Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

The wolf tone in the cello:Acoustic and holographic

studiesFirth, I. M.

journal: STL-QPSRvolume: 15number: 4year: 1974pages: 042-056

http://www.speech.kth.se/qpsr

STL-QPSR 4/1974

111. MU SI CAL 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 in the cello has been inves- tigated by Sonograph analysis and computer calculated F. F. T. Odd harmonics of the tone a r e shown to be split into pa i r s of vibration where- a s even harmonics remain single confirming the suggestion that the wolf tone i s caused because of beating of pa i r s of equally forced vibra- tions in the string. I

Interference holographic studies of the motion of the top plate of the / cello show that such pa i rs of vibration a r e caused because of strong coupling between string and top plate a t the frequency of the f i r s t top plate mode. Deflection of top plate under a static force h a s been meas- ured and has given an absolute value for top plate impedance.

Pa ramete r s of the cello have been measured by bowing the instru- ment a t the wolf note and determining the ranges over which i t will wolf. Vibrating m a s s , stiffness and impedance of the top plate thence have been calculated. The wolf note criterion introduced by Schelleng is shown to be a valid one in describing the wolfing character is t ics of the cello.

Introduction

The violin family of instruments suffers f rom a very famous defect,

the wolf tone. In many instruments of this family, violin, viola and cello,

there is one particular tone elicited by the bow which sounds quite dif-

ferent f rom the r e s t , but the wolf tone is mos t well known, o r notorious,

in the cello. In general this tone is classified a s unmusical and in most

instruments i s found a t the frequency of the f i r s t top plate resonance;

in celli i t is found a t about Fg o r I?$ on the G s t r ing , o r a t the same note

on the C string, that i s on the two heaviest strings. The wolf tone has

been described a s a jarring note, a n impure and wheesy sound, a note of

rapid fluctuation in loudness, and a s a cyclic stuttering response to the

bow. In some instruments a wolf tone is found a t the octave above the I I I

f i r s t top plate resonance and in some others wolf tones a r e associated

with a i r resonances of the a i r cavity. This study is concerned with the

well known wolf tone in the cello which occurs 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 la ter in this report.

*visitor a t the Dept. of Speech Communication, April-Sept. 1974.

STL-QPSR 4/1974 43.

In previous studies the occurrence of the wolf note in the cello has

been shown by d i rec t frequency analysis to be caused by the beating of

two equally forced vibrations in the string('). The fundamental sinusoidal

component of vibration of the string has been shown to be split a t the wolf

note into a pair of sinusoidal oscillations separated by an interval equal to

that of the stuttering frequency of the wolf note. In mos t celli higher

par t ia ls a r e often a l so split into pa i rs , their separation being related to

that of the fundamental pair .

This repor t descr ibes experiments undertaken in this laboratory

while on Study Leave f rom the Univcrsity of St. Andrews, The work was

ca r r i ed out in o rde r to gather m o r e data about (a) the frequency content

of the wolf tone, (b) the mechanical action of the cello in the region of the

wolf note, and (c) about the parameters of the cello, in particular about

those which have been suggested a s useful in describing the wolfing be-

haviour of this instrument. Such measurements were made using equip-

ment not available in St. Andrews but forming the complement of the

Department of Speech Communication in Stockholm. In particular the

technique of interference holography, which can be used for recording

the modes of vibrating surfaces, has been applied to the cello for the

f i r s t time.

There have been previous investigations of the cello ( 2 p 3 9 4 ) . The ex-

planation frequently accepted for the physical occurrence of the wolf tone

i s that of Raman, that there i s a cyclic alternation between vibrations a t

fundamental and octave of the string. In a detailed analysis of the action

of the instruments of the violin family in t e r m s of elementary electr ical

c i rcui t theory ~ c h e l l e n ~ ( ~ ) has considered the wolf tone in the cello. By

