1. Intro to Fourier Series
By Jordan Kearns (W&L ‘14)
& Jon Erickson (still here )

2. 220 Hz (A3) Why do they sound different? Instrument 1 Instrument 2
Sine Wave
Why do they sound different?

3. Waveform (A = 220 Hz)
To = 4.5 ms
Piano
Guitar
Pure Sine Wave

4. Overtones and Music Perception
Overtones occur at integer multiples of the fundamental frequency when an object vibrates. The addition of these tones at regular intervals is musical to the human ear.
Example: Fundamental (1st Harmonic): 220Hz
1st Overtone (2nd Harmonic): 440Hz
2nd Overtone (3rd Harmonic): 660Hz
Video produced by Brandon Pletsch
Univ. of Georgia Medical School
URL:

5. Frequency Spectrum
Piano
Guitar

6. Fourier Series
Periodic functions (e.g., sound vs time) can be mathematically represented as some superposition of sine and cosine waves!

8. Time Domain
Frequency Domain

10. Why you should change strings
A quick experiment with a spectrogram
Old
New

11. Frequency Spectra for Different Instruments
Same pitch played, but
TIMBRE
is entirely unique

12. Orthogonality
m = 1, n = 2, T = 0.2
m = 2, n = 3, T = 0.2

13. More orthogonality m = 3, n = 1, T = 0.2 m = 2, n = 2, T = 0.2
Integrate over one period: m = n is the only case where any of these is non-zero. Allows us to extract an’s and bn’s

14. Waveform
To = 4.5 ms
Piano
Guitar
Sine Wave

15. Piano: Component Sine Waves
Microphone Signal Amplitude
Time

16. Piano: Component Sine Waves
Composite Wave
(From Previous Slide)
Original Piano Wave
Look how close with only three sine waves!!!
Try it yourself:

17. Biomedical Example Gastrointestinal Rhythmic Activity
GI electrical signals (voltage vs. time)
V(t) t

18. Fourier series representation
EXAMPLE recorded signal
Fourier series representation
2|cn|
Frequency (Hz)
Peaks occur at: 0.059, 0.293, 0.527, 0.820, 1.113, … Hz

19. Harmonic Motion in Guitar

20. Piano C chord (2nd inversion)
C major chord
G4 (388)
E5 (657) C5
1171
G5 (775)
1314
1564

21. Modes of Vibration: Standing Waves

22. Spectrogram: Piano

23. Fourier Series and Superposition
Any wave (sound) can be mathematically represented as some combination of sine waves.
Wave= SineWave1 + SineWave2 + SineWave3+…
𝑓 𝑡 = 𝑎 1 sin 2𝜋∗𝑓∗𝑡 + 𝑎 2 sin 2𝜋∗2𝑓∗𝑡 + 𝑎 3 sin 2𝜋∗3𝑓∗𝑡 +…
Fourier Series = Frequency Spectrum lets us see the component frequencies that make up the unique sound!

25. Frequency Decomposition: Pure Sine Wave
T = 2ms
f = 1/T
f = 500Hz

26. Frequency Decomposition: Pure Sine Wave
T = 1ms
f = 1/T
f = 1000Hz

27. Composite Wave I

28. Composite Wave II