Air and structural modes of a harpsichord
WilliamR. Savage
Department
ofPhysics
andAstronomy,
TheUniversity
oflowa,IowaCity,Iowa52242
Edward L. Kottick
SchoolofMusic,The University
ofIowa,Iowa City,Iowa 52242
Thomas J. Hendrickson
DepartmentofPhysics,
Gettysburg
College,Gettysburg,
Pennsylvania
17325
Kenneth D. Marshall
TheUniroyalGoodrich
TireCompany,
Research
andDevelopment
Center,Brecksville,
Ohio44141
(Received17May 1991;accepted
for publication30 December1991)
The acoustical
behaviorof a harpsichord
modeledafter 17th-century
Flemishprototypes
was
studiedusingboth experimentaland analyticaltechniques.The vibrationalmodesof its
enclosedair volumeweremeasuredandfoundto correspond
closelyto thosepredictedby the
JoandJ0solutionsto the Besselequationfor a wedgeshapedspace.A modalanalysisof the
completeharpsichord
revealedthat thesoundboard
has36 vibrationalmodesovera frequency
rangeof 0 to 600 Hz, and that thereare numerousmodeswherethe instrument'scasehasa
significant
amplitudeof motion.Additionalinformationisreportedshowingthattheacoustic
outputof theharpsichord
isreasonably
fiat overa frequency
rangeof 50-2000Hz. It is
concluded that the resonance behavior of both the soundboard and the enclosed air are
importantto thetonequalityof theharpsichord,
andthatitsgenerallyuniformacoustic
output
resultsfromtheexcitationof a largenumberof woodandair modesby thestringpartials.
PACS numbers:43.75.Mn, 43.75.Gh
eigenfunction
(mode shapecoefficient),
LISTOFSYMBOLS
soundboard
o
•
c
2[
Ca
circular
frequency,
radians
viscous
damping
factor
velocity
ofsound,
m/s
wavelength,
m
eigenfunction
(modeshapecoefficient),
air
The harpsichordis a pluckedstringinstrumentwhose
tone-producing
mechanism
isactivatedfromoneor two keyboards.Harpsichordsnormally haveone,two, or threesets
(sometimescalledchoirsor ranks) of strings:if one set,it
will be at normal (8') pitch;if two, eitherboth will be at 8'
pitch, or one will be at 8' pitch and the secondan octave
higher(4' pitch); if three,twowill beat 8' pitchandthethird
at 4'. Someverylargeharpsichords
may havea 16' choiras
well. Eachsetof stringshasa setof jacks.Thesesit on the
distalendsof the keys,and their plectra (traditionally bird
quill,butnowusuallyDelrinplastic)plucktheappropriate
stringswhenthekeysaredepressed.
The 8' stringstransmit
theirvibrationsto thesoundboard
throughthe8' bridge;a 4'
setof strings,if present,requires
a separate,
shorterbridge.
HarpsichordbuildinggenerallyfollowedeitherNorthem or SouthernEuropeanconstructionalpractices,although the productsof some regional schoolsshow influencesof both.The instrumentusedfor the investigations
describedin thisreportis Northern, modeledafter 17th-cenJ. Acoust.Sec. Am. 91 (4), Pt. 1, April 1992
effective
angleof theair cavity,20 deg
heightof air cavity, 18 cm
effectivelengthof the air cavity, 170cm
number of radial nodal surfaces inside the air
cavity
turyFlemishexamples.
• It wasbuiltin 1976by thesecond
INTRODUCTION
2180
h
L
m
author,frompartsandplanssupplied
byZuckermannHarpsichords(seeFig. 1). It hasonekeyboardwith 52 keys,and
oneseteachof 8' and4' stringsandjacks.Its 8' rangeisfrom
G t (49 Hz) toD6 ( 1176Hz).2Depressing
a keyraises
both
setsof jacks;however,a setcan be turned"on" or "off' by
movingitsregisterslidesin or out,causingtheplectraeither
to engageor to miss the strings.The instrumentcan be
playedwith the8' stringsalone,the4' alone,or bothtogether. In normal useit sitson a standwith its lid proppedup
approximately45 deg.We call this instrumentour "acoustics harpsichord."We have marked it, photographedit, recordedit, wired it, shakenit, hammeredit, drilled it, blasted
it with sound,filled it with sand,and dismantledit. We have
evenplayedit.
Developedsome500yearsago,the harpsichord
wasin
continualusethroughthe end of the 18th century;but its
rigidlevels&loud andsoftsoundsdid not suitthe requirementsof theclassicalperiodfor flexibility.By the endof the
18thcenturyit wassuperseded
by themoreexpressive
piano.
It hasenjoyeda revivalin thiscentury,andis nowaccepted
0001-4966/92/042180-10500.80
¸ 1992 AcousticalSociety of America
2180
SOUND
BENTSIDE
4' HITCHPIN-...
RAIL
4' 81RIDGE ....
CUTOFF
BAR
----
J•3TTOM
ß
LOWER
LOWER
BEL•
RAIL
Previousstudieson theacousticsof harpsichords
canbe
found in papers by Kellner (1976), Fletcher (1977),
Spencer( 1981), and Kottick ( 1985); and of its relative,the
clavichord,by Thwaites( 1981) and Thwaitesand Fletcher
( 1981). Theseare valuablearticles,but they are limitedeither to a few aspectsof the acousticalbehaviorof harpsichords,or to purelytheoreticalconsiderations
without experimentalverification.Therefore,since1976,the first two
authorsof this paper,with considerable
assistance
from the
secondtwo, haveattemptedto betterunderstandthe mysteriesof this instrument.We have gatheredexperimentaldata
on soundboardand air resonancesand their interactionby
meansof responsecurvesand Chladni patterns.We have
testedour acousticsharpsichordwith and without strings,
with and without the bottom, and with the soundboardboth
in and out of the case.The propertiesof the air cavityhave
beenmeasuredwith the soundboardrenderedimmobile, and
LEGS
ME
•KEYFRAME
FIG. 1. Schematicviewof the Flemish (acoustics)harpsichord.
with the bellyrailslotboth openand closed.
Our understanding
of the harpsichord's
behaviorwas
enhancedby fieldwork performedin 1980,whenresponse
curvesand Chladni patternswere obtainedfor 39 harpsichords,bothnewand antique(Kottick, 1985). Until 1986,
the informationwe gatheredon the vibrationalbehaviorof
the instrumentwaslimitedto the studyof the forceand motion at one location at a time. Since then, the use of modal
as the appropriateinstrumentfor keyboardmusicwritten
beforeca. 1750.The harpsichordseemsdestinedto remain
with usaslongaswecontinueto enjoythesoundsof "acoustic" instruments.
