More properties of resonant curves

Each peak in a resonant curve can be characterized by a center frequency.  The drawing on the left shows three resonance curves, and each circled frequency is a center frequency. The resonant frequencies for the vowels listed in your textbook on page 101 are those center frequencies.

A second property of resonant curvesis bandwidth--how wide are the peaks. Bandwidth is measured by going down 3 decibels from the peaks. You can see an example in the picture on the right. You can see the center frequency, marked f0. f1 is the lower cutoff frequency, and f2 is the upper cutoff frequency.  The distance between f1 and f2 is the bandwidth.

Most of the bandwidths for the lower frequencies of speech are around 50 Hz. Bandwidth can be plotted as a function of the center frequency. After about 1800 Hz, bandwidth starts increasing, as you can see in this drawing. To test your understanding of this, answer the following question:

At which resonant frequency would you likely find a bandwidth of over 50 Hz?

Let's see what the formants do with the vowels of English. Figure 5.27 in your textbook (page 108) shows the first 2 formant frequencies of the vowels of English. You can see that for /i/, the vowel sound in "heat," F1 is low and F2 is high.  Here is another version of frequency for the quadrilateral vowels.

From this drawing, we can see that

F1 is inversely proportional to tongue height.  (if the vowel is high, then F1 is low), and

F2 is directly proportional to tongue advancement (the pinching of the node in the front part of the oral cavity.)

The first 2 formant frequencies are more important in identifying a particular vowel.  The third formant frequency helps the vowel sound more natural (not computer generated, for example).  The lowest 2 formants are necessary and sufficient for discriminating among vowels.

Summarizing the Source Filter Theory

Now we're ready to put it all together.  The sound source from the larynx is relatively independent of the vocal tract.  The output spectrum depends on the combination of both the larynx and the vocal tract.  The resonant properties of the vocal tract will modify the components of the sound source.

The frequency components (harmonics) that are near the resonant frequencies (formants) will be emphasized.

In this drawing, the transfer function is the horizontal line. If the harmonic is at the frequency of one of the three peaks,it is enhanced or emphasized.  If not, it is surpressed. Remember that harmonics do NOT equal fromants. The vowel /i/ has a low f1, but relatively high f2 and f3 compared to the other vowels.  Producing the vowel /i/ will lower F1 because we are constricting the antinode.  With this vowel, thought, we are also constricting at the node, which tends to raise F2 and F3.

Laryngeal sources and vocal tract resonances are essentially independent of each other.  We can produce the same vowel at different fundamental frequencies, and different vowels at the same fundamental frequency.

A mathematical summary of the source filter theory

If we wanted to write an equation for the source filter theory it would be this:

The source X the transfer function X the radiation factor = the output spectrum

The source is -12db/octave (the slope of the frequencies as seen on the output spectrum)

The transfer function is the formant frequencies + the bandwidths (the slope of the line across the frequencies).

The radiation factor adds 6 db per octave.  This is not too interesting, and remains fairly constant.  The face acts as an acoustic baffle and radiates the sound outward.  The face emphasizes, or restores, higher frequencies.

You can see the formula here that produces an output spectrum: