Investigations in the amplitude of sounded piano tones
Caroline
Palmer
Psychology
Department,
OhioStateUniversity,
1885NeiL4venue,Columbus,
Ohio43210
dudith C. Brown
Massachusetts
Instituteof Technology
MediaLaboratory,Cambridge,Massachusetts
02319
andPhysics
Department,Wellesley
College,Wellesley,
Massachusetts02181
(Received8 November1990;acceptedfor publication27 March 1991)
The relationshipbetweenfinal hammervelocityandmaximumamplitudeof radiatedpiano
soundwasinvestigated.
Pianotoneswith varyinghammervelocitieswereproducedby a
computer-monitored
acousticpianocontainingopticalsensors
andsolenoids,
and the sounded
toneswererecordedand digitizedfor analysis.Maximum amplitudesoverthe durationof the
soundedtoneswere linearlyproportionalto pianohammervelocitiesfor a rangeof frequencies
and hammervelocities.Changesin roomacoustics
did not alter the linearrelationship.
Measurementsof maximumamplitudesof individualtonesand combinedtones(dyads) also
indicateda linearrelationshipbetweenthe sumof the maximumamplitudesof the individual
tonesand the maximumamplitudeof the dyads.Thesefindingsindicatethat the principleof
superposition
holdsfor peakamplitudesof soundedpiano tones.Findingsare discussed
with
regardto productionand perceptionof musicaldynamics.
PACS
numbers: 43.75.Mn
INTRODUCTION
Hall's modelof stringvibrationamplitudes,1987a).Careful
measurements
of the amplitudeof initial stringpulsesreTherelationship
between
production
ofmusicaldynam- vealeda linearitywithfinalhammervelocity(Askenfeltand
icsin pianoperformance
and its subsequent
perception
is Jansson,
1988),suggesting
that,if soundboard
displacement
largelyunstudied,with a few exceptions(Fletcheret al.,
isproportionalto stringmotion,thentheamplitudeof radi1962;Nakamura,1987).Musicaldynamics
referto changes ated soundwould be proportionalto hammer velocityas
in perceived
loudness,
whicharisefromaninterplayofinten- well. Althoughrecentstudiessuggestthat the lineartheory
sity,timbral,andtemporalchanges
affordedby a particular cannotaccountfor spectralchangesseenin the upperparmusicalinstrument(Handel, 1989;LerdahlandJackendoff, tials of string motion (Hall and Askenfelt, 1988; Hall,
1983).Thepiano,a musicalinstrument
witha largerangeof
1990), the nonlinearcomplianceof pianohammersmaynot
intensityandtemporalchangebut a relativelyfixedtimbral substantiallyaffectthat part of the spectrumwhich domirange,providesan excellentdomainfor studyof the mecha- natesmaximum amplitude.The experimentsdescribedbenismsleadingto changes
in musicaldynamics.Preliminary low addressthe questionof linearitybetween(final) hamto an understanding
of the musicians'useof dynamics,we
mer velocitiesand the maximum(or peak) amplitudeof
needto examinethe relationshipbetweenradiatedpiano radiated sound.
soundandhammervelocities,
thepianist'sprimarymeansof
Relativeamplitudechanges
duringtheattachportionof
controllingloudness.
We describe
hereinvestigations
of the
musicalsoundsmaybethemostimportantperceptualcueto
influenceof hammervelocityon the amplitudeof sounded onset time and timbrai identification, as well as to loudness.
pianotones.
Perceptualexperiments
indicatethat listenersare mostsenFactorscharacterizing
themechanisms
of productionof
sitiveto the initial portion of a soundedtone in their perceppianosoundhavebeenwell documented
(Weinreich,1977; tion of timbre (Grey, 1977) andrelativeonsettimes(Rasch,
Hall, 1987a;Hall and Clark, 1987;Askenfeltand Jansson, 1978;Vos and Rasch, 1981). For most acousticmusicalin1988; Boutilion, 1988; Hall and Askenfelt, 1988). Sound
struments,the amplitudechangesover the duration of the
generationin the pianodependson manyfactors,mostimradiated sound,with the greatestchangein the attack, or
portantof whicharethemotionof thestringscausedby the
time from zero amplitudeto peak amplitude (Apel, 1973).
impact of hammersand the impedanceradiation characterIn piano sound, the attack contains the most change in specisticsof thesoundboard.
