Audibility of Inharmonicity in String Instrument Sounds, and Implications to Digital Sound Synthesis

Abstract

Listening tests were conducted in order to find out the audibility of inharmonicity in musical sounds produced by string instruments, such as the piano or the gui-tar. The threshold of the audibility of inharmonicity was measured as a function of the inharmonicity coef-ficient at five fundamental frequencies. It was found that the detection of inharmonicity is strongly depen-dent on the fundamental frequency. A simple model is presented for estimating the threshold as a function of the fundamental frequency. The need to implement inharmonicity in digital sound synthesis is discussed.

Full text

AUDIBILITY OF INHARMONICITY IN STRING INSTRUMENT SOUNDS, AND
IMPLICATIONS TO DIGITAL SOUND SYNTHESIS
Hanna Järveläinen, Vesa Välimäki, and Matti Karjalainen
Helsinki University of Technology
Laboratory of Acoustics and Audio Signal Processing
P.O.Box 3000, FIN-02015 HUT, Finland
hanna.jarvelainen@hut.fi, vesa.valimaki@hut.fi
matti.karjalainen@hut.fi
http://www.acoustics.hut.fi
ABSTRACT
Listening tests were conducted in order to find out the
audibility of inharmonicity in musical sounds produced
by string instruments, such as the piano or the gui-
tar. The threshold of the audibility of inharmonicity
was measured as a function of the inharmonicity coef-
ficient at five fundamental frequencies. It was found
that the detection of inharmonicity is strongly depen-
dent on the fundamental frequency. A simple model
is presented for estimating the threshold as a function
of the fundamental frequency. The need to implement
inharmonicity in digital sound synthesis is discussed.
1. INTRODUCTION
The frequencies of the partials of string instrument
sounds are not exactly harmonic. This is due to stiff-
ness of real strings, which contributes to the restoring
force of string displacement together with string ten-
sion. The stretching of the partials can be calculated
in the following way [1]:
(1)
(2)
where is the partial number, is Young’s modulus,
is the diameter, is the length and is the tension of
the string, and would be the fundamental frequency
of the string completely without stiffness.
Inharmonicity is not necessarily unpleasant. Fle-
tcher, Blackham, and Stratton [1] pointed out that a
slightly inharmonic spectrum added certain “warmth”
into the sound. They found that synthesized piano
tones sounded more natural when the partials below
middle C were inharmonic.
The effect of mistuning one spectral component in
an otherwise harmonic complex is well known. Moore
and his colleagues [2] reported that the thresholds for
detecting mistuning decreased progressively with in-
creasing harmonic number and increasing fundamen-
tal frequency. In their experiment, mistuning was ex-
pressed as percentage of the harmonic frequency, and
the test tones were complex tones with 12 harmonics
at equal levels.
Moore’s group also showed that mistuning is heard
in different ways depending on the harmonic number
[2]. Shortening the stimulus duration produced a large
impairment in performance for the higher harmonics,
while it had only little effect on the performance for
the lower harmonics. It was reasoned that particu-
larly for long durations beats provide an effective cue,
but for short durations many cyclesof beats cannot be
heard. For the lower harmonics, beats were generally
inaudible, and the detection of mistuning appeared to
be based on hearing the mistuned component stand out
from the complex. The thresholds varied only weakly
with duration.
Scalcon et al. [3] studied the bandwidth of correct
positioning of the partials of synthesized piano tones.
They found cutoff frequencies abovewhich it was un-
necessary to imitate inharmonicity. For low tones the
relevant bandwidth was smaller than for higher tones,
but on the other hand, many more partials were in-
cluded in the frequency range at low tones. They also
stated that the effect of inharmonicity was unimportant
on the highest part of the keyboard.
We studied the audibility of inharmonicity as a func-
tion of the inharmonicity coefficient with funda-
mental frequency and sound duration as parameters.
The aim is to find general rules for the need to imple-
ment inharmonicity in digital sound synthesis. If in-
harmonicity were ignored, computational savings could
be achieved, since for instance in digital waveguide
modeling, an additional allpass filter is needed to im-
plement inharmonicity [4], [5], [6].

