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Audibility of Initial Pitch Glides in String Instrument Sounds
Hanna J¨arvel¨ainen, Vesa V¨alim¨aki
Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology
email: hanna.jarvelainen@hut.fi
Abstract
Listening experiments were made to measure the detection
thresholds for initial pitch glides in string instrument sounds,
where a rapid decline of pitch is caused by tension modu-
lation during the attack. Realistic sounding synthetic tones
were generated by additive synthesis. The frequency decay of
the glide was defined through the overall decay rate of am-
plitude, simulating the behavior of real instruments. It was
found that on the ERB frequency scale, the thresholds re-
mained roughly constant at approximately 0.1 ERB with vary-
ing fundamental frequency. Thus, any pitch glide weaker than
the given threshold remains inaudible for most listeners and
could be left unimplemented in digital sound synthesis.
1 Introduction
High-quality sound synthesis is possible with the modern
synthesis methods, such as physical modeling (Smith 1998)
and sinusoidal modeling (Serra and Smith 1990). However,
implementing all details of the sound is computationally costly.
It would be desirable to leave such features, whose effects are
not perceived by the listener, unimplemented.
A rapid descent of pitch during the attack is character-
istic to many plucked and struck string instruments in forte
playing. It can be detected for instance in the clavichord
(V¨alim¨aki et al. 2000), the guitar (Tolonen et al. 2000), and
the kantele – a traditional Finnish string instrument (V ¨alim¨aki
et al. 1999). The primary cause of the pitch descent is the
varying string tension as a consequence of finite string dis-
placement after plucking or striking the string (Legge and
Fletcher 1984). In the clavichord, the effect is boosted by
the mechanical aftertouch. The string tension can be directly
controlled by the player through key pressure.
Fig. 1 shows a fundamental frequency estimate obtained
from a recorded electric guitar tone by the autocorrelation
method (Tolonen et al. 2000). The estimate decreases
exponentially with time from 499 to 496 Hz, giving a glide
extent of approximately 3 Hz.
The tension modulation can be implemented in physical
modeling, for instance, by a special filter structure with signal-
dependent fractional delay elements (Tolonen et al. 2000),
(V¨alim¨aki et al. 1998), but ignoring it would bring remark-
able computational savings.
The detection and discrimination of frequency glides has
been previously studied from a more theoretical viewpoint.
Still the underlying mechanism remains unclear. It was sug-
gested by Madden and Fire (1997) that the detection is based
on changes in the low-frequency side of the excitation pattern.
Moore and Sek (1998) argued that at least both sides of the
excitation pattern should be compared, and that for low cen-
ter frequencies the time-related cues, such as phase locking
could have an effect as well. The studies agree that the detec-
tion and discrimination of glides is little affected by duration,
center frequency, or direction (up or down). However, the
previous results are of little help for the synthesis of instru-
ment tones, since the range of center frequencies was from
0.5 to 6.0 kHz and the shape of the glide as a function of time
was unnatural to real instruments.
The objective of this study is to set perceptually motivated
guidelines for the need to implement the initial pitch glide in
string instrument synthesis. The test tones were synthesized
and the pitch glides were defined in a way typical of string
instruments. The results of two listening experiments are re-
ported.
0 0.2 0.4 0.6 0.8 1
−1
0
1
Level
0 0.2 0.4 0.6 0.8 1
496
497
498
499
F0 (Hz)
Time (s)
Figure 1: Waveform of a single tone played on the electric
guitar (top) and its short-time fundamental frequency esti-
mate, which shows a typical descent (bottom).
Reprinted from Proceedings of the International Computer Music Conference. Havana, Cuba, 17-23 September 2001, pages 282-285.
2 Listening tests
The detection thresholds for initial pitch glides were mea-
sured in two separate listening experiments. In both experi-
ments, the independent variable was the transition span ,
i.e., the extent of the pitch glide; in addition, fundamental fre-
quency was used as a parameter in experiment I and decay
time constant in experiment II.
2.1 Test sounds
The test sounds were generated using additive synthesis
for easy control of the pitch glides. Each tone consisted of
its 20 lowest harmonics and had a duration of 2.0 seconds.
An exception was made in the highest tone (659 Hz), for
which only 16 harmonics could be generated before meeting
the Nyquist limit of 11 kHz at the sampling rate of 22.05 kHz.
