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A SMARTER WAY TO FIND PITCH
Philip McLeod, Geoff Wyvill
University of Otago
Department of Computer Science
pmcleod, geoff @cs.otago.ac.nz
ABSTRACT
The ’Tartini’ project at the University of Otago aims to
use the computer as a practical tool for singers and in-
strumentalists. Sound played into the system is analysed
fast enough to create useful feedback for teaching or, at a
higher level, for practising musicians to refine their tech-
nique. Central to this analysis is the accurate determina-
tion of musical pitch.
We describe a fast, accurate and robust method for find-
ing the continuous pitch in monophonic musical sounds.
We employ a special normalised version of the Squared
Difference Function (SDF) coupled with a peak picking
algorithm. We show how to implement the algorithm effi-
ciently. Inherent in our method is a ’clarity’ estimate that
measures to what extent the sound has a tone. This has al-
ready found application in showing defects in a violinist’s
bowing technique.
1. INTRODUCTION
Over the last three years at the University of Otago we
have been investigating ways to use a computer to anal-
yse sound and provide useful, practical feedback to musi-
cians, both amateur and professional. We have informally
dubbed this activity the Tartini Project, named for the vi-
olinist and composer Guiseppe Tartini. In 1714, Tartini
discovered that when two related notes were played simul-
taneously on a violin a third sound was heard. He taught
his students to listen for this third sound as a device to
ensure that their playing was in tune.
We are using visual feedback from a computer simi-
larly to provide useful information to help musicians im-
prove their art. Our system can help beginners to learn
to hear musical intervals and professionals to understand
some of the subtle choices they need to make in expressive
intonation.
Pitch is the perception of how high or low a musi-
cal note sounds, which can be considered as a frequency
which corresponds closely to the fundamental frequency
or main repetition rate in the signal. Estimation of f0has
quite a history. It is used in speech recognition and music
information retrieval, and in handheld ’tuners’ that help
developing musicians to tune their instruments. Existing
algorithms for pitch estimation include the Average Mag-
nitude Difference Function (AMDF), Harmonic Product
Spectrum (HPS), Log Harmonic Product Spectrum, Phase
Vocoder, Channel Vocoder, Parallel Processing Pitch De-
tector [6], Square Difference Function (SDF) [1], Cepstral
Pitch Determination [5], Subharmonic-to-harmonic ratio
[8] and Super Resolution Pitch Detector (SRPD) [4].
We have already demonstrated that we can produce use-
ful feedback in real time to musicians [3]. In particu-
lar, we have successfully displayed the shape of a profes-
sional violinist’s vibrato and helped at least one amateur
violinist to develop smoother changes in bow direction.
Once f0is known, a full harmonic analysis of the sound
becomes possible in real time and we can display many
other aspects of a sound that are useful to a musician. In
this paper, we deal only with the ’McLeod Pitch Method’
(MPM), our latest and much improved method of finding
the fundamental pitch.
MPM runs in real time with a standard 44.1 kHz sam-
pling rate. It operates without using low-pass filtering so
it can work on sound with high harmonic frequencies such
as a violin and it can display pitch changes of one cent re-
liably. MPM works well without any post processing to
correct the pitch. Post processing is a common require-
ment in other pitch detectors.
The Tartini system has an option to equalise the lev-
els of the signal to the sensitivity of the inner ear. Stan-
dard equal-loudness curves [7] are used, tending to reduce
low frequencies not perceived well relative to frequencies
around 3700 Hz heard best. This helps move from a direct
fundamental frequency estimate towards something more
correlated with pitch.
Existing pitch algorithms that use the Fourier Domain
suffer from spectral leakage. This is because the finite
window chosen in the data does not always contain a whole
number of periods of the signal. The common solution to
this is to reduce the leakage by using a windowing func-
tion [2], smoothing the data at the window edges. This
requires a larger window size for the same frequency res-
olution. A similar problem happens in some time domain
methods, such as the autocorrelation, where a window
containing a fractional number of periods, produces max-
ima at varying locations depending on the phase of the in-
put. MPM however, introduces a method of normalisation
which is less affected by edge problems. Keeping track of
terms on each side of the correlation separately.
To explain our fast calculation of the Normalised Square
Difference Function (NSDF) it is necessary first to de-
scribe the relationship between an Autocorrelation Func-
tion (ACF) and the Squared Difference Function (SDF).
