Mathematical model for the vibrato

As her PhD defended at the Public University of Navarre, telecommunications engineer Ixone Arroabarren has analysed the vibrato, one of the most important tools of classical singers.

The study applies both to the teaching of singing in music as well as to the medical treatment of voice pathologies. It has put forward a mathematical model for the production of the voice that can be used both in the medical study/detection of pathologies of the vocal chords and speech as well as the teaching of the art of singing. This PhD has been developed within the framework of the research project awarded by CEIN as the best Project for the Transference of Research Result.

Amongst these differences, the engineer points out, that the vibrato is an exclusively musical characteristic which is of great use to the classical singer because, on the one hand, it enables the unmasking of possible off-key notes and, on the other, it makes sure the listener does not have the sensation that they are listening to the same sound. Furthermore, the theme itself has been of great interest to many researchers in areas such as physiology and musicology.

From an acoustic perspective, the vibrato in classical singing can be defined as a regular fluctuation in the fundamental frequency of the pitch or signal, the timbre and/or the volume of a single note. Nevertheless, the origin of these variations and their relation with mechanisms of voice production are still enigmatic.

Ixone Arroabarren’s thesis studies this theme in depth with aim of carrying out a complete characterisation of the vibrato in the art of classical singing, starting from the measuring and the performance of its most relevant acoustic characteristics, and ending with an analysis of its origin and relation to the mechanisms of voice production. In brief, what we are doing is to relate what we perceive acoustically what is generated physiologically. In this way, we offer an explanation of the collateral effects which we knew were there but the exact origin of which was unknown.

To carry out this study a number of Signal Processing tools have been used – “the most suitable in each case, given that the overall study of the vibrato has implicitly involved the resolution of very different problems, from calculating the instantaneous frequency of non-stationary signals to estimating the source by means of Inverse Filtering.

As an end result of the researcher’s study, she puts forward a mathematical model of voice production that can be used both for the study and medical treatment of vocal chord and speech pathologies as well as for learning the art of singing.

This model of vibrato production has permitted relating the most important acoustic characteristics – fundamental frequency, timbre and volume, with the most relevant elements in voice production at the level of acoustics, glottal source and response of the vocal tract. In this way we have demonstrated that the features of both elements do not show substantial changes during vibrato, only the fundamental frequency of the glottal excitation varying.

All this enables two models of signal production of the vibrato to be put forward. A Non-Interactive Model of Vibrato Production, has enabled relating the most important acoustic characteristics – variations in fundamental frequency, timbre and volume, with and Response of Vocal Tract elements in voice production. With this it has been shown that variations in fundamental frequency generated in the Glottal Source are the cause of the variations in timbre and volume, dependant on both elements of voice production.

Besides, there is an Interactive Model of Vibrato Production, which enables us to state that the variations in amplitude and frequency of the harmonics of the acoustic signal can be used to obtain more information about the mechanisms of voice production. Moreover, this model admits the inclusion of additional effects, such as synchronic variations of the Response of the Vocal Tract, which may be related to similar effects identified by other authors through physiological studies.