2.1 Six kinds of auditory dispersion effects

In Fig. 1 we see examples of how phonological categories can be dispersed along one-dimensional auditory continua. This figure helps us to illustrate the six kinds of dispersion effects. Most languages tend to be dispersed like the cases we discuss here; cases of real or apparent undispersed inventories are discussed at the end of this section.

Figure 1

Observed auditory dispersion.

2.2 Non-OT accounts of auditory dispersion

Auditory dispersion in vowel inventories was first modelled explicitly by Liljencrants & Lindblom (Reference Liljencrants and Lindblom1972), who showed that two-dimensional vowel spaces (auditory height and backness) with a minimum probability of perceptual confusion look like real attested vowel inventories (some problems of detail were addressed by Lindblom Reference Lindblom, Ohala and Jaeger1986 and Schwartz et al. Reference Schwartz, Boë, Vallée and Abry1997).

Ten Bosch (Reference Bosch1991) compared several techniques for optimising the distances between vowels, and found that the strategy of maximising the minimal distance worked best. If we start out with a random set of vowels and then iteratively move apart the two vowels that are closest to each other, we ultimately obtain an evenly dispersed inventory.

A possible criticism of this work is that the resulting vowel inventories are not symmetric enough. For instance, real vowel inventories tend to be structured in such a way that back vowels tend to have the same height contour as front vowels, yet there is no force in these models according to which such symmetries are enforced, and indeed the vowel inventories that result from the simulations of these authors tend to be asymmetric (Boersma Reference Boersma1998: 357). To remedy this problem, the simulations would have to be extended with tricks to incorporate an efficient use of available auditory, articulatory and/or phonological features.

A problem that is much more difficult to remedy is the inherent teleology in these models. In order to arrive at an optimal vowel inventory, a model typically starts with a random non-optimal vowel inventory, and tries to move towards an optimal end result by making small changes to the locations of the vowels in the vowel space, where every change has to be optimising, i.e. every change has to improve the auditory distinctiveness of the whole system. Even at the most concrete level of modelling, then, these models work with teleological devices.

Non-teleological accounts of auditory dispersion have been proposed as well. Blevins (Reference Blevins2004: 285–289) sketches how within a framework where listeners store auditory events as exemplars in episodic memory and subsequently reuse these exemplars in production, following Pierrehumbert (Reference Pierrehumbert, Bybee and Hopper2001), speakers may tend to choose exemplars that are little likely to be perceived as anything but the intended category. An explicit account of the details of such a scheme is provided by Wedel (Reference Wedel2004, Reference Wedel2006). We discuss this in detail in §7.2.

Few of the non-OT accounts mentioned in this section make contact with phonological theory by modelling the interaction of the simulated inventories with phonological rules or constraints. Only exemplar models are likely to provide such a link: it seems perfectly possible to combine Wedel's (Reference Wedel2006) dispersion model with his lexical analogy model (Wedel Reference Wedel2007), which, for example, implements violable metrical constraints as potentially conflicting biases that lead to the emergence of Gordon's (Reference Gordon1999) frequency-informed regular patterns of weight-stress interaction. In the next section we discuss attempts to integrate the dispersion idea into a tried and tested framework for phonological theory, namely Optimality Theory.

2.3 OT accounts of auditory dispersion

The original proposal of Optimality Theory by Prince & Smolensky (Reference Prince and Smolensky1993) handled inventories by constraint interaction and the device of ‘richness of the base’. As in the featural markedness accounts mentioned above, a marked phonological element was only allowed to surface in a language if the unmarked counterpart of that phonological element also surfaced in that language. Prince & Smolensky's approach is therefore problematic for the same reason as mentioned earlier: it cannot account for the ‘excluded centre’ effect, where marked segments appear without their unmarked counterparts, as noted by Flemming (1995: 37, Reference Flemming2004: 235, Reference Flemming, Pisoni and Remez2005: 164, Reference Flemming2006: 250). So an OT account requires something more than just markedness and faithfulness constraints.

Flemming (Reference Flemming1995) translated Lindblom's dispersion idea, and specifically ten Bosch's idea of maximising the minimum distance, into OT by introducing MinDist constraints, which explicitly militate against inventories with small auditory distances between its members. Dispersion constraints such as Flemming's MinDist, as well as the reformulations by Padgett (Reference Padgett, Hume and Johnson2001) and Sanders (Reference Sanders2003), work very well in formalising the dispersion idea. However, they are explicitly teleological with respect to auditory dispersion. Furthermore, an empirical problem is that these constraints evaluate multiple inputs at a time: they can be said to evaluate whole inventories (Flemming 1995: 33–35, Boersma Reference Boersma1998: 361, McCarthy Reference McCarthy2002: 226–227) or even entire languages (Padgett Reference Padgett2003a: 311, Flemming Reference Flemming2004: 268). These constraints are therefore hard to reconcile with the single-input constraints introduced by Prince & Smolensky, and the tableaux are hard to reconcile with tableaux that basically evaluate the processing of a single form in production (Prince & Smolensky Reference Prince and Smolensky1993) or comprehension (Smolensky Reference Smolensky1996). Flemming's and Padgett's general defence is that phonological theory is about possible languages, rather than about processing single forms. We feel that this standpoint underestimates the power of Optimality Theory as a decision mechanism: when used to evaluate single inputs, OT can be and has been applied successfully to processes such as production, comprehension and acquisition. If dispersion effects can be shown to emerge in OT even from modelling single-form processing, OT will not have to be invoked separately to evaluate the entire language.

