CHOICE OF STRINGS FOR INSTRUMENTS OF THE VIOLIN FAMILYUSING SIMPLIFIED THEORETICAL ASPECTS

Gerhard H. Muller and Helmut A. MullerMuller-BBM GmbH, Robert-Koch-Str. 11, D-82152 Planegg, Germany

Abstract: In choosing the strings, the violin makers have to take many aspects into consideration,because it influences the static forces within the instrument, the overall loudness, the balance betweenthe different strings, the attack and the timbre. In general, the instrument makers have to try differentstrings to find an appropriate set, taking into account the user's objections as well as his or her capability.Using the basic formulas for the eigenfrequencies, the vibrating forces acting on the bridge as well asinsights gained by sophisticated studies by many scientists, it becomes clear that in any case acompromise between the different aspects and their consequences is necessary for getting an optimum. Italso becomes evident that precise descriptions of the strings by their manufacturer, including the massper length andlor necessary tensile force and bending stiffness, makes the work of the violin maker mucheasier. Finally, it will be shown how the behaviour of the instrument represented by the mobility of itsbridge influences the sound of the strings.

INTRODUCTION

It is the intention of this paper to show, based on the knowledge about a vibrating string, howsimple rules for a violin-maker can be deducted from old scientific insights and how useful they are. Inaddition, in many cases it becomes clear why violin-makers carry out certain tests which they. derivedfrom experiences.

SELECTION OF STRINGS

Many of the viewpoints taken by instrument makers for choosing the strings can be derived fromthe basic formula for the eigenfrequencies of a string. A totally flexible string under tension, rigidly fixed

on both ends, has the following resonance- or eigenfrequencies: in = (nI2!) .JF I m' . It can vibrate in itsbasic frequency (n = 1) and all whole-numbered multiples called in its natural notes. These vibrationsdepend on the length 1, the tensile force F, and the mass per unit length m' of the strings.

The dependence on its length I is used for playing the instrument. The dependence on the force F isused for tuning the strings. The dependence on the mass per unit length m' is applied for adjusting thetensile force and the loudness in such a way that it does not differ too much for the four strings of a vio­lin, Viola or cello.

The radiated sound of an instrument depends on the sound energy transmitted into the corpus bythe vibrating strings and the radiation efficiency of the instrument. The transmitted energy into the sound~a~iating corpus of the instrument is proportional to the square of the vibrating forces acting on thefldge. In a first approximation, the peak value of the transverse fluctuating force Ft acting at the bridge

amOunts to Ft = JF. rn' (l/XB) VB = F 1(2 .fi .XB) VB·

b It is obvious, and in accordance with the experience of players, that the loudness depends on theow velocity VB and the position of the bow XB measured from the bridge.

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Adjusting the permanent forces on the instrument

Due to the necessary tension of the strings, there are enormous forces acting on the instrumentswhich have, due to acoustical reasons, very thin plates. TABLE 1 gives the ranges of the total tensileforces for the three classical string instruments together with the notes, frequencies fi and lengths I of thefree vibrating strings.

TABLE 1. Data of classical string instruments tuned in fifths

Instrument Pitch Length of the free Total tensile forcefrequency (Hz) vibrating strings I (mm) (N)

Violin 9 d' a' e"~325 ~160 - 280

~f 196 293 440 660

Viola c 9 d' a' ~370 ~180 - 290

3;£~f 130 196 293 440

Cello c G d a ~690 ~350 - 640

¥ r- EI

f 65 98 146 220

If one used the same strings within one instrument, there would be large differences in the tensileforces for the strings tuned in fifths. The d' string of a violin e. g. would have to be tightened more thantwo times, the a' string more than five times, and the e" string more than eleven times as much as the gstring. Therefore, the total force would not act parallel to the axis of the instrument. Another drawbackwould be that the loudness of the single strings would differ greatly.

The same force for the different tuned strings with the same length I could be achieved if the massper unit length m' of the strings behave like the inverse ratio of the squared frequencies or the squarednumerical intervals. This means m'2Im'\ = (fj/h)2.

