EXPERIMENTAL STUDI ES OF THE ACOUS TICS OF CL ASSIC AND FLAME NCO GUITARS

w. Bartolini and P.A. Bartolini

To study t he behavior of classic and Flamenco guitars particular l y i n the lower two octaves of t hei r ran ge, we built the experimental enclosure s hown in Fig. 1. The ribs or

sides are 2.25 inches t hic k and weigh 11 . 5 pou nd s . The soundboard and back plates are glued to 0.5 i nc h plyw ood pieces which can be bo l te d to the sides. The resulting rib height is 3.25 inches, which is s hallower tha n t hat of mos t guitars. We thought this

would enhance acoustic coupling between the top and back plates . Fi0ures 5, 6, 7 and 8 show mo re detailed views of t hese plates. The wood t hicknesses

chosen were . 062 inch f or the top and . 064 inch f or the back plate. These are about 10% thinner t ha n those we had measured in s ome Flamenco guitars. Th is choice made the total stiffness of each plate fa r more dep enden t on the strutting than on the plate t hi c kness

and therefore more easi l y modifiable. Th e strutting f rom the waist strut to the top of th e ins trument is signif i ca nt ly stiffer in e it her pl ate than that of real i nstruments in

order to s hi ft the resonances of t hese areas out of t he reg i o n of interest. Figu re 3 shows our moving coi l driver attached to a real instrument on the test stand.

The drive r was made from an old field coil (electrodynamic) speaker. The speaker cone was replaced by a sma l ler cone which termi na ted in a steel stem as shown in Fi g. 9. Fig . 10 shows th e external suspension sp id er. The original suspension sp ider i s attached at thP joint between the voice coil and the cone and is fastened to the center polepiece of the armature. It i s accessi ble through the holes i n the cone. The mass of t he voice coil,

co ne and stem is 5.4 grams. This driver allowed us to obtai n f r eq uency response curves from the experimental enclosure, real instruments and top and back pla t es in the process

of construction. The electronics used consisted of an audio oscillator and amp li fier to dr iv e the voice

co il and a n AC voltmeter to measure the output from t he microp ho ne . The amplifier output voltage was he ld co nstant. A freq ue ncy meter a l lowed measurements of t he oscil l ator out­

put to 0. 1 Hz accuracy (10 sec . gate) . All frequency response curves were taken wit h the driver attac hed to the center of t he bridge. When test i ng rea l instruments the mass of the driver plus bridge clamp was 10.75 grams. The microphone was a l ways positioned 20

inches above a point midway betwee n the lower edge of the so und hole and the bridge. In ­struments were he l d in the test s t and cradled in foam in a 3 point suspens i on: 2 points at the lower bout of the instrument and l point at t he first position underneath the neck.

The strings were damped with f elt st rip s . Our earliest frequency response t ests were carried out indoo rs i n conditions that were

far from a nechoic . The resulting frequency response curves showed mostly room resonances ( Fig . 13). Our fir st driver consisted of an iron s lu g fastened to the bridge rai l and driven by a solenoid co il surrounding it. The second harmon i c distortion that resu l ted

from using a permeable sl ug caused "ghost" peaks at hal f t he f requency of any large peak. Our third instrument was made with redwood back and sides and its third peak, s how n in Fi g. 13 at 245Hz, started out at hi gher frequency and lower amplitude. The change was

accompl ished by trimming t he height of the hip strut after the i nstrument was finished, that is by l owering the resonant f requency of the lower back plate.

In these early frequency response tests we found that in the l ower two octaves of t he gu i ta r range the areas in mot i on were confined to ellipt i ca l patc hes cen t e red on t he bridge, and hip and waist struts of the back. The waist strut resonanc e was of ten in the 300-350 Hz region for classic instruments and in the 250Hz region for some Flamenco instr um ents. Depe ndi ng on th e strength of the back seam cover some of these instruments