2. LISTENING TESTS
Subjects were required to distinguish between a com-
plex tone whose partials were exactly harmonic and an
otherwise identical complex tone whose partials were
mistuned. The threshold of audible inharmonicity was
measured as a function of at fundamental frequen-
cies 55 Hz ( ), 82.4 Hz ( ), 220 Hz ( ), 392 Hz
( ), and 1108.7 Hz ( ).
2.1. Test sounds
The test signals were generated using additive synthe-
sis to enable accurate control of the frequency and am-
plitude of each partial. The sampling rate was 22.05
kHz. The spectrum of the tones had a lowpass charac-
ter of the form 1/frequency, which is similar to spectra
of many string instruments. The decay of all partials
was exponential with a time constant seconds.
The initial phase of each partial was chosen randomly.
The tones contained all partials of the fundamen-
tal frequency up to 10 kHz. A constant cutoff fre-
quency was chosen because it was impossible to use
the same number of partials for every fundamental fre-
quency. Up to 50 partials would be important to the
perception of inharmonicity at low bass tones [3]. How-
ever, less than ten partials could be generated for the
highest tone before meeting the Nyquist limit. Realiz-
ing that the variable number of partials might affect the
audibility of inharmonicity, we reasoned that a con-
stant cutoff would still be the most practical solution.
A constant upper limit was also needed to keep the
spectral width of all test sounds equal. Galembo and
Cuddy showed [7] that without control of the spec-
tral width, inharmonicity changes the balance between
high and low frequencies and creates an impression of
sharpening.
Based on initial listening, we found that the thresh-
old of audibility varies over frequency. Thus, the range
of the inharmonicity factor must also vary as a func-
tion of frequency due to the chosen test method. A
range of was found for each fundamental frequency
to cover the probable threshold of audibility. The val-
ues of were uniformly spaced within the range.
A pitch increase due to inharmonicity was heard
at the highest tone. The subjects might listen to pitch
differences instead of timbre and ignore the effect of
inharmonicity unless there was an audible change of
pitch. To prevent this, the harmonic references of
were slightly tuned up to match the pitch of each in-
harmonic test sound. The pitches were adjusted until
no annoying difference was heard.
The effect of duration of the tone on the audibility
of inharmonicity was also studied. Two tone lengths
were tested, 1.5 s and 300 ms. The short samples were
generated by cutting off most of the decay phase of the
longer sounds. The time constant remained the same
for both durations.
2.2. Subjects and test method
Four subjects participated the final listening test. The
listeners were 20-30 years old, and all of them had mu-
sical training in some string instrument, either the pi-
ano or the guitar. None of them reported any hearing
defects. One of the listeners was the author HJ. The
sound samples were played through headphones from
a Silicon Graphics workstation using the GuineaPig
software [8]. Before the test the subjects were allowed
to practise until they made firm judgments.
The method of constant stimuli [9] was used. The
subjects heard pairwise a perfectly harmonic reference
sound and a possibly inharmonic sound, and the task
was to decide whether they sounded equal or different.
Eight values of (including ) were used for
each fundamental frequency, and each sound pair was
judged four times. The playback order of the sound
pairs was randomized, and the harmonic reference was
the first sound within a pair twice and the second one
twice.
3. TEST RESULTS
The psychometric function was approximated by count-
ing how many times each subject had regarded a sound
pair as “different” [9]. If there were zero “different”
judgments out of the four trials, the subject had no-
ticed the possible inharmonicity 0% of the time, and
by making four “different” judgments100% of the time.
The 50% threshold was chosen for the threshold of
audibility. If the threshold was not directly seen in
the data, it had to be interpolated between higher and
lower percentages. If it was spread over two values of
, the mean was calculated between them.
The audibility thresholds are shown as a function
of fundamental frequency for each subject in fig. 1.
The sample duration was 1.5 seconds. The answers
of the four subjects were roughly normally distributed.
The mean thresholds over the listeners at different fun-
damental frequencies are shown in table 1.
Table 1: Inharmonicity coefficient at mean thresholds av-
eraged over the subjects for the different fundamental fre-
quencies.
at mean threshold Standard deviation
0.00000058 0.00000021
0.0000013 0.00000038
0.000033 0.000019
0.000055 0.000027
0.0012 0.00081
The thresholds show a strong linear trend when a
logarithmic scale is used both for the frequency and
the . The highest note was judged with least ac-
curacy, and the effect of inharmonicity was also small-

