The wolf tone in the cello: Acoustic and holographic studies

Dept. for Speech, Music and Hearing
Quarterly Progress and
Status Report
The wolf tone in the cello:
Acoustic and holographic
studies
Firth, I. M.
journal:
volume:
number:
year:
pages:
STL-QPSR
15
4
1974
042-056
http://www.speech.kth.se/qpsr
STL-QPSR 4/1974
111. MU SICAL ACOU STICS
A.
THE WOLF TONE IN THE CELLO: ACOUSTIC AND HOLOGRAPHIC
STUDIES
I.M. ~ i r t h *
Abstract
The frequency spectrum of the wolf tone i n the cello h a s been investigated by Sonograph a n a l y s i s and computer calculated F. F. T . Odd
h a r m o n i c s of the tone a r e shown to be split into p a i r s of vibration wherea s even h a r m o n i c s r e m a i n single confirming the suggestion that the
wolf tone i s caused because of beating of p a i r s of equally forced vibrations i n the string.
I n t e r f e r e n c e holographic studies of the motion of t h e top plate of the
cello show that such p a i r s of vibration a r e caused because of strong
coupling between s t r i n g and top plate a t the frequency of the f i r s t top
plate mode. Deflection of top plate under a s t a t i c f o r c e h a s been m e a s u r e d and h a s given a n absolute value for top plate impedance.
I
/
P a r a m e t e r s of the cello have been m e a s u r e d by bowing the i n s t r u m e n t a t the wolf note and determining the r a n g e s o v e r which i t will wolf.
Vibrating m a s s , stiffness and impedance of the top plate thence have been
calculated. The wolf note c r i t e r i o n introduced by Schelleng is shown to
be a valid one i n describing the wolfing c h a r a c t e r i s t i c s of the cello.
Introduction
The violin family of i n s t r u m e n t s s u f f e r s f r o m a v e r y famous defect,
the wolf tone.
In many i n s t r u m e n t s of this family, violin, viola and cello,
t h e r e is one p a r t i c u l a r tone elicited by the bow which sounds quite diff e r e n t f r o m the r e s t , but the wolf tone is m o s t well known, o r notorious,
i n the cello.
In g e n e r a l t h i s tone i s classified a s unmusical and i n m o s t
i n s t r u m e n t s i s found a t the frequency of the f i r s t top plate resonance;
in celli i t i s found a t about F g o r I?$
on the G s t r i n g , o r a t the s a m e note
on the C s t r i n g , that i s on the two heaviest s t r i n g s .
The wolf tone h a s
been d e s c r i b e d a s a jarring note, a n i m p u r e and wheesy sound, a note of
rapid fluctuation i n loudness, and a s a cyclic stuttering r e s p o n s e to the
bow.
In some i n s t r u m e n t s a wolf tone is found a t the octave above the
I
I
f i r s t top plate resonance and in s o m e o t h e r s wolf tones a r e associated
with a i r resonances of the a i r cavity.
T h i s study is concerned with the
well known wolf tone i n the cello which o c c u r s a t the pitch of the f i r s t
top plate resonance.
Fig. 111-A- I shows the waveform of a wolf tone
studied l a t e r i n this r e p o r t .
*visitor a t the Dept. of Speech Communication, April-Sept.
1974.
STL-QPSR 4/1974
43.
I n previous studies the o c c u r r e n c e of the wolf note i n the cello h a s
been shown by d i r e c t frequency a n a l y s i s to be caused by the beating of
two equally f o r c e d vibrations i n the string(').
T h e fundamental sinusoidal
component of vibration of the s t r i n g h a s been shown to be split a t the wolf
note into a p a i r of sinusoidal oscillations s e p a r a t e d by a n i n t e r v a l equal to
that of the stuttering frequency of the wolf note.
I n m o s t celli higher
p a r t i a l s a r e often a l s o split into p a i r s , t h e i r s e p a r a t i o n being r e l a t e d to
t h a t of the fundamental p a i r .
T h i s r e p o r t d e s c r i b e s e x p e r i m e n t s undertaken i n t h i s l a b o r a t o r y
while on Study Leave f r o m the Univcrsity of St. Andrews,
The work was
c a r r i e d out i n o r d e r to g a t h e r m o r e data about (a) the frequency content
of the wolf tone, (b) the m e c h a n i c a l action of the cello i n the region of the
wolf note, and (c) about the p a r a m e t e r s of the cello, i n p a r t i c u l a r about
those which have been suggested a s useful in d e s c r i b i n g the wolfing behaviour of t h i s i n s t r u m e n t .
