another way totransfer energy – and it doesn't involve a transfer ofmatter. The other way to transfer energy is by using awave. A wave is a transfer of energy without atransfer of matter . If you and a friend hold onto bothends of a rope you can get energy down to her simplyby moving the rope back and forth. Although therope has some motion, it isn't actually transferred toher, only the energy is transferred.

A tsunami (tidal wave generally caused by anearthquake) hit Papua New Guinea in the summer of1998. A magnitude 7 earthquake 12 miles offshoresent energy in this 23-foot tsunami that killedthousands of people. Most people don't realize thatthe energy in a wave is proportional to the square ofthe amplitude (height) of the wave. That means that ifyou compare the energy of a 4-foot wave that youmight surf on to the tsunami, even though thetsunami is only about 6 times the height, it wouldhave 62 or 36 times more energy. A 100-foot tsunami

(like the one that hit the coast of the East Indies inAugust 1883) would have 252 or 625 times moreenergy than the four-foot wave.

Streaming through the place you're in right nowis a multitude of waves known as electromagneticwaves. Their wavelengths vary from so small thatmillions would fit into a millimeter, to miles long.They’re all here, but you miss most of them. Theonly ones you're sensitive to are a small group thatstimulates the retinas of your eyes (visible light) anda small group that you detect as heat (infrared). Theothers are totally undetectable. But they’re there.

WAVES, SOUND, AND THE EARAnother type of wave is a sound wave. As small

in energy as the tsunami is large, we usually need anear to detect these. Our ears are incredibly awesomereceptors for sound waves. The threshold of hearing issomewhere around

†

1¥10-12Watts /meter 2 . Tounderstand this, consider a very dim 1-watt night-light. Now imagine that there were a whole lot morepeople on the planet than there are now – about 100

AFigure 1.1: Most people think of the ocean whenasked to define or describe a wave. The recurringtumult is memorable to anyone who has spent t imeat the beach or been out in the surf. But waves occurmost places and in many different forms, transferringenergy without transferring matter.

WWWAAAVVVEEESSS AAANNNDDD SSSOOOUUUNNNDDD

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times more. Assume therewas a global populationof 1 trillion (that's amillion, million) people.If you split the light fromthat dim bulb equallybetween all those people,each would hold a radiantpower of

†

1¥10-12Watts.Finally, let's say that oneof those people spread thatpower over an area of onesquare meter. At thissmallest of perceptiblesound intensities, theeardrum vibrates lessdistance than the diameterof a hydrogen atom! Well,it's so small an amount ofpower that you can hardlyconceive of it, but if youhave pretty good hearing,you could detect a soundwave with that smallamount of power. That'snot all. You could alsodetect a sound wave athousand times morepowerful, a million timesmore powerful, and even abillion times morepowerful. And, that's before it even starts to getpainful!

I have a vivid fifth grade memory of my goodfriend, Norman. Norman was blind and the first andonly blind person I ever knew well. I sat next to himin fifth grade and watched amazed as he banged awayon his Braille typewriter. I would ask him questionsabout what he thought colors looked like and if hecould explain the difference between light and dark.He would try to educate me about music beyond top-40 Pop, because he appreciated and knew a lot aboutjazz. But when it came to recess, we parted and wentour separate ways – me to the playground and him tothe wall outside the classroom. No one played withNorman. He couldn’t see and so there was nothing forhim. About once a day I would look over at Normanfrom high up on a jungle gym of bars and he wouldbe smacking one of those rubbery creepy crawlersagainst the wall. He would do it all recess … everyrecess. I still marvel at how much Norman could get

out of a simple sound. He didn’t have sight so he hadto compensate with his other senses. He got so muchout of what I would have considered a very simplisticsound. For him the world of sound was rich anddiverse and full. When I think of sound, I alwaysthink first of Norman. He’s helped me to look moredeeply and to understand how sophisticated the worldof sound really is.

What about when more than one wave is presentin the same place? For example, how is it that youcan be at a symphony and make out the sounds ofindividual instruments while they all play togetherand also hear and understand a message beingwhispered to you at the same time you detectsomeone coughing five rows back? How do the soundwaves combine to give you the totality as well as theindividuality of each of the sounds in a room? Theseare some of the questions we will answer as wecontinue to pursue an understanding of music andmusical instruments.

Figure 1.2: The ear is an astonishing receptor for sound waves. A tthe smallest of perceptible sound intensities, the eardrum vibratesless distance than the diameter of a hydrogen atom! If the energy i na single 1-watt night-light were converted to acoustical energy anddivided up into equal portions for every person in the world, i twould still be audible to the person with normal hearing.


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TWO TYPES OF WAVESWaves come in two basic types, depending on

their type of vibration. Imagine laying a long slinkyon the ground and shaking it back and forth. Youwould make what is called a transverse wave (seeFigure 1.3). A transverse wave is one in which themedium vibrates at right angles to the direction theenergy moves. If instead, you pushed forward andpulled backward on the slinky you would make acompressional wave (see Figure 1.4).Compressional waves are also known as longitudinalwaves. A compressional wave is one in which themedium vibrates in the same direction as themovement of energy in the wave

Certain terms and ideas related to waves are easierto visualize with transverse waves, so let’s start bythinking about the transverse wave you could makewith a slinky. Imagine taking a snapshot of the wavefrom the ceiling. It would look like Figure 1.5. Somewave vocabulary can be taken directly from thediagram. Other vocabulary must be taken from amental image of the wave in motion:

CREST: The topmost point of the wave medium orgreatest positive distance from the rest position.

TROUGH: The bottommost point of the wavemedium or greatest negative distance from the restposition.

WAVELENGTH (l ): The distance from crest toadjacent crest or from trough to adjacent trough orfrom any point on the wave medium to the adjacentcorresponding point on the wave medium.

AMPLITUDE (A): The distance from the restposition to either the crest or the trough. Theamplitude is related to the energy of the wave. As the

energy grows, so does the amplitude. This makessense if you think about making a more energeticslinky wave. You’d have to swing it with moreintensity, generating larger amplitudes. Therelationship is not linear though. The energy isactually proportional to the square of the amplitude.So a wave with amplitude twice as large actually hasfour times more energy and one with amplitude threetimes larger actually has nine times more energy.

The rest of the vocabulary requires getting amental picture of the wave being generated. Imagineyour foot about halfway down the distance of theslinky’s stretch. Let’s say that three wavelengths passyour foot each second.

Figure 1.4: A compressional wave moves t othe right while the medium vibrates in thesame direction, right to left.

fi

fi

Figure 1.3: A transverse wave moves to theright while the medium vibrates at rightangles, up and down.

Amplitude

Amplitude

Wavelength

Wavelength

Crest

Trough

Restposition

Figure 1.5: Wave vocabulary

vocabulary

David Lapp


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Figure 1.6: The energy of the vibrating tuning forkviolently splashes water from the bowl. In the air, theenergy of the tuning fork is transmitted through the air andto the ears.

FREQUENCY (f): The number of wavelengths topass a point per second. In this case the frequencywould be 3 per second. Wave frequency is oftenspoken of as “waves per second,” “pulses per second,”or “cycles per second.” However, the SI unit forfrequency is the Hertz (Hz). 1 Hz = 1 per second, soin the case of this illustration, f = 3 Hz.

PERIOD (T): The time it takes for one fullwavelength to pass a certain point. If you see threewavelengths pass your foot every second, then thetime for each wavelength to pass is

†

13 of a second.

Period and frequency are reciprocals of each other:

†

T =1f

and

†

f =1T

SPEED (v) Average speed is always a ratio ofdistance to time,

†

v = d / t . In the case of wave speed,an appropriate distance would be wavelength, l . Thecorresponding time would then haveto be the period, T. So the wavespeed becomes:

†

v =lt

or

†

v = lf

SOUND WAVESIf a tree falls in the forest and

there’s no one there to hear it, does itmake a sound? It’s a commonquestion that usually evokes aphilosophical response. I could argueyes or no convincingly. You willtoo later on. Most people have avery strong opinion one way or theother. The problem is that theiropinion is usually not based on aclear understanding of what sound is.

I think one of the easiest waysto understand sound is to look atsomething that has a simplemechanism for making sound. Thinkabout how a tuning fork makessound. Striking one of the forkscauses you to immediately hear atone. The tuning fork begins to act

somewhat like a playground swing. The playgroundswing, the tuning fork, and most physical systemswill act to restore themselves if they are stressed fromtheir natural state. The “natural state” for the swing,is to hang straight down. If you push it or pull it andthen let go, it moves back towards the position ofhanging straight down. However, since it’s movingwhen it reaches that point, it actually overshoots and,in effect, stresses itself. This causes another attemptto restore itself and the movement continues back andforth until friction and air resistance have removed allthe original energy of the push or pull. The same istrue for the tuning fork. It’s just that the movement(amplitude) is so much smaller that you can’t visiblysee it. But if you touched the fork you could feel it.Indeed, every time the fork moves back and forth itsmacks the air in its way. That smack creates a smallcompression of air molecules that travels from thatpoint as a compressional wave. When it reaches yourear, it vibrates your eardrum with the same frequencyas the frequency of the motion of the tuning fork.You mentally process this vibration as a specifictone.


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A sound wave is nothing more than acompressional wave caused by vibrations . Next timeyou have a chance, gently feel the surface of a speakercone (see Figure 1.7). The vibrations you feel withyour fingers are the same vibrations disturbing theair. These vibrations eventually relay to your ears themessage that is being broadcast. So, if a tree falls inthe forest and there’s no one there to hear it, does itmake a sound? Well … yes, it will certainly causevibrations in the air and ground when it strikes theground. And … no, if there’s no one there tomentally translate the vibrations into tones, thenthere can be no true sound. You decide. Maybe it is aphilosophical question after all.

CHARACTERIZING SOUNDAll sound waves are compressional waves caused

by vibrations, but the music from a symphony variesconsiderably from both a baby’s cry and the whisperof a confidant. All sound waves can be characterizedby their speed, by their pitch, by their loudness,and by their quality or timbre.

The speed of sound is fastest in solids (almost6000 m/s in steel), slower in liquids (almost1500 m/s in water), and slowest in gases. Wenormally listen to sounds in air, so we’ll look at thespeed of sound in air most carefully. In air, soundtravels at:

†

v = 331 ms + 0.6 m / s

°CÊ

Ë Á

ˆ

¯ ˜ Temperature

The part to the right of the “+” sign is thetemperature factor. It shows that the speed of soundincreases by 0.6 m/s for every temperature increase of1°C. So, at 0° C, sound travels at 331 m/s (about 740

mph). But at room temperature (about 20°C)sound travels at:

†

v = 331 ms

+ 0.6 m / s°C

Ê

Ë Á

ˆ

¯ ˜ 20°C( )

=

†

343 ms

( This is the speed you shouldassume if no temperature is given).

It was one of aviation’s greatestaccomplishments when Chuck Yeager, onOct. 14, 1947, flew his X-1 jet at Mach1.06, exceeding the speed of sound by 6%.Regardless, this is a snail’s pace compared tothe speed of light. Sound travels through airat about a million times slower than light,which is the reason why we hear sound

echoes but don’t see light echoes. It’s also the reasonwe see the lightning before we hear the thunder. Thelightning-thunder effect is often noticed in bigstadiums. If you’re far away from a baseball playerwho’s up to bat, you can clearly see the ball hitbefore you hear the crack of the bat. You can considerthat the light recording the event reaches your eyesvirtually instantly. So if the sound takes half a secondmore time than the light, you’re half the distancesound travels in one second (165 meters) from thebatter. Next time you’re in a thunderstorm use thismethod to estimate how far away lightning isstriking. Click here for a demonstration of the effectof echoes.

Do you get it? (1.1)Explain why some people put their ears on railroadtracks in order to hear oncoming trains

Figure 1.7: The front of a speaker cone faces upwardwith several pieces of orange paper lying on top of i t .Sound is generated when an electric signal causes thespeaker cone to move in and out, pushing on the airand creating a compressional wave. The ear can detectthese waves. Here these vibrations can be seen as theycause the little bits of paper to dance on the surface o fthe speaker cone.


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Do you get it? (1.2)The echo of a ship's foghorn, reflected from the cliffon a nearby island, is heard 5.0 s after the horn issounded. How far away is the cliff?

The pitch of sound is the same as thefrequency of a sound wave. With no hearing losses ordefects, the ear can detect wave frequencies between 20Hz and 20,000 Hz. (Sounds below 20 Hz areclassified as subsonic; those over 20,000 Hz areultrasonic). However, many older people, loud concertattendees, and soldiers with live combat experiencelose their ability to hear higher frequencies. The goodnews is that the bulk of most conversation takesplace well below 2,000 Hz. The average frequencyrange of the human voice is 120 Hz to approximately1,100 Hz (although a baby’s shrill cry is generally2,000 - 3,000 Hz – which is close to the frequencyrange of greatest sensitivity … hmm, interesting).Even telephone frequencies are limited to below3,400 Hz. But the bad news is that the formation ofmany consonants is a complex combination of veryhigh frequency pitches. So to the person with highfrequency hearing loss, the words key, pee, and teasound the same. You find people with these hearinglosses either lip reading or understanding aconversation by the context as well as the actualrecognition of words. Neil Bauman, a hearing expertat www.hearinglosshelp.com , offers the followinginformation:

“Vowels are clustered around thefrequencies between about 300 and 750 Hz.Some consonants are very low frequency,such as j, z, and v – at about 250 Hz. Otherssuch as k, t, f, s, and th are high frequencysounds and occur between 3,000 and 8,000Hz. All of these consonants are voiceless onesand are produced by air hissing from aroundthe teeth. Most people, as they age, beginlosing their hearing at the highest frequenciesfirst and progress downwards. Thus, theabove consonant sounds are the first to belost. As a result, it is most difficult to

distinguish between similar sounding wordsthat use these letters. Furthermore, thevowels are generally loud (they use about95% of the voice energy to produce). Theconsonants are left with only 5% to go aroundfor all of them. But it is mostly the consonantsthat give speech its intelligibility. That is whymany older people will say, ‘I can hear peopletalking. I just can't understand what they aresaying.’”

One important concept in music is the octave –a doubling in frequency. For example, 40 Hz is oneoctave higher than 20 Hz. The ear is sensitive over afrequency range of about 10 octaves: 20 Hz Æ 40 HzÆ 80 Hz Æ 160 Hz Æ 320 Hz Æ 640 Hz Æ1,280 Hz Æ 2,560 Hz Æ 5,120 Hz Æ 10,240 Hz Æ20,480 Hz. And within that range it can discriminatebetween thousands of differences in sound frequency.Below about 1,000 Hz the Just NoticeableDifference (JND) in frequency is about 1 Hz (at theloudness at which most music is played), but thisrises sharply beyond 1,000 Hz. At 2,000 the JND isabout 2 Hz and at 4,000 Hz the JND is about 10 Hz.(A JND of 1 Hz at 500 Hz means that if you wereasked to listen alternately to tones of 500 Hz and501 Hz, the two could be distinguished as twodifferent frequencies, rather than the same). It isinteresting to compare the ear’s frequency perceptionto that of the eye. From red to violet, the frequency oflight less than doubles, meaning that the eye is onlysensitive over about one octave, and its ability todiscriminate between different colors is only about125. The ear is truly an amazing receptor, not onlyits frequency range, but also in its ability toaccommodate sounds with vastly different loudness.

The loudness of sound is related to theamplitude of the sound wave. Most people have somerecognition of the decibel (dB) scale. They might beable to tell you that 0 dB is the threshold of hearingand that the sound on the runway next to anaccelerating jet is about 140 dB. However, mostpeople don’t realize that the decibel scale is alogarithmic scale. This means that for every increaseof 10 dB the sound intensity increases by a factor often. So going from 60 dB to 70 dB is a ten-foldincrease, and 60 dB to 80 dB is a hundred-foldincrease. This is amazing to me. It means that we canhear sound intensities over 14 orders of magnitude.This means that the 140 dB jet on the runway has aloudness of 1014 times greater than threshold. 1014 is100,000,000,000,000 – that’s 100 trillion! It meansour ears can measure loudness over a phenomenallylarge range. Imagine having a measuring cup thatcould accurately measure both a teaspoon and 100trillion teaspoons (about 10 billion gallons). The earis an amazing receptor! However, our perception isskewed a bit. A ten-fold increase in loudness doesn’t

www.hearinglosshelp.com


11

sound ten times louder. It may only sound twice asloud. That’s why when cheering competitions aredone at school rallies, students are not very excited bythe measure of difference in loudness between afreshmen class (95 dB) and a senior class (105 dB).The difference is only 10 dB. It sounds perhaps twiceas loud, but it’s really 10 times louder. (Those lungsand confidence grow a lot in three years!)

Do you get it? (1.3)When a passenger at the airport moves from inside awaiting area to outside near an airplane the decibellevel goes from 85 dB to 115 dB. By what factor hasthe sound intensity actually gone up?

The quality of sound or timbre is thesubtlest of all its descriptors. A trumpet and a violincould play exactly the same note, with the same pitchand loudness, and if your eyes were closed you couldeasily distinguish between the two. The difference inthe sounds has to do with their quality or timbre.The existence of supplementary tones, combined withthe basic tones, doesn’t change the basic pitch, butgives a special “flavor” to the sound being produced.Sound quality is the characteristic that gives theidentity to the sound being produced.

DETAILS ABOUT DECIBELSIt was mentioned earlier that the sensitivity of

the human ear is phenomenally keen. The thresholdof hearing (what a young perfect ear could hear) is

†

1¥10-12Watts /meter 2 . This way of expressing soundwave amplitude is referred to as Sound Intensity (I).It is not to be confused with Sound Intensity Level(L), measured in decibels (dB). The reason whyloudness is routinely represented in decibels ratherthan

†

Watts /meter 2 is primarily because the earsdon’t hear linearly. That is, if the sound intensitydoubles, it doesn’t sound twice as loud. It doesn’treally sound twice as loud until the Sound Intensity isabout ten times greater. (This is a very roughapproximation that depends on the frequency of thesound as well as the reference intensity.) If the soundintensity were used to measure loudness, the scalewould have to span 14 orders of magnitude. That

means that if we defined the faintest sound as “1”, wewould have to use a scale that went up to100,000,000,000,000 (about the loudest sounds youever hear. The decibel scale is much more compact(0 dB – 140 dB for the same range) and it is moreclosely linked to our ears’ perception of loudness.You can think of the sound intensity as a physicalmeasure of sound wave amplitude and sound intensitylevel as its psychological measure.

The equation that relates sound intensity to soundintensity level is:

†

L = 10 log I 2

I1

L ≡ The number of decibels I2 is greater than I1I2 ≡ The higher sound intensity being comparedI1 ≡ The lower sound intensity being compared

Remember, I is measured in

†

Watts /meter 2 . It islike the raw power of the sound. The L in thisequation is what the decibel difference is betweenthese two. In normal use, I1 is the threshold ofhearing,

†

1¥10-12Watts /meter 2 . This means that thedecibel difference is with respect to the faintest soundthat can be heard. So when you hear that a busyintersection is 80 dB, or a whisper is 20 dB, or a classcheer is 105 dB, it always assumes that thecomparison is to the threshold of hearing, 0 dB. (“80dB” means 80 dB greater than threshold). Don’tassume that 0 dB is no sound or total silence. This issimply the faintest possible sound a human withperfect hearing can hear. Table 1.1 provides decibellevels for common sounds.

If you make I2 twice as large as I1, then

†

DL @ 3dB . If you make I2 ten times as large as I1,then

†

DL = 10dB . These are good reference numbers totuck away:

†

Double Sound Intensity @ +3dB

10 ¥ Sound Intensity = +10dB

Click here to listen to a sound intensity level reducedby 6 dB per step over ten steps. Click here to listento a sound intensity level reduced by 3 dB per stepover ten steps.


12

Source of soundSoundIntensityLevel (dB)

SoundIntensity

†

Wm2

Ê

Ë Á

ˆ

¯ ˜

Threshold of hearing 0

†

1¥10-12

Breathing 20

†

1¥10-10

Whispering 40

†

1¥10-8

Talking softly 60

†

1¥10-6

Loud conversation 80

†

1¥10-4

Yelling 100

†

1¥10-2

Loud Concert 120

†

1Jet takeoff 140

†

100

Table 1.1 Decibel levels for typical sounds

FREQUENCY RESPONSE OVER THEAUDIBLE RANGE

We hear lower frequencies as low pitches andhigher frequencies as high pitches. However, oursensitivity varies tremendously over the audiblerange. For example, a 50 Hz sound must be 43 dBbefore it is perceived to be as loud as a 4,000 Hzsound at 2 dB. (4,000 Hz is the approximatefrequency of greatest sensitivity for humans with nohearing loss.) In this case, we require the 50 Hz soundto have 13,000 times the actual intensity of the4,000 Hz sound in order to have the same perceivedintensity! Table 1.2 illustrates this phenomenon ofsound intensity level versus frequency. The lastcolumn puts the relative intensity of 4,000 Hzarbitrarily at 1 for easy comparison with sensitivity atother frequencies.

If you are using the CD version of thiscurriculum you can try the following demonstration,which illustrates the response of the human ear tofrequencies within the audible range Click here tocalibrate the sound on your computer and then clickhere for the demonstration.

Frequency(Hz)

SoundIntensityLevel (dB)

SoundIntensity

†

Wm 2( )

RelativeSoundIntensity

50 43

†

2.0 ¥10-8 13,000100 30

†

1.0 ¥10-9 625200 19

†

7.9 ¥10-11 49500 11

†

1.3¥10-11 8.11,000 10

†

1.0 ¥10-11 6.32,000 8

†

6.3¥10-12 3.93,000 3

†

2.0 ¥10-12 1.34,000 2

†

1.6 ¥10-12 15,000 7

†

5.0 ¥10-12 3.16,000 8

†

6.3¥10-12 3.97,000 11

†

1.3¥10-11 8.18,000 20

†

1.0 ¥10-10 62.59,000 22

†

1.6 ¥10-10 10014,000 31

†

1.3¥10-9 810

Table 1.2: Sound intensity and soundintensity level required to perceivesounds at different frequencies to beequally loud. A comparison of relativesound intensities arbitrarily assigns4,000 Hz the value of 1.

ExampleThe muffler on a car rusts out and the decibel levelincreases from 91 dB to 113 dB. How many timeslouder is the leaky muffler?

The “brute force” way to do this problem would be tostart by using the decibel equation to calculate thesound intensity both before and after the muffler rustsout. Then you could calculate the ratio of the two.It’s easier though to recognize that the decibeldifference is 22 dB and use that number in the decibelequation to find the ratio of the sound intensitiesdirectly:

†

L = 10 log I 2

I1

†

22dB = 10 log I 2

I1 fi 2.2 = log I 2

I1

Notice I just dropped the dB unit. It’s not a real unit,just kind of a placeholder unit so that we don’t haveto say, “The one sound is 22 more than the othersound.” and have a strange feeling of “22 … what?”

