3.1. Characterising the production process

Language production in generative phonology is often assumed to be a modular feed-forward process (Pierrehumbert Reference Pierrehumbert, Gussenhoven and Warner2002; Bermúdez-Otero Reference Bermúdez-Otero and Lacy2007). This type of model is understood as a kind of abstract assembly line: a lexical item is chosen and then is modified through a series of specialised stations until it reaches the end point as a phonetic object that can be pronounced. Since assembly lines are successive, each station is essentially insensitive to the history of the objects it receives. To make this metaphor more concrete, we can imagine that the Lexicon places URs on a conveyor belt which takes them to the Phonology station to be worked on. At the Phonology station, URs are transformed into SRs, and SRs are placed back on the conveyor belt to be taken down the line to the Phonetics station. The Phonetics station receives each SR with no knowledge of its previous history. The role of Phonetics in this instance is to transform each SR into a corresponding phonetic form (e.g., a gradient representation containing acoustic/articulatory instructions). In this example, Phonology acts as an intermediary between the Lexicon and Phonetics. Consequently, when two identical SRs are derived from distinct URs, the Phonetics station must treat those SRs exactly the same way.

Imagine, instead, that Phonology was not a station that a lexical item had to pass through during the production process, but rather a target design that the phonetic module was given alongside a lexical item. In this metaphor, the lexical item is a set of materials, the phonology is a blueprint for what the assembled form should look like, and the Phonetic station is the module which is doing the assembling. The phonology still operates in the same way as in modular feed-forward models: given a UR as an input, it returns an SR as its output. Only now this process does not strictly precede phonetic implementation (cf. Gafos & Benus Reference Gafos and Benus2006). This characterisation of the production process situates the phonology in a way that allows it to maintain its primary role of determining the surface form of an UR. It also allows the Phonetics station simultaneous access to both the UR and the phonological instructions on how to modify it. As we explain in more detail later, by ‘phonological instructions’, we simply mean a map from URs to SRs. No other history of a phonological derivation or evaluation is visible to the Phonetics station.Footnote ⁴ The main point here, however, is that under this architecture, the lexical form is not invisible to the Phonetics Station.

Crucial to our analysis is the view that each module can be thought of as a function (Roark & Sproat Reference Roark and Sproat2007; Heinz Reference Heinz, Hyman and Plank2018). In the modular feed-forward model, the phonological module is a function that maps a UR to an SR and the phonetic module is a function that maps an SR to a PR. The BMP continues to view the phonological module as a function that maps a UR to an SR, but views the phonetic module as a higher-order function that takes the phonological module function as an input. In addition, to generalise over all lexical items, we consider the entire lexicon to be an input to the phonetic module instead of a single UR.Footnote ⁵ The phonetic module is therefore a function with at least two inputs: the lexicon and the phonological module; and one output: a set of PRs $\{PR\}$ . The two contrasting models are shown in Figure 1. The next section provides a formal definition of the BMP.

Figure 1 Visual comparison of the architecture for modular feed-forward models and the BMP. Each box represents a function/module. Solid lines represent the inputs to each function while dashed lines represent the outputs of each function.

3.2. From assembly line to blueprint: function (de)application

While giving the phonetic module direct access to the lexicon and phonology may seem like a large departure from the feed-forward model, the BMP can be related directly to the feed-forward model via function application. We also show that the BMP is an abstraction of feed-forward models under the constraint that the representation output by the phonological module includes the input to the phonological module (Prince & Smolensky Reference Prince and Smolensky1993; Goldrick Reference Goldrick2000; Revithiadou Reference Revithiadou, Blaho, Bye and Krämer2008; van Oostendorp Reference van Oostendorp2008). Our analyses rely on the function type each module computes. Our notation follows from Pierce (Reference Pierce2002) which derives from the lambda calculus (Church Reference Church1932, Reference Church1933). Therefore, we begin with a basic introduction to functions and function types.

A function maps one or more elements in a set A to elements in a set B such that each a in A maps to at most one element in b in B. For a function f that maps elements from set A to set B we write $f::A\rightarrow B$ . The phonology function above (or P for short) would therefore be written as $P::UR\rightarrow SR$ . In prose this means ‘the phonology function P maps URs to SRs’. Note the phonology function P is agnostic as to the particulars of the representations of $UR$ and $SR$ . For example, they could be continuous, discrete, or some combination. $P::UR\rightarrow SR$ simply means that the phonological module takes a UR-type thing and returns an SR-type thing.

