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Simply add the effect of each wave at every moment

Green + Blue = RedThe resultant frequency is the Greatest Common Factorof the component frequencies: fnew = GCF(f1,f2)

Beating

f1= 5f2= 5.5fcomb= .5

f1= 20f2= 21fcomb= 1

f1= 500f2= 510

f1= 500f2= 505

f1= 500f2= 501

The beat frequency is the difference of the component frequencies

The beat frequency is the difference of the component frequencies

4th harmonic4th harmonic f4 = 4 f1f4 = 4 f1

3rd harmonic3rd harmonic f3 = 3 f1f3 = 3 f1

2nd harmonic2nd harmonic f2 = 2 f1f2 = 2 f1

HarmonicsWhat is the greatest common factor of...

(10, 20, 30, 40, ...)?

(150, 300, 450, 600, ...)?

(440, 880, 1320, 1760, ...)?

fundamentalfundamental f1f1

perceived pitch is always the fundamental frequency

perceived pitch is always the fundamental frequency

Superposition of many harmonicssawtooth wave

f =500 Hz (A = 1) f +2f (A2 = 1/2)

f +2f +3f (A3 = 1/3) f +2f +3f+4f+5f (A5 = 1/5)

10 harmonics

Superposition of many sine wavessquare wave

f =500 Hz (A = 1) f +3f (A3 = 1/3)

f +3f+5f (A5 = 1/5) f +3f+5f+7f+9f (A9 = 1/9)

10 harmonics

A sound is the sum of its partsPiano

Fundamental only

Harmonics 1 & 2

Harmonics 1, 2, 3

Harmonics 1-4

Harmonics 1-5

Harmonics 1-6

Full sound