In the two-mass model control of the fundamental frequency can be exercised in two ways. The first approach is to add a "tension parameter" Q (Ishizaka & Flanagan 1972:1279), the second is to correct the vocal folds stiffness directly and according to muscular action.
In the first method, the mass and thickness of the folds are scaled down by the factor Q, whereas the tensions k are scaled up, "causing the fundamental frequency to vary proportionally with Q. Phase difference, duty ratio, and the glottal area waveforms are essentially uninfluenced by Q, and the amplitudes of glottal area and glottal flow decrease gradually with increasing Q" (Ishizaka & Flanagan, 1972:1259). As is known from several experiments the Open Quotient increases with the increase in fundamental frequency. This effect has been included in the model by modification of the coupling tension kc. The kc variation was empirically determined to be proportional to Q2. The Q factor is related to the stretching of the vocal folds (by contraction of the cricothyroid muscle) and contracting the vocalis muscle (see section 27.3 for details). The modifications of the model which are closely related to Q are in accordance with the experimental measuremnts: "Variation of the duty ratio with frequency falls into the range measured in inverse filtering experiments" (Ishizaka and Flanagan, 1972:1260).
The simulation of the EGG waveform for a varying Q factor was performed for neutral /e:/ with Q varying from 0.8 to 1.4 in 0.1 steps and other parameters remaining unchanged (exactly as in section 27.1). In particular, the subglottal pressure Ps was set to 8 cm H2O. The increase in the Q factor leads to an increase in F0 from 115 to 190 Hz. The simulated EGG waveforms exhibit the expected dependencies and correlate well with the modelled glottal airflow.
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The relations between the glottal flow Open Quotient and the Open Quotient of the EGG are depicted in Fig. 52. The vocal folds open longer when the muscular tension increases. The same tendency can be observed in the EGG waveform, however, the results by the second measure of the Open Quotient (section 12) are strongly related to the opening and closing angles of the folds. If the closing angle thetac and the opening angle thetao are relatively (very relatively) wide (thetac > 1º and thetao > 2.5º), a complementary effect is observed, i.e. the EGG duty ratio (especially OQII) increases with increasing muscular tension. This should not be overestimated; it is to be expected that higher muscular tension causes rather narrower angles of the folds opening and closing. However, the appropriate physiological measurements are not given in the literature.
The skewness of the modelled glottal airflow pulse decreases with increased pitch as can be see in Fig. 50a and Fig.53a. This is in accordance with the results of the glottal modelling of Ishizaka and Flanagan (1972), the body-cover models of Titze (1992, 1994) and the inverse filtering measurements of Koreman (1996:157-159), but contradictory to the results of Pierrehumbert's inverse filtering (1989) and Sluijter's speech signal measurements (1995:125). Pierrehumbert (1989) explained the difference between the experimental outcome and the two-mass model with the increase in the coupling stiffness kc between the vibrating masses, which in turn is related to the tension in the vocalis muscle (thyroarytenoid muscle). The experiments conducted with the two-mass model do not confirm this explanation. In the two-mass model the stiffness kc changes with Q2 and the skewness of the glottal pulse decreases with the increase of F0 and OQ. A better explanation for the effect is the simultaneous action of two sources: the contraction of muscles and the increased subglottal pressure. Higher muscular tension effects the increase in the glottal airflow duty ratio, while the higher skewness of the glottal pulse is caused by an increased subglottal pressure. The increase of the subglottal pressure has already been correlated with the linguistic marking of word stress, while muscular tension has been anticipated to correlate with the prosodic function of sentence intonation. This hypothesis is in accordance with Pierrehumbert's results (1989), because the examined intonation contours (F0 changes) are superimposed on the word stress.
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Other results of the EGG modelling support this hypothesis, too. As shown in Fig. 54b, the steepness of the contact rise and fall does not depend on the increased tension of the vocal folds. This is in agreement with the outcome our experiments (chapter III), where the influence of intonation patterns on vowels in stressed positions was measured. This allows us to formulate the thesis that the increase in subglottal pressure is positively correlated with the steepness of the EGG pulse flanks, whereas the rise of the duty ratio of the EGG waveform depends rather on the increase in muscular tension (increased stiffness of the vocal folds). Both mechanisms also cause a rise of the fundamental frequency.