This we do in Sections 2 and 3. The fast calculation de-
pends on a standard method for ACF [6]; how this is used
is described in Section 6. The NSDF automatically gen-
erates an estimate of the clarity of the sound, describing
how tone-like it is. This is basically the value of the cho-
sen maximum of the function (section 7).
2. AUTOCORRELATION FUNCTION
There are a two main ways of defining the Autocorrelation
Function (ACF). We will refer to them as type I, type II.
When not specified we are referring to type II.
We define the ACF type I of a discrete signal xtas:
rt(τ) =
t+W−1
X
j=t
xjxj+τ(1)
where rt(τ)is the autocorrelation function of lag τcalcu-
lated starting at time index t, where Wis the initial win-
dow size, i.e. the number of terms in the summation.
We will define the ACF type II as:
r0
t(τ) =
t+W−1−τ
X
j=t
xjxj+τ(2)
In this definition the window size decreases with increas-
ing τ. This has a tapering effect, with a smaller number of
non-zero terms being used in the calculation at larger τ.
Note that ACF Type I and Type II are the same for a zero
padded data set i.e. xk= 0, k > t +W−1.
3. SQUARE DIFFERENCE FUNCTION
Again we define two types of discrete signal Square Dif-
ference Functions (SDFs). The SDF of Type I is defined
as:
dt(τ) =
t+W−1
X
j=t
(xj−xj+τ)2(3)
and the SDF Type II is defined as:
d0
t(τ) =
t+W−τ−1
X
j=t
(xj−xj+τ)2(4)
As in Type II ACF, the window size decreases as we in-
crease τ. In both types of SDFs minima occur when τis
a multiple of the period, whereas in the ACFs maxima oc-
curred. These do not always coincide. If we expand out
Equation 4 we see that there is an ACF inside the SDF
calculation.
d0
t(τ) =
t+W−τ−1
X
j=t
(x2
j+x2
j+τ−2xjxj+τ)(5)
If we define
m0
t(τ) =
t+W−τ−1
X
j=t
(x2
j+x2
j+τ)(6)
we can see that
d0
t(τ) = m0
t(τ)−2r0
t(τ)(7)
When using the Type II ACF it is common to divide
r0
t(τ)by the number of terms as a method of counteract-
ing the tapering effect. However this can introduce ar-
tifacts, such as sudden jumps when large changes in the
waveform pass out the edge of the window. Our normali-
sation method provides a more stable correlation function
even down to a window containing just two periods of a
waveform.
4. NORMALIZED SQUARE DIFFERENCE
FUNCTION
Once the SDF has been calculated at time t, we have the
central problem of deciding which is the τthat corre-
sponds to the pitch. This does not always correspond with
to overall minimum, but is usually one of the local min-
ima. Without a grasp of the range of the values it is dif-
ficult to decide which minimum it corresponds to. We
have discovered a useful way of normalizing the values
to simplify this problem. We define a Normalised Square
Difference Function (NSDF) as follows:
n0
t(τ)=1−m0
t(τ)−2r0
t(τ)
m0
t(τ)(8)
=2r0
t(τ)
m0
t(τ)(9)
The greatest possible magnitude of 2r0
t(τ)is m0
t(τ)i.e.
|2rt(τ)|<=m0
t(τ). This puts n0
t(τ)in the range of -1 to
1, where 1 means perfect correlation, 0 means no corre-
lation and -1 means perfect negative correlation, irrespec-
tive of the waveform’s amplitude. From equation 9, we
see this becomes the same as normalising the autocorre-
lation in the same fashion. Notice that m0
tis a function
of τ, minimising the edge effects of the decreasing win-
dow size. Having these normalised values simplifies the
problem of choosing the pitch period, as the range is well
defined. We refer to the process of choosing the ’best’
maximum as peak picking, and our algorithm is shown in
Section 5. But another reason for normalisation is that it
enables us to define a clarity measure. Clarity is discussed
in Section 7.
An important property we have found useful in a time
domain pitch detection algorithm is what we call the Sym-
metry property. This means there are the same number of
evenly spaced samples being used from either side of time
tfor a given τ, and that these samples are symmetric in
terms of their distances from t. This maximises cancella-
tions of frequency deviations from opposite sides of time
t, creating a frequency averaging effect.