est. At the mean threshold the value of was more
than 2000 times higher than that for the lowest tone.
102103
10−6
10−4
10−2
Frequency Hz
Inharmonicity coefficient B
Results of the four subjects
Figure 1: The individual thresholds for the four listeners
at , , , , and . Sample duration was 1.5
seconds.
A straight line was fitted to the mean threshold
values in the least-squares sense. This way a simple
formula was derived that could be used to model the
audibility threshold as a function of fundamental fre-
quency: (3)
The natural logarithm was used. The fitted line is il-
lustrated in fig. 2. The thresholds of one subject were
especially near the estimated line (subject J, see figure
2). The thresholds of two other subjects differ from
the line at , where they were able to detect smaller
inharmonicity than at . The nonmonotonic behav-
ior is not surprising, since the subjects reported that
they used several different cues to detect inharmonic-
ity. The performance can depend on the existence of
certain cues at different fundamental frequencies and
the subject’s sensitivity to the particular cue such as
beating or roughness.
3.1. Effect of duration
The test was repeated using shorter sound samples (300
ms). The thresholds of the four subjects (see fig. 3)
showed again a linear trend, and a straight line was fit-
ted to the results as before. The following model was
derived:
(4)
Both models as well as the average thresholds over
all subjects in both cases are shown in fig. 4. The
slopes of the models suggest that at low fundamen-
tal frequencies the detection of inharmonicity becomes
harder and at high fundamental frequencies somewhat
102103
10−6
10−4
10−2
Frequency Hz
Inharmonicity coefficient B
o− Straight line fit to average data over all subjectso− Straight line fit to average data over all subjects
x−− Subject J
Figure 2: The straight line fitted to average data over all
subjects (solid line), and the thresholds of subject J (dashed
line).
102103
10−6
10−4
10−2
Frequency Hz
Inharmonicity coefficient B
Short data: Results of the four subjects
Figure 3: The individual thresholds for the four listeners at
, , , , and . Sample duration was 300 ms.
easier when the samples become shorter. To test the
significance of the linear models, t-tests were performed
on the slopes [10]. The steeper slope of the long dura-
tion model was tested against the slope of the short du-
ration model, and the slope of the short duration model
was tested against zero slope. Both results were highly
significant, suggesting that the two slopes are greater
than zero ( (slope=0) = 0.0028) and that the slope of
the long duration model is steeper than that of the short
duration model ( (slopes are equal) = 0.0008).
4. DISCUSSION
Typical values of for piano strings lie roughly be-
tween 0.00005 for low bass tones and 0.015 for the
high treble tones [11]. Though the sounds are less in-

