This page contains demonstration material for the Master's Thesis entitled "Physics-Based Sound Synthesis of the Piano", written by Balázs Bank.
 

Acoustical properties of the piano

Inharmonicity

Additive synthesis of C2 note using 64 exponentially decaying sinusoids

inharmonic
harmonic

The bridge and the soundboard

The bridge is hit at the position A#4 with an impulse hammer, the pressure was recorded near to the soundboard (The signals presented here are the pressure signals, they are not deconvolved with the force signal)

response (three times)
response - sustain pedal down

The velocity of the bridge and the nearfield pressure recorded in the same time, played one after the other

C2 bridge velocity - pressure
A#4 bridge velocity - pressure

Bridge acceleration, velocity and sound pressure for Schumann's Träumerei (one can hear that the measured piano was quite out of tune...)

bridge acceleration
bridge velocity
pressure

Note C2 is played at piano, mezzo forte and forte dynamic level, but it is scaled to the same RMS value, hence the timbral differences are easily observed

The piano model

Modeling beating and two stage decay with a parallel resonator bank

The beating and two stage decay is modeled by a parallel resonator bank whose parameters can be determined by analyzing the sound of real pianos, as described in Section 6.4. However, since the measurement of all piano notes was not possible (it will be done in the autumn), another method had to be chosen. It is based on Gabriel Weinreich's '77 JASA paper. Weinreich presents the formulas which calculate the decay times and the frequencies of the two normal modes from a given terminating impedance. The impedance of the bridge was measured and these expressions were used to calculate the parameters of the resonators (see the last paragraph of Section 6.4.3). This method was found to be robust and stable, but the results do not coincide with the ones measured from sound pressure. This is because Weinreich's two-mode model is a simplification of the real word. However, the method gives physically meaningful parameters, therefore better sonic output than when no resonators are used. This is especially true for the high notes.

A#4 without resonators
A#4 with 4 resonators
A#4 with 8 resonators

C2 without resonators
C2 with 8 resonators
C2 with 16 resonators

One can hear that this works much better for the high notes than for the low ones. I still think that 8 or 16 partials with beating should be enough, but the parameters of the resonators should be more carefully adjusted. I guess that the beating should be overemphasized compared to the results of the theoretical model.

Music example

Here 8 resonators were used for the lowest notes, 4 for the midrange and 2 for the high range. The soundboard model is still the 2000 tap FIR filter, I have to work on the FDN a little more to make it better... but first I go for holiday.

The MIDI file is still that Beethoven sonata, I promise I will find something else soon.

Beethoven Appasionata sonata 3rd movement with resonators

It sounds a bit dull, I still have to tune the hammer parameters.
The computation complexity is about the half of the model below, since here the second and third string models are replaced with a parallel resonator bank.

The past:

Demo for a Hungarian Student Conference (Nov. 1999) and for the HUT Pythagoras Seminar (Dec. 1999)

1. Hammer model + digital waveguide with frequency independent damping
2. + Frequency dependent damping (lowpass filter in the delay loop)
3. + Inharmonicity (allpass filter in the delay loop)
4. + Beating (two independent, slightly mistuned waveguides)
5. + Soundboard (filtered by a 2000 tap FIR filter)

note C2
note C5

Dynamics in the piano model

C2 p, mf, f (scaled to the same RMS value)

What if we would pluck the piano string in the middle?

Plucked C2

Final Demo

One, two, or three independent digital waveguides were used, corresponding to the number of strings in the real piano. The parameters of the string and hammer were interpolated between the measured values. Artificial reverberation was added to the sound to create more natural impression.

Beethoven Apassionata Sonata 3rd movement