Linear scaling of vowel-formant ensembles (VFEs) in consonantal contexts

Abstract

There are familiar terms such as “contour” and “trajectory” to refer to a vowel formant frequency as a function defined on the time axis, but there is no readily understood term for the analogous idea of how a formant behaves on the “vowel axis”. For this we introduce the concept of a vowel-formant ensemble (VFE) as the set of values realized for a given formant (e.g., F₂) in going from vowel to vowel among a speaker's vowel phonemes for a fixed time frame in a fixed CVC context. The VFE affords a simple description of our development: we observe that D.J. Broad and F. Clermont's [J. Acoust. Soc. Am. 81 (1987) 155] formant-contour model is a linear function of its vowel target and that as a consequence all its VFEs for a given speaker and formant number are linearly scaled copies of one another. Are VFEs in actual speech also linearly scaled? To show how this question can be addressed, we use F₁ and F₂ data on one male speaker's productions of 7 Australian English vowels in 7 CVd contexts, with each CVd repeated 5 times. Our hypothesized scaling relation gives a remarkably good fit to these data, with a residual rms error of only about 14 Hz for either formant after discounting random variations among repetitions. The linear scaling implies a type of normalization for context which shrinks the intra-vowel scatter in the F₁F₂ plane. VFE scaling is also a new tool which should be useful for showing how contextual effects vary over the duration of the syllable's vocalic nucleus.

Résumé

“Contour” et “trajectoire” sont devenus des termes familiers qui, pour toute voyelle, servent à décrire, sur l'axe des temps, l'évolution des fréquences propres à chacun des formants. Par contre, il y aurait lieu d'établir des vocables analogues permettant de préciser le profil de ces fréquences sur “l'axe des voyelles”. On introduit, donc, le concept de vowel-formant ensemble (et l'acronyme VFE qui en découle) afin de pouvoir regrouper, de voyelle à voyelle, les fréquences d'un formant (e.g., F₂) qui sont obtenues, à un instant fixe de l'axe des temps, pour le même locuteur et dans le même contexte syllabique CVC. Notons que le concept de VFE contient à lui seul toute la démarche adoptée ici, à savoir que notre modélisation précédente des trajectoires des formants (Broad et Clermont, J. Acoust. Soc. Am. 81, 1987, 155–165) repose sur une fonction linéaire de la cible des voyelles et, de ce fait, suggère l'hypothèse que des relations linéaires devraient aussi servir à caractériser les VFEs propres à un locuteur et chacun des formants à la fois. De telles relations sont-elles vérifiables sur des échantillons réels de parole? On aborde cette question pour les fréquences des formants F₁ et F₂ de 7 voyelles de l'anglais australien qui ont été prononcées par un locuteur masculin, 5 fois de suite, dans 7 contextes syllabiques du type CVd. Nonobstant les variations aléatoires inhérentes aux 5 répétitions, l'application de notre hypothèse aux voyelles en question engendre un écart quadratique moyen qui ne dépasse pas la valeur, remarquablement faible, de 14 Hz pour F₁ et F₂. Les relations linéaires ainsi obtenues se prêtent à une normalisation par rapport au facteur contexte, que l'on démontre par une réduction de la dispersion intra-voyelle dans l'espace planaire F₁F₂. Les relations dérivées du concept de VFE constituent également un nouvel outil devant permettre la mise en évidence des effets de différents contextes au travers des noyaux vocaliques de syllabes.

Introduction

A goal in the study of speech communication is to better understand the link between the continuous stream of physical events in speech and the corresponding discrete sequence of phonetic units. One of the problems we face in establishing this link at the acoustic level is that formant contours for vowels in context consist mainly of transitions and have steady states that are only fleetingly realized. At any instant the formants are functions not only of the current vowel, but also of its preceding and following contexts.

Fig. 1 illustrates what happens to a formant (such as F₂) for a set of monophthongal vowels in some fixed consonant–vowel–consonant (CVC) context (bVd, for example). Just as the contour for each individual vowel makes its transition from the initial consonant to the syllable center and then its transition to the following consonant, the set of contours taken as a whole starts from a relatively compressed pattern at the initial-consonant boundary, moves to a more widely spaced pattern in the syllable center and then becomes more compressed again as it approaches the final consonant.

It is this systematic variation in the intervowel spacing for vowels in CVC context that is the topic of this paper and our hypothesis for it will be more easily stated if we first define a new concept, that of the vowel-formant ensemble.

