Proceedings Fonetik 2009 - Institutionen för lingvistik

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Proceedings, FONETIK 2009, Dept. of Linguistics, Stockholm UniversityFor each sample the time courses of F1, F2 andF3 were traced by clicking the mouse along theformant tracks. Swell automatically produced atwo-column table with the sample’s time andfrequency values.The value of α was determined after rearrangingand generalizing Eq (1) as follows:(F n (t) - F nT )/(F nL - F nT ) = e -αt (3)and taking the natural logarithm of both sideswhich produces:ln[(F n (t) - F nT )/(F nL - F nT )] = -αt (4)Eq (4) suggests that, by plotting the logarithmof the Fn(t) data – normalized to vary between1 and zero – against time, a linear cluster of datapoints would be obtained (provided that thetransition is exponential).A straight line fitted to the points so that itruns through the origin would have a slope of α.This procedure is illustrated in Figure 2.Figure 3. Measured data for 5 repetitions of [da](black dots) produced by male speaker. In red: Exponentialcurves derived .from the average formantspecificvalues of locus and target frequencies andtime constants.ResultsHigh r squared scores were observed (r2>0.90)indicating that exponential curves were goodapproximations to the formant transitions.Figure 2. Normalized formant transition: Top: linearscale running between 1.0 and zero; (Bottom):Same data on logarithmic scale. The straight-linepattern of the data points allows us to compute theslope of the line. This slope determines the value ofthe time constant.Figure 3 gives a representative example of howwell the exponential model fits the data. Itshows the formant transitions in [da]. Measurementsfrom 5 repetitions of this syllablewere pooled for F1, F2 and F3. Time constantswere determined and plugged into the formantequations to generate the predicted formanttracks (shown in red).Figure 4. Formant time constants in V:CV andVC:V words plotted as a function of formant frequency(kHz). F1 (open triangles), F2 (squares) andF3 (circles). Each data point is the value derivedfrom five repetitions.The overall patterning of the time constants isillustrated in Figure 4. The diagram plots timeconstant values against frequency in all V:CVand VC:V words. Each data point is the valuederived from five repetitions by a single maketalker. Note that, since decaying exponentialsare used, time constants come out as negativenumbers and all data points end up below thezero line.20
Proceedings, FONETIK 2009, Dept. of Linguistics, Stockholm UniversityF1 shows the highest negative values and thelargest range of variation. F2 and F3 are seen tooccupy a limited range forming a horizontalpattern independent of frequency.A detailed analysis of the F1 transition suggestspreliminarily that VC transitions tend tobe somewhat faster than CV transitions; VC:data show larger values than VC measurements.Figure 5.Vowel duration (left y-axis) and .F2 timeconstants (right y-axis) plotted as a function ofnumber of syllables per word.Figure 5 shows how the duration of the vowel[ɑ:] in [dɑ:(g)-] varies with word length. Usingthe y-axis on the left we see that the duration ofthe stressed vowel decreases as a function ofthe number of syllables that follow. This compressioneffect implies that the ‘speaking rateincreases with word length.The time constant for F2 is plotted along theright ordinate. The quasi-horizontal pattern ofthe open square symbols indicates that timeconstant values are not influenced by the rateincrease.DiscussionNon-linear acoustic mappingIt is important to point out that the proposedmeasure can only give us an indirect estimate ofarticulatory activity. One reason is the nonlinearrelationship between articulation andacoustics which for identical articulatorymovement speeds could give rise to differenttime constant values.The non-linear mapping is evident in thehigh negative numbers observed for F1. Do weconclude that the articulators controlling F1(primarily jaw opening and closing) move fasterthan those tuning F2 (the tongue front-backmotions)? The answer is no.Studies of the relation between articulationand acoustics (Fant 1960) tell us that rapid F1changes are to be expected when the vocal tractgeometry changes from a complete stop closureto a more open vowel-like configuration. Suchabrupt frequency shifts exemplify the nonlinearnature of the relation between articulationand acoustics. Quantal jumps of this kind lie atthe heart of the Quantal Theory of Speech (Stevens1989). Drastic non-linear increases canalso occur in other formants but do not necessarilyindicate faster movements.Such observations may at first appear tomake the present method less attractive. On theother hand, we should bear in mind that thetransformation from articulation to acoustics isa physical process that constrains both normaland disordered speech production. Accordingly,if identical speech samples are compared itshould nonetheless be possible to draw validconclusions about differences in articulation.Figure 6: Same data as in Figure 4. Abscissa: Extentof F1, F2 or F3 transition (‘locus’–‘target’ distance).Ordinate: Average formant frequency rate ofchange during the first 15 msec of the transition.Formant frequency rates of change arepredictable from transition extents.As evident from the equations the determinationof time constants involves a normalizationthat makes them independent of the extent ofthe transition. The time constant does not sayanything about the raw formant frequency rateof change in kHz/seconds. However, the dataon formant onsets and targets and time constantsallow us to derive estimates of that dimensionby inserting the measured values intoEqs (1) and (2) and calculating ∆Fn/∆t at transitiononsets for a time window of ∆t=15 milliseconds.21
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