Such m e a s u r e m e n t s w e r e m a d e using equip-
m e n t not available in St. Andrews but forming the complement of the
Department of Speech Communication i n Stockholm.
I n p a r t i c u l a r the
technique of i n t e r f e r e n c e holography, which can be u s e d f o r r e c o r d i n g
the m o d e s of vibrating s u r f a c e s , h a s been applied to the cello f o r the
f i r s t time.
T h e r e have been previous investigations of the cello ( 2 p 3 9 4 ) . The explanation frequently a c c e p t e d f o r the physical o c c u r r e n c e of the wolf tone
i s that of R a m a n , that t h e r e i s a cyclic a l t e r n a t i o n between vibrations a t
fundamental and octave of the string.
I n a detailed a n a l y s i s of the action
of the i n s t r u m e n t s of the violin family i n t e r m s of e l e m e n t a r y e l e c t r i c a l
c i r c u i t t h e o r y ~ c h e l l e n ~h(a~s )considered the wolf tone i n the cello.
By
t r a n s f o r m i n g the cello a t the wolf tone into a n equivalent e l e c t r i c a l c i r cuit comprising a r e s o n a n t t r a n s m i s s i o n line (the string) coupled through
a t r a n s f o r m e r (the bridge) to a s e r i e s resonant c i r c u i t with r e s i s t i v e
l o s s e s (the body), Schelleng a r g u e s that a s the two p a r t s have n e a r l y the
s a m e frequency a t the wolf tone they will a c t i n the s a m e well known
fashion a s o t h e r coupled e l e c t r i c a l r e s o n a n t c i r c u i t s .
With excitation of
the equivalent c i r c u i t by a g e n e r a t o r (the bow) placed i n the t r a n s m i s s i o n
line t h e r e should be t h r e e f r e q u e n c i e s a t which the r e a c t i v e p a r t of the
impedance p r e s e n t e d to the g e n e r a t o r i s z e r o , and that a t two of t h e s e
steady oscillations a r e possible, each of which will be equally forced.
STL-QPSR 4/1974
44.
I t i s only by exciting these two together that the bow can elicit a note at
this particular position on the stopped string, usually F$ on the G string.
By a consideration of the harmonic content of the string motion Schelleng
suggests that not only should the fundamental vibration of the string be
s p l i t but a l s o a l l odd harmonics.
Secondly, a wolf note criterion i s in-
troduced and i t is suggested that ratio of the c h a r a c t e r i s t i c impedance
of the string to that of the body together with the Q of the body of the instrument a t the f i r s t top plate resonance gives a m e a s u r e of the propensity of a n instrument to wolf.
On the b a s i s of these p a r a m e t e r s the electrical analogue predicts the
heavier strings of a n instrument to be in g r e a t e r danger of wolfing, and
that the cello i s m o r e likely to have a wolf tone than viola o r violin, p r e dictions both in a c c o r d with players' experience.
Measurements made
i n these studies have been aimed a t obtaining a better frequency analysis
of the wolf tone and testing the adequacy of the wolf note criterion.
The
two celli which have been used i n these studies a r e described in Table
111-A-I and the m e a s u r e m e n t s undertaken on each indicated.
Table 111-A-I.
Descriptions of Celli and Experiments
Cello I
Made and owned by
H e r r Stig Kgllhager,
Stockholm
Date: 1974
Cello 2
Maker unknown
Owner - 1 . M . F i r t h
Age > 40 y e a r s
-
Steel Covered
Strings by P r i m
Tested in-theWhite
Frequency analysis by
(a) sonograph, and
(b) F F T .
(c) Measurements of
normal mode f r e quencie s by holography, and
(d) a i r tones by
ionophone.
Flexocor , heavy, Frequency analysis by
Steelcovered
(a) Sonograph.
Strings by
(b) Measurements of
Pirastro
normal frequencies
by holography, and
(c) a i r tones by ionophone.
(d) Mechanical action of
top plate a t wolf
tone.
(e) Measurement of
p a r a m e t e r s of cello,
and
(f) evaluation of wolf
note criterion.
I
I
Fig. 111-A-I .
Waveform of wolftone of cello 1.
P a r t of tone a n a l y s e d by sonograph.
0-1 s e c
I
time
F i g . 111-A-3. Sonograph of wolftone of F i g
with a n a l y s e r bandwidth = 2
0.1 s e c
-
Fig. III-A-2.