Because
of itsshapetheharpischord
wouldseemto have
muchin commonwith thepiano,particularlysincetheearliestexamplesof the latterwereessentially
harpsichords
with
actionsthat struck,ratherthanpluckedthe strings.But the
resemblanceis specious.The modern piano has a thick
soundboard,massivesides,no bottom, and doesnot enclose
a volumeof air, while the harpsichordhas a thin soundboard, somewhatflexible sidesand bottom, and an enclosed
air mass.In theseways,the harpsichord
morecloselyresemblestheguitarthanthepiano--aninstrumentwith whichit
evenseemsto sharethe presence
of "soundholes."Like the
relationship
betweenthe harpsichordand the piano,however,theseapparentsimilaritiesareoutweighedby sharpdifferences.The guitaris symmetricalin shape,but the harpsichord is decidedly asymmetrical,both in shape and in
barring(ribbing). While the guitarisinternallyunobstructed, the interior of the harpsichordis heavilybraced.The
ratio of the area of the rose hole to the enclosed volume of air
is far largerin the guitarthan in the harpsichord.Furthermore,Northernharpischords
usuallyhavea sizableopening
just behindthe keyboard(the bellyrail slot) whosearea is
muchgreaterthan that of its rosehole.The guitar is a compact instrument,and its vibrationalmodescan be excited
overa widerangeof frequencies
throughits relativelysmall
bridge. This cannot happen with a harpsichord,whose
bridgemay be from 1.5-2.0 m in length.Finally, the sheer
sizeof the harpsichordhasdiscouragedexperimentalstudy
of its vibrationalbehavior.It is not an easyinstrumentto
suspend,
mount,shake,blastwith sound,or subjectto holographicinterferometry.
2181
J. Acoust.Soc. Am.,VoL91, No. 4, Pt. 1, April1992
analysistechniquesto studythe harpsichordas an input/
outputsystemhasresultedin a moreglobaldescription,and
a greaterunderstandingof its dynamicbehavior.
The work describedin thispaper,therefore,represents
a
blendingof the experimentaland mathematicalprocesses
necessary
to study the generalacousticalbehaviorof the
harpsichord.
Assuch,it suggests
a frameworkfor thefuture
studyof otherharpsichords,
andfor instruments
with similar physical
properties
suchasthevirginal,thebentside
spinet and the early piano.
I. AIR MODES
Our understanding
of the air cavity'sresonances
and
their contributionto our acoustics
harpsichord's
properties
wasincreasedby boththeoreticalandexperimentalstudies.
An approximate
mathematical
modelprovided
uswithvaluable cluesto the interpretationof the experimentalresults
obtained for the instrument itself. The tests on the acoustics
harpsichord
werecarriedout at the Universityof Iowa, eitherin the"soundroom"--a resonantroomwith a relatively
flatresponse--in
theacoustics
laboratoryin theDepartment
of Physicsand Astronomy,or in the anechoicchamber
housedin the Departmentof Speechand Hearing.
The air cavityof the acousticsharpsichordis bounded
on top by a flexiblesoundboard
about2.5 mm thick,andon
the bottomandsidesby boardsabout12mm thickto which
aregluedseveralbraces(theseandsubsequent
relationships
aremadeclearby Fig. 1). The wallsof thiscavityconsistof a
spineabout 152cm long,a cheekabout57 cm long,a bentsideapproximately
parabolicin shape,an angledtail about
27 cm long, and upper (12 mm thick) and lower (15 mm
thick) bellyrailsabout77 cm long.The depthof the inside,
from bottom to soundboard,is 18 cm. At the keyboardend
of this asymmetricalboxis a horizontalopening,the bellySavageeta/.: Airand structuralmodesof a harpsichord
2181
rail slot, resultingfrom the offsetbetweenthe upperand
lowerbellyrails.The slot,spanningthe distancefrom spine
to cheek,is 4.3 cm wide. The two rowsof jacks are located
directlybehindit. To allow accesss
to the instrument'sinterior we cut four portholesin the spineand providedthem
with coverplates.
The acoustics
harpsichord'sshapein plan view led usto
speculatethat the propertiesof its air cavity might be approximatedby a wedge-shaped
space--likea piecesliced
from a roundof cheese.We foundthat a 20-degwedgecut
froma circularcylinder18cmin heightand 170cmin radius
doesindeedexhibitsimilarproperties,and the waveequation in cylindricalcoordinates
canbesolvedexactlyfor this
keyboardendof theacoustics
harpsichord.
The frequencies
of the air modesmaybecalculatedfrom
f=
(7)
whereL is the effectivelengthof the cylindricalair cavity.
Valuesofx are chosento satisfythe boundaryconditionsat
the keyboard,r = L; valuesof x>0 for whichJo(x) = 0 if
the slot is open,valuesofx > 0 for whichJo(x) is a maxima
or a minima if the slot is dosed, or intermediate values of
x > 0 for a slot with an end correction.
Air modeswithbothr and&dependence
obeythedifferentialequation,
rS{rSR(r)/c•r}/ø•r
t-k•r2+ ø92(I)(•)/ø9•62
- 0. (8)
geometry.
R(r)
V2p(r,t)
= cI2• 2p(r,t)
(1)
Harmonic solutionsof the wave equationcan be found
by the methodof separationof variables.The space-depen-
dentpart of the waveequationis
These air modes have one or more radial nodal surfaces
insidetheair cavity.Ther-dependent
and•-dependentequationscanbeseparated
by theintroduction
of a constant
n2.
The resultingdifferentialequationsare
ß•-• -!+n•(•)=0
I/ro•{rcTR(r)/cTr}/cTrt l/r • a•2•( .aS)/&62
R(r)
-t
(c/2rrL )x,
•(qb)
c•
Z(z)
t-k•=0,
(2)
Air cavity modes with only z dependence(modes
formed in the vertical direction between the soundboard and
the casebottom) obeythe differentialequation
t- k2Z(z) = 0.
ra{raa
(
c7r ] -I-[kZr2--n2lR(r)
:0,
(10)
wheren isa constant
whosevaluedepends
ontheangleof the
where k = o/c = 2rr/A.
c•2Z(z)
(9)
and
(3)
wedge,½•,,andthe numberof internalradial nodalsurfaces,
in. The requirementthat q)(½) be a maximumon the sidesof
the wedgeleadsto the equationfor n,
n = m(180/•,,).
