Experimental Studies of the Acoustics of Classical and Flamenco

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 Flam enco 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 piec es 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 a nd 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
stiffn ess 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 trum ent is signif i ca nt ly stiffe r 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 o ur moving coi l driver attached to a real instrument on the test stand.
The drive r was made from a n 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 acc essi 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 o utput 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 conditio ns 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 245H z, 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
..75
76
Fi g . 1 3
guitar no. 2
inside
moving 1ron
driver
8.6
zoo
300
>
guitar no. 3
outside
mov1ng iron
driver
E
.j...J
;.zo
.j...J
;::,
0
Q)
c:::
0
..r:
c..
0
s....
u
.,...
:::E
0
100
zoo
300
guitar no. 3
10
outside
mov1ng coil
driver
zoo
Frequency
. 300
Hz
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 bolting 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 in ch long tube which presented
a very lar ge acoustic mass and i mpedance to the cavity .
All determinations of mechanical mass for each plate were done by measuring the frequency 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 behavior. 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 experimental errors and gave rise to other experiments described later on. In 1963 John
Schelleng 1 had 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
78
Figure 14
Back plate resonance
vs.
added mass
ko
~
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zo
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180
170
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A
J
t/9/r.,
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FE£ LEI{
Back plate compliance
vs.
curvature
£QUil.. l{j~IUM
PO$/TION
100
50
30
70
CW?VATURE
-
..SA61TTA ( mt/5) IN
IZ INCH
90
..SPAN
79
Figure 15
Equivalent circuits of experimental enclosure
I
3.
........
2.
1.
c
~
--- ----]
-- ~-_)
Mp
I
4.
-----~_-_]
£~
fc,
~Mp
f~ fs ~._J::
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 ncre ases 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 e s, goes throug h a min i mum when th e res onant frequenc y 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 a nd 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 ep arat 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 sec on 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 it ative behavior (dot t ed li ne) . Si nce the behav i or of the t hree peaks in the experimenta 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 son a 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
lik e 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 a nd 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 larg e c ha nge i n th e distance from t he port to th e ne ar est 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 sasse mbli 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 ff erent port diameters.
Si nce the comp li a nce of the air vo l ume was consta nt, c hanges i n resonan t fr equ en cy meant
80
Fig. 16
20
top
10
30
4
Ma =35kg/m
Mm=80grams
top+ rigid back
20
>
E
back
Ma =22 kg/m4
Mm=70grams
100
back+ rigid top
200
Frequency
300
Hz
400
81
4 0 ~------------------------------------~
Fig. 17
M' = 0 g.
20
>
E
+-'
::I
eo~~~~==~~--------~~--------~~~
100
::I
0
200
400
300
(])15
1::
0
...s:::
Q_
0
t;
......
M'=28g.
10
::E
5
100
200
frequency
M'
--
.
fI
I
--
300
f2. .
--
400
Hz.
II
f3
--.,-~
I
0 g.
110
225 I 292
e.c.
111
207
297
18g.
109
215
257
e.c.
110
196
278
208
249
191
273
-·
......_.
_..~
- ·-
-·-
--
- -
.
28g.
e.c.
108
1
109 1
I
-
-
82
Fig.18
10
I
o~~~~~~==r==;~-r--.-~.--,---r--.---~-,---t
100
200
/
300
I
30
I
l
I
I
I
s~,
---with soundpost
·-··-········equiv. circuit
1
I
>
E
I
4
+J
:::::l
0.
!
!
I /
20
I!
1/
i
+J
:::::l
0
<lJ
;t
Peak II
s::
/I
0
Peak III
s
\
f,
..s::
0.
0
j
s...
u
.,....
j
\:
I
,..
/I
t\
I
,
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:..
....
.
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10
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I
'···.................. ·""'····
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Peak I
.
I
I
. y. /
.
I
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....··
.•
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p\"
\
\
I
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.• •
.