Equation 4 can be made to hold this property by simply
shifting the center to time t, yielding
d0
t(τ) =
t+(W−τ)/2−1
X
j=t−(W−τ)/2
(xj−xj+τ)2(10)
0 200 400 600 800 1000 1200
−1
−0.5
0
0.5
1
1.5 NSDF Delay Space
Delay τ (in samples)
NSDF Correlation n(τ)
Figure 1. A NSDF graph showing the highest key max-
imum at 650, whereas the pitch has a period at 325. The
graph is sprinkled with other unimportant local maxima.
5. PEAK PICKING ALGORITHM
The algorithm so far gives us correlation coefficients at
integer τ. We will choose the first ’major’ peak as repre-
senting the pitch period. This is not always the maximum,
which is considered the fundamental frequency. Firstly,
find all of the useful local maxima. These are maxima
with τwhich potentially represent the period associated
with the pitch. We will refer to these as key maxima. As
can be seen from Figure 1, if we just take all the local
maxima we get a lot of unnecessary peaks which are not
of much use. We find that taking only the highest max-
imum between every positively sloped zero crossing and
negatively sloped zero crossing works well at choosing
key maxima. The maximum at delay 0 is ignored, and we
start from the first positively sloped zero crossing. If there
is a positively sloped zero crossing toward the end with-
out a negative zero crossing, the highest maximum so far
is accepted, if one exists.
In the example from Figure 1 this leaves us with three
key maxima. It is possible to get some spurious peaks as
key maxima: for example if the value at τ= 720 had
crossed through the zero line then it would add another
maximum to our key list. But these are normally a lot
smaller than the other key maxima, so are not chosen in
the later part of the algorithm.
Parabolic interpolation is used to find the positions of
the maxima to a higher accuracy. This is done using the
highest local value and its two neighbours.
From the key maxima we define a threshold which is
equal to the value of the highest maximum, nmax, multi-
plied by a constant k. We then take the first key maximum
which is above this threshold and assign its delay, τ, as
the pitch period. The constant, k, has to be large enough
to avoid choosing peaks caused by strong harmonics, such
as those in Figure 2, but low enough to not choose the un-
wanted beat or sub-harmonic signals. Choosing an incor-
rect key maximum causes a pitch error, usually a ’wrong
octave’. Pitch is a subjective quantity and impossible to
get correct all the time. In special cases, the pitch of a
given note will be judged differently by different, expert
listeners. We can endeavour to get the pitch agreed by the
user/musician as most often as possible. The value of k
can be adjusted to achieve this, usually in the range 0.8 to
1.0.
The pitch period is equal to the delay, τ, at the chosen
0 200 400 600 800 1000 1200
−1
−0.5
0
0.5
1
1.5 NSDF Delay Space
Delay τ (in samples)
NSDF Correlation n(τ)
Figure 2. A graph showing the NSDF of a signal with a
strong second harmonic. The real pitch here has a period
of 190. But close matches are made at half this period.
key maximum. The corresponding frequency is obtained
by dividing the sample rate by the pitch period (in sam-
ples). We turn this into a note on the even tempered scale
using:
note =log10(f /27.5)
log10(12
√2) (11)
These correspond to notes on the midi scale, and contain
decimal parts representing fractions of a semitone.
6. EFFICIENT CALCULATION OF SDF
To calculate the SDF by summation takes O(W w )time,
where wis the desired number of ACF coefficients. By
splitting d0
t(τ)into the two components m0
t(τ)and r0
t(τ),
we can calculate these terms more efficiently. The ACF
can be calculated in approximately O((W+w)log(W+
w)) time by use of the Fast Fourier Transform [6]. The
ACF part of the SDF, r0
t(τ), can be calculated as follows:
1. Zero pad the window by the number of NSDF val-
ues required, w. We use w=W/2.
2. Take a Fast Fourier Transform of this real signal.
3. For each complex coefficient, multiply it by its con-
jugate (giving the power spectrial density).
4. Take the inverse Fast Fourier Transform.
The two terms of m0
t(τ)from Equation 6 can each be
calculated incrementally, by simply using the result from
τ−1, and subtracting the appropriate x2
tstarting (when
τ= 0) with both sums equal to the total sum squared of
the whole window, which we already have in r0
t(0).
