Musical fundamental frequency tracking using a pattern
recognition method")
Judith C. Brown
Physics
Department,14reliesIcy
College,Wellesley,
MassachusettsO1281
andMediaLab,Massachusetts
Instituteof Technology,
Cambridge,
Massachusetts
02139
(Received12March 1991;accepted
for publication15May 1992)
In a recentarticle [J. C. Brown,"Calculationof a ConstantQ SpectralTransform,"J. Acoust.
Soc.Am. 89, 425-434 ( 1991) ], the calculationof a constantQ spectraltransformthat givesa
constantpatternin thelogfrequencydomainfor soundswith harmonicfrequencycomponents
hasbeendescribed.
This propertyhasbeenutilizedin calculatingthecross-correlation
function
of spectraof soundsproducedby musicalinstruments
with the idealpattern,whichconsists
of
one'sat thepositions
of harmonicfrequencycomponents.
Therefore,thepositionof thebest
approximationto the "ideal" patternfor the spectraproducedby theseinstrumentshasbeen
determined,and in sodoingthe fundamentalfrequencyfor that soundhasbeenobtained.
Resultsarepresented
for scalesproduced
by thepiano,flute,andviolinaswellasfor arpeggios
playedby a widevarietyof instruments.
PACS numbers:43.60.Gk, 43.75.Yy
I. BACKGROUND
INTRODUCTION
Previous work in the field of musical fundamental
fre-
quencytrackingwasreviewedrecentlyin an articledescribing a frequencytracker operating in the time domain
(Brown, 1991). Sincethis work is basedon a calculation in
the frequencydomain,we will limit our backgrounddiscussionto studiesthat havebeendoneon musicalsystems
in the
frequencydomainexceptwhere directly relevantto this
work.
Most of the effortsat musicalfrequencytrackinghave
takenplacein the frequencydomain(Terhardt, 1979;TerhardtetaL, 1982;Amuedo,1985;ChafeandJaffe,1986) and
In a recentarticle (Brown, 1991), we describeda calcu-
lationthatserves
asthebasisforthefrequency
trackerwhich
will be discussed.
In this calculation,a spectraltransform
equivalent
to a 1/24thoctavefilterbankiscarriedoutevery
15msonthedigitizedsoundfroma musicalinstrument.
The
frequencycomponents
thushavelogarithmicspacing.As
described(seeFig. I in Brown, 1991), for a soundwith harmonicfrequencycomponents,
theseFouriercomponents
havea spacing
in thelogfrequency
domainwhichisindependentofthefundamental
frequency.
Forexample,
thespacing
between the fundamental
and the second harmonic
is
have used a method of attack similar to that of the Schroeder
log(2), that betweenthe secondand third components
is
(1968) histogrammethod.After the calculationof a fast
Fourier transform,a hypothesisis assertedfor eachfrequen-
1og(3/2), and soon.
Sincethis patternis constantfor harmonicfrequency
cy componentof all possiblefundamentalfrequencies
for
which it couldbe a harmonic;i.e., eachfrequencycomponent is dividedby integersand the resultsare enteredin a
table.The entriesareweighted,anda decisionismadebased
on criteria involvingthe numberof components
and their
weights.The frequencyis chosenwhichmostcloselymeets
previouslydeterminedcriteria.
A similar frequencytracker (Piszczalskiand Galler,
1979) took ratiosof pairsof components
to form their hypotheses
for the fundamentaland thenproceededas above.
components,
the"pattern"with l's at theappropriate
spacingscanbeconvolved
(or crosscorrelated
) withthespectral
transform,anda maximumshouldoccurat thepositionof
the fundamental.
Note that this methodaccomplishes
the
Duifhuis et al. (1982) studied speechsegmentsusing a
of the analyzedsound.This "asserts"the appropriatefunda-
methodwhich mostcloselyapproaches
that of this article.
Followingan FFT, theykepta maximumof sixpeaksand
mentalasa hypothesis
(weightedwith thevalueof thecrosscorrelationfunctionat that point) but it is simultaneously
asserting
thatsamefundamental
foreachof thecomponents
of thesoundwhicharein theappropriate
position.Sorather
thandividing,choosing
a weightingsystem,andkeepingentries in a table, we obtainone numberfor eachfrequency
corresponding
to thesumof all thefrequency
components
of
the soundthat are in the correctpositionto be multiplesof
that frequency(harmonicsof that fundamental).Thus in a
very elegantand completeway we are obtainingthe results
then used a "harmonic
sieve" to determine which of these
peaksbestfit thelogarithmicspacingobtainedwith harmonic frequencycomponents.
