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?

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.