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outside mov1ng coil driver

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Fi g . 13

77

showed another resonance which was an elliptica l patch between the waist and hip struts of the back . We hoped that t he experimental enclosure would show some simple way of

understanding the behavior of these simple resonances and perhaps a way to control their

placement in fi nished instruments. The behavior of the instrument as a Helmholtz resonator could also be studied by bolt­

ing rigid panels and appropriate spacers to the sides (Fig . 2). For these measurements

the rigid cavity was excited with the P.A. driver shown in Fig. 11. This driver was coup l ed to the cavity through a . 188 i nch diameter by 0 . 5 inch long tube which presented

a very large acoustic mass and i mpedance to the cavity . All determinations of mechanical mass for each plate were done by measuring the fre­

quency sh i ft due to the addition of some extra mass to the plate (Fig . 14) . In the ear l y stages of strutting the back pl ate, we mapped a large fraction of the peak for different added masses to insure that there were no mu l tiple resonances or other objectionable be­

havior. In some of these the upper peak is quite skew, but better than previous tries which showed t hat it was composed of two closely spaced resonances. This behavior dis ­

appeared as we strengthened and added struts. The variation in mechanical mass with add­ed mass shown at the top of the graph is quite real and was a major stumbling bl ock later. The discrepancy between masses calculated from the peak frequency and from the centroid

at a lower amplitude adds another uncertainty to the mass determinations . After this we always used the centroid of the pe3k at -3 db . Fig. 14 also shows the compliance of the unstrutted redwood plate measured with a partial l y distributed l oad. It is sufficiently

linear over the usual amplitude of motion of t he plate for us to rule out contributions to tone color from non-linearity of the springing .

Once the top and back plates were strutted so they exh i bited only one major resonance in the 80 to 300Hz region, we could proceed to measure the behavior of the enclosure in

the following basic configurations: 1) the rigid enclosure as a Helmholtz resonator, 2 & 3) each vibrating panel (top & back) with its rigid counterpart and 4) both vibrating panels together. Th i s configuration should be cal l ed the augmented bass reflex. These

circuits can be either acoustic or mechanical analogs. We chose to investigate them as acoustic analogs . The early measurements of the Helmholtz resonance suffered from exper­

imental errors and gave rise to other experiments described later on. In 1963 John Schelleng1had published an equivalent circuit for the viol i n in which all body resonances

were in parallel with the soundboard. Our experiments showed that for some tunings of the back plate the cavity resonance was "shorted out" by the back. This implied that the back resonances were in para ll el with the cavity instead of the soundboard .

The bass ref l ex configurations (each vibrating panel with its rigid counterpart) gave the expectable response with two resonances (Fig. 16) . The acoustic masses shown are cal ­

culated from the mechanical masses . We knew the thickness and density of all the compo­nents as we built up the vibrating panels and cou l d calculate for a given mechanical mass

the effective area in motion. When both vibrating plates were bolted to the sides and tested, the response showed three resonances (Fig .. 17) . The frequency and amp li tude of

each resonance changes when extra mass i s added to either top or back . In this case, mass was added to the back. Just as in guitar #3, lowering the resonant frequency of the lower

back lowered the frequency of the third resonance and raised its amplitude. The values labeled e.c . were obtained from a "real'' equivalent circuit. The rea l capacitors behaved

well, but the inductance of the rea l inductors varied with the current and their Q-value was too low. Al though there were systematic errors, qualitatively the circuit seemed to work .

Figure 18 shows the height vs. frequency behavior of each of the three peaks as the

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resonant freq uency of the bac k i s l owered . The i ndepende nt varia bl e, the extra mass added

to the back to lower its frequency, i s not shown. The solid l i nes show th at t he heig ht of the t hird peak i ncreases and i ts frequency decreases as t he resona nt freq uency of the back is low e r ed . The heig ht of t he mi dd le peak decreas es, goes throug h a min i mum when th e res ­onant frequency of t he bac k i s eq ual to t hat of th e r igid cavity and t hen rises again. The he i gh t and pos i tio n o f t he lowe st pea k s i mp l y dec r ease. Qua li t at i ve l y the same beh av ­

io r occurs when the t op and back are joined by a soundpost at one end of the bridge (dash­ed l i ne) . The s eparat i on betwee n th e second and t hi r d peaks i ncreases. The r eso nant fre ­

quen cy of the back plate needed to yield equal outp uts f rom the secon d a nd th i rd peaks is con siderably l owe r with a so un dpost tha n wi t hout. Th e "real" equ i valen t c ircu i t shows t he same qual itative behavior (dot t ed li ne) . Si nce the behav i or of the t hree peaks in the ex­

perimenta l enc l osure an d i n real instruments l ooked qui t e like t he Meinel spectra of Stradivarius violins, we described t hi s experiment to t he ASA in June 1 966 wi th Ca rl een