The fact that Flemming, Padgett and Sanders handle dispersion effects by dedicated inherently teleological means (the dispersion constraints) in a synchronic grammar is sometimes seen as unproblematic (e.g. Hayes & Steriade Reference Hayes and Steriade2004: 27), but a theory in which these effects arise automatically as side effects of more general independently needed devices should be preferred by Occam's razor, if such a theory exists. Padgett (Reference Padgett2003b: 80) realises this shortcoming and suggests that dispersion constraints may just express abstract observations about inventories while at the same time the real underlying mechanism may be more concrete and perhaps not explicitly goal-oriented. We agree, and in the present paper we provide just such an underlying mechanism within OT (in §7.6 we show that it also works in other constraint-based frameworks). Providing the underlying mechanism is necessary because we seek the locus of explanation in an acquisition bias on the part of the learner; formalising acquisition has to be done with the constraints that are in the learner's brain, not with constraints that describe behaviour at a higher level of abstraction.

In the present paper, we employ neither dispersion nor faithfulness constraints to account for dispersion effects. We show that dispersion effects can instead arise within a number of generations as the automatic result of cue constraints, which are independently needed to model language-specific perception (e.g. Escudero & Boersma Reference Escudero and Boersma2004), and articulatory constraints, which are independently needed to model articulatory effort in phonetic implementation (e.g. Boersma Reference Boersma1998, Kirchner Reference Kirchner1998). The only assumption that we need to add to the pre-existing work on OT phonetics is that the speaker and the listener use the same grammar. The point is that the same constraints are used bidirectionally, i.e. both by the listener in comprehension and by the speaker in production. The following section illustrates this in more detail.

5.1 The English auditory language environment

For simplicity we assume here that the auditory difference between the two English sibilants is wholly caused by a difference in their spectral means. The spectral mean of /s/, for example, will vary between speakers, vowel environments and acoustic and auditory conditions, as well as between replications by the same speaker. A listener will be able to normalise away some of this variation, but not all of it. Since the non-normalisable variation has multiple sources, the remaining distribution of spectral means for all the /s/ tokens that a listener hears is likely to have a bell-like shape, perhaps similar to the Gaussian shape in Fig. 6. We fix the average spectral mean of all normalised /s/ tokens in the listener's language environment at a slightly arbitrary value of 30·5 Erb (6656 Hz). Likewise, we assume for /ʃ/ a similar Gaussian distribution with an average spectral mean of 25·5 Erb (3478 Hz).Footnote ⁸ For both bell-shaped curves in Fig. 6 we choose a characteristic width (standard deviation) of 1·13 Erb.

Figure 6

Distributions for tokens of the English sibilants /ʃ/ and /s/ along the spectral mean.

The two distributions in Fig. 6, then, are the material that an adult listener has to work with, and that an infant learner has to make sense of. We assume somewhat arbitrarily that relevant spectral mean values for sibilants universally lie between 20·0 Erb (1789 Hz) and 36·0 Erb (16177 Hz).

5.2 The adult as an optimal listener

We explicitly assume that adult English listeners have learned to minimise the probability of misunderstanding, i.e. that they have become optimal listeners. To minimise the probability of misunderstanding, the optimal listener must show maximum-likelihood behaviour (Fisher Reference Fisher1922), i.e. she must classify an auditory event as the category that is most likely to have been intended by the speaker. For sibilants, the optimal English listener's decision criterion must be the frequency where the two curves in Fig. 6 intersect: she must perceive all tokens with a spectral mean below 28·0 Erb (4765 Hz) as /ʃ/, and all tokens above 28·0 Erb as /s/.

To model this correlation of spectral mean values and surface categories, we follow Escudero & Boersma (Reference Escudero and Boersma2004) in using a complete set of arbitrary cue constraints, which connect ‘all’ spectral mean values between 20·0 and 36·0 Erb to both sibilant categories. The constraints are summarised in (1).

1. (1)

   

For perception, the constraint *[20·0 Erb]/s/ can be read as ‘an auditory spectral mean of [20·0 Erb] should not be perceived as the surface phonological category /s/’; for phonetic implementation, the constraint can be read as ‘a surface form /s/ should not be realised with a spectral mean of [20·0 Erb]’. For simplification, we discretise the spectral mean range into 161 steps of 0·1 Erb. Although the cochlea, the auditory nerve and the brain perform much more granular discretisations, a granularity of 0·1 Erb is fine enough to oversample any of the three local effects that we will discuss (see §§5.3 and 7.5 for arguments and proof that this discretisation is legitimate and consistent). Combining these 161 spectral mean values with the two sibilant categories results in a total of 322 cue constraints.

In connecting every possible spectral mean frequency between 20·0 Erb and 36·0 Erb to both sibilant categories, the cue constraints in (1) are very different from usual OT constraints, which either tend to express typological trends directly (e.g. markedness constraints) or tend to express universally preferred relations (e.g. faithfulness constraints). That is, the set in (1) has no preference for certain spectral mean values, for certain sibilants, or for connecting certain spectral mean values to certain sibilants, i.e. the constraints are not restricted to actually occurring spectral mean values or to actually occurring combinations of spectral means and sibilant categories. It is the ranking of these constraints that will have to be responsible for making the correct connections for English. From Fig. 6 we see, for example, that an optimal listener of English should perceive [26·6 Erb] as /ʃ/. A possible ranking that achieves this perception is given in the perception tableau in (2).

1. (2)

   

In tableau (2) we have given all possible candidates (only two, because we only consider the English sibilants), but we have restricted ourselves, for reasons of space, to only six of the 322 cue constraints. We can do this because 320 of our constraints have no preference for or against perceiving [26·6 Erb] as /s/ or /ʃ/. Tableau (2) illustrates that the correct classification of [26·6 Erb] relies solely on the ranking *[26·6]/s/≫*[26·6]/ʃ/. In the same way, we can establish rankings for the 160 remaining spectral mean values, e.g. *[31·4]/ʃ/≫*[31·4]/s/; the constraint *[28·0]/s/ will be ranked at approximately the same height as *[28·0]/ʃ/. We have now shown that the 322 constraints can be ranked in such a way that we can model an optimal listener for the distributions of Fig. 6.