(11)
For rn = 1 and •, = 20 deg,n = 9, and the solutionof
Solutions
with maximumpressure
at z = 0 andz = h,
whereh is the heightof the air cavity,are givenby
Eq. (8) istheBessel
functionJ9.Here,Joiszeroat theapex
of thewedge,r = 0 andincreases
slowlyuntilr isquitelarge.
Onceagain,the frequencies
of the air modeswill be determinedby the boundaryconditionat the keyboardr = L.
with A = 2h/n and n = 1,2,3..... The frequenciesof the
The boundaryconditionat the keyboardisnot obvious.
modes are
For zeropressure
to occurhere,a pressure
wavemustbeable
to escapefrom the air cavity--not an easytask.At the slot
f, = (c/2h)n.
(5)
If weassign
a valueofh = 18cm,thefrequency
of thelowest the upperand lower bellyrails,jacks, touchrail,and other
standingwaveis 945 Hz, and the highermodefrequencies componentspresenta substantialinertlyeimpedance,thus
makingtheslotappearto benearlyclosed.We wouldthereare integermultiples.The bracinginsidethe air cavitymay
foreexpectto findmodesthatcorrespond
to valuesOfJoand
changethe frequencies
somewhatby alteringthe verticaldisJ9
with
maximum
pressure
at
the
keyboard.
Figure2 shows
tancesof travelin the cavity,so945 Hz shouldbe considered
some
representative
Jo
and
J9
air
cavity
modes
for a closeda lower bound for the modes.
closedair cavity.
Air cavitymodeswith only r dependence
obeythe difBy spreadingmore than 70 poundsof sandover its surferential equation
face,weimmobilized
thesoundboard
sowecouldstudythe
modesof theair cavitywithoutmajorinterference
from the
Z = .4 cos(2rrz//D,
(4)
(1
• a{rc•R(r)/0r)•
•rr .l+k2R(r)
=0.
(6)
The solutionof this equationis the BesselfunctionJo-
[ Equation(6) maybeconverted
to a standardformbymaking the substitution
x = kr, wherek = 2rr/A andf= c/•.. ]
In thisapproximation,themodefrequencies
dependonlyon
the lengthof the air cavity and the boundaryconditionsat
the ends.The air pressureat the apex of the wedge,r = 0,
which corresponds
to the tail of the acousticsharpsichord,
will alwaysbe a maximum.Therefore,the modefrequencies
will be determinedby the boundaryconditionat r = L, the
2182
d. Acoust.Soc. Am., Vol. 91, No. 4, Pt. 1, April 1992
surroundingstructure. However, while the wood vibrations
were substantiallysuppressed
by the sand,they were not
completelyeliminatedand contributedslightlyto someof
theair modepeaks.Theair modeswerefoundbyexcitingthe
cavitywitha loudspeaker
mountedeitherat thetail or in the
spine.Sweepinga frequencyrange from 30-1000 Hz, we
recordedthe internal pressurepatternswith six miniature
probemicrophones
whoseoutputswentto a chartrecorder.
Then,with theBesselfunctionmodelasa guide,thepressure
patternswere usedto classifythe air modesas membersof
Savage ota/.: Air and structuralmodes of a harpsichord
2182
.••
'NOOE
LINE
Jo(2)- 122Ha
•
JO(3) = 224 H!
,/ ZERO
{
J9(I }- 341
H•
showsthe calculatedfrequenciesfor the Jo and J9 Bessel
functionmodeland the measuredfrequencies
for the immobilized soundboard.There is good agreementbetweenthe
predictedresultsand the measureddata. Becausethe acousticsharpsichordis not shapedlike a true wedgeits effective
lengthcouldnot beeasilydetermined.The testdatashowed
that thefirstmodeof theJofamilywith highpressure
at both
tail and keyboardoccurredat around 122 Bz. The calculationswereperformedusingan effectivelengthof the air cavity that matchedthe requiredlengthfor that mode(L = 170
We alsolookedat the acoustics
harpsichordin its normal state,with thesoundboardunencumbered
by sand.Takingintoaccounttheexpectedloweringof thefrequencies
due
to the reducedboundaryrigidity of the soundboard,thereis
reasonableagreementbetweenthe measureddata and the
predictionsfrom the Besselfunctionmodel.
ERO
Js(
2}-48?
Ht
FIG. 2. Representative
Joand J, air modespredictedfrom the cylindrical
wedgemodelfor a closedbellyrailgap.The dottedlinesshowlocations
of
the nodal surfaces.
The test results also showed evidence for the existence of
a fairly weak mode near 76 Hz with a well-developed
insidethe air cavityand nearzeropreseither the Joor the J9 family, and to assignthe appropriate quarter-wavelength
mode numbers.Additional criteria for these assignments sureat the bellyrailslot. This is indicativeof the develop-
werethat a plot of frequencyversusmodenumberfor both
theJoandJ9familyshouldbeapproximately
linearanduniformlydistributedoverthe frequencyrange[seeEq. (7) ].
As expected,
thedominantmodeshadpressure
maxima
at the bellyrailslotwhichbehavedlike a closedend.Table I
ment ofa Helmholtz mode, but sincethe effectivearea of the
slotisrelativelylargethefrequency
of themodeisstillrather
high. If the sizeof the openingweregraduallyreducedthe
frequencyof this modewould progressively
tiecrease.For a
smallopening,the pressurein the cavitywouldeventually
become uniform and a "classic" Helmholtz
mode would be
present.
We have been able to show that the Bessel function modTABLE I. Mode frequencies
of theacoustics
harpsichordpredictedby the
Besselfunctionmodeland measuredexperimentally.In the "blocked"and
"free" columnsnotethat frequencies
associated
with doublepeakswere
observed
for certainmodes.The Bessel
frequencies
werecalculatedfor an
air cavitylengthofL= 170cm. Measureddataisfor theacoustics
harpsichord with an immobilized soundboard (blocked) and for a free sound-
board(free). The 62-Hz modefor the freesoundboardisprobablya soundboard mode.
el, whichapproximates
the shapeof the air cavityof the
acoustics
harpsichord
by a cylindricalwedge,successfully
predictstheexistence
of twofamiliesof modesrelatedto the
BesselfunctionsJ,)andJ9 and predictstheir frequencies
as
well. The closeagreementbetweenthe calculatedand observedfrequencies
stronglysupports
thevalidityof boththe
modeland its usefulness
in interpretingthe data.
Acoustics
Bessel
Mode no.
