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### Harmony

This shows a superposition (in black) of two waves of frequecies f1 (red) and f2 (green).

Try these experiments:

• 3 and 5 (the original setting). This illustrates composition of a note and its fifth. The result "explains" why they sound together well.
• 15 and 25 (same but from a greater perspective) You can see the pattern of harmony. (Try also 30 and 50).
• 15 and 27 (second sound "off"). Now you see "disharmony."
• 15 and 17 (close but different frequencies) show "beats." Try also 30 with 31,32,33, etc.
• 10 and 20 give the "octave," while 10 and 30 give the fifth in the second octave.
• 5 and 40 --- can you guess the result of this mix?

Some math:

Let the wave at your ear be a superposition of two simple waves

y    =    sin k1 t   +   sin k2 t

where k1 and k2 are two different frequencies.

Using the well-known identity, we can express this as

y    =    2 cos dt   sin kt
where
d   =   ( k1 -- k2 ) / 2       (half the difference between the frequencies)
k   =   ( k1 + k2 ) / 2      (average frequency)

For instance, values k1 = 30 and k2 = 32 give

y  =   2 cos t   sin 31 t

i.e., a sound of 31 Herz (average frqn'cy), the volume of which varies as cos t, giving 2 beats per second.