However, such a language-specific ranking of 322 constraints is rather uninformative when it comes to predicting what kinds of sibilant-categorisation strategies are possible in the languages of the world. For one thing, the set of 322 constraints seems to generate an incorrect factorial typology: the traditional OT idea of factorial typology would predict the existence of languages with completely undispersed categories, and even the existence of categories with massively non-contiguous auditory correlates.Footnote ⁹ Such languages have not been shown to exist. Apparently, the idea of factorial typology must be incomplete, if cue constraints are to be the correct way to model perception. A general explanation for apparent gaps in the factorial typology is the idea that if UG allows grammars that are not attested in reality, then those grammars might be unlearnable or unstable over the generations (Boersma Reference Boersma2003b). We will therefore model both the acquisition and the evolution of these grammars, and show that undispersed and discontinuous categories are unstable over multiple generations of learners. The following subsection starts by illustrating how the acquisition of English sibilants is modelled with a simple learning procedure and algorithm.

5.3 Learning to become an optimal listener of English

We describe here the situation when a child already has correct lexical representations, but not yet an adult-like prelexical perception. That is, she already knows which lexical items have underlying |ʃ| and which have underlying |s|, but she does not know in the adult-like way of §5.2 what spectral mean values occur with which of the two surface sibilants.

During this acquisition period, the child will receive many tokens of /ʃ/ and /s/ drawn from the distributions in Fig. 6. Sometimes she will make a mistake, as in (3), which occurs when an adult speaker talking to the child pronounces an intended |ʃ| with a reasonable spectral mean of 26·6 Erb.

1. (3)

   

At this arbitrarily chosen point during her acquisition period, the child of tableau (3) happens to have the non-optimal ranking *[26·6]/ʃ/≫*[26·6]/s/ and therefore perceives the incoming auditory event [26·6 Erb] as the sibilant category /s/, as indicated by the pointing finger. However, the speaker had intended to transmit the lexical symbol |ʃ|, and the semantic and pragmatic context may lead the child's comprehension system to realise this (perhaps because the recognised lexical item was sheep). As a result, the child's lexicon can subsequently act as a ‘teacher’ or ‘supervisor’ and ‘tell’ her that she should have perceived /ʃ/ instead of /s/. This new knowledge by the child is indicated by the check mark in the tableau.

When a perception tableau such as (3) contains the child's own winning form (3a) as well as a form that she considers correct (3b), and the two forms are different, the child can conclude that she has made a mistake. As a result, she can take action by taking a ‘learning step’. A good strategy for executing a learning step is the Gradual Learning Algorithm (Boersma Reference Boersma1997, Boersma & Hayes Reference Boersma and Hayes2001), which is indicated by the arrows in the tableau: all constraints favouring the correct category are moved up, and all constraints favouring the child's own ‘incorrect’ winner are moved down.Footnote ¹⁰ As a result, the two cue constraints for the value [26·6 Erb] will be re-ranked slightly in the direction of the arrows. This process is called lexicon-driven perceptual learning (Boersma Reference Boersma1997, Escudero & Boersma Reference Escudero and Boersma2004).

The arrows in (3) represent small steps along a continuous scale of ranking. After having heard a number of [26·6 Erb] events that should have been perceived as /ʃ/, the learner will have swapped the rankings of *[26·6]/ʃ/ and *[26·6]/s/. From then on, the learner will perceive the auditory form [26·6 Erb] correctly as the category /ʃ/. The same successful learning applies to all other auditory forms for which one of the curves in Fig. 6 is close to zero, i.e. for all forms below approximately [26·7 Erb] or above approximately [29·3 Erb]. It now becomes clear why we called the Gradual Learning Algorithm a ‘good strategy’: the learner has become a maximum-likelihood listener for those auditory values. In the region where the curves in Fig. 6 overlap, something slightly different happens: the listener will necessarily continue to make some mistakes in prelexical perception, simply because e.g. an incoming token of [27·5 Erb] was intended as |ʃ| 75% of the time, but as |s| 25% of the time, as the curves show. In Stochastic Optimality Theory (Boersma Reference Boersma1997, Boersma & Hayes Reference Boersma and Hayes2001), where the ranking of every constraint is subject to a bit of additive noise at evaluation time, the listener is likely to vary her perceptual decisions. The Gradual Learning Algorithm then leads to a situation of probability matching: the learner will end up in an equilibrium situation in which she perceives [27·5 Erb] as /ʃ/ 75% of the time, but as /s/ 25% of the time. This works because the constraint *[27·5]/s/ will end up being ranked just above *[27·5]/ʃ/. This strategy of probability matching in perception, which is nearly as good as the maximum-likelihood strategy, has been shown to emerge automatically from the lexicon-driven learning mechanism both for one-dimensional continua (Boersma Reference Boersma1997) and for two-dimensional continua (Escudero & Boersma Reference Escudero, Boersma, Arunachalam, Kaiser and Williams2003).

There is one technical correction that has to be made. The above description of the learning algorithm involves changing the ranking of only one cue constraint, as if constraints did not influence their neighbours. A more realistic view of the auditory continuum has to take into account the fact that the properties of the basilar membrane in the inner ear are such that when a hair cell at a certain frequency is excited, several hair cells within a frequency distance of about 1 Erb will also be excited (Moore & Glasberg Reference Moore and Glasberg1983). This correlation between the firings of adjacent hair cells is reflected in the tonotopic map in the auditory cortex (Romani et al. Reference Romani, Williamson and Kaufman1982), and presumably in whatever neural device determines the auditory spectral mean. If we acknowledge this mutual correlation between adjacent spectral means and therefore assume a correlation between adjacent cue constraints, the learning algorithm must push bell-shaped dents into the shape of the cue-constraint curves. Following the simulations by Boersma (Reference Boersma1997: 51), we implement these dents as Gaussians with a standard deviation (an ‘effective resolution’) of 0·5 Erb. We have seen, for example, that the ranking of the constraint *[26·6 Erb]/s/ rises by 0·01 in (3). Our vicinity correction now implies that the constraints *[26·5 Erb]/s/ and *[26·7 Erb]/s/ will rise by 0·0098, the constraints *[26·4 Erb]/s/ and *[26·8 Erb]/s/ will rise by 0·0092, and so on; the constraints *[25·6 Erb]/s/ and *[27·6 Erb]/s/ will still rise by 0·01 exp (−0·5(1·0/0·5)²)=0·0014. The non-zero effective resolution guarantees a consistent discretisation of the spectral mean continuum by ensuring that the simulation results will be independent of the granularity as long as the spacing (which is 0·1 Erb in this paper) is at least several times smaller than this resolution (see §7.5).