This methodwaslater refinedby
Scheffcrs (1983).
'•Thc nomenclature
"fundamentalfrequencytracker" or "frequency
tracker" is usedrather than "pitch tracker" becausethe editor wishesto
observethe psychoacoustical
distinctionbetweenpitch as a perceived
quantityand frequencyasa physicalquantity.
1394
samepurposeas the histogrammethod;the valuesof the
cross-correlation
functionbeingsimilarto the the sumsof
tableentries.Here, however,the valuescorrespond
to all
possible
harmoniccomponents.
For exampleastheconvolution
iscomputed,
eachcomponentof theharmonicpatternwill fall oneachcomponent
J. Acoust.
Soc.Am.92 (3),September
1992 0001-4966/92/091394-09500.80 ¸ 1992Acoustical
SocietyofAmerica
1394
thattheprevious
researchers
approached
withthehistogram
method.There is a computationaladvantageaswell in that
wesimplyaddthecomponents
withtheappropriate
spacing.
Finallythisfrequency
trackersolvestheproblemof the
"missingfundamental"in muchthe samemanneras that
hypothesized
for humans.It is essentially
comparingthe
harmonics
presentto a templateandfindingthebestmatch.
Thisisconsistent
with thepatternmatchingtheory(Gerson
andGoldstein,1978) of humanpitchperception.
II. RESULTS
spectrumincludesa strongfundamental,
andthecrosscorrelationexhibitsan evenstrongerand more unambiguous
peakthanforthesynthesized
datawhichwererepresented
in
Fig. 1. We will representsomeof the calculations
for the
violinthat arethe mostdifficultto interpretafterthe general
presentation
of our resultsandthe discussion
of adjustable
parameters
fi3rthisfrequencytracker.
With any patternmatchingmethod,the crosscorrelationestablishes
mostunambiguously
thepositionof thepattern that is soughtthe closerin shapeit is to the known
"ideal"pattern.(DudaandHart, 1973) Thusthenumberof
components
in the ideal patternshouldmatchthe average
number of non zero Fourier componentsfor a particular
instrument.The effectof varying thesecomponentsin the
patternis thusan adjustable
parameterto be optimizedfor
Beforepresenting
the resultson musicalinstruments,
it
is instructiveto considerthismethodwith singleframes.In
general,thecalculation
wascarriedouton 15msof analyzed
sound.In Fig. 1, the signalanalyzedconsisted
of a sound
generated
in software
with20harmonics
ofequalamplitude. each instrument.
The graphsof Figs. 3-5 demonstratethe frequency
The lowergraphof Fig. 1 is the spectraltransformof this
signalin thelogfrequencydomain.The patternthat is con- trackingresnltson soundsproducedby a flute,piano,and
volvedwith this transformconsistof l's with the samespac- violin as examplesof wind, keyboard,and string instruWe haveplottedmidinoteversustime
ingasthatof thesepeakssincethissignalwassynthesized
to ments,respectively.
where
middle
C
(C4)
is representedby midinote60, and
havetheidealshape.The cross-correlation
functionisgiven
each
semitone
corresponds
to a valueof onemidinotehigher
in the top of Fig. 1. Clearlythe largestpeakof the cross
(or
lower).
Graphs
of
the
spectrathat were usedfor these
correlationisat thepositionof thefundamental
for thisidecalculations
are
found
in
Brown
( 1991). Eachpoint in these
alizedspectrum,
andit canbechosen
easilybya peakpicker.
thepeakof a cross-correlation
calculation
It shouldalsobe noted,however,that the cross-correla- graphsrepresents
on
an
analysis
frame
similar
to
that
of
Fig.
2
corresponding
tionfunctionhaspeaksat thepositionof one-halfof andtwo
15msofsound.Eachinstrument
isplaying
timesthefundamental.Thesepeaksgiveriseto octaveerrors toapproximately
of a sequential
set
(a problemfor all frequencytrackers).With our method, a scalesothatperfectresultswouldconsist
thesourceof theproblemisthattheevenpeaksof thepattern of horizontal lines rising by one or two midinotescorreto a half or a wholestepin thescale.Thuserrors
lineup with thespectrumfor thefrequencyanoctavebelow sponding
madeby the frequencytrackerare easilydistinguished
as
that of the fundamentalso,if enoughpeaksare includedin
thepattern,thecross-correlation
functionwill havethesame pointsoff theappropriatehorizontalline.
The spectrumof the soundproducedby the flute convalueat thispositionasat thatof thetruefundamental.
For
the frequency
an octaveoverthat of the fundamental(the sistsof a fundamentaland a few upperharmonicsthat disecond
harmonic),thepeaksof thepatternarealignedwith minishin amplitudein a regularfashion.In Fig. 3, weshow
in the harthe evenpeaksof the spectrumagaingivingriseto a large theeffectof varyingthe numberof components
value of the cross-correlation function. The octave errors
monicpatternfromthreecomponents
to fivewheretheoptiof fourharmonics.
The fluteisplaying
thusproduced
arethechiefsource
oferrorforourfrequency mumpatternconsists
a C majorscalefromC4 to E6. Errorsoccuronnotechanges
tracker. In a later section,we will describea meansof elimiwhenthereismorethanonenotepresent.Most of theerrors
natingtheerroroccurring
on thesecond
harmonic.
An exampleof thecross-correlation
functionfor a digi• for the flute occur on the lowest notes when there are too few
in the pattern (lowestcurve). This is due to
tizedmusicalsoundisgivenin Fig.2 forthesoundproduced components
by a violin.Thisis a particularlyfavorable
exampleasthe little energyin the lowerharmonicsfor thesenotes.
CROSS
CORRELATION
FUNCTION
FIG. 1. Spectrumversuslog frequeucy
fora soundconsisting
of 20 harmonics
of
equalamplitude(below) and thecrosscorrelation function of this spectrum
with a functionconsisting
of l's with the
SPECTRUM
RE
1395
(
)
J. Acoust.Soc. Am., Vol. 92, No. 3, September 1992
JudithC. Brown:Musicalfundamentalfrequencytracking
1395
SS
CORRELATI
FUNCTIO
FIG. 2. Spectrumversuslogfrequencyfor the
noteC4 produced
bya violin(below)andcrosscorrelationfunction(above) of this spectrum
with the samefunctionasthat describedfor Fig.
SPECTRUM
I
1.
I
.I
I
FREQUENCY(HZ} •
For the highernotesthe errorson notechangesare almostall low by an octave.Theseerrorsoccurbecause,
when
the patternhasits lowestcomponentat the positionof an
octavebelowthefundamental
of thesound,theevencomponentsof thepatternpickup all theharmonics
of the sound.
The oddcomponents
of thepatternarein a positionto pick
up any soundfrom the previousnote'sdecay,givingan advantageoverthepatternat thepositionof thefundamental.
Only by limitingthenumberof components
in thepatternto
match thoseof the particularinstrumentstudiedcan this
errorbeeliminated.In practicethisisnota seriousproblem,
•
as none of theseerrors occurredfor more than one frame;
theycouldbeeasilyeliminated
by havingtwoframesagree.
The resultsfor a pianosoundarefoundin Fig. 4 where
thequalitativebehaviorissimilarto thatof theflute,andthe
optimum number of componentsin the cross-correlation
patternisthesame.The totalnumberof errorsfor thepiano,
with the optimumnumberof components,
is three as opposedto six for the flute.
Preliminaryfundamentalfrequencytrackingresultson
the violin were reportedat the Syracusemeeting(Brown,
1989)of theAcousticalSocietyof America.The spectrumof
o
z
TIME
FIG. 3. Fundamental
frequency
trackingresultsfora flutescalefromC,4toE6 usingcrosscorrelation
witha patternconsisting
of threecomponents
(below),
four components(center), and fivecomponents(above).
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J. Acoust.Soc. Am.,Vol.92, No. 3, September1992
JudithC. Brown:Musicalfundamentalfrequencytracking
1396
...............
pq
TIME
FIG. 4. Fundamental
frequencytrackingresultsfor a pianoscalefrom G3 to G5 usingcrosscorrelationwith a patternconsisting
of threecomponents
(below),fourcomponents
(center),andfivecomponents
(above}.
of the violin is quitedifferentfrom thoseof the flute and
piano.There is a strongformantin the regionof 3000 Hz,
whichgivesrise to extremelystrongupperharmonics.In
Fig. 5, thefrequencytrackingresultsarefoundfor thisviolin
with 6 to 20 components
in the cross-correlation
pattern.
The optimumnumberof components
is 11, but the results
are not terribly sensitiveto this parameteras long as it is
reasonably
closeto theoptimum.
To clarifythesourceof errorswiththismethod,wehave
chosenfourframesfor theviolinfromtheregionof thenote
transitionfrom E5 to F$5 (Fig. 6). All notetransitionsare
difficultfor a frequencytracker sincethere are two notes
present.For eachof theseframeswehavegraphedthespectrum,thecross-correlation
function,andthepeakchosento
representthe fundamentalfrequency.Time increasesfrom
the bottom three graphsto the top three. The first two
frames(lowestsixgraphs)are in error by an octavebelow
the correct fundamental.
We have indicated the harmonicsfor F#5 with small
arrows. Harmonics three, four, and five fall into the formant
regionandarethusstronglyamplified.Sincepositions
onthe
pattern are closelyspacedfor the higher harmonics,when
1397
J. Acoust.
Sec.Am.,Vol.92, No.3, September
1992
the patternlinesup on the (winning) frequencyan octave
belowthefrequencyof E5, contributions
from thesehigher
harmonicsfrom F$5 are sufficientto makethisthe winning
note.This happensagainfor the nextframe.In thethird and
fourthframes(top sixgraphs)the higherharmonicsof E5
haveessentiallydied out, and F•5 wins as it should.
Asidefromthenumberof components
in thecross-correlationpattern,theotheradjustable
parameteristhetuning
of the centerfrequencies
for the binsfor the calculationof
the spectrum.In Fig. 7, we have variedthe tuningover a
semitonein stepsof 1% of thecenterfrequencyfor theviolin.
For example,the curvemarked0.97 hascenterfrequencies
3% lower than those in the curve marked 1.00. Note that the
frequency
trackeris verysensitive
to thisparameterwith an
obviousdeterioriationin performancewith eventhe small
changeto 0.99 from the optimum 1.00.It is suggested
that
thistuninganalysisbecarriedout for eachA to D converter
used,sincefrequencyshiftscanoccurduringthe process
of
sampling.
As a finaltestfor thisfrequency
trackerwehaveapplied
it to a "musicalobstaclecourse"consisting
of fourascending
notesplayedvery slowly (about I s per note) by eachof a
JudithC. Brown:
Musical
fundamental
frequency
tracking
1397
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TIME
FIG. 5. Fundamental
frequency
trackingresultsfor a violinscalefrom03 to G5 usingcrosscorrelation
withpatternsconsisting
of components
with the
numbervaryingfrom 6 to 20. The numberof components
is indicatedon the curve.
of eightharmonicsin thepatternwaschosen,anda graphof
thefrequencytrackingresultsisfoundin Fig. 8. The notesin
cannotbe optimizedfor eachinstrumentaswasdonepre- the soundfilewere splicedtogetherso thereis silencefor a
viouslysopoorerresultsmightbeexpected.
A compromise few framesbetweennotes;thusthesepointsoff the horizonviolin, viola, cello, clarinet, alto sax, tenor sax, trumpet,
trombone,and French horn. The cross-correlation
pattern
1398
J. Acoust.Soc.Am.,Vol.92, No.3, September1992
JudithC. Brown:Musicalfundamental
frequency
tracking
1398
SPECTRUM
t
t
t
t
t
WINNING
PEAK
CROSS
CORRELATION
SPECTRUM
WINNING
PEAK
•
_ _
'
CROSS
CORRELATION"
-
-
SPECTRUM
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FREQUENCY(HZ}•
FIG.6.Fourframes
fortheviolintransition
fromE5toF•5.Foreach
isgraphed
thespectrum
versus
logfrequency,
thecross-correlation
function,
andthe
peakchosenasthefundamental
frequency.
tal linesbetweennotesdo not representerrors.