†

102.2 =I 2

I1 fi I 2 = 102.2 I1 fi

†

I2 = 158I1

So the muffler is actually 158 times louder thanbefore it rusted out.


13

Do you get it? (1.4)A solo trumpet in an orchestra produces a soundintensity level of 84 dB. Fifteen more trumpets jointhe first. How many decibels are produced?

Do you get it? (1.5)What would it mean for a sound to have a soundintensity level of -10 dB

Another factor that affects the intensity of thesound you hear is how close you are to the sound.Obviously a whisper, barely detected at one metercould not be heard across a football field. Here’s theway to think about it. The power of a particularsound goes out in all directions. At a meter awayfrom the source of sound, that power has to cover anarea equal to the area of a sphere (4πr2) with a radiusof one meter. That area is 4π m2. At two meters awaythe same power now covers an area of4π(2 m)2 = 16π m2, or four times as much area. Atthree meters away the same power now covers an areaof 4π(3 m)2 = 36π m2, or nine times as much area.So compared to the intensity at one meter, theintensity at two meters will be only one-quarter asmuch and the intensity at three meters only one-ninthas much. The sound intensity follows an inversesquare law, meaning that by whatever factor thedistance from the source of sound changes, theintensity will change by the square of the reciprocalof that factor.

ExampleIf the sound intensity of a screaming baby were

†

1¥10-2 Wm 2 at 2.5 m away, what would it be at

6.0 m away?The distance from the source of sound is greater by afactor of

†

6.02.5 = 2.4 . So the sound intensity is decreased

by

†

12.4 2( )

= 0.174 . The new sound intensity is:

†

1¥10-2 Wm 2( ) 0.174( ) =

†

1.74 ¥10-3 Wm 2

Another way to look at this is to first considerthat the total power output of a source of sound is itssound intensity in

†

Watts /meter 2 multiplied by thearea of the sphere that the sound has reached. So, forexample, the baby in the problem above creates asound intensity of

†

1¥10-2W /m2 at 2.5 m away.This means that the total power put out by the babyis:

†

Power = Intensity ¥ sphere area

†

fi P = 1¥10-2 Wm2

Ê

Ë Á

ˆ

¯ ˜ 4p 2.5m( )2[ ] = 0.785 W

Now let’s calculate the power output from theinformation at 6.0 m away:

†

P = 1.74 ¥10-3 Wm2

Ê

Ë Á

ˆ

¯ ˜ 4p (6.0m)2[ ] = .785 W

It’s the same of course, because the power outputdepends on the baby, not the position of the observer.This means we can always equate the power outputsthat are measured at different locations:

†

P1 = P2 fi I1( ) 4pr12( ) = I2( ) 4pr2

2( )

fi

†

I1r12 = I 2r2

2


14

Do you get it? (1.6)It’s a good idea to make sure that you keep thechronic sound you’re exposed to down under 80 dB. Ifyou were working 1.0 m from a machine that createda sound intensity level of 92 dB, how far would youneed move away to hear only 80 dB? (Hint: rememberto compare sound intensities and not sound intensitylevels.)


15

ACTIVITYORCHESTRAL SOUND

You can hear plenty of sound in a concert hallwhere an orchestra is playing. Each instrumentvibrates in its own particular way, producing theunique sound associated with it. The acoustical powercoming from these instruments originates with themusician. It is the energy of a finger thumping on apiano key and the energy of the puff of air across thereed of the clarinet and the energy of the slam ofcymbals against each other that causes theinstrument’s vibration. Most people are surprised tolearn that only about 1% of the power put into theinstrument by the musician actually leads to thesound wave coming from the instrument. But, as youknow, the ear is a phenomenally sensitive receptor ofacoustical power and needs very little power to bestimulated to perception. Indeed, the entire orchestraplaying at once would be loud to the ear, but actuallygenerate less power than a 75-watt light bulb! Anorchestra with 75 performers has an acoustic power ofabout 67 watts. To determine the sound intensity

level at 10 m, we could start by finding the soundintensity at 10 m:

†

I =67W

4p 10m( )2 = 0.056 Wm2 .

The sound intensity level would then be:

†

L = 10 log0.056 W

m2

1¥10-12 Wm2

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

= 107.5dB .

Table 1.3 lists the sound intensity level of variousinstruments in an orchestra as heard from 10 m away.You can use these decibel levels to answer thequestions throughout this activity.

OrchestralInstrument

Sound IntensityLevel (dB )

Violin (at its quietest) 34.8Clarinet 76.0Trumpet 83.9Cymbals 98.8Bass drum (at itsloudest)

103

Table 1.3: Sound intensity l eve l s(measured at 10 m away) for variousmusical instruments

1. How many clarinets would it take to equal the acoustic power of a pair of cymbals?


16

2. If you had to replace the total acoustic power of a bass drum with a single light bulb, what wattage wouldyou choose?

3. How far would you have to be from the violin (when it’s at its quietest) in order to barely detect its sound?

4. If the sound emerges from the trumpet at 0.5 m from the trumpet player’s ear, how many decibels does heexperience during his trumpet solo?


17

5. If the orchestra conductor wanted to produce the sound intensity of the entire orchestra, but use only violins(at their quietest) to produce the sound, how many would need to be used? How about if it were to be donewith bass drums (at their loudest?

6. The conductor has become concerned about the high decibel level and wants to make sure he does notexperience more than 100 dB. How far away from the orchestra must he stand?

7. If he doesn’t use the one bass drum in the orchestra, how far away does he need to stand?


18

WAVE INTERFERENCE

T’S INTRIGUING THAT at a lively partywith everyone talking at once, you can hearthe totality of the “noise” in the room andthen alternately distinguish and concentrate onthe conversation of one person. That isbecause of an interesting phenomenon of

waves called superposition. Wave superpositionoccurs when two or more waves move through thesame space in a wave medium together. There are twoimportant aspects of this wave superposition. One isthat each wave maint ains its own identity and isunaffected by any other waves in the same space . Thisis why you can pick out an individual conversationamong all the voices in the region of your ear. Thesecond aspect is that when two or more waves are inthe same medium, t he overall amplitude at any pointon the medium is simply the sum of the individualwave amplitudes at that point . Figure 1.8 illustratesboth of these aspects. In the top scene, two wave

pulses move toward each other. In the second scenethe two pulses have reached the same spot in themedium and the combined amplitude is just the sumof the two. In the last scene, the two wave pulsesmove away from each other, clearly unchanged bytheir meeting in the second scene.

When it comes to music, the idea of interferenceis exceptionally important. Musical sounds are oftenconstant frequencies held for a sustained period. Soundwaves interfere in the same way other waves, butwhen the sound waves are musical sounds (sustainedconstant pitches), the resulting superposition cansound either pleasant (consonant) or unpleasant(dissonant). Musical scales consist of notes (pitches),which when played together, sound consonant. We’lluse the idea of sound wave interference when webegin to look for ways to avoid dissonance in thebuilding of musical scales.

I

Figure 1.8: Wave superposition. Note in the middle drawing that the wave shape is simply thearithmetic sum of the amplitudes of each wave. Note also in the bottom drawing that the twowaves have the same shape and amplitude as they had before encountering each other.


19

ACTIVITYWAVE INTERFERENCE

In each of the following two cases, the wave pulses are moving toward each other. Assume that each wave pulsemoves one graph grid for each new graph. Draw the shape the medium would have in each of the blank graphsbelow.

20

“Music should strike fire from the heart of man, and bringtears from the eyes of woman.”– Ludwig Van Beethoven

CHAPTER 2RESONANCE, STANDING WAVES,

AND MUSICAL INSTRUMENTSHE NEXT STEP in pursuing the physics of music and musicalinstruments is to understand how physical systems can be made to vibratein frequencies that correspond to the notes of the musical scales. Butbefore the vibrations of physical systems can be understood, a diversionto the behavior of waves must be made once again. The phenomena of

resonance and standing waves occur in the structures of all musical instruments. Itis the means by which all musical instruments … make their music.

Why is it that eight-year-old boys have such anaversion to taking baths? I used to hate getting in thetub! It was perhaps my least favorite eight-year-oldactivity. The one part of the bathing ritual that madethe whole process at least tolerable was … makingwaves! If I sloshed back and forth in the water withmy body at just the right frequency, I could createwaves that reached near tidal wave amplitudes. Toolow or too high a frequency and the waves would dieout. But it was actually easy to find the rightfrequency. It just felt right. I thought I had discoveredsomething that no one else knew. Then, 20 yearslater, in the pool at a Holiday Inn in New Jersey witha group of other physics teachers, I knew that mydiscovery was not unique. Lined up on one side of thepool and with one group movement, we heaved ourbodies forward. A water wave pulse moved forwardand struck the other side of the pool. Then it reflectedand we waited as it traveled back toward us, continuedpast us, and reflected again on the wall of the poolright behind us. Without a word, as the crest of thedoubly reflected wave reached us, we heaved ourbodies again. Now the wave pulse had greateramplitude. We had added to it and when it struck theother side, it splashed up on the concrete, drawing theamused and, in some cases, irritated attention of theother guests sunning themselves poolside. As a childpushes her friend on a playground swing at just righttime in order to grow the amplitude of the swing’smotion, we continued to drive the water waveamplitude increasingly larger with our rhythmicmotion. Those who were irritated before were nowcurious, and those who were amused before were nowcheering. The crowd was pleased and intrigued by

something they knew in their gut. Most had probablyrocked back and forth on the seat of a car at astoplight with just the right frequency to get theentire car visibly rocking. Many had probably had theexperience of grabbing a sturdy street sign andpushing and pulling on it at just the right times toget the sign to shake wildly. They had all experiencedthis phenomenon of resonance.

To understand resonance, think back to thediscussion of the playground swing and tuning forkrestoring themselves to their natural states after beingstressed out of those states. Recall that as the swingmoves through the bottom of its motion, itovershoots this natural state. The tuning fork doestoo, and both of the two will oscillate back and forthpast this point (at a natural frequency particular to thesystem) until the original energy of whatever stressedthem is dissipated. You can keep the swing movingand even increase its amplitude by pushing on it atthis same natural frequency. The same could be doneto the tuning fork if it were driven with an audiospeaker producing the fork’s natural frequency.Resonance occurs whenever a physical system isdriven at its natural frequency. Most physical systemshave many natural frequencies. The street sign, forexample can be made to shake wildly back and forthwith a low fundamental frequency (first mode)or with a higher frequency of vibration, in thesecond mode. It’s easier to understand how musicalinstruments can be set into resonance by thinkingabout standing waves.

Standing waves occur whenever two waves withequal frequency and wavelength move through amedium so that the two perfectly reinforce each other.

T

SSSTTTAAANNNDDDIIINNNGGG WWWAAAVVVEEESSS

21

In order for this to occur the length of the mediummust be equal to some integer multiple of half thewavelength of the waves. Usually, one of the twowaves is the reflection of the other. When theyreinforce each other, it looks like the energy isstanding in specific locations on the wave medium,hence the name standing waves (see figure 2.1).There are parts of the wave medium that remainmotionless (the nodes) and parts where the wavemedium moves back and forth between maximumpositive and maximum negative amplitude (theantinodes).

Standing waves can occur in all wave mediumsand can be said to be present during any resonance.Perhaps you’ve heard someone in a restaurant rubbinga finger around the rim of a wineglass, causing it tosing. The “singing” is caused by a standing wave inthe glass that grows in amplitude until the pulsesagainst the air become audible. Soldiers are told to“break step march” when moving across small bridgesbecause the frequency of their march may be the

natural frequency of the bridge, creating and thenreinforcing a standing wave in the bridge and causingthe same kind of resonance as in the singing wineglass. A standing wave in a flat metal plate can becreated by driving it at one of its natural frequencies atthe center of the plate. Sand poured onto the platewill be unaffected by and collect at the nodes of thestanding wave, whereas sand at the antinodes will bebounced off, revealing an image of the standing wave(see figure 2.2).

Resonance caused the destruction of the TacomaNarrows Bridge on November 7, 1940 (see Figure2.3). Vortices created around its deck by 35 - 40 mphwinds resulted in a standing wave in the bridge deck.When its amplitude reached five feet, at about 10 am,the bridge was forced closed. The amplitude of motioneventually reached 28 feet. Most of the remains ofthis bridge lie at the bottom of the Narrows, makingit the largest man-made structure ever lost to sea andthe largest man-made reef.

Figure 2.1: A time-lapse view of a standing wave showing the nodes and antinodes present. It i sso named because it appears that the energy in the wave stands in certain places (the antinodes).Standing waves are formed when two waves of equal frequency and wavelength move through amedium and perfectly reinforce each other. In order for this to occur the length of the mediummust be equal to some integer multiple of half the wavelength of the waves. In the case of th i sstanding wave, the medium is two wavelengths long (4 half wavelengths).

ANTINODES

NODES


22

Figure 2.2: A flat square metal plate is driven in four of its resonance modes. These photographsshow each of the resulting complicated two-dimensional standing wave patterns. Sand poured ontothe top of the plate bounces off the antinodes, but settles into the nodes, allowing the standingwave to be viewed. The head of a drum, when beaten, and the body of a guitar, when played,exhibit similar behaviors.


23

The Tacoma Narrows Bridge on November 7 ,1940. Workers on the construction of thebridge had referred to it as “Galloping Gertie”because of the unusual oscillations that wereoften present. Note nodes at the towers and i nthe center of the deck. This is the secondmode.

Vortices around its deck caused by winds o f35 – 40 mph caused the bridge to begin risingand falling up to 5 feet, forcing the bridge t obe closed at about 10 am. The amplitude o fmotion eventually reached 28 feet.

The bridge had a secondary standing wave i nthe first mode with its one node on thecenterline of the bridge. The man in the photowisely walked along the node. Although thebridge was heaving wildly, the amplitude at thenode was zero, making it easy to navigate.

At 10:30 am the oscillations had f inallycaused the center span floor panel to fal lfrom the bridge. The rest of the breakupoccurred over the next 40 minutes.

Figure 2.3: The Tacoma Narrows Bridge collapse (used with permission from University of WashingtonSpecial Collection Manuscripts).

http://www.lib.washington.edu/specialcoll/tnb/


24

Since the collapse of the Tacoma NarrowsBridge, engineers have been especially conscious ofthe dangers of resonance caused by wind orearthquakes. Paul Doherty, a physicist at theExploratorium in San Francisco, had this to sayabout the engineering considerations now made whenbuilding or retrofitting large structures subject todestruction by resonance:

“The real world natural frequency of largeobjects such as skyscrapers and bridges isdetermined via remote sensoraccelerometers. Buildings like the Trans-America Pyramid and the Golden GateBridge have remote accelerometers attachedto various parts of the structure. When windor an earthquake ‘excites’ the building theaccelerometers can detect the ‘ringing’(resonant oscillation) of the structure. TheGolden Gate Bridge was ‘detuned’ by havingmass added at various points so that a standingwave of a particular frequency would affectonly a small portion of the bridge. TheTacoma Narrows Bridge had the resonanceextend the entire length of the span and onlyhad a single traffic deck which could flex. TheGolden Gate Bridge was modified after theNarrows Bridge collapse. The underside ofthe deck had stiffeners added to dampentorsion of the roadbed and energy-absorbingstruts were incorporated. Then massadditions broke up the ability of the standingwave to travel across the main cablesbecause various sections were tuned todifferent oscillation frequencies. This is whysitting in your car waiting to pay the toll youcan feel your car move up a down when alarge truck goes by but the next large truckmay not give you the same movement. Thatfirst truck traveling at just the right speed mayexcite the section you are on while a truck ofdifferent mass or one not traveling the samespeed may not affect it much. As for thehorizontal motion caused by the wind, thesame differentiation of mass elements underthe roadbed keeps the whole bridge from goingresonant with Æolian oscillations.Just envision the classic Physics experiment

where different length pendulums are hungfrom a common horizontal support. Measuredperiodic moving of the support will make only

one pendulum swing depending on the periodof the applied motion. If all the pendulums hadthe same oscillation you could get quite amotion going with a small correctly timedforce application. Bridges and buildings nowrely on irregular distributions of mass to helpkeep the whole structure from moving as aunit that would result in destructive failure.Note also on the Golden Gate the secondarysuspension cable ‘keepers’ (spacers) arelocated at slightly irregular intervals todetune them. As current structuralengineering progresses more modifications ofthe bridge will be done. The new super bridgein Japan has hydraulically movable weightsthat can act as active dampeners. What isearthquake (or wind) safe today will besubstandard in the future.”

The collapse of the Tacoma Narrows Bridge isperhaps the most spectacular example of thedestruction caused by resonance, but everyday thereare boys and girls who grab hold of street sign polesand shake them gently – intuitively – at just the rightfrequency to get them violently swaying back andforth. Moreover, as mentioned previously, all musicalinstruments make their music by means of standingwaves.

To create its sound, the physical structure ofmusical instruments is set into a standing wavecondition. The connection with resonance can be seenwith a trumpet, for example. The buzzing lips of thetrumpet player create a sound wave by allowing aburst of air into the trumpet. This burst is largelyreflected back when it reaches the end of the trumpet.If the trumpet player removes his lips, the soundwave naturally reflects back and forth between thebeginning and the end of the trumpet, quickly dyingout as it leaks from each end. However, if the player’slips stay in contact with the mouthpiece, the reflectedburst of air can be reinforced with a new burst of airfrom the player’s lips. The process continues,creating a standing wave of growing amplitude.Eventually the amplitude reaches the point whereeven the small portion of the standing wave thatescapes the trumpet becomes audible. It is resonancebecause the trumpet player adds a “kick of air” at theprecise frequency (and therefore also the samewavelength) of the already present standing soundwave.

MMMUUUSSSIIICCCAAALLL IIINNNSSSTTTRRRUUUMMMEEENNNTTTSSS

25

INTRODUCTION TO MUSICAL INSTRUMENTS

F YOU DON’T play a musical instrument,it’s humbling to pick one up and try to createsomething that resembles a melody or tune. Itgives you a true appreciation for the musicianwho seems to become one with hisinstrument. But you certainly don’t have to be

a musician to understand the physics of music or thephysics of musical instruments. All musicians createmusic by making standing waves in their instrument.The guitar player makes standing waves in the stringsof the guitar and the drummer does the same in theskin of the drumhead. The trumpet blower and fluteplayer both create standing waves in the column of air

within their instruments. Most kids have done thesame thing, producing a tone as they blow over thetop of a bottle. If you understand standing waves youcan understand the physics of musical instruments.We’ll investigate three classes of instruments:

• Chordophones (strings)• Aerophones (open and closed pipes)• Idiophones (vibrating rigid bars and pipes)

Most musical instruments will fit into these threecategories. But to fully grasp the physics of thestanding waves within musical instruments andcorresponding music produced, an understanding ofwave impedance is necessary.

IFigure 2.4:Chordophones aremusical instrumentsin which a standingwave is initiallycreated in the stringsof the instruments.Guitars, violins, andpianos fall into thiscategory.

Figure 2.5:Aerophones aremusical instrumentsin which a standingwave is initiallycreated in the columnof air within theinstruments.Trumpets, flutes, andoboes fall into thiscategory.

Figure 2.6:Idiophones aremusical instrumentsin which a standingwave is initiallycreated in the physicalstructure of theinstruments.Xylophones,marimbas, and chimesfall into thiscategory.

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MUSICAL INSTRUMENTS ANDWAVE IMPEDANCE

The more rigid a medium is the less effect theforce of a wave will have on deforming the shape ofthe medium. Impedance is a measure of how muchforce must be applied to a medium by a wave in orderto move the medium by a given amount. When itcomes to standing waves in the body of a musicalinstrument, the most important aspect of impedancechanges is that they always cause reflections.Whenever a wave encounters a change in impedance,some or most of it will be reflected. This is easy tosee in the strings of a guitar. As a wave moves alongthe string and encounters the nut or bridge of theguitar, this new medium is much more rigid than thestring and the change in impedance causes most of thewave to be reflected right back down the string (goodthing, because the reflected wave is needed to createand sustain the standing wave condition). It’s harderto see however when you consider the standing wave

of air moving through the inside of the tuba. How isthis wave reflected when it encounters the end of thetuba? The answer is that wave reflection occursregardless of how big the impedance change is orwhether the new impedance is greater or less. Thepercentage of reflection depends on how big thechange in impedance is. The bigger the impedancechange, the more reflection will occur. When thewave inside the tuba reaches the end, it is not asconstricted – less rigid, so to speak. This slightchange in impedance is enough to cause a significantportion of the wave to reflect back into the tuba andthus participate and influence the continuedproduction of the standing wave. The part of the wavethat is not reflected is important too. The transmittedportion of the wave is the part that constitutes thesound produced by the musical instrument. Figure 2.7illustrates what happens when a wave encountersvarious changes in impedance in the medium throughwhich it is moving.

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Figure 2.7: Wave pulse encountering medium with different impedanceA. Much greater impedance fi Inverted, large reflection.B. Slightly greater impedance fi Inverted, small reflection.C. Much smaller impedance fi Upright, large reflection.D. Slightly smaller impedance fi Upright, small reflection.

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“After silence that which comes nearest to expressing theinexpressible is music.”Aldous Huxley

CHAPTER 3MODES, OVERTONES, AND HARMONICS

HEN A DESIRED note (tone) is played on a musical instrument,there are always additional tones produced at the same time. Tounderstand the rationale for the development of consonant musicalscales, we must first discuss the different modes of vibrationproduced by a musical instrument when it is being played. It doesn’t

matter whether it is the plucked string of a guitar, or the thumped key on a piano,or the blown note on a flute, the sound produced by any musical instrument is farmore complex than the single frequency of the desired note. The frequency of thedesired note is known as the fundamental frequency, which is caused by the firstmode of vibration, but many higher modes of vibration always naturally occursimultaneously.

Higher modes are simply alternate, higherfrequency vibrations of the musical instrumentmedium. The frequencies of these higher modes areknown as overtones. So the second mode producesthe first overtone, the third mode produces the secondovertone, and so on. In percussion instruments, (likexylophones and marimbas) the overtones are notrelated to the fundamental frequency in a simple way,but in other instruments (like stringed and windinstruments) the overtones are related to thefundamental frequency “harmonically.”

