Grand vs Upright Pianos
Popular version of paper 5aMUe17
Wavelet analysis is useful for extracting patterns and thus analyzing signals. Although the Fourier analysis can reveal different features of a signal, it is less appropriate for describing transient phenomena and sudden sound changes. Wavelet analysis is capable of highlighting different attributes of a signal. Different types of wavelets are thus used in the present study in order to compare the sound produced by grand pianos with that produced by upright pianos. It was found that with the use of the discrete wavelet transform it is possible to distinguish between the sound of grand and upright pianos.
At the beginning of the twentieth century the Haar wavelets were introduced. They are composed of a positive pulse succeeded by a negative pulse. Then the Gabor transform was created which has similarities with the Fourier transform. In 1984 Jean Morlet was capable of decomposing a signal into wavelet elements and then recomposing them into the original signal. In 1987 Ingrid Daubechies created another family of wavelets that can be easily applied in other sciences. Nowadays, wavelets are mainly used for compression of data as well as for de-noising.
Grand pianos produce better balanced sound and are less probable to produce after-sound fluctuations. Due to the bigger mass of the grand piano dampers, the damping effect is efficient also for lower frequencies. Grand pianos generate sound with less inharmonicity as they are bigger with longer bass strings. As far as the way pianos are placed inside a room is concerned, grand pianos are usually free from any direct obstacle though upright pianos are usually placed against a wall that consequently reflects the produced sound.
The recordings performed involve four fully assembled grand pianos as well as four fully assembled upright pianos of different brands. The sample rate and dynamic level used are 44.1 kHz and forte, respectively. One microphone was used and was placed in the middle of the grand piano bend as well as in the middle of the open upright piano top at a distance of approximately 0.25 m from the soundboard for both cases.
There are significant differences between Fourier and wavelet analysis. The fact that wavelets are localized both in scale via dilations as well as in time via translations whereas Fourier basis functions are localized only in frequency can be proven advantageous in many cases. Moreover, functions with sharp spikes and discontinuities can be represented with fewer wavelet basis functions, which makes them appropriate for data compression.
Wavelet analysis uses a fully scalable window for the observation of a signal, which is shifted along the signal many times. For every new cycle a slightly shorter or longer window is used. Then the corresponding spectrum is calculated and the results of this calculation are time-scale representations of different resolutions. Fourier analysis has the shortcoming that the time information is lost in the frequency domain and whereas Short-Time Fourier Transform (STFT) keeps also time information, the observation window used is the same for all frequencies. As a result, wavelet analysis is more appropriate for processing burst-like signals such as piano tones as it keeps time information and uses a variable sized observation window.
By observing the Discrete Wavelet Transform (DWT) of C4 with onset and decay played by four grand and four upright pianos as well as analyzed with Daubechies order 40 wavelet it can be concluded that with the use of level 8 it is possible to distinguish between a grand and an upright piano as the corresponding DWT seems to fade out much faster for a grand than for an upright piano. Other piano tones near C 4 have produced similar results. Symlet order 10 wavelet as well as Coiflet order 5 wavelet have been used in the same way and have produced similar results. The following figure presents the first second of the DWT at level 8 of C 4 played by eight pianos and analyzed with Daubechies order 40, Symlet order 10, and Coiflet order 5 wavelets.