The Effect of Inharmonic Partials on Pitch of Piano Tones
Piano tones have partials whose frequencies are sharp relative to harmonic values. A listening test was conducted to determine the effect of inharmonicity on pitch for piano tones in the lowest three octaves of a piano. Nine real tones from the lowest three octaves of a piano were analyzed to obtain frequencies, relative amplitudes, and decay rates of their partials. Synthetic inharmonic tones were produced from these results. Synthetic harmonic tones, each with a twelfth of a semitone increase in the fundamental, were also produced. A jury of 21 listeners matched the pitch of each synthetic inharmonic tone to one of the synthetic harmonic tones. The effect of the inharmonicity on pitch was determined from an average of the listeners’ results. For the nine synthetic piano tones studied, pitch increase ranged from approximately two and a half semitones at low fundamental frequencies to an eighth of a semitone at higher fundamental frequencies.
The effect of inharmonic partials on pitch of piano tonesa)
Brian E. Andersonb) and William J. Strong
Department of Physics and Astronomy, N-283 ESC, Brigham Young University, Provo, Utah 84602
~Received 4 November 2004; revised 17 January 2005; accepted 8 February 2005!
Piano tones have partials whose frequencies are sharp relative to harmonic values. A listening test
was conducted to determine the effect of inharmonicity on pitch for piano tones in the lowest three
octaves of a piano. Nine real tones from the lowest three octaves of a piano were analyzed to obtain
frequencies, relative amplitudes, and decay rates of their partials. Synthetic inharmonic tones were
produced from these results. Synthetic harmonic tones, each with a twelfth of a semitone increase
in the fundamental, were also produced. A jury of 21 listeners matched the pitch of each synthetic
inharmonic tone to one of the synthetic harmonic tones. The effect of the inharmonicity on pitch was
determined from an average of the listeners’ results. For the nine synthetic piano tones studied, pitch
increase ranged from approximately two and a half semitones at low fundamental frequencies to an
eighth of a semitone at higher fundamental frequencies. © 2005 Acoustical Society of America.
@DOI: 10.1121/1.1882963#
PACS numbers: 43.75.Cd, 43.66.Hg @ADP# Pages: 3268–3272
I. INTRODUCTION
Several studies have investigated various subjective behaviors
of complex inharmonic pianolike tones. Fletcher
et al. ~1962! found, through subjective testing, that inharmonicity
in piano tones contributes a sense of warmth. In 1983,
Peters et al. ~1983! had listeners match components in complex
harmonic pianolike tones to a single sine wave to determine
the effect on pitch. Moore et al. ~1985a! found, through
subjective testing, that a complex tone consisting of one
slightly mistuned partial resulted in a linear residue pitch
shift for small mistunings. Moore et al. ~1985b! also used
this type of subjective testing to determine detection thresholds
and the importance of each partial. They suggested that
inharmonicity is audible for lower piano tones. Moore et al.
~1993! determined listeners’ psychoacoustic abilities to hear
individual partials in inharmonic complex tones. Conklin
~1996! wrote a series of tutorial papers, the third of which is
a summary of research investigations on the subject of inharmonicity.
Rocchesso et al. ~1999! studied ‘‘The influence of
accurate reproduction of inharmonicity on the perceived
quality of piano tones.’’ And most recently, Galembo et al.
~2001! determined the perceived effect of having various
kinds of starting phases for synthetic tones.
This study investigates the effect of inharmonicity on
pitch by having listeners match complex inharmonic pianolike
tones ~based on measured tones! to complex harmonic
pianolike tones ~based on ideal string theory!. Nine piano
tones from a standard upright piano were recorded and analyzed.
The analysis determined various parameters for as
many partials as could be resolved for each of the ninerecorded
tones. From this analysis, a synthetic inharmonic
tone and a set of harmonic tones with different fundamental
frequencies were produced for each of the nine tones. A jury
of listeners matched each synthetic inharmonic tone to one of
the synthetic harmonic tones among the corresponding set.
The effect of inharmonicity on pitch was determined from
the average fundamental frequency of harmonic tones
matched to an inharmonic tone.
II. TONE GENERATION
A. Recording
In order to study the effects of inharmonicity, a readily
available Yamaha Upright Piano ~P22 L. OAK! was used for
the study. A Larson Davis 2540 pressure microphone was
used with a Larson Davis 900B Preamplifier. The 900B preamplifier
was then connected to a Larson Davis 2200C Microphone
Power Supply and Instrumentation Amplifier. The
output of the 2200C went into an Aphex Amplifier 124A. A
Panasonic DAT recorder was used to record the output signal
from the amplifier.