When a musical instrument’s overtones areharmonic, there is a very simple relationship betweenthem and the fundamental frequency. Harmonics areovertones that happen to be simple integer multiplesof the fundamental frequency. So, for example, if astring is plucked and it produces a frequency of110 Hz, multiples of that 110 Hz will also occur atthe same time: 220 Hz, 330 Hz, 440 Hz, etc will allbe present, although not all with the same intensity.A musical instrument’s fundamental frequency and allof its overtones combine to produce that instrument’ssound spectrum or power spectrum. Figure 3.1shows the sound spectrum from a flute playing thenote G4. The vertical line at the first peak indicates itsfrequency is just below 400 Hz. In the musical scaleused for this flute, G4 has a frequency of 392 Hz.Thus, this first peak is the desired G4 pitch. Thesecond and third peaks are also identified with verticallines and have frequencies of about 780 Hz and1,170 Hz (approximately double and triple the lowestfrequency of 392 Hz). This lowest frequency occurring

when the G4 note is played on the flute is caused bythe first mode of vibration. It is the fundamentalfrequency. The next two peaks are the simultaneouslypresent frequencies caused by the second and thirdmodes of vibration. That makes them the first twoovertones. Since these two frequencies are integermultiples of the lowest frequency, they are harmonicovertones.

When the frequencies of the overtones areharmonic the fundamental frequency and all theovertones can be classified as some order of harmonic.The fundamental frequency becomes the firstharmonic, the first overtone becomes the secondharmonic, the second overtone becomes the thirdharmonic, and so on. (This is usually confusing formost people at first. For a summary, see Table 3.1).Looking at the Figure 3.1, it is easy to see that thereare several other peaks that appear to be integermultiples of the fundamental frequency. These are allindeed higher harmonics. For this flute, the thirdharmonic is just about as prominent as thefundamental. The second harmonic is a bit less thaneither the first or the third. The fourth and higherharmonics are all less prominent and generally followa pattern of less prominence for higher harmonics(with the exception of the eighth and ninthharmonics). However, only the flute has thisparticular “spectrum of sound.” Other instrumentsplaying G4 would have overtones present in differentportions. You could say that the sound spectrum ofan instrument is its “acoustical fingerprint.”

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Mode ofvibration

Frequency name (forany type ofovertone)

Frequency name(for harmonicovertones)

First Fundamental First harmonicSecond First overtone Second harmonicThird Second overtone Third harmonicFourth Third overtone Fourth harmonic

Table 3.1: Names given to the frequenciesof different modes of vibration. Theovertones are “harmonic” if they areinteger multiples of the fundamentalfrequency.

TIMBRE, THE QUALITY OF SOUNDIf your eyes were closed, it would still be easy to

distinguish between a flute and a piano, even if bothwere playing the note G4. The difference in intensitiesof the various overtones produced gives eachinstrument a characteristic sound quality ortimbre (“tam-brrr”), even when they play the samenote. This ability to distinguish is true even betweenmusical instruments that are quite similar, like theclarinet and an oboe (both wind instruments usingphysical reeds). The contribution of total soundarising from the overtones varies from instrument toinstrument, from note to note on the same instrumentand even on the same note (if the player produces thatnote differently by blowing a bit harder, for example).In some cases, the power due to the overtones is lessprominent and the timbre has a very pure sound to it,

Figure 3.1: The “sound spectrum” of a flute shows the frequencies present when the G4 note i splayed. The first designated peak is the desired frequency of 392 Hz (the “fundamentalfrequency”). The next two designated peaks are the first and second “overtones.” Since these andall higher overtones are integer multiples of the fundamental frequency, they are “harmonic.”(Used with permission. This and other sound spectra can be found athttp://www.phys.unsw.edu.au/music/flute/modernB/G4.html )

392 H z

784 H z

1176 H z

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http://www.phys.unsw.edu.au/music/flute/modernB/G4.html


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like in the flute or violin. In other instruments, likethe bassoon and the bagpipes, the overtonescontribute much more significantly to the powerspectrum, giving the timbre a more complex sound.

A CLOSER LOOK AT THEPRODUCTION OF OVERTONES

To begin to understand the reasons for theexistence of overtones, consider the guitar and how itcan be played. A guitar string is bound at both ends.If it vibrates in a standing wave condition, thosebound ends will necessarily have to be nodes. Whenplucked, the string can vibrate in any standing wavecondition in which the ends of the string are nodesand the length of the string is some integer number ofhalf wavelengths of the mode’s wavelength (seeFigure 3.2). To understand how the intensities ofthese modes might vary, consider plucking the stringat its center. Plucking it at its center will favor anantinode at that point on the string. But all the oddmodes of the vibrating string have antinodes at thecenter of the string, so all these modes will bestimulated. However, since the even modes all havenodes at the center of the string, these will generallybe weak or absent when the string is plucked at thecenter, forcing this point to be moving (see Figure3.2). Therefore, you would expect the guitar’s soundquality to change as its strings are plucked at differentlocations.

An additional (but more subtle) explanation forthe existence and intensity of overtones has to dowith how closely the waveform produced by themusical instrument compares to a simple sine wave.We’ve already discussed what happens when a flexiblephysical system is forced from its position of greateststability (like when a pendulum is moved from itsrest position or when the tine of a garden rake ispulled back and then released). A force will occur thatattempts to restore the system to its former state. Inthe case of the pendulum, gravity directs thependulum back to its rest position. Even though theforce disappears when the pendulum reaches this restposition, its momentum causes it to overshoot. Nowa new force, acting in the opposite direction, againattempts to direct the pendulum back to its restposition. Again, it will overshoot. If it weren’t forsmall amounts of friction and some air resistance,this back and forth motion would continue forever. Inits simplest form, the amplitude vs. time graph ofthis motion would be a sine wave (see Figure 3.3).Any motion that produces this type of graph isknown as simple harmonic motion.

Not all oscillatory motions are simple harmonicmotion though. In the case of a musical instrument,it’s not generally possible to cause a physicalcomponent of the instrument (like a string or reed) tovibrate as simply as true simple harmonic motion.

Instead, the oscillatory motion will be a morecomplicated repeating waveform (Figure 3.4). Here isthe “big idea.” This more complicated waveform canalways be created by adding in various intensities ofwaves that are integer multiples of the fundamentalfrequency. Consider a hypothetical thumb piano tinevibrating at 523 Hz. The tine physically cannotproduce true simple harmonic motion when it isplucked. However, it is possible to combine a1,046 Hz waveform and a 1,569 Hz waveform withthe fundamental 523 Hz waveform, to create thewaveform produced by the actual vibrating tine. (The1,046 Hz and 1,569 Hz frequencies are integermultiples of the fundamental 523 Hz frequency and donot necessarily have the same intensity as thefundamental frequency). The total sound produced bythe tine would then have the fundamental frequency of523 Hz as well as its first two overtones.

mode 1

mode 2

mode 3

mode 4

mode 5

Figure 3.2: The first five modes of avibrating string. Each of these (and all otherhigher modes) meets the following criteria:The ends of the string are nodes and thelength of the string is some integer numberof half wavelengths of the mode’swavelength. The red dashed line indicatesthe center of the string. Note that if thecenter of the string is plucked, forcing th i sspot to move, modes 2 and 4 (and all othereven modes) will be eliminated since theyrequire a node at this point. However, modes1, 3, and 5 (and all other odd modes) w i l lall be stimulated since they have antinodesat the center of the string.

String center


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The remainder of this section providesvisualization for the timbre of a variety of musicalinstruments. Each figure provides two graphs for aparticular instrument. The first graph is a histogramof the power spectrum for that musical instrument.The first bar on the left is the power of thefundamental frequency, followed by the overtones. Becareful when comparing the relative strength of theovertones to the fundamental and to each other. Thescale of the vertical axis is logarithmic. The top ofthe graph represents 100% of the acoustic power ofthe instrument and the bottom of the graph represents

80 dB lower than full power (

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10-8 less power). Thesecond graph is a superposition of the fundamentalwave together with all the overtone waves. Threecycles are shown in each case. Notice that wheregraphs for the same instrument are shown that thisacoustical fingerprint varies somewhat from note tonote – the bassoon almost sounds like a differentinstrument when it goes from a very low note to avery high note.

You can click on the name of the instrument inthe caption to hear the sound that produced bothgraphs.

Figure 3.5: E5 played on the VIOLIN . Note the dominance of the fundamental frequency. Themost powerful overtone is the 1st, but only slightly more than 10% of the total power. Thehigher overtones produce much less power. Of the higher overtones, only the 2nd overtoneproduces more than 1% of the total power. The result is a very simple waveform and thecharacteristically pure sound associated with the violin.

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Figure 3.3: The sine wave pictured hereis the displacement vs. time graph of themotion of a physical system undergoingsimple harmonic motion.

Figure 3.4: The complicated waveformshown here is typical of that produced by amusical instrument. This waveform can beproduced by combining the waveform of thefundamental frequency with waveformshaving frequencies that are integermultiples of the fundamental frequency.


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Figure 3.6: F3 played on the CLARINET . Note that although all harmonics are present, the1st, 3rd, 5th, and 7th harmonics strongly dominate. They are very nearly equal to each other i npower. This strong presence of the lower odd harmonics gives evidence of the closed pipenature of the clarinet. The presence of these four harmonics in equal proportion (as well asthe relatively strong presence of the 8 th harmonic) creates a very complex waveform and theclarinet’s characteristically “woody” sound

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Figure 3.7: B4 played on the BAGPIPE . There is a strong prominence of odd harmonics(through the 7 th). This is a hint about the closed pipe nature of the bagpipe’s soundproduction. The 3rd harmonic ( 2nd overtone) is especially strong, exceeding all other modefrequencies in power. Not only is the third harmonic more powerful than the fundamentalfrequency, it appears to have more power than all the other modes combined. This gives thebagpipe its characteristic harshness.

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Figure 3.8: F2 played on the BASSOON . Note that the fundamental frequency produces farless than 10% of the total power of this low note. It’s also interesting to note that the firstfive overtones not only produce more power than the fundamental, but they eachindividually produce more than 10% of the total power of this note. This weak fundamentalcombined with the dominance of the first five harmonics creates a very complex waveformand the foghorn-like sound of the bassoon when it produces this note.

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Figure 3.9: B4 played on the BASSOON . The power spectrum is much different for this highnote played on the same bassoon. Over two octaves higher than the note producing thepower spectrum in Figure 3.8, the first two harmonics dominate, producing almost all thepower. The third and fourth harmonics produce far less than 10% of the total power and a l lthe remaining higher harmonics produce less than 1% combined. The result is a s implewaveform with an almost flute-like sound.

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Figure 3.11: D5 played on the TRUMPET . The fundamental frequency and the first twoovertones dominate the power spectrum, with all three contributing over 10% of the totalpower. The next five harmonics all contribute above 1% of the total power. However, evenwith the much greater prominence of the 3rd through 7 th overtones, when compared to thebassoon’s B4 note, the waveform graphs of the trumpet D5 and the bassoon B4 aresurprisingly prominent.

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Figure 3.10: Bb3 played on the TRUMPET . Like the F2 note on the bassoon, the fundamental

frequency here produces less than 10% of the total power of this low note. Also like thebassoon’s F2, the first five overtones here not only produce more power than thefundamental, but they each individually produce more than 10% of the total power of th i snote. There are significant similarities between the two power spectra, but the seemingsubtle differences on the power spectra graph become far more obvious when eithercomparing the two waveform graphs or listening to the notes.

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ACTIVITYINTERPRETING MUSICAL INSTRUMENT POWER SPECTRA

In this activity, you will view the power spectrum graphs for a pure tone as well as three musical instruments: aflute (open end), a panpipe (closed end), and a saxophone. The power spectra are presented randomly below. For eachpower spectrum you will be asked to:

• Match the power spectrum to one of four waveform graphs.• Make a general statement about the timbre of the sound (pure, complex, harsh, clarinet-like, etc.).• Match the power spectrum to the correct musical instrument or to the pure tone.

1. a. Which of the waveforms shown at the end ofthis activity corresponds to this powerspectrum? Explain.

b. Describe the timbre of this sound. Explain.

c. Predict which, if any, of the musical instruments listed above produced this power spectrum. Explain.

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4. a. Which of the waveforms shown at the end ofthis activity corresponds to this power spectrum?Explain.



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BEGINNING TO THINK ABOUT MUSICAL SCALES

OW THAT YOU know a bit aboutwaves and even more so about thesound waves that emerge frommusical instruments, you’re ready tostart thinking about musical scales.In some cultures, the music is made

primarily with percussion instruments and certaincommon frequencies are far less important thanrhythm. But in most cultures, musical instrumentsthat produce sustained frequencies are more prominentthan percussion instruments. And, for theseinstruments to be played simultaneously, certainagreed upon frequencies must be adopted. How didmusicians come up with widely agreed upon commonfrequencies used in musical scales? How would youdo it? Take a moment to consider that question beforeyou move on. What aspects of the chosen frequencieswould be important in the development of your scale?

Use the space below to make some proposals forwhat you believe would be important as you begin tobuild a musical scale.

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Now show your proposals to a neighbor and have that person provide feedback concerning your ideas. Summarizethis feedback below.


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BEATS

NE WAY TO tune a guitar is tocompare the frequencies of variouscombinations of pairs of the guitarstrings. It’s easy to tell whether twostrings are different in frequency by aslittle as a fraction of one hertz. And,

it’s not necessary to have a great ear for recognizing aparticular frquency. This type of tuning is dependenton the type of wave intereference that producesbeats. In the same way as two slinky waves willinterfere with each other, either reinforcing each otherin constructive interference or subtracting fromeach other in destructive interference, soundwaves moving through the air will do the same. Withsound waves from two sources (like two guitarstrings), constructive interference would correspond toa sound louder than the two individually anddestructive interference would correspond to a quietersound than either, or perhaps … absolute silence (ifthe amplitudes of the two were the same). Figure3.12 attempts to illustrate this. Two tuning forkswith slightly different frequencies are sounded at thesame time in the same area as a listening ear. Theclosely packed black dots in front of the tuning forksrepresent compressions in the air caused by thevibrations of the forks. These compressions havehigher than average air pressure. The loosely packedhollow dots in front of the tuning forks represent“anti-compressions” or rarefactions in the air, alsocaused by the vibrations of the forks. Theserarefactions have lower than average air pressure.

When the top tuning fork has produced 17compressons, the bottom has produced 15. If the timeincrement for this to occur were half a second, aperson listening to one or the other would hear afrequency of 34 Hz from the top tuning fork and30 Hz from the bottom tuning fork. But listening tothe sound from both tuning forks at the same time,the person would hear the combination of the twosound waves, that is, their interference. Notice thatthere are regions in space where compressions fromboth tuning forks combine to produce an especiallytight compression, representing a sound amplitudemaximum – it’s especially loud there. There are otherlocations where a compression from one tuning forkis interfering with a rarefaction from the other tuningfork. Here the compression and the rarefactioncombine to produce normal air pressure – no sound atall. At locations in between these two extremes ininterference, the sound amplitude is either growinglouder or quieter. You can see in the bottom of thediagram that there is a rhythm of sound intensityfrom loud to silent to loud, over and over. This is thephenomenon of beats. In the half second that thediagram portrays there are two full cycles of thisbeating, giving a beat frequency of 4 Hz. The earwould perceive the average frequency of these twotuning forks, 32 Hz, getting louder and quieter fourtimes per second (note that this beat frequencyis the difference in the frequency of thetwo tuning forks) .

O

Si lent Si lentLoud Loud

Compression Rarefaction

Figure 3.12: Beats. The two tuning forks have slightly different frequencies. This causes the soundwaves produced by each one to interfere both constructively and destructively at various points. Theconstructive interference causes a rise in the intensity of the sound and the destructive interference causessilence. This pattern is repeated over and over causing the phenomenon of beats.


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The beat frequency decreases as the two frequenciescausing the beats get closer to each other. Finally, thebeat frequency disappears when the two frequencies areidentical. This is why it is so easy to tune two guitarstrings to the same frequency. You simply strumboth strings together and then tune one until no beatfrequency is heard. At that point, the two frequenciesare the same. Click on the following links to heardemonstrations of beats caused by variouscombinations of audio frequencies:

300 Hz and 301 Hz300 Hz and 302 Hz300 Hz and 305 Hz300 Hz and 310 Hz

CRITICAL BANDS ANDDISSONANCE

In addition to considering the issue of beats whenchoosing frequencies for a musical scale, there is alsothe issue of critical bands. When sound enters theear, it ultimately causes vibrations on the basilarmembrane within the inner ear. Different frequenciesof sound cause different regions of the basilarmembrane and its fine hairs to vibrate. This is howthe brain discriminates between various frequencies.However, if two frequencies are close together, thereis an overlap of response on the basilar membrane – alarge fraction of total hairs set into vibration arecaused by both frequencies (see figure 3.13).

When the frequencies are nearly the same, they can’tbe distinguished as separate frequencies. Instead anaverage frequency is heard, as well as the beatsdiscussed above. If the two frequencies were 440 Hzand 450 Hz, you would hear 445 Hz beating at 10times per second. If the lower frequency were kept at440 Hz and the higher one were raised slowly, therewould come a point where the two frequencies werestill indistinguishable, but the beat frequency wouldbe too high to make out. There would just be aroughness to the total sound. This dissonance wouldcontinue until finally the higher frequency wouldbecome distinguishable from the lower. At this point,further raising the higher frequency would cause lessand less dissonance. When two frequencies are close

enough to cause the beating and roughness describedabove, they are said to be within a critical band on thebasilar membrane. For much of the audible range, thecritical band around some “central frequency” will bestimulated by frequencies within about 15% of thatcentral frequency.

SUMMARYWhen two tones with similar frequencies, f1 and

f2, are sounded in the same space, their interferencewill cause beats, the increase and decrease of perceivedsound intensity – a throbbing sensation. Theperceived frequency is the average of the twofrequencies:

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f perceived =f1 + f2

2The beat frequency (rate of the throbbing) is thedifference of the two frequencies:

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fbeat = f1 - f2

ExampleIf two people stood near each other and whistled,one with a frequency of 204 Hz and the other with afrequency of 214 Hz, what would people near themhear?

The observers would hear a pitch that was the averageof the two frequencies, but beating at a frequencyequal to the difference of the two frequencies:

Given: f1 = 204 Hzf2 = 214 Hz

Find: The perceived and beat frequencies

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f perceived =f1 + f2

2=

204Hz + 214Hz2

= 209Hz

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fbeat = f1 - f2 = 204Hz - 214Hz = 10Hz

When considering the frequencies to use for amusical scale, critical bands should be considered.Two frequencies that stimulate areas within the samecritical band on the basilar membrane will produceeither noticeable beats (if they are similar enough toeach other) or other dissonance undesireable in music.

The observers would hear the frequency of209 Hz getting louder and softer 10 times persecond.

440 Hz 880 Hz 440 Hz 450 Hz

Figure 3.13: The closer two frequencies areto each other, the more overlap there w i l lbe in the response of the basilar membranewithin the inner ear.

David Lapp

Beats Movie

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“The notes I handle no better than many pianists. But thepauses between the notes – ah, that is where the art resides.”– Arthur Schnabel

CHAPTER 4MUSICAL SCALES

HE PHOTOGRAPH OF the piano keyboard in Figure 4.1 shows severalof the white keys with the letter C above them. The keys are all equallyspaced and striking all keys in succession to the right from any one C keyto the next C key would play the familiar sound of “Do – Re – Me – Fa– So – La – Ti – Do.” That means every C key sounds like “Doe” in the

familiar singing of the musical scale. T

Figure 4.1: The repeating nature of the musical scale is illustrated on a piano keyboard.

C4 D4 E4 F4 G4 A4 B4 C5

261.63 Hz 293.66 Hz 329.63 Hz 349.23 Hz 392.00 Hz 440.00 Hz 493.88 Hz 523.25 Hz

C1 C2 C 3 C4 C5 C6 C7 C8

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The reason for the equally spaced positioning ofthe C’s is because there is a repetition that takes placeas you move across the piano keyboard – all the Ckeys sounds alike, like “Do.” And every keyimmediately to the right of a C key sounds like“Re.” So if you want to understand the musicalscale, you really only need to look at the relationshipbetween the keys in one of these groupings from C toC (or from any key until the next repeated key).

The particular frequencies chosen for the musicalscale are not random. Many keys sound particularlygood when played together at the same time. To helpshow why certain pitches or tones sound goodtogether and why they are chosen to be in the scale,one grouping of notes has been magnified from thephotograph of the piano keyboard. The seven whitekeys (from C4 to B4 represent the major diatonicscale). The black keys are intermediate tones. Forexample, the black key in between D4 and E4 ishigher frequency (sharper) than D4 and lower frequency(flatter) than E4. This note is therefore knownsynonymously as either “D4 sharp” (D4

#) or “E4 flat”(E4

b). Including these five sharps or flats with theother seven notes gives the full chromatic scale.The frequency of the sound produced by each of thekeys in this chromatic scale (as well as for the firstwhite key in the next range) is shown on that key. Itwill help to refer back to this magnified portion ofthe keyboard as you consider the development of themusical scale.

CONSONANCE AND SMALLINTEGER FREQUENCY RATIOS

“All art constantly aspires towardsthe condition of music.”– Walter Pater

The opposite of dissonance is consonance –pleasant sounding combinations of frequencies.Earlier the simultaneous sounding of a 430 Hz tuningfork with a 440 Hz tuning fork was discussed. If the430 Hz tuning fork were replaced with an 880 Hztuning fork, you would hear excellent consonance.This especially pleasant sounding combination comesfrom the fact that every crest of the sound waveproduced by the 440 Hz tuning fork would be in stepwith every other crest ofthe sound wave producedby the 880 Hz tuningfork. So doubling thefrequency of one tonealways produces a secondtone that sounds goodwhen played with the

first. The Greeks knew the interval between these twofrequencies as a diapson. 440 Hz and 880 Hz soundso good together, in fact, that they sound … thesame. As the frequency of the 880 Hz tone isincreased to 1760 Hz, it sounds the same as when thefrequency of the 440 Hz tone is increased to 880 Hz.This feature has led widely different cultures tohistorically use a one arbitrary frequency and anotherfrequency, exactly one diapson higher, as the first andlast notes in the musical scale. Diapson meansliterally “through all.” If you sing the song,Somewhere Over the Rainbow, the syllables Some-where differ in frequency by one diapson. Asmentioned above, frequencies separated by onediapson not only sound good together, but they soundlike each other. So an adult and a child or a man and awoman can sing the same song together simply bysinging in different diapsons. And they’ll do thisnaturally, without even thinking about it.

The next step in the development of the musicalscale is to decide how many different tones toincorporate and how far apart in frequency they shouldbe. If a certain frequency and another one twice ashigh act as the first and last notes for the scale, thenother notes can be added throughout the range. Tworeasonable constraints are that the frequencies chosenwill be fairly evenly spaced and that they will soundgood when played together.

