Extracting the Modal Parameters of the Horizontal and
Vertical Transverse Polarisations of a Guitar String
Bertrand Scherrer, Philippe Depalle
To cite this version:
Bertrand Scherrer, Philippe Depalle. Extracting the Modal Parameters of the Horizontal and
Vertical Transverse Polarisations of a Guitar String. 9 pages. 2009. <hal-00392757v1>
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Submitted on 4 Jul 2009 (v1), last revised 4 Jan 2009 (v2)
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Extracting the Modal Parameters of the
Horizontal and Vertical Transverse Polarisations
of a Guitar String
Bertrand Scherrer, Philippe Depalle,
Sound Processing and Control Laboratory (SPCL),
CIRMMT – McGill University, Montréal, Canada
[email protected]
June 8, 2009
Abstract
This study presents first results for the audio-based identification of the type
of stroke used on the classical guitar. The link between the type of stroke and
the modal parameters of the horizontal and vertical transverse polarisations of the
string is revealed through physical considerations. We then describe in detail the
specialized sound analysis framework we developed and explain the reasons for
certain analysis strategies and parameter choices. Finally we present some first
results on the extraction of the modal parameters of the transverse polarisations
and outline our future research directions.
1
Introduction
Musicians can often infer how an instrument was played simply by listening to a
recording. This implies that the information about the instrumental gesture [1] is embedded in the sound signal structure and could be extracted armed only with two microphones and a computer performing signal analysis. In this paper, we present some
first results on the identification of the type of stroke on the classical guitar. The core of
our approach is to use prior knowledge on the acoustics of the guitar to devise a sound
analysis strategy.
Such physically-informed methods have been published for the retrieval of the
plucking position along a guitar string. Most recently, [2] used the fact that the effect of the plucking position can be modeled by a first order feedforward comb filter
whose delay is proportional to the relative plucking position [3]. Using a weighted
least square algorithm it was then possible to estimate the feedfoward delay that would
best match the observed partials amplitude pattern.
The present study is concerned with the extraction of another instrumental gesture:
the type of stroke. Indeed, performers resort to different strokes according to the sound
they want to obtain. Typically, the rest stroke is used to produce a “strong, full sound"
whereas the free stroke is used when a “softer" sound is sought for [4]. The rest stroke,
is characterized by the fact that, after the pluck, the finger tends to rest on the next
string. The free stroke, is just the contrary: at the end of the pluck, the plucking finger
1
clears the next string. In more quantitative terms, the difference between these strokes
actually resides in the angle with which the string is left to vibrate at the end of the
interaction between the plucking finger and the string: the angle of release (θr ).
In Section 2, we present the physical considerations that relate the angle of release
to the modal parameters of the transverse polarisations of the string. Then, in Section 3,
we describe the analysis framework we developed to actually extract polarisation modal
parameters from recordings. Section 4 reveals the first results of modal parameter
identification for different angles of release based on sound recordings. Finally, Section
5 recapitulates our observations and outlines future research directions.
2
From rest/free stroke to modal parameters of string
polarisations
Fig. 1 illustrates how the rest and free strokes can be characterized by their angle of
release. Fig. 1 (a) features the trajectories of a point along the string during the fingerstring interaction for both types of stroke as measured in [5]. The circled dots marked
lf ree and lrest represent the points at which the string is released for the free and rest
stroke respectively. We can see that the different kinds of plucks result in differences
in angle of release θr,rest and θr,f ree . Let us now draw the link between the angle of
release and the modal parameters of the transverse polarisations of the string.
(a) Trajectory of a string before release
for rest and free stroke (from [5]).
(b) Coordinate system.
Figure 1: Comparing rest and free strokes.
