HE DEVELOPMENT AND HISTORY of the vio-lin covers much the same time frame as contemporaryphysics. While Newton was developing his classicalmechanics and formulating how the planets move in thesky, Antonio Stradivari was creating some of the worlds

greatest and most treasured violins in Cremona, Italy. Thisinstrument had evolved from the viol familyabout one hundred years earlier and was givenits final form by another Italian maker,Gasparo da Salo, and his student GiovanniMaggini. Like the modern-day electric guitar,this evolution was driven by the need for alouder sound to accommodate public func-tions. It was an instrument of the masses,unsophisticated and brash. The viol remainedthe refined favorite of the royal courts.

Salos new design was seen and copied by a Cremonesemaker, Andrea Amati, whose sons continued their fatherslead and perfected the violins design. The question ofwhether Stradivari was an apprentice in the Amati shop restson a single label in one of his violins attributing its style toAmati. What we know for sure is that both lived in Cremonain the second half of the 1600s and that Stradivari began mak-ing violins in the Amati style. He soon branched out,however, and went on to a long and illustrious career duringwhich he produced over a thousand instruments.

By 1750 the violin had gained wide acceptance. Indeed,most famous composers of the day produced works that fea-tured this instrument. It was their demands that led to somelast modifications. The definition of pitch was raised and theneck (and hence the string length) was lengthened, allowing

20 SUMMER 1998

A Physicist in the World of Viby WILLIAM ATWOOD

A physicist applies well-known

principles to the construction

of violins and in the process comes

to a better understanding

of techniques for shaping their sound.

BEAM LINE 21

a greater range in notes.Unfortunately, the originaltechnique of simply nail-ing and gluing the neckonto the body proved too weak to counterbalance the pull ofthe strings. In response, in the late 1700s the maker Man-tegazza developed a method of mortising the neck into thesides and top block for added strength. The modern violinhad been born. However, violins made during the next twocenturies failed to achieve the success of the earlier Italianmakers, leading many to believe that a secret or secrets hadsomehow been lost.

Yet, progress was made during this period in understand-ing how bowed instruments produced their characteristicsound. The German physicist Hermann von Helmholtz iscredited with explaining bowed string motion. The condi-tions of fixed length and tension and of continuous sound(periodicity) mean that the only frequencies present could beintegral multiples of the fundamental (that is, the lowest al-lowed vibration mode). Musically this means that for a givennote, its octave, the fifth above that, the next octave, thethird above that, and so on are part of the sound. Helmholtzrecognized that the timbre of the violin sound must be veryrich in these harmonics, and he became curious as to howthe moving string produced them. The string, he determined,does not just wag back and forth upon being agitated by thebow. Rather a sharp kink makes round-trips between thebridge and the nut, flipping over at each reflection (see theabove illustration). As the kink passes by the contact pointof the bow on its way to the bridge, it first releases the gripof the bow hairs on the string and then an instant later grabsthem again. The result is a continuous plucking action oftenreferred to as slip stick. The kink travels in an apparent

lins KinkBridge Nut

An example of Helmholtz string motionwhere a kink appears to rotate aroundthe string, alternately bouncing off thebridge and the nut.

22 SUMMER 1998

profound influence on the sound butalso are structurally critical as thedownward force of the bridge owingto the tension of four strings is abouttwenty pounds.

The violin body is basically a res-onator driven by the vibrations fromthe bridge induced by the aforemen-tioned Helmholtz-type string mo-tion. However, only those frequen-cies that approximately match thenatural resonances of this systemwill result in any appreciable sound.