The recording was done in a large classroom that had
been acoustically treated to reduce reverberation. Recordings
were done at times when ambient levels could be maintained
without interruptions. With this recording environment and
the recording setup, it was possible to maintain an overall
S/N ratio of at least 30 dB. The Yamaha Upright Piano has a
removable cover that allowed for better exposure of the
soundboard on the player’s side of the piano. The microphone
was placed about 6 in. from the soundboard.
Tones were recorded in the lower three octaves of the
piano where the effects of inharmonicity are greater. Nine
tones ~A0, C1, E1, A1, C2, E2, A2, C3, and E3! were individually
played and recorded. An effort was made to strike
each piano key with the same amount of force. Tones were
recorded until their overall sound pressure level decayed to
the level of ambient noise in the room. The resulting recorded
tones had durations ranging from 10 to 15 s. Finally,
the tones recorded on the DAT were digitally transferred
onto a CD.
a!Portions of this work ~preliminary results! were presented in ‘‘The effect of
inharmonic partials on pitch of piano tones,’’ 143rd Meeting: Acoustical
Society of America, Pittsburgh, PA, June 2002.
b!Present address: Graduate Program in Acoustics, The Pennsylvania State
University, P.O. Box 30, State College, PA 16802. Electronic mail:
bea3@email.byu.edu
3268 J. Acoust. Soc. Am. 117 (5), May 2005 0001-4966/2005/117(5)/3268/5/$22.50 © 2005 Acoustical Society of America
B. Analysis
The tones were played on a CD player whose output was
directly connected to the input of a Hewlett-Packard dynamic
signal analyzer and analyzed in order to obtain the
desired partial frequencies. Some of the piano tones had as
many as 56 resolvable partials that were obtained in the
analysis. In this analysis, it was assumed that the partial frequency
values remained constant over time, which, from the
tones’ spectrograms, proved to be a valid assumption.
The recorded tones were then imported into MATLAB
for further analysis. The resolvable frequency values of the
partials obtained from the dynamic signal analyzer were used
in the MATLAB analysis. The analysis determined the relative
initial amplitudes and decay rates for as many partials as
could be resolved for each of the nine-recorded tones. An
attack portion was also determined for each tone analyzed.
Fourier analysis was used to determine the amplitude of each
partial, over the period of the fundamental frequency, repeated
for 10 s. A linear curve fit was applied to a logarithmic
representation of the amplitude versus time data for each
partial, from which the initial amplitude and decay rate were
obtained. The attack portion of each recorded tone was determined
as the time it took from the onset of the tone to
reach peak amplitude. The initial phase fn for each partial
was assigned a random value between 0 and 2p. The results
of the analysis for each of the nine tones consisted of frequencies
of the partials, f n , amplitudes of the partials, An ,
decay rates of the partials, d n , and initial phases, fn .
C. Synthesis
Based on the analysis, a synthetic inharmonic tone was
produced for each of the nine tones analyzed. Each synthetic
inharmonic tone was generated based on the following equation,
yinharmonic5Ane2d nt sin~2p f nt1fn!, ~1!
where the variables are the results from the analysis.
A series of 10 to 30 harmonic tones was generated for
comparison with each inharmonic tone. Partial frequencies
of the harmonic tones were integer multiples of the fundamental.
Each synthetic harmonic tone was generated based
on the following equation,
yharmonic5Ane2d nt sin~2pn f 1t1fn!, ~2!
where the variables, with the exception of the fundamental
frequency f 1 , are results from the analysis. The fundamental
frequency for each harmonic tone in the series was incremented
in a ratio of 1.005 ~approximately a twelfth of a
semitone!.
A linearly increasing ~from zero to the maximum amplitude!
attack portion, determined from the analysis procedure,
was applied to each synthetic tone. For each of the synthetic
harmonic piano tones, an audio CD track was created consisting
of the corresponding synthetic inharmonic tone presented
first followed by the synthetic harmonic tone. Much
of the synthesis procedure was chosen based on previous
work done by Anderson ~2002!.
D. Inharmonicity coefficient
In 1964, Fletcher published a theoretical derivation of an
equation governing the shift in partial frequencies of piano
strings due to the inherent stiffness in piano strings. Fletcher
~1964! gave the following equation for predicting partial frequencies
once the fundamental frequency, which itself is affected
by the stiffness, is known:
f n5n f 1F~11Bn2!
~11B! G1/2
, ~3!
where B represents the inharmonicity coefficient. Using this
equation, an estimate for the inharmonicity coefficient was
determined, using the method of least squares, for each of
the nine measured tones.
The least-squares curve fit for the inharmonicity coefficient
included a heavier weighting on the lower partials. For
the nth partial, the least squares error weighting W(n) is
computed as
W~n!5N112n, ~4!
where N is the total number of partials for the given tone.