Understanding what makes the difference between the sound of the rest and free
strokes requires the understanding of how the vibrational forces of the string are transformed into body vibrations through the bridge [6]. This knowledge can be accessed
via the mechanical admittance matrix of the bridge. If we only consider the horizontal
(along ~y ) and vertical (along ~z) transverse polarisations of the string, this matrix will
characterize, as a function of frequency, how much the bridge moves along ~y and ~z as
a result of the forces exerted by the two polarisations. The presence of cross terms in
the matrix indicates the presence of coupling between the horizontal and vertical polarisations. Adapting a reasoning presented by [7], one finds that the angle of release of
the string will modify the coupling between resonances of the horizontal and vertical
polarisations.
In other words, according to the type of pluck, the modal parameters of the resonances of the horizontal and vertical polarisations will be different. In the next section,
2
we present the analysis framework we use to extract these modal parameters from a
recording.
3
Description of the Analysis Framework
The different parts of our analysis framework are detailed in the following and Fig. 2
gives an overview of the whole processing chain.
Figure 2: Polarisation identification process.
3.1
Preliminary Analysis Steps
The first step in the analysis is to center and cut the signal (Fig. 2, ).
1 Typically, we
want to analyze the sound after the onset transient while staying close enough to it so
that both polarisations are still present.
Then, we estimate the fundamental and the higher partials (F0 , F1 , . . . , Fm ) from
scc , the centered and cut signal (Fig. 2, ).
We use Audiosculpt [8] to perform a
2
standard additive analysis (STFT, peak picking, etc.) combined with a partial tracking
focused on the harmonics of the fundamental frequency.
Finally, based on different frequency-based criteria (Fig. 2, ),
3 we decide on which
partials to perform further analysis. The frequencies of the selected partials are noted
Fn in the rest of the paper. In practice, we discard the partials that are too close from
the main body modes (frequencies under 250 Hz). This is due to the fact that we only
want to focus on string resonances and not a mix of body and string resonances. We
also decide to restrict our analysis to partials that are of strong enough amplitude. In
other words, we will tend to focus on partials close to the maxima of the “plucking
filter" [2]. Fig. 3, illustrates the effect of the plucking position on the amplitude of the
partials. The tone was produced using the rest stroke, 10 cm from the bridge, on the
open E4 (330Hz) string, of length 65cm.
3.2
Fine Analysis of the Partials
We use a “high resolution” (HR) spectral analysis method (the well-known ESPRIT
method [9] ) to perform a finer analysis of the signal around each partial frequency
Fn . For that purpose, we need to pre-process scc (Fig. 2, ).
4 First we bandpass filter
scc using a complex, linear phase, filter of central frequency ν0 = Fn and bandwidth
δν. The filter is obtained using the window method (a Blackman window is used as
in [10]). Note that the resulting signal sn is now complex and only contains positive
frequencies. Also, since the complexity of ESPRIT increases tremendously when it is
3
Guitar Pluck Spectrum
80
Plucking Filter
Magnitude of Guitar Pluck
60
40
Partial 10
A (dB) →
20
0
−20
−40
−60
−80
−100
0
500
1000
1500
2000
2500
f (Hz) →
3000
3500
4000
4500
5000
Figure 3: Influence of the plucking position on the amplitude of the partials, for the
open string E4, of length 65cm, plucked at 10 cm from the bridge, using a rest stroke.
fed with more than 1000 samples at 44.1kHz [11] we need to down-sample the signal.
So, after the bandpass filter, the signal is modulated to DC and down-sampled by a
factor M = 1/(2δν) [11].
Next, the frequencies, damping factors, amplitudes and phases of K components
are estimated (Fig. 2, ).
5 In order to obtain reliable results, we need to choose appropriate parameters for the ESPRIT method. There is an optimal value for the “pencil
parameter” p [12] but we have to carefully choose the model order K. Theoretically,
an order of two would be enough since we are looking for only two transverse polarisations in a complex signal. As usual with parametric methods, however, it is good to
slightly overestimate the order of the model. Note that, if we increase the number of
components of the model, we also increase the number of components to choose from
when identifying the horizontal and vertical polarisations. In practice we use K = 6.
3.3 Component Selection
For a given sn , we are only interested in two out of the K components estimated at
the previous stage. This means we need to perform some post-processing on the data
(Fig. 2, ).