In order to see exactly what happens in perception, and to be able to predict the effect of lexicon-driven perceptual learning on phonetic implementation, we cannot limit ourselves to the present description of the learning mechanism. Instead, we have to compute what listeners-speakers will do, and since any case involving more than two constraint curves is impossible to solve analytically, a computer simulation is the designated route.

5.4 Simulation of the acquisition of English perception

The learning of an English perception grammar can be simulated by computer. Our virtual learner has the 322 cue constraints in (1), and starts out having them all ranked at the same height of 100·0; as a result, the learner has an initial state where she perceives every incoming spectral mean value as /s/ 50% of the time and as /ʃ/ 50% of the time. At the same time we assume that this initial virtual learner already has correct lexical representations for all words with /s/ and /ʃ/. Although this combination of fully random prelexical perception and perfect lexical storage is obviously unrealistic, a more realistic modelling is very likely to exhibit effects that work in the same directions as the ones we will find here, although their size may differ (see §7.3 for discussion).

During the simulated acquisition period, our learner hears 1 million /ʃ/ and 1 million /s/ tokens in random order and with spectral mean values that are randomly drawn from the distributions in Fig. 6. Each token is also labelled as /s/ or /ʃ/ by the learner's lexicon. When hearing a token, the learner will perceive it into either category. In our simulations we set the standard deviation of Stochastic OT's evaluation noise (per tableau random variation in ranking) to a constant value of 2·0.

It can happen that the learner's perceived category is identical to the category the lexicon tells her she should have perceived. In such a case, the learner does not change her grammar. But if the perceived category is different from the one her lexicon says is correct, our learner changes the ranking of her cue constraints according to the scheme in tableau (3). The Gradual Learning Algorithm is here taken to have a constant plasticity (re-ranking step) of 0·01.

While listening to the 2 million tokens, our learner's cue constraints gradually move away from 100·0. The final perception grammar and three intermediate stages are shown in Fig. 7, where the constraints for the sibilant /ʃ/ are connected by a solid curve, and those for /s/ by a dotted curve. We can see, for instance, that for a spectral mean value of 24·0 Erb (2896 Hz), the dotted curve ends up having a ranking value of 105·0, and the solid curve one of 95·0. This means that the cue constraint *[24·0 Erb]/s/ ends up being ranked much higher than *[24·0 Erb]/ʃ/, and thus the token [24·0 Erb] is most often perceived as /ʃ/ in the final perception grammar (not always, because of the evaluation noise). In general, the lowest curve determines which category is perceived most often at any specific spectral mean value in Fig. 7.

Figure 7

A virtual learner acquiring the perception of the two sibilant categories of English: perception grammars after 2,000, 20,000, 200,000 and 2,000,000 input tokens.

A complete OT analysis of a problem generally involves two parts, namely a description of the constraint ranking and a description of what outputs the grammar assigns to all of its inputs. While the ranking of all 322 constraints is given in Fig. 7, a description of the workings of the grammar would involve giving 161 input–output pairs, perhaps in the form of 161 perception tableaux that are analogous to tableau (2). And even these 161 tableaux would not suffice, because as a result of the evaluation noise of Stochastic OT it is fully possible that the same spectral mean value is sometimes perceived as /ʃ/, and sometimes as /s/. A full account of how the 161 spectral mean values are handled by the perception grammar therefore involves giving for each of the 161 spectral mean values the probability that it is perceived as /ʃ/ and the probability that it is perceived as /s/. An estimate of these is given in Fig. 8. This figure has been computed by running each of the 161 spectral mean values through the final perception grammar 100,000 times (with an evaluation noise of 2·0, as during learning), and noting how often the output was /ʃ/; dividing each of the 161 results by 100,000 yields an estimate of the probability that each spectral mean is classified as /ʃ/.Footnote ¹¹ The probability of classifying it as /s/ is estimated in an analogous way.

Figure 8

Identification curves for the virtual listener of Fig. 7 (final state).

In Fig. 8 we see that the probability of perceiving /ʃ/ is greatest for those spectral values x for which the curve of *[x Erb]/ʃ/ lies furthest below the curve *[x Erb]/s/ in Fig. 7. This is a general property of Stochastic OT: the further constraint A is ranked above constraint B, the smaller the chance that B will overcome A at evaluation time.

Figure 8 confirms that the Gradual Learning Algorithm leads to optimal perception if the learner is given sufficient information. In the regions where she has heard a large number of spectral mean tokens (i.e. between approximately 23 and 33 Erb, as can be seen from inspecting Fig. 6), Fig. 8 shows that the learner has become an optimal (probability-matching) listener. For instance, the spectral mean of 27·5 Erb has (according to Fig. 8) a 75% probability of being perceived as /ʃ/ and a 25% chance of being perceived as /s/; hence, the odds of perceiving 27·5 Erb as /ʃ/ are three times as high as the odds of perceiving it as /s/; this ratio of 3 to 1 corresponds exactly to the relative heights of the two curves in Fig. 6 at [27·5 Erb].

However, the learner does not become optimal if she is given too little information. In the left periphery of the auditory space (around 21 Erb), the probability that such a spectral mean value was intended as /ʃ/ is very much higher than that it was intended as /s/, and this would predict that a probability-matching listener perceives such spectral mean values as /ʃ/ 100% of the time. Nevertheless, Fig. 8 shows that in this region the learner's perception grammar varies between perceiving /ʃ/ and /s/. This imperfection arises because the learner has not heard enough peripheral tokens to drag the two curves apart in this region.Footnote ¹² The same holds for the right periphery of the auditory space (around 35 Erb).

