2. Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 8 Harmonic Series Unit 1 Session 8 Harmonic Series

3. Physics 1251 Unit 1 Session 8 Harmonic Series Foolscap Quiz: What is the lowest frequency component of the following waveform? Repeat period P =10 ms =0.010 sec; f = 1/P = 100. Hz

4. Physics 1251 Unit 1 Session 8 Harmonic Series Recall: 80/20 The lowest (or fundamental) frequency f 1 of the components of a waveform is equal to 1/P repeat.

5. Physics 1251 Unit 1 Session 8 Harmonic Series 1′ Lecture: A harmonic series is a Fourier Frequency Spectrum in which the frequency of each of the components is an integral multiple of the fundamental frequency: A harmonic series is a Fourier Frequency Spectrum in which the frequency of each of the components is an integral multiple of the fundamental frequency: f 1 = 1 f 1, f 2 = 2 f 1, f 3 = 3 f 1, f 4 = 4 f 1 etc.

6. Physics 1251 Unit 1 Session 8 Harmonic Series “Harmonic” implies an integer ratio of frequency. f 1 = 1 f 1, f 2 = 2 f 1, f 3 = 3 f 1, f 4 = 4 f 1 etc. f 1 = 1 f 1, f 2 = 2 f 1, f 3 = 3 f 1, f 4 = 4 f 1 etc. e.g. 100 Hz, 200 Hz, 300 Hz, 400 Hz…

7. Peregrination in Existential Physics: Move you hand up and down in sinusoidal motion. Move you hand up and down in sinusoidal motion. Now diminish the range of motion. Now diminish the range of motion. Now increase the range of motion. Now increase the range of motion. You have changed the amplitude of the oscillation. You have changed the amplitude of the oscillation. Physics 1251 Unit 1 Session 8 Harmonic Series

8. Peregrination in Existential Physics: Move you hand up and down in sinusoidal motion. Move you hand up and down in sinusoidal motion. Now move you hand faster but keep the same range of motion. Now move you hand faster but keep the same range of motion. Now move you rhand more slowly. Now move you rhand more slowly. You have changed the frequency of the oscillation. You have changed the frequency of the oscillation. Physics 1251 Unit 1 Session 8 Harmonic Series

9. Peregrination in Existential Physics: Move both of your hands up and down in sinusoidal motion, starting them at the same time. Move both of your hands up and down in sinusoidal motion, starting them at the same time. Now let one hand precede the other. Now let one hand precede the other. You have changed the phase of the oscillation. You have changed the phase of the oscillation. Physics 1251 Unit 1 Session 8 Harmonic Series

10. 80/20 Important terms: Amplitude A: amount or size of a component wave. Amplitude A: amount or size of a component wave. Frequency f : number of oscillations per second. Frequency f : number of oscillations per second. Phase φ : the relative starting point of a component wave. Phase φ : the relative starting point of a component wave.

11. Physics 1251 Unit 1 Session 8 Harmonic Series Amplitude: the size of the excursion from the average. Amplitude

12. Physics 1251 Unit 1 Session 8 Harmonic Series Frequency, Amplitude, and Phase of a sine wave: Amplitude Amplitude

13. Physics 1251 Unit 1 Session 8 Harmonic Series Frequency the number of oscillations per second Higher frequency Lower frequency

14. Physics 1251 Unit 1 Session 8 Harmonic Series Frequency, Amplitude, and Phase of a sine wave: Frequency Frequency

15. Physics 1251 Unit 1 Session 8 Harmonic Series Phase: the relative starting point of an oscillation. Phase

16. Physics 1251 Unit 1 Session 8 Harmonic Series Frequency, Amplitude, and Phase of a sine wave: Phase Phase

17. Physics 1251 Unit 1 Session 8 Harmonic Series 80/20 Important terms: Amplitude A: amount or size of a component wave. Amplitude A: amount or size of a component wave. Frequency f : number of oscillations per second. Frequency f : number of oscillations per second. Phase φ : the relative starting point of a component wave. Phase φ : the relative starting point of a component wave. p = A 1 sin( 2π f 1 t + φ 1 ) + A 2 sin( 2π f 2 t + φ 2 ) + A 3 sin( 2π f 3 t + φ 3 ) + A 4 sin( 2π f 4 t + φ 4 ) … Amplitude Frequency Phase

18. Physics 1251 Unit 1 Session 8 Harmonic Series How does a square wave sound? What about a different waveform with the same harmonic series but different phases? FrequencyPhase

19. Physics 1251 Unit 1 Session 8 Harmonic Series The waveform is determined by the frequency spectrum and phase spectrum. Frequency Phase Waveform

20. Physics 1251 Unit 1 Session 8 Harmonic Series 80/20 Ohms’ Law for acoustics: Humans cannot directly sense the phase of a sound wave. Different waveforms with the same frequency spectra (but different phase spectra) sound the same to us.

21. Physics 1251 Unit 1 Session 8 Harmonic Series ♬☺ Listen to difference in Amplitude Amplitude Frequency Frequency (Phase) (Phase) Waveform Waveform

22. Physics 1251 Unit 1 Session 8 Harmonic Series Correlation of physical with musical characteristics: Amplitude—Intensity (A 2 ) Amplitude—Intensity (A 2 ) Frequency Frequency Waveform (phase + harmonic series) Waveform (phase + harmonic series) ♫ Loudness ♫ Pitch ♫ Timbre

23. Physics 1251 Unit 1 Session 8 Harmonic Series 80/20 Pitch is based on the ratio of frequencies. 80/20 A given frequency ratio is called an interval. 80/20 An octave is the pitch interval that corresponds to a ratio of f 2 / f 1 =2 / 1.

24. Physics 1251 Unit 1 Session 8 Harmonic Series Summary: A harmonic series is the sum of sine wave components each frequency of which is an integral multiple of the fundamental frequency. A harmonic series is the sum of sine wave components each frequency of which is an integral multiple of the fundamental frequency. Ohm’s Law : humans do not sense the phase of sound directly. Ohm’s Law : humans do not sense the phase of sound directly. Intensity determines loudness. Intensity determines loudness. Frequency determines pitch. Frequency determines pitch. Waveform (harmonic series) determines timbre. Waveform (harmonic series) determines timbre. An octave interval corresponds to a frequency ratio of 2 : 1. An octave interval corresponds to a frequency ratio of 2 : 1.