Constraints for ChoosingFrequencies for a Musical Scale

• Even Spacing • Consonance when Played

Together

The Greek mathematician, Pythagoras,experimented plucking strings with the same tension,but different lengths. This is easy to do with amonochord supported by a movable bridge (see figure4.2). He noticed that when two strings (one twice aslong as the other) were plucked at the same time, theysounded good together. Of course he didn’t knowanything about the difference in the frequenciesbetween the two, but he was intrigued by thesimplicity of the 2:1 ratio of the lengths of the twostrings. When he tried other simple ratios of string

lengths (2:1, 3:2, and 4:3)he found that he also gotgood consonance. We nowknow that if the tensionin a string is keptconstant and its length ischanged, the frequency ofsound produced when the

L2L1

Figure 4.2: A monochord . The movablebridge turns its one vibrating string into twovibrating strings with different lengths butthe same tension.


42

string is plucked will be inversely proportional to itschange in length – twice the length gives half thefrequency and one-third the length gives three timesthe frequency. So a frequency of 1,000 Hz soundsgood when sounded with2,000 Hz

†

2,0001,000 = 2

1( ) , 1,500 Hz

†

1,5001,000 = 3

2( ) , and1,333 Hz

†

1,3331,000 @ 4

3( ) . One of the theories given forthis consonance is that the frequencies will neither besimilar enough to cause beats nor be within the samecritical band.

We’re now in a position to pull together theideas of modes, overtones, and harmonics to furtherexplain the consonance of tones whose frequencies areratios of small integers. Recall that one theory forconsonance is that simultaneously soundedfrequencies will neither be similar enough to causebeats nor be within the same critical band. A secondtheory for this consonance is that many of theovertones of these two frequencies will coincide andmost of the ones that don’t will neither cause beatsnor be within the same critical band.

Theories for ConsonanceBetween Two Frequencies withSmall Integer Ratios

• The frequencies will neitherbe similar enough to causebeats nor be within the samecritical band.

• Many of the overtones ofthese two frequencies willcoincide and most of theones that don’t will neithercause beats nor be within thesame critical band.

The following two examples illustrate these theories for theconsonance between two frequencies with small integer ratios.

Think of a musical instrument that producesharmonic overtones (most instruments do). Let’sconsider two fundamental frequencies it can produce:f1, = 100 Hz and another frequency, f2, = 200 Hz. f2has a ratio with f1 of 2:1. The table to the right andgraph below show the harmonics for each frequency.Notice that all of the harmonics of f2 are identical to aharmonic of f1.

Frequency 1 (Hz) Frequency 2 (Hz)

f1 = 100 f2 = 200

2f1 = 200 2f2 = 400

3f1 = 300 3f2 = 600

4f1 = 400

5f1 = 500

6f1 = 600

0 f1 2f1 3f1 4f1 5f1 6f1

0 f2 2f2 3f2


43

You can see similar circumstances betweenanother combination of the instrument’s fundamentalfrequencies: f1 = 100 Hz and f2, = 150 Hz.

†

f2f1

= 150Hz100Hz = 3

2 , so f2 has a ratio with f1 of 3:2. Thetable to the right and graph below show theharmonics for each frequency. The match ofharmonics is not quite as good, but the harmonics off2 that don’t match those of f1 are still differentenough from the harmonics of f1 that no beats areheard and they don’t fall within the same critical band.

Frequency 1 (Hz) Frequency 2 (Hz)

f1 = 100 f2 = 150

2f1 = 200 2f2 = 300

3f1 = 300 3f2 = 450

4f1 = 400 4f2 = 600

5f1 = 500

6f1 = 600

0 f1 2f1 3f1 4f1 5f1 6f1

0 f2 2f2 3f2 4f2


44

ACTIVITYCONSONANCE

Do the same types of graphs as shown on the previous page, but for pairs of frequencies that are in ratios of othersmall integers:

†

f2

f1=

43

and f2

f1=

54

. Make each graph long enough so that there are at least eight harmonics

showing for f2.

Compare the level of consonance that you believe exists between each of the above pairs of frequencies. Alsocompare the consonance between each of these two pairs of frequencies with those pairs shown on the previous page.Finally, rank the four pairs of frequencies in the order you believe them to be in from most to least consonant andexplain your ranking.

RANKING OF FREQUENCY RATIOS FROM MOST TO LEAST CONSONANT

FrequencyRatio

Consonance

Most

Least

EXPLANATION OF RANKING


45

A SURVEY OF HISTORIC MUSICAL SCALES

OU SHOULD NOW be able toverify two things about musicalscales: they are closely tied to thephysics of waves and sound and theyare not trivial to design. Many scaleshave been developed over time and in

many cultures. The simplest have only four notes andthe most complex have dozens of notes. Three of themost important scales are examined here: thePythagorean Scale, the Scale of Just Intonation, andthe Even Temperament Scale.

THE PYTHAGOREAN SCALEIt has already been mentioned that Pythagoras

discovered the consonance between two strings (orfrequencies, as we know now) whose lengths were inthe ratio of two small integers. The best consonancewas heard with the 2:1 and 3:2 ratios. So Pythagorasstarted with two strings, one twice as long as theother and the tones from these strings were defined asthe highest and lowest tones in his scale. To produceintermediate tones, he used ratios of string lengthsthat were 3:2. To get this ratio a string length couldbe multiplied by 3/2 or divide by 3/2. In either case,the original string and the new string would havelength ratios of 3:2. There’s one catch though. Inmultiplying or dividing a length by 3/2, the newlength might be shorter than the shortest string orlonger than the longest string. However, this reallyisn’t a problem since a length that is either too longor too short can simply be cut in half or doubled inlength (even repeatedly) in order to get it into thenecessary range of lengths. This may seem a bitcavalier, but remember that strings that differ by aratio of 2:1 sound virtually the same. This soundsmore complicated than it really is. It’s easier tounderstand by simply watching the manner in whichPythagoras created his scale.

Let’s give the shortest string in Pythagoras’ scalea length of 1. That means the longer one has a lengthof 2. These two strings, when plucked, will producefrequencies that sound good together. To get the firsttwo additional strings we’ll multiply the shorterlength by 3/2 and divide the longer length by 3/2:

†

1•

†

32

=32

†

2 / 32

= 2 •

†

23

=43

In order of increasing length, the four string lengthsare 1, 4/3, 3/2, 2. These also represent the numbersyou would multiply the first frequency in yourmusical scale by in order to get the three additionalfrequencies. These are represented (to scale) in thefollowing graph.

If 100 Hz were chosen as the first note in this simplemusical scale, the scale would consist of thefollowing frequencies: 100 Hz, 133 Hz, 150 Hz, and200 Hz. This simple four-note scale is thought tohave been used to tune the ancient lyre. Pythagorashowever expanded the notes of this simplistic scaleby creating two new string lengths from theintermediate lengths, 4/3 and 3/2, making sure thatthe string length ratio continued to be 3:2.

†

43

/ 32

=43

•

†

23

=89

†

32

•

†

32

=94

Neither 8/9 nor 9/4 are between 1 and 2 so they mustbe adjusted by either multiplying or dividing theselengths by two (recall this type of adjustment willlead to tones that sound virtually the same).

†

89

•

†

2 =169

†

94

/2 =98

The string lengths (or frequency multipliers) in orderof increasing size is now 1, 9/8, 4/3, 3/2, 16/9, 2.These are represented (to scale) in the followinggraph.

This scale, with five different notes, is known as thepentatonic scale and has been very popular inmany cultures, especially in Eastern music. Butmodern Western music is based on seven different

Y1

†

43

†

32 2

1

†

98

†

43

†

32

†

169 2


46

notes, so we’ll go through the process once more,adjusting the last two lengths so that there are twoadditional lengths in a 3:2 ratio.

†

169

/ 32

=169

• 23

=3227

†

98

• 32

=2716

The final collection of string lengths (or frequencymultipliers) in order of increasing size is now 1, 9/8,32/27, 4/3, 3/2, 27/16, 16/9, 2. These are represented(to scale) in the following graph.

This is one version of the Pythagorean Scale(other ways of building the scale lead to a differentorder of intervals). Whether you think of the intervalson the graph as string lengths or as frequencies, youcan easily verify with a pocket calculator that thereare many ratios of small integers meaning thatcombinations of many of the frequencies lead to highlevels of consonance. You should also notice that theintervals are not uniform. There are mostly largerintervals, but also two smaller intervals (between thesecond and third note and between the sixth andseventh note). In every case, to get from onefrequency to the next across a large interval, the firstfrequency must be multiplied by 9/8 (for example,

†

32 /27 ⋅ 9 /8 = 4 /3 and 4 /3 ⋅ 9 /8 = 3/2). In bothcases of moving across the smaller intervals, the firstfrequency must be multiplied by 256/243. Lets lookat these two intervals more carefully:

†

98

= 1.125 and 256243

= 1.053

Looking at the decimal representations of these twointervals shows that the larger is an increase of a littleover 12% and the smaller is an increase of just lessthan half that (5.3%). The larger of these intervals isknown as a whole tone, W and the smaller isknown as a semitone , s (semi, because it’sabout half the increase of a whole tone).(Whole tones and semitones are synonymous with

the terms whole steps and half steps). Going upPythagoras’ scale then requires a series of differentincreases in frequency:

W s W W W s W

If you were to start with a certain frequency, calculateall the others in the scale and then play them insuccession it would sound like the familiar “Do – Re– Me – Fa – So – La – Ti – Do” except that it wouldstart with Ray. The more familiar string of tonesbeginning with “Doe” (the note “C”) would thenhave to have the following increases in frequencyintervals:

W W s W W W s C1 D E F G A B C2Do Re M e Fa So La Ti Do

This is familiar to musicians, but most non-musicians have probably not noticed the smallerincrease in pitch when going from “Me” to “Fa” andfrom “Ti” to “Do.” Try it. Do you hear the differencein the intervals?

There are seven different notes in the Pythagoreanscale (eight if you include the last note, which is onediapson higher than the first note, and thus essentiallythe same sound as the first). In this scale, the eighthnote has a ratio of 2:1 with the first note, the fifthnote has a ratio of 3:2 with the first note, and thefourth note has ratio of 4:3 with the first note. Thisis the origin of the musical terms the octave theperfect f ifth , and the perfect fourth . (Click onthe interval name to hear two notes separated by thatinterval.) This is why, for example, “G” sounds goodwhen played with either the upper or the lower “C.” Itis a fifth above the lower C and a fourth below theupper C.

Ci D E F G A B Cf

1

†

98

†

8164

†

43

†

32

†

2716

†

243128

2

Table 4.1: The Pythagorean ScaleIntervals for a C major scale. Multiplyingthe frequency of a particular “C” by oneof the fractions in the table gives thefrequency of the note above that fraction.

1

†

98

†

3227

†

43

†

32

†

2716

†

169 2


47

It’s also interesting to look at the ratios betweenindividual notes. Table 4.2 shows the full list offrequency intervals between adjacent tones.

Note change Frequency ratioC Æ D 9/8D Æ E 9/8E Æ F 256/243F Æ G 9/8G Æ A 9/8A Æ B 9/8B Æ C 256/243

Table 4.2: Pythagorean Scaleinterval ratios. Note there aretwo possible intervalsbetween notes: 9/8 (a wholetone) and 256/243 (asemitone).

Do you get it? (4.1)Assuming C5 is defined as 523 Hz, determine theother frequencies of the Pythagorean scale. Showwork below and fill in the table with the appropriatefrequencies.

THE JUST INTONATION SCALEThe scale of Just Intonation or Just Scale

also has a Greek origin, but this time not from amathematician, but from the spectacular astronomer,Ptolemy. Like Pythagoras, Ptolemy heard consonancein the frequency ratios of 2:1, 3:2, and 4:3. He alsoheard consonance in the frequency ratio of 5:4. Hefound that groups of three frequencies soundedparticularly good together when their ratios to eachother were 4:5:6. His method of generating thefrequency intervals of what we now know as the Cmajor scale was to group the notes of that scale intotriads each having the frequency ratios of 4:5:6.

4 5 6Ci E GG B DF A Cf

Let’s start with Ci and give it a value of 1. Thatautomatically makes Cf = 2. In order to get theCi:E:G frequency ratios to be 4:5:6 we can representC1 as 4/4. Then E would be 5/4 and G would be 6/4,or 3/2. So now we have the first three frequencyratios:

Ci D E F G A B Cf

1

†

54

†

32

2

With the next triad (G,B,D) we’ll start with G = 3/2and multiply this ratio by 4/4, 5/4, and 6/4 to get thenext set of 4:5:6 frequency ratios.

†

G =32

• 44

=128

=32

†

B =32

• 54

=158

†

D =32

• 64

=188

=94

This last value for D is more than double thefrequency for C1, so we have to divide it by two toget it back within the octave bound by Ci and Cf.Therefore,

†

D = 9 /4( ) / 2 = 9 /8 . Now we have twomore frequency ratios:

Note C5 D E F G A B C6

Freq.(Hz)

523


48

Ci D E F G A B Cf

1

†

98

†

54

†

32

†

158

2

With the last triad (F,A,Cf) we’ll have to start withCf because it’s the only one we know the value of. Inorder to the get the next set of 4:5:6 frequency ratios,this time we’ll have to multiply Cf’s value by 4/6,5/6, and 6/6.

†

F = 2 • 46

=86

=43

†

A = 2 • 56

=106

=53

†

C f = 2 • 66

=126

= 2

The complete scale of frequency ratios is:

Ci D E F G A B Cf

1

†

98

†

54

†

43

†

32

†

53

†

158

2

Table 4.3: Just Scale Intervalsfor a C major scale. Multiplyingthe frequency of a particular “C”by one of the fractions in thetable gives the frequency of thenote above that fraction.

It’s also interesting to look at the ratios betweenindividual notes. For example, to get from C to D,the C’s frequency must be multiplied by 9/8, but toget from D to E, D’s frequency must be multiplied by10/9. Table 4.4 shows the full list of frequencyintervals between adjacent tones.

Note change Frequency ratio

C Æ D 9/8D Æ E 10/9E Æ F 16/15F Æ G 9/8G Æ A 10/9A Æ B 9/8B Æ C 16/15

Table 4.4: Just Scale intervalratios. Note there are threepossible intervals betweennotes: 9/8 (a major wholetone), 10/9 (a minor wholetone), and 16/15 (asemitone).

The largest of these ratios, 9/8 (12.5% increase), isthe same as the Pythagorean whole tone. In the Justscale it is known as a major whole tone. There isanother fairly large interval, 10/9 (11.1% increase),known as a minor whole tone. The smallest,interval, 16/15 (6.7% increase), while slightlydifferent from the smallest Pythagorean interval, isalso called a semitone.

Some of the names of small integer ratios offrequencies that produce consonance have already beendiscussed. These and others are listed, in order ofdecreasing consonance, in the Table 4.5.


49

FrequencyRatio

Interval Interval name

2/1 C Æ C Octave

3/2 C Æ G Perfect fifth

4/3 C Æ F Perfect fourth

5/3 C Æ A Major sixth

5/4 C Æ E Major third

8/5 E Æ C Minor sixth

6/5 A Æ C Minor third

Table 4.5: Just Scale intervalratios and common names.Click on any of the names to

hear the interval played.

Tables 4.2 and 4.4 give the interval ratiosbetween notes within one octave of either thePythagorean or the Just scales. However, if thefrequencies within a particular octave are too low ortoo high, they can all be adjusted up or down one ormore octaves, either by starting with the first “C” ofthe desired octave and multiplying by each of thefrequency ratios or simply by multiplying eachfrequency in the above scale by the appropriate integer(this integer would be 2 if you wanted to produce thenext higher octave).

Do you get it? (4.2)C4 is the frequency or note one octave below C5(523 Hz). Calculate the frequencies of the notes in theJust scale within this octave.

A PROBLEM WITH TRANSPOSINGMUSIC IN THE PYTHAGOREAN ANDJUST SCALES

One of the problems with both the Pythagoreanand the Just scales is that songs are not easilytransposable. For example, if a song were written inthe key of C (meaning that it starts with the note, C)and you wanted to change it so that it was written inthe key of F, it wouldn’t sound right. It wouldn’t beas easy as transposing all the notes in the song up bythree notes (C Æ F, F Æ B, etc.) because of thedifference in intervals between various notes. Let’ssay the first two notes of the song you want totranspose are C and F and you want to rewrite thesong in the key of F. Using Table 4.2, the increase infrequency for the first note (C Æ F) is:

†

98

• 98

• 256243

= 1.33.

If the newly transposed song is to have the samebasic sound, then every note in the song shouldchange by that same interval, but going from F Æ Bis actually an interval of:

†

98

• 98

• 98

= 1.42.

The Just scale has the same problem. One attempt ata solution would be to add extra notes (sharps andflats) between the whole tones on both scales so thatthere were always choices for notes between thewhole tones if needed for transposing. So going fromC to F or from F to B would be an increase of fivesemitones in both cases. But there is still theproblem of the two different whole tone intervals inthe Just scale. And in both scales, there is theproblem that the semitone intervals are not exactly

Note C4 D4 E4 F4 G4 A4 B4 C5

Freq.(Hz)


50

half the interval of the whole tones, so the idea ofcreating additional notes between the whole tonesdoesn’t fully solve the problem of being able toseamlessly transpose music. There are also mistunedinterval issues as well with both scales that will bedealt with later, but first we’ll look at the perfectsolution to the transposing problem – the EqualTempered scale.

THE EQUAL TEMPERAMENTSCALE

The Equal Temperament scale attempts to correctthe frequency spacing problem without losing thebenefits of the special intervals within the twoprevious scales. It includes the seven notes of theprevious scales and adds five sharps (for a total of 12semitones), but it places them so that the ratio o fthe frequencies of any two adjacent notes i sthe same. This is not as simplistic as it may seemat first glance. It is not the same as taking thefrequency interval between two C’s and dividing it by12. This would not lead to the condition that twoadjacent frequencies have the same ratio. For example:

†

13/1212 /12

=1312

but 14 /1213/12

=1413

and 1312

=1413

Instead, it means that multiplying the frequency of anote in the scale by a certain number gives thefrequency of the next note. And multiplying thefrequency of this second note by the same numbergives the frequency of the note following the second,and so on. Ultimately, after going through thisprocess twelve times, the frequency of the twelfthnote must be an octave higher – it must be twice thefrequency of the first note. So if the multiplier is “r”,then:

†

r ⋅ r ⋅ r ⋅ r ⋅ r ⋅ r ⋅ r ⋅ r ⋅ r ⋅ r ⋅ r ⋅ r = 2

†

fi r12 = 2 fi r = 212 = 1.05946

Click here to see a table of all frequencies of the notesin the Equal Temperament Scale.

ExampleThe note, D, is two semitones higher than C. If C6 is1046.5 Hz, what is D7 on the Equal TemperamentScale?

Solution:• Think about the problem logically in terms of

semitones and octaves.

D7 is one octave above D6, so if D6 can befound then its frequency just needs to bedoubled. Since D6 is two semitones higherthan C6, its frequency must be multiplied twiceby the Equal temperament multiplier.

• Do the calculations.1.

†

D6 = 1046.5Hz ¥ (1.05946)2

†

= 1174.7Hz

fi

†

D7 = 2D6 = 2(1174.7)Hz

=

†

2349.3Hz

or alternately

†

D7 = 1046.5Hz ¥ (1.05946)14

=

†

2349.3Hz

With the extensive calculations in this section itwould be easy to lose sight of what the purpose ofthe last few pages was. Let’s step back a bit. Youknow music when you hear it. It is a fundamentallydifferent sound than what we call noise. That’s partlybecause of the rhythm associated with music, but alsobecause the tones used sound good when playedtogether. Physically, multiple tones sound goodtogether when the ratio of their frequencies is a ratioof small numbers. There are many scales thatendeavor to do this, including the Just scale, thePythagorean scale, and Equal Temperament scale, alldiscussed above. In Western music the EqualTemperament scale is the most widely used. Itstwelve semitones all differ in a ratio of

†

212 from eachadjacent semitone. All notes of the major scale areseparated by two semitones except for E and F, and Band C. These two pairs are separated by one semitone.The # indicates a sharp (see Table 4.6).


51

1 2 3 4 5 6 7 8 9 10 11 12C C# D D# E F F# G G# A A# B

Table 4.6: The notes of the 12-tone EqualTemperament scale. All notes are separatedby two semitones (a whole tone) except for Eand F , and B and C . The # indicates a“sharp.”

Now remember the ancient discovery of frequencyratios of small numbers leading to pleasant soundswhen the two frequencies are played together. Thishas not been entirely sacrificed for the purpose ofmathematical expediency. It turns out that

†

212( )7

= 1.498 which is about 0.1% different from1.5 or 3/2 (the perfect fifth). So in the EqualTemperament scale, the seventh semitone note aboveany note in the scale will be close to a perfect fifthabove it. That means C still sounds good with the Gabove it, but D# also sounds just as good when playedwith the A# above it (see figure 4.3). The next bestsounding combination, the perfect fourth, occurswhen two notes played together have a frequency ratioof 4/3. It conveniently turns out that

†

212( )5

= 1.335(less than 0.4% different from a perfect fourth). Sothe fifth semitone higher than any note will be higherby a virtual perfect fourth (see figure 4.3).

Do you get it? (4.3)In one suggestion for a standard frequency, C4 is256 Hz. In this particular standard, what would be thefrequency of E4 on each of the following scales?a. Pythagorean:

b. Just:

c. Equal Temperament

C C #Db D D #Eb E F F #Gb G G #Ab A A #Bb B C

Octave

Perfect Fifth

Perfect Fourth Perfect Fourth

Perfect Fifth

Figure 4.3: The chromatic scale with various consonant musical intervals


52

A CRITICAL COMPARISON OFSCALES An important question is how large can thepercentage difference from perfect intervals be beforethe difference is detected or unacceptable? A moreprecise method for comparing frequencies is with theunit, cents. Musically, one cent is 1/100 of an EqualTemperament semitone. And since there are 12semitones per octave, one cent is 1/1200 of anoctave. But remember that this is still a ratio offrequencies, so it’s not as simple as saying that onecent is equivalent to 2 divided by 1200. Rather, anytwo frequencies that differ by one cent will have thesame frequency ratio. So after the frequency ratio “R”of one cent is multiplied by itself 1,200 times, theresult is 2.

†

fi R(1cent)( )1200= 2

†

fi R(1cent) = 21

1200 = 1.000578

The frequency ratio equivalent to two cents would be:

†

fi R(2cents) = 21

1200Ê Ë Á ˆ

¯ ˜ 2

11200Ê

Ë Á ˆ

¯ ˜ fi R(2cents) = 2

21200

Finally, the frequency ratio of I cents would be:

†

R = 2I

1200

And the corresponding number of cents for aparticular frequency ratio can be found by taking thelogarithm of both sides of this equation:

†

log R = log 2I

1200Ê Ë Á ˆ

¯ ˜ fi log R =

I1200

log 2

fi

†

I =1200logR

log2

ExampleThe perfect fifth is a frequency ratio of 1.5. Howmany cents is this and how does it compare with thefifths of the Just, Pythagorean, and Equal Temperedscale?