Whena pianokeyisdepressed,
the
tral energy,with all contributingharmonicsdecayingrapidpianoactiontransmitsenergyfrom the pianist'sfingersto
ly after peakamplitude.The role of the peakamplitudedurthe appropriatehammersand strings.A hammeris thrown
ingtheinitial portionof a soundisimportantin perceptionof
againstthe strings,settingtheminto vibration.Stringvibradynamic emphasisin both music and speech(Sundberget
tionsare thentransmittedvia thebridgeto the soundboard, al., 1983;Carlsonet al., 1989), suggestingcommonalitiesin
the main sourcesof radiatedsound(Boutilion, 1988;Hall,
auditory perceptionunderlyingthe two domains.
1990;Rossing,1990).
We describeseveralexperimentsthat addressthe relaSeveralstudiesof pianostringmotionsuggest
that amtionship between amplitude of radiated piano sound and
plitudesof stringvibrationsare proportionalto the final
hammer velocity,the pianist'smain methodof controlling
hammer velocity (Askenfelt and Jansson,1988; see also
dynamiclevel.Thisrelationshipwasstudiedwith theaidof a
60
J. Acoust.Sec.Am.90 (1), July1991
0001-4966/91/070060-07500.80
@ 1991 Acoustical
Societyof America
60
computer-monitored
acoustic
grandpiano,whosesolenoids addition,the frequencyrangewaschosenover7 octaves(a
allowedprecisecontrolandgeneration
of pianotoneswith frequencydifferenceof approximately2000 Hz acrossthe
varyingfrequencies
andhammervelocities.
We presentevi- rangeof 33-2096Hz), allowingstudyof sounded
amplitude
denceconsistentwith Hall's (1987a) model of a linear relaacrossa widerangeof hammervelocitiesandfrequencies.
A
tionshipbetweenstringvibrationamplitudesand hammer set of measurementsof C4 (middle C) sounded at l-m/s
velocity,suggesting
that the maximumor peakamplitudeof
hammervelocitywererecordedat the beginningof eachexthe radiatedpianosoundis proportionalto the hammerveperiment,to allow comparisons
acrossrecordingsessions
locity as well.
(becausethe gain settingof the recordingequipmentwas
readjusted
at thebeginning
of eachexperiment).
I. EQUIPMENT
The pianowascalibrated
beforeeachrecording
for seven
different
hammer
velocities
at
each
frequency.
The
reliA 9-ft 6-in. BosendorferSE computer-monitored
conability
of
the
recording
equipment
was
determined
by
examcert grand piano,containingoptical sensorsand solenoids,
for
allowedpreciserecordingand playbackof hammerveloc- ining the standarderror of repeatedmeasurements
sounds
produced
with
the
same
hammer
velocity
and
freities.Infrared photodetectors
in the piano,placednear the
andendof eachrecording
session.
positionof impactof eachhammerheadona string,measure quencyat thebeginning
The
consistency
of
the
repeated
measurements,
represented
the final velocitywith which the hammer approachesthe
by the standarderror,indicatedlessthan5% difference
for
string.Solenoidscalibratedfor eachkey allowedprecisereproductionof eachhammervelocity.The final hammervelocityandkeypress
onsetandoffsettimesfor eachnoteevent
werecontrolledby an IBM/PC computer.
The computer-monitored
piano sat in a rectangular
room,approximately9.5X 3.7X 3.7 m in length,width,and
height,with paintedconcretewalls,a tiled floor,and suspendedacoustic
tileceilingconsisting
of largepipesandother irregulargeometricalshapes.The room was furnished
with a computerworkstationand electronicsrack in additionto thegrandpianoandrecordingequipmentusedin the
experiment.The reverberationtime of the room was measuredbyrecordinganddigitizinga handclapin theroomand
analyzingitsdecayafterthecompletion
of theexperiments.
the set of hammer velocitiesand frequenciesbeing mea-
sured.