Realistic sounding test tones were created by copying the
initial amplitudes and decay characteristics of real tones. The
acoustic guitar was taken as a reference, for its behavior is
well-known and plenty of analysis data are available. The ini-
tial phases of the harmonics were randomized. The amplitude
decay was controlled by two parameters, the time constant
of the overall decay and a frequency-dependent damping co-
efficient , which corresponds to the feedback coefficient of
the one-pole loop filter in a digital waveguide string model.
The initial amplitudes as well as the decay parameters were
chosen according to (V¨alim¨aki and Tolonen 1998).
The independent variable was the extent of the pitch glide
in Hz. The pitch contour decreased exponentially with time
from the highest value towards the steady state fun-
damental frequency . The time constant of the frequency
descent was 50% of the overalltime constant of amplitude
decay (Legge and Fletcher 1984).
In experiment I, the thresholds were measured at four dif-
ferent fundamental frequencies to cover the whole pitch range
of the acoustic guitar: (659.26 Hz), (349.23 Hz),
(196 Hz), and (116.5 Hz). The overall decay time con-
stant and frequency-dependent damping coefficient were
kept constant.
The motivation forthe second experiment is the known re-
lation of the time constant of amplitude decay and time con-
stant of pitch decay in string instrument tones (Legge and
Fletcher 1984). The perceptual tolerances for amplitude de-
cay have been published previously(Tolonen and J¨arvel¨ainen
2000), giving the allowable deviation of the time constant
from the reference value. However, varying the time con-
stant even withinthis tolerance affects the time constant of the
pitch decay and hence the duration of the pitch glide, which
can cause a shift in the detection threshold. This was studied
by fixing the fundamental frequency to 196 Hz and varying
the time constant . Three values were used for the time
constant according to the previous results on the perceptual
tolerances: 100%, 80%, and 60% of the reference value 0.39
0 0.5 1 1.5 2
−1
−0.5
0
0.5
1
Level
0 0.5 1 1.5 2
348
350
352
354
F0 (Hz)
Time (s)
Figure 2: Waveform of a synthetic guitar tone (top) and
its fundamental frequency with a 5-Hz pitch glide (bottom).
s, which is 50% of the measured time constant of the overall
amplitude decay (0.77 s).
2.2 Subjects and test method
Five subjects participated in both experiments. They were
20-30 years old, and all had previous experience in psychoa-
coustic listening tests. None of them reported any hearing de-
fects, and they were allowed to practise before the test. The
sound samples were played through headphones from a com-
puter.
A standard tone without a pitch glide and a stimulus tone
with a pitch glide were presented to the subject sequentially
in random order, and the task was to judge whether the sounds
were same or different. Five values of glide extent were
used for each fundamental frequency in experiment I and four
for each time constant in experiment II. Each trial was judged
four times together with as many corresponding fake trials
(two standard tones, no stimulus) in random order.
A detected difference was either a hit or a false alarm,
depending on whether the trial actually included the stimulus
tone or not. A measure of correct answers was derived
for each condition from the proportion of hits and false alarms
as follows (Yost 1994):
(1)
The function has values between 0.50, which corresponds to
chance level with equal proportions of hits and false alarms,
and 1.0, which requires 100% hit proportion and no false
alarms. The detection threshold was estimated by finding the
midpoint (i.e., the 75% point) of this function. If the thresh-
old was not directly evident in the data, it was interpolated
between the nearest higher and lower scores.
3 Results
3.1 Effect of fundamental frequency
The effect of fundamental frequency on the detection of
pitch glides was studied in experiment I. The detection thresh-
olds increasedmonotonically withfundamental frequency. The
mean thresholds were 3.1 Hz, 4.4 Hz, 5.4 Hz, and 11.7 Hz for
, , , and , respectively. The situation turns upside
down when the thresholds are expressed on the logarithmic
scale, see Fig. 3. For , the median of individual thresh-
olds is 52 cents (1/100 of a semitone) – more than half of a
semitone, while for the highest tone it is 30 cents.
116.5 196 349 659
20
40
60
80
100
∆ f at threshold (cents)
Fundamental frequency (Hz)
Figure 3: Listening test results with fundamental frequency
as a parameter. Boxplot of (cents) at threshold.
The results were roughly normally distributed,but the er-
ror variance between different tones was typically unequal.
However, these differences were reasonably equalized by a
transformation from the linear frequency scale to the audi-
torily motivated ERB (Equivalent Rectangular Bandwidth)
scale as follows (Glasberg and Moore 1990):
(2)
where is frequency in kHz. The analysis of variance (ANO-
VA) was now performed on the data presented on the ERB
scale (Lehman 1991). The result was nonsignificant (
), indicating that the threshold could well be the same for
each . The results are summarized in Fig. 4. The figure
presents a boxplot of the results with themedian and the 75%
and 25% quartiles. The mean thresholds are between 0.083
and 0.122 ERB.