The frequency response can bemeasured by vibrating the bridgewith a pure sine wave (no harmon-ics). This is accomplished by at-taching a lightweight transducer tothe bridge and sweeping the drive fre-quency while keeping the amplitudeconstant. The relative sound out-put can then be recorded (see left il-lustration on the opposite page).While the data in this plot are in de-tail complex, some overall featuresare easily understood. First there isa forest of resonances extending fromabout two hundred hertz up to aboutfifteen kilohertz. Although there aremany sharp gaps, most notes gener-ally sit sufficiently close to a reso-nance to be excited. The lower endof this range is quite curious: the bot-tom few notes on a violin (G at 196Hz and G# at 208 Hz) fall off the edgeof the response envelope. Somehowour ears are able to infer these notesfrom the higher harmonics they con-tain. Also, investing money in homeaudio equipment with responsesabove fifteen kilohertz wont makethe violins sound better!

Each note on a violin has its ownparticular sound. The fundamentalas well as its harmonics will fall invarious places in the response

clockwise direction for down bowsand counter clockwise for up bows.As a result, when drawn across a lev-el string, the bow tends to skate to-wards the bridge on an up-bow and to-wards the fingerboard on a down-bow.

Following the Second World War,a renaissance in violin making beganwith luthiers bent on recapturing theprecision and artistry of former yearsand a new interest from scientistsabout how the violin works. Sever-al new schools dedicated to the con-struction of stringed instrumentsnow exist in the United States. Andthere are organizations with refereedjournals focused mainly on the sci-ence of the instrument. In this stim-ulating environment, a young and en-thusiastic group of makers is argu-ably producing violins that may infact rival the old Italian instruments.

FUNDAMENTALSOF A WORKING VIOLIN

The illustration at the top of the pageis a cross section of the violin at thelocation of the bridge. The body is es-sentially an empty, thin-walled box.The top plate, or belly, that pro-duces most of the sound is typicallytwo to three millimeters thick whilethe back is similar in thicknessaround the edges but can be five mil-limeters or more near the center. Thewalls are approximately one mil-limeter thick except for the liningsthat reinforce the glue joint to thebelly and the back. Under the footof the bridge corresponding to the Estring, a small dowel mechanical-ly couples the top to the back. Un-der the other foot is a longitudinalstiffener called the bass bar. Both ofthese components not only have a

Bridge

Bass Bar

Sound Post

f Hole

A cross section of the violin at thebridge.

BEAM LINE 23

spectrum and be amplified accord-ingly. In the right-hand figure abovethe waveform of the sound of a vio-lin being bowed on the open E stringis shown fit to a harmonic series. Forthis note the fifth and sixth har-monics are three and two times larg-er than the fundamental. The appar-ent rapid oscillations are the resultof these harmonics, slowly interfer-ing with each other while the (slow)overall envelope modulation is at thefundamental frequency of 660 Hz.Yes, we hear E, but we also veryquickly identify the note as comingfrom a violin. It is the presence ofthese harmonics that identifies thesource of the sound for us. If we re-peated the above exercise for othernotes, similar results will be ob-tained; however, the harmonic con-tent (which harmonics are impor-tant) will change.

To achieve good projection, onewants positive pressure waves to beproduced simultaneously from thetop and back of a violin. You mightvisualize this condition as the violinalternately swelling and shrinking.Jack Fry, another noted physicist,pointed out in the 1970s that thearrangement of sound post and bassbar inside a violin resulted in thistype of radiator. But certainly not allof the resonances can be so simple.Various places on the surfaces of thetwo plates will be moving in and outof phase and hence form higher pole

radiator patterns. For each note, var-ious radiation patterns will empha-size various harmonics when mea-sured at different locations. AsGabriel Weinreich recently observed,perhaps this is part of the intrigu-ing aspect of the violin sound. . .itseems to dance and sparkle, comingall at once from some place and atthe same time from no place inparticular!

APPLYING PHYSICSTO MAKING VIOLINS

A physicists approach to learningabout a new physical system oftenstarts with fundamentals such ashow big, how small, what controlsthe overall behavior, etc. John Schel-leng pursued this type of dimensionalanalysis and found an overall figureof merit for violin wood: the ratioof the density to the velocity ofsound. One can intuitively under-stand this ratio is important by ob-serving that if, for instance, the topweighs a lot, it will require a hardshove to get it moving. But, just be-ing light is not sufficient. We wantthe entire top plate to movenot justthe local area under the feet of thebridge. Hence, the stiffness is also ofprime importance.