We havealsoanalyzedthesesoundswith a frequency
trackerusingthe methodof narrowedinvertedautocorrelation (Brown andPuckette,1987;BrownetaL, 1989;Brown,
1991) in orderto comparethesetwomethods(Fig. 8). The
autocorrelation
resultsare foundin Fig. 9. Figures8 and 9
represent
on theorderof 2300framesor discretefrequency
trackingeventsandhadto besampledto becompressed
on
to a singlegraph.They are sampledat a rate of 4 to 1; soa
given point representsthe resultsof the analysisof four
frames.Eachnoteisheldfor approximately
1 swith a frame
sizeof 16mswhichgives64 eventspernote.
The autocorrelation
frequencytracker missesfewer
framesin theregionsbetweennotesbecause
it hasa mechanismfor not reportinga noteif the valueof the the function
differssufficiently
from its idealvalue.The pointsoff the
horizontallinesfor thismethodrepresent
singleerrors.The
points on the time axis are the ones which have been
dropped.However,this frequencytracker had more difficulty with tuningproblems(pointsjust off the horizontal
line or a splitwithin a horizontalline into two linesa semitoneapart). The patternrecognitionfrequencytrackerdoes
betterduringtheplayingof thenote,i.e., thehorizontallines
areunbrokenwith theexceptionof the firstnote.The points
that are off duringthe silences
couldeasilybe droppedfor
both methodsby addingan amplitudedetector.This wasnot
done,nor did we requiretwo or more framesto agree,as is
1399
J. Acoust.
Sec.Am..Vol.92.No.3. September
1992
usualfor frequencytrackers,aswe did not want to eliminate
the basesfor comparisons
for eachof the adjustableparameters.
III. DISCUSSION
AND CONCLUSIONS
Essentiallyall errorswith the patternrecognitionmethod were octave errors. We are able to eliminate these errors
on the high sideby a rather ingeniousmethodsuggested
by
StevenHaflich. A cross-correlation
pattern is used with
componentspacingcorresponding
to 2f 3f 4f, 6f... that is,
twiceasmanycomponents
asin theusualcasebut with alternatingsignsfor thesecomponents.
Thus the positiveeven
components
of thispatternwill lineup withharmonicspectral components
asbefore;in thispositionthe negativecomponentsof thepatternhavenoeffectastheywill fall between
components
of thesound.The valueof the cross-correlation
functionwill be the sameat the positionof the fundamental
as with the previouspattern.Now however,whenthe pattern isalignedwith its lowestcomponenton the secondharmonic (positionof the octave up error) of the musical
sound,eachof thecomponents
of thepatternmatchesa componentof the sound.Sinceeveryother componentof the
patternhasa differentsign,thesumgivesa lowvalueof the
cross-correlation
functionandeliminates
thisfrequency
asa
candidate.
This methodof suppressing
the cross-correlation
peak
for the secondharmonicworkedextremelywell aspredictJudith
C. Brown:
Musical
fundamental
frequency
tracking
1399
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FIG.7.Effect
onthefrequency
tracking
results
ofvarying
thecenter
frequencies
inthecalculation
ofthespectrum
versus
logfrequency.
Tunings
range
from
97% to 103%of thestandard
tuning.
1400
J.Acoust.
Soc.Am.,Vol.92,No.3,September
1992
Judith
C.Brown:
Musical
fundamental
frequency
tracking
1400
z
83.
•
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TIME
FIG. 8. Fundamental
frequency
trackingresultsusingthemethodof patternmatching
for nineinstruments
playingarpeggios.
ed. We did not include these results for two reasons. First the
in the pattern. This adds to the calculationtime, and de"optimized"frequencytracker worked extremelywell on
creases
therangeof frequencies
whichcanbeexamined.
thesounds
studiedwithoutit; andsecond,thedisadvantage
Our patternrecognition
methodhasproduced
excellent
of thismethodis that it doublesthe numberof components trackingresultsfor the musicalsoundsin thisstudy.While
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FIG.9.Fundamental
frequency
tracking
results
using
themethod
ofnarrowed
autocorrelation
fornineinstruments
playing
arpeggios.