Typical window sizes we use for a 44100 Hz signal are
512, 1024, 2048 or 4096 samples. with 75% overlap in
time, i.e. incrementing tby W/4.
7. THE CLARITY MEASURE
We define clarity as a measure of how coherent a note
sound is. If a signal contains a more accurately repeating
waveform, then it is clearer. This is similar to the term
voiced, used in speech recognition. Clarity is independent
of the amplitude or harmonic structure of the signal. As
a signal becomes more noise-like, its clarity decreases to-
ward zero. The clarity is simply taken as the correlation
value of the chosen key maximum. If no key maxima are
found, it is set to zero.
We use the clarity measure in combonation with the
RMS power to weight the alpha value (translucency) of
the pitch contour at a given point in time. This maximises
the on-screen contrast, displaying the pitch information
most relevant to the musician. The clearer the sound the
larger the weight and the louder the sound the larger the
weight. This ensures that background sounds and non-
musical sounds are not cluttering the display, but are faded
into the background. Sounds below the noise threshold
level are completely ignored.
8. CONCLUSION
The MPM algorithm can provide real-time pitch contours.
With its ability to extract pitch with as little as two peri-
ods, smaller window sizes can be used than in other algo-
rithms. Smaller window sizes allow for better representa-
tion of a changing pitch, such as that during vibrato. Tar-
tini works well on a range of instruments including string,
woodwind, brass and voice.
Tartini emphasises the importance of a loud and clear
pitch by adjusting the alpha of the pitch contours (Section
7), thus hiding away unwanted background or pitch-less
sounds. This can be seen in figure 3(a) where the pitch
contour fades out at either end. Also the rate and steadi-
ness of vibrato can be seen. This direct feedback allows a
musician to see where they are going wrong, or to get the
effect they want. Breaks in playing, for example when a
violinist changes bow direction, can be seen as a break in
the contour. A violinist can practice shortening the break,
helping to improve the bowing gesture. Figure 3(b) is a
screen-shot showing the strength of each harmonic as a
track, during a descending violin scale. This is done using
the pitch as a basis for the harmonic analysis. A number
of musicians and singers have shown great interest in the
system. Tartini can be downloaded from www.tartini.net.
9. ACKNOWLEDGEMENTS
We would like to thank our professional musicians, Mr
Kevin Lefohn (violin) and Miss Judy Bellingham (voice)
for numerous discussions and providing us with samples
of good sound. Also we thank Dr Don Warrington of the
Physics Department, University of Otago for his advice
and encouragement .
10. REFERENCES
[1] Cheveigne, A. ”YIN, a fundamental frequency
estimator for speech and music”, Journal of
the Acoustical Society of America, Vol 111(4),
pp 1917-30, April 2002.
[2] Harris. F, ”On the Use of Windows for Har-
monic Analysis with the Discrete Fourier
Transform”, Proc. of the IEEE, Vol 66, No 1,
Jan 1978.
Figure 3. Two screen-shots from Tartini. (a) A pitch con-
tour and log-RMS plot of a Violin vibrato about a D on the
5th octave. (b) Harmonic tracks from a descending scale
beside their equivalent key on a keyboard.
[3] McLeod. P, Wyvill. G, “Visualization of Musi-
cal Pitch”, Proc. Computer Graphics Interna-
tional, Tokyo, Japan, July 9-11, 2003, pp 300-
303.
[4] Medan. Y, Yair. E, and Chazan. D, ”Su-
per Resolution Pitch Determination of Speech
Signals”, IEEE Tans. Signal Processing, Vol
39(1), pp 40-48, 1991.
[5] Noll. A, ” Cepstrum Pitch Determination”,
Journal of the Acoustical Society America, Vol
41(2), pp 293-309, 1967.
[6] Rabiner. L, Schafer. R, Digital Processing of
Speech Signals, Prentice Hall, 1978
[7] Rossing. T, The Science of Sound, 2nd ed. Ad-
dison Wesley, 1990.
[8] Sun. X, “Pitch determination and voice quality
analysis using subharmonic-to-harmonic ra-
tio”, Proc. of IEEE International Conference
on Acoustics, Speech, and Signal Processing,
Orlando, Florida, May 13-17, 2002.