If the strings are made out of homogenous material, such as gut, and if they have a circular cross

section, the mass per length amounts to rn' = p·A = p·n·crI4, where p is the density of gut (~ 1,3 g/cm\A the cross section and d its diameter. In this case, the diameters have to have the ratio of the frequen­cies, or the musical intervals, if the same tensile force is to be achieved for strings with the same length:d\/d2 = f2/f\. If the instruments are tuned in fifths, the diameter of the higher tuned string has to be 2/3 ofthe lower one. This simple ratio for homogenous strings may be the reason why some manufacturersprint the diameters of strings on the bag.

TABLE 2 gives the data of a commercial set of gut strings from which it can be seen that the men­tioned ratios of diameters and masses per length m' are more or less in accordance with this simple rule.

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TABLE 2. Strings made out of gut (Pirastro Chorda Violin) and metal covered (Pirastro Eudoxa )for violins (free vibrating length 1=325 mm)

String Basic frequency Diameter Mass per length Tensile forceJi(Hz) d(mm) (PM)*) m' (g/m) F(N)

g gut 196 1,4 28 2,0 32

d' " 293 1,0 19 ~ 1,0 35

a' " 440 0,7 14 ~ 0,5 44

e" " 660 0,6 11~ 0,3 62

total: 173

g Nylon/silver 196 0,8 16 2,2** 35

d' Nylonlalu 293 0,9 18 1,1 ** 38

a' Gutlalu 440 0,7 14 % 0,8** 67

e" Steel/alu 660 0,3 6 0,4** 64

total; 204

*) PM: PIrastromaB, 1 PM = 1/20 mm **) estImated

-

For covered strings, however, the diameter is not at all a measure for m'. m' depends on the mate­rial of the cover. So for example the silver covered g' string given in TABLE 2 is thinner than the d'string with its alu cover. Nevertheless, the silver covered g string has twice the weight of the d' string.

So it would make sense not to indicate the diameter but the mass per length or the necessary tensileforce for helping the instrument builder to choose the adequate and optimum string.

Adjusting the loudness of the different strings

If one assumes that the same loudness of the different strings can be achieved by adjusting thetransverse forces on the bridge in such a way that they are equal for all different strings, when the in­strument is bowec/ with the same velocity VB and dis\imce XB from the bridge, the tension on the highertuned strings would have to be enlarged by a factorh/fi. In case of fifths, this would be a factor of 1.5.

TABLE 3. Different objectives and their influence on strings tuned in fifths

Change in % of the higher-tuned string relative to the lowerone

Objective Tensile force F Transverse Force Ft Mass per length m'

same permanent force F 0% (unchanged) -34 % -56%

same vibrating force Ft 50% 0% (unchanged) -33 %

commercial set of strings (gut) d'/g 9 % -26% -50 %f-------------f------------- -------------

according to TABLE 2 a'/d' 26% -21 % -50%f-------------f------------- -------------

e"/a' 41 % -8% -40%

commercial set of strings (covered) d'/g 9% -26% -50 %f-------------f------------- -------------

according to TABLE 2 a'/d' 76 % 13% -27%f-------------f------------- -------------

e"/a' -4% -31 % -50 %

Taking a violin, this would mean that the d' string would have to be tightened 1.5 times more, the a', string 2.25 times more, and the e" string 3.38 more than the g string. Fortunately, the radiation efficiency

of the instrument is increasing with frequency so that the increase can be smaller than

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deducted under this simple assumption. The relations between tensile forces of the strings F, transverseforces Ft and masses per length are given in TABLE 3.

Checking whether strings are flawless

In earlier times, the violin-maker either had to fabricate the strings himself or to choose an appro­priate piece of a raw string. So he was looking for a sufficiently long perfect piece. Sufficiently longmeant somewhat longer than the free vibrating string between bridge and the support at the peg box. Un­der "perfect" he understood a constant thickness, a constant mass per length and a uniform bending stiff­ness. For checking the thickness, he passed the strings through his fingers. So he could feel even smallfluctuation in thickness. For controlling the mass distribution, he made it twang by a sudden stretchwatching the vibration pattern. If the vibration pattern was looking like arches joining each other, themass distribution was uneven. The homogeneous stiffness was checked by forming a loop. A loop with akind of an edge indicated a nonhomogeneous stiffness distribution.