\
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\
I
'
~
\
•
..J
.... J
I
0+---~--~·--~--r--..--..--.--~--~--~---.--~--~--~
100
200
Frequency
300
Hz
Fig . 19
COUPLIN~
TE~MS
+
+
+
,_,z.
\..UpA
l.
WB
+ ZCBCsMpMsP
+ 2 C~Cs Mp Mse
+2 CeC~MpMpB
- Z C~Cs rv\~t> Msp
2.
+ CAC~ Msp
+ cA c~ tv'\;~
+C a c~ "";a
+C..aC~M~~
84
Fig. 20.
(f SPACC.~S (10% OF fiOL.UHE)
{
Experimental Enclosure 2
(Hl)
C)
0
zeo~.b
X
\
\
\
I
I
I
I~
·707
~I
OF
z~ 8.(,
0
L£Nc,TH
INTERNAL DIMENSIONS
6.5 in X 12. 0 i n. X 1. 63 in.
X
,\/8
~/4. 4 X
t\!32
AT 260 Hz
zc,4.3
0
2S5.4
0
X
l(
vs. GUITAR CAVITY
,l../6.3
X
~ /4.8
A!26
X
AT 150 Hz
x
=
P.A. DRIVER (Fig. 11)
I
POR.T
......~-----
Y4 1N. THRCAD£D ROD
T
D
ALUHINUM
/
""'!lEX. NUT,J /
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Fig. 21
/5ACI( WALL LOADING -
MBW --
.z
J8 w =
w
SEE F/6,, 20
;,aJ51V
-
AP
(o.Z5 w-% -.01) /A
1)
0
&
/. 0'/
/NCI-1 f'ORT
2.50 tNCH PORT
./
0~--------r-------~---------.--------.--------,,-------~
0
5
4
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;m 4 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;m 4 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 ra dius circle (see Fig . 22).
Accurate measurements of the Helmholtz resonance of the experimental enclosure yielded
the results shown in Table 1. Th e se 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)
- v
76.03 X 10-9 m5;n
MBw from Figure 21
CAB-~
Rib height
fPA
(inches)
3.25
2 . 75
2 . 25
(Hz)
143 . 9
152 .6
162.6
exp. Mp
(Kg/m 4 )
Mp+MBw
(Kg/m 4 )
difference
16 . 35
16.78
17.44
16.34
ln. 7S
1 7. 41
-.06
( %)
-. 1 B
- . 17
86
Fig. 22
TOP - Spruce
II
·~o
••
•'••
1.58±.05 mm
I
••
II
,,II
MMS
77.7 grams with driver
=
Driver
MMS
=
+
clamp = 8.0 grams
1.3r
2
+ 2.9r + 23.3
r=12.6 em
+10%
-- - ·1:
I I
II
1:
II
,,
II
I'
••
I'
II
II
••
,,
,...
,,
/
II
I
/
,,'•
,,
,,II
,,,,
,,
II
' ,,
I
r
_,.L
I
II
'I
II
.,.,
.,
II
'•
'I
n
-'\., .......
"" '
"
•''
"",,
•'
·'
'•
It
II
'I
-
'I
lj
:•
_,_
.,..
II
MMB
'
-
=
74.0 grams
MMB = 1.4r 2 + 5.6r + 0.9
r
= 13.9 em
MAB
,,
I
ll
1.63 :t.05mm
,,••
II
n
BACK - Redwood
•,
II
=
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
MsB-2.5
Kg/m 4
fs=207.3 Hz
Msp=0.9 Kg/m 4
Peak I I
Peak I
calculated
% error
calculated
% error
experimental
experimental
(Hz)
(Hz)
(Hz)
(Hz)
+0.3
240.6
241.23
125.5
123.72
-1.4
4
~\ -33 Kg/m , all other values the same .