Solution:• Do the calculations.

†

I =1200 log R

log 2=

1200 log 1.5( )log 2

=

†

702cents

• Make the comparisons.Just scale: identical.Pythagorean scale: identical.Equal Tempered scale: 2 cents less.

Of course, all of this is only useful if there issome understanding of how many cents a musicalscale interval can differ from one of the perfectintervals before it becomes detected or unacceptable.With that in mind, a musician with a good ear caneasily detect a mistuning of 5 cents and a 10 to 15cent deviation from perfect intervals is enough to beunacceptable, although at times musicians will oftenintentionally deviate by this much for the purpose ofartistic interpretation. Table 4.7 compares severalimportant ideal musical intervals with their equaltempered approximations.

Interval Frequencyratio

Frequencyratio (cents)

Equal Temp.scale (cents)

Octave 2 : 1 1200 1200Fifth 3 : 2 702 700

Fourth 4 : 3 498 500Major sixth 5 : 3 884 900Major third 5 : 4 386 400Minor sixth 8 : 5 814 800Minor third 6 : 5 316 300

Table 4.7: A comparison of ideal intervalswith their approximations on the EqualTempered scale. Note the large deviationsfrom ideal for the thirds and sixths.


53

Because of its equal intervals, the EqualTemperament scale makes transposing music verysimple. For example, someone with only a marginalunderstanding of the scale could easily take a melodywritten in the key of C and rewrite it in the key of Fby simply increasing every note’s frequency by fivesemitones. This is a major advantage over the scaleswith unequal intervals. But a close look at Table 4.7illustrates why many musicians take issue with theapproximations necessary to make these intervalsequal. Although its octave is perfect and its fifth andfourth differ from the ideal by only 2 cents, the equaltempered sixths and thirds all differ from the idealanywhere from 14 to 16 cents, clearly mistuned andnoticeable by anyone with a good ear.

The frequency interval unit of cents is especiallyuseful in analyzing the musical scales. In thefollowing example, Table 4.2 is redrawn with thefrequency intervals expressed in cents rather thanfractions. This leads to a quick way to analyze thequality of the various consonant intervals. Using

†

I = 1200 log R / log 2I to convert the intervals gives:

Notes Frequencyinterval (cents)

Ci 0D 204E 408F 498G 702A 906B 1110Cf 1200

Table 4.8: Pythagoreanscale interval ratiosexpressed in cents

It’s clear that the interval between Ci and G is 702cents – a perfect fifth. It’s almost as clear to see thatthe interval between D and A

†

906cents - 204cents = 702cents( ) is also a perfectfifth. The additive nature of the cents unit makes iteasy to judge the quality of various intervals. Table4.8 shows the values of various important intervalsin the Pythagorean scale.

Interval Intervalname

Frequencyratio (cents)

Ci Æ Cf Octave 1200Ci Æ G Fifth 702D Æ A Fifth 702E Æ B Fifth 702F Æ C Fifth 702Ci Æ F Fourth 498D Æ G Fourth 498E Æ A Fourth 498G Æ Cf Fourth 498Ci Æ E Major third 408F Æ A Major third 408G Æ B Major third 408

Table 4.9: An evaluation of thePythagorean intervals. Note theabundance of perfect fifths andfourths, but also the verypoorly tuned thirds.

Table 4.9 shows why the Pythagorean scale is soimportant. Within the major scale, there are fourperfect fifths and four perfect fourths (there are manymore of both if the entire chromatic scale is used).However, the three major thirds differ by 22 cents

†

408cents - 386cents( ) from the perfect major third,noticeably sharp to most ears. This is the reason thatPythagoras felt the major third was dissonant. Theminor third is a problem in the Pythagorean scale aswell. Going from E to G is an increase of 294 cents(see Table 4.8), but the perfect minor third, 6/5, is316 cents (see Table 4.7). So the Pythagorean majorthird is 22 cents sharp and the minor third is flat bythe same amount. This 22-cent interval is actually21.5 cents (due to rounding errors) and is known asthe syntonic comma, d. One way to deal with thisparticular mistuning is to compromise the position ofthe E. Decreasing it a bit would help the consonanceof both the major third (C to E) and the minor third(E to G). This and similar adjustments made to othernotes in the scale are known as meantone tuning.There are different types of meantone tuning, but themost popular appears to be quarter-commameantone tuning. In this version, every note,except for C, is adjusted by either 1/4, 2/4, 3/4, 4/4,or 5/4 of the syntonic comma (see Table 4.10)


54

PythagoreanNote

Quarter-commaMeantoneadjustment

C noneD

†

- 24 d

E

†

-dF

†

+ 14 d

G

†

- 14 d

A

†

- 34 d

B

†

- 54 d

C none

Table 4.10: Quarter-comma Meantoneadjustments to thePythagorean scale.

“If I were not a physicist, I would probably be a musician. Ioften think in music. I live my daydreams in music. I see mylife in terms of music.”– Albert Einstein

“When I hear music, I fear no danger. I am invulnerable. Isee no foe. I am related to the earliest times, and to the latest.”– Henry David Thoreau


55

ACTIVITYCREATE A MUSICAL SCALE

Now that you’ve had some time to look at various musical scale constructions as well as to consider the meritsof these various scales, it’s time to try your hand at it. In this activity, you will design an equal-tempered musicalscale. Your goal is to discover one that has both a reasonable number of total intervals as well as many consonantintervals. You may choose any number of intervals except for the commonly used twelve intervals.

1. What is the number of intervals in your equal-tempered scale? _________

2. What is the size of your equal-tempered interval? Express this as a decimal interval increase as well as thenumber of cents per interval.

3. In the space below, indicate the presence of the following consonant intervals in your scale: fifth, fourth,major third, minor third, major sixth, and minor sixth. State how many of your equal-tempered intervals areequivalent to these consonant intervals. (Do not consider that a particular consonant interval exists unless itis within 0.01 of the perfect interval.)

Fifth:

Fourth:

Major third:

Minor third:

Major sixth:

Minor sixth:


56

4. Evaluate each of the consonant intervals in your scale, considering a 5-cent or less deviation to be perfectand a 6 – 15 cent deviation to be acceptable.

5. Compare the quality of your equal tempered scale to that of the 12 tone equal tempered scale. Consider boththe presence and quality of the consonant intervals as well as the number of total intervals.


57

ACTIVITYEVALUATING IMPORTANT MUSICAL SCALES

INTRODUCTIONYou’ve seen that the process of building or choosing a particular musical scale is not trivial. It should be clear

now that there is no such thing as perfect tuning. This was shown for the 12-tone Equal Temperament scale inTable 4.7 and for the Pythagorean scale in Table 4.9. During this activity you will be asked to evaluate the pros andcons of three types of scales: the Just Scale, the Quarter-comma Meantone scale, and variants of the Equal TemperedScale.

CALCULATIONS (EXPLAIN THE PROCESS YOU’RE USING THROUGHOUT ANDSHOW ALL CALCULATIONS CLEARLY)THE JUST SCALE

1. Transform Table 4.3 so that the Just scale frequency intervals are expressed in cents.

Just scalenotes

Frequencyinterval (cents)

Ci

DEFGABCf

2. Now create a table in which you identify all fifths, fourths, major thirds and minor thirds in the Just scale.What are the best and worst aspects of the Just scale when compared to the Pythagorean scale and 12-toneEqual Tempered scale?


58

THE QUARTER-COMMA MEANTONE SCALE3. Use Tables 4.8 and 4.10 to create a table showing the frequency intervals in the Quarter-comma Meantone

scale

Quarter-commaMeantone scale

notes

Frequencyinterval(cents)

Ci

DEFGABCf

4. Now create a table in which you identify all fifths, fourths, major thirds and minor thirds in the Quarter-comma Meantone scale. What are the best and worst aspects of the Quarter-comma Meantone scale whencompared to the Pythagorean scale, Just scale, and 12-tone Equal Tempered scale?


59

OTHER EQUAL TEMPERED SCALESThe 12-tone Equal Temperament scale is a compromise between the consonance of perfect intervals and the

utility of equal intervals. Some have wondered why the fifth and the fourth work so well, but the other intervalsdon’t. The reason is simply that in deciding to use 12 tones, the spacing between some notes just happens to be veryclose to that of the consonant intervals. Other equal interval temperaments have been proposed over time, includingthose with 19, 31, and 53 tones. Remember that although these have 19, 31, and 53 notes, respectively, the rangefrom the first note until the note following the last note is still one octave (1200 cents). The first step is todetermine the interval between notes in the three scales.

5. What is the interval (in cents) between notes on each of the following equal temperament scales?a. 19 tone b. 31-tone c. 53-tone

6. Now determine the number of steps in each of these Equal Temperament scales required for the followingintervals: perfect fifth, perfect fourth, major third, major sixth, minor third, and minor sixth. Finally,indicate the deviation of each of these intervals from the ideal interval as shown in column 3 of Table 4.7.

19-tone EqualTemperament

IntervalNumberof stepson thescale

Deviationfrom idealinterval(cents)

PerfectFifth

PerfectFourthMajorthird

Majorsixth

Minorthird

Minorsixth




PerfectFifth


Majorsixth

Minorthird

Minorsixth




PerfectFifth


Majorsixth

Minorthird

Minorsixth


600

7. Rank the three Equal Tempered scales against each other in terms of the quality of their tuning of the majorconsonant intervals.

8. Thinking of the piano keyboard, explain carefully what the chief problem of these Equal Temperamentscales are.

9. What are the best and worst aspects of the best of the three Equal Tempered scales when compared to theQuarter-comma Meantone scale, Pythagorean scale, Just scale?

61

Cel lo

Viola

Vio l in

C1 C2 C 3 C4 C5 C6 C7 C8(Hz) 33 65 131 262 523 1046 2093 4186

Harp

“Words make you think a thought. Music makes you feel afeeling. A song makes you feel a thought.”– E.Y. Harbug

CHAPTER 5CHORDOPHONES

(STRINGED INSTRUMENTS)

HE YEAR WAS 1968, the event was Woodstock, one of the bands wasThe Grateful Dead, the man was Jerry Garcia, and the musical instrumenthe used to make rock and roll history was the guitar – a chordophone(stringed instrument). Instruments in this class are easy to pick out. Theyhave strings, which either get plucked (like guitars), bowed (like violins),

or thumped (like pianos). It includes all instruments whose standing waveconstraint is that at each end of the medium there must be a node.Technically this includes drums, but because of the two dimensional nature of thevibrating medium, the physics becomes a lot more complicated. We’ll just look attrue strings.

T

SSSTTTRRRIIINNNGGGEEEDDD IIINNNSSSTTTRRRUUUMMMEEENNNTTTSSS

62

THE FIRST MODEThe simplest way a string can vibrate in a

standing wave condition is with the two requirednodes at the ends of the string and an antinode in themiddle of the string (see Figure 5.1).

This is the first mode. The length of the string (inwavelengths) is half a wavelength (see Figure 5.1).That means that for the string length, L:

†

L =12

l fi l = 2L .

Now frequency, in general, can be foundusing

†

f = v /l , so for the first mode of a stringedinstrument:

†

f1 =v

2L.

WAVE SPEEDS ON STRINGSWave speed on strings depends on two factors:

the tension in the string and the “linear mass density”of the string. Tightening or loosening the stringswith the tuners can change their tension . It takesmore force to pluck a taut string from its restingposition. And with more force acting on the tautstring, the more quickly it will restore itself to itsunplucked position. So, more tension means aquicker response and therefore, a higher velocity forthe wave on the string.

The wave velocity can also be affected by the“heaviness” of the string. Strictly speaking, it’s thelinear mass density of the string that causes thiseffect. Linear mass density is the amount of mass per

length of string (in

†

kg /m ). The greater this density,the greater the overall mass of a particular string willbe. Most people have noticed that the strings on aguitar vary in thickness. The thicker strings havegreater mass, which gives them more inertia, orresistance to changes in motion. This greater inertiacauses the thicker, more massive strings to have aslower response after being plucked – causing a lowerwave velocity.

It should be clear that higher tension, T, leads tohigher wave speed (see Figure 5.2), while higherlinear mass density, m, leads to lower wave speed (seeFigure 5.3). The two factors have opposite effects onthe string’s wave velocity, v. This is clear in theequation for string wave velocity:

†

v =Tm

Recall the frequency for the first mode on a string is

†

f1 = v / 2L . Combining this with the expression forthe string’s wave velocity, the fundamental frequencyof a stringed instrument becomes:

†

f1 =

Tm

2L

This complicated looking equation points out threephysical relationships that affect the fundamentalfrequency of a vibrating string. Since string tension isin the numerator of the equation, frequency has adirect relationship with it – if tension increases, sodoes frequency. However, since both linear massdensity and length are in denominators of theequation, increasing either of them decreases thefrequency. The following graphic illustrates thesethree relationships

†

T ↑ fi f ↑

m ↑ fi f Ø L ↑ fi f Ø

String length, L

l

Figure 5.1: First mode of vibration.This is the simplest way for a string t ovibrate in a standing wave condition.This mode generates the fundamentalfrequency.


63

THE SECOND MODENow lets look at the next possible

mode of vibration. It would be thenext simplest way that the string couldvibrate in a standing wave pattern withthe two required nodes at the end of thestring (see Figure 5.4)

We can figure out the frequency of thesecond mode the same way as before.The only difference is that the stringlength is now equal to the wavelengthof the wave on the string. So, for thefrequency of the second mode of astring:

†

f =vl

fi f2 =

Tm

L

You should notice that this is exactlytwice the frequency of the first mode,

†

f2 = 2 f1 . And, when you pluck theguitar string, both modes are actuallypresent (along with many even highermodes, as discussed earlier).

String length, LFigure 5.4: Second mode o fvibration. This is the nextsimplest way for a string t ovibrate in a standing wavecondition. This modegenerates the first overtone.

Figure 5.2: Changing a string’s tension changes i t sfrequency of vibration. When a tuner either tightens orloosens a string on a violin its frequency of vibrationchanges. The equation for the fundamental frequency of avibrating string,

†

f1 = T /m( ) / 2L , shows the connectionbetween string tension and frequency. Since tension is in thenumerator of the square root, if it increases, so will thestring’s frequency.

Figure 5.3: Changing a string’s linear mass density, m ,changes its frequency of vibration. Two strings with the sametension, but different mass will vibrate with differentfrequency. The equation for the fundamental frequency of avibrating string,

†

f1 = T /m( ) / 2L , shows the connectionbetween linear mass density and frequency. Since linear massdensity is in the denominator of the square root, if i tincreases, the string’s frequency decreases. The thicker andheavier strings on the violin are the ones that play the lowerfrequency notes.


64

Do you get it? (5.1)a. In the space below, draw the string vibrating in

the third mode:

b. Now write the equation for the frequency of thethird mode. Explain how you arrived at thisequation.

c. In the space below, draw the string vibrating inthe fourth mode:

d. Now write the equation for the frequency of thefourth mode. Explain how you arrived at thisequation.

e. Finally, look for a pattern in these fourfrequencies and write the equation for the nth modefrequency.

Figure 5.5: A two-stringed musical instrumentfrom Java with very large tuners. See movie

David Lapp

below.


65

SOUNDPRODUCTION INSTRINGEDINSTRUMENTS

If you were to remove astring from any stringedinstrument (guitar, violin,piano) and hold it taut outsidethe window of a moving car,you would find very littleresistance from the air, evenif the car were moving veryfast. That’s because thestring has a very thin profileand pushes against muchless air than if you held a marimba bar of the samelength outside the car window. If you stretched thestring in between two large concrete blocks andplucked it you would hear very little sound. Not onlywould the string vibrate against very little air, butalso because of the huge impedance difference betweenthe string and the concrete blocks, its vibrationswould transmit very little wave energy to the blocks.With so little of its energy transmitted, the stringwould simply vibrate for a long time, producing verylittle sound. For the non-electric stringed instrumentto efficiently produce music, its strings must coupleto some object (with similar impedance) that willvibrate at the same frequency and move a lot of air.To accomplish this, the strings of guitars, violins,pianos, and other stringed acoustic instruments allattach in some fashion to a soundboard. Figure 5.6shows the strings of a guitar stretched between thenut and the bridge. Figure 5.7 shows a magnifiedimage of the bridgeattachment to the top ofthe guitar. As a stringvibrates it applies a forceto the top of the guitar,which varies with thefrequency of the string.Since the impedancechange between thestring and the bridge isnot so drastic as thatbetween the string andthe concrete blocks,much of the wave energyof the string istransmitted to the bridgeand guitar top, causingthe vibration of a muchgreater surface area. This,in turn, moves atremendously greateramount of air than the

string alone, making the vibration clearly audible.Electric stringed instruments use a different

method for amplification. Try strumming an electricguitar with its power turned off and the soundproduced will be similar to that of the taut stringbetween the two concrete blocks. But turn on thepower and the sound is dramatically increased. Thestring of an electric guitar vibrates over the top of upto three electromagnetic pickups. The pickup consistsof a coil of wire with a magnetic core. As the string(which must be made of steel) vibrates through thepickup’s magnetic field, it changes the flux of themagnetic field passing through the core. And sincethe flux change is at the same frequency as thevibrating string, this becomes a signal, which can beamplified through a loud speaker. The string is notcoupled directly to any soundboard, and will thusvibrate for a far longer time than that of its acousticcousin.

Figure 5.7: Tension from the stretched guitar strings causes a downwardforce on the bridge, which moves the entire face of the guitar at thesame frequency as that of the strings. This larger movement of airgreatly amplifies the nearly inaudible sound of the strings alone.

Figure 5.6: The energy used to pluck the strings of the guitaris transferred to the top of the guitar as the strings vibrateagainst the bridge.

Bridge Nut

GGGUUUIIITTTAAARRR PPPHHHYYYSSSIIICCCSSS

66

INVESTIGATIONTHE GUITAR

INTRODUCTIONIt is believed that the origin of the guitar was in

Egypt more than 3,000 years ago. The modern guitarpictured above has its strings tuned to E2, A2, D3, G3,B3, and E4. In order to get intermediate frequencies,the strings are “fretted.” Pressing a finger down on the

space above the fret changes the length of a particularstring with negligible change to the tension in thestring. The new string length is measured from thefret to the bridge. The length of the unfretted strings(from nut to bridge) is 0.65 m.

Nut

Bridge

Unfrettedstringlength= 0.650 m

Fingerposit ionabovefret

Frettedstringlength


67

CALCULATIONS (EXPLAIN THE PROCESS YOU’RE USING THROUGHOUT ANDSHOW ALL CALCULATIONS CLEARLY)

1. The frequency of C2 is 65.406 Hz. Use this frequency and the even temperament scale to calculate thefrequencies of all the unfretted stringsE2:

A2:

D3:

G3:

B3:

E4:

2. Give some justification, based on the idea of consonant intervals, for choosing these particular notes for thetuning of the six strings on the guitar.

3. Calculate the speed of the second harmonic wave on the E2 string.

4. The linear mass density of the A2 string is

†

0.0085kg /m . Calculate the tension in this string.


68

5. Make measurements on the photograph to the right to calculate the frequency of the E2string fretted as shown to the right. You should notice that the frequency is now thesame as A2 string. How close are you to A2?

6. If you had a tuning fork that had the frequency of E2, you could use it to make sure yourE2 string was tuned perfectly. If the E2 string were out of tune by say, 3 Hz, you wouldhear a beat frequency of 3 Hz when sounded together with the tuning fork. Then youcould tighten or loosen the string until no beats were heard. What could you do then tomake sure the A2 string was tuned correctly?

7. Now calculate the length for the E2 string and the A2 string to have the frequency of D3.On the photograph to the right, show where you would fret both strings (indicate in thesame way as shown on the E2 string).

8. Finally, calculate the string lengths and corresponding fret positions for the:a. D3 string to have the frequency of G3.

b. G3 string to have the frequency of B3.

c. B3 string to have the frequency of E4.

E2 D3 B3 A2 G3 E4


69

9. a. How many semitones higher is A2 from E2? __________

b. How many frets did you have to go down on the E2 string to get the frequency of A2? __________

c. Use your answers to “a” and “b” to make a statement about how you think the placement of frets isdetermined. As evidence for your hypothesis, include information from your answers in problems 7 and 8.

The photographs in the following three questions show popular chords guitar players use. The white dots stillrepresent the places where strings are fretted. All strings are strummed except those with an “x” over them.

10. a. List the notes played when this chord is strummed:

_____ _____ _____ _____ _____ _____

b. Why does this chord sound good when its notes are played?


_____ _____ _____ _____ _____ _____



_____ _____ _____ _____


E2A2D3G3B3E4

E2A2D3G3B3E4

E2A2D3G3B3E4

XX

PPPRRROOOJJJEEECCCTTT

70

BUILDING A THREE STRINGED GUITAR

OBJECTIVE:To design and build a three stringed guitar based on the physics of musical scales and the

physics of the vibrating strings.

MATERIALS:• One 80 cm piece of 1” x 2” pine• Approximately 10 cm of thin wood molding• Three #4 x 3/4” wood screws• Three #14 screw eyes• Approximately 3 meters of 30 - 40 lb test fishing line• One disposable 2 - 4 quart paint bucket• Seven thin 8” plastic cable ties• Small wood saw• Screwdriver• Small pair of pliers• Wood glue• Metric ruler or tape

PROCEDURE

1. Sand the corners and edges of the 80 cm piece of pine. This will be the neck of theguitar.

2. Screw the wood screws into one end of the neck. They should be evenly spaced andapproximately 2 cm from the end of the board. The heads should stick up just enough toget a loop of fishing line around them.

3. Put the screw eyes into the other end of the board (in line with the wood screws),making sure that they do not interfere with each other when they are rotated(they will need to be staggered). Screw them in only enough so that theyare stable. They will be tightened later when the fishing line isattached.

4. Cut two 4-cm long pieces of molding. Glue each of these to theneck 3-cm in from each of the two sets of screws (on the side of thescrews closer to the middle of the neck). These are the nut and thebridge.

5. Measure the distance in between the nut and the bridge. Use thisdistance along with the intervals of one of the musical scales discussedto calculate the placement of frets. The frets will be used to shorten thelengths of the strings so that they can produce all the frequencies within the majorscale of one octave.

6. Place a cable tie at each fret position. Tighten it with pliers and cut off the excess end of thecable tie.

7. Cut strings from the fishing line in 1-meter increments. Tie a knotted loop into one end of eachstring.


71

8. For each string, place the knotted loop over the wood screw and pull the other end tight, wrapping itclockwise several times around the appropriate screw eye. While maintaining the tension in the string, loopit through the screw eye and tie a knot.

9. If necessary, cut notches in the bottom of the paint bucket edges so that the neck can lie flush against thepaint bucket bottom. Position the paint bucket about equidistant between the bridge and the fret closest tothe bridge. Glue the bottom of the paint bucket to the neck. Let dry.