A final test of a set of strings is called checking the pureness of the fifths ("Quintenreinheit"). Forthis test, the strings are tuned perfectly in fifths. Then they are played by shortening all of them by thesame amount. The intervals should again be fifths. If this is not the case, it may come from defectivestrings but also from a bridge not perfectly aligned, a faulty finger board position and so on.

QUOTATION OF THE SOUND

Judging strings by plucking

When the strings are mounted on an instrument, they are first tested by the instrument maker or themusician plucking them and listening to the reverberating tone and its colour. If one tries to understandthis procedure using measurements, it becomes immediately obvious that the observations of this simpletest are very confusing because they depend on many factors. Nevertheless, comparing measurementresults and audible observations, some conclusions are evident.

The reverberation is determined by the energy losses due to internal friction of the string, the en,.ergy losses at the fixings points and the transmission of energy mostly via the bridge into the sound ra­diating corpus. All these effects depend on frequency. The reverberation times defined according tow. C. Sabine as the time T in which the sound pressure level decends by 60 dB after plucking the stringare given in TABLE 4 for the strings according to TABLE 2 and 3. For determining the values of T, adecay of 30 dB, beginning with the maximum level, had been rated.

TABLE 4. Reverberation times T of the plucked strings given in TABLE 2

Gut strings Covered strings

String open T (s) fmgered T (s) open T (s) fmgered T (s)

g 0.4 - 2.2* 0.5 - 1.4* 1.0 - 2.5** 0.8 - 2.4*

d' 2.6** 2.2* 2,7** 1.5

a' 0.8 - 2.9* 0.4 - 1.2* 1.2 - 3.0* 0.3 - 1.2*

e" 0.9 - 2.2* 0.3 - 0.8* 1.2 - 2.4** 0.3 - 2.0*

* sagging decay curve ** arched decay curve

Comparing the reverberation times of the free vibrating (open) strings and played by pressing themon the finger board (fingered), it becomes clear that most of the energy is transmitted via the bridge intothe corpus.

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Some of the decay curves in a dB scale are not straight lines but curved either sagging or arched.The sagging curves are caused by differently decaying overtones. The different kinds of decay curves areshown in FIGURE 1.

.~

o 0.5 1.0 s 1.5 0 0.5 1.0 s 1.5

»

FIGURE 1. Sound level decay of plucked strings

FIGURE la shows a decay where all the partial tones have the same decay rate. Listening to such adecay, one gets the impression that it does not change timbre. FIGURE Ib represents cases in which thepartial tone with the slowest decay starts with the dominating level and detennines the reverberation.This sounds like a decaying "pure tone" immediately after the pluck. FIGURE lc gives the case in whichthe basic frequency of the g string is radiated very poorly and has an extremely long reverberation. Whenthe fast decaying high overtones surpass the slowly decaying basic frequency, a hanging decay curveoccurs. If one listens to this case, only the long reverberating basic frequency can be heard after a certaintime. FIGURE Id with its bumpy decay indicates that one eigentone of the string - normally its basic fre­quency - is close to an intensive resonance frequency of the corpus - mostly its Helmholtz resonance.The two coupled resonators act like coupled systems: The vibrating energy bounces from the not radiat­ing string into the radiating resonator and vice versa. In this case, the decay sounds somewhat fluctuat­ing. .

As mentioned before, the sound of plucked strings is influenced by many factors and therefore use-ful for judging different qualities of the instrumeq,ts: So, e. g., a long reverberation indicates a low energytransmission into the instrument via the bridge; together with a loud sound this means a good radiation ofthe instrument. A fluctuating decay is a hint for a reson~ce with an eigenfrequency close to the fre­quency of the string. A short reverberation may be caused by a high energy loss via the bridge. Togetherwith a small loudness, it represents an insufficient radiation. In addition, a cello with shortly reverberat­ing strings may probably have a "Wolfston" etc.

