A case study: the vowel "A"

In the page Simple models for the voice we showed that knowing the position of the first two peaks (called formants) in the frequency response of the vocal tract is enough to determine which vowel is being pronounced, and to distinguish it from the others. In particular, we used a numerical simulation to calculate the position of the formants from measurements of the cross-section of the vocal tract.

drawing (indicative only)	X-ray measurements	frequency response

Now let us focus on the vowel A alone, and see whether we can gain a more concrete understanding of the shape of its spectrum, and whether we can relate it to the most evident geometric feature of the vocal tract, that is, a narrower section followed by a wider one.

spectrum of the vowel A (in red) and its envelope (in green)

If we look at the spectrum of the sound A, we see that it consists of a sequence of narrow peaks in harmonic ratio (in red in the figure alongside), modulated by a smoother shape with less pronounced and more widely spaced peaks (highlighted in green). The red peaks are due to the action of the glottis, which produces an almost triangular wave made up of a harmonic series of components. Of course this structure does not depend on the vowel being pronounced, but changes, for example, with the particular note being sung (even in speech, vowels have a reference "note"). It is the green structure that we must focus on, because it does not depend on the action of the glottis but on the filtering effect of the vocal tract. The green curve, whose most prominent peaks are marked by the arrows, is the distinctive signature of each vowel.

We can see that the "signature" of A consists of two rather close peaks at about 1000 Hz, followed by a broad antiresonant region (that is, without peaks) up to about 3000 Hz. Both these features are clearly visible in the drawing.

Now let us go back to our model but, this time, let us simplify it further to understand it better. We repeat the simulation assuming that the two sections of tube are cylindrical. We also neglect the fact that part of the sound is radiated from the mouth. To get a better view, we run many simulations, varying the relative length of the two sections (see the animation below). The resonances change systematically at each position (middle animation). If we plot them as a function of the position of the junction point between the two tubes, we obtain a symmetric pattern (rightmost figure). Now we have to interpret it.

Geometry	Acoustic response	Resonances

Resonances of the two tube sections uncoupled (in green and light blue), and coupled (in red)

The pattern is easy to interpret if we superimpose on it the natural frequencies of the two tube sections, calculated as if they were disconnected from each other.

Indeed, if we follow the path of the sound wave from the glottis through the narrow section of tube towards the mouth, we see that the propagation conditions change abruptly at the point where the tube widens. The wave is partly reflected back towards the glottis, and partly transmitted into the wider section of tube. This is a phenomenon of refraction in one dimension, which modifies the properties of the transmitted wave. For an extended illustration of the phenomenon, see the pages refraction in strings and refraction in pipes.

We note that the resulting frequencies of the whole tube match well the frequencies of the individual uncoupled tubes as long as we stay far from the crossings. For example, for x between 0 and 5 cm the fundamental frequency of the system essentially coincides with the frequency of the right-hand section alone, since the fundamental of the left-hand section is much higher. For x greater than 11 cm the situation is reversed, and the fundamental essentially corresponds to a resonance of the left-hand section alone.

When the frequencies of the two uncoupled sections are identical, the light blue and green curves cross. For example, for ${\displaystyle x=8}$ the right-hand and left-hand sections have the same length, and therefore the same fundamental (and the same harmonics). However, precisely at these points the red curve departs from the light blue and green curves.

The reason lies in the coupling between the two sections, which plays a more important role the more influential the interference between the wave reflected in the narrow section and the wave transmitted into the wide section becomes.

It is beyond the scope of this page to prove analytically that the distance between the actual frequencies is greater the more strongly the tubes are coupled, but we can do a quick numerical check: if we exaggerate the difference between the radii of the tubes, their coupling decreases, and so the overall response must look more like that of two independent tubes. In other words, if the diameters of the two tubes are very different, the red frequencies will get closer to the crossing points.

Indeed, to decouple the two sections while leaving them connected, it is enough to mismatch their impedances (see Impedance matching), and this, from an acoustic point of view, simply means making the cross-section of one much larger than that of the other. Obviously this configuration does not correspond to any real vocal tract, but it serves simply as a check.

Looking at the result of the calculation, we clearly see that the thick red curve, corresponding to the complete tract, comes closer to the green and light blue curves, corresponding to the two separate tubes, and, in particular, that the frequencies get closer to each other at the crossing points.

Geometry	Acoustic response	Resonances
A limiting case of the model tract. By making the cross-section of the first section much smaller than that of the second (that is, by mismatching the acoustic impedances), the response must look more like that of the two sections treated as separate tubes.	The impedance of one of the two tube sections is now much greater than that of the other, and indeed very small peaks appear superimposed on a much larger profile.	Indeed, the response of the complete tract comes closer to the responses of the two sections and, in particular, the frequencies get closer at the crossing points.