It is interesting to see for which spectral mean values the probabilities are maximal. The probability-matching criterion would predict that the probabilities would be maximal in the regions near 20 and 36 Erb, but the scarcity of such tokens has moved the point of maximal separation quite far toward the centres of the two categories, i.e. towards 25·3 and 30·7 Erb. The spectral mean values for which the two curves are furthest apart are approximately 24·9 and 31·1 Erb. These locations can thus be said to represent the least confusable spectral mean tokens, and will turn out to be relevant when we consider how the listener of Fig. 7 will pronounce /ʃ/ and /s/ herself.

5.5 The near-optimal English listener's preferred production: the prototype effect

In a bidirectional model of phonetics, the cue constraints and rankings used by the listener in perception are also used in her phonetic implementation. When implementing an articulation, the cue constraint *[20·0 Erb]/s/ is read as ‘an /s/ should not be produced with a spectral mean of [20·0 Erb]’, and so on.

In (4) we see how the virtual learner of §5.4 would now produce an /s/, if only the cue constraints and rankings of Fig. 7 were involved.

1. (4)

   

The candidate [31·1 Erb] in (4) wins because the curve of the cue constraints for /s/ in Fig. 7 is lowest at this value (which is also where the two curves are maximally separated). If we compare this phonetic output to the token with the highest frequency in the distribution of Fig. 6, which has 30·7 Erb, we can see that our speaker produces an /s/ that has shifted slightly by 0·4 Erb from the /s/ that is most commonly produced by her surroundings.

We now show that the shift is actually larger than the 0·4 Erb seen in (4). As a result of the evaluation noise of Stochastic OT, the winner in (4) will not always be 31·1 Erb. Instead, the winner will be the spectral mean value whose cue constraint (for /s/) happens to be lowest-ranked when evaluation noise is added to the ranking of each constraint. If we apply the same evaluation noise as in perception, namely with a standard deviation of 2·0, all spectral mean values with a ranking not much higher than that of *[31·1 Erb]/s/ are also likely to win. In Fig. 7 we see that the curve of *[x Erb]/s/ has a low plateau in the whole region between, say, 30·0 and 32·8 Erb, and these are all quite likely to win. The right-hand side of Fig. 9 shows the distribution of spectral values that we obtain by running the category /s/ 10 million times through production tableaux with the rankings of Fig. 7. The average spectral mean of the /ʃ/ category is 24·69 Erb, that of the /s/ category 31·35 Erb. The size of the shift (from the original 25·5 and 30·5 Erb) towards the periphery is therefore approximately 0·83 Erb.

Figure 9

Output distributions of the inverted perception grammar (distributions of ‘prototypes’).

What is behind this shift? From the discussion at the end of §5.4 we can infer that a probability-matching learning algorithm works in such a way that the constraints for values that are least likely to have been intended by the speaker as anything other than the dominant category in that region tend to be lowered furthest. Very roughly, the learning algorithm causes cue constraints to end up ranked lowest in auditory regions where the learner has heard the largest number of least confusable tokens. Teleologically speaking, our speaker prefers to produce an /s/ that is more peripheral than the average token that she has heard herself, because the auditory distance of such a token from the competing /ʃ/ is larger. What we observe here is the prototype effect in OT (Boersma Reference Boersma, Fanselow, Féry, Vogel and Schlesewsky2006). This phenomenon has been attested in experiments in the laboratory: when speakers of a language are asked to choose the best auditory instance of a sound category, they choose a more peripheral token than they would actually produce themselves (Johnson et al. Reference Johnson, Flemming and Wright1993), apparently because they choose the token that is least likely to present any other category than the one requested. It is important to note that although this explanation is couched in teleological terms (as if the listener-speaker knows about this), the underlying mechanism is not explicitly goal-oriented at all (and does not even know about auditory distance): the prototype effect occurs in OT simply because people employ in production the same cue constraint rankings that have optimised their prelexical perception.

The conclusion must be that learners will end up preferring more peripheral tokens than they have heard on average in their language environment. This would predict a sound shift if real learners really did that. But the phonetic implementation process of real speakers is not determined solely by their cue constraints. Articulatory considerations will be seen to keep the prototype effect within bounds.

5.6 Really speaking involves more constraints: the articulatory effect

Something is wrong with the assumption in §5.5 that phonetic implementation involves cue constraints alone. In real production the speaker is restricted by articulatory constraints as well. For instance, we can define the constraint *[31·2 Erb] as the articulatory constraint whose ranking reflects the articulatory effort associated with producing a spectral mean of 31·2 Erb. In the following we assume the independently plausible hypothesis (§§2.1, 4) that articulatory effort is minimal (though definitely not zero) for central auditory values of the spectral mean, and that more peripheral auditory values are harder to produce. If this hypothesis is correct (it at least explains the central tendency in single-sibilant inventories in §2.1), the articulatory constraints for peripheral auditory values must be ranked higher than those for more central auditory values.

In the production of a sound, articulatory constraints interact with the cue constraints. This is illustrated with the production grammar in Fig. 10, where articulatory constraints are represented with a thick grey curve, following a parabolic shape reminiscent of effort curves found in biophysics (Hoyt & Taylor Reference Hoyt and Taylor1981). The cue constraints have the same rankings as in the perception grammar in Fig. 7, and are again represented as solid and dotted curves. Following Kirchner (Reference Kirchner1998), we assume for simplification that the ranking of articulatory constraints is fixed and not influenced by language-specific learning.Footnote ¹³

Figure 10

A production grammar for the two sibilant categories in English.

The interactions of articulatory and cue constraints in production become clear in the tableau in (5), where our learner tries to articulate /s/.