II. MEASUREMENTS
AND RESULTS
A. Peak amplitude proportional to hammer velocity
The firstexperiment
investigated
the relationship
of finalhammervelocityto peakamplitudein theradiatedpiano
sound.Two variableswerealtered:frequencyandfinalhammervelocity.Sevenfrequencies
(usingC'srangingfromC1
to C7, whereC4 = middleC) werepairedwithfivehammer
velocities,
including0.25, 0.5, 1, 2, and4 m/s. Ten repetitions of each combinationof velocity and frequencywere
timeof 0.3 s for theconditionsof thefirstexperiment(Hall,
generatedby computer,resultingin 350 measurements.
Each tone wassoundedfor 500 ms, followedby 1500ms of
silence.
Thiswasfollowedbythesounding
of thenextrepeti-
1987b).
tion. After the radiated sound was recorded and transferred
During the recordingsessions,
the pianolid was open
and the musicstandwasdown;no attemptsweremadeto
to computer,
numerical
analyses
wereproduced.
Theamplitudevalueswerethen scaledto a commonrangefor all experimentsreported,usingtheC4 calibrationmeasurements
recordedat thebeginning
ofeachsession.
Notethattheabsoluteslopevaluesarearbitrarybothbeforeandafterthescaling,butthepercentage
difference
in slopes
remainsthesame.
Comparisons
will onlybedrawnbetween
slopes
withinexperiments.
Figure1contains
thefirst240msofsound
fortwotones,
Based on this measurement,we estimated a reverberation
muffle the radiated sound with acoustic material. Record-
ingsof the radiatedsoundweremadewith a NeumannKM84cardioid(directional)microphoneplacedneara pianist's
head position,centeredover the middle of the piano keyboardapproximately0.5 m from, and directedtoward,the
soundboard
edge.Thusthemeasurements
of peakamplitude
were made at one positionin the nearfieldunder normal
roomconditions.The microphonewasconnectedto a Sony
MX-P21 mixer, which passedthe signal to a 16-bit Sony
digitalaudiotaperecordersamplingat 48 kHz. The sound
wastransferred
in analogformto computerm•dredigitized.
To reducethe requiredcomputerspace,analyseswerecomparedat samplingratesof 48 and 32 kHz. Spectralanalyses
revealedlittle to no power in the frequencyrange over 16
kHz; because
therewasnolossin information,analyses
were
conductedon the informationsampledat 32 kHz.
Piano hammer velocitiesin the rangeof 0.25 to 4 m/s
were chosento representmusicaldynamicmarkingsfrom
pianissimoto forte (an approximatedifferenceof 25 dB
acrossthe rangeof 45-70 dB sound-pressure
level). The
range of hammer velocitieswas based on common values
recordedin performances
on this piano (over 90% of hammer velocitiesfrom a setof pianoperformancesof Western
classicalexcerptsfell within this range;Palmer, 1988). In
61
J. Acoust.Sec. Am.,Vol. 90, No. 1, July1991
C3 soundedat the hammer velocitiesof 2 m/s (mezzoforte)
and4 m/s (forte).Thepeakamplitude,
defined
asthemaximumoftheabsolute
values
ofamplitude
overtheduration
of
eachtone,isapproximately
doubled
withthedoubling
ofthe
hammervelocity.
Theposition
ofpeakamplitudes
wasstable
across
trials;it tendedto shiftlessthanthreeperiodsbetween
repeatedmeasurements
of thesamefrequency.
The peakamplitudeof eachtone wasdeterminedby
computer,
bychoosing
thelargest
absolute
valueof thesamplesoverthe500-msdurationof eachtone.Figure2 shows
themeanpeakamplitude
across
tenmeasurements
for each
combinationof hammervelocityand frequency.Note that
thepeakamplitude
increases
linearlywithhammervelocity
with the exceptionof the smallesthammervelocityof 0.25
m/s, whichyieldedamplitudessimilarto thoseof 0.5 m/s.