It is interesting to plot the den-sity versus velocity for various typesof wood (see illustration next page).Along straight lines emanating from

90

80

70

600.30.2 0.5 1 2

Frequency (kilohertz)3 5 10

Am

plit

ude

(d

ecib

els)

These two plots illustrate aspects of vio-lin sound. On the left is the frequency re-sponse of a Guarnerius del Ges violin.A dense forest of resonances coversthe frequency band from approximately220 Hz to 15 kHz. On the right, thewaveform of a single note (E) played ona violin is shown. The pronounced rapidoscillations are due to the fifth and sixthharmonics, while the slow, overall enve-lope modulation is at fundamental fre-quency (660 Hz). (Courtesy CarleenHutchins)

0 4 6 82

Time (m sec)

0

10

20

30

10

20

30

Am

plit

ude

(a

rb. u

nit

s)

Ala

n D

. Ise

lin

24 SUMMER 1998

A modern mechanical engineermight view the violin as an exampleof thin shell technology. Violin platesbegin as thick pieces of wood. The fi-nal cross-sectional shape shown inthe figure on page 22 is arrived atthrough a two step-carving process.First the outside is shaped. The curve(called the arching pattern) is usu-ally copied from one of the old Ital-ian masters. While these arching pat-terns may appear the same to theuntrained eye, there can be a widevariety in the maximum height andhow full the curve is. Height is ameasure of how much the center sec-tion puffs up from the plane of thesides (usually 1416 mm). The backof the instrument can have a differ-ent height than the front. The full-ness of the arch refers to how farthis puff extends toward the edges.Swoopy arches start their descentcloser to the center. Full arches (de-scent closer to the edge) tend to bestronger and more resistance to long-term deformation owing to the sta-tic loading produced by the strings.

After shaping the outside, the in-side is scooped out. Heres where thefun begins! How thick should thewood be and what thickness patternshould be used. This is complicat-ed and still only poorly understood.Since the material is wood and there-fore anisotropic with variations inphysical properties from piece topiece, any recipes based purely ondimensions will fail. Somehow onemust use the vibrational propertiesof the individual plate to guide thethinning (or graduating) process.

Traditionally makers have use atechnique based on tap tones.Holding the free plate off center inthe upper area with the thumb and

the origin the ratio is con-stant: large values beinggood. Balsa wood seemsto be the material ofchoice! Of course werenot even suggesting thisoption since, as alreadymentioned, the static load-ing of twenty pounds fromthe downward force of thebridge would tax thestrength of balsa. Not farbehind, however, isspruce. This is the mate-rial of choice in practicallyall musical instrumentsthat have a wooden sound

board (pianos, guitars, harps, violins)and for the same reason (high stiffnessper unit weight) was the material ofchoice for aircraft before aluminumwas readily available (HowardHughes famous Spruce Goose).

However only the top of a violinis made of spruce. The rest is usu-ally made from curly maple, a highfigured hard wood. The above chartreveals that this material wasnt cho-sen on the basis of sound productionbut rather for its beauty and perhaps,more fundamentally, its strength (theaxial pull of the four strings combineto over sixty pounds). It also mattershow the wood is cut from the tree.Detailed experiments have revealedthat the velocity of sound decreasesas the alignment of the fibers deviatesfrom being parallel to the surface. Infact not only does the sound velocitygo down, but the rate at which vi-brations are damped-out increases.This becomes a noticeable effect fordeviations as small as a few degrees.Traditional makers have always pre-ferred wood split from logs rather thansawed. Now we know why.