1. (5)

   

The candidate [31·1 Erb], which was the winner in the tableau without articulatory constraints in (4), no longer wins, because its articulation involves more effort than that of [30·7 Erb], which is the new winner. Loosely speaking, candidates below [30·7 Erb] are too indistinctive and candidates above [30·7 Erb] are too hard (note that in this tableau *[30·8 Erb] outranks *[30·7 Erb]/s/ despite the fact that in the region of 30·7 and 30·8 Erb the articulatory curve in Fig. 8 lies below the cue curve for /s/; there is no contradiction: in stochastic OT, constraints that are as closely ranked as these two will be ranked in the opposite order in a non-negligible fraction of the evaluations).

While the bidirectional use of cue constraints causes the categories to drift apart auditorily (§5.5), the presence of the articulatory constraints checks this expansion and drives the production distributions back towards the centre of the spectral mean continuum. Figure 11 shows the production distributions, estimated by running each sibilant category 10 million times through the grammar of Fig. 10 with an evaluation noise of 2·0.

Figure 11

The production distributions for the two sibilant categories in English.

The average spectral mean of the /ʃ/ category is 25·31 Erb, and that of the /s/ category is 30·70 Erb. When we compare this with the averages of Fig. 6 (25·5 and 30·5 Erb) and Fig. 9 (24·69 and 31·35 Erb), we conclude that the articulatory constraints have effectively prevented the categories from drifting apart auditorily: our English learner produces nearly the same average spectral means as her parents. Apparently, the articulatory effect has cancelled out the prototype effect. This balance of powers was first noted and modelled by Boersma (Reference Boersma, Fanselow, Féry, Vogel and Schlesewsky2006).

There is an important difference between the distributions in Fig. 6 and Fig. 11. In Fig. 6 the standard deviations of the two distributions are 1·13 Erb, and in Fig. 9 they are 1·03 and 1·06 Erb. In Fig. 11, however, the standard deviations are only 0·85 and 0·86 Erb: the distributions are narrower. Apparently, the articulatory effect has caused, in addition to the shift, an entrenchment of the produced distribution when compared with the distribution heard. This sharpening is an important property of our model, which will be seen to help to make the distributions stable over the generations (§5.7) and to allow the model to be compared favourably to other models (§7.4).

The near-exact cancelling of the articulatory and prototype effects is a direct result of our choice of parameters: knowing that English has a stable sibilant system, we chose the height of the articulatory curve to be the one in Fig. 10, not a higher or lower one. In this sense, the parameters of the model have been established on the basis of language data. Therefore we have to question the equal sizes of the prototype effect and the articulatory effect, and will do so in §§5.8 and 5.9. First, however, we show that the language simulated in §§5.4–5.6 is indeed stable over the generations.

5.7 Simulating sound change: a stable language

Are the average spectral means of the English sibilants, namely 25·5 and 30·5 Erb, indeed stable over the generations, or do they slowly drift? We test this by simulating the acquisition of the two sibilants over nine more generations.

Some care has to be taken in how the produced spectral mean values of the first generations of learners are fed to the second generation. We cannot transmit the distribution of Fig. 11 directly from speaker to learner, because the only variation in the spectral mean values of Fig. 11 is due to decision noise in production. That is, the standard deviation of 0·85 Erb only reflects the evaluation noise in the production tableaux. The listener will be confronted with this source of variation (and not be able to normalise it away), but also with some non-normalisable between-speaker variation (because real learners will hear multiple speakers), some random variation within the speaker's muscle system (which is independent from the speaker's evaluation noise), the background noise in the air and the noise in the learner's ear. We represent these influences on the spectral mean together as transmission noise, with a standard deviation of 0·80 Erb. More precisely, the second generation of learners will be presented with spectral mean tokens computed from the grammar in Fig. 10, where to each token produced we add a value drawn randomly from a Gaussian curve with a mean of zero and a standard deviation of 0·80 Erb. The resulting distribution of spectral mean input values for generation 2 is given in Fig. 12, which was estimated by running each sibilant category 10 million times through the grammar of Fig. 10 with an evaluation noise of 2·0 and a transmission noise of 0·80 Erb. The standard deviations are approximately 1·17 Erb.

Figure 12

The sibilant environment of the second generation of learners.

Our second generation is thus represented by a learner who acquires a perception grammar in the way described in §5.4, on the basis of outputs of generation 1 modified by transmission noise (we again assume that the learner can distinguish the categories from each other). This learner then turns into a speaker by including articulatory constraints, as described in §5.6. We then feed tokens produced by this speaker, modified by 0·80 Erb transmission noise, to a new learner, our third generation. This learner acquires again a perception grammar and a production grammar; her transmitted output is used as input to the fourth learner, and so on.

In total we simulate this acquisition process for 10 learners in a row, with the simplifying assumption that every learner stands for a whole generation of speaker-listeners. Every learner receives 1 million /ʃ/ and 1 million /s/ tokens, and has a plasticity of 0·01 and an evaluation noise of 2·0.

Over the generations, the English spectral mean values stabilise at about 25·40 and 30·60 Erb, and the standard deviations stabilise at about 1·18 Erb. This is shown in Fig. 13, where the black curves connect the single-speaker averages and the grey areas represent the standard deviations of distributions like those in Figs 6 and 12 (i.e. including transmission noise).

Figure 13

The spectral means of the two English sibilants and their stable learning over 10 generations.

Our simulated English (Fig. 6) turns out to be boringly constant (Fig. 13). There is no real change between the output of generation 1 and the output of generation 10 (except that the distributions are no longer Gaussian, but slightly skewed, as in Fig. 12). The next question is whether a language with different auditory distributions than Fig. 6 is equally stable, or whether it instead drifts away from its initial distributions.

5.8 Simulating sound change: a language with an exaggerated contrast is unstable

Here we simulate the learning of a language with a much more extreme two-sibilant contrast than English, i.e. [ʂ] vs. [&sdsqrb;], with spectral means of 23·0 and 33·0 Erb. Except for these initial distributions, everything else (including the shape and height of the articulatory constraint curve of Fig. 10) is the same in this simulation as in the one of §§5.4–5.7.

