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The frequency content of musical tones yields numerous informations related to pitch and tone quality.
In the case of pianos, spectral analysis can also be used with success for estimating important physical data:
- String tension (derived from the estimation of the fundamental frequency)
- Young's modulus of strings (derived from measurements of the inharmonicity)
For the present project, particular attention was given to the spectral envelope, averaged over the total duration of the tone. Comparisons were also done between the spectral envelopes of the initial and final portions of the tones, respectively, in order to account for the evolution of the spectrum with time. Relevant descriptors, such as the inharmonicity and the Spectral Center of Gravity (or spectral centroid) were extracted from the estimation of the spectral envelope.
Inharmonicity curve for the note F3 (piano J.B. Streicher 1851). In pianos, the ratio (fn/nf1) where fn is the frequency of the string's partial of rank n, and f1 the fundamental frequency, varies almost linearly with n^2. From the slope of the curve, an estimation of the Young's modulus of the string can be derived.
This figure shows a comparison of spectral analysis performed on the same note F#2 played on two different pianos (N. Streicher 1819 and J.B. Streicher 1873). The vertical axis is the pressure (in dB) normalized with respect to the maximum value. In fact, for a similar striking force, the JBS73 sounds louder. However, because of this better acoustical efficiency, the sound is also damped more rapidly. This damping primarily affects the high frequencies. As a consequence, we can see on the figure that the normalized spectrum of the JBS73 tone (in red) is lower than the N19 one (in black) above 4 kHz. As a consequence, the tone color is clearly different.
This figure is an example of sonagraphic representation for the string G6 of the Gert Hecher piano (GH05). It shows the evolution of the spectrum with time. The amplitude (in dB) is represented by the color scale (on the right). In this particular case, a number of high-level components can be seen between 0 and 4 kHz, especially during the first 400 ms of the tone. High-resolution spectral analysis show that these components are soundboard modes excited by the hammer pulse. They are very audible in transients, and contribute to the timbre of piano notes, especially in the treble range.
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