To test the linear relationshipbetweenpeakamplitudeand
hammervelocity,a linearregression
modelwasfit, predictC. PalmerandJ. C. Brown:Amplitudeof soundedpianotones
61
10ooo
C3 at 2 m/s
A
M
5OO0
P
T
U
D
E
-5000
!
I
120
240
TIME (ms)
C3 at 4 m/s
A
M
10000
P
L
I
0
T
U
D
E
that of C6, asseenin Fig. 2. The variabilityof the measurementsmadeat thisfrequencywasmuchhigherthanfor other frequencies,
indicatingthat the rangeof hammervelocitiesproducedby thissolenoidwasmorevariablethan those
of the othersolenoids.
With the exceptionof C6, frequency
did not significantly
alter the proportionality
betweenpeak
amplitudeand hammervelocity,andthe greatestdifference
betweenslopevalueswas lessthan 15%.
Although peak amplitudewas highly correlatedwith
final hammer velocity,other soundcharacteristicssuchas
intensityintegratedover someportion of the tone may be
morecloselyrelatedto musicaldynamiclevel.To investigate
this hypothesis,final hammervelocitywascomparedwith
othercharacteristics
of the radiatedsound.Simplecorrelationswerecomputedbetween:hammervelocityandtheintegralof theabsolute
valueof amplitudefromthebeginning
of
sounduntil time of peakamplitude(within 50 ms for all but
the lowestfrequencyand hammervelocitycombinations);
hammervelocitysquaredand the integralof the intensity
(amplitudesquared)up to timeof peakamplitude;andthe
same measurements
-10000
-20000
120
240
TIME (ms)
FIG. 1. Samplewaveforms
fromsounded
tones:Beginning
section(240
ms)of toneC3 sounded
at 2- and4-m/shammervelocity.
Peakamplitude
marked with arrow (see text).
for the entire duration
of the tones.
Noneof thesevariablesshowedas largea correlationwith
hammervelocity(or hammervelocitysquared)asdid peak
amplitude.
Thus peakamplitudeof radiatedpianosoundwaslinearlyproportionalto hammervelocityfor a givenfrequency
in thesemeasurements.
Onepossible
limitationof theexperimentalprocedureconcernedthe roomacoustics;
becausethe
walls and floor were reflectant, the reverberant conditions
may have affectedthe linear relationship.The next study
attemptedto replicatethe linearityfindingunderdifferent
ingthepeakamplitudefromthehammervelocities,
omitting
0.25m/s. The linearregression
modelprovidedan excellent
room acoustic conditions.
fit for eachfrequency,
shownin Fig. 2, with a multiple B. Change in acoustic environment
regression
coefficient
greaterthan0.99 (p < 0.001) for each
frequency.
Figure2 alsoindicatesthat the slopesassociated
with
eachfit are not orderedin termsof frequency;
that is, the
proportionality
doesnot appearto be affectedby the frequencyrange.To test for differences
amongslopes,95%
confidence
intervals
werecalculated
foreachslope.Theonly
slopethat significantly
differedfrom the others(by 40%)
30000
E 25000
c1, c5
20000
A
C7
C4, C2, C3
M 15000
P
L
[
T
U
D
E
10000
5000 -
1,00
2.00
HAMMER
3.00
4.00
VELOCITY (m/s)
FIG. 2. Peakamplitude
asa functionofhammervelocityforsevenfrequencieswith fittedlinearregression.
62
ume-to-surface ratio. The reverberation time for this acous-
tic environment,basedon measurements
of a handclap(recordedand digitized under theseconditions)wasestimated
to be approximatelyhalf of its previousvalueof 0.3 s.
Insteadof the C frequencies,
F-sharpsweremeasured
from octaves 1-7. The stimulus durations and hammer ve-
P
A
K
To extendthe previousfindings,changes
weremadein
room acousticsand in the rangeof frequencies
measured.
