# Why are vocal cord vibrations quasi periodic and not periodic?

I read* that one reason that f0 varies over time owes to the quasi-periodic nature of vocal chord vibrations.

But what does f0 variation over time have to do with the quasi-periodicity of the signal?

• The title of this question (which refers to the movements of articulators) asks something fundamentally different from the body of the question (about the relationship between F0 and periodicity in the acoustic signal). Please edit one or the other so that they match. Aug 20 '15 at 14:45

Well, this is really a basic mathematics or physics question. The frequency of a sound is "F0" if and only if "1/F0" is the smallest possible period of time for which

pressure (t) = pressure(t+1/F0)

for each moment of time "t". To be able to define the frequency means to define the periodicity of the pressure waves (sound are oscillations), and the frequency (in Hz) is the inverse of the periodicity (in seconds). For example, a tone with the frequency 500 Hz is periodic and the oscillating pressure gets repeated every 1/500=0.002 second.

However, no such exact periodicity exists for the human voice – which also means that "F0" isn't quite well-defined, either. The vibrations of the vocal cord are complicated and not periodic. They are close to periodic ones but not quite periodic: they are "quasi-periodic". It means that some frequency "dominates" but it cannot be determined exactly. The vibrations of the vocal cord are aperiodic because different points of the vocal cord vibrate independently and the vibrations can't be divided to simple harmonics because they affect each other, due to "nonlinearities" of the equations governing the vocal cord.

Ideally, one could consider sounds that are periodic, with the periodicity "1/F0". Such sounds may be written as "Fourier sums" (Fourier series, Fourier expansions) over the frequency "F0" and the "higher harmonics" such as "2*F0", "3*F0", and so on. Each of these higher harmonics would have some coefficient which would describe the "color" of the sound.

However, the human voice isn't quite periodic so the Fourier series aren't enough. We need a full-fledged "Fourier transform" in which all frequencies, and not just a multiple of "F0", are represented. Some frequencies around "F0" (and partly its multiples) make higher contributions but the other frequencies can't be completely neglected.

• You wrote "The vibrations of the vocal cord are aperiodic because different points of the vocal cord vibrate independently and the vibrations can't be divided to simple harmonics because they affect each other, due to "nonlinearities" of the equations governing the vocal cord." this seems to ascribe the inability to divide vibrations into harmonics into equations that govern the vocal cord. While referring to equations may be a useful heuristic, I wonder if you know of the acoustic or physiological explanation for this phenomena? Thanks a lot for the detailed explanation by the way... Aug 21 '15 at 7:19
• Thanks a lot for your interest but I am not sure whether the kind of explanation I would give you would be of the genre you want. It is ultimately a physics question. Almost all equations - for observable quantities in the real world - are ultimately non-linear in character. The linear equations are a "measure zero", infinitely unlikely subset that is only good as idealizations. So the real question isn't whether nonlinearities are there or not. They always exist, the only question is how large they are. Aug 21 '15 at 7:28
• An equation that would easily produce a frequency and higher harmonics would have to be linear, like $(\partial_t^2 - \partial_x^2) p = 0$, the wave equation. But in the real case, there are terms of order $p^2$ as well, and moreover, the vocal cord is not really one-dimensional but 3-dimensional, so one has to consider all the possible frequency eigenvalues and they're not multiples of a fundamental one - even if the nonlinearities were absent. Aug 21 '15 at 7:30
• It's a matter of applied mathematics to understand the nonlinearities, their origin, and their consequences. See e.g. this paper robots.ox.ac.uk/~sjrob/Pubs/nolisp05.pdf - the nonlinearities make the dependence of pressure p on time complicated. It's a little bit like 3 bodies in Newton's physics. If you imagine the Sun and 2 heavy planets orbiting it, they will act on each other and the motion quickly becomes pretty chaotic and largely unpredictable (the 3-body problem can't be "solved" in terms of "elementary" functions). Aug 21 '15 at 7:31
• The pressure of air as a function of time generated by vocal cords "somewhat" resembles the location (or x-coordinate) of a planet that is orbiting around 2 other massive, mutually interacting objects. It will be similar to a simple orbit but in detail, it won't be quite periodic. Aug 21 '15 at 7:34