JUdging strings by bowing

Bowing the strings gives a hint on the "pureness" of the row of overtones. This check can normallynot be carried out by plucking the strings because of the relatively short duration of available samples.This limits the narrowest bang width not only from the analyzer but also from the human ear. For testingthe preciseness of the overtones, it is better to use a bowed tone, which can be played sufficiently long.As can be seen in FIGURE 2, there is quite a difference between the frequency spectrum measured witha band width of 0.5 Hz for a bowed, relatively stiff g'-gut string, and a relatively flexible g coveredstring, measured on the same instrument.

Since the row of frequencies is determined by the periodicity of the exciting stick/slip process, thefrequency rows in both cases are just the same. But a small impreciseness of the eigenfrequencies of thestrings caused by its stiffness leads to a somewhat broader bandwidth and a higher "background" noise

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:~ 0 =2 [ ~ ~ [ Gut string :~l 0 =2 :1:, Covered string70 -- -- .... --------- ---1-~ ~--1-------------------:------------'------ 70 ------------- .. - ---1-': ~--j--------- -- --------1'---------- .. -------

0=1 i;; i 0=7 i 0=1 i: :0=5 0=7 'n=8 0=960 --------------- ---j-, ;--j--------------.----~------------------- 60 ------- ------- --or; :-- --------------- ---1'------------------

i; ;0~5 ,n=1 8 0=9 : : : 0=6 :50 ------- -- J : ~ ; -------- L -----', 50 ------- ------- +: ;-- ------- ------- --+-- ------- -------

.. --1-: :-- i- n = 6----- --- i -- - 40" -------f ..J. ~ ~ __. ---L . ~_~_ 10

" " ~ r 11: : ", ~ ",' :100

0

~_ ~:~,{\,],:;,:;i. :,:_::::::::_:••:_ bl],:"11~I,Jl,~,::-:1':" 1~um.': ~u :ut ,m' loin ~W 1. ~r ~'TI~jl

o 500 1000 1500 Hz 2000 0 500 1000 1500 Hz 2000

FIGURE 2. Sound level spectra of a bowed gut- and a covered-string

caused by the corrections necessary to keep the row of frequencies in the order of the row of naturaltones. If one listens to the difference, in these cases one hears a more "rough" tone with the stiff gutstring than with the more flexible covered string.

ADAPTATION OF THE BRIDGE TO THE STRINGS

600 Hz 800400200

bridge, damped, T=4,3 s.-

~-' , .. - -

11 IU,

I~IIn mI' I I

i~ "1600 Hz 800 0400200

- .. bridge with mass, T=11,9s_, .- -

~ ...\ "\ ,

~ -.~

IullIW f11 b~ UI ,~ I~ I

111 ".1

600 Hz 800 0400200oo '1l1 I

Finally we want to demonstrate the extremely high influence of the bridge on the behaviour of thestrings and the tones which can be heard and judged.

In a test model, one end of a G-silver covered cello string had been mounted "rigid" whereas theother end at a relatively flexible "bridge". The mobility of the "bridge" was changed by adding a massand also by damping. In FIGURE 3, a comparison of the measurement results can be seen: It is obviousthat the string ending on a very flexible support vibrates rather imprecise (sounding like having a"Wolfston") whereas it sounds more "precise" if the mobility is reduced. If the mobility is flattened bydamping, the overtones of the strings become more or less uniform.

90 --r--,--,.---r-----.--r---.,...-,.--,1---+--+ flexible bridge, T=4,9 s_

dB1--+.+.1---:1--+----11--+-++---1

FIGURE 3. Influence ofthe mobility of the "bridge" on the vibration of a bowed string

Of course the reverberation of the strings changes in accordance with the different energy flowthrough the bridge point. To study the influence of the bridge on a completed instrument, different addi­tional "dampers" may be used.

LITERATURE FOR FURTHER INFORMATION

1. Cremer, L., The Physics ojViolin, Cambridge, Mass., MlT Press, 19842. Rossing, T. D., The Science ojSound, Reading, Mass., Addison-Wesley Publishing Company, 19823. Hutchins, C. M., Benade, V_ (Ed.), Research Papers in Violin Acoustics, New York, Acoustical

Society of America, 1997

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