I
time
Sonograph of wu!ftma of cello 1 on O rtring.
Effective bandwidth of rnrlyaer = 5 Nc.
Z
111-A-2
Hz.
beat. rnax.
beat. rnin.
Harmonic
F i g . 111-A-4.
C o m p u t e r a n a l y s i s of (a) beat m a x i m u m and (b) beat m i n i m u m
of wolftone of c e l l o 1 ( s a m e a s F i g . 111-A-1) by F a s t F o u r i e r
T r a n s f o r m calculation. (c) H a r m o n i c content of beat m a x i m u m
(X) and b e a t m i n i m u m (I)
versus harmonic number.
S.P.L.
<
frequency
F.F.T. Analyis
F i g . 111-A- 5.
Computer calculated F a s t F o u r i e r
Transform analysis f o r three periods
of t h e wolftone i n c e l l o 1. H a r m o n i c s
1, 3 show m o s t r e g u l a r f l u c t u a t i o n s ,
a n d 2 , 6 , 8 , 10 g r e a t e s t c o n s t a n c y
o v e r a p e r i o d of wolf.
STL-QPSR 4/1974
46.
Over three repetition cycles of the wolf, Fig. 111-A-5 indicates that
harmonics numbered 2, 6, 8 , 10 show the greatest consistency in intensity, and those numbered 1 and 3 the most regular fluctuations. In this
particular wolf, produced in cello 1, it appears that the f st and 3rd
harmonics determine the periodicity of the wolf; they show the same r e petition rate (that of the intensity fluctuation of the tone) and appear to
change in phase with one another.
It i s to be s t r e s s e d that investigating
1
the wolf through measurements of the fine structure of i t s harmonic content (i. e. searching for pair s of vibrations), o r through calculating the
instantaneous spectrum a t different times during the repetition period of
I
the wolf tone (i. e. searching for differences in harmonic content) a r e
analogous measurements.
They both lead to the same conclusion, that
the perceived wolf tone results from the creation in the string by the bow
of p a i r s of equally forced vibrations.
The components of each pair a c t
together to give an instantaneous frequency which i s acceptable to the
string, and each component remains substantially unchanged through a
whole beat cycle.
These investigations indicate further that for a n
easily elicited wolf, such a s that of cello I, i t i s the odd harmonics
which a r e split into p a i r s of vibrations wh.ereas the even remain practically
unchanged.
Holographic studies were made to ascertain the normal plate modes
of top and back plate of cello 1 , and acoustical measurements using the
STL-Ionophone were made to determine resonance frequencies of a i r
modes of the cello cavity(6) and these a r e listed in Table 111-A-11.
1
It i s
noted that the frequencies of T i , B i , A1 a r e a l l close to the wolf tone
region but further holographic studies with plate tone testing and string
excitation testing, similar to those reported below for cello 2 , showed
that the wolfing action of cello 1 occurs because of coupling between
string and top plate a t the frequency of the T i mode.
Frequency Analysis: Cello 2
Measurements were made on the most easily bowed and most regular
wolf tone which could be obtained on this cello.
Such a wolf was obtained
by loading the plates of the cello with 34 g added to the back plate a t the
position of the sound post and 5 . 6 g added to the top plate between f-hole
and foot of the bridge on the b a s s bar side.
These additions a r e justified
i
47.
STL-QPSR 4/1974
Table 111-A-11.
Normal Mode Frequencies of P l a t e s and A i r of Celli
Mode
Top plate
Ti
T2
Back plate
Bi
B2
B3
B4
B5
Cello 1
Frequency,Hz
Q
Cello 2
Frequency, Hz
Q
178
295
32 0
39 0
B6
A i r cavity
later.
A0 (Helmholtz) 90
A1 (1st A i r
188
tone)
The Sonograph of Fig. 111-A-6 shows the detailed s t r u c t u r e of the
fundamental of the wolf tone, the analysis being m a d e on a tape recording
of the wolf tone speeded u p by 16 t i m e s by Sonograph.
The wolf tone h a s
a complicated frequency content a t fundamental, four strong peaks being
p r e s e n t close to 187 Hz.
split into two components.
Other Sonographs showed the 2nd harmonic to
This wolf h a s clearly a different c h a r a c t e r to
that of cello 1.
