In the early years of the 19th century, the French mathematician Jean-Baptiste-Joseph Fourier presented a realization about the composite nature of the partial tones, or modes of vibration in a composite sound.

Any periodic vibration however complicated can be represented as the superimposition of pure harmonic vibrations, whose fundamental frequency is given by the repetition rate of the periodic vibration.¹⁸⁹

Fourier came to the conclusion that the vibrations of sound and their subsequent harmonics and overtones are mathematically related by the fundamental vibration or prime tone.¹⁹⁰

Fourier discovered that individual sinusoidal¹⁹¹ waves illustrated the qualities of each partial tone within the whole composite tone.This discovery established that it is the interaction between the fundamental tone and its upper harmonic partial tones that produces the quality and tonal colour of sound. The Fourier Theory of harmonic analysis has subsequently established a method for the analysis of how partial tones interact within a sound. ‘Fast Fourier Transform’ (FFT) is a computerized application of the Fourier Theory, which presents a visual illustration of the qualities of each partial frequency. The rates of decay are visually plotted in both loudness (dB) and duration (ms), dB/ms. This visualised graphic array illustrates the transient elements of the harmonic spectrum of the sound over time in spectrograms and decay curves. ¹⁹⁰ To visually display and examine the differences in the sound qualities of the Stuart and Steinway pianos, I have used the computer software ‘Fuzz measure Pro 3’which implements F.F.T. to produce illustrations of the piano sounds

The spectrograms below, Spectrogram 3.1 of the note C2 65.406 Hz is produced by Fuzz measure Pro 3.2.The Stuart piano sound (left) and the Steinway piano sound (right) are illustrated by the amplitudes, frequencies and decay rates of their partial frequencies. The notes C2 were struck at velocity 81, ff by the calibrated key striker. ¹⁹³

¹⁸⁹ ibid p.127
^{190     5}Roederer,127.
¹⁹¹ sinusoidal =sine wave depictions of simple tone, harmonic, partial tone, overtone, Fourier components.
^{192     2}Campbell& Greated, 19.
¹⁹³ For details about the electronic key striker see Chapter 2.