The second type of vocal folds tension modification follows the body-cover model (Titze, 1994) and provides a more detailed view of the physiology of fundamental frequency control. Pierrhumbert's (1989) remark that "in the mid to high region of the pitch range, the dominant mechanism for F0 control appears to be cricothyroid contraction and relaxation" can be applied to the model. Generally, the stiffness of the vocal folds is controlled mainly by the actions of the cricothyroid and thyroarytenoid muscles (Titze, 1994:202-209). The theory of the F0 control mechanism introduced by Titze (1988, 1994) explains F0 variations in terms of the body-cover model of the vocal folds. The F0 changes in modal phonation with normal voice effort are primarily caused by the action of the thyroarytenoid muscle. The muscle can stiffen the body of the vocal folds while slackening the cover. That is why the F0 changes depend on the type of vibrating tissue, because the vibrating tissue is primarily muscular in the cross-section (the body). It is to be expected that the effective stiffness of the vibrating mass will rise due to the thyroarytenoid contraction. If, however, the non-muscular part of the folds (the cover) vibrates, then the effective stiffness will presumably be lowered because of the contraction of the thyroarytenoid muscle (Titze, 1994:206). This causes positive or negative changes in F0 depending on the type of vibrating tissue. Titze (1994:206) relates the changes in the fundamental frequency to the relative thickness of that part of the muscular layer which is vibrating:
where da/d is the ratio of the depth of the TA muscle in vibration to the total depth in vibration, sigmaam denotes the maximum possible stress of the thyroarytenoid, sigmap is the effective passive stress of all tissue in vibration (the stress during the total inactivity of the thyroarytenoid). Lm is the length of the mucous part of the folds, rho is the tissue density (1.04 kg/m3) and aTA represents the activity of the thyroarytenoid muscle. Titze (1994:208) gives the average dependency of F0 on muscle activity and notes that in speech production "cricothyroid and thyroarytenoid activity are both relatively low (<50 percent of maximum), and changes in F0 with increased thyroarytenoid activity are all positive, in agreement with average correlations". It can be concluded that in the normal speech phonation modus, the rise in F0 results from higher thyroarytenoid activity while cricothyroid activity is low.
Muscle activities and their influence on fundamental frequency are usually modelled in the continuous model (Berry & Titze, 1996; Titze & Talkin, 1979) or the 3-mass model (Story & Titze, 1995). However, Lucero (1993, 1996) shows that the different mechanical properties of body and cover can be modelled even using the standard configuration of the two-mass model. Taking his simplification into consideration, the stiffness of the vibrating masses m1 and m2 then depends on the contraction of the cricothyroid and thyroarytenoid muscles, in the sense that their action can be regarded as additive in the lower mass and contrary (differential) in the upper mass. Stiffness is then expressed as k1=CT+TA and k2=CT-TA, where CT and TA represent the weighted activities of the cricothyroid and thyroarytenoid muscles (Lucero, 1996). During modal phonation, TA is relatively high and CT relatively low, which results in high values for k1 and low values for k2. k1 resembles the kb parameter of the body-cover model, in that "increasing of the body stiffness (i.e. contracting the TA) leads to lower body layer amplitude and higher pitches" (Story & Titze, 1996:1256).
These relations are also included
in our model, as the influence of muscular tension on the simulated EGG waveform
is investigated. As before, the model settings are kept constant,
while only the tension parameters k1, k2 and kc
are modified. Neutral [
] is modelled and the parameters
are set to the values given in section 27.1. In particular, the opening and
closing angles were set to thetao = 1.0º and
thetac = 0.5º.
The tensions are set to the following values:
| k1 (kdyn) | k2 (dyn) | kc (kdyn) |
|---|---|---|
The results of the simulation are depicted in Fig. 54a and b. Generally, the results are similar to those produced by Q factor modelling. The simulated increased tension of the cricothyroid and thyroarytenoid muscles causes the increase of the Open Quotient in both Ug and EGG waveforms. The steepness of the EGG slopes remains almost unchanged, whereas the skewness of the glottal flow decreases.
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Titze (1990) has formulated the dependencies of the contact area on the motion of the folds6. Compared to the nominal (typical) vocal folds configuration, the simulated EGG pulses are
6 Titze's conclusions are based on the continuous model of vocal folds motion.