10. Choose the string that will produce the lowest frequency. Rotate the eyehook screw attached to it(clockwise) while plucking the string in order to achieve the desired tone.

11. Fret the lowest frequency string so that it is a fifth higher than its fundamental frequency. Using beats, tunethe next string to this frequency.

12. Fret the second string so that it is a fourth higher than its fundamental frequency. Again use beats to tunethe third string to this frequency.

13. Play the guitar. With the open strings you can play an octave, a fifth, and a fourth. You can also play tunesand other intervals by using the frets.

72

“The Irish gave the bagpipes to the Scotts as a joke, but theScotts haven't seen the joke yet.”– Oliver Herford

CHAPTER 6AEROPHONES

(WIND INSTRUMENTS)

TANDING WAVES ARE created in the column of air within windinstruments, or aerophones. Most people have a more difficult timevisualizing the process of a wave of air reflecting in a flute or clarinet asopposed to the reflection of a wave on the string of a guitar. On the guitar,for example, it’s easy to picture a wave on one of its strings slamming into

the bridge. The bridge represents a wave medium with obviously differentimpedance than that of the string, causing a significant reflection of the wave. It maynot be as obvious, but when the wave of air in a wind instrument reaches the end ofthe instrument, and all that lies beyond it is an open room, it encounters animpedance change every bit as real as the change seen by the wave on the guitarstring when it reaches the bridge. The openness beyond the end of the windinstrument is a less constricted environment for the wave (lower impedance), andbecause of this change in impedance, a portion of the wave must be reflected backinto the instrument.

S

Bb Tuba

Trombone

FlutefF

C1 C2 C 3 C4 C5 C6 C7 C8(Hz) 33 65 131 262 523 1046 2093 4186

TrumpetfFBassoon

fF ClarinetfF

WWWIIINNNDDD IIINNNSSSTTTRRRUUUMMMEEENNNTTTSSS

73

To initially create the sound wavewithin the aerophone, the player directs astream of air into the instrument. This airstream is interrupted and chopped intoairbursts at a frequency within the audiblerange. The interruption is accomplished byvibrating one of three types of reeds: amechanical reed, a “lip reed,” or an “airreed.”

THE MECHANICAL REEDInstruments like the clarinet, oboe,

saxophone, and bagpipes all havemechanical reeds that can be set intovibration by the player as he forces an airstream into the instrument. Most of us haveheld a taut blade of grass between theknuckles of our thumbs and then blown airthrough the gaps on either side of the grassblade. If the tension on the grass is justright and the air is blown with the necessaryforce, the grass will start shrieking. The airrushing by causes a standing wave to beformed on the grass blade, the frequency ofwhich is in the audible range. You canchange the pitch of the sound by movingthe thumbs a bit so that the tension isvaried.

The mechanical reeds in windinstruments (see Figure 6.2) can be set intovibration like the grass blade, except thatthe length of the tube largely governs thefrequency of the reed. Figure 6.3 illustratesthe vibration mechanism for the mechanicalreed/tube system.

Figure 6.1: Theclarinet is in thewoodwind classof aerophones.The player blowsacross a smallreed, whichcauses the reed tovibrate. Thevibrating reedallows bursts ofair into the bodyof the clarinet.The bursts of airare responsiblefor the resultingstanding wave ofair that becomesthe distinctivesound producedby the clarinet.

Figure 6.2: A clarinet mouthpiece and reed. By holding her mouth in just the right position andwith just the right tension, the clarinet player causes the reed to vibrate up and down againstthe mouthpiece. Each time the reed rises, creating an opening above the mouthpiece, a burst o fair from the player enters the clarinet. The length of the clarinet largely controls the frequencyof the reed’s vibration.

WIND INSTRUMENTS

74

LIP AND AIR REEDSNot all wind instruments use a mechanical reed.

Brass instruments like the trumpet, trombone, andtuba use a “lip reed” (see Figure 6.4). Although thelips are not true reeds, when the player “buzzes” hislips on the mouthpiece of the instrument they causethe air stream to become interrupted in the same wayas the mechanical reed does. The same type offeedback occurs as well, with low-pressure portions ofthe sound wave pulling the lips closed and high-pressure portions forcing the lips open so that anotherinterrupted portion of the air stream can enter theinstrument.

1. The puff of air the player initially b lowsthrough the instrument begins to pull thereed toward the body of the instrument andcreates a region of high pressure thatmoves toward the end of the instrument.

2. When the high-pressure region reachesthe lower, normal air pressure at the end o fthe instrument it is largely reflected. Thiscauses the reflected pulse to be a negativeor low-pressure pulse and has the effect o fpulling the reed toward the body of theinstrument, closing the gap sharply.

3. The low-pressure pulse is reflected fromthis closed end of the instrument andmoves back to the other end. When i treaches the higher, normal air pressure, i tlargely reflects again, this time as a high-pressure pulse.

4. When the high-pressure pulse reachesthe reed, it forces it open and allows theair from the player to enter and reverse thedirection of the high-pressure pulse. Thispattern of feedback makes it easy for theplayer to keep the reed frequency at thesame frequency as that of the pressure waveinside the instrument.

FIGURE 6.3: VIBRATION MECHANISM FOR THE MECHANICAL REED/TUBE SYSTEM

Figure 6.4:While thetrumpetplayer’s lipsare not a true“reed,” whenthey buzzagainst themouthpiece,they providethe samefrequency ofairbursts as amechanicalreed.

WIND INSTRUMENTS

75

The last method for interrupting the air stream ofa wind instrument is with an “air reed” (see Figure6.5). As the player blows a steady air stream into themouthpiece of the recorder, the air runs into the sharpedge just past the hole in the top of the mouthpiece.The air stream gets split and a portion of the airenters the recorder, moving down the tube andreflecting back from its open end, as in the case ofother wind instruments. However, rather thaninterrupting the air stream mechanically with awooden reed or with the lips of the player, thereflected air pulse itself acts as a reed. The low andhigh-pressure portions of this sound wave in therecorder interrupt the player’s air stream, causing it tooscillate in and out of the instrument at the samefrequency as the standing sound wave. Other windinstruments that rely on the “air reed” include flutes,organ pipes and even toy whistles. This, by the way,is the mechanism that people use when they use theirlips to whistle a tune.

Regardless of the type of reed used, windinstruments all create sound by sustaining a standingwave of air within the column of the instrument. Theother major distinction between wind instruments iswhether there are two ends open (open pipes) or onlyone end open (closed pipes).

OPEN PIPE WIND INSTRUMENTSRecorders and flutes are both examples of open

pipe instruments because at both ends of theinstrument there is an opening through which air canmove freely. Since the air at both ends of the columnis relatively free to move, the standing waveconstraint for this class is that both ends of theair column must be a displacementantinode.

The simplest way a column for air in an openpipe to vibrate in a standing wave pattern is with thetwo required antinodes at the ends of the pipe and anode in the middle of the pipe (see Figure 6.6). Thisis the first mode of vibration.

The length of the pipe (in wavelengths) is

†

1/2l(think of it as two quarters joined at the ends).Therefore:

†

L =12

l fi l = 2L (same as for strings).

We can find the frequency like we did before byusing

†

f = v /l . Thus, for the first mode of an openpipe instrument:

†

f1 =v

2L.

The speed, v, of waves in the pipe is just the speed ofsound in air, much simpler than that for the string.The frequency of a particular mode of an open pipedepends only on the length of the pipe and thetemperature of the air.

Now let’s look at the next possible mode ofvibration. It is the next simplest way that the columnof air can vibrate in a standing wave pattern with thetwo required antinodes at the ends of the pipe (seeFigure 6.7).

Pipe length, LFigure 6.6: First mode of vibration.This is the simplest way for acolumn of air to vibrate (in an openpipe) in a standing wave condition.This mode generates the fundamentalfrequency.

Figure 6.5: The “air reed.” A portion o fthe air stream entering the recordermoves down the tube and reflects backfrom its open end, as in the case o fother wind instruments. However, ratherthan interrupting the air streammechanically with a mechanical reed orwith the lips of the player, the reflectedair pulse itself acts as a reed. The l o wand high-pressure portions of this soundwave in the recorder interrupt theplayer’s air stream, causing it t ooscillate in and out of the instrument atthe same frequency as the standing soundwave.

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76

We can figure out the frequency of this second modethe same way as before. The only difference is thatthe pipe length is now equal to one wavelength of thesound wave in the pipe. So for the frequency of thesecond mode of an open pipe:

†

f =vL

fi f2 =vsound

L.

You should notice that, as with the modes of thestring, this is exactly twice the frequency of the firstmode,

†

f2 = 2 f1 .

Do you get it? (6.1)a. In the space below, draw the air vibrating in the

third mode:

b. Now write the equation for the frequency of thethird mode. Explain how you arrived at thisequation.

c. In the space below, draw the air vibrating in thefourth mode:

d. Now write the equation for the frequency of thefourth mode. Explain how you arrived at thisequation.

e. Now look for a pattern in these four frequenciesand write the equation for the nth mode frequency.Explain how you arrived at this equation.

CLOSED PIPE WIND INSTRUMENTSThe trumpet

and the clarinet areboth examples ofclosed pipe windinstruments,because at one endthe player’s lipsprevent the freeflow of air. Sincethe air at the openend of the columnis relatively free tomove, but isconstricted at theclosed end, thestanding waveconstraint forclosed pipes is thatthe open end o fthe air columnmust be adisplacementantinode andthe closed endmust be a node.

Pipe length, LFigure 6.7: Second mode of vibration.This is the next simplest way for astring to vibrate in a standing wavecondition. This mode generates thefirst overtone.

Figure 6.8: The Panpipe i sa closed pipe instrumentpopular among Peruvianmusicians.

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The simplest way a column of air in a closedpipe can vibrate in a standing wave pattern is with therequired antinode at the open end and the required nodeat the closed end of the pipe (see Figure 6.9). This isthe fundamental frequency, or first mode.

The length of the pipe (in wavelengths) is

†

(1/4)l .So, for the first mode of a closed pipe:

†

L =14

l fi l = 4L .

We can find the frequency as we did before byusing

†

f = v /l . Thus, for the first mode of a closedpipe instrument:

†

f1 =vsound

4L.

Now let’s look at the next possible mode ofvibration. It is the next simplest way that the columnof air can vibrate in a standing wave pattern with therequired antinode at the open end and the required nodeat the closed end of the pipe (see Figure 6.10).

We can figure out the frequency of this nextmode the same way as before. The pipe length is nowequal to

†

3/4 the wavelength of the sound wave in thepipe:

†

L =34

l fi l =43

L .

And the frequency of the next mode of a closed pipeis:

†

f =vl

fi f =vsound

43

L fi f =

3vsound4L

.

You should notice a difference here between themodes of strings and open pipes compared to themodes of closed pipes. This second mode is threetimes the frequency of the fundamental, or first mode.This means that this harmonic is the third harmonic.The second harmonic can’t be produced with thestanding wave constraints on the closed pipe. This isactually true for all the even harmonics of closedcylindrical pipes. However, if the closed pipe has aconical bore or an appropriate flare at the end (likethe trumpet), the spectrum of harmonics continues tobe similar to that of an open pipe.

Do you get it? (6.2)a. In the space below, draw the air vibrating in the

mode after the third:

b. Now write the equation for the frequency of thenext higher mode after the third. Explain how youarrived at this equation.

c. In the space below, draw the air vibrating in themode two higher than the third:

Pipe length, LFigure 6.9: First mode of vibration.This is the simplest way for a columnof air to vibrate (in a closed pipe) in astanding wave condition. This modegenerates the fundamental frequency.

Pipe length, L

Figure 6.10: Second mode of vibration.This is the next simplest way for astring to vibrate in a standing wavecondition. This mode generates the firstovertone.

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78

d. Now write the equation for the frequency of themode two higher than the third. Explain how youarrived at this equation.

e. Now look for a pattern in these four frequenciesand write the equation for the nth mode frequency.Explain how you arrived at this equation.

THE END EFFECTEverything said about open and closed pipes is

basically true so far. However, there is one little issuethat needs to be dealt with. Otherwise, the music youmake with any aerophone you personally constructwill be flat – the frequency will be too low. Amusician with a good ear could tell there was aproblem. The problem is with the open ends of thesepipes. When the standing wave in the column of airreaches a closed end in a pipe there is a hardreflection. However, when the same standing wavereaches the open end of a pipe, the reflection doesn’toccur so abruptly. It actually moves out into the air abit before reflecting back. This makes the pipesacoustically longer than their physical length. This“end effect” is equal to 61% of the radius of the pipe.This end effect must be added to the length of theclosed pipe and added twice to the length of the openpipe.

ExampleLet’s say you wanted to make a flute from one-inchPVC pipe. If the lowest desired note is C5 on theEqual Temperament Scale (523.25 Hz), what lengthshould it be cut?

Solution:• Identify all givens (explicit and implicit) and

label with the proper symbol.Given: f1 = 523.25 Hz

n = 1 (Lowest frequency)v = 343 m/s (no temperature given)

r =

†

0.5inch( ) 2.54cm1inch

Ê

Ë Á

ˆ

¯ ˜

†

= 1.27cm = .0127m

• Determine what you’re trying to find.Length is specifically asked forFind: L

• Do the calculations.

1.

†

f1 =v

2Lacoust. fi Lacoust. =

v2 f1

=343m / s

2 523.5 1s( )

†

= 0.328m

2. 0.328 m is the desired acoustic length ofthe pipe, which includes the end effect onboth ends of the pipe. Therefore, the pipemust be cut shorter than 0.328 m by twoend effects.

†

Lphys. = Lacoust. - 2(end effect)

†

fi Lphys. = 0.328m - 2(.61¥ .0127m)

=

†

0.312m

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79

Do you get it? (6.3)You want to make a 4.0-cm diameter tube, closed atone end, that has a fundamental frequency of 512 Hzand the temperature is 30° C.a. What length will you cut the tube?

b. If you blew across the tube a lot harder to producethe next mode, what would be the new frequency?

c. Now if the bottom of the tube were cut off so thatthe tube was open at both ends, what would be thenew fundamental frequency?

CHANGING THE PITCH OF WINDINSTRUMENTS

In equations for both open and closed pipe windinstruments, the variables that can change thefrequency are the number of the mode, the speed ofsound in air, and the length of the pipe. It would bedifficult or impossible to try to control the pitch ofan instrument by varying the temperature of the air,so that leaves only the number of the mode and thelength of the pipe as methods for changing the pitch.Some wind instruments, like the bugle, have asingle, fixed-length tube. The only way the bugleplayer can change the pitch of the instrument is tochange the manner in which he buzzes his lips, andso change the mode of the standing wave within thebugle. The standard military bugle is thus unable toplay all the notes in the diatonic scale. It typically isused to play tunes like taps and reveille, which onlyrequire the bugle’s third through sixth modes: G4, C5,E5, G5. In order to play all notes in the diatonic orchromatic scale, the tube length of the windinstrument must be changed. The tromboneaccomplishes this with a slide that the player canextend or pull back in order to change the length ofthe tube. Other brass instruments, like the trumpetand tuba, accomplish this change in length withvalves that allow the air to move through additionaltubes, thereby increasing the overall length of thestanding wave. Finally, the woodwinds change tubelength by opening or closing tone holes along thelength of the tube. An open hole on a pipe, if largeenough, defines the virtual end of the tube.

MORE ABOUT BRASSINSTRUMENTS

The trumpet, trombone, and French horn are allclosed pipes with long cylindrical sections and shouldtherefore only be able to produce odd harmonics. Thelength of a Bb trumpet is 140 cm. A closed cylindricalpipe with the same length produces a fundamentalfrequency of 61 Hz. It’s higher modes are odd integermultiples of this first harmonic (see Table 6.1).However, as with all brass instruments, themouthpiece and the bell have a significant effect onthe resonant frequencies. The cylindrical piece of pipewithout a bell or mouthpiece will reflect all of its

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80

standing wave modes at the same point – the end ofthe pipe. But add a bell to the pipe and the modes willreflect at different points. The lower the frequency ofthe mode, the earlier it “sees” the flare of the bell. Sothe lower frequency modes become shorter inwavelength as a result of the bell. This shorterwavelength increases the frequencies of the lowermodes (see Figure 6.11). The mouthpiece also has aneffect. It is approximately 10 cm long and has itsown fundamental frequency of about 850 Hz. Thisfrequency is known as the popping frequency becauseof sound that “pops” from the mouthpiece if it isremoved from the trumpet and struck against thehand. However, the mouthpiece retains some of itsidentity even when it is inserted into the trumpet. It’spresence affects the frequencies of the trumpet’shigher modes, decreasing their frequency and alsoincreasing their prominence in the total spectrum ofthe trumpets sound. Together, the bell andmouthpiece cause the sound production of thetrumpet, trombone, and French horn to be like that ofan open pipe (having all harmonics instead of just theodd ones – see Table 6.1). The presence of thesemodified resonance modes provides greater feedback tothe player and enhances his ability to “find” aparticular mode. The lowest note the Bb trumpet isdesigned to play is Bb

3 (233 Hz). The actual“harmonic” frequencies that a high quality Bb trumpetis able to produce are shown in table 6.1.

Mode Frequency withina closed 140 cmcylindrical pipe(Hz)

Frequencywithin the Bb

trumpet (Hz)

1 61 not playable2 184 2303 306 3444 429 4585 551 5786 674 6957 796 8148 919 931

Table 6.1: Resonant mode frequencies for aclosed 140 cm cylindrical pipe vs. thoseobtained by the Bb trumpet, also 140 cmlong, but with a bell and mouthpiece(Berg, Stork)

Clearly the frequencies are neither truly harmonic norare they notes in the scale of equal temperament, butthey are close enough that a good trumpet player can“lip up” or “lip down” the frequency with subtle lipchanges as he listens to other players in a band ororchestra.

Table 6.1 shows that the interval between thetwo lowest notes produced by the trumpet isapproximately a fifth (ªBb

3 to ªF4). That leaves sixmissing semitones. In order to play these missingnotes, the player uses the valves on the trumpet (seeFigure 6.12). When the valves are not depressed, airflows only through the main tube. However, when avalve is pressed down, the air is forced to flowthrough an additional tube linked to that tube (Figure6.13). The three tubes lengthen the trumpet by anamount that changes the resonant frequency by asemitone, whole tone, or a minor third (threesemitones). By using valves in combination, thetrumpet can be lengthened by an amount that changesthe resonant frequency by four, five, or six semitones.

Figure 6.11: The bell of a brass instrumentcauses lower modes to reflect prior t oreaching the end of the instrument. Thissmaller wavelength for the lower modesincreases their frequencies, forcing them t oapproach the harmonicity of an open pipe.

Figure 6.12: Each trumpetvalve has two pathsthrough which air canflow. When the valve i snot depressed, it allows airto flow through theprimary tube. When thevalve is depressed, air i sforced through differentchambers that divert i tthrough an additionallength of tube. Thisadditional length causesthe standing sound wave t obe longer and its frequencyto be lower.

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81

MORE ABOUT WOODWINDINSTRUMENTS

The woodwinds are so named because originallythey were mostly constructed from wood or bamboo.Wood is still preferred for many modern woodwinds,however metal is used in constructing flutes andsaxophones and plastic is used to make recorders. Tochange the pitch of the woodwind, tone holes alongthe side of the instrument are covered and uncoveredto produce the desired pitch. The simplest way tolook at the function of a tone hole is that, if it isopen, it defines the new end of barrel of theinstrument. So, a single pipe can actually be turnedinto eight different acoustic pipes by drilling sevenholes along the side of the pipe. The length of anyone of these eight virtual pipes would simply be thedistance to the first open hole (which the wave sees asthe end of the pipe). Consider making the placementof the holes so that the standing waves produced hadfrequencies of the major scale. If a tone were generatedin the pipe with all the holes covered and then theholes were released one by one, starting with the oneclosest to the actual end of the pipe and workingbackward, the entire major scale would be heard.

It’s not as easy as it sounds though. Choosingthe position of a hole, as well as its size, is not astrivial as calculating the length of a pipe to produce aparticular frequency and then drilling a hole at thatpoint. Think about the impedance difference the wavein the pipe experiences. It’s true that when thestanding wave in the instrument encounters an openhole it experiences a change in impedance, but if thehole were a pinhole, the wave would hardly notice itspresence. On the other hand, if the hole were as largeas the diameter of the pipe, then the wave wouldreflect at the hole instead of the true end, becausethere would be no difference between the two and thehole would be encountered first. So the open holeonly defines the new end of the pipe if the hole isabout the same size as the diameter of the pipe. As

the hole is drilledsmaller and smaller,the virtual (oracoustic) length ofthe pipe approachesthe actual length ofthe pipe (see Figure6.14). Structurallyit’s unreasonable todrill the holes aslarge as the diameter.And if the bore of theinstrument werelarger than thefingers, then drillinglarge holes wouldrequire other

engineering solutions to be able to fully plug thehole (see Figure 6.15).

It gets even more complicated. Even the presenceof closed holes has an effect on the standing wave.The small amount of extra volume present in thecavity under the closed hole (due to the thickness ofthe pipe) causes the pipe to appear acoustically longerthan the actual length of the pipe. And don’t forgetthe end effect at that first open hole. Even thepresence of the open holes past the first one have aneffect. If they are spaced evenly they will tend toreflect lower frequencies more strongly than higherones. Indeed, the presence of these open holes leads toa cutoff frequency. Above this critical frequency,sound waves are reflected very little, giving thewoodwinds their characteristic timbre.

Figure 6.13: Pressing one or more of the trumpet’s three valvesprovides additional tubing to lengthen the standing sound wave. Thisphoto transformation of the trumpet (courtesy of Nick Deamer, WrightCenter for Innovative Science Education) helps to visualize the role o feach valve on the trumpet.

Aco

ustic

leng

th

Aco

ustic

leng

th

Aco

ustic

leng

th

Aco

ustic

leng

th

Figure 6.14: A hole drilled on the side o fa pipe changes the acoustic length of thepipe. The larger the hole, the closer theacoustic length will be to the holeposi t ion.

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82

So the question is, can the equations for thefrequencies of standing waves within an open orclosed pipe be used to determine the position and sizeof tone holes? The answer is … no, not really. Theconsequences of so many different factors leads to

complicated equations that give results that are onlyapproximations. Actual woodwind construction isbased on historic rules of thumb and lots of trial anderror.

Figure 6.15: From the recorder to the clarinet to the saxophone, tone holes go from small andsimple to small and complex to large and complex. As the instrument grows in length anddiameter, the tone holes get further apart and must also grow in diameter. Compare the s impletone holes of the recorder, which can be easily covered with the player’s fingers to the tone holesof the saxophone, which must be covered with sophisticated multiple, large diameter hole closersystems.

