1. PK 6 Oct 2010 Some fundamental concepts
Periodicity and aperiodicity
Fundamental frequency
Harmonics
Formants and spectra

2. Periodicity and aperiodicity
Periodic waveforms repeat themselves
Aperiodic waveforms do not repeat themselves
The frequency of the periodic cycle is called the Fundamental Frequency, F0.

3. Periodicity and aperiodicity
periodic wave
aperiodic wave

4. Periodicity and aperiodicity
The waveforms on the previous slide were taken from the following waveform in Praat: my pronunciationof "fish":
f i S

5. PK 6 Oct 2010 Some fundamental concepts
Periodicity and aperiodicity
Fundamental frequency
Harmonics
Formants and spectra

6. The frequency of the periodic cycle is called the Fundamental Frequency, F0.
The pink shaded area is one cycle, = seconds,
= 1/ = c.p.s = Hz.

7. PK 6 Oct 2010 Some fundamental concepts
Periodicity and aperiodicity
Fundamental frequency
Harmonics
Formants and spectra

8. Complex periodic waves and harmonics
A periodic wave is made up of a (potentially very large) number of simple sine waves of different amplitudes.
There is a simple mathematical relation between the fundamental frequency and the component sine waves:
The F0 is the Greatest Common Demoninator (GCD)* of all the component sine waves – the largest number that divides exactly into the frequency of all the components.
Another way of saying this is that the sine waves are harmonics of the F0. A harmonic is any whole-number multiple of the F0.
The second harmonic is twice the F0, the third is three times, the nth is n times. (There is no “first harmonic”)
* also called Greatest/Largest Common Factor/Divisor

9. PK 6 Oct 2010 Some fundamental concepts
Periodicity and aperiodicity
Fundamental frequency
Harmonics
Formants and spectra

10. Spectrum (pl. spectra)
We calculate the number and amplitude of the component sine waves (the harmonics) in a complex periodic waveform using Fourier Analysis.
This will give us a spectrum of the waveform.

11. Spectrum of a waveform Ladefoged Elements p.94
This vowel has been synthesized from a small number of harmonics (sine waves)
The harmonics around 520 and 1400 have the greatest amplitude: these are the
first and second formants.

12. Spectrum of a waveform Ladefoged Elements p.97 Actual
not showing the actual harmonics, just the outline.

13. The density of the harmonics depends on the F0
Ladefoged Elements p.94
(b) 100 Hz
(c) 120 Hx
(d) 150 Hz

14. Spectrum of Ladefoged´s FLEECE vowel
The density of the harmonics depends on the F0
Spectrum of Ladefoged´s FLEECE vowel
Low F0

15. Spectrum of Ladefoged´s FLEECE vowel
The density of the harmonics depends on the F0
Spectrum of Ladefoged´s FLEECE vowel
F0 higher

16. Spectrum of Ladefoged´s FLEECE vowel
The density of the harmonics depends on the F0
Spectrum of Ladefoged´s FLEECE vowel
and higher ....

17. Spectrum of Ladefoged´s FLEECE vowel
The density of the harmonics depends on the F0
Spectrum of Ladefoged´s FLEECE vowel

18. Spectrum of Ladefoged´s FLEECE vowel
The density of the harmonics depends on the F0
Spectrum of Ladefoged´s FLEECE vowel

19. Spectrum of Ladefoged´s FLEECE vowel
The density of the harmonics depends on the F0
Spectrum of Ladefoged´s FLEECE vowel
very high F0

20. Spectra of various waveforms

21. Harmonics and formants
Formants are the peaks where the harmonics have the greatest amplitude.