6 This step is detailed further, for a given sn , in Fig. 4. The poles estimated
by the HR analysis are represented as circles in the z-plane. The radius of a circle
corresponds to the amplitude of the pole it represents. Note that the damping factor
δmax and frequency Fn are normalized by the sampling rate Fs .
ˆ
A first sorting is done by discarding the poles ẑi = e(δ̂i +j2πfi ) that are obviously
irrelevant: too damped (δ̂i < δmax ) or diverging (δ̂i > 0) and too far from the partial
frequency Fn (|fˆi − Fn | > ∆f ). From these poles, we only keep the poles with high
enough amplitude (within -20 dB of the component of highest amplitude).
Next, we have to identify the poles corresponding to the horizontal and vertical
polarisations from the poles that passed the first sorting steps (within the grey area in
Fig. 4). Here, we use two criteria derived from the physics of the guitar: |δv | > |δh |
and fh < fv . The first criterion simply takes into account the fact that the top plate
of the instrument is more efficient at absorbing energy in the vertical (~z in Fig. 1 (b))
direction than in the horizontal direction (~y in Fig. 1 (b)). Hence the damping of the
vertical polarisation (δv ) will be greater than that of the horizontal polarisation (δh ).
The second criterion is based on observations by Woodhouse in [13]: unlike the vertical
polarisation, the horizontal polarisation is not completely stopped at the frets, bridge
4
Figure 4: Sorting Poles from HR analysis around Fn .
saddle, and nut. This leads to a slight lateral displacement of the string in the horizontal
plane, and thus an increased vibrating length.
The pair of poles that meets these criteria is then identified as the horizontal, vertical
polarisation pair estimate. In Fig. 4 the pair of poles selected is highlighted with a
thicker border. The horizontal polarisation is highlighted in lighter grey. In case there
are more than one pair satisfying the physics-based criteria, we chose to take the one
with greatest energy.
4
4.1
Tests and Results
Sound Data Base
To test our method, we asked a classically trained guitarist to play around 20 “identical" plucks for four different angles of release. We included free and rest strokes as
well as some more exaggerated plucks: strictly parallel and strictly perpendicular to
the soundboard. The strings that were not played were damped to avoid sympathetic
vibrations. All the tones were performed by plucking the string with the middle finger,
with an angle of attack of 45 degrees, at a constant plucking position of 10 cm from
the bridge. The guitar used was an Alhambra Model 1C (entry-level study guitar). The
recording was performed in a controlled acoustic environment (hemi-anechoic room)
and the microphones were positioned 50 cm away from the guitar. The performer was
instructed to keep the guitar in such a way that the microphones would aim at the junction between the tone hole and the fret board with a 45 degree angle. The analog to
digital conversion was performed using an RME Micstasy at 44.1kHz, 16 bits.
4.2
Results of the Estimation of the Modal Parameters of the Transverse Polarisations
Fig. 5, and 6 display the estimated vertical (inverted triangles) and horizontal (squares)
polarisations of partial 10 in a frequency-damping plane for the E4 (330Hz) and B3
(248Hz) strings respectively. The four different types of strokes (parallel, perpendicular, rest and free) are differenciated by their colour. For a given sound the transverse
polarisations are linked by a dotted line. Out of the 25 partials we extracted on each of
5
the 20 occurences of a given stroke, we only included the results obtained for partial
10.
For both strings, and especially for the E4 string, we can observe that there are
some distinct groups corresponding to angles of release despite a few outliers in each
category. Fig. 7 and 8 display the average location in the frequency-damping plane
of both polarisations as well as the standard deviation in each dimension (rectangle in
dotted lines), for the four angle of release categories, for both strings.
A first observation is that there is a difference in variance between the rest and
free strokes: the free strokes tend to have a bigger variance. This is maybe due to
the fact that the performer has less constraints on the exact displacement of his finger
when pulling the string up than when he is pushing the string down stopping on the
next string. Also there seems to be a trend in the ordering of the types of strokes in
frequency: perpendicular, parallel, rest stroke and free stroke. The exact reasons for
this are not quite clear yet but we believe that the answer could be found with further
physical study of the coupling between the two transverse polarisation via the bridge.