As just described, Fig. 1 shows variations of a given formant along the two dimensions of time and vowel category for a fixed CVC context. To look at the variations with time while keeping the vowel fixed amounts to selecting a vowel in the figure and following the course of its formant trajectory from beginning to end. For such variation along the time axis we have readily understood terms such as “contour”, “trajectory” and “transition”, terms for which the notion of the time axis is already implicit.

But as illustrated by the vertical line in the figure, we can also look at how the different vowels are distributed for some fixed frame (relative time position in the syllable). Unfortunately, we have no readily understood term for this idea of how the formant varies on the “vowel axis”. In the absence of a ready-made term for this, we now introduce the concept of the vowel-formant ensemble (VFE) by which we mean the set of formant frequencies realized for the set of vowels for the given frame and context. In the figure the vertical slice represents a vowel-formant ensemble.

In this paper our focus will be on the vowel-formant ensemble rather than on individual vowel contours, particularly on how the ensembles for different time frames and contexts are scaled in relation to one another.

The hypothesis we explore is the simplest scaling relation we can imagine, namely, that for a fixed formant (such as F₂) all a speaker's VFEs will be linearly scaled copies of one another across CVC contexts and relative time frames in the syllable, i.e., that all these VFEs will be geometrically similar to one another.

Our approach is to first motivate the hypothesis by showing how our earlier time-domain model for formant contours (Broad and Clermont, 1987; cited below as BC87) predicts the linear scaling of VFEs and then to show how this prediction can be tested.

The hypothesis itself is illustrated in Fig. 2 where the top two panels show families of contours for the same formant and the same set of vowels but in two different CVC contexts. A frame is selected from each context and its corresponding vowel-formant ensemble is marked by a vertical line. These ensembles from the two contexts are transferred to the bottom panel, which has the same vertical scale (representing formant frequency) as the top two plots. The horizontal arrangement of the ensembles in the bottom panel is arbitrary, and is planned simply to fit the picture to reasonable proportions. The fact that the vowel placements subdivide the two ensembles in the same proportions, i.e., that the ensembles are geometrically similar to each other, is shown by the fact that the lines connecting identical vowels in the two ensembles all intersect at the same point. (The location of this point in and of itself is meaningless, as can be seen by how it could be moved around by adjusting the arbitrary horizontal placement of the two ensembles.) That the ensemble from Context A and the one from Context B are selected arbitrarily illustrates our hypothesis: that all pairs of a speaker's vowel-formant ensembles for a given formant will be similar to each other, i.e., all these VFEs will be linearly scaled copies of one another across contexts and time frames.

The vowel-formant contours in the different contexts in Fig. 2 have different shapes and different displacements. As the diagram suggests, however, these complexities in the individual contours may give way to a simpler pattern when our point of view shifts from the time axis to the vowel-formant ensemble.

In Section 2 we start from one of the time-domain models we developed in BC87. It embodies the simple properties of (1) additivity of effects from initial and final consonants, (2) per-consonant similarity of transition shapes, and (3) scaling of transitions by the differences between vowel targets and consonant loci. The model involves parameters and functions which are indexed by the vowel and by the initial and final consonants. The only element of the model that depends on the vowel category is the vowel target, which enters the model only as a first-order factor. These properties imply that within a given formant number (such as F₂) the vowel-formant ensemble from any frame and any context in the model is a linearly scaled version of the model's target ensemble and that as a result all the VFEs for the given formant in the model are linearly scaled versions of one another.

We finish Section 2 by showing how this theoretical result can be connected to data through some simple numerical operations. F₁ and F₂ data on a single speaker's vowels spoken in different CVd contexts are used in Section 3 to illustrate the linear scaling relation and test it statistically using interrepetition variation as a baseline. For this example dataset the hypothesized linear scaling of its VFEs cannot be confirmed in the strictest statistical sense, but its implied rms departure from linearity is only about 14 Hz for each formant. Therefore the F₁ and F₂ VFEs for this dataset are only nearly linearly scaled. If not strictly true, then our hypothesis of linear scaling will be seen to be a remarkably good approximation for this dataset.

In Section 3 we also adapt the linear scaling relation to a form of normalization of vowel-formant ensembles for context and show how vowels become better separated on an F₁F₂ plot. Limitations and applications of the scaling relation are discussed in Section 4 while our conclusions are summarized in Section 5.

It is noteworthy that none of the data operations for analyzing the linear scaling relation require the estimation of any of the parameters or functions that make up the original formant-contour model. This follows from the fact that the result arises from the linear structure of the model and not from any specific implementation of it. As a consequence, the linear scaling of VFEs can be studied as a phenomenon in its own right without reference to the model which predicted it.