Modelling The Decay of Piano Sounds
Tian Cheng, Simon Dixon, Matthias Mauch
[email protected]
3. Partial Detection
• Given inharmonicity coefficient B,
the partial frequencies are given
by
fn =nf0(1+Bn2)(1/2).
• Parameters B and f0 are
estimated in a parametric nonnegative matrix factorization
framework1.
(a)
(b)
0.5
0.5
0.5
0
0
0
2
1
2
Amplitude (dB)
−15
0
1000
(b)
−2
10
−3
10
−4
30
40
50
4. Decay Modelling
• Linear regression
(b)
−5
−5
−30
−10
−10
−40
1
1.5
Observation
Model
−20
Observation
Model
Time (s)
2
−80
0
1
2
Time (s)
• In a grand piano the string is set into
a vertical motion after being struck.
Due to the coupling of bridge and
soundboard, the plane of vibration
gradually rotates from vertical to
horizontal motion which decays
more slowly, referred to as a typical
double decay of the piano.
• Detuning between strings causes a
coupled oscillation. If the detuning is
small, the coupled motion will result
in a double decay. When the
detuning is large, beats appear. In
this case, the decay becomes a
decaying periodic curve.
Energy (dB)
1
−15
−15
Observation
Model
1
2
−20
0
−50
Observation
Model
1
2
−60
0
Time (s)
Observation
Model
1
2
Time (s)
(b)
0
0
−20
−20
−20
−40
−40
−40
−60
−60
−60
Observation
Model
1
−80
2
0
Time (s)
Observation
Model
−80
2
0
1
Time (s)
• Decay for different dynamics
the 1st partial
Observation
Model
1
0
2
−100
20
Time (s)
(b)
0
−20
−40
−60
−80
0
−20
−40
−40
−60
Observation
Model
1
−80
2
0
Time (s)
1
Time (s)
−80
2
0
120
f
m
p
40
60
80
100
120
the 3rd partial
0
f
m
p
−50
−100
20
40
60
80
100
120
the 4th partial
f
m
p
−50
Observation
Model
1
2
Time (s)
Fig. 3. (a) the 22nd partial of note D♭1 (34.6Hz); (b) the 10th partial of
note G1 (49Hz); and (c) the 10th partial of note A1 (55Hz).
• The decay response of the piano shows various decay
rates along the frequency range.
• Dynamics have no significant effect on the decay rate.
Future work: use decay modelling for offset detection and
100
0
−60
Observation
Model
80
0
−100
20
0
−20
60
−50
(c)
0
40
the 2nd partial
• Non-linear curve fitting
(a)
f
m
p
−50
Fig. 2. (a) the 7th partial of note B♭0 (29.1Hz); (b) the 4th partial of
note B♭2 (116.5Hz), (c) the 1st partial of note E5 (659.3Hz).
6. Conclusions
• We model the decay of piano notes based on piano
acoustics theory.
• The use of non-linear models provides a better fit to
the data, especially for low-pitch notes.
other music analysis applications.
(c)
0
−80
0
Energy (dB)
Time (s)
−60
0
8
Decay rate (dB/s)
(a)
−60
7
• Decay response
• Multi-phase linear regression2
0
4
5
6
Note group number
0
0.5
3
−20
0
1.5
2
−40
1
−40
2
0.5
1
Fig. 1. (a) the 3rd partial of note D1 (36.7Hz); (b) the 2nd partial of
note A♭1 (51.9Hz), (c) the 30th partial of note D♭2 (69.3Hz).
−40
1
(c)
−60
−40
0
0.6
−80
0.5
−20
−30
0.7
−20
0
f3
m3
p3
fL
mL
pL
120
−20
R2
0.8
100
−10
1
0.5
0
Observation
Model
100
80
0
90
60
0
1.5
80
40
1
70
0.9
Time (s)
0.5
60
MIDI note index
1
0.5
Mixed
model
0.878
0.831
0.800
10
−20
0
0.5
1500
Frequency (Hz)
2
1.5
1
500
Linear
model
0.727
0.686
0.643
• Adjacent 11 notes are grouped
together for detailed investigation.
Average R2s of different note groups
show the biggest improvement
occurring on notes in the low register.
20
1
Observation
Model
Noise Level
0
1
1512
1423
1226
−100
1.5
−10
(a)
−0.5
0
f
m
p
−120
1.5
−0.5
0
NP
−5
Decay rate (dB/s)
1
Dynamics
0
Energy (dB)
−0.5
0
0
Energy (dB)
(c)
• We compare the average coefficient
of determination, R2, between the
linear and mixed model (best among
the three models). The mixed model
has a better fit to the data by 15
percentage points.
Log frequency (MIDI note index)
Frequency (kHz)
Amplitude
2. Background
• Double decay and beats are
well-known phenomena caused
by the interaction between
strings and soundboard.
5. Results
(a)
B (log scale)
1. Introduction
• We investigate piano acoustics
and compare the theoretical
temporal decay of individual
partials to recordings of realworld piano notes from the RWC
Music Database.
• The energy of piano tones
decreases over time, causing
problems for piano analysis,
such as offset detection. We
would expect that modelling
decay will help to improve
performance of piano analysis
applications.
−100
20
40
60
80
100
120
the 5th partial
0
f
m
p
−50
−100
20
40
60
80
100
Log frequency (MIDI note index)
120
Acknowledgement
Tian Cheng is supported by a China Scholarship Council
PhD Scholarship. Matthias Mauch is funded by a Royal
Academy of Engineering Research Fellowship. This work
is supported by an EPSRC Platform Grant EP/E045235/1.
1 https://code.soundsoftware.ac.uk/projects/inharmonicityestimation
2 http://staff.washington.edu/aganse/mpregression/mpregression.html