Thesameroomusedin theprevious
studywasnowequipped
withheavycurtainscovering2/3 of eachwall (fromceiling
to floor) to decrease
the reverberation
time. A secondpiano
hadalsobeenaddedto theroom,furtherdecreasing
thevol-
d.Acoust.
Sec.Am.,Vol.90,No.1,July1991
locitieswerethe sameasin the previousmeasurements,
except that the 0.25-m/s hammer velocity was not included
(becausethe previous measurementshad shown no differ-
ence from the 0.5-m/s hammer velocity), and a hammer
velocityof 3 m/s wasincluded.Five repetitionsof eachhammer velocityand frequencywererecorded,becausethe variability of the previousmeasurements
waslow.
Figure3 showsthepeakamplitudeasa functionof hammer velocity,for eachfrequency.Again, peak amplitudeis
linearlyproportional
to hammervelocity.A multipleregressionmodelwasfit for eachfrequency,predictingthe peak
amplitudefrom the hammer velocityfor the 25 measurements.The linesin Fig. 3 demonstratethe fitsfor eachfreC. PalmerandJ. C. Brown:
Amplitude
ofsounded
pianotones
62
30000
Fsl
Fs4
Fs3
Fs2
Fs5
Fs6
2•ooo
20000'
Fs7
•5ooo.
The previousexperimentsemployedfrequencieswith
simpleintegerratios(2:1 or an octaveapart). In thisstudy,
frequencies
werechosenthat differedin ratio.The common
musical intervals of the minor third and the fifth were in-
cluded,alongwith the tritone as an exampleof an interval
with fewoverlappingharmonics.Thesethreeintervalswere
chosento representa rangeof frequencyratiosbetweenthe
individual
ioooo.
tones: fifths with the most common
harmonics
(3:2 ratio) of these three intervals, minor thirds with next
most common harmonics (6:5 ratio), and the tritone with
5000-
o
o
1.00
2.00
HAMMER
3.00
4.00
VELOCITY (m/s)
the fewestcommonharmonics.If frequencyratiosaffectthe
peakamplitudesof thecombinedtones,thentherelationship
betweenpeakamplitudeandhammervelocitymaydifferfor
each interval.
Each dyad containedC4 (middle C) asits base,played
at a hammer velocityof I m/s (a dynamiclevelof mezzociesunder
altered
roomacoustic
conditions.
withfittedlinearregression. forte). Thus the three dyads in this study were: C4/Ef4,
FIG. 3.Peakamplitude
asa function
ofhammer
velocity
forseven
frequen-
C4/Fs4, and C4/G4. Hammer velocitiesfor the Ef4, Fs4,
quency.The linearregression
modelprovidedan excellent
fit,witha multiplecorrelation
coefficient
> 0.99(p<0.001)
for eachfrequency.
Figure3 alsoindicatesthat the linearrelationship
was
not affectedby frequencyrange,because
the slopesassociated with eachlinearenotorderedin termsof frequency.
To
testfor anydifferences
amongslopes,99% confidence
intervalswereagaincalculated
for eachslope.Again,theslopes
did not differsignificantly
(the largestdifference
in slopes
was18%), with the exception
of Fs7,whoseslopeis lower
thanthatof theotherfrequencies.
Thustherangeof differencesamongslopeswassimilarto that of the firstexperi-
and G4 werethe sameasin the firstexperiment:Thesewere
0.5, 1, 2, and 4 m/s. Ten recordingswere made for each
individualtoneandfor eachof the threedyadsat eachof the
four hammer velocities.
First, the relationshipbetween peak amplitude and
hammervelocityfor the individualtoneswasexamined.Fig-
ure 4 depictsthe peakamplitudeas a functionof hammer
velocity,for the three individual tones(Ef4, Fs4, and G4).