6

7

5

4

3

Vel

ocit

y (

x105

cm

/sec

)

2

1

00 0.40.2

1815 12 10

87

65

4

3

0.80.6Density (g/cc)

1.0 1.2

1

2

3

4

5

6

7

8

9 10 12

1113

14 15

16 17

1819

1 Balsa 8 Pine 15 Sycamore2 Poplar 9 Larch 16 Rosewood3 Norway Spruce 10 Walnut 17 Oak4 Poplar 11 Amer. Mahogany 18 Pear5 Linden 12 Ash 19 Pernambuco6 Padouk 13 Norway Maple7 White Fir 14 Sycamore Maple

Data from Barducci and Pasqualini

The velocity of sound versus the densityis plotted for different types of wood.This ratio (velocity/density) is a goodfigure of merit for acoustical quality of amaterial, high values being best. Thedata points indicate typical values for aparticular wood; however, a variation of30 percent in both sound velocity anddensity is not uncommon.

BEAM LINE 25

forefinger and tap-ping near the cen-ter with the otherhand produces adistinct pitch thatcan be perceived asthe plate nears itsfinal thicknesses.Keeping the sameholding point andtapping at the cen-ter near the bot-tom reveals yetanother pitch.When old Italianinstruments are disassembled forrepair, their tap tones are often ex-amined. In particular, a famousmaker/restorer named Simone Sac-coni, who worked on many of the ex-isting Stradivari violins, reported thatall had tap tones between F and F#,strongly suggesting that this was partof the equation that evolved in Italythree hundred years ago.

Carleen Hutchins, who studiedunder Sacconi, developed a newmethod for measuring tap tone whilemaking instruments. She showedthat they corresponded to eigen-modes of the free plate, which can beseen clearly using the technique ofChladni patterns. Hutchins sus-pended a violin plate horizontallyabove a loud speaker connected to anaudio signal generator. When blackglitter is sprinkled on the surface,at resonance, the patterns shown inthe drawing on the right appear. Thenodal lines are the places where theplate is not moving and hence theglitter tends to accumulate. The tra-ditional tap-tone technique requiredholding the plate on a nodal line andtapping at an anti-node. This simpleprocess is in wide use today.

Tuning the tap tones has now beenreduced to a science, but free vio-lin plates are not an end in them-selves. The plates are eventually gluedto the sides, changing the boundaryconditions from free to somewherebetween hinged and clamped. Inves-tigations into the vibration modes forplates both free and clamped haveshown that the thickness of woodnear the edges of a plate is critical indetermining the eigenmodes for theclamped condition but has little ef-fect on the free plate. Wood removalnear the center controls the latter.

SETUP

Completing the violin after graduat-ing the plates involves a great dealof precision woodworking and thencomes the varnishing. The next phasein which physics can play a part isin the setup. This includes shapingthe fingerboard and bridge, fitting thesound post, adjusting the tailpiece,and so on. Much of how the finalproduct will sound is determined atthis stage.

The bridge of the instrument con-veys the vibrations from the strings Mode 1 Mode 2 Mode 5

The three main eigenmodes for a freeviolin top plate are shown below. Thefrequencies associated with thesemodes are approximately an octaveapart.

26 SUMMER 1998

While there has been littlequantitative work to detail whatsound a mode-matched instrumentmakes as opposed to an unmatchedone, both players and listeners pre-fer the former. The frequency of theair cavity mode is determined most-ly by the volume of the box and thesize and shape of the f holes. Littlecan be done to alter this bottle noteand indeed one finds little variationfrom instrument to instrument. Aclosely related note can be found byhumming into an f hole and findingthe pitch at which the body resonates(usually between C to C#). The firstbending mode has two transversenodal lines running across the widestsections of the upper and lower ar-eas. The pitch of this eigenmode canbe determined by holding the in-strument upside down at the widestpoint in the area next to the tailpieceand lightly tapping on the scroll. Itturns out that the fingerboard can bemodified to vary this note. By addingweight at the end near the bridge orby thinning the fingerboard in thearea where it joins the neck, the notecan be lowered.