Partial 10 at 3310.24 Hz
0
δ (s−1)→
−5
−10
V polar., Para. Stroke
H polar., Para. Stroke
V polar., Perp. Stroke
H polar., Perp. Stroke
V polar., Rest Stroke
H polar., Rest Stroke
V polar., Free Stroke
H polar., Free Stroke
−15
3302
3307
f (Hz) →
3312
3317
Figure 5: Horizontal and vertical polarisation for partial 10 of string 1 (E4: 330Hz),
in the frequency-damping plane. The portion of sound used for the analysis spanned
from 0.15s to 0.55s after the onset transient.
5
Conclusion
We have presented some first results for the audio-based identification of the type of
stroke on the classical guitar. We showed the direct realtionship between the type
of stroke and the angle of release of the string. We then proceeded to draw the link
between this angle and the modal parameters of the transverse polarisations of the
string. We outlined the analysis framework we developed to estimate these modal
parameters and gave the first results of our analyses. At least for two strings out of
the three investigated, some well chosen partials exhibit a grouping of horizontal and
vertical polarizations according to the type of pluck.
The future directions for our research include some improvements to the analysis
framework. We are thinking of implemeting an automatic detection of the end of the
transient. Also, we are planning on thoroughly evaluating the effect of certain analysis
parameters to improve the reliablity of the pole estimation produced by ESPRIT. At a
more theoretical level, we are also working on formulating an explicit correspondance
6
Partial 10 at 2479.50 Hz
0
−1
δ (s )→
−5
−10
−15
2471
V polar., Para. Stroke
H polar., Para. Stroke
V polar., Perp. Stroke
H polar., Perp. Stroke
V polar., Rest Stroke
H polar., Rest Stroke
V polar., Free Stroke
H polar., Free Stroke
2476
f (Hz) →
2481
2486
Figure 6: Horizontal and vertical polarisation for partial 10 of string 2 (B3: 248Hz),
in the frequency-damping plane. The portion of sound used for the analysis spanned
from 0.15s to 0.55s after the onset transient.
Partial 10 at 3310.24 Hz
0
δ (s−1)→
−5
−10
−15
3302
mean V polar., Para. Stroke
mean H polar., Para. Stroke
mean V polar., Perp. Stroke
mean H polar., Perp. Stroke
mean V polar., Rest Stroke
mean H polar., Rest Stroke
mean V polar., Free Stroke
mean H polar., Free Stroke
3307
f (Hz) →
3312
3317
Figure 7: Mean location of the horizontal and vertical polarisation for partial 5 of
string 1 (E4: 330Hz), in the frequency-damping plane. The portion of sound used for
the analysis spanned from 0.15s to 0.55s after the onset transient.
between the modal parameters of the transverse polarisations and the angle of release,
based on an acoustical model of the coupling between the transverse polarisations.
This final step currently constitutes the missing link between the type of stroke and the
modal parameters of the transverse polarisations of the string.
6
Acknowledgments
Thanks go to Federico O’Reilly Regueiro for producing the sounds used in this work.
The authors would also like to acknowledge the support provided by Center for Interdisciplinary Research in Music and Media Technology (CIRMMT) and NSERC.
7
Partial 10 at 2479.50 Hz
0
δ (s−1)→
−5
−10
−15
2471
mean V polar., Para. Stroke
mean H polar., Para. Stroke
mean V polar., Perp. Stroke
mean H polar., Perp. Stroke
mean V polar., Rest Stroke
mean H polar., Rest Stroke
mean V polar., Free Stroke
mean H polar., Free Stroke
2476
f (Hz) →
2481
2486
Figure 8: Mean location of the horizontal and vertical polarisation for partial 10 of
string 2 (B3: 248Hz), in the frequency-damping plane. The portion of sound used for
the analysis spanned from 0.15s to 0.55s after the onset transient.
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9