I
calculation
122
harpsichord
blocked
free
224
4
324
424
117
113
instrument sat on its stand in the normal manner. The
223
soundboard
wasmarkedoff in a 5-cmgrid, with the lines
runningparallelandperpendicular
to thegrainof thesound-
215
board. Measurements were taken at the intersections of the
two setsof lines,and alsoon the sidesand bottomof the case,
323
286
330
296
thewrestplank,
andonthelid. In all,605measurement
loca-
407
404
tions were identified.
442
To examine the motion of the acousticsharpsichord,
5
524
503
547
508
two miniature accelerometers were attached to the underside of the soundboard with a thin film of accelerometer
6
625
645
596
mountingwax. One wasplacednear the tail, directlybeneaththe 8' bridge,and the other was mountednear the
cutoffbar (seeFig. 1). The vibrationalmodeswereexcited
by tappingthe acoustics
harpsichord
4 to 8 timesat each
!
341
330
349
2
487
500
432
3
605
586
453
2183
Weconducted
a modalanalysis
3to determinethevibrational behaviorof the acoustics
harpsichord.The unstrung
62
233
3
MODES
78
127
2
II. STRUCTURAL
J. Acoust.Sec. Am.,Vol. 91, No. 4, Pt. 1, April1992
549
556
location, at intervals of severalseconds,with a small impact
hammerequipped
witha forcecell.Thedatasignals
fromthe
accelerometers
andthehammerweretaperecorded(thefrequencyresponse
of the tape recorderlimitedthe analysis
Savageat a/.: Airand structuralmodesof a harpsichord
2183
rangeto 0-600 Hz) andthenprocessed
by a minicomputer- thewith-graindirectionandalignthemselves
with the stiffbasedvibrationanalyzer.The analyzerlow-passfilteredthe
eningelements.
analog data and convertedthem to digital format, deterThe soundboard's
outeredgeis taperedalongthe sides
mined the frequencyresponsefunctionsfor the acoustics for approximately10cm, andis gluedbetweenthe liner and
harpsichord--the
output/inputresponse
foreachpairoftest molding,and to the top of the upperbelly rail. To a first
location--and calculatedthe frequency,dampingfactor,
approximation,
therefore,the soundboard
canbe regarded
andcaseare inteand relativemotionof all locationson the acoustics
harpsi- asa clampedplate.Sincethe soundboard
chordforeachmodeof vibration.4
TableII presents
a brief grally connected,they will vibrateas a combinedsystem,
descriptionof the first ten modeshapesand comparesthe
Hence, althoughrigid when comparedto the soundboard,
relative motion of the case to that of the soundboard. The
the casemustbeexpectedto demonstrate
somemotion,and
most of the modes of vibration should exhibit a nodal line
modal propertiesdata are availablefrom the authorsupon
request.
somedistanceinward from the edgeof the soundboard.
An understandingof the constructionalfeaturesof the
The stiffnessof the soundboardwasdeterminedby statisoundboard,and how they might be expectedto affect its
cally deflectingit at variouslocationsand recordingthe apvibrationalbehavior,will help clarify the modal analysis. pliedforceandresultingdisplacement.
The resultsshownin
The soundboard
is a thin orthotropicspruceplate,approxi- Fig. 3 indicatethat the soundboardpossesses
areasof both
matelytriangular,with the grainrunningin the longdirechighandlow stiffness.
The portionsof unbracedsoundboard
tion. The with-grainelasticmodulusis 10to 16timesthat of
are leaststiff;thoseregionsstiflenedby thebridgesare more
the cross-grain
direction,resultingin a with-grainwaveveso;and the areasaroundthe 4' hitchpinrail and the cutoff
locity 3 to 4 timesgreaterthan that of the crossgrain. If the
barand ribsare the stiffest.Onemightthereforeexpectnodsoundboard
werefreeandunbraced,we wouldexpectto find
al linesto passthroughor nearthe4' hitchpinrail and/or the
certainmodalpatternsthat describeellipseswith the long cutoffbar, and alsopossiblythe bridges.The latter is less
axis being 3 to 4 timesthat of the short axis. The spatial likely, however,sincethe bridgesmusttransmitthe vibraorientation would be generally parallel to the spine of the
tional energyof the stringsto the soundboardwith some
case.
ease,and a bridgehavinga largenumberof nodallocations
In real life the soundboard is neither free nor unbraced.
wouldnotbeableto performits taskefficiently.
The 8' and4' bridgesaregluedto itstop,andthe4' hitchpin
The unstiffened
portionsof the soundboardnearthe 4'
rail, cutoffbarandribsto its underside(seeFig. 1). Eachof
and 8' bridgeswould be expectedto exhibitconsiderable
vithesecomponents
contributesmassandstiffness;
in particu- brationalmotion.This shouldresultin the developmentof
lar, they increasethe cross-grainstiffness,
causingthe modal
half-wavelength
ellipsesadjacentto the bridges.At higher
ellipsesto "round-out" somewhatand to rotate away from
frequencies
theellipses
shouldoccurmoreor lessuniformly
TABLE
II.
Descriptions
of
the
soundboard
and
case
mode
shapes
IP
=in-phle
motion;OP = out-of-phase
motion.Maximumsoundboard
motionis
+ 100.
ModeFrequency
Motion
D•
no.
(Hz)
of case
ription
I
27. I
- 20.0
"Drumhead"motionof soundboard,
OP withrigidbodymotionof thecase
2
56.8
- 35.0
Lengthwise
dipole.Nodallinecrosses
fromspinetobentside
nearmidpointofboard.
Maximummotionat tailof 8' bridgeandOP withremainder
of 8' and4' bridge.Case
3
60.3
-- 68.0
Similartomode2 exceptthenodallinefollowsthe4' hitchpinrail.Casemotionlikemode.
4
78.5
-- ! 1.0
Againsimilar,butthenodallinefollowsthe4' bridge.8' bridgemovesasa unitandOP
with thecutofftriangle.First twistingmotionof thecase.
5
102.4
-- 1.0
pitchesaboutan axisbetweenthe front and rear stands.
"Tripole"mode.
Maximum
motion
attipofS'bridge
IP with4' bridge.
CenterorS'bridge
OP. Bothbridges,
4' hitchpinrailandcutoffbarall movewithconsiderable
freedom.Case
movesvery little.
6
109.6
-- 17.0
7
114.3
18.0
8
149.4
25.0
9
169.4
-- 8.0
10
185.2
-- 14.0
Tripole, slightlyshiftedfrom mode5. Casemotion not distinct.
Tripole.Threeregions
withverystrong,nearlyequalamplitudes
of motion.Firstvertical
bendingof the case.
2184
Dipoleat thetail,tripoleat upperpartofboard.Bridges,
ribs,cutoffbarand4' hitchpin
railall vibratewithconsiderable
strength.