Action of the Top P l a t e a t the Wolf Tone
The motion of the top plate can be i n f e r r e d f r o m two types of m e a s u r e ment: plate excitation o r string excitation testing with microphone pickup,
and t i m e a v e r a g e interference holography of the top plate vibrations.
M e a s u r e m e n t s using the f i r s t technique show that with plate tone testing,
in which the plate i s driven n e a r the f-hole on the b a s s b a r side with a
,
signal swept in frequency, the usually single peak of the f i r s t top plate
I
mode ( T i ) changes to a double peak when the G string i s stopped in the
range of the wolf tone, Fig. 111-A-7.
The mutation of sound emission in
this type of t e s t a t the wolf tone h a s been r e c o r d e d previously(i) i n another cello and shows the extensive coupling between body and string.
The r e c o r d e d dip c o r r e s p o n d s to a z e r o of the input admittance a t the
driving point and o c c u r s in this type of t e s t because the string i s absorbing energy a t this frequency.
I
F i g . 111-A-6.
0.1 sec
I
time
Sonograph of wolftone of c e l l o 2 on G
s t r i n g . ( T o p p l a t e t 5 . 6 g , back l a t e
t 34 g. ) A n a l y s e r bandwidth = 2
Hz.
I n t e n s i t y s e c t i o n shown.
z
-
Frequency
(a)
F i g . 111-A-7.
(b)
P l a t e excitation t e s t i n g of cello 2. CDA s t r i n g s open,
but d a m p e d , with G s t r i n g stopped i n r a n g e of wolf.
(a) Single peak a t 192 Hz with G s t r i n g damped b e c o m e s
(b) a double peak (183, 196 Hz) with s h a r p dip i n S P L a t
191 H z .
STL-QPSR 4/1974
48.
When the string i s vibrated by placing a horseshoe magnet about the
G string a t the point of bowing and passing a n a . c. c u r r e n t , swept in
frequency, through the m e t a l string emission of sound can likewise be
m e a s u r e d in the range of the wolf tone and i t i s found that sound emission
takes place over a broad frequency band showing two peaks, but no z e r o
between.
The acoustical m e a s u r e m e n t s of these two m e a n s of excitation
can be compared with r e s u l t s which would be obtained by measuring the
secondary c u r r e n t i n the equivalent electrical circuit proposed by
~ c h e l l e n ~when
' ~ ) the generator i s placed f i r s t in the secondary circuit
(the body) and second, in the p r i m a r y (the string).
Diagrams of Fig.
111-A-8 indicate the action of the equivalent circuit; h e r e c u r r e n t
velo-
city of top plate oc sound p r e s s u r e output.
The action of the top plate can be studied by using time average interference holography when the instrument i s plate excitation, o r string excitation tested a t the wolf.
By the u s e of a n additional speckle interfero-
m e t e r i n the holography equipment, the vibrational frequency can be adjusted easily to peaks o r dip in plate tone testing, f o r a s Fig. 111-A-9
demonstrates the peaks and dip m e a s u r e d acoustically correspond to
l a r g e motion o r no motion of the plate respectively.
The peaks c o r r e s -
pond to the f i r s t top plate mode, T i , s e e Fig. 111-A- 11 and demonstrate
that only a single mode of vibration of the top plate t a k e s p a r t i n the production of the wolf tone.
Damping the G string, which is stopped to a
length sounding a t 185 Hz the frequency of the dip, r e i n s t a t e s the vibration of this mode to i t s normal value and the vibration of the plate then
only shows a single peak.
With string exci tation testing the top plate
mode T i r e m a i n s present o v e r a wide frequency range
that i s , t h e r e i s no appreciable dip.
180 to 195 H z ,
Fig. 111-A-10 shows the holographic
r e c o r d over this frequency range; the motion of the plate i s in a c c o r d
with the broad flat frequency sound output w5ich is recorded in string
excitation testing.
That the fringe pattern obtained by holography i s that
of the mode T i confirms the coupling between string and this mode of
vibration.
F o r r e f e r e n c e , the frequencies of the normal modes of vibra-
tion of cello 2 a r e given in Table 111-A-11; the frequencies of the modes
T i , B i , A1 a r e significantly different in this cello.
Holographic and
acoustical studies on both celli indicate, therefore, that i t i s the coupling
between string and the mode T i which i s important for the production of
the wolf in the cello.
172 Hz
Peak
F i g . 111-A-9.