Eachof the threetoneswasfit by a linearregression
analysis,
predictingthe peak amplitudefrom hammervelocity;the
linesin Fig. 4 indicatethosefits.The linearrelationshipseen
previouslywas replicated,with a significantfit (multiple
regression
coefficient= 0.99,p < 0.001) for eachfrequency.
ment, and the absolute differences in measurements are atNext, the superpositionof the individual tonesin the
tributabletodifferences
in gainsettings
duringtherecording dyadswasexamined.The dyads'peakamplitudeswerecomsessions
(the rangeof measurements
from the firstsetting paredwith the peak amplitudeof the individuallysounded
wasapproximately
half that of the secondsetting,and the
tones.For eachdyad, the sumof the peakamplitudesof its
absolute
difference
in slopes
in Fig.2 wasapproximately
half
individual componenttoneswas calculated.Figure 5 disthatof theslopedifferences
in Fig. 3).
playsthemeanpeakamplitudefor thedyadsasa functionof
the sum of the mean peak amplitudesfor the individual
Peakamplitudeof radiatedpianosoundwasagainlinanalyearlyproportional
to hammervelocityfora givenfrequency. tones.The linearrelationshipwasfit with a regression
Theseresultsreplicatethoseof the previousexperiment, sis,and the regressionfitsfor eachdyadare shownin Fig. 5.
with changes
in roomacoustics
and frequencies.
It is not
clear, however,that multiple (simultaneously
sounded)
toneswouldholdthesamerelationship
between
peakampli30000
tudeandhammervelocity.The nextstudywasdesigned
to
testthis hypothesis.
C. Principleof superposition
P
E
A
K
25000
Fs4
20000
G4
Ef4
The peak amplitudeof soundeddyads(two simulta- A
neously
sounded
tones)werecompared
withthepeakampli- M •5ooo
P
tudesof theirindividualcomponent
tonesin thenextstudy. L
Thequestion
addressed
waswhethertheprinciple
of super- I •oooo
T
positionholdsforpeakamplitudes;
isthepeakamplitudeof
U
dyadsproportional
to thesumof thepeakamplitudes
of the
D
5000
individualtones?
Thiswouldimplythattheresponse
of the E
soundboard
islinear;itsvibrationin response
to a dyadisthe
sum of the vibrationsproduced by the individual tones
soundedseparately.To addressthis question,radiated
soundfrom commonmusicaldyadsand their individual
component tones was recorded.
63
J. Acoust.
Sec.Am.,VoLg0,No.1, July1991
0
1.00
2.00
HAMMER
3.00
4.00
VELOGITY (m/s)
FIG. 4. Peakamplitudeasa functionof hammervelocityfor threeindividual toneswith fitted linear regression.
C. PalmerandJ. C. Brown:
Amplitude
of sounded
pianotones
63
(within three periods), and amplitudesover severalsurroundingcycleswereapproximately
equalto the peakamplitude,
as
shown
in
the
example
given
below.
C4G4
Figure6 demonstrates
boththesounded
dyad(top) and
20000
thesuperposition
(middle) of theindividualtonesof C4 and
15000
G4 at l-m/s hammervelocity,summedsampleby sample.
Thisexample
represents
an extremecasein whichthepeak
amplitudes
of
the
individual
toneswerefar apartin time.
10000
The relativephaseof the individualtonesin the computed
superposition
wasdetermined
by shiftingthewaveforms
of
the individualtonesby threesamples
with respectto each
other (a phaseshiftof approximately
0.1 ms, whichis less
0
5000
10000
15000
20000
25000
30000
SUM OF PEAK AMPL. OF INDIVIDUAL
TONES
than the 1.25-mstemporalresolutionof the hammermovethewaveforms
donotvarymuchin ampliFIG. 5. Peakamplitude
ofdyadsasa function
ofsummed
peakamplitudes ments).Because
of individualtonesfor threefrequencycombinations,
with fittedlinear
tudefrom the maximumoverthe criticaltime range,the
20000
25000
C4Fs4
C4Ef4
5000
regression.
peak amplitude for the sum of the individual tonesis still
The multipleregression
coefficientis greaterthan 0.99 in
eachfit (p < 0.01) and the slopesrangedfrom 0.90 to 0.95,
proportional
to thepeakamplitudeof thedyads(Fig. 5).
To measurehow closelythe principleof superposition
was obeyedfor the entire waveform, the differencebetween
thecomputedsumof theindividualwaveforms
andthedyad
supporting
the principleof superposition.