Hut c hins chairing .4

To cope with t he systemat i c errors of the "real" equiva l ent ci rcu i t , we had to solve the equa ti on for the circuit we had proposed . This turned out to be a bi - cubic eq uat i on i n UJ . I t has t hree real zeroes or sol utio ns whic h are the peak f requencies . I t in ­

cludes three terms that the "rea l " equivalent e l ectrical circ uit co ul d no t inc l ude : M58 ,

t he mutual mass or coupl i ng rrom so un dbo a rd to back, MSP ' soundboard to so un dho l e (p or t )

and MPB' soundho l e to back. Fi gure 20 shows experimental enclos ure 2, whi c h i n a n abst rac t way re s emb l es a guitar .

Near it s r e sona nt freq ue ncy th e physical dimensio ns of the box i n te rm s of t he wav e lengt h

were quite sim i la r to t hose of t he guitar. The r e lat i ve posi t ioni ng of th e po rt was just like that of the gu it ar . Th i s enclosure could be changed eas il y in a number of d i f f e r ent

ways i n an attempt to understa nd t he He l mhol t z resonanc e of th e guita r . The upper pair of compariso ns shows that we were able to disassemb l e and reassem ble t he en cl osure wi t h good repeatabi li ty . The low e r pa i r wi t h th e soundhole off-ce nter s hows a l a r ge change in the resonan t f r equency f or a large c ha nge i n th e distance from t he port to th e nearest s ide wa 11 .

So we mod i fied the enclos ure by mak i ng i t deeper and adding a plate that cou l d be pos ­itioned at d i f fe r ent di s ta nces from the port wi t hou t di sassembli ng t he e nclos ure or chang­

ing its vol ume. We measured t he resonant f requency of t he enc l os ur e while varying the distance between the pl ate and the port . Thi s was do ne for two di fferent port diameters. Si nce the comp li a nce of the air vo l ume was constant, c hanges i n resonan t fr equen cy meant

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w changes in the acoustic mass of the port.

~hen the extra masses calculated from these measurements are normalized by the square root of the area, they can be approximated very closely by the expression shown in Figure 21. This equation implies an additional 1.32 kg;m4 for the mass of the port due to the presence of the back plate 3.25 i nc hes away. It also yie ld s a value of 2.47 kg;m4 for the coupling term betwee n so undboar d and back since t he soundboard effecti ve area turned out to be that of a 5.0 inch radius circle (see Fig . 22).

Accurate measurements of the Helmholtz resonance of the experimental enclosure yielded the results shown in Table 1. Th ese measurements show that this guitar shape and sound­hole placement result in the same behavior as that of an ideal Helmholtz resonator even though the length of the guitar body is nearly A /5 at this frequency and the value of lh~ md~~ uf lhe port was taken as that of a zero th1ckness hole 1n an 1nt1n1te plane . We

TABLE 1

Mp = 15.05 Kg;m 4 (zero thickness ho 1 e in infinite baffle)

MBw from Figure 21 - v CAB-~ 76.03 X 10-9 m5;n

Rib height fPA exp. Mp Mp+MBw difference

(inches) (Hz) (Kg/m 4) (Kg/m 4) ( %)

3.25 143 . 9 16 . 35 16.34 -.06 2. 75 152 .6 16.78 ln. 7S -. 1 B 2 . 25 162.6 17.44 1 7. 41 - . 1 7

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TOP - Spruce

1.58±.05 mm

MMS = 77.7 grams with driver

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2 MMS = 1.3r + 2.9r + 23.3

r=12.6 em

+10%

BACK - Redwood

1.63 :t.05mm

MMB = 74.0 grams

' MMB = 1.4r2 + 5.6r + 0.9

r = 13.9 em

MAB = 20.1 Kg/m 4 +10%

87

did not realize this until quite recently . This measurement was a check done for com­pleteness but neglected because errors in the determination of the acoustic mass of the back made it necessary to assume different values of CA and Mp in order to fit the data. It seemed reasonable that such a complex shape would not behave as an ideal resonator.

Figure 22 summarizes the top and back data. The equation for the mechanical mass has a square term for the plate, a linear term for the strutting, which is considered radial, and a constant which is either the bridge plus the driver clamp, or the holding screws for the mass added to the back plate . The coupling between either plate and the port was computed as r~ = 2';, d where dis the distance between radiating elements.