NOSE FLUTE PHYSICS

83

INVESTIGATIONTHE NOSE FLUTE

INTRODUCTIONThe nose flute is a curious musical instrument. Actually it is only part of a musical instrument, the remainder

being the mouth cavity of the player. To play it, the rear of the nose flute is placed over the nose and open mouth ofthe player and air is forced out of the nose. The nose flute directs this air across the mouth in the same way that onemight direct air over the top of a soda bottle to produce a tone. The result is a clear, pleasing, flute-like tone. Clickhere to listen to the nose flute. These are available at a variety of sites, including one selling them for only $.99 pernose flute: http://www.funforalltoys.com/products/just_for_fun_3/nose_flute/nose_flute.html

Nose cover

Mouth cover

Front View

Rear View

http://www.funforalltoys.com/products/just_for_fun_3/nose_flute/nose_flute.html

NOSE FLUTE PHYSICS

84

QUESTIONS AND CALCULATIONS (MAKE CLEAR EXPLANATIONSTHROUGHOUT AND SHOW ALL CALCULATIONS CLEARLY)

1. What type of musical instrument classification fits the nose flute? Be specific, indicating the vibratingmedium and how the pitch of the nose flute is changed.

2. If you and another person nearby both played nose flutes, could you produce beats? How would you do it?

3. What are the similarities and differences between whistling and playing the nose flute?

4. Is the pitch range for the nose flute fixed or different for various players? Explain.

NOSE FLUTE PHYSICS

85

5. If you played the nose flute, what is the lowest theoretical note on the Equal Tempered Scale that youpersonally would be able to get? (You will need to make a measurement to answer this question.)

6. What two things could you do to produce a tone an octave and a fifth above the lowest theoretical frequencycalculated in the previous problem?

SOUND PIPE PHYSICS

86

INVESTIGATIONTHE SOUND PIPE

INTRODUCTIONThe sound pipe is a fun musical toy. To operate the sound pipe, you simply hold one end and twirl the sound

pipe in a circle. The movement of the end not being held causes a low-pressure region in the air. This is due to theBernoulli Effect and is also the explanation for how the perfume atomizer works. In the case of the atomizer, airpuffed past the tube connected to the perfume creates a low-pressure region above the tube, thus causing the perfumeto rise into the path of the puffed air. In the case of the sound pipe, the low-pressure region caused by its motiondraws air into the pipe. Depending on how fast it is twirled, it is possible to make four different audible frequencies.The inside diameter of the sound pipe is 2.5 cm. Both ends of the sound pipe are open.

QUESTIONS AND CALCULATIONS (SHOW ALL WORK)1. What type of musical instrument classification fits the sound pipe? Be specific, indicating the vibrating

medium.

2. What is the relationship between the four possible audible frequencies?

3. When you twirl this sound pipe, what is the lowest pitch you could produce?

74 cm

SOUND PIPE PHYSICS

87

4. What are the other three frequencies this sound pipe is capable of producing? What notes do the fourpossible frequencies represent? You can listen to all four frequencies here .

5. In order to most easily use the sound pipe to play music, you would really need several sound pipes ofdifferent lengths. Let’s say you wanted to be able to play all the notes within a C major scale.a. First determine what length to cut from this sound pipe so that it produces the frequency of the C

closest to the fundamental frequency of this sound pipe.

b. Now calculate what the lengths the other six sound pipes would have to be in order to produce a fullC-major scale (assume the fundamental frequency will be used for each pipe).

SOUND PIPE PHYSICS

88

6. Imagine you had two sound pipes identical to the one in the photograph. If 1.0 cm were cut from one ofthem, what specifically would be heard if they were both twirled so that they were resonating in the secondmode?

7. Another company produces a sound pipe of the same length, but only 2.0 cm in diameter. If the soundpipes from both companies were twirled so that they produce standing waves vibrating in the third mode,what specific sound would be heard?

TOY FLUTE PHYSICS

89

INVESTIGATIONTHE TOY FLUTE

INTRODUCTIONThe inexpensive toy flute pictured here is a

slightly tapered metal tube, open at both ends. Thereis also an opening near where the mouth is placed aswell as six tone holes. The length of the flute fromthe opening on top near the mouth to the end of theflute is 29.5 cm. The diameter of the end of the flute

is 1.0 cm and it increases to 1.4 cm at the position ofthe tone hole closest to the mouth. The walls of thepipe are very thin – less than 0.1 cm. The diametersof the tone holes in order, starting from the oneclosest to the mouth are: 0.6 cm, 0.6 cm, 0.6 cm,0.5 cm, 0.7 cm, and 0.5 cm.

CALCULATIONS (EXPLAIN THE PROCESS AND REASONING YOU’RE USINGTHROUGHOUT. SHOW ALL CALCULATIONS CLEARLY)

1. a. What is the lowest note the flute is designed to play?

b. How would this note change if the flute were played outside on a hot day with the temperature of 37°Cinstead of inside at room temperature?

2. How many tones could be produced with this flute if only the first mode were used? Explain.

Tone holes

29.5 cm

Mouth Position

TOY FLUTE PHYSICS

90

3. Let’s assume for this question that the tone holes were all the same diameter as the pipe (at the positionthat the tone hole is drilled in the pipe). If played only in the first mode, could the flute produce a tone anoctave or more higher than the lowest possible frequency?

4. When this actual flute is played in the first mode, with all the tone holes uncovered, it produces a note 11semitones higher than the note produced when all the tone holes are covered. Explain why the flute isunable to play notes an octave or more higher than the lowest possible frequency.

5. What would you have to do with this flute (without drilling anymore holes) in order to produce tones anoctave or more higher than the lowest possible frequency?

6. It is possible (although it doesn’t sound very pleasant) to produce the third mode with this flute. What isthe highest note the flute is capable of playing?

TRUMPET PHYSICS

91

INVESTIGATIONTHE TRUMPET

INTRODUCTIONThe trumpet is a closed pipe with a

predominantly long cylindrical section. If it werecylindrical throughout its length it would only beable to produce odd harmonics. However, as with allbrass instruments, the mouthpiece and the bell have asignificant effect on the resonant frequencies. The bellincreases the frequency of the lower modes and themouthpiece decreases the frequency of the highermodes. The mouthpiece also increases the prominenceof particular frequencies. Together, the bell andmouthpiece cause the sound production of the trumpetto be like that of an open pipe. The Bb trumpet is 140cm and the lowest note it is designed to play is Bb

3(233 Hz). The actual “harmonic” frequencies that ahigh quality Bb trumpet is able to produce are shownin the table below:

Mode Freq. (Hz)2 2303 3444 4585 5786 6957 8148 931

Frequencies obtained bythe Bb trumpet (Berg,Stork)

Clearly the frequencies are not truly harmonic nor arethey notes in the Equal Temperament Scale, but theyare close enough that a good trumpet player can “finetune” the frequency with subtle lip changes as helistens to other players in a band or orchestra.

The table shows rather large intervals (theinterval between the second and third harmonic isabout a fifth). In order to play smaller intervals, theplayer uses the valves on the trumpet. When thevalves are not depressed, air flows through them inthe main tube. However, when a valve is presseddown, the air is forced to flow through an additionaltube section. The three tubes lengthen the trumpet byan amount that changes the resonant frequency asfollows:

• Valve 1 (valve closest to player’s mouth) – 1whole tone.

• Valve 2 – 1 semitone• Valve 3 – minor third (three semitones)

By using valves in combination, the trumpet can belengthened by an amount that changes the resonantfrequency by four, five, or six semitones.

Bel l

Mouthpiece

Valves

TRUMPET PHYSICS

92

CALCULATIONS (EXPLAIN THE PROCESS AND REASONING YOU’REUSING THROUGHOUT. SHOW ALL CALCULATIONS CLEARLY)

1. If the bell and mouthpiece of the trumpet were not present and the trumpet was still 140 cm long,what would be the frequencies and approximate notes of the first four modes?

2. There is a frequency, known as the pedal tone that is not normally played on the trumpet. It is thefundamental frequency (first harmonic) for the trumpet. What is this frequency?

3. Use the table on the previous page to discuss, as quantitatively as you can, how closely the modesof the trumpet are to being harmonic.

4. With the bell and mouthpiece in place, what is the effective length of the trumpet?

TRUMPET PHYSICS

93

5. Recall that the ratio of frequencies between notes that are separated by one semitone is 1.05946.So from semitone to semitone, the frequencies increase or decrease by 5.946%. The ratios betweenlengths of pipes that play consecutive semitones have a similar relationship. Calculate the lengthsof each of the three tubes that can be activated by the valves of the trumpet.

6. The trumpet valves can be used in combination to change the resonant frequencies by more thanthree semitones. Use the results from the previous problem to determine the extra lengths oftubing in the trumpet when multiple valves are pressed at the same time to lower the resonantfrequency by four, five, or six semitones.

7. Now assume that the trumpet actually had six valves and the trumpet could be lengthened by anamount that changed the resonant frequency by one two, three, four, five, or six semitones.Calculate the length of individual tubes that would change the resonant frequencies by four, five,and six semitones.

TRUMPET PHYSICS

94

8. Compare the extra tube lengths (producing true semitone intervals) that you calculated in the lastproblem with the sums of tube lengths that the trumpet actually uses to make four, five, and sixsemitone interval adjustments.a. Why is there a discrepancy?

b. Without making any adjustments, would the notes produced by the trumpet sound flat or wouldthey sound sharp when making four, five, and six semitone interval adjustments? Why?

9. The photograph to the rightshows the three valves, the extratube sections connected to each ofthem, and two adjusters for thelengths of the tube sectionsconnected to the first and thirdvalves. Why would there be a needfor these adjusters?

10. Indicate which mode is played and which valve(s) would be pressed to produce each of thefollowing notes.a. G4

b. F3

c. E5

d. C4

Adjusters for the lengths of thefirst and third valve tube sections.

Tube sections connectedto the three valves.

TRUMPET PHYSICS

95

11. The brass pipe pictured to the righthas few of the trumpet’s attributes.There is no mouthpiece, no bell, andno valves, but it is a brass tube thatyou could buzz your lips into. Whatare the four lowest notes that wouldbe possible to blow on this pipe?

12. Now assume that the pipe is fitted with a bell and transformed into the horn pictured below.a. What would happen to each of the frequencies of the notes calculated in the previous problem?

b. What other change in the sound production of the horn would occur?

11.0 cm

PROJECT

96

BUILDING A SET OF PVC PANPIPES

OBJECTIVE:To design and build a set of panpipes based on the physics of

musical scales and the physics of the vibrating air columns.

MATERIALS:• One 10-foot length of 1/2” PVC pipe• Small saw or other PVC cutter• Fine sand paper• Eight rubber or cork stoppers• Approximately 1 meter of thin, 1-inch wide wood trim• Strong, wide, and clear strapping tape• Metric ruler or tape

PROCEDURE

1. Decide which notes the panpipes will play. The panpipespictured to the right consist of the C4 major scale plus aC5 note (C4, D4, E4, F4, G4, A4, B4, C5) on the EqualTempered scale.

2. Decide which scale you will use and then calculate thefrequencies your panpipe will have. Do all calculations andcheck work before making any cuts .

3. Calculate the length of each of your pipes to the nearestmillimeter, assuming the panpipes will be used in the firstmode. Keep in mind the end effect as well as the fact thatthe stoppers will not only block the bottoms of the pipes,but stick up inside of them a bit too. Before making yourlength calculations you should check how far the stoppersenter the pipe when they are snuggly in place.

4. Carefully cut each of the pipes. Make sure that each pipeis cut precisely to the number of millimeters calculated.Mistakes in this part of the procedure will be audibly detectable and may not be correctable.

5. Sand the rough edges of the pipes caused by cutting them.

6. Insert a stopper into one end of each of the pipes. Gently blow over the top of each pipe, listening for thosethat may be too flat or too sharp. A stopper may be moved a small distance in or out of a pipe in order totune it slightly.

7. Lay the pipes side by side, with their tops all flush. Measure the distance across the set of pipes. Cut fourpieces of wood trim to this length. Sand the edges of the trim.

8. Place two pieces of cut trim on each side of the set of pipes; one pair 3-4 cm below the tops of the pipesand another pair 3-4 cm above the bottom of the shortest pipe. Wrap strapping tape tightly (several times)around each pair of trim strips. Wrap one piece of tape around the panpipe between the two pairs of woodtrim.

9. Play the panpipes! You can play tunes and experiment by listening to octaves, fifths, fourths, and otherintervals.

97

“Music should strike fire from the heart of man, and bringtears from the eyes of woman.”– Ludwig Von Beethoven

CHAPTER 7IDIOPHONES

(PERCUSSION INSTRUMENTS)

T’S PRETTY HARD to pass by a set of wind chimes in a store and not givethem a little tap. And few of us leave childhood without getting a child’sxylophone for a gift. The sounds produced when pipes or bars are tapped ontheir sides are fundamentally different from the sounds produced by theinstruments in the previous two categories. That’s because the frequencies of

higher modes in vibrating pipes and bars are not harmonic. Musical instrumentsconsisting of vibrating pipes or bars are known as idiophones.

BARS OR PIPES WITH BOTH ENDSFREE

In a bar whose ends are free to vibrate, a standingwave condition is created when it is struck on its side,like in the case of the marimba or the glockenspiel.The constraint for this type of vibration i sthat both ends of the bar must be

antinodes. The simplest way a bar can vibrate withthis constraint is to have antinodes at both ends andanother at its center. The nodes occur at 0.224 L and0.776 L. This produces the fundamental frequency(see Figure 7.1).

I

Timpani

Marimba

C1 C2 C 3 C4 C5 C6 C7 C8(Hz) 33 65 131 262 523 1046 2093 4186

Steel Pan (tenor)fF Xylophone

PPPEEERRRCCCUUUSSSSSSIIIOOONNN IIINNNSSSTTTRRRUUUMMMEEENNNTTTSSS

98

The mode of vibration, producing the next higherfrequency, is the one with four antinodes includingthe ones at both ends. This second mode has a node inthe center and two other nodes at 0.132 L and 0.868 L(see Figure 7.2).

The mathematics used to describe this particularvibration of the bar is pretty complicated, so I’ll justpresent the result. (Fletcher and Rossing in ThePhysics of Musical Instruments, Vol. 2, pp 56 – 64give a full mathematical development). If the bar isstruck on its side, so that its vibration is like thatshown, the frequency of the nth mode of vibration willbe:

†

fn =pvK8L2 m2

Where: v = the speed of sound in the material ofthe bar (Some speeds for commonmaterials are shown in Table 7.1.)

Material Speed of sound, v (m/s)

Pine wood 3300Brass 3500Copper 3650Oak wood 3850Iron 4500Glass 5000Aluminum 5100Steel 5250

Table 7.1: Speed of sound forsound waves in various materials(Aski l l )

L = the length of the barm = 3.0112 when n = 1, 5 when n = 2,

7 when n = 3, … (2n + 1)

†

K =thickness of bar

3.46 for rectangular bars

or

K =

†

(inner radius)2 + (outer radius)2

2for tubes

Do you get it? (7.1)a. In the space below, draw the third mode of

vibration for a copper tube (both ends free), withan outer diameter of 2.5 cm, an inner diameter of2.3 cm and a length of 50 cm.

b. Now calculate the frequency of the second modefor this bar.

Figure 7.1: First mode of vibration. Thisis the simplest way for a bar or pipe t ovibrate transversely in a standing wavecondition with both ends free. This modegenerates the fundamental frequency.

Bar length, L

Figure 7.2: Second mode of vibration.This is the next simplest way for a baror pipe to vibrate in a standing wavecondition with both ends free. Thismode generates the first overtone.

Bar length, L


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BARS OR PIPES WITH ONE ENDFREE

Another type ofvibration for bars iswhen one of the ends isclamped, like in athumb piano. The freeend is struck or plucked,leading to a standingwave condition inwhich the constraintis that the clampedend is always anode and the freeend is always anantinode. Thesimplest way the barcan vibrate is with no additional nodes or antinodesbeyond the constraint. This produces the fundamentalfrequency (see Figure 7.3).

The next mode of vibration, producing the nexthigher frequency, is the one with two antinodes andtwo nodes including the node and antinode at each end(see Figure 7.4).

The expression for the nth frequency of theclamped bar looks identical to that of the bar with freeends. The only difference is in the value of “m”. If thebar is plucked or struck on its side, so that itsvibration is like that shown, the frequency of the nth

mode of vibration will be:

†

fn =pvK8L2 m2

Where: m = 1.194 when n = 1, 2.988 when n = 2,5 when n = 3, … (2n - 1)

And all other variables are defined identicallyto those of the bar with free ends equation.

Do you get it? (7.2)a. In the space below, draw the third mode of

vibration for an aluminum bar, with a thicknessof 0.75 cm and a length of 50 cm.

b. Now calculate the frequency of the second modefor this bar.

Figure 7.4: Second mode of vibration.This is the next simplest way for a baror pipe to vibrate in a standing wavecondition with both ends free. Thismode generates the first overtone.

Bar length, L

Bar length, L

Figure 7.3: First mode of vibration. Thisis the simplest way for a bar or pipe t ovibrate transversely in a standing wavecondition with only one end free. Thismode generates the fundamental frequency.


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As mentioned earlier, the frequencies of themodes of transversely vibrating bars and pipes aredifferent from those of vibrating strings and aircolumns in that they are not harmonic. This becomesobvious when looking at the last two equations fortransverse vibration frequency of bars and pipes. Inboth cases,

†

fn µ m2 , where fn is the frequency of thenth mode and m is related to the specific mode. Fortransversely vibrating bars and pipes with free ends:

†

f2

f1=

52

3.01122 = 2.76 and

†

f3

f1=

72

3.01122 = 5.40.

Do you get it? (7.3)Use the space below, to calculate the relationshipsbetween the frequencies of different modes oftransversely vibrating pipes and bars. Complete thetable below with your results.

TOWARD A “HARMONIC”IDIOPHONE

It was shown earlier that a transversely vibratingbar with both ends free to move has a second modevibration frequency 2.76 times greater than that of thefirst mode. The third mode has a frequency 5.40 timesgreater than that of the first mode. These areobviously not harmonic overtones. Recall, that theinterval for one semitone on the 12-tone EqualTempered scale is

†

1.05946. And

†

1.05946( )12= 2 .

This relationship can be used to find the number of12-tone Equal Tempered semitones that separate themodes of the transversely vibrating bar:

†

log 1.05946( )12= log 2 fi 12 log 1.05946( ) = log 2

†

fi log 2log 1.05946( )

= 12 .

Or more generally, the number of 12-tone EqualTempered semitones for any interval is equal to:

†

Equal Tempered semitones =log interval( )log 1.05946( )

So for the transversely vibrating bar the intervalbetween the first and second mode is:

†

log 2.76( )log 1.05946( )

= 17.6 semitones .

And the interval between the first and third mode is:

†

log 5.40( )log 1.05946( )

= 29.2 semitones .

These clearly do not match the 12 semitones ofthe octave, the 24 semitones of two octaves or the 36semitones of three octaves, but there are othercombinations of consonant intervals that could beconsidered. 19 semitones would be equivalent to anoctave plus a fifth and 17 semitones would beequivalent to an octave plus a fourth, either of whichwould be consonant. And 29 semitones would beequivalent to two octaves plus a fourth. This is prettyclose to the 29.2 semitones of the third mode, but it’sreally a moot point because the third mode ends updying out so quickly anyway. The real concern is forthe second mode which is much more persistent. Thesecond mode is not only inharmonic; it isn’t evenmusically useful as a combination of consonantintervals. This causes unmodified idiophones to haveless of a clearly defined pitch than harmonic

Both Ends Free One End FreeMode Multiple

of f1

Mode Multipleof f1

1 1 1 12 2.76 23 5.40 34 45 5


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instruments. However, a simple modification can bemade to the bars of xylophones and marimbas tomake the second mode harmonic.

Figures 7.1 and 7.2 show that the center of thetransversely vibrating bar is an antinode in the firstmode and a node in the second mode. Carving outsome of the center of the bar makes it less stiff anddecreases the frequency of the first mode. However, ithas little effect on the second mode, which bends theparts of the bar away from the center. An experiencedmarimba builder can carve just the right amount of

wood from under the bars so that the first modedecreases to one-quarter of the frequency of the secondmode (see Figure 7.5). The xylophone maker carvesaway less wood, reducing the frequency of the firstmode to one-third the frequency of the second mode.Both modifications give the instruments tones thatare clearly defined, but the two octave differencebetween the first two modes on the marimba gives ita noticeably different tone than the xylophone’soctave-plus-a-fifth difference between modes.

Figure 7.5: The second mode o fxylophone and marimba bars i smade harmonic by carving woodfrom the bottom center of the bar.This lowers the fundamentalfrequency of the marimba bars t oone-quarter the frequency of thesecond mode and lowers thefundamental frequency of thexylophone bars to one-third thefrequency of the second mode.

David Lapp

View a 20 minute movie about the Javanese Gamelan. Click the box above.

HHHAAARRRMMMOOONNNIIICCCAAA PPPHHHYYYSSSIIICCCSSS

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INVESTIGATIONTHE HARMONICA

INTRODUCTIONMany people may consider the harmonica to be a

wind instrument. But the harmonica makes all itssound by means of reeds, bound at one end, that aredriven to vibrate by the breath of the player. There areno resonant tubes or pipes. In the photograph aboveyou can see ten holes in the front of the harmonicathrough which you can either blow air into or drawair from.

The harmonica’s history can be traced back to amusical instrument called the “aura.” In 1821, 16-year-old Christian Friedrich Buschmann registered theaura for a patent. His instrument consisted of steelreeds that could vibrate alongside each other in littlechannels. Like the modern design, it had blow notes.However, it had no draw notes. Around 1826 amusical instrument maker named Richter changed the

design to its modern style – ten holes with 20 reeds,and tuned to the diatonic scale (Buschmann’s designused the chromatic scale).

For as little as $2.00 apiece educators maypurchase inexpensive harmonicas from Hohner, Inc.