Wrestplank
bendsvertically.
Strongdipoleat tailendof 8' bridge.Twoweakerdipoles
alongspineandbentside.
Maximummotionbetween8' and4' bridges.Combinedbendingandtwistingof case.
Similarto mode8, butorientation
isdifferent.Firstlateralbendingof case.
J. Acoust.Sec.Am.,VoL91, No.4, Pt. 1, April1992
Savageeta/.: Airandstructuralmodesof a harpsichord
2184
STIFFNESS
(104 newtons/meier)
on eithersideof the 8' bridge,a transverse
dipoleacrossthe
4' hitchpinrail, anda monopolein the upperleft-handcornerof theboard.The higherfrequencymodesillustratemore
complexsituations,
with a varietyof monopoles,
dipoles,
tripoles,and higher-ordercombinations.
Mode 14 showsa
dipolebetweenthe 8' bridgeand the casee,anotherdipole
betweenthe 8' bridgeandthe4' hitchpinrail, a third dipole
alongthe8' bridgenearthebentside,
anda tripoleassociated
with the4' bridgeandthe left-handcorner.Modes 18and 25
demonstrate
evenmorecomplicated
situations,
andin mode
34 practicallyall the vibration energyis concentratedbetween the 8' bridgeand the case,a regionof low stiffness.
Other modes can be found where maximum motion occurs
,6.9
\\
nearthe 4' bridge.
As expected,the thinner, nonstiffened
regionsof the
soundboardexhibitedthe largestamplitudesof motion for
the higherfrequencymodes.The locationof maximummotion neveroccurredpreciselyon the 8' or 4' bridge.From a
musicalpoint of view this is desirable.If the vibratingportionof a stringterminatedat anexceptionally
flexibleareaof
a bridge,the waveimpedance
of the stringand the soundboard would approacheach other and the energyof the
stringwouldbe transferredto the soundboard
too readily.
The resultwouldbea quicklydampednote--a ratherunmusical "thunk." Therefore, a certain amount of stiffnessis re-
FIG. 3. Static stiffness at various locations on the soundboard of the acous-
ticsharpsichord.
alongthelengthof thebridges,
andadjacent
ellipses
should
quiredat the bridges.They mustmove,but not too much.
An importantresultof the modal studiesof the acousticsharpsichordis the realizationthat the soundboardpossessesa large number of vibrational modes;36 modesbetween0 and 600 Hz. This amountsto an averagemodal
densityof onemodeper 16.8Hz. The importanceof a high
modal densityto the tonequality of the acousticsharpsichord will be discussed later.
occurwith alternatephaseformingmultipoles.
III. OUTPUT RESPONSE OF THE ACOUSTICS
From this descriptionof the acousticsharpsichord's
HARPSICHORD
constructional
featuresitsexpected
behaviorcanbesummed
The musicalworld insiststhat the treble,tenor, and bass
up as follows:(1) the soundboard
is a thin, orthotropic,
stiflened,clampedplate,and the modalpatternsshouldderegistersof the harpsichordmaintain some individuality,
scribeellipticallyshaped
regions
eitheralignedwiththelong andit prefers
thateveryharpsichord
havesomething
unique
axis of the soundboard or inclined in the direction of the
about its tonal character. Both characteristics are heard as
stiffeners,
(2) nodallinesshouldgenerallyfollowthestiffen- part of theoutputresponse
of the instrument.Sincethe soft,
ing elements,(3) modalellipsesshoulddevelopin the recompliantnatureof the soundboard
woodassures
a multigionsof thesoundboard
adjacentto thebridges,
and(4) the tude of resonances,
it will respondin someway to virtually
caseshouldparticipatein many,if notmost,of thevibration- any frequency. But since every soundboardis different,
al modes,and nodallinesshouldappearnearthe clamped harpsichords
vary in their outputresponse.
The varietyin
edgesof the soundboard.
the peaksin response
curves(seeKottick, 1985), undoubtThe test resultsshownin Fig. 4, representative
of the
edly indicatesindividualqualities.
completefamily of 36 vibrationalmodesmeasuredfrom 0We would like to perceivea fairly consistentloudness
600 Hz, correspond
well to the expectedbehavior.Mode 1
levelovertheentireplayingrangeof a harpsichord,
sothat
describes
a "monopole"typeof motionwherethecomplete treble, tenor, and bassregistersbalanceeach other. Thus
soundboard
movesasa unit. For theflow-frequency
modes, evena merelydecentharpsichord
mustnotonlyrespondto a
largeregions
ofthesoundboard
nearthebridgeformellipti- broadrangeof frequencies,
it mustdo so with an acoustic
calareas,andthe nodallinestendto followthe4' hitchpin outputof greateramplitudeat lowerfrequencies,
whereour
rail andcutoffbar. Accordingly,mode3 is a lengthwise
diearsarenot veryefficient,andwith lessamplitudeat higher
polewitha nodallinenearthe4' hitchpinrail, and,ascanbe
frequencies,
wherewe hearmuchmoreacutely.Although
seenin Fig. 5, thecasemovesin a pitchingmotionaboutthe
we do havesomedata that demonstratethe consistency
of
location of the front and back legs of the stand. Mode 5
the loudness
levelof oneparticularinstrument,
s oneneed
includesa lengthwise
dipolealongthe8' bridgeanda mono- only listento any well-constructed,
well-regulated
harpsipolenearthe 4' bridge.Mode 8 describes
a dipoleat the tail
chordto havethepointdrivenhome(seeFig. 6).
2185
J. Acoust.Soc.Am.,Vol.91, No. 4, Pt. t, April1992
Savageeta/: Airandstructuralmodesof a harpsichord
2185
I''
/'
MODE
I -27.1
MODE 5-102.4
Hz
I • ½'
Ha
MODE 8 - 149.4 H•
FIG. 4. Vibrational modes of the acoustics
harpsichord
soundboard.
Contourlinesare
for 25, 50, 75, and95 unitsof relativemotion
( + is upwardmotion, -
is downward
motion). Numericalnotationsdesignatepositions of local maximum or minimum. - - are the nodal lines.
-,37
•
/._[•..•+30
•-
MODE14- 253.1Hz
' -/T
MODE 25-449.7
Hz
I
-56
.•"•c'%•=•,•"*"•
+1•0
-• 59
M•E
3•-573.9
.
Hz
We mustnow ask if there is a sufficientlylarge number
of air and/or soundboardresonances
excitedby playingthe
acoustics
harpsichord
to producea reasonably
uniformbut
monotonically
decreasing
acousticoutputoveritsfrequency
range.We mightalsoaskif the air and soundboard
modes
interact,or couple,in a physicalsense.