The equation of Figure 19 was programmed on a TI59 and the results of the computations are shown in Figure 23. The figures in parentheses are experimental points . The errors are larger than our estimate of 0.5% accuracy in determining the resonant frequencies and show some systematic behavior. The resonant frequency of the back plate varies fr om 20% above the soundboard frequency to 10% above the cavity resonance, a range of half an octave. The three resonant peaks span a range of frequencies for which the longest di­mension of the cavity changes from 0.3 A to 0 . 9 A. Since all parameters were assumed independent of frequency, we consider the discrepancies between experimental and calcu ­lated values quite satisfactory.

Fig. 23

TOP PLATE WITH DRIVER WITH RIGID BACK PANEL

experimental (Hz) 125.5

- 4 ~\ -33 Kg/m , 12 5. 5

fs=207.3 Hz

Peak I calculated

(Hz) % error experimental

(Hz) 123.72 -1.4 240.6

all other values the same . 124.58 -0.7 240.6

- 4 4 MsB-2.5 Kg/m Msp=0.9 Kg/m

Peak I I calculated % error

(Hz) 241.23 +0.3

239.54 -0.5 This value of Ms improves the calculated peak spacing and is within our experimental error.

BACK PLATE WITH DRIVER WITH RIGID TOP PLATE

MB=24.24 Kg/m 4 f 8=223.5 Hz Mp==l6 . 35 Kg/m 4 fPA==l43.85 Hz M58=2.5 Kg;m 4 MPB=0.3 Kg/m 4

Peak I Peak I I experimental calculated % error experimental calculated % error

123.0 122.27 -0.6 265.7 262.7 -1.0

TOP PLATE WITH DRIVER AND BACK PLATE

M5=33 Kg/m 4 fs=207.3 Hz Mp=l6.35 Kg/m 4 fPA=l43.85 Hz

11sB=2.5 Kg;m 4 Msp"'0.9 Kg/m4 MPB=0.3 Kg/m 4

Added mass 118 fB Peak I Peak I I Peak III

(grams) (Kg/m4 ) (Hz) % error % error % error

0 20.0 239.5 - 1.8 (112.6) -0.3 (229.4) -4.4 (304.1) 13.6 26.6 219.0 -1.8 +0. 1 -2.2 28. 1 35.8 1 99. 7 -0.9 +0.9 -0.9 42.7 46.9 182.85 -0.2 +1. 6 +0.3 55.9 58.4 170.75 +0 . 5 + 1 . 6 -0.7 68.85 69.6 160. 15 +0. 7 ( l 09. 7) + 1 . 5 ( l 7 6. 8) -0.4 (243.8)

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The values of the acoust i c mass of the back are calculated from the data shown in Fig. 24 . The upper curve shows the resonant frequency of the back vs. added mass . This data and the data for Figure 18 were taken on the same day. The lower curve shows t he varia ­tion in the mechanical mass calcu l ated from the upper curve and the added mass. With

these values of the mechanical mass and the express i on for effective radius from Figure 22, we calculated the acoustic masses M8 , shown in Figure 23. The early, and incorrect, calculations of acoustic mass assumed a constant effective area and an increase in mech­

anica l mass simp l y proportional to the added mass . The effect of adding mass to the plate was to decrease the effective area in motion, causing further increases in th e acoustic masses.

Figure 25 shows the vi olin spectra obtained by Meinel . 2 The lower three resonances are those of the augmented bass reflex . These and Guarneri spectr um obtained by Jesus Alonzo Moral 3 show t hat the Cremona vio l in makers adjusted their instruments to obtain

nearly equal output from the second and third resonances . The experimenta l enclosure could also be fitted wi th a neck and strings and played

(Fig. 4) . Wi th a crude constant picking arrangement, we tested a few configurations of

the experimenta l enclosure, measuring the peak amplitude of the microphone output with an oscilloscope . Loading the top reduces the eff i ciency drast i cally (Fig. 26 upper).

The back can be removed with litt l e effec t when its resonant frequency i s high (no added mass) Fig . 26 lower. The evenness of the bass range can be affected by changing the resonant freq uency of the back (Fig. 27 lower). Even the presence of the performer shows up in these picked spectra (Fig . 27 upper) .