The resulting weighting is a linearly decreasing function.
The results for the curve fitted B values may be found in
Table I. Each curve fit result represented the increasing partial
frequency values fairly accurately, with a slight tendency
to underestimate the lower partials and overestimate the
higher partials. The B values were determined in order to
investigate the correlation between the pitch shift due to inharmonicity
and the inharmonicity coefficient for each measured
piano tone.
III. METHOD
A. Subjects
Twenty-one subjects volunteered from among the Physics
167 ‘‘Descriptive Acoustics of Music and Speech’’ course
offered at Brigham Young University. Students who participated
in the study were given extra credit towards their grade
in that course. The average age of participants was 20.7
years, with 76% female; 33% had taken a recent hearing test.
Participants had an average of 6.6 years of piano playing
experience, with an average of 9.1 years total musical instrument
experience. Table II shows the data for the listeners,
listed according to piano playing experience.
TABLE I. Curve-fitted inharmonicity coefficients for each of the nine measured
piano tones.
Piano note
Inharmonicity
coefficient
A0 0.000 453
C1 0.000 319
E1 0.000 231
A1 0.000 144
C2 0.000 130
E2 0.000 108
A2 0.000 111
C3 0.000 129
E3 0.000 110
J. Acoust. Soc. Am., Vol. 117, No. 5, May 2005 B. E. Anderson and W. J. Strong: The pitch of inharmonic piano tones 3269
B. Stimuli
Listeners were given an audio CD player and a set of
Sony Studio Monitor/Dynamic Stereo Headphones ~model
MDR-7506! to playback the tone pairs. Listeners set the volume
levels according to their comfort levels. Listeners were
placed in a quiet, isolated, semi-anechoic chamber facility
located at Brigham Young University. Listeners were free to
listen to tone pair CD tracks in any order they chose and as
many times as needed. Each listener was instructed to try and
find a tone pair, for each of the nine sets, that they perceived
as having the closest pitch match. If no match was found
among a given set of tone pairs, they were instructed to
indicate whether the harmonic tones needed to be higher or
lower in pitch in order to find a match; these responses were
not included in further analysis.
IV. RESULTS
Figure 1 shows a scatter plot of individual responses and
their average for each tone. ~Overlapping individual responses
are not apparent in the figure.! The average pitch
shift due to inharmonicity was greater in lower frequency
piano tones. It can also be seen that the spread or deviation is
much greater for lower tones. Table III tabulates the average
results of the listening tests. Table III also shows deviation
values.
In order to compare the listeners’ results for the perceived
shift in pitch due to inharmonicity with the least
squares fitted values for the inharmonicity coefficients, each
set of results was normalized by their respective mean value.
A plot of the two normalized sets of results may be found in
Fig. 2. Also contained in Fig. 2 is a plot of the normalized
relative standard deviation of listeners’ results.
A plot of the average perceived cents sharp values divided
by the fitted inharmonicity coefficients versus fundamental
frequency results in an exponentially decreasing
function. This function was least squares curve fitted to an
equation of exponential form with an added offset constant.
FIG. 1. Results for pitch matching study. Open circles represent unique
listeners’ answers and closed circles represent the average match for each
piano tone set.
TABLE II. Data for volunteer listeners who participated in inharmonicity
perceptual study.
Listener
rank
Age
~years! Sex
Taken
recent
hearing
test
Years
experience
playing the
piano
Total years
of musical
experience
1 21 F N 15 15
2 20 F N 15 15
3 20 F N 14 14
4 20 F Y 13 13
5 19 F N 12 12
6 20 F Y 11 11
7 24 F N 10 101
8 19 F N 10 10
9 20 F N 8 10
10 22 M N 6 12
11 19 F N 6 8
12 20 F Y 5 8
13 17 F Y 5 8
14 18 F Y 5 7
15 20 F Y 2 5
16 20 F Y 1 8
17 22 M N 0 13
18 24 M N 0 101
19 25 M N 0.5 0.5
20 25 M N 0 3
21 19 F N 0 0
Average 20.7 76% F 33% Y 6.6 9.1
TABLE III. Average results of listening tests. The actual fundamental and the average matched fundamental
frequency values are given in Hertz. Standard deviation values are determined from the matched fundamental
frequency values. The relative standard deviation values are defined as the standard deviation values divided by
their respective mean values ~then multiplied by 100 to represent a percentage!.
Piano
note
Actual
fundamental
~Hz!
Average
perceived
fundamental
~Hz!
Cents sharp
((Ratio21)/0.059)
Standard
deviation
~Hz!