Hohner Contact :Johnna CossaboonMarketing Communications ManagerHohner, Inc./HSS1000 Technology Park DriveGlen Allen, VA 23059804-515-1900 ext. 3043804-515-0840 FAX


103

QUESTIONS AND CALCULATIONS

1. If the chrome cover is taken off the harmonica, the reed plates can be seen, joined to the comb. The topphotograph shows the top of the harmonica. Notice that you can see the channels in which the blow reedsvibrate. The reeds are attached tothe underside of the metal plateand if you look carefully, youcan see the point of attachment.The free end of the blow reed isthe end farthest from where themouth blows. If the harmonicais now turned over, it appears asthe bottom photograph and youcan see the draw reeds above thechannels in which they vibrate.You can also see the points ofattachment. The free end of thedraw reed is the end closest towhere the mouth blows.Plucking the reeds produces asound that is hardly audible.a. Use the photographs to

explain how the reeds aredriven to vibrate and whatthe mechanism is for the production of sound from the harmonica?

b. Why must the reeds be mounted on the particular sides of the reed plates that they are attached to?

comb Reed point of attachment

Topview

Bottomview


104

2. A close look at the draw reedsshows that they all have evidenceof being filed (the blow reeds dotoo). Some of the reeds are filedup near the point of attachmentand others have been filed at theirfree ends. Some reeds are filed inboth regions. The file markshave the appearance of beingrandom, but they are actually intentional. Filing theend of a reed causes it to behave like a shorter reed.Filing the reed near the point of attachment causes itto behave like a thinner reed.a. What does filing the end of a reed do to the

frequency of the reed when it is vibrating?

b. What does filing the reed near the point of attachment doto the frequency of the reed when it is vibrating?

c. In the magnified section, you can see that the reed on the left has been filed at both the end and near thepoint of attachment. The end shows a greater degree of filing. Propose a scenario to explain why bothparts of the reed would be filed and why the end would have a greater degree of filing done on it.


105

3. The lowest note on this harmonica is C4. The length of this reed is 1.60 cm. How long would a reedidentical to those in this harmonica need to be to play C3?

4. The blow and draw notes for the C major harmonica are shown on the photograph on the first page of thisactivity. There are three ways to look at the organization of notes on the harmonica. One way is as asystem of chords (two or more consonant notes played at the same time). A second way to look at the notestructure is by thinking of it as based around the major scale. Finally, the harmonica can be thought of interms of an abundance of octaves. Explain how each of these viewpoints is valid. Remember that you canget your mouth around multiple holes and you can also use your tongue to block air from moving throughunwanted channels.a. A system of chords

b. The major scale

c. An abundance of octaves


106

5. The figure below shows an E major harmonica. The first draw hole note is an E. Use your answers from theprevious question to decide on and fill in the other blow holes and all the draw holes. Use the space belowthe figure to fully explain your rationale.

E

E

MMMUUUSSSIIICCC BBBOOOXXX PPPHHHYYYSSSIIICCCSSS

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INVESTIGATIONTHE MUSIC BOX ACTION

INTRODUCTIONThe heart of any music box is the action. One part of the action consists of a small metal cylinder with tiny

metal projections attached to it. The other part of the action is a flat piece of metal with a set of thin metal tines,each of which can be plucked by the metal projections on the cylinder as it turns. In most cases the drum is attachedto a coiled spring, which can be wound up to provide energy for the drum to rotate for a few minutes. When theaction is firmly attached to the music box and wound up, it will play a clearly audible and recurring tune during thetime the drum is in motion. Click here to hear this music box action. An inexpensive action can be purchased for$3.75 or only $1.85 each when ordering 50 or more at: http://www.klockit.com/product.asp?sku=GGGKK&id=0309301913064849755057

Rotating metalcylinder withmetal stubs.

Metal platewith bendablemetal tines.

Spring-drivendrive turns themetal cylinder

Gap betweenthe cylinderand the tines istoo small forthe metal stubsto pass freely.

http://www.klockit.com/products/dept-166__sku-GGGKK.html


108

QUESTIONS AND CALCULATIONS (SHOW ALL WORK)1. What type of musical instrument classification fits the music box action? Be specific, indicating the

vibrating medium.

2. What is the means for changing the pitch of the music box?

3. How many different tones are possible with this particular music box action? Explain.

4. Each tine actually creates a pitch one semitone different from its adjacent tines. The longest of the tinescreates a C5 pitch. If the tines were all uniformly thick, what percentage of the length of the longest tinewould the shortest tine be?


109

5. The photograph to theright shows theunderside of the metalplate holding the tines.The measurementincrements aremillimeters. How doesthe percentage of thelength of the shortesttine compare to yourcalculation in theprevious problem.

6. Explain the reason forthe discrepancy.

For the last two problems, assume the metal is steel and the tines are uniformly thick.7. What would be the thickness of the longest tine?

8. What note would the shortest tine actually play?


110

BUILDING A COPPER PIPE XYLOPHONE

OBJECTIVE:To design and build a xylophone-like musical

instrument based on the physics of musical scales andthe physics of the transverse vibrations of bars andpipes.

MATERIALS:• One 10-foot long piece of 1/2” copper pipe• Pipe cutter• Rubber foam self stick weather seal• Metric ruler or tape

PROCEDUREThe 10-foot piece of copper is long enough to

make a full C5 (or higher) major scale, plus one pipean octave higher than the lowest note. There will stillbe pipe left over to use as a mallet. To determine thelengths of the pipes, you will use the equation for thefrequency of a pipe with transverse vibrations:

†

fn =pvL K8L2 m2

1. Measure the inner and outer diameter of the pipe to the nearest half-millimeter. Use the corresponding radiito calculate the radius of gyration (K)

2. Decide what notes the xylophone will play. The xylophone pictured above consists of the C5 major scaleplus a C6 note (C5, D5, E5, F5, G5, A5, B5, C6) on the Equal Tempered scale. If a full chromatic scale ormore than one octave is desired a higher scale must be chosen.

3. Calculate the frequencies you will use. Do all calculations and check work b efore making any cuts .

4. The speed of sound in copper is 3650 m/s. Calculate the length of each of your pipes to the nearestmillimeter, assuming the first mode of vibration.

5. Carefully cut each of the pipes. Make sure that each pipe is cut precisely to the number of millimeterscalculated. Mistakes in this part of the procedure are not correctable and will be audibly detectable.

6. Measure in 22.4% of its length from both ends of each pipe and place a mark. These are the positions of thenodes in the standing wave for the first mode of vibration.

7. Cut two 3-cm pieces of the foam weather strip for each pipe and attach at the nodes of the pipes. Place eachpipe on a surface so that it rests on the weather seal.

8. Cut two 20-cm pieces of copper pipe from what is left of the 10-foot piece. These will be the mallets forthe xylophone. Wrap one end of each mallet several times with office tape. The more tape used, themellower the tone.

9. Play the xylophone! You can play tunes and experiment by listening to octaves, fifths, fourths, and otherintervals.

This “xylophone,” made from a s ingle10-foot length of 1/2” copper pipe playsthe C5 major scale.

REFERENCESAUTHOR TITLE PUBLISHER COPY-

RIGHTCOMMENTS

Askill, John Physics of Musical Sounds D. Van Nostrand Company,Inc., New York

1979 Chapter 5 has the nicest inductive approach to scale buildingof the bunch. Sound spectra provided for dozens ofinstruments. Lots of good practice questions at the ends of thechapters.

Backus, John The AcousticalFoundations of Music

W. W. Norton & Company,Inc., New York

1969 This is one of the “classics” on musical acoustics. It iscomprehensive, but very readable. Backus gives considerabledetail to the sound production of many instruments.

Benade, Arthur H Fundamentals of MusicalAcoustics

Oxford University Press,New York

1976 This is the most classic book on musical acoustics. Chapters17 – 24 give an exhaustive, yet very readable treatment ofmany musical instruments. Benade’s experimental results arewidely used as references in other books, but the treatment ismore complete here.

Berg, Richard EStork, David G

The Physics of Sound Prentice Hall, Inc., NJ 1982 Chapter 9 covers musical scale development very completely.Chapters 10 – 14 also do a very complete conceptual coverageof various classes of musical instruments.

Fletcher, Neville HRossing Thomas D

The Physics of MusicalInstruments

Springer-Verlag, New York 1998 Especially technical treatment from first principles of mostclasses of musical instrument. Its mathematical rigor makes itunsuitable for those without an extensive and strongbackground in calculus.

Gibson, George NJohnston, Ian D

“New Themes and Audiencesfor the Physics of Music”

Physics Today January2002

A seven-page article discussing the development of auniversity musical acoustics course for non-science majors.

Hall, Donald E Musical Acoustics (thirdedition)

Brooks/Cole, Pacific Grove,CA

2002 A very thorough modern textbook with an excellent breadth ofand depth into musical topics. End of chapter exercises andprojects are perfect for generating ideas. Highly recommended.

Hopkin, Bart Musical Instrument Design See Sharp Press, Tucson, AZ 1996 An excellent resource for those interested in building a widerange of musical instruments with lots of “tricks of the trade.”It is not just a “cookbook” style of construction, but explainsthe physics as well.

Hopkin, Bart Making Simple MusicalInstruments

Lark Books, Asheville, NC 1995 Great illustrations and photographs complement thiscollection of about 20 very eclectic instruments. Step-by-stepinstructions, as well as difficulty level, accompany eachinstrument.

111

Hutchins, Carleen M(providesintroduction)

The Physics of Music:Readings from ScientificAmerican

W. H. Freeman andCompany, San Francisco

1978 Five slim chapters on different classes of instruments that areboth conceptually and technically very strong and appropriatefor the high school level.

Johnston, Ian Measured Tones: TheInterplay of Physics andMusic

Institute of PhysicsPublishing, London

1989 Pretty standard content for a book on musical acoustics, butnot as thematic and predictable as other similar works.Johnston gives considerably more attention to the historicaldevelopment of the physics of music.

Moravcsik, Michael J Musical Sound: AnIntroduction to the Physicsof Music

Paragon House Publishers,New York

1987 This book assumes the reader has NO knowledge of music,physics, or any math higher than simple computation. It is agreat comprehensive reference for the non-science reader.

Neuwirth, Erich Musical Temperaments Springer-Verlag/Wien, NewYork

1997 A thin volume (70 pages) that provides a very completecoverage of the characteristics of musical scales, but nothingelse. Comes with a CD-ROM (PC only) containing soundfiles.

Rossing, Thomas DMoore, Richard FWheeler, Paul A

The Science of Sound (thirdedition)

Addison Wesley, Reading,MA

2002 This is an exhaustive “must have” resource. The Science ofMusic and Musical Instruments would describe the bookbetter. Well suited for high school through graduate school.Highly recommended

Sethares, William A Tuning, Timbre, Spectrum,Scale

Springer-Verlag, London 1997 Chapter 3 (Musical Scales) does an excellent andmathematically exhaustive treatment of various scales. Therest is beyond the scope of the high school student.

Taylor, Charles Exploring Music: TheScience and Technology ofTones and Tunes

Institute of PhysicsPublishing Inc., Philadelphia

1992 Excellent explanation of the shortcomings of various scales.Excellent survey of the physics of many instruments.

Waring, Dennis Making Wood FolkInstruments

Sterling Publishing Co.,Inc., New York

1990 Chapters for simple stringed, wind, and percussioninstruments as well as a chapter for more complex stringedinstruments. Lacking physical explanations, it is fairly“cookbook” in style. A good source for ideas though.

White, Harvey EWhite, Donald H

Physics and Music:The Science of MusicalSound

Saunders College/ Holt,Rinehart and Winston,Philadelphia

1980 This is really the Physics of Waves with a strong emphasison Music. Good, comprehensive coverage, but the DopplerEffect and Interference seem a bit out of place here.

Yost, William A Fundamentals of Hearing Academic Press, San Diego 2000 Technical and exhaustive treatment of all aspects of hearing.Teachers may be most interested in the physiology of the ear,which is treated here in much greater depth than in a typicalacoustics book

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PHYSICS OF MUSIC AND MUSICAL INSTRUMENTS WEBSITESWebsite Address Description Comments

http://asa.aip.org/discs.html Auditory Demonstrationscompact Disc

This is the least expensive ($26) place to find this CD-ROM of very usefulauditory demonstrations. It includes 39 demonstrations and is produced by theAcoustical Society of America.

http://hyperphysics.phy- astr.gsu.edu/hbase/sound/soucon.h tml#c1

The sound and hearing portion ofGeorgia State University’sHyperphysics project

Nice tutorials on many topics within sound and hearing, including much on thephysics of music. Concise, navigable, and some interactivity. Students wouldbenefit from exploring this site.

http://www.angelfire.com/tx/myq uill/Harmonica.html

Diatonic Harmonica Reference A comprehensive treatment of the harmonica

http://www.bashthetrash.com Musical instrument constructionfrom recycled/reused materials

This is a non-technical, but useful site to get some ideas for building differenttypes of musical instruments from very inexpensive items (many you wouldnormally throw away) around most any house.

http://www.carolenoakes.co.uk/In strumentLibrary.htm

Photographs of musicalinstruments

There are many high quality photographs here of musical instruments by theirclass. While there are no descriptions, all images are clickable to reveal largerversions

http://www.exhibits.pacsci.org/m usic/Instruments.html

Musical instrument families This is a nice glossary of information about the various musical instrumentfamilies that includes the following sections for each family:

• How the sounds are made• How the pitch is changed• Examples of instruments in the family

http://www.klockit.com/product.a sp?sku=GGGKK&id=0309301913 064849755057

Music box actions A page from Klockit.com where music box actions can be purchased for $3.75each or as low as $1.85 each when purchased in lots of 50 or more.

http://www.mfa.org/artemis/collec tions/mi.htm

Online collection of musicalinstruments from the BostonMuseum of Fine Arts

A wide variety of musical instruments from many cultures can be viewed here.Images are high resolution and can be zoomed.

http://www.physicsclassroom.co m/Class/sound/soundtoc.html

A tutorial on sound waves andmusic from The PhysicsClassroom

A fairly detailed set of tutorials geared for the high school audience. Goodreinforcement with many animations and multiple choice, check-yourselfproblems.

113

http://asa.aip.org/discs.html

http://www.angelfire.com/tx/myquill/Harmonica.html

http://hyperphysics.phy-astr.gsu.edu/hbase/sound/soucon.html#c1

http://home.earthlink.net/~jbertles/

http://www.carolenoakes.co.uk/InstrumentLibrary.htm

http://www.exhibits.pacsci.org/music/Instruments.html


http://www.mfa.org/artemis/collections/mi.htm

http://www.physicsclassroom.com/Class/sound/soundtoc.html

http://www.ehhs.cmich.edu/~dhav lena/

Homepage of an amateur musicalinstrument maker

There are dozens of plans here for making simple musical instruments. There arealso lots of links to other musical instrument building sites.

http://www.phys.unsw.edu.au/mu sic/

Homepage for the University ofSouth Wales Music Acousticswebsite

An excellent site for in depth study of many classes of instruments and individualinstruments. There are many sound files (mp3’s and wav’s) that are available asdemonstrations. Many, many good links as well.

http://www.Pulsethemovie.com/ Homepage for the large screenmovie, Pulse

An excellent movie that explores the rhythmic music from many differentcultures. It will change the way in which you look at and describe music. Thewebsite includes a 25-page curriculum guide probably most appropriate for middleschool students.

http://www.stomponline.com/ Homepage for the stageperformance, STOMP

The performers do a 95-minute show celebrating rhythm. As the website says,“STOMP is a movement, of bodies, objects, sounds - even abstract ideas. Butwhat makes it so appealing is that the cast uses everyday objects, but in non-traditional ways.” The website also has educational materials related to the show.

http://www.synthonia.com/artwhi stling/

A history of and physicaldescription of human whistling.

This is a mostly historical, but comprehensive coverage of Kunstpfeifen(“artwhistling”). Very little is written about one of the most available and usedmusical instruments – the human whistle. This is a nice overview.

http://www.tuftl.tufts.edu/mie/ Homepage for the MusicalInstrument Engineering Programat Tufts University

The Musical Instrument Engineering Program at Tufts University is a both aresearch and teaching program. It contains useful information for those interestedin developing a program for designing musical instrument and understandingmusical acoustics.

http://www.windworld.com Homepage for ExperimentalMusical Instruments

Experimental Musical Instruments is an organization devoted to unusual musicalsound sources. There is a quarterly newsletter, an online store with many books,CD’s, and supplies, lots of interesting articles and an especially impressive list oflinks to those active in the field of experimental musical instruments.

114

http://www.ehhs.cmich.edu/~dhavlena/

http://www.phys.unsw.edu.au/music/flute/modernB/G4.html

http://www.Pulsethemovie.com/

http://www.stomponline.com/

http://www.synthonia.com/artwhistling/

http://www.tuftl.tufts.edu/mie/

http://www.windworld.com/

PHYSICS OF MUSIC RESOURCE VENDORS

VENDOR PRODUCT AND COST COMMENTS

Arbor ScientificPO Box 2750Ann Arbor, MI 48106-2750800-367-6695http://www.arborsci.com/Products_Pages/So und&Waves/Sound&WavesBuy1B.htm

Sound Pipes1-11 $2.75 each12 $1.95 each

These open-ended corrugated plastic pipes are about 30” long and 1” in diameter. Afundamental tone is produced when the sound pipe is twirled very slowly. Twirlingat progressively higher speeds allows for the production of the next four harmonics(although the advertised fifth harmonic is quite hard to get). These are a lot of funand very useful for talking about harmonics and the physics of open-end pipes.

Candy Direct, Inc.745 Design Court, Suite 602Chula Vista, CA 91911619-374-2930http://www.candydirect.com/novelty/Whistle- Pops.html?PHPSESSID=f5df897f67e7d525ea dbcedfe4703e96

Whistle Pops32 $16.45

These are candy slide whistles that really work. However, since they are candy(except for the slide), once you use them you have to either eat them or dispose ofthem.

Century Novelty38239 Plymouth RoadLivonia, MI 48150800-325-6232http://www.CenturyNovelty.com/index.aspx? ItemBasisID=444478&IndexGroupID=27&Ite mID=259422 for siren whistlesorhttp://www.CenturyNovelty.com/index.aspx? ItemBasisID=824&IndexGroupID=27&ItemI D=744 for two tone whistles

2” Siren Whistles1-11 $0.59 each12-35 $0.40 each36+ $0.30 each

Two tone Whistles1-11 $0.12 each12-35 $0.10 each36+ $.06 each

These 2” long Siren Whistles are fun to play with, can be used to discuss theproduction of sound, and can be used to discuss the siren effect that harmonicas andaccordions use to produce tones.

The Two Tone Whistles are 2.5” long and almost an inch wide. They are cheaplymade, but can still be used when discussing closed pipes, beats, and dissonance.

Creative Presentation Resources, Inc.P.O. Box 180487Casselberry, FL 32718-0487800-308-0399http://www.presentationresources.net/tfe_mus ic_noisemakers.html

Groan Tubes$1.29 each

Plastic Flutes$0.99 each

The Groan Tubes are 18 inches long and have a plug inside that can slide down thetube whenever it is inverted. The plug is equipped with a reed that vibrates as airmoves past it as the plug slides down the tube. The tube is closed acoustically atone end and open at the other. The changing pitch produced as the plug moves downthe tube makes it useful for investigating the relationship between pitch and lengthin a closed pipe.

http://www.arborsci.com/Products_Pages/Sound&Waves/Sound&WavesBuy1B.htm

http://www.candydirect.com/novelty/Whistle-Pops.html?PHPSESSID=f5df897f67e7d525eadbcedfe4703e96

http://www.CenturyNovelty.com/index.aspx?ItemBasisID=444478&IndexGroupID=27&ItemID=259422

http://www.CenturyNovelty.com/index.aspx?ItemBasisID=824&IndexGroupID=27&ItemID=744

http://www.presentationresources.net/tfe_music_noisemakers.html

Hezzie Group3322 Sleater Kinney Road NEOlympia, WA 98506360-459-8087http://www.hezzie.com/cgi- bin/shop.pl/page=hezzieinstruments.html

Plastic Slide Whistles1-23 $1.2524 $20.00

The Slide Whistles are very cheaply made and do not work very well as slidewhistles. However, if the end of the whistle is cut off and the slide removed, thewhistles can be used to explore the differences between open and closed pipes (seethis article).

Johnna CossaboonMarketing Communications ManagerHohner, Inc./HSS1000 Technology Park DriveGlen Allen, VA 23059804-515-1900 ext. 3043

Harmonicas$2.00 each

(This is an educational pricethat you can receive bycontacting the MarketingCommunications Managerdirectly)

These harmonicas are exceptionally well made for the cost. They can be dissected toinvestigate the physics of vibrating bars. Although the harmonica’s reeds are drivencontinuously with air, as opposed to the plucked bars in a thumb piano the physicsis very similar. Additionally, the process of tuning a bar can be investigated, sincenearly all the reeds show evidence of scraping at the base or the tip in order to flattenor sharpen the note. Finally, musical intervals can be investigated while studyingthe rationale for the placement of the notes.

KlockitN2311 County Road HP.O. Box 636Lake Geneva, WI 53147800-556-2548http://www.klockit.com/products/dept- 166__sku-GGGKK.html

Music Box Actions1-4 $3.75 each5-9 $2.9510-24 $2.5525-49 $2.2550+ $1.85

The Music Box Actions are an excellent resource for both introducing the physics ofsound production, the use of a sound board for amplification, and for showing therelationship between the length of a vibrating bar and the corresponding frequency ofvibration. There are many tunes to choose from and orders of mixed tunes can bemade. Choose actions with housings that can be removed.

Talking Devices Company37 Brown StreetWeaverville, NC 28787828-658-0660http://www.talkietapes.com/

Talkie Tapes (sample packs)2 $2.0050 $20.0075 $30.00100 $40.00125 $50.00Lots of other orderingoptions are also available

These Talkie Tapes are 18” long red plastic strip audio recordings. They have ridgeson one side and running a thumbnail over the ridges causes an audible message to beheard (“happy birthday,” for example). Doing this while holding the tape against aplastic cup or piece of paper causes considerable amplification. These are verysurprising for those who have never seen them before and an excellent tool fordiscussing sound production, sound-board amplification, and the physics ofphonograph records..

The Nash Company2179 Fourth StreetSuite 2-HWhite Bear Lake, MN USA 55110http://www.nashco.com/noseflutes.html

4 $3.408 $6.4012 $9.2524 $18.0048 $35.00100 $70.00

Nose Flutes have physics that is similar to the slide whistle. The one-piece plasticdevice fits over the nose and partially open mouth. Blowing gently through the nosecauses air to be directed across the opening of the mouth. Changing the shape(length) of the mouth (which is the resonant cavity in this wind instrument) causes achange in pitch. This provides for a very tactile reinforcement of the relationshipbetween the length of a closed pipe and its corresponding resonant frequency.Extensions can also be made to the physics of whistling.

http://www.hezzie.com/cgi-bin/shop.pl/page=hezzieinstruments.html


http://www.talkietapes.com

http://www.nashco.com/noseflutes.html

“In most musical instrumentsthe resonator is made ofwood, while the actual soundgenerator is of animal origin.In cultures where music isstill used as a magical force,the making of an instrumentalways involves the sacrificeof a living being. That being’ssoul then becomes part of theinstrument and in times thatcome forth the “singingdead” who are ever presentwith us make themselvesheard.”- Dead Can Dance

The Physics of Music and Musical Instruments

Documents

mus ic box

linear mass

skugggkkamp

comma meantone

standing wave

experimental

tacoma narrows

perceptible