The physical difference in the waveform of a C# note produced on a violin and a C# note produced on a trumpet is not frequency; it is timbre. Timbre is the characteristic tone color of an instrument's voice. The rapid onset of a trumpet's C# occurs in about twenty milliseconds, while the slower onset of a violin's takes considerably longer in the hundreds of milliseconds range. The overtones of the various frequency harmonics and the interplay that occurs due to superposition of differing phases of sound waves causing each to produce a unique waveform following the onset. This difference is visible in the structure of the resulting waveform.
Timbre is a multidimensional property of sound waves. It fluctuates with room, player, and instrument. This leaves scientists at a loss of quantitative scientific measurement of tone quality through timbre. While tone can be analyzed for quality in its entirety through the resultant waveform, timbre cannot. Timbre can be analyzed by looking at the number and frequency of harmonic, constructive interference, but measurement of timbre's quality is restricted to qualitative descriptions consisting of words which approximate those used in musical texts. (Radocy and Boyle 1988)
Timbre is mainly confined to the initial sonic disturbance or attack of the note. This is where it is most apparent in the waveform due to differences in phase of the frequencies. A violin string plucked with a player's fingernail emphasizes the higher overtones, thus creating a "bright" tone quality while a soft attack, such as that of a bow, emphasizes the fundamental pitch. (see figure 3) Tone is more dependent on standing waves, which are commonly referred to as overtones.
Tone is recognized in regularity of vibration. A simple tone has only one frequency, while a complex tone consists of two or more simple tones, called partial tones. The partial tone of lowest frequency is called the fundamental tone. (see figure 4) Other tone components include the upper partial tones, or overtones. A harmonic occurs when the frequency of the overtone is an derivative multiple of the frequency of the initial frequency. The second harmonic is the first overtone, the third harmonic is the second overtone, and so on. The tone of a sound is a function of the amplitudes of the harmonics and not of the phase relationships between them.