1401
J. Acoust.
Soc.Am..Vol.92, No.3, September
1992
JudithC. Brown:
Musical
fundamental
frequency
tracking
1401
singleexamples
of musicalinstruments
canbe atypical,the
spectraof the soundschosenvariedfrom a simplespectrum
consisting
of a strongfundamentalwith a few higherharmonicsto an extremelycomplexspectrumwherethe fundamentalwasoftenweakand mostof the energywasconcentratedin higherharmonicsover2000Hz. The success
of our
frequencytracker in analyzingthesesoundsindicatesthe
ability to dealwith a widevarietyof musicalsounds.Finally,
weemphasize
that essentially
perfectresultscouldhavebeen
obtainedforthesounds
in thisstudysimplybyrequiringthat
resultsfrom two successive
framesagree.
ACKNOWLEDGMENTS
Brown, I. C., and Puckette,M. S. (1987). "Musical Informationfrom a
NarrowedAutocorrelation
Function"Proceedings
of the1987InternationalConference
on ComputerMusic,Urbana,Illinois,84-88.
Brown,J. C., and Puekette,M. S. (1989). "Calculationof a NarrowedAutocorrelationFunction,"J. Acoust.Soc.Am. 85, 1595-1601.
Brown,
J.C. (1989)."Musical
PitchTracking
Based
onaPattern
RecognitionAlgorithm,"J.Acoust.Soc.Am. Suppl.1 85,S79.
Brown,J.C. ( 1991}."Calculation
ofa Constant
Q Spectral
Transform,"
J.
Acoust. Sac. Am. 89, 425-434.
Brown,1.C.,andZhang,B. (1991)."MusicalFrequency
UsingtheMethodsof Conventional
and'Narrowed'Autocorrelation,"
J. Acoust.Soc.
Am. 89, 2346-2354.
Chafe,C.,andJaffe,
D. (1986)."Source
Separation
andNoteIdentification
in Polyphonic
Music,"Proc.ICASSP,Tokyo.
Duda,R. O., andHart,P. E. (1973).PatternClassification
andScene
Anab
ysis(Wiley, New York }.
of
I am verygratefulto StevenHaflichfor suggesting
this Duifhuis,H., Willeros,L. F., andSluyter,R. 3. (1982). "Measurement
PitchinSpeech:
An Implementation
ofGoldstein's
TheoryofPitchPerprojectaswellasfor manyhoursof extremelyhelpfulconception,"J. Acoust.SOc.Am. 71, 1568-1580.
versations.
I wouldlike to thankthe Marilyn Brachman Gerson,
A., andGoldstein,
J.L. ( 1978}."Evidence
foraGeneral
Template
in CentralOptimalProcessing
for Pitchof Complex
Tones,"J. Acaust.
HoffmanCommitteeof WellesleyCollegefor a fellowship
for releasedtime during which much of this work was accomplished.I am indebtedto IRCAM (Institut de Re-
SOe.Am. 68, 498-510.
Piszezalski,
M., andGaller,B. F. (1979). "Predicting
MusicalPitchfrom
Component
FrequencyRatios,"I. Acoust.Soc.Am. 66, 710-720.
Seheffers,
M. T. M. (1983)."Simulation
ofAuditoryAnalysis
ofPiteh:An
Elaborationon the DWS PitchMeter," J. Acoust.So:. Am. 74, 1716-
chercheet Coordination
Acoustique/Musique)
for their
hospitalityand the useof their facilitiesand to Wellesley
1725.
Collegefora Sabbatical
leaveduringwhichthewritingtook Sehoeder,
M. R. (1968)."PeriodHistogram
andProduct
Spectrum:
New
place.I wouldalsoliketo thankKennethMalskywhocreateda soundfile
for an experiment
described
in thispaper.
Amuedo, J. (1985). "Periodicity Estimation by Hypothesis-Directed
Search,"ICASSP-IEEEInternational
Conference
onAcoustics,
Speech,
and SignalProcessing,
395-398.
1402
J. Acoust.Soc. Am.,Vol.92, No. 3, September1992
Methodsfor Fundamental-Frequency
Measurements,"
J. Aeoust.Soe.
Am. 43, 829-834.
Terhardt,E. (1979}. "Calculating
VirtualPitch,"Hear.Res.1, 155-182.
Terhardt,E., Stoll,G., andSwarm,
M. (1982)."Algorithms
forExtraction
ofPitchandPitchSailartec
fromComplex
TonalSignals,"
J.Acoust.
Soe.
Am. 71, 679-688.
JudithC. Brown:Musicalfundamentalfrequencytracking
1402