To assistin thisdiscussion,we haveconstructed
the followinglogicpath.
(1) The pluckedstringprovidesenergyto the sound-
FREQUENCY (H•)
FIG. 5. Motionof acoustics
harpsichord
casefor modes3 and7. -- -- -- is
theundeformed
shape,- - - is themaximumpositivedeformedshape.
2186
d. Acoust. Soc. Am., Vol. 91, No. 4, Pt. 1, April 1992
FIG. 6. Acousticoutputof a Hyman FrenchDoubleharpsichord.Notes
playedon theback8' strings,oneat a time.
Savageet aL: Airand structuralmodesof a harpsichord
2186
boardat nearharmonic
partials.Thereareat least20significantpartialspernotein thebassregister,
fewerin themiddle,
STRING
2-
•,
3-
•
mode around 76 Hz, the test results found little evidence that
the air modesradiate soundout the bellyrail slot. However,
the air modesmight affectthe overallradiatedsoundby in-
fluencingthe motionof the soundboard
throughmassloading,by actingasan internalspring,or throughphysicalinteraetion(coupling).
5-
6-
7-
8-
9-
300
400
o
C•
,•
0---- &
65
&
,•
O• 7ø,
D) 148
F,
88
F•
176
&
•o
o
E• 83
E• 166
A
o
F#• 9•
F#• 186
MO
--f ½,½,
dS.
0
4
e
4'
•
4t
•
ß
4s
A
o
0
----0-------4
0
o---
•8'
G• 208
ß
II-
I•-
•
a
117
A
o
Kk3
2O0
300
4O0
FREQUENCY
';1
$0AIR MODESi
BOUND8OARD
.,
a
8'
o
As 220
Second,the locationon the bridgewherethe stringter-
ed by the samestringpartial,a physicalinteraction,or coupling,of the modesmay occur.Couplingcan existif the
mode-overlap
integraldoesnot vanish.
A
o
O-
Gs1•6
(12)
the soundboardor air mode (Fig. 3).
(5) If both an air and a soundboardresonanceare excit-
n
o
,•
o
board or air resonance:
or air resonance.
The excitationforce shouldbe applied
withinthehalf-amplitudeportionof themodeshape;that is,
thebridgepinshouldlie at or abovethe50% contourlinefor
o
'-
0
criteriaare satisfied.First, the frequencyof a stringpartial
shouldfall within the half-powerbandwidthof the sound-
minates should not occur at near a node for the soundboard
600
o--•o
o
soundboard and/or air resonances? This will occur if two
Af----2•f,.
500
I0-
to at least some extent. But are there cases where a resonance
will bestronglyexcited;thatis,instances
wherea stringpartial providesa greaterthan averageamountof energyto a
200
55
iio
C• 130
(4) Since there are numerous air and soundboard reson-
ances,any stringpartial will excitea numberof resonances
100
124
4-
tion even in the absence of a soundboard resonance.
(3) With the possibleexceptionof the "Helmholtz"
0
49
HtI
98
and only a few at the trebleend (Kottick, 1987).
(2) Sincethestringdoesnotcommunicate
directlywith
the enclosedair, the only way it can excitean air modeis
throughthe motionof the soundboard.The motionof the
soundboard
mightexcitean air resonance
by forcedexcita-
F-I
MODES
2
U
U
3
4
4'
•
8'
o
4'
5OO
600
( Ha )
lJilii
õ
6
7
111II IIili III1 II111111111
IIIII
I
l 4. 5
5
6
8 9
7
II 14 16
I1• 202,5 •5 26 •9 :51 35
I0 12
15 17
19 21 Z4
27 50 •Z 5G
15
22
Z8
35
FIG. 7. Ability of stringpartialsto exciteair (Jo) and soundboardresonances: --A--,
no effect; --O--,
modeexcited;•
identification
soundboard mode excited; --&--,
air
bothair andsoundboardmodesexcited.Locationand
numbers of the air and soundboard modes shown on the bot-
tom scale.Pairsofazrowsfor theJ. air modesindicatethenearbyfrequenciesfoundfor the acousticharpsichord(seeFig. 2 andTable I}.
(13)
Usingthislogicpathweexaminedthepartialsbetween0
and 600 Hz for the lowest8' and 4' stringson the acoustics
harpsichord.The results,presentedin Fig. 7, showthat a
significantnumberof air and soundboardresonances
will be
stronglyexcitedby mostof the strings.For example,the 8'
A, string, with a fundamentalfrequencyof 55 Hz, excites
soundboardresonances
by partials 1, 2, 3, 5, and 11; an air
mode by partial 9; and both air and soundboardmodesby
partials4, 6, and 8. Only partials7 and 10 fail to excitea
resonance.
The other stringsare not generallyas activeas
samestring partial (solid dots). In thesecasesthere is the
possibilityfor an air-wood interactionor coupling.As stated
above,for couplingto existthe mode-overlapintegralmust
not vanish.This calculationwas performedfor all possible
air-soundboard
interactions(both the Jo and J9 air modes
the direct resultof the excitationof a large number of air and
soundboardresonances
by the string partials.
since we have not performed any experimentsto investigate
were considered), and the results are shown in Table IIl.
The strengthof the air-soundboard
couplingis givenas a
percentof a "perfect"match;that is, if the modeshapesfor
the air andthe soundboard
werepreciselythe samethecoupling-coefficient
wouldequal 100.
thisexample,butih everycaseat leastoneair or soundboard
Interpretingtheseresultsis difficult.There are several
resonanceis strongly excited. Although this analysiswas
instances
wherethecouplingcoefficient
appearsto berather
restrictedto 600 Hz, it may be assumedthat at higherfresizable;for example,the "Helmholtz" mode at 76 Hz and
quenciesthe stringpartialswill stronglyexcitea similarly the fourth soundboardmodeat 78.5 Hz seemto befairly well
largenumberof air and soundboardresonances.
Therefore, coupled.The question.however,is how high mustthe coufroma qualitativepointof view,weconcludethatthegener- pling coefficientbe beforewe can reliablyclaim that it is
ally uniformacousticoutputof the acoustics
harpsichordis
musicallysignificant?
At the presenttime, we do not know,
Figure7 alsoshowsthat therearea numberof situations
whereboth an air and a soundboardmodeare excitedby the
2187
d. Acoust.Soc. Am., Vol. 91, No. 4, Pt. 1, April 1992
these matters. The data are certainly suggestive,but we
chooseto withholdjudgmenton the importanceof coupling
to some future
time.