185 Hz
Dip
(b)
R e c o n s t r u c t e d i m a g e s of t h e top p l a t e of t h e c e l l o 2
by t i m e a v e r a g e i n t e r f e r e n c e h o l o g r a p h y when p l a t e
excitation t e s t e d i n t h e wolf tone region. CDA s t r i n g s
open, but d a m p e d , with G s t r i n g i n r a n g e of wolf.
(a) Singe1 peak a t 185 H z with G s t r i n g d a m p e d b e c o m e s
(b) a double peak (172, 201 H z ) with s h a r p d i p a t 185 H z
showing l i t t l e v i b r a t i o n ( s e e F i g . 111-A-7).
(Top p l a t e + 5. 6 g , Back p l a t e
34 g . )
+
Fig. 111-A-10. Vibration of top plate of cello 2 when string excitation tested. CDA strings open,
but damped, with G string in range of wolf and excited with a current through
string and magnet at bowing position. Series of reconstructed i m a g e s in the
range of the wolf. (Top plate + 5 . 6 g , Back plate
34 g. )
+
Fig. 111-A-11.F i r s t top plate mode Ti of cello 2 with v a r i o u s additional m a s s e s
placed a t f-hole on b a s s b a r side. (Back plate
34 g . )
+
STL-QPSR 4/1974
49.
Top and Back P l a t e Modes of Cello
I n these studies i t h a s been found useful to add m a s s e s to top and back
plate of the cello i n o r d e r to (a) adjust the instrument so that a regularly
pulsating wolf could be easily elicited by bowing, and (b) so that m e a s u r e m e n t s could be made of effective vibrating m a s s and stiffness of the top
plate.
In view of the importance of the mode T i for the production of a
wolf tone a s e r i e s of holographic m e a s u r e m e n t s were c a r r i e d out to a s certain the influence on vibrational patterns of the f i r s t few modes of the
cello and on their vibrational frequencies.
M a s s e s w e r e added to the
top plate between f-hole and bridge foot on the b a s s b a r side (the position
of maximum displacement of the mode T i ) and on the back plate directly
above the sound post.
The b e s t wolf was obtained in cello 2 with the addi-
tion of 5 . 6 g to the top and 34 g to the back.
The r e c o r d s to Fig. 111-A- I I show (a) that a m a s s added to the back
plate a t the sound post enhances and sharpens the top mode T i through
making the bridge foot a t the sound post side l e s s moveable, and (b) the
subsequent addition of m a s s to the top plate does not change the vibra-
I
tional pattern of this mode, t h e r e being only a reduction i n vibrational
amplitude with i n c r e a s e d m a s s .
I t i s concluded that a point m a s s in this
position on the top plate adds to i t s effective vibrating m a s s changing i t s
frequency accordingly.
Frequency m e a s u r e m e n t i n this s e r i e s of experi-
m e n t s was a c c u r a t e to )5 Hz so that the vibrating m a s s could not be found
f r o m these measurements.
These m a s s added to the top plate o r back plate affects higher modes
of the top plate to a l e s s e r extent.
In general i t h a s been found that the
m a s s added to back plate (34 g) sharpens modes but does not change
vibrational pattern o r frequency.
M a s s added to top plate changes the
frequency of the modes T2, T3, T4 to a v e r y s m a l l extent, for i t is fixed
to the plate a t inactive a r e a s f o r these modes.
Fig. 111-A-11 demonstrates
the effect of added m a s s f o r these higher modes.
Investigations w e r e f u r t h e r c a r r i e d out on the back plate.
The modes
of vibration were a s c e r t a i n e d by driving the back plate a t positions of high
vibrational amplitude (antinodal regions) and modes w e r e explored f i r s t l y
with the speckle interferometer and photographed using time average int e r f e r e n c e holography.
The effect of the additional m a s s of 34 g placed
on the back plate a t the position of the sound post was a s s e s s e d , Fig.
111-A- 13.
The m a s s produced little change in vibrational frequency, even
5 0.
STL-QPSR 4/1974
f o r the m o d e s B l and B2 f o r which i t is a t a position of considerable
motion, and only f o r B i was the mode p a t t e r n changed.
A t o t h e r back
m o d e s , the m a s s again s e r v e d to s h a r p e n the m o d e s making them m o r e
distinct (that i s , enhancing the definition of the m o d e , o r i n o t h e r w o r d s
reducing the inactive a r e a s between regions of motion).