Relative
standard
deviation
A0 26.750 30.597 243.8 13.71 44.81
C1 32.125 35.161 160.2 3.39 9.63
E1 40.375 41.649 53.5 3.29 7.89
A1 54.125 55.131 31.5 2.53 4.59
C2 64.500 65.094 15.6 1.99 3.06
E2 81.375 82.162 16.4 2.21 2.69
A2 108.625 109.416 12.3 1.15 1.05
C3 128.500 129.559 14.0 1.27 0.98
E3 163.250 164.830 16.4 1.76 1.07
3270 J. Acoust. Soc. Am., Vol. 117, No. 5, May 2005 B. E. Anderson and W. J. Strong: The pitch of inharmonic piano tones
The resulting coefficients for the curve fit resulted in the
following equation,
CentsSharp5B@2713exp~20.06813f !112.5#
310 000, ~5!
where CentsSharp is the perceived cents sharp value, B is the
inharmonicity coefficient, and f is frequency in Hz. Figure 3
shows the average cents sharp values divided by the fitted
inharmonicity coefficients versus fundamental frequency,
along with the optimum exponential curve fit result. The ‘‘error
bars’’ represent the standard deviation among listeners’
results. It should be noted that for frequencies above around
60 Hz, the relationship between the perceived cents sharp
value and the inharmonicity coefficient becomes equivalent
to a linear relationship according to the following equation,
CentsSharp~ f.60 Hz!5125 0003B. ~6!
An analysis of each individual listener’s results was
done in order to determine the dependence on listeners’ musical
experience. The absolute value of the bias ‘‘error,’’ between
each listener’s answer and the average value for the
given tone, was determined. These values were then averaged
across the nine piano tone sets for each listener to determine
an average bias error value ~relative to the average
answers! for the given listener. The standard deviation of the
bias errors was also determined and averaged across each
listener to create a relative standard deviation value. A plot
of the bias results and the deviation results may be found in
Fig. 4.
V. DISCUSSION
From Fig. 2 it is apparent that the average amount of
pitch shift perceived by the jury of listeners is strongly correlated
with the inharmonicity coefficient values in the upper
seven piano tones studied ~E1, A1, C2, E2, A2, C3, and E3!.
For the lower two piano tones, A0 and C1, the increase in
pitch shift relative to the increase in inharmonicity rises significantly.
This study also found an increase in deviation of listeners’
perceived matching results for lower fundamental frequency
piano tones. This might have some explanation in the
well-known psychoacoustic phenomenon that pitch perception
ability decreases at lower frequencies.
It was intuitively compelling to find that a given listener’s
ability to find a pitch match did not depend on their
piano musical training. Any given listener tended to have
relatively equal probability of perceiving the average of the
total listener’s pitch matches.
VI. CONCLUSIONS
The perceived pitch due to inharmonicity in piano tones
correlated with the inharmonicity coefficient for a given piano
tone, although the correlation was less pronounced for
the lowest two tones. The deviation among listeners’ pitch
matches increased at lower frequencies and showed correla-
FIG. 2. Normalized average results for pitch matching study, normalized
results for inharmonicity coefficient determination, and normalized relative
standard deviation results. A value of 1.0 on this plot represents the average
value for each data set.
FIG. 3. Filled circles represent average perceived cents sharp values divided
by the curve fitted inharmonicity coefficients ~scaled by 10 000!. The solid
line represents the least-squares exponential curve fit to the filled circle
values. The ‘‘error bars’’ represent the standard deviation among listeners’
results.
FIG. 4. Comparison across individual listeners of average listener’s bias
deviation from the average responses and average, across all nine piano
tones, listener’s relative standard deviation of bias values.
J. Acoust. Soc. Am., Vol. 117, No. 5, May 2005 B. E. Anderson and W. J. Strong: The pitch of inharmonic piano tones 3271
tion with pitch shift. A given listeners’ musical experience of
playing the piano did not correlate with the ability to perceptually
match piano tones.
The correlation between the pitch shift and the inharmonicity
coefficient for the higher frequency piano tones should
not be a surprising result. The significant rise in the pitch
shift trend relative to the inharmonicity coefficient values for
the lowest two tones, however, is a surprising result. The
increased pitch shift found in the lower two piano tones
should be further studied and extended to the entire lowest
octave on a piano ~roughly A0 to G]1!. The pitch shift found
in piano tone A0 was determined to be nearly 250 cents, two
and a half steps, or two and a half semitones.
ACKNOWLEDGMENTS
Portions of this project were supported by the Brigham
Young University Acoustics Research Endowment Fund and
by the Brigham Young University Physics and Astronomy
Department. Thanks are also due to Michael Thompson, who
helped on the project while at Brigham Young University.
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3272 J. Acoust. Soc. Am., Vol. 117, No. 5, May 2005 B. E. Anderson and W. J. Strong: The pitch of inharmonic piano tones