transforming the cello a t the wolf tone into an equivalent e lectr ical c i r -

cuit comprising a resonant t ransmission line (the string) coupled through

a t ransformer (the bridge) to a se r i e s resonant circuit with resis t ive

losses (the body), Schelleng argues that a s the two par t s have nearly the

same frequency a t the wolf tone they will ac t in the same well known

fashion a s other coupled electr ical resonant circuits. With excitation of

the equivalent circuit by a generator (the bow) placed in the t ransmission

line there should be three frequencies a t which the reactive par t of the

impedance presented to the generator i s zero , and that a t two of these

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

sp l i t but a lso a l l odd harmonics. Secondly, a wolf note criterion is in-

troduced and i t is suggested that ratio of the character is t ic impedance

of the string to that of the body together with the Q of the body of the in-

strument a t the f i r s t top plate resonance gives a measure of the propen-

sity of an instrument to wolf.

On the basis of these parameters the electrical analogue predicts the

heavier strings of an instrument to be in greater danger of wolfing, and

that the cello i s m o r e likely to have a wolf tone than viola o r violin, pre-

dictions both in accord with players ' experience. Measurements made

in 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 in these studies a r e described in Table

111-A-I and the measurements undertaken on each indicated.

Table 111-A-I. Descriptions of Celli and Experiments

Cello I Made and owned by Steel Covered Frequency analysis by H e r r Stig Kgllhager, Strings by P r i m (a) sonograph, and Stockholm Tested in-the- (b) F F T . Date: 1974 White (c) Measurements of

normal mode f re - quencie s by holo- graphy, and

(d) a i r tones by ionophone.

Cello 2 Maker - unknown Flexocor , heavy, Frequency analysis by Owner -1 .M. Fi r th Steelcovered (a) Sonograph. Age > 40 yea r s Strings by (b) Measurements of

P i ra s t ro normal frequencies by holography, and

(c) a i r tones by iono- phone.

(d) Mechanical action of top plate a t wolf tone.

(e) Measurement of I I

parameters of cello, and

(f) evaluation of wolf note criterion.

Fig. 111-A- I . Waveform of wolftone of cello 1. P a r t of tone analysed by sonograph.

I

0.1 sec time

I 0-1 sec time

Fig. 111-A-3. Sonograph of wolftone of Fig 111-A-2 with analyser bandwidth = 2 Hz. Z

- Fig. III-A-2. Sonograph of wu!ftma of cello 1 on O rtring.

Effective bandwidth of rnrlyaer = 5 Nc.

beat. rnax.

beat. rnin.

Harmonic

Fig . 111-A-4. Computer ana lys is of (a) beat maximum and (b) beat minimum of wolftone of cel lo 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) Harmonic content of beat maximum (X) and beat minimum (I) v e r s u s harmonic number .

S.P.L.

< frequency

F.F.T. Analyis

Fig . 111-A- 5. Computer calculated F a s t F o u r i e r T r a n s f o r m ana lys i s f o r t h r ee per iods of the wolftone in ce l lo 1. Harmon ic s 1, 3 show m o s t r egu l a r f luctuations, and 2 , 6 , 8 , 10 g rea t e s t constancy ove r a per iod 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 inten-

sity, and those numbered 1 and 3 the most regular fluctuations. In this

particular wolf, produced in cello 1, i t appears that the f st and 3rd

harmonics determine the periodicity of the wolf; they show the same re-

petition rate (that of the intensity fluctuation of the tone) and appear to

change in phase with one another. It i s to be stressed that investigating 1 the wolf through measurements of the fine structure of i t s harmonic con-

tent (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 pairs of equally forced vibrations. The components of each pair act

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 an

easily elicited wolf, such a s that of cello I, i t is the odd harmonics

which a r e split into pairs 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 1 modes of the cello cavity(6) and these a r e listed in Table 111-A-11. It i s

noted that the frequencies of T i , B i , A1 a r e al l close to the wolf tone

region but further holographic studies with plate tone testing and string i 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 bass bar side. These additions a r e justified

STL-QPSR 4/1974 47.