The intercepts
wascalculated.
Thewaveformphases
werematchedbyshiftweresmalland negative;the deviations
of the intercepts
ingthe dyad5 samples(lessthan0.16 ms) with respectto
fromzeroandslopes
from 1 areduein partto differences
in
thecomputed
sum.Thedyadwasthensubtracted,
sampleby
phaseof the two superposed
waveforms.The observed
addisample,fromthesumof theindividualtones.Figure6 (bottivityisprimarilyduetothefactthatthepeakamplitudes
for
tom) depictsthe difference(sum - dyad) betweenthe two
eachwaveform
occurred
at approximately
thesameposition
waveforms.
The smalldifference
indicates
thattheprinciple
of superposition
isobeyedwith verysmalldeviations.This is
consistent
with the findingsof Fig. 5, showingsmalldeviations in peak amplitudesof dyads predictedfrom the
summedpeakamplitudesof individualtones.
A
M 2000P
D. Superpositionin a different frequency range
T
Previousmeasurements
of string-hammerinteraction
indicatethatstring-hammer
contactdurationchanges
with
frequencyand hammervelocity (Askenfeltand Jansson,
1988;Hall andAskenfelt,1988), and the energypresentin
the upperharmonicsincreases
with hammervelocity.The
questionof how thesechangesaffectmaximumamplitude
wasaddressed
by studyingtheprincipleof superposition
in a
differentfrequencyand hammervelocityrange.Measure-
E
-4000
C4 + G4
ments were made of the same common musical intervals: the
minor third, the fifth, and the tritone. Individual tones were
chosenfrom 2 octaves(Ef3, Fs3, G3, EfS, Fs5, O5), and
dyadswerecreatedby pairingeachindividualtonewith the
C in itsoctave(Ef3 waspairedwithC3;Ef5 waspairedwith
4OOO
C5). Hammer velocitieswere 0.5, 1, 1.5, 2, 3, and 4 m/s;
eachindividualtoneanddyadwaspairedwith eachhammer
velocityto create72 combinations.
C3 and C5 werealways
Difference
presented at I m/s (a dynamic level of mezzoforte). Ten
recordingswere made for eachsound.
P
L
I
A linearregression
analysistestedwhetherthepeakamplitudeof thedyadswasproportional
to thesumof thepeak
amplitudes
of theindividualtones.Figure7 displays
thefits,
which were good (multipleregression
coefficients> 0.99,
0
T
U
D -2000
E
-4000
t
120
I
240
TIME (ms)
FIG. 6. Waveformsfor sounded
dyadC4G4 at hammervelocityof l m/s
(top), sum of individual tonesC4 and G4 at I m/s (middle), and their
difference(sumof individualtonesrainusdyad) ( bottom).
64
J. Acoust.
Soc.Am.,Vol.90, No.1,July1991
p < 0.001). The peak amplitudeof dyadsis proportionalto
the sum of the peak amplitudesof the individualtonesdespitethe changesin frequencyand hammer velocities.With
one exception(the fit for C5G5), the slopesdid not differ
(ranging from 0.81-0.96) and the interceptswere again
C. PalmerandJ. C. Brown:
Amplitude
ofsounded
pianotones
64
P
E
A
K
30000
tion, from hammers,to strings,to bridge, and to soundboard, approximateslinearity for peak amplitudesin the
rangestudiedhere.This findingis at firstsurprisingbecause
25000
A
M
P
C3Ef3
C5Fa5,
J
20000
L
I
T
U
C5Ef5///
/.C5G5
I•.{•3G3
15000
D
E
the radiated sound reflects the sum of the normal modes of
vibrationofbothstringsandsoundboard
andimpliesa linear
couplingfor thosemodescontributingto peak amplitude
measurements.However, we haveexamineda setof hammer
velocitiesfor a givenfrequency,andalthoughthehammer's
velocitywasvaried,exactlythe samemodesof a particular
0
y
5000
string
wereexcited.In addition,becausestringresonances
A
D
with periodsshorterthan the contactdurationof hammer
S
0
with stringare only weakly excited(Hall, 1990), the fre5000
10000
15000
20000
25000
30000
SUM OF PEAK AMPL. OF INDIVIDUAL TONES
quencies
usedheregeneratedlittle energyin the upperharmonicsfor eachsoundedevent,and peakamplitudewaspriFIG. 7. Peakamplitudeof dyadsasa functionof sumof peakamplitudes
of
individualtonesforextended
frequency
range,withfittedlinearregression. marily determinedby the first few spectralcomponents.