The tailpiece also plays an im-portant role. The relative lengths ofthe strings between the bridge andthe nut at the far end of the finger-board and between the bridge and thetailpiece should be six to one. Thislength can be approximately tunedusing a ruler; however, a knowledgeof harmonics gives a simpler andmore accurate technique. A stringwhich is one-sixth the full lengthwill sound a note at the fifth abovethe second octave. Given that violinsare tuned in fifths means that byplucking the string behind the bridge

to the top. In some sense it may beviewed as an audio filter. Its tradi-tional and intricate shape was mostlikely arrived at through trial and er-ror with an eye on the esthetics andan ear on the resulting sound. Ob-servations of which bridges soundgood suggested an experiment us-ing similar tuning techniques tothose used for free plates. A high-power audio tweeter and audio gen-erator revealed that the best bridgeshad a simple bending eigenmodepitched at high F (2800 Hz).

After a final fitting with strings,the vibrational modes of the com-plete instrument can be investigatedand adjusted. When the lowest aircavity resonance and the lowestbody-bending mode have similar fre-quencies, the violin comes alive.

Both Helmholtz and Drell seem fascina-ted with bowed string motion. Here Drellis engaged in a real time analysis of thebows slip-stick interaction with thestring.

BEAM LINE 27

and comparing it to the note of thenext higher string plucked in front ofthe bridge an octave should be heard.The pitch of the note behind thebridge may be adjusted by changingthe length of the cord (called the tail-gut) fastening the tailpiece to the end-pin of the instrument. But the tail-piece also has resonances which havebeen found to have optimal values.Changing the tail-gut length changesthese pitches as well. The only re-maining free parameter is to adjustthe weight of the tailpiece: heavierlowers its resonance pitches whilelighter raises them.

PLAYING

One of the appealing attributes of theviolin is the variety of sounds. Thissound palette is manipulated throughthe bowing technique. The mainvariables are the speed of the bow,the downward pressure on the string,and the point of contact on the stringwith the bow hair. (To a lesser extentthe amount of hair which is in con-tact with the string also plays a role.)All this occurs essentially in a three-dimensional space. A two-dimen-sional projection of this bowing spaceis illustrated in the figure on theright. The softest sound is obtainedwith a slow, light bow halfway be-tween the fingerboard and the bridgewhile the loudest notes are producedwith a fast, heavy bow at a similarcontact point. The most variation inspeed and pressure can be found inthe middle of the range. Here too isthe largest range of possible contactpoints. Harsh, biting sounds may beproduced with the bow near thebridge, while soft, fluffy sounds are

obtained with the contact point near-er the fingerboard. By adjusting thesethree variables, the violinist is chang-ing not only the music dynamic butalso the timbre of the sound.

An obvious question is what hap-pens if you go outside the allowedregion in the bowing space? You pro-duce either an annoying scratchingsound or an equally unpleasantwhistling. The edges of this space aregiven by the boundaries for theHelmholtz-type slip stick actionas discussed in the introduction. Pre-ferred violins have a large bowingspace. These instruments allow theconcert artist the greatest range insound color and dynamic. A corol-lary is that cheap violins have a smallbowing space. This explains how thebeginning student, with little bowcontrol, may sound atrocious on a$200 fiddle provided by the parentswhile the teacher (with a great dealof bow control) can sound reasonablygood!

Physics or physics training doesnot give any intrinsic advantage toviolin making. Music is essentiallyesthetic and producing an instrumentnecessitates highly developed wood-working skills. However, physics canbe a tool for understanding what pa-rameters of the instrument controlvarious aspects of its sound, therebyproviding a guide for shaping it. It canalso inspire experiments with nar-row focus aimed at resolving specificissues in violin making. And, final-ly, good experimental science, specif-ically the recording of detailed noteson each instrument, can pay offhandsomely when trying to deter-mine what works or doesnt work.

Finger Board(Fluffy)

Bridge(intense)

Middle

Middle

Pressure(light) (heavy)

mf

ppp

fff

Spee

d(s

low

)(f

ast)

The bowing space for the violin is illus-trated above. The speed that the bow isdrawn across the string together with thedown pressure changes the musicaldynamic (shown inside the white dia-mond). The point of contact between thebow and string changes the color ofsound and has its largest range in themiddle (shown outside the whitediamond).