240.6
239.54
-0.5
12 5. 5
124.58
-0.7
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
experimental
123.0
Mp==l6 . 35 Kg/m 4
f 8 =223.5 Hz
Peak I
calculated
122.27
% error
fPA==l43.85 Hz
experimental
265.7
-0.6
M58 =2.5 Kg;m 4 MPB=0.3 Kg/m 4
Peak I I
calculated
% error
262.7
-1.0
TOP PLATE WITH DRIVER AND BACK PLATE
M5 =33 Kg/m 4
11sB=2.5 Kg;m 4
Added mass
(grams)
0
13.6
28. 1
42.7
55.9
68.85
fs=207.3 Hz
Mp=l6.35 Kg/m 4
4
Msp"'0.9 Kg/m
118
(Kg/m 4 )
20.0
26.6
35.8
46.9
58.4
69.6
fB
(Hz)
239.5
219.0
1 99. 7
182.85
170.75
160. 15
fPA=l43.85 Hz
4
MPB=0.3 Kg/m
Peak I
% error
- 1.8 (112.6)
-1.8
-0.9
-0.2
+0 . 5
+0. 7 ( l 09. 7)
Peak I I
Peak III
% error
% error
-0.3 (229.4)
+0. 1
+0.9
+1. 6
+1 . 6
+1 . 5 ( l 76. 8)
-4.4 (304.1)
-2.2
-0.9
+0.3
-0.7
-0.4 (243.8)
88
z40~--------------------------------------------------------------------------,
Fig. 24
• BAct< fo
'f...
MM6
?11
ADDED
MAss
M'
C A LCUL.. A T£ D FROM
FR£QU£.NCY SHIFT
2.00
:t
>-
,s.,
\J
~~
~
~
~
~
~
"
~
~
150
(,()
X
ss
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 mechanica 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) .
89
Fig. 25
l
J~----~L-_L~~~~~~~~~-----
1
bar
I
J
J
J
t
I
191
I
2511
I
I
I
5110
1/XJO
NXlJ
Sttadlvarlw, 1717
I
I!JI
0
lS/1
'
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 ( 1 967) 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 patter ns 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.
90
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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 cons t ructio n of guitars 5, 7 and 8 ( 1 970- 1972)we developed a n emp iri ca l fo rmula for the stiffness of the traditional soundboard (Fig . 35). There is a f our f old increase i n st iff ness
when the fan struts are glued to the soundboard in the traditio na l curvature of approximatel 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 str ut 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 incre asing 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
fre q 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 ic eab 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
95
Fig . 31
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Fig.
Fig. 32
33
CLASSIC-FLAMENCO
•
0
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GUITAR
•
0
•
BRIDGE
•
0
between b r i d ge and sound ho 1e i s 1 . 8 whi c h i s too weak s i n c e t hi s so und board was not b ui 1t
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 instrume nt 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 centerl i ne at the treb l e side of the soundboard. The f an strut thickness es 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
97
COMPLIANCE MEASUREMENT
Fig. 34
FORCE (WEIGHT) APPLIED AT OR
VERY NEAR POINT OF MEASUREMENT
INSTRUMENT HELD RIGIDLY AT SIDES
5. 71
.OO}tN.
t
lb
x10-C, m/n
Q
25. I
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Fig. 35
SOUNDBOARD STIFFNESS
at the center of the bridge
0
Soundboard clamped at lining
Load distributed over bridge area
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Load concentrated at center
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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 is "tight". Frequency response and coml)liances of this instr ument
5
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 -
99
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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
1
so that the ~ 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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END CORRECTION
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REFERENCES
l.
2.
3.
4.
John C. Schelleng, "The Violin as a Circuit", J. Acoust . Soc . Am.~ . 326-338 (1963)
G. Meinel, "Scientific Principles of Violin Making", Akust. Z. .§., 147-161 (1960)
Jesus Alonzo Moral, "Eigenmodes and Qualities of Violins", CAS International
Conference on Musical Acoustics, DeKalb, Illinois (1982)
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.