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' -Figure 28 shows similar measurements done on a de la Chica and a Ramirez Flamenco

guitar . The back peaks (h i p and wa i st struts) of the de la Chica, although small, are clear l y visible. Most Flamenco guitars show a single l arge peak for the f irst overtone of the soundboard. The Chladni pattern s hows four vibrating regions or patches around the bridge. The amplitude of thi s s i ngle peak sometimes exceeds that of the lower ma in resonance . I n classic instrume n t~ thh pectk Sefictr·ates into two peaks, one in wh i ch the vibration of the pa t c hes at the ends of the bridge is dominant and the other in which the

vibration of t he patches above and below the bridge is dominant. Figure s 29 upper and lower and Figure 30 show class i c guitar frequency response plots

which do not have strong contributions from reso na nces of t he back plate . Our experience with guitar building at t hi s stage ( 1967) was bringing out the need for more knowledge of

the sou ndboard behavior. These instr um ents did not have triple resonances involv i ng the

bac k pl ate, but t hey were very fi ne in str uments. We started our search into soundboards by reviewing some Ch l adn i patterns we had obtained earl i er. Figure 31A shows patterns ob­tained by drivi ng the unstrutted plates from underneath with a loudspeaker while they were

clamped between two heavy wooden rings. Grain orientation is vertical. With the speaker driving the pl ate from less than an inch away, the 2 patc h mode at 260 Hz cou l d be excited,

although weakly. Figures 31B, C and 0 show patterns obtained with rectangular surrounds clamped onto a fir plate. The resonant freq uencies of t he lowest 4 modes are shown for different rectangular shapes . The first colum n i n Figure 31B correspo nds to the aspect

ratio closest to t hat of the guitar soundboard. With few exceptions the sequence of strong resonant patterns i s t hat of an increasing odd number of patches: l, 3, 5, 7. The

2 patch pattern does not usua l ly occur at a f requency intermediate between the 1 patch (fundamen t a l ) and 3 patch modes. At l ow frequencies it seems to be a weak mode of vibra­tion for these shapes a nd thicknesses. The thickness and t aper of the spruce plate is

shown at the right hand margin. The fir pl ate t hi c kne ss was . 070 i nc h. Figure 32 shows the spruce circu l ar plate, with a bridge added, but still without str utt i ng. The 3 patch pattern now occurs at severa l different freque nc ies. The 5 and 7 patch pa t terns are also present. The fir ~ i r~u la r platA shows the s~me kind of hehavior .

At this point in our search my Flamenco teacher brought back from Spain a 1924 Santos Hernandez. Thi s was a very fine, very responsive Flamenco instrument. I stared at it

for many hours wh i le ta king lessons and eventual l y realized what the builder had done. Figure 33 shows the classic -Fl amenco guitar bridge . The curvature of the soundboard has been exaggerated to show that the weak po i nt s at the ends of the tie rail are purposely

weak. The f l exure of t he bridge at these points en hances the 3 and 5 patch modes.

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92

Fig. 28

a~

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93

Fig. 29 J: WtT7ES 1 KoHNO

11/ti/C.?

1000

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FReauC:NCY f/i!.

5

0~~~------------~--------------~------------~--------------~----------~ /00 200 JOO 400 500

r~EQU£NC y - /-12:

Fig. 30

5

l·k~MAN HAVSLf<

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rREQU£NCY- H2

Aft er guitar #3, we had star ted to measure the compliance at th e center of t he bridge

i n an effort to re late it to my teacher's eva lua tion of "hard " and "soft" i nstr uments (Fig. 34). Since dial indicators were avai l ab l e in t housandt hs of an inc h and scale weight sets were me t r ic, we grew accustomed t o measuring compliances in tho usa ndths of an

in c h per 100 gram weight . I t is a mixed unit which is easy to read, requires a minimum of computat ion and the value 1 mi l / 100 g i s near t he center of the range of comp l iances mca~ u rcd on classic and Fl ame nco guita r s . The conversion factors are shown for complete ­

ness. Al l compliance measurements which follow are give n in mils/ lOO g. We began to l ook more closely into the stiffness of soundboards and during the con­

s t ructio n of guitars 5, 7 and 8 ( 1970- 1972)we developed a n emp iri ca l fo rmula for the stiff­

ness of the traditional soundboard (Fig . 35). There is a f our f old increase i n st iffness when the fan struts are glued to the soundboard in the traditio na l curvature of approx­

