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Two successive C keys are
separated by what is called an octave which corresponds to a ratio of two in frequency. That is, the
frequency of a C note is exactly double the frequency of the C immediately below (to the left of) it
and exactly half the frequency of the C immediately above it. Two sounds are perceived to be very
similar if they are separated by an octave and the only explanation for
this is that that is how mother nature has made it!
The concept of harmony is closely
related to the notion of harmonics. Consider a string fixed at both ends
and vibrating at a fundamental frequency . From basic physics, the upper
harmonics of the string are at integral multiples off, namely, 2f,
fie, 4f etc. Harmonious tones have common harmonics and this implies that the tones have
fundamental frequencies that are related as a ratio of two integers. A
high degree of harmony is associated with ratios involving powers of 2 (2:1, 4:1, 8:1 etc.) and
small integers (e.g.. 3:2).