F r o m t h e s e investigations, F i g s . 111-A-11, 111-A-12, 111-A-13, t h r e e
conclusions can be made: (a) M a s s added a t the position of the sound
post on the back plate (and t h i s m e a n s p e r h a p s a l s o to the f r o n t plate a t
the sound post) does not influence mode p a t t e r n s o r frequencies to any
I
significant extent but s e r v e s to s h a r p e n the definition of the lowest modes.
I n this study such a m a s s h a s been u s e d to obtain a ' b e t t e r ' wolf.
(b) A
point m a s s added to the top plate between f-hole and bridge foot on the
b a s s b a r side adds to the vibrating m a s s of the f i r s t top plate mode a n d
can be u s e d t h e r e f o r e to m a s s load this plate (i. e. lower i t s frequency)
without a l t e r i n g the vibrational c h a r a c t e r of t h i s mode.
Such a point load
i s a useful adjunct in m e a s u r i n g c e r t a i n p a r a m e t e r s of the cello.
1
dictated m o r e by boundary conditions imposed a t the edges of the p l a t e s
I
where they a r e glued to the r i b s , and by the position of the sound post
1
(c)
T h e s e e x p e r i m e n t s indicate that the vibrational p a t t e r n s of m o d e s is
than by the distribution of m a s s (and p e r h a p s stiffness) of the wood of
the plates.
F u r t h e r evidence f o r this l a s t conclusion c o m e s f r o m Table
111-A-I1 i n which the Q ' s of s i m i l a r m o d e s of vibration of wood and a i r
a r e noted f o r two radically different celli; one in-the-white with new
glued joints, and the o t h e r , old and with v e r y poor joints, not only loose
but leaky: the Q of T 1 r i s e s with a g e of joint w h e r e a s that of A1 falls.
P a r a m e t e r s of Cello: Wolf Note C r i t e r i o n , Static Deflection and
Absolute M e a s u r e of Body Impedance
In o r d e r to evaluate the u s e f u l n e s s of the wolf note c r i t e r i o n i n t r o duced by Schelleng i t i s n e c e s s a r y to m e a s u r e p a r a m e t e r s of the cello,
especially impedance of s t r i n g and body.
By applying a s t a t i c f o r c e to
the bridge of the cello a static deformation r e s u l t s .
By double exposure
holography the deflection of the top plate can be measured'?).
Fig.
111-A-14 shows a s e t of r e c o n s t r u c t e d i m a g e s of cello 2 which r e c o r d
static deflection of the p l a t e s a s a s e t of f r i n g e s . A s this method of r e cording i s m o s t sensitive to motion perpendicular to the top plate i t can
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TOP + 2
Back + 5
T4
TOP + 2
Back + 5
F i g . 111-A- 12. Top plate m o d e s T 2 , T 3 , T4 of c e l l o 2 with
different m a s s e s added to top and back plates.
B2
Top + 2
Back+5
83
T0p+2
Back + 34
F i g . 111-A-13. Back plate m o d e s B l , B2, B3 of c e l l o 2
with different m a s s e s added to back plate.
52.
STL-QPSR 4/1974
The wolf tone can be used a s a n indicator of the frequency of the mode
T i ; the cello i s bowed a t a fixed position on the s t r i n g and the r a n g e of
s t r i n g length o v e r which a wolf can be obtained m e a s u r e d . Both the
length of the G s t r i n g and the frequency of the s t r i n g a t the e x t r e m i t i e s
of the range can be noted f o r added m a s s , m .
Fig. 111-A-15 shows a
plot of the frequencies of the e x t r e m i t i e s of the range of the wolf tone
v e r s u s applied m a s s , m .
The peak frequency of the mode T i c a n be
considered a s centred in the range of wolfing s o that (df/dm) a t m = 0
can be obtained. I n cello 2 (df/dm) m=O = - 1 . 0 H Z / ~a t the frequency of
8
T i , 189 Hz, so that M = 94. 0 g , S = 1 . 3 10 dyn c m and the impedance
of the body
5
= 1 . 1 1 10 c g s 0 . F r o m a c o u s t i c a l m e a s u r e m e n t s the
T h i s value of impedance i s to be
4
compared with that m e a s u r e d absolutely by static deflection, 8 . 7 10 cgs Q .
Q of this mode i s 37 (Table 111-A-11).
F o r the C G D s t r i n g s on cello 2 it h a s been found through bowing that
t h e r e e x i s t s a r a n g e of stopped s t r i n g length in which a wolf can be
elicited.
Ranges a r e changed by a l t e r i n g p a r a m e t e r s of the cello.