Thus the linear relationshipbetweenpeak amplitudeand
hammervelocitymaybemorepredictablethanit wouldfirst
0
10000
F
small and negative.Someof the data contributingto the
C5G5 slopewereinadvertentlyerasedfrom the computer,
and the data had to berecoveredfrom tapeand redigitized.
This error may accountfor the notable differencein the
C5G5 slope;however,the samelinearrelationshipbetween
thesumof peakamplitudesofindividualtonesandthepeak
amplitudeof the dyadsstill held.
To examinefurther the effectsof frequencyrangeand
hammervelocityon upperharmonics,a spectralanalysis
was conducted on the waveforms.
Results of the Fourier
transformfor a sampledyad (C5Fs5) are shownin Fig. 8,
andindicaterapiddecayin theamplitudesof lowerharmon-
ics,andlittle to noenergyin theupperharmonics.
Thusthe
proportionality
of peakamplitudesremainedunaffected
by
thesmallchangesin relativecontributionsof upperharmonics.
IlL DISCUSSION
First, thesestudiesindicate a linear relationshipbetweenthe peakamplitudeof individualtonesand the hammer velocitiesthat producedthem.This findingimpliesthat
the seriesof eventsin piano mechanismsof soundproduc-
appear.
The spectralanalyses
described
aboveindicatedthatthe
energypresentat higherharmonicsdroppedoff quicklyin
time and the peak amplitudeof radiated soundwas composedprimarilyof the fundamentalanda smallercontributionof thelowerharmonics.
Thus,althoughchanges
in hammer velocityand hammer-stringcontactdurationalter the
relative contributionsof upper harmonicsto the sounded
tone (Hall andAskenfelt,1988), theymay not substantially
alter the peak amplitude.Whereasa linear theory cannot
accountfor the rapiddecrease
in amplitudeof higherstring
modes,the smallmagnitudesof thesespectralcomponents
had little effecton the peakamplitudesobservedhere.
Second,thesestudiesindicatethat the principleof superpositionholds for peak amplitudesof dyads obtained
fromthepeakamplitudes
of twoindividualtones.Thisresult
held true over a rangeof frequencyand hammervelocity
combinations.Together with the linearity seen between
hammervelocityand peak amplitudefor individualtones,
thisfindingisencouraging
for thestudyof perceiveddynamics in pianoperformance,
becauseit suggests
a functional
mappingof individualhammervelocities
to peakamplitudes
of soundedchords.Hammer velocityis oneof the few parametersundera pianist'sdirectcontrol;therefore,
it isimportant that combinationsof hammer velocitiescan be
mappedto amplitudecharacteristics
of radiatedsoundthat
arecrucialfor listeners.Futurework mayextendthelinearity andsuperposition
findingsto a morenaturalmusicalsettinganda perceptual
domain.Thesefindingsmayaid investigationof therelationship
between
thepianist's
useof force
andtimingandthemusicaldynamics
perceived
bythelistener.
ACKNOWLEDGMENTS
This workwassupported
in partby an OhioStateUni-
versitySmall Grant and Grant IR29-MH45764from
NIMH to the firstauthorandby a Marilyn BrachmanHoff-
man Fellowshipto the secondauthor.We thankDonald
Hall, William Strong,and an anonymous
revicwcrfor comments on an earlier draft and the members of the Music and
FIG. 8. Graphicaldepiction
ofspectralanalysis
ofdyadC5Fs5waveformat
4-m/s hammervelocity.
65
J. Acoust.Sec.Am.,VoL90, No.1, July1991
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