Since a true difference in the mean thresholds for each
note is relatively unlikely, the data were collapsed across fun-
damental frequency. The sample mean across all subjects and
fundamental frequencies is 0.10 ERB, which could be consid-
ered an estimate of the constant detection threshold of pitch
glides within the pitch range of the acoustic guitar.
3.2 Effect of decay rate
The listening test data from experiment II were processed
in the same way as the data from the first experiment. Again
116.5 196 349 659
0.05
0.1
0.15
∆ f at threshold (ERB)
Fundamental frequency (Hz)
Figure 4: Listening test results with fundamental frequency
as a parameter. Boxplot of (ERB) at threshold.
the ANOVA revealed no significant differences among the
means of the different conditions ( ). This suggests
that relatively great variations in the time constant of the pitch
glide have no effect on detecting the glide. The results are
shown in Fig. 5. The boxplot is shown in the top figure and
the mean thresholds on the ERB scale in the bottom figure.
Of course, it is likely that the glide duration would show
some effect if the glides were short enough. Such behavior
would be unnatural to string instruments, but it is an interest-
ing task to extend this study to synthetic sounds with greater
variation in the decay characteristics. A third, rather infor-
mal test was conducted to study extremely fast and slowly
decaying sounds. The time constant was now varied with
five linear steps between 20% and 180% of the original value
of 0.39 s, i.e., between 0.08 s and 0.7 s. The procedure was
the same as before, but only three subjects participated. The
shortest caused a considerable increase in the thresholds
while the other thresholds were similar to the previousresults.
The mean threshold for = 0.08 s was 10.8 Hz. For the four
greater values of the mean thresholds were between 5.6 Hz
and 7.7 Hz.
4 Conclusions and future work
The thresholds for detecting initial pitch glides in string
instrument tones were measured. It was found that when
expressed on the ERB scale, the thresholds remain roughly
constant with fundamental frequency within the pitch range
of the acoustic guitar. Furthermore, no significant effect was
detected for glide duration, which was controlled through the
time constant of the pitch decay. This suggests that a constant
threshold of 0.1 ERB could be proposed for string instrument
sounds at least within the studied range. Thus any pitch glide
weaker than this could be left unimplemented.
For instance, the electric guitar tone in Fig. 1 exhibits an
initial pitch glide of 3 Hz at a fundamental frequency of 466
Hz, i.e., about 0.06 ERB. Thus it would probably remain in-
audible to most listeners. On the other hand, a kantele tone
presented in (V¨alim¨aki et al. 1999) shows a pitch glide of
0.05
0.1
0.15
0.2
Time constant τ of pitch glide (s)
0.23 0.31 0.39
0.23 0.31 0.39
0.05
0.1
0.15
0.2
Time constant τ of pitch glide (s)
∆f at detection threshold (ERB)
Figure 5: Listening test results with decay time constant as a
parameter. Top: Boxplot with the median and 75% and 25%
quartiles, bottom: mean thresholds on the ERB scale.
almost 0.3 ERB, which should be clearly audible.
Although the effect of glide duration was insignificant
in the range typical of string instruments, the thresholds in-
creased for very fast decaying sounds. For these short sounds,
there was less time to listen to the steady pitch . This sug-
gests that the subjects were using end point detection, i.e.,
comparing the absolute frequencies at both ends of the glide,
to detect the glides. Both Madden and Fire (1997) and Moore
and Sek (1998) prevented their subjects from using such cues,
which might partly explain why the thresholds from this study
are lower than what they measured. On the other hand, in mu-
sical context the expectance of a certain pitch may well con-
nect the detection of a glide to absolute frequency,and this is
why our study enabled the end point cues.
The contradiction between glide detection for short and
long tones may also concern real instrument sounds, not only
synthetic sounds with unrealistic decay characteristics. Cut-
ing off the steady part of the tone might have a similar ef-
fect on the detection threshold as shortening the decay time
showed in this study. This calls for some more experiments.
The results from this study can be applied in digital sound
synthesis, where computational savings can be achieved by
ignoring the pitch glides whenever they are inaudible. Coding
is another field of application. The new structured methods of
sound representation (Vercoe et al. 1998) make it desirable to
control the perceptual features of sounds separately.
Acknowledgments
This work was supported by the Pythagoras graduate school,
Nokia Research Center, and the Academy of Finland.
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