Savage ota/.: Air and structuralmodes of a harpsichord
2187
TABLEIii. Calculation
oftheair-soundboard
modal
coupling
forthe
acousticsharpsichord.Perfectcoupling--identicalair and soundboard
modeshapes--would
resultin a couplingcoefficient
of 100.
Air mode
Helmholtz--76
Soundboard
mode(s)
Hz
1-122
2-224
3-324
4-424
Couplingcoefficient
4- 78.5 Hz
28.9
6-109.6
16.1
7-114.3
4.5
11-207.7
12-213.1
13-215.0
14-253.4
1.3
0.2
12.6
17-295.1
18-335.8
0.8
19-349.4
15.0
2.7
4.4
23-401.8
8.3
24-433.6
2.5
m above the soundboardto record the acousticsharpsichord'soutputresponse(seeKottick, 1985).
Removingthe bottom preventsthe formationof air
modeswith only a minimal effecton the structuralvibrationsof the soundboard.
Figure8 showsa graphof the logarithm of the ratio of the responses
of the acousticsharpsichordversusfrequency,both with and without its bottom.
Peaksin thisgraphat 88, 110,196,310,and466 Hz lie close
to frequencies
for whichlargeair-soundboard
modalcouplingsareshownin TableIII at air modesfrequencies
of 76,
122,224, 324,424, and487 Hz. When playedin thisconfigurationthemusicianreportedthat the acoustics
harpsichord
withoutitsbottom"wasstill playable,but is soundstinny."
Theseanecdotalcasestudiesand limited quantitative
dataillustratethat the air cavitymodesareunquestionably
importantto the tonequality of the acousticsharpsichord.
25-449.7
19.6
IV. CONCLUSIONS
5-524
28-493.9
29-511.6
30-527.0
9.1
10.0
6.8
6-625
35-587.5
8.7
36-605.1
12.5
18-335.8
0.6
19-349.4
2.8
This studyhasrevealedthat theair cavityof theacousticsharpsichord
canbeapproximated
asa wedge-shaped
cylindrical volumeand analyzedby conventionalwave equation methods.Here Jo and J9 Besselsolutionsaccurately
predictthemodesof theenclosed
air. TheJoair modeswith
pressure
maximaat boththetail andkeyboardof theacousticsharpsichord
weredominant,buttherewasa weakHelmholtz-likemodeat 76 Hz with low pressureat the keyboard.
The air modesdo not contributegreatlyto the soundof the
instrumentby radiatedenergyout thebellyrailslot,but they
do influencethe soundby affectingthe motionof the soundboard.While it wasnot possibleto predictthe air modefrequencieswith completeaccuracy,we havenevertheless
arrived at someunderstandingof how the air behaves.
The modalanalysisof the completeinstrumentuncov-
1-341
2-487
3-605
20-365.0
1.3
26-470.9
29.9
27-483.8
22.8
28-493.9
3.2
29-511.6
9.7
34-573.9
16.3
35-587.5
16.2
36-605.1
25.1
ered a wealth of information.
The soundboard has an aver-
agemodaldensityof onemodeper 16.8Hz overthe rangeof
0-600 Hz. At lower frequencies,
the modal patternsare tyifledby ellipsesthat encompassed
largeareasof the soundThe couplingissuenotwithstanding,there is other evidenceto suggestthat the air modesare important. We examined the influenceof the air cavity resonanceson the tonal
qualitiesof the acousticsharpsichordthrough two experiments. In the first, the bellyrail slot was completely,then
partiallyclosed,in orderto alter the boundaryconditionsat
the keyboardend, and, therefore,the ability of the air cavity
to communicate
with thesurroundingenvironment.Playing
the acousticsharpsichordwith the bellyrail slot completely
closed,the musicianreportedthat the low F at 88 Hz "had
lost its guts," and the low C at 65 Hz became"very strong
and resonant."When the slot was openedby 1/8 in., "the
low C was still strong, the low F was still weak, and the low E
(at 83 Hz) was alsoweak." When it was openedfurther, to
3/8 in., "the C wasstill strong,the E and F werestill weak,
and the G was also weak."
In a somewhatmorequantitativeexperiment,we investigatedthe responseof our acousticsharpsichordwith and
without its bottom.A piezoelectricvibratorwas usedto excite the 8' bridge.Placedon thebridgepin for eachstringin
turn,it applieda sinusoidal
excitationwith a frequencyequal
to thefundamentalof the string.A microphonewasplaced3
2188
d. Acoust.Soc. Am., Vol. 91, No. 4, Pt. 1, April 1992
-150 1(30 150 2OO250 300 350 400 450 500
FREQUENCY ( Hz )
FIG. 8. Logarithmof the ratio of the outputresponses
of theacousticsharpsichord with and without
the bottom attached.
Savage otal.: Air and structuralmodes of a harpsichord
2188
board,with nodallinesthat generallyfollow the stifferelementssuchasthe4' hitchpinrail and thecutoffbar and ribs.
At higherfrequencies
the modal patternsbecomeincreasinglycomplexanddescribe
compactellipsesthatoccurmore
or less uniformly along the unreinforcedregionsof the
soundboard. We also found that the case of the acoustics
ysisequipment.The cooperationof all theseorganizations
madethis work possible.
•Since
NorthernandSouthern
European
styles
ofharpsichord
building
varied considerably,
any inferences
drawn from this studycan referto only
Northerninstruments,
andevenhereonlyin a generalsense.Southerninstrumentsmaybeexpectedto behavesomewhatdifferently.
•This instrument has a so-called "short octave" in the bass.Rather than
harpsichordexhibitsa significantdegreeof motionat many
soundtheirnormallyexpected
pitches,the Bi key produces
the noteG•,
frequencies.
Whilethisactivitymaynotbeof musicalimportheC2produces
thenoteA•, andtheD2 produces
thenoteB• (or B•).
tance,it is probablysignificantto the "feel" of the acoustics Thus, practicallyall the bottomoctaveis diatonicrather than chromatic.
Theactualpitches
canbeseenin Fig.7. It shouldbenotedthatshortoctave
harpsichord
to theplayer.
keyboards
werethenormonearlyharpsichords,
andonoccasion
couldstill
It wasfurtherdiscovered
thatthestringpartialsstrong- befoundwellintothe 18thcentury.The lackof a chromaticbasswasrarely
ly excitedmanyair andwoodresonances.