Figure 14 shows the result. For this ‘exaggerated English’, the second generation more or less learns the oversized range, although they reduce it to a much more moderate contrast of 24·83~31·18 Erb. Apparently, the articulatory effect outweighs the prototype effect for this generation; this comes as no surprise, since the articulatory curve in Fig. 10 (in the regions of 24·83 and 31·18 Erb) approaches the cue-constraint curve. The third generation has already shifted the system towards an unmarked articulatorily-perceptually balanced [ʃ] and [s]. Within two generations, the learners have changed the exaggerated English into plain English.

Figure 14

The development of the two sibilants in ‘exaggerated English’ over 10 generations.

The conclusion is that the articulatory and prototype effects can act independently, and work together to establish, non-teleologically, an optimal balance between articulatory ease and auditory contrast.

5.9 Simulating sound change: a language with a confusable contrast is unstable

If the optimal balance achieved in Figs 13 and 14 is characteristic of two-sibilant inventories in general, then the 25·4~30·6 Erb inventory should emerge for every possible initial inventory. The two cases that yet have to be investigated in this respect are the case of an initial ‘confusable’ English, where the categories are closely spaced, and the case of an initial ‘skewed’ English, where the categories are not positioned symmetrically around 28·0 Erb (as they are in Figs 13 and 14).

The two cases are combined in the simulation of Fig. 15, a language with initial sibilants at 28·0 and 32·0 Erb, which is both skewed (one sibilant has a central, the other a high spectral mean) and relatively confusable (the difference is only 4·0 Erb, rather than the 5·0 Erb of the earlier simulations).

Figure 15

The development of the two sibilants in ‘skewed and confusing English’ over 15 generations.

The first generation of learners moves the spectral mean values apart (to 26·8 and 32·2 Erb), immediately reaching an English-like distance of about 5 Erb. Apparently, the prototype effect plays a very fast role here; this confirms the repulsive dynamic nature of the present model and shows that the model will be able to handle phonetic enhancement and chain shifts. The articulatory effect is also seen to play a role, because the amount the lower category moves down (i.e. to the left in the figure) is larger than the amount the top category moves up. This effect works quite slowly, however: it takes quite a number of generations to obtain a perfectly symmetric inventory. The cause of the slowness is that the upper category ‘wants’ to move down but that this is hampered both by the repulsive force from the lower category and from the fact that the lower category itself experiences an upward bias from the articulatory effect. Nevertheless, a symmetric English-like inventory is eventually reached.

We can conclude from Figs 13–15 that our model predicts that, independently of the situation in generation 1, the inventory always evolves towards the same two auditory values. In other words, all stable languages with two sibilants are like English (or French, or German).

So we see that optimal dispersion indeed happens. Figs 13–15 all show the effect of the excluded centre (§2.1): the region around 28 Erb is avoided in languages with two sibilants. The next step is to look at other kinds of sibilant inventories: which of the dispersion effects will we see?

6.1 The Polish three-sibilant inventory: a chain shift

Polish used to have the sibilants /ʃ s&phosupj; s/ (Carlton Reference Carlton1991). If the dispersion principle is correct, and if the spectral mean is the only auditory cue that distinguishes these three sibilants, such an inventory cannot be stable. The simulation in Fig. 16 shows what happens if we start out with a language whose sibilants have spectral means of 25·5, 30·3 and 30·7 Erb (in order to make our point, we have taken values for /s&phosupj;/ and /s/ that are closer together than they probably actually were in medieval Polish).

Figure 16

The development of the three sibilants in Polish over 15 generations.

The first striking phenomenon that Fig. 16 shows is that the /s&phosupj;/~/s/ contrast is phonetically enhanced: the palatalised alveolar (the middle category) lowers its spectral mean beyond 29 Erb, i.e. into the [ɕ] region. This is reported to have happened in real Polish in the 13th century (Stieber Reference Stieber1952, Carlton Reference Carlton1991). The second thing that happens is that the laminal postalveolar (the sound on the left in Fig. 16) shifts down towards the apical postalveolar (‘retroflex’) [ṣ], which is reported to have happened in real Polish in the 16th century (Rospond Reference Rospond1971, though this sound is usually transcribed as postalveolar /ʃ/; see the discussion in Hamann Reference Hamann2004). Our simulation (when compared with the English simulations) confirms Jones' (Reference Jones2001) proposal that this second shift was caused by the first, i.e. that we are observing a contrast-enhancing push chain here: the /ʃ/ category is shifted down as a result of the approach by the lowering /ɕ/. Furthermore, we can observe the equally contrast-enhancing shift (again proposed by Jones Reference Jones2001) of the alveolar /s/ towards a more peripheral [&sdplus;], its present-day location (Puppel et al. Reference Puppel, Nawrocka-Fisiak and Krassowska1977). This simulation thus explains the present Polish sibilant system /ṣ ɕ &sdplus;/, which has spectral means very close to the 24·46, 28·08 and 31·59 Erb found here (the three female speakers recorded for the study by Zygis & Hamann Reference Zygis and Hamann2003 have average values of 24·40, 27·82 and 31·29 Erb). The grammar acquired by the fifteenth generation in Fig. 16 is given in Fig. 17; the articulatory constraint curve is the same as in Fig. 10. As before, this grammar is bidirectional. When given a certain spectral mean value, the grammar will most likely perceive it as the category for which the cue constraint at that spectral mean value is the lowest of the three. When given a category, the grammar will produce a spectral mean value where both the relevant cue constraint curve and the articulatory constraint curve are low; the average realised spectral mean values are 24·46, 28·08 and 31·59 Erb, as mentioned above. We have thus established an account of the Polish sibilant inventory without using dispersion constraints: only the cue and articulatory constraints of Fig. 17 are necessary (cf. Padgett & Zygis Reference Padgett and Zygis2003, who did use dispersion constraints for this example).

Figure 17

A production grammar for the three sibilant categories in Polish.