imatel y 0. 1 inch across the lower bout. A clamping plate attac hed to the heavy sides of the exp e rime ntal enclosure a l lowed us to test so un dboards and backs in the process o f con ­struction (Fig. 12). Figures 36 and 37 sh ow our ninth guitar with a cedar top and a Koh no

type symmetr ic fan brace in early stages of co ns tr uc tion . The hor i zo nt a l strut s that ter­minate the fan brace in the Kohno fan preve nt dimpling of cedar tops by t he t i ps of the fan struts. The horizontal strut under the bridge is abo ut .03 inch thick and prevents soundboard cra c ki ng wi thout serious l y affecting i ts performance.

In order to simp l ify record in g the increasing numb e r of freq ue ncy response curves taken,

we scanned the r egion from 80 t o 800 Hz f or the · hig hest peak reading and then recorded the freq ue ncy and microp hone o utput of any peak greater tha n TO% of the maximum. Figure 39A shows the frequency response and compliance of our ninth instrument . The fan and bridge

compl i ances are symmetric. The third resonance is not iceab l e, bu t a l i ttle weak . Figure 39B s nows th e frequency response and compliance of the 1924 Santos Hernandez Fl amenco. This inst r ument was ve ry ligh t , we ig hi ng only 2.25 l bs. The comp li a nces at t he br i dge center or e nds are twice that of our ni nth instrument . The compliance at the midpo in t

f f

,,... 3 r-

leo yes-

95

Fig . 31

s 0 CfS

.I "

·I~

I

(If·-~

®sao~

/10 ~

8 Fig. 32

Fig. 33 CLASSIC-FLAMENCO GUITAR BRIDGE

• • • • • • 0 0 0

between b r i d g e and sound h o 1 e i s 1 . 8 w hi c h i s too weak s i n c e t hi s so und board was not b u i 1 t to withstand the higher tension of mode r n nylo n s t r i ngs. The soundboard has d i pped at this point about l /8 i nch. The so und i s excellent in the midrange and up t o A4 . The treble thins ou t beyond that frequency. The asymmet ry between t he compliances o f the

t reb le and bass en ds of t he bridge i s 15 %. Fi gu r e 38 shows a Ch l adn i pattern fo r our tenth guitar in early s t ages of co nst ruct ion.

This instrument was built with an asymmetric brac ing i n which the waist strut a nd hor ­i zonta l struts of the Kohno fa n hav e been ti lted symmetrically toward the bridge center­

l i ne at the treb l e side of the soundboard. The f an strut thicknesses vary from . 070 inc h at the bass to . 170 inch at t he t reb l e s i de of the fan. See also Fig . 40. This was a very balanced ins tr um en t w1th the hig he st br1dge compliance asymmetry we know of. The tap

t ones at the ends of the bridge are an octave apart . The bass ra nge of this instrume nt

Fig. 35

SOUNDBOARD STIFFNESS at the center of the bridge

Soundboard clamped at lining Load distributed over bridge area

Sum of stiffness of fan struts supported at ends Load concentrated at center

Bridge supported at ends Load concentrated at center

97

Fig. 34 COMPLIANCE MEASUREMENT

FORCE (WEIGHT) APPLIED AT OR VERY NEAR POINT OF MEASUREMENT INSTRUMENT HELD RIGIDLY AT SIDES

.OO}tN. t lb

5. 71 x 10-C, m/n .00/tN.

Q _,

25. I X /Q m/n /00 ~

0 I I I I I

I I I I \ I I I I I \

I I I I \ I I I I \ I I I I \

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98

Fig. 36 Fig. 37 Fig. 38

does not have "hot spots" and is most satisfying l y complex. The coupled osci ll ations of so und board, enclosure and back are well worth the effort of making the second and third resonances near l y equal. The c l arity of t he third string was vastly i mproved over pre ­vi ous instrume nts and is probab l y due to the hi gh compliance of the bridge. The asymmetry b2tween the ends of t he bridge shou ld have been sma l ler. The upper treb l e range of the instrument, above E

5 is "tight". Frequency response and coml)liances of this instrument

are shown in Figure 39C. Figure 390 shows the frequency response and compliances of a Kohno 20 with a very

strong main peak and yet a resonant back (hip and waist) and even a 5 patch resonance. It has almost 40% asymmetry between bridge ends. Figure 40A is a comparison of four classic guitars. The Kohno and the Ramirez on the right are both concert instruments. Both have 40- asymmetry between bridge ends. It is interesting to compare the compliance at diff ­erent points in the soundboard between these instruments. Symmetric soundboards tend to produce a very full and sweet sound. In many cases this is accompanied by a lack of clar­ity or pitch definition at the very peak of loudness of many notes, especially in the treble range of the instrument. The effect is reminiscent of the G3 or G# 3 notes that co­incide with the second resonance, but far more subtle. Asymmetric soundboards avoid this problem.

