2. A.Diederich– International University Bremen – USC – MMM Spring 2005 1 Sound waves cont'd –Goldstein, pp. 331 – 339 –Cook, Chapter 7

3. A.Diederich– International University Bremen – USC – MMM Spring 2005 2 Additive synthesis –Fundamental frequency (or first harmonic): starting frequency for a complex sound –Harmonics: pure tones, each of which has a frequency that is a multiple of the fundamental

4. A.Diederich– International University Bremen – USC – MMM Spring 2005 3 Frequency spectrum Fundamental or first harmonic Second harmonic Third harmonic frequency: line's position amplitude: line's height

5. A.Diederich– International University Bremen – USC – MMM Spring 2005 4 Example –Fundamental or first harmonic: –220 Hz, given amplitude –Third harmonic: –660 Hz, 1/3 of amplitude –Fifth harmonic: –1100 Hz, 1/5 of amplitude –Sum all of three:

6. A.Diederich– International University Bremen – USC – MMM Spring 2005 5

7. 6 12 harmonics top down bottom up

8. A.Diederich– International University Bremen – USC – MMM Spring 2005 7 Waveforms with 12 equal-amplitude sinusoids using cosine/Schroeder/random phase, at frequencies of 880, 440, 220, 110, 55, and 27.5 Hz

9. A.Diederich– International University Bremen – USC – MMM Spring 2005 8 Waveform and amplitude spectra. Periodic waveforms A through D have line spectra, the others either continuous spectra (E and F) or a band spectrum (G).

10. A.Diederich– International University Bremen – USC – MMM Spring 2005 9 Average spectral shape

11. A.Diederich– International University Bremen – USC – MMM Spring 2005 10

12. A.Diederich– International University Bremen – USC – MMM Spring 2005 11 "Holy" spectra

13. A.Diederich– International University Bremen – USC – MMM Spring 2005 12 The width of critical bands as a function of center frequency

14. A.Diederich– International University Bremen – USC – MMM Spring 2005 13 Schematic representation of the frequency (heavy lines) corresponding to the tone sensation evoked by the superposition of two pure tones of nearby frequencies f 1 and f 2 = f 1 +  f

15. A.Diederich– International University Bremen – USC – MMM Spring 2005 14

16. A.Diederich– International University Bremen – USC – MMM Spring 2005 15 All three instruments playing the note G3 with a fundamental frequency of 196 Hz..

17. A.Diederich– International University Bremen – USC – MMM Spring 2005 16 Demo Missing fundamental 300 Hz+ 600 Hz + 900 Hz + 1200 Hz600 Hz + 900 Hz + 1200 Hz

18. A.Diederich– International University Bremen – USC – MMM Spring 2005 17 Sound composition and timbre: Helmholtz’s (1863) summary of the various subjective feelings pertaining on the composition of a complex sound