Table 111-A-11. Normal Mode Frequencies of P la tes and Air of Celli

Cello 1 Cello 2 Mode F r e q u e n c y , H z Q Frequency, Hz Q

Top plate T i T2

Back plate B i 178 B2 295 B 3 32 0 B4 39 0 B5 B6

A i r cavity A0 (Helmholtz) 90 A1 (1st Air 188

tone)

la te r . The Sonograph of Fig. 111-A-6 shows the detailed s t ructure of the

fundamental of the wolf tone, the analysis being made on a tape recording

of the wolf tone speeded up by 16 t imes by Sonograph. The wolf tone has

a complicated frequency content a t fundamental, four strong peaks being

present close to 187 Hz. Other Sonographs showed the 2nd harmonic to

split into two components. This wolf has clearly a different character to

that of cello 1.

Action of the Top Plate a t the Wolf Tone

The motion of the top plate can be inferred f rom two types of measure-

ment: plate excitation o r string excitation testing with microphone pickup,

and t ime average interference holography of the top plate vibrations.

Measurements using the f i r s t technique show that with plate tone testing,

in which the plate i s driven near the f-hole on the bass bar 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 tes t a t the wolf tone has been recorded previously(i) in an-

other cello and shows the extensive coupling between body and string.

The recorded dip corresponds to a zero of the input admittance a t the

driving point and occurs in this type of tes t because the string is absorb-

ing energy a t this frequency.

I 0.1 sec I time

Fig. 111-A-6. Sonograph of wolftone of cel lo 2 on G str ing. (Top plate t 5.6 g , back la te t 34 g. ) Analyser bandwidth = 2 Hz. Intensi ty sect ion shown. z

- Frequency

(a ) (b)

Fig. 111-A-7. P l a t e excitation test ing of cello 2. CDA s t r i ngs open, but damped, with G s t r i ng stopped in range of wolf. (a) Single peak a t 192 Hz with G s t r ing damped becomes (b) a double peak (183, 196 Hz) with sha rp dip in S P L a t

191 Hz.

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 an a . c. current , swept in

frequency, through the metal string emission of sound can likewise be

measured in the range of the wolf tone and i t is found that sound emission

takes place over a broad frequency band showing two peaks, but no zero

between. The acoustical measurements of these two means of excitation

can be compared with resul ts which would be obtained by measuring the

secondary current in 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 pr imary (the string). Diagrams of Fig.

111-A-8 indicate the action of the equivalent circuit; he re current velo-

city of top plate oc sound p ressu re output.

The action of the top plate can be studied by using time average inter-

ference holography when the instrument i s plate excitation, o r string ex-

citation tested a t the wolf. By the use of a n additional speckle interfero-

me te r in the holography equipment, the vibrational frequency can be ad-

justed easily to peaks o r dip in plate tone testing, for a s Fig. 111-A-9

demonstrates the peaks and dip measured acoustically correspond to

large motion o r no motion of the plate respectively. The peaks corres-

pond to the f i r s t top plate mode, T i , see Fig. 111-A- 11 and demonstrate

that only a single mode of vibration of the top plate takes par t i n the pro-

duction of the wolf tone. Damping the G string, which is stopped to a

length sounding a t 185 Hz the frequency of the dip, reinstates the vibra-

tion 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 remains present over a wide frequency range 180 to 195 Hz,

that i s , there i s no appreciable dip. Fig. 111-A-10 shows the holographic

record over this frequency range; the motion of the plate i s in accord

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 reference, 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

185 Hz Dip (b)

Fig. 111-A-9. Recons t ruc ted i m a g e s of the top plate of the cel lo 2 by t ime ave rage i n t e r f e r ence holography when plate excitation tes ted i n the wolf tone region. CDA s t r i ngs open, but damped, with G s t r i ng i n range of wolf. (a) Singe1 peak a t 185 Hz with G s t r i ng damped becomes (b) a double peak (172, 201 Hz) with s h a r p dip a t 185 Hz

showing l i t t l e v ibra t ion ( s ee Fig. 111-A-7). (Top plate + 5. 6 g, Back plate + 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 images 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 various additional m a s s e s placed a t f-hole on bass bar side. (Back plate + 34 g . )

STL-QPSR 4/1974 49.