102103
10−6
10−4
10−2
Frequency Hz
Inharmonicity coefficient B
o− Straight line fit to average long sample data
o−− Average long sample data
x− Straight line fit to average short sample data
x−− Average short sample data
Figure 4: The linear models of audibility of inharmonicity
as a function of fundamental frequency for long and short
durations (solid lines), and the corresponding mean thresh-
olds (dashed lines).
harmonic in the bass range, inharmonicity is detected
more easily than at the highest tones. Compared to the
audibility thresholds, inharmonicity would be clearly
audible at low fundamental frequencies. In the treble
range, audibility is questionable. At , the thresh-
old of audibility is closer to the possible values of
in real instruments. Furthermore, the highest tone was
judged less accurately than the others and even non-
monotonously, i.e., the performance weakened locally
when inharmonicity was increased.
Thus it would be necessary to implement the effect
of inharmonicity in digital sound synthesis systems at
least in the bass range. In the treble range computa-
tional savings could be achieved by omitting the all-
pass filter responsible for the effect of inharmonicity.
There can be several causes for the better perfor-
mance at lower frequencies. The subjects told that they
were using beats as a cue. Beats were mostly audible
at low fundamental frequencies. When most of the de-
cay phase was cut off, the performance weakend in the
bass range but improved in the treble range, where an-
other cue was possibly used.
The test results are consistent with the general as-
sumption that the effect of inharmonicity is greater
when more partials are present [7]. This could also be
a cause for the differences in performance. However,
the number of partials in our hearing range decreases
with increasing fundamental frequency also in sounds
of real musical instruments, so it would make no sense
to isolate these factors from each other. Inharmonicity
can be influenced also by many other features [7]. To
study all these in a formal listening experiment would
be too laborous.
Our future objective is to find more general rules
for the need to implement the effect of inharmonic-
ity in digital sound synthesis systems. More accurate
models are needed for the effect of duration, number
of partials and spectral width, relative level of the par-
tials, and different decay rates between them.
5. ACKNOWLEDGMENTS
This work was supported by the Academy of Finland
and the Pythagoras Graduate School. The authors ex-
press sincere thanks to the test subjects and also to
Alexander Galembo for kindly providing his recent
publications.
6. REFERENCES
[1] H. Fletcher, E. D. Blackham, and R. Stratton,
“Quality of piano tones,” J. Acoust. Soc. Am.,
vol. 34, no. 6, pp. 749–761, 1962.
[2] B. C. J. Moore, R. W. Peters, and B. C. Glasberg,
“Thresholds for the detection of inharmonicity
in complex tones,” J. Acoust. Soc. Am., vol. 77,
no. 5, pp. 1861–1867, 1985.
[3] F. Scalcon, D. Rocchesso, and G. Borin, “Subjec-
tive evaluation of the inharmonicity of synthetic
piano tones,” in Proc. Int. Comp. Music Conf.
ICMC’98, pp. 53–56, 1998.
[4] D. A. Jaffe and J. O. Smith, “Extensions of the
Karplus-Strong plucked-string algorithm,” Com-
puter Music Journal, vol. 7, no. 2, pp. 56–69,
1983.
[5] A. Paladin and D. Rocchesso, “A dispersive res-
onator in real time on MARS workstation,” in
Proc. Int Comp. Music Conf. (ICMC’92), (San
Jose, CA), pp. 146–149, Oct. 1992.
[6] S. Van Duyne and J. O. Smith, “A simplified ap-
proach to modeling dispersion caused by stiff-
ness in strings and plates,” in Proc. Int. Comp.
Music Conf. ICMC’94, pp. 407–410, 1994.
[7] A. Galembo and L. Cuddy, “String inharmonic-
ity and the timbral quality of piano bass tones:
Fletcher, Blackham, and Stratton (1962) revis-
ited.” Report to the 3rd US Conference on Mu-
sic Perception and Cognition, MIT, Cambridge,
MA, July - August 1997.
[8] J. Hynninen and N. Zacharov, “GuineaPig – a
generic subjective test system for multichannel
audio.” Presented at the 106th Convention of the
Audio Engineering Society, May 8-11 1999, Mu-
nich, Germany, preprint no. 4871, 1999.
[9] J. P. Guilford, Psychometric Methods. McGraw-
Hill, 1956.
[10] J. Milton and J. C. Arnold, Introduction to prob-
ability and statistics. McGraw-Hill, 1990.
[11] H. A. Conklin, “Generation of partials due to
nonlinear mixing in stringed instruments,” J.
Acoust. Soc. Am., vol. 105, no. 1, pp. 536–545,
1999.