It is best to study both the frequency response curve and compliance of the sou ndboard at different points to improve guitar designs. In the absence of good vibration testing facilities, comp l iance testing will provide a simple diagnostic for control and experi ­mentation that is unmatched by other met hods. At the very l east it allows the luthier to build the lightest soundboard that will just barely not deform from string pull . Figure 408 is a compar i son of steel str i ng guitars ranging from an inexpensive instrument (upper left) clockwise to a concert instrument that has a lot of volume and projection and an almost classic sound (lower left). The Alvarez Yairi bass strut was shaved a f ter purchase.

To calculate better fret spacing, we measured the fractional pitch change to length change ratio for many strings (Fig. 41A) . This simple device allowed one string to be stretched and compared to another string used as reference. Two strings of the same kind and gage were tuned in unison very accurately and then one of the strings was stretched until the beat rate could be measured accurately {4 to 5 beats per second). The length is calculated from the screw pitch and the wing nut rotation. The fractional change in pitch divided by the fractional change in length is shown as the ratio R for nylon strings. The value for the third string is significantly higher than the average of the rest of the strings. This creates intonation problems. To attempt a solution, we assumed that the stretch of the string being depressed between frets by a finger was equivalent to its be -

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Fig. 42 FRET POSITION CALCULATIONS FOR NYLON STRINGS INCLUDING STRING STRETCH

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ing depressed by an edge all the way to the fretboard (Fig. 418). If it did not turn out

to be exactly true, we could always adjust the fret height. The object of the calculation (Fig. 42) is to compare the open string length L2 to the

fretted length which is made up of 4 segments: 01 (nut to preceding fret), 02 {preceding fret to fretboard at the mi ddle of the fret space), 03 (fretboard to fret being ca l c ul ated)

and 04 (fret to bridge). So S is the ratio of the stretc hed to t he unstretched le ngth and T is the fractional stretch. One way to make the correction is to modify the exponent N so that the 1~ is raised to a fractional power, which corrects for the stretch . Now, since R is the measured pitch to l ength change ratio, T x R is the fractional pitch change

at that fret. If that fraction became as large as one semitone ( 1~ ) then the fret location would move back to the previous fret. B then replaces the exponent N and cor­rects the spacing for string stretch.

4 calculation using this method shows most of the difference between scales calculated for R = 35 (third string) and R = 26 (average o+ other strings) occurring in the first fret. Fret to fret spacing remains nearly identical (to . 001 inch) beyond the first fret.

This fortunate result allows the string compensations to be built into the nut (Fig. 44). The first fret is cut to the R = 36 calculation and the third string pivots at the forward (fretboard) edge of the nut. The other strings are made to pivot further back from the fretboard by filing the nut. The amount of correction depends on fret height. The results are very clean inner octaves throughout the fretboard.

102

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REFERENCES

l. John C. Schelleng, "The Violin as a Circuit", J. Acoust . Soc . Am.~ . 326-338 (1963) 2. G. Meinel, "Scientific Principles of Violin Making", Akust. Z . .§., 147-161 (1960) 3. Jesus Alonzo Moral, "Eigenmodes and Qualities of Violins", CAS International

Conference on Musical Acoustics, DeKalb, Illinois (1982) 4. W. Bartolini, "Equivalent Circuit of the Guitar" (abstract), J . Acoust . Soc . Am. 12·

1219-20 (1966)

AC KN0\4L EDGM ENTS Our deepest thanks to Antony Bartolini for his patience and help with computation and

to Mariano Cordoba, Ed Carr, Warren White, Ray Jewell, Jim Wittes, John Hume, Herb Robson and others who contributed musical knowledge, instruments to test, electronic gear, advice and encouragement.