Top and Back Plate Modes of Cello

In these studies i t has been found useful to add ma sse s to top and back

plate of the cello in order to (a) adjust the instrument so that a regularly

pulsating wolf could be easily elicited by bowing, and (b) so that measure-

ments could be made of effective vibrating m ass and stiffness of the top

plate. In view of the importance of the mode T i for the production of a

wolf tone a se r i es of holographic measurements were carr ied out to as -

certain the influence on vibrational patterns of the f i rs t few modes of the

cello and on their vibrational frequencies. Masses were added to the

top plate between f-hole and bridge foot on the bass bar side (the position

of maximum displacement of the mode T i ) and on the back plate directly

above the sound post. The best wolf was obtained in cello 2 with the addi-

tion of 5 . 6 g to the top and 34 g to the back.

The records 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 ss moveable, and (b) the I subsequent addition of mas s to the top plate does not change the vibra-

tional pattern of this mode, there being only a reduction in vibrational

amplitude with increased mass . I t i s concluded that a point m as s in this

position on the top plate adds to i t s effective vibrating ma s s changing i t s

frequency accordingly. Frequency measurement in this ser ies of experi-

ments was accurate to )5 Hz so that the vibrating m ass could not be found

f rom these measurements.

These mass added to the top plate o r back plate affects higher modes

of the top plate to a l e s se r extent. In general i t has been found that the

mass added to back plate (34 g) sharpens modes but does not change

vibrational pattern o r frequency. Mass added to top plate changes the

frequency of the modes T2, T3, T4 to a very small extent, for i t is fixed

to the plate a t inactive a r ea s for these modes. Fig. 111-A-11 demonstrates

the effect of added mass for these higher modes.

Investigations were further carr ied out on the back plate. The modes

of vibration were ascertained by driving the back plate a t positions of high

vibrational amplitude (antinodal regions) and modes were explored firstly

with the speckle interferometer and photographed using time average in-

terference 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 assessed, Fig.

111-A- 13. The mass produced little change in vibrational frequency, even

STL-QPSR 4/1974 5 0.

for the modes B l and B2 for which i t is a t a position of considerable

motion, and only fo r B i was the mode pattern changed. At other back

modes, the m a s s again served to sharpen the modes making them m o r e

distinct (that i s , enhancing the definition of the mode, o r in other words

reducing the inactive a r e a s between regions of motion).

F r o m these investigations, Figs. 111-A-11, 111-A-12, 111-A-13, three

conclusions can be made: (a) Mass added a t the position of the sound

post on the back plate (and this means perhaps a l so to the front plate a t

the sound post) does not influence mode patterns o r frequencies to any I

significant extent but serves to sharpen the definition of the lowest modes.

In this study such a m a s s has been used to obtain a ' bet ter ' wolf. (b) A

point m a s s added to the top plate between f-hole and bridge foot on the

bass bar side adds to the vibrating m a s s of the f i r s t top plate mode and

can be used therefore to m a s s load this plate (i. e. lower i t s frequency)

without a l ter ing the vibrational character of this mode. Such a point load

i s a useful adjunct in measuring certain parameters of the cello. (c) 1 These experiments indicate that the vibrational pat terns of modes is

dictated m o r e by boundary conditions imposed a t the edges of the plates I

where they a r e glued to the r ibs , and by the position of the sound post 1 than by the distribution of m a s s (and perhaps stiffness) of the wood of

the plates. Fur ther evidence for this l a s t conclusion comes f rom Table

111-A-I1 in which the Q ' s of s imilar modes of vibration of wood and a i r

a r e noted for two radically different celli; one in-the-white with new

glued joints, and the other , 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 age of joint whereas that of A1 falls.

P a r a m e t e r s of Cello: Wolf Note Criterion, Static Deflection and Absolute Measure of Body Impedance

I In o rde r to evaluate the usefulness of the wolf note cr i ter ion intro-

duced by Schelleng i t i s necessary to measure pa ramete r s of the cello,

especially impedance of string and body. By applying a static force to

the bridge of the cello a static deformation resul ts . By double exposure

holography the deflection of the top plate can be measured'?). Fig.

