Chapter 1. Physical Sound
When an instrument is played or a note sang, different frequencies are generated: the fundamental frequency (or fundamental note) and overtones, also called harmonics. The basic reason why this happens is because a vibrating string (attached at both ends) has very precise ways of vibrating to accommodate for the physical constraints of the system (the string material, how the ends are fixed and the medium in which it vibrates: air, water, etc.). These different ways are “modes” which possess unique points with zero vibration on the string called “nodes”. The number of these nodes tells you what harmonic we are talking about.
Octaves happen every time the number of nodes is a power of 2: 1 (because 20=1), 2, 4, 8, 16, etc. Alternatively, it can be defined as the difference in frequency between a base frequency and its double. A semitone (or half-step) is defined as 1/12th of an octave.
Table 3 First Five Harmonics for Standard A
Note that the first overtone is the second harmonic – it is only a matter of definition; they do represent the same frequency. Harmonics in between those perfect intervals become increasingly complex. Most plucked stringed instruments do not generate natural harmonics past the 5th; only bowed and percussion instruments can generate harmonics past 20 kHz.
Figure 3 Harmonics: The first order harmonic (n = 0) has no nodes; the second order harmonic (n = 1) has one node in the middle of the string; the next even order harmonic nodes are marked with markers of decreasing size up to fourth order harmonics. The doubling of the frequency for each subsequent octave becomes obvious.
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