Harmonics, Timbre & The Frequency Domain

1. Harmonics, Timbre & The Frequency Domain

2. Real Instruments • Real instruments do not normally produce pure tones • Instead the sound produced by hitting a single note has: • a fundamental frequency • some extra frequencies

3. Harmonics • Frequencies can be harmonically related • These are called harmonics • They are related in whole number multiples

4. Harmonics • If extra frequencies are harmonics the sound produced will be perceived as a single pitch at the fundamental frequency • One fundamental frequency and several harmonics

5. Harmonics • The harmonics will be multiples of the fundamental frequency • The fundamental is the largest common divisor of the harmonics

7. Harmonics • The composite wave has the same frequency as the fundamental • The fundamental is frequency at which the entire waveform vibrates • The brain perceives the composite waveform as a sound that has the same pitch as the fundamental

8. Fundamental Tracking • The ability of human brain to track the fundamental frequency of a sound • Occurs even when the fundamental waveform is not present • This is because the wave will repeat at the fundamental frequency and the brain detects this

9. f0 = 1/2f1 = 1/3f2 f1 2.5 f0 pressure f2 2 1.5 1 0.5 0 time -0.5 t2 -1 t1 -1.5 -2 t0 -2.5

10. f1 = 2 and f2 are integer multiples of f0 • t0 = 2t1 = 3t2 • substitute in t = 1/f to get: • f0 = 1/2f1 = 1/3f2 • So: f1 = 2f0 and f2 = 3f0

11. For Example • Say: f1 = 440Hz • 1/2f1 = 1/3f2 : f1 = 2/3f2 : 3f1 = 2f2 : f2 = 3/2f1 • So: f2 = 3/2 * 440 = 660Hz • f0 = 1/2f1 = 220Hz • Which is the highest common divisor of 440 and 660

12. Harmonic Series

13. Harmonic Series

14. Frequency Domain Representation • Used to sound waves being represented in terms of time and amplitude • Known as Time Domain Representation • Frequency Domain Rep shows frequency and amplitude

15. dB 1 0 seconds - 1 dB 1 0 100 200 300 400 500 600 frequency - 1 Frequency Domain Representation A 440Hz sine wave shown in the time domain (above) and the frequency domain (below).

16. Frequency Domain Plots • Both real instruments and synthesisers normally produce more complex waves • This means additional frequency components or overtones • All of these frequencies (including the fundamental) are called partials

17. Frequency Domain Plots • The frequency domain content of a wave is represented by plotting each partial on the x-axis • The height of each line indicates the strength of each frequency component

18. Trumpet amplitude frequency

19. Clarinet amplitude frequency

20. Trumpet

21. Timbre ‘Characteristic quality of sounds produced by each particular voice or instrument, depending on the number and character of overtones.’ Oxford English Dictionary

22. ‘A common timbre groups tones played by an instrument at different pitches, loudnesses, and durations. No matter what note it plays, for example, we can always tell when a piano is playing. Human perception separates each instrument’s tones from other instrument tones played with the same pitch, loudness, and duration. No one has much trouble separating a marimba from a violin tone of the same pitch, loudness and duration. Of course a single instrument may also emit many timbres, as in the range of sonorities obtained from saxophones blowed at different intensities.’ Roads (1996, p 544)

23. Harmonics Again • Remember: a harmonic is a sound that is an integer multiple of the fundamental frequency • So: while the fundamental carries out one cycle (or period), a harmonic of this will carry out an exact number of whole cycles

25. Harmonics Again Still • So a fundamental plus several harmonics produces a composite waveform that is? • Periodic: it repeats itself exactly • If the added waveforms were not harmonics the waveform would not be periodic

26. Timbre ‘Characteristic quality of sounds produced by each particular voice or instrument, depending on the number and character of overtones.’ Oxford English Dictionary

27. Harmonics and Musical Instruments • Most instruments produce overtones • Generally the overtones are nearly harmonic but not quite • Because they are not exact harmonics the sound wave produced is not periodic (but quasi-periodic)

28. Harmonics and Musical Instruments • The trumpet has strong harmonics (brash, full and brassy) • The clarinet has weaker ones (pure, smooth flute like) • In-harmonic frequencies are non-periodic (metallic and percussive)

29. Trumpet amplitude frequency

30. Clarinet amplitude frequency

32. Amplitude Variations • Timbre is also strongly affected by variations in amplitude over time • This is why amplitude envelopes are so commonly used in electronic sound synthesis

33. Trumpet

34. Formants • A formant is a peak of energy in an absolute frequency region • Responsible for the timbre of the human voice and many real instruments • Formant peaks stay the same when pitch of sound is changed

35. Wave Shapes • Wave shapes like triangle waves, and sawtooth waves don’t have formants • This is because they have a shape that always remains the same • They have identical relationships among their frequency components no matter what pitch they are

36. Amplitude 1 51 71 91 111 31 131 frequency Triangle Wave

37. Triangle Wave • Sound produced will be mostly between 1st and 7th harmonics • So one of 70Hz will have it’s most prominent frequencies 70 - 490Hz • And one of 500Hz will have it’s most prominent frequencies 500 - 3500Hz

38. Sound Sources with Formants • Like the human voice (e.g. saying ah) will change shape depending on their frequency

39. Humans Saying “Ahh” • On average the following frequencies are emphasised when a man says “ahh”: 730Hz, 1090Hz, and 2440Hz • No matter what the pitch the man says ah at • Female “ahh”s are on average: 850Hz, 1220Hz and 2810Hz

40. Formants Are Caused by Physical Characteristics • The formants of a particular instrument or voice are determined by resonance chambers • Human voice formants vary according to nasal, oral and pharyngeal cavities

41. Formants Are Caused by Physical Characteristics • A guitar has formants based on shape character and dimensions of resonance chamber • Formants are what make a human voice recognisable or give an instrument a particular sound