When we compare Fig. 16 to Fig. 15, we see all the dispersion effects discussed in §2.1. First, the centre category (around 28·0 Erb) is back in the picture. Second, the dispersion in Fig. 16 is even: both pairs of adjacent categories are removed from each other by approximately 3·6 Erb. Third, we see that the size of the auditory space is greater when there are three categories (7·1 Erb) than when there are two (5·2 Erb). Fourth, from the widths of the grey areas we see that the stable variation within a category is somewhat smaller when there are three categories (namely 1·12 Erb for the middle category and 1·15 Erb for the peripheral categories) than when there are two (1·18 Erb, according to §5.7). Fifth, we have observed a push chain when a lowering mid category caused the bottom category to lower as well. Sixth, we will observe a drag chain if we remove the bottom category from the final state in Fig. 16: from the simulation in §5.9 we know that the original mid category will move down to 25·4 Erb, allowing the top category to move down from 31·6 to 30·6 Erb. We conclude that we have faithfully modelled all aspects of auditory dispersion by assuming bidirectionality of cue constraints and a U-shaped articulatory effort curve, as in the production grammars in Figs 10 and 17.

6.2 A four-sibilant inventory

The largest inventory we consider is one with four sibilants, like Toda in Fig. 5. Independently of where the centres of the categories lie in the environment of the learners of the first generation, the centres of the categories evolve towards values around 24·4, 26·5, 29·5 and 31·6 Erb, i.e. the distances between the categories are again smaller than in the three-sibilant case, while the total space taken up has again increased (although very little). The spacing between the categories is somewhat greater in the middle than at the edges, probably because at the edges the limiting effect of the articulatory constraints is greater (this can be seen as a more precise formulation of §2.1.3). The final standard deviations are 1·13 Erb for the two outer categories and 1·22 Erb for the two inner categories, i.e. a bit larger than in the three-sibilant case.

We conclude that all dispersion effects identified in §2.1 remain valid in larger inventories (except the unexpectedly high within-category variation in the four-sibilant inventory, for which we have no explanation).

6.3 A one-sibilant inventory

The smallest inventory we consider is one with a single sibilant. The first generation of learners already turns up with a category centred at the centre of the auditory continuum, i.e. 28·0 Erb, and the standard deviation will be and stay a gigantic 3·03 Erb. The cause of this situation is that the learners do not learn: if the only category along the continuum is the unspecified sibilant /S/, they will perceive any spectral mean value as /S/, and therefore never make a mistake. As a result, all cue constraints stay ranked at 100·0, and in production the decision about which auditory value to pronounce is determined partly by which cue constraint happens to be lowest ranked, partly by the articulatory constraints.

The reader may object that the predicted one-shot shift to the centre is unrealistic, yet this is what we predict will happen in the absence of other phonological entities. In practice it will be very difficult to find such a situation: the single (retracted apico-alveolar) sibilant of Iberian Spanish, for instance, must cope with the existence of a rather strident /θ/ in the same language.

6.4 Can non-contiguous categories be learned?

The present model does not involve any representation of auditory distance. That is, the learner represents nothing more than 161 values along the spectral mean continuum, and does not necessarily know that they are ordered in a natural way. For instance, the virtual learner never needs to know that the spectral mean associated with the 37th value along the continuum (i.e. 23·6 Erb) lies in between those of the 27th value (22·6 Erb) and the 47th value (24·6 Erb). As a result, the constraint set of our virtual learner can easily represent an inventory where category 1 has spectral means around 26·5, 28·5 and 30·5 Erb, whereas category 2 has spectral means around 25·5, 27·5 and 29·5 Erb (as was mentioned in note 9). The question naturally arises, however, whether such discontiguous categories are not only representable, but learnable as well.

The answer turns out to be that discontiguous categories are ‘semi-learnable’: if the learner's environment contains a discontiguous category, then the learner's final category may still be discontiguous, but less so than the one in her environment, and within a number of generations the language will have changed into one with only contiguous categories.

An example is shown in Fig. 18. The initial inventory has a simple (monomodal) category (which we arbitrarily label /ʃ/) centred at 28·0 Erb with the usual relative frequency of occurrence of 50%, plus a bimodal category (which we label /s/) with a peak centred at 31·0 Erb with a relative frequency of 37·5% and a peak centred at 25·0 Erb with a relative frequency of 12·5%; all three peaks have a standard deviation of 1·13 Erb. Thus, both categories are equally likely, and the upper part of /s/ is three times more likely than the lower part.

Figure 18

The demise of a bimodally distributed category. Black=/ʃ/, white=/s/.

Figure 18 shows that the following generations continue to associate the /s/ category with two peaks. However, the lower peak shrinks slowly, and has almost disappeared after generation 11. After 20 generations, the English inventory has arisen again.

We conclude that in our model non-contiguous categories are learnable, but not stable over the generations. The ultimate cause of the development into monomodal distributions can be explained with Fig. 18 in the following way. The fact that the upper peak of /s/ is taller than the lower peak causes an asymmetry in the regions of confusion. That is, the region of confusion between the lower peak and /ʃ/ lies somewhat further from the centre of the continuum (namely around 26·0 Erb, which is 2·0 Erb away) than the region of confusion between the lower peak and /ʃ/ does (around 29·5 Erb, which is only 1·5 Erb away). As a result, /ʃ/ will shift down (i.e. to the left), which then pushes the lower peak of /s/ down into an articulatory more effortful region (push chain) and allows the upper peak of /s/ to come down into a less effortful region (drag chain). As a result, the articulatory bias against the lower peak of /s/ becomes greater than the articulatory bias against the upper peak, and this causes a slight preference in production for selecting tokens from the upper peak. This process is self-reinforcing, because it moves the boundaries between the three categories down.

We do not yet know how to assess our predicted relative learnability of bimodal distributions. If there exist mechanisms that cause such distributions to arise, we must be able to observe them in real languages, because such distributions are predicted to take ten generations to become monomodal. Whether actual languages do have this kind of transitory allophony in sibilants remains to be seen.Footnote ¹⁴