The
p a r a m e t e r s of importance a r e :
(a) Impedance of top plate. This can be a l t e r e d by attaching a
m a s s to the top plate (frequency of T i i s lowered);
(b) Impedance of s t r i n g . A l t e r e d by (i) using a s t r i n g of different
l i n e a r density, o r by (ii) using a different tension i n a p a r t i c u l a r
s t r i n g (the open s t r i n g pitch can be lowered). The full range
on any s t r i n g can be m e a s u r e d i n t e r m s of the r a n g e of length
of s t r i n g o v e r which wolfing o c c u r s , the impedance of plate,
and the impedance of the string; o r m o r e p r a c t i c a l l y , the range
of length of the s t r i n g , the m a s s added to the top plate and the
frequency to which the open s t r i n g i s tuned giving a t h r e e
dimensional plot of the range.
In o r d e r to investigate the n a t u r e of wolfing r a n g e s t h r e e exploratory
m e a s u r e m e n t s w e r e c a r r i e d out.
So that m e a s u r e m e n t s w e r e a s r e p r o -
ducible a s possible only one s t r i n g w a s on the bridge, the one being
bowed to obtain the wolf.
T h i s s t r i n g w a s positioned i n the G s t r i n g
notch a t nut and bridge and was bowed s t r a i g h t a c r o s s the top plate a t a
distance of 3. 5 c m f r o m the bridge.
The r a n g e s in which the cello
could be m a d e to wolf w e r e investigated (a) with a G P i r a s t r o Flexocor
s t r i n g of m e a s u r e d l i n e a r density 11
= 6 . 8 lo-' g crn - 1 with a n addi-
G
tional m a s s of 31. 1 g on the top plate and 34 g on the back f o r v a r i o u s
tensions of the s t r i n g , (b) a s i n (a) but with a m a s s of 2 g fixed to the top
plate, and (c) with a D P i r a s t r o Flexocor s t r i n g of m e a s u r e d l i n e a r den2
-1
= 4. 6 10
sity p,
with a m a s s of 2 g on top plate and 34 g on
g cm
,
-
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FREQUENCY
STL-QPSR 4/1974
back.
53.
The wolfing ranges a r e shown in Fig. 111-A-16.
In each c a s e
t h e r e i s a l a r g e range of string length over which the instrument can be
m a d e to wolf easily when the tension i s n o r m a l , but a s the tension of the
string i s l e s s e n e d this range reduces to become eventually a single length
of the string.
With lessened tension t h e r e i s i n c r e a s e d difficulty i n
producing the wolf by bowing.
The r e s u l t s of Fig. 111-A-I6 suggest that i t might be possible to comp a r e the p a r a m e t e r s which have been suggested a s describing the wolfing
c h a r a c t e r i s t i c s of a n instrument a t different body and string impedances.
The points on the graph a t which the wolfing r a n g e s collapse to a single
value of s t r i n g length might r e p r e s e n t equality of wolfing c h a r a c t e r i s t i c s
-
the position a t which the cello j u s t w o 1f s
scribing these points might be compared.
- and p a r a m e t e r s de-
In p a r t i c u l a r , the wolf note
c r i t e r i o n introduced by Schelleng should be equivalent a t these points
and should have a value which r e p r e s e n t s the transition f r o m ' s a f e from
wolfing' to ' danger f r o m wolfing' ( s e e ref. (5), Fig. 9).
The j u s t w o 1f i n g points of Fig. 111-A- 16 can be u s e d to calculate
(a) the effective vibrating m a s s of the T i m o d e , and (b) the ratio of l i n e a r
densities of G and D s t r i n g s , quantities which have been m e a s u r e d in
other ways.
P r e s u m e that the stiffness, S, of the top plate r e m a i n s
constant over the range of Fig. 111-A-16.
F i r s t l y , consider the j u s t w o 1f i n g points for the G string with
the additional m a s s e s m
1
= 31. 1 g, m 2 = 2g. A t these points p r e s u m e
that the ratio of c h a r a c t e r i s t i c impedances
-1
a r e equal,
A s the s t r i n g tension T i s r e l a t e d to the open string length lo and i t s
tuned frequency fo by
{c
= 210fo
which i s in a g r e e m e n t with the value obtained f r o m the data of Fig.
111-A- 15.
LENGTH OF STRING, cm
tr
m
0:
c 7,
tu
*Ow
E "
0;:
;
5. 2.