111-A-14 shows a set of reconstructed images of cello 2 which record

static deflection of the plates a s a se t of fringes. A s this method of r e -

cording i s most sensitive to motion perpendicular to the top plate i t can

TOP + 2 Back +5

T 4 TOP + 2

Back + 5

Fig. 111-A- 12. Top plate modes T2 , T3, T4 of ce l lo 2 with different m a s s e s added to top and back plates.

B2 Top + 2 Back+5

8 3 T0p+2 Back + 34

Fig . 111-A-13. Back plate modes B l , B2, B3 of ce l lo 2 with different m a s s e s added to back plate.

STL-QPSR 4/1974 52.

The wolf tone can be used a s an indicator of the frequency of the mode

T i ; the cello is bowed a t a fixed position on the string and the range of

string length over which a wolf can be obtained measured. Both the

length of the G str ing and the frequency of the string a t the extremit ies

of the range can be noted for added m a s s , m . Fig. 111-A-15 shows a

plot of the frequencies of the extremit ies of the range of the wolf tone

ve r sus applied m a s s , m . The peak frequency of the mode T i can be

considered a s centred in the range of wolfing so that (df/dm) a t m = 0

can be obtained. In cello 2 (df/dm) = - 1 .0 H Z / ~ a t the frequency of m=O 8 T i , 189 Hz, so that M = 94. 0 g , S = 1 . 3 10 dyn cm and the impedance

5 of the body = 1.11 10 cgs 0 . F r o m acoustical measurements the

Q of this mode i s 37 (Table 111-A-11). This value of impedance i s to be 4

compared with that measured absolutely by static deflection, 8 .7 10 cgs Q .

F o r the C G D str ings on cello 2 it h a s been found through bowing that

there exists a range of stopped string length in which a wolf can be

elicited. Ranges a r e changed by altering parameters of the cello. The

parameters of importance a re :

(a) Impedance of top plate. This can be a l te red by attaching a m a s s to the top plate (frequency of T i i s lowered);

(b) Impedance of string. Altered by (i) using a string of different l inear density, o r by (ii) using a different tension in a particular string (the open string pitch can be lowered). The full range on any string can be measured in t e r m s of the range of length of string over which wolfing occurs , the impedance of plate, and the impedance of the string; o r m o r e practically, the range of length of the string, the m a s s added to the top plate and the frequency to which the open string i s tuned giving a three dimensional plot of the range.

In o rde r to investigate the nature of wolfing ranges three exploratory

measurements were ca r r i ed out. So that measurements were a s repro-

ducible a s possible only one string was on the bridge, the one being

bowed to obtain the wolf. This string was positioned in the G string I I

notch a t nut and bridge and was bowed straight a c r o s s the top plate a t a I

distance of 3. 5 cm f rom the bridge. The ranges in which the cello

could be made to wolf were investigated (a) with a G P i ra s t ro Flexocor - 1 string of measured linear density 11 = 6.8 lo-' g crn with an addi- G

tional m a s s of 31. 1 g on the top plate and 34 g on the back for var ious

tensions of the string, (b) a s in (a) but with a m a s s of 2 g fixed to the top

plate, and (c) with a D P i ra s t ro Flexocor string of measured l inear den- - 2 -1

sity p, ,, = 4. 6 10 g c m with a m a s s of 2 g on top plate and 34 g on

FREQUENCY

STL-QPSR 4/1974 53.

back. The wolfing ranges a r e shown in Fig. 111-A-16. In each case

there i s a la rge range of string length over which the instrument can be

made to wolf easily when the tension is normal , but a s the tension of the

string i s lessened this range reduces to become eventually a single length

of the string. With lessened tension there i s increased difficulty i n

producing the wolf by bowing.

The resu l t s of Fig. 111-A-I6 suggest that i t might be possible to com-

pare the parameters which have been suggested a s describing the wolfing

character is t ics of an instrument a t different body and string impedances.

The points on the graph a t which the wolfing ranges collapse to a single

value of s t r ing length might represent equality of wolfing character is t ics

- the position a t which the cello j u s t w o 1 f s - and parameters de-

scribing these points might be compared. In par t icular , the wolf note

cr i ter ion introduced by Schelleng should be equivalent a t these points

and should have a value which represents the transition from ' s a fe from

wolfing' to ' danger f rom wolfing' (see ref. (5), Fig. 9).