It wasconcluded seenas a drawback;composersavoidedthosenotes,buildersusedfewer
that the acoustics
harpsichord's
generallyuniformacoustic stringsandjacks, and the instrumentmaintainedits slendershape.For
further information see Kottick, 1987.
outputresultedfrom the activationof manyof thesereso- •'he term"modalanalysis
refersto theprocess
of describing
thedynamic
nances.Informationwasalsopresented
showingthat in cerresponse
of a structurethrougha setof mathematicalrelationships
known
tain casesthereis couplingbetweenthe air and soundboard asthemodalproperties:
theresonantfrequency,
dampingfactor(ratioof
theviscous
dampingvalueto thecriticaldampingvalue),andtherelative
modes,but the musicalimportanceof the fact remainsunclear.However, anecdotalobservationsand somelimited ex-
perimentalresultsindicatethat air modesare of morethan
passing
importance
to thetonequalityof theacoustics
harpsicord.
motionof the structureat .selected
locations.The modalpropertiescanbe
established
eitheranalytically,usingfiniteelementtechniques,
or experimentally,as wasdone here.Additionalinformationon the mathematical
foundations
of modalanalysis,testingtechniques,
data analysismethods,
andtheproperapplication
of modalpropertydatacanbefoundin Ewins
( 1986} and Marshall ( 1985, 1986).
Althoughthe informationreportedin thispaperrepre- 4Themathematical
algorithmusedto calculatethe frequency
response
sentsthe culminationof more than 13 yearsof studyingthe
functionsis somewhatmorecomplicatedthan indicatedhere,but the deacoustics
harpsichord,a completeunderstanding
of the inscriptioncaptures
theessence
of theprocess.
Seethereferences
mentioned
in footnote 3 for additional details.
strument would entail the ability to calculate its radiated
SFigure
6 shows
whatmightbetermed
the"loudness
curve"fora William
soundfrom firstprinciples.For at leastonetypeof harpsi- HymanFrenchdouble-manual
harpsichord.
A sound-pressure
levelmeter
chord,this paperrepresents
a smallstepin that direction. wasmounted3 m abovethe instrument,andeachnotewasplayedfromthe
keyboardon a single8' stringby therear-mostsetofjacks.The presumpAlthoughwe havelearneda greatdeal aboutthe waysin
which the dynamicbehaviorof the acousticsharpsichord tion wasthat for eachnotethe vibratingstringwouldexcitewhateverair
and soundboardmodeswere availableto it. The figureshowsthat the
relatesto its tonequalities,much remainsto be understood, acousticoutputisgreatestat thelowerfrequencies
and,witha fewnotable
particularlyin regardto the interplayof the air and sound- exceptions,
decreases
asthefrequency
increases.
The averagerateof deboardresonances.
Othermissingingredients
are an absolute cline in the acousticoutput•sapproximately2 dB/oct. This decreasein
level seemstrivial; so much so, that we might expectthe
measureof the strengthsof the stringpartials,the string- sound-pressure
bassto beoverwhelmed
by thetreblein perceivedloudness.
However,the
bridge-soundboard
terminationimpedances
at all stringlobassnotesof a goodharpsichord
are richin partialsandmakea strongly
cations,completemodaldatafor the air resonances
and,for
distinctivesound,whilethe treblenotes,with fewpartials,arelesscolored
andmoresinusoidal.
Thisdecrease
inopulence
ofsoundfrombassto treble
correlationpurposes,
an accuratemeasureof thesoundradimustaccountfor at leastsomeof the perception
of loudness.
Thesedata
ated by the acoustic harpsichord. Further studies are
weremadein 1980,at the shopof ZuckermannHarpsichords,
while the
plannedto addresstheseissues.
secondauthorwason a fieldassignment
to measurethe response
curvesof
Modern sciencehasmade possiblemodestadvancesin
newandantiqueharpsichords.
It shouldbenotedthat theworkwasdone
underfieldconditions,
with little concernfor roommodesor radiationpatour understanding
of the acousticsharpsichord,but it has
terns.To facilitatetheexperimental
measurements,
thelid, whichnormalnotgivenusmuchinsightintothemethodsthemasterbuildly reflectsa greatdealof soundout towardthelistener,wassetat 90 degto
ersof the pastusedto designandimprovetheir instruments. theinstrument,ratherthanat themoreusual45 deg.The Frenchharpsichord mentionedis identifiedasHyman KI0 in Kottick (1985).
Their magnificentharpsichords,
andthefinemoderninstrumentsinspiredby them, challengescientistsand builders Ewins,D. J. (1986). Modal Testing:Theoryand Practice(ResearchStudalike to uncover more of the secrets of their rich tradition.
ies,Letchworth,Hertfordshire,England).
Fletcher,N. H. (1977). "Analysisof theDesignandPerformance
of Harpsichords,"AcousticaXXXVII,
ACKNOWLEDGMENTS
AND DEDICATION
The authorsherebyacknowledgethe many contributionsof our recentlydeceasedcolleague,Dr. William R. Savage,without whosededicationand leadershipthis research
would not havebeenaccomplished.
We dedicatethis paper
to his memory.We also expressour appreciationto the
GraduateCollege,the Departmentof Physicsand Astronomy, and the Schoolof Music at The University of Iowa for
researchassignments,
equipment,financialsupport,and encouragement;
to GettysburgCollegefor sabbaticalleaveand
researchgrants;and to the Uniroyal GoodrichTire Companyfor the useof their computersandelectronicdata anal2189
J. Acoust.Soc. Am.,Vol. 91, No. 4, Pt. 1, April1992
139-147.
Kellner, H. A. (1976). "TheoreticalPhysics,The Harpsichord,and Its
Construction:A Physicists's
Annotations,"Das MusikinstrumentXXV,
187-194.
Kottick, E. L. (1985). "The Acousticsof the Harpsichord:Response
Curvesand Modesof Vibration,"CiaipinSoc.J. XXXVIII, 55-77.
Kottick, E. L. (1987). TheHarpsichordOwnersGuide(The Universityof
North Carolina,Chapel1till, NC).
Marshall,K. D. (1985), "Modal Analysisof a Violin," J. Acoust.Soc.Am.
77, 695-709.
Marshall, K. D. (1986). "Modal Analysis:A Primer on Theory and Practice," Catgut Acoust.SOc.46, 7-17.
Spencer,M. (1981). "HarpsichordPhysics,"GalpinSoc.J. XXXIV, 2-20.
Thwaites,S. (1981}. "SomeAcousticsof a Clavichord,"Catgut Acoust.
SOc.38, 29-33.
Thwaites, S., and Fletcher, N.H. ( 1981}. "Some Notes on the Clavichord,"
J. Acoust. SOc.Am. 69, 1476-1483.
Savageotal.: Airaridstructuralmodesof a harpsichord
2189