3
,.@Q
3
09
109
s.5.
r rr
9. s r,
2r m
:
ca
: p a
map:
p
r.
-
g
R
m o o
5 R'd
PI 0
;o^-
z
2
q p ?
el-
STL-QPSR
54.
4/1974
Secondly, c o m p a r e the r a t i o of c h a r a c t e r i s t i c impedances a t the
j u s t w o 1f i n g
points of the G and D s t r i n g s with additional m a s s
F r o m d i r e c t m e a s u r e m e n t s of the weight and length of t h e s e s t r i n g s
'
D = 0.68.
'
G
I t a p p e a r s t h e r e f o r e , that (a) the wolf note c r i t e r i o n i s relevant to
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the d e s c r i p t i o n of the o c c u r r e n c e of a wolf i n a n i n s t r u m e n t , and that
(b) the
j u s t w o 1f i n g
points of Fig. 111-A- I 6 r e p r e s e n t a n equality
of strength i n the wolf tones, a t which, f o r one i n s t r u m e n t , a r e the same.
The impedance of the s t r i n g s a t the points of j u s t w o 1f i n g with
2 g added to the top plate a r e f o r the
G string
and f o r the
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D string
An a v e r a g e s t r i n g impedance a t j u s t w o 1f i n g
for cello 2 i s 6 0 3 Q
a n d hence the r a t i o of c h a r a c t e r i s t i c impedances a t t h i s adjustment h a s
,
a value
Together with the m e a s u r e d Q of 37 f o r the top plate mode T i this r a t i o
p l a c e s t h i s cello oh the dividing line between the safe and dangerous
r e g i o n s a s calculated by Schelleng (ref.
('I,
Fig. 9).
Conclusion
The application of t i m e a v e r a g e i n t e r f e r e n c e holography together
with speckle holography i s f e a s i b l e to i n s t r u m e n t s the s i z e of the cello
using a n optical bench of m o d e s t dimensions and a l a s e r of low power.
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5 5.
STL-QPSR 4/1974
A p a r t f r o m observing the n o r m a l modes of vibration of a n i n s t r u m e n t
the method is admirably suited to investigating p a r t i c u l a r a s p e c t s of
behaviour ( o r , i n t h i s study, misbehaviour !) of a n i n s t r u m e n t o v e r a
n a r r o w frequency range.
Double exposure holography c a n be u s e d to obtain a n absolute value
of the input impedance of the body of a n i n s t r u m e n t a t its f i r s t top plate
made.
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The addition of m a s s e s to the plates of a n i n s t r u m e n t a t selected
positions does not radically d i s t u r b the lowest m o d e s of vibration, and,
in p a r t i c u l a r , does not a l t e r the mode shape of the f i r s t top plate mode.
The wolf tone i s caused by splitting of harmonic vibrations of the
s t r i n g into p a i r s .
In simple, e a s i l y played, wolf tones odd h a r m o n i c s
a r e split, w h e r e a s even h a r m o n i c s a r e not a l t e r e d significantly i n a
period of the wolf.
Such splitting o c c u r s because of coupling between
the fundamental vibration of a stopped s t r i n g of the cello and i t s f i r s t
top plate mode T 1.
The wolf note c r i t e r i o n introduced by Schelleng i s relevant to describing the wolf note.
of the
I t s value for a n i n s t r u m e n t together with the Q
TI mode d e t e r m i n e whether a n i n s t r u m e n t will wolf. I n s t r u m e n t s
can be adjusted to a j u s t w o 1 f i n g condition by change of tension of
s t r i n g s , o r vibrating m a s s of top plate.
Such adjustments a r e useful to
m e a s u r e m e n t s of p a r a m e t e r s of i n s t r u m e n t s .
P a r a m e t e r s of celli can be obtained by simple m e a s u r e m e n t s i n which
bowing and listening a r e the important experimental techniques.
Acknowledgments
The work c a r r i e d out in the Department of P h y s i c s , University of
St. Andrews, Scotland, h a s had the support of the Royal Society through
a Grant In Aid of Scientific Investigation, and of the Science R e s e a r c h
Council.
Both a r e gratefully acknowledged.
Study Leave f r o m the University of St. Andrews was spent in the
Department of Speech Communication, KTH with the kind p e r m i s s i o n of
P r o f e s s o r Gunnar Fant.
The visit was m a d e possible through a n a w a r d
by the Royal Society under the European Scientific Exchange P r o g r a m m e .
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