The j u s t w o 1 f i n g points of Fig. 111-A- 16 can be used to calculate

(a) the effective vibrating m a s s of the T i mode, and (b) the ratio of l inear

densit ies of G and D str ings, quantities which have been measured in

other ways. P resume that the stiffness, S, of the top plate remains

constant over the range of Fig. 111-A-16.

F i rs t ly , consider the j u s t w o 1 f i n g points for the G string with

the additional m a s s e s m = 31. 1 g, m 2 = 2g. At these points presume 1

that the ratio of character is t ic impedances 11- a r e equal,

A s the s t r ing tension T i s re lated to the open string length lo and i t s

tuned frequency fo by {c = 210fo

which i s in agreement with the value obtained f rom the data of Fig.

111-A- 15.

tr m 0: c 7, tu

* O w E " 0;: ; 5. 2. ,.@Q 3

3 109 09 s.5. r rr r 9. s ,

2 r m ca : : p a

map: p g r. - R

m o o 5 R 'd PI 0 ;o^- z

2 q p ?

el-

LENGTH OF STRING, cm

STL-QPSR 4/1974 54.

Secondly, compare the ratio of character is t ic impedances a t the

j u s t w o 1 f i n g points of the G and D str ings with additional m a s s

F r o m direct measurements of the weight and length of these s t r ings

'D = 0.68. - 'G

I t appears therefore, that (a) the wolf note cr i ter ion is relevant to I the description of the occurrence of a wolf in an instrument, and that

(b) the j u s t w o 1 f i n g points of Fig. 111-A- I 6 represent an equality

of strength in the wolf tones, a t which, fo r one instrument, a r e the same.

The impedance of the s t r ings a t the points of j u s t w o 1 f i n g with

2 g added to the top plate a r e for the

G string

and for the I

D string

An average string impedance a t j u s t w o 1 f i n g for cello 2 i s 6 0 3 Q

and hence the ratio of character is t ic impedances a t this adjustment has ,

a value

Together with the measured Q of 37 for the top plate mode T i this ratio

places this cello oh the dividing line between the safe and dangerous I

regions a s calculated by Schelleng (ref. ('I, Fig. 9).

Conclusion

The application of t ime average interference holography together

with speckle holography i s feasible to instruments the s ize of the cello

using a n optical bench of modest dimensions and a l a s e r of low power.

STL-QPSR 4/1974 5 5.

Apart f rom observing the normal modes of vibration of an instrument

the method is admirably suited to investigating particular aspects of

behaviour (o r , in this study, misbehaviour ! ) of a n instrument over a

narrow frequency range.

Double exposure holography can be used to obtain a n absolute value

of the input impedance of the body of an instrument a t its f i r s t top plate

made. I

The addition of m a s s e s to the plates of an instrument a t selected

positions does not radically disturb the lowest modes of vibration, and,

in particular, does not a l te 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

string into pairs . In simple, easily played, wolf tones odd harmonics

a r e split, whereas even harmonics a r e not a l te red significantly in a

period of the wolf. Such splitting occurs because of coupling between

the fundamental vibration of a stopped string of the cello and i t s f i r s t

top plate mode T 1.

The wolf note criterion introduced by Schelleng i s relevant to de-

scribing the wolf note. I t s value for an instrument together with the Q

of the TI mode determine whether an instrument will wolf. Instruments

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 ings, o r vibrating m a s s of top plate. Such adjustments a r e useful to

measurements of parameters of instruments.

Pa ramete r s of celli can be obtained by simple measurements in which

bowing and listening a r e the important experimental techniques.

Acknowledgments

The work car r ied out in the Department of Physics, University of I

St. Andrews, Scotland, has had the support of the Royal Society through I

I a Grant In Aid of Scientific Investigation, and of the Science Research I

Council. Both a r e gratefully acknowledged.

Study Leave from the University of St. Andrews was spent in the

Department of Speech Communication, KTH with the kind permission of

P ro fesso r Gunnar Fant. The visit was made possible through an award

by the Royal Society under the European Scientific Exchange Programme.