Functions with more than one argument are written similarly. Addition can be thought of as a function with two arguments: $add(x)(y) = x + y$ . Its function type would then be written as: $add :: \mathbb {R}\rightarrow \mathbb {R}\rightarrow \mathbb {R}$ . When reading function types with multiple arguments, everything to the left of the rightmost arrow is an argument and everything to the right of the rightmost (non-bracketed) arrow is the output. The function type of $add$ can therefore be understood as a map from two real naumbers to a single real number. In §3.3, we will discuss how this notation relates to the common $add(x,y) = x + y$ notation.

Our analysis below relies on two other concepts: higher-order functions and the notion of function application. Functions like the ones described above are first-order functions. These are contrasted with higher-order functions. A higher-order function is a function that either takes as an input another function or returns a function as its output. An example of a higher-order function that takes a function as part of its input is the $map$ function.

Given two inputs f and $\vec {x}$ , where f is a function of type $f :: X\rightarrow Y$ that takes things of type X as its input, and $\vec {x}$ is an array of length n that contains x’s $[x_1, \dots , x_n]$ , $map(f)(\vec {x})$ applies function f to every individual element of $x\in \vec {x}$ and returns the array $[f(x_{1}),\ldots ,f(x_{n})]$ . To give a concrete example, consider the function $add1(x) = x+1$ and the array of integers $[-23,1,9,307]$ . If we were to provide both of these as the input to the $map$ function, we would end up with $map(add1,[-23,1,9,307]) = [-22,2,10,308]$ . The $map$ function is not limited to numerical data types/functions and works just as well over strings. For example, for all strings w, let $redup(w)=ww$ . Then $map(redup,[a, ba, cab])=[aa, baba, cabcab]$ . To summarise, the function type of $map$ is given by $map :: (X\rightarrow Y)\rightarrow [X]\rightarrow [Y]$ .

We now move to a discussion of function application. Function application is the act of applying a specific function to an argument, but it can also be thought of as a higher-order function itself. The two arguments for function application would be one of type X and the other of type $X\rightarrow Y$ (i.e., a function that maps X type things to Y type things). Given these two arguments it would output something of type Y. For the overall type we would therefore write $function\text{-}application :: X\rightarrow (X\rightarrow Y)\rightarrow Y$ . The notion of function application is important for our analysis because it allows us to relate the BMP to the modular feed-forward model.

We now apply these ideas to architectures of language production. Throughout the remainder of this section the following abbreviations are used: L, P and A as functions representing the Lexicon, Phonology and and Phonetics (Articulation or Acoustics); $UR$ , $SR$ and $PR$ to represent Underlying Representations, Surface Representations, and Phonetic Representations. The proposed types are listed in Table 1.Footnote ⁶

Table 1 Types

This paragraph describes the steps that turn the modular feed-forward model into the BMP. To start, the phonetic module in the modular feed-forward model has the following type:

This idealises the phonetic module as a map from SRs to PRs. Given a UR, the phonology P, and the definition of function application from above, one can decompose $SR$ into $UR\rightarrow (UR\rightarrow SR)$ :

Next, $(UR\rightarrow SR)$ is just another way of representing the phonological module:

To complete this reconceptualisation, we change $UR$ to L in order to generalise over the entire lexicon. By doing so, the output is now a set of PRs rather than a single specific representation. This gives us a new type for the phonetics function:

The phonetic module is therefore a higher-order function with two arguments: the lexicon and the entire phonological module (a function). As is the case in the modular feed-forward model, the phonology still maps an underlying form to a surface form. Additionally, in both the BMP and the modular feed-forward model an underlying form is ultimately transformed into a PR. The main difference is that the phonology is no longer an intermediary between the lexical form and the phonetic module. Instead, the phonology and the lexical form are both input to the phonetic module.

If it is not clear yet, why we call this model the Blueprint Model, consider this. For every n-ary function, there is an equivalent $(n+1)$ -ary relation. Since phonology is a unary function (i.e., it has one input, a $UR$ ), it can also be envisioned as a binary relation consisting of $UR$ and $SR$ pairs $\langle UR,SR\rangle $ . This latter perspective highlights the fact that we can view phonology not as a module that directly shapes the phonetic output, but instead as a set of instructions that informs the phonetic module as to how a given lexical item should be pronounced. In other words, in the same way, one would query a blueprint, the phonetic module queries the phonology as to how a $UR$ should be pronounced.

The derivation shown above does not exhaustively represent all the factors that determine production. It simply shows how the BMP relates to the feed-forward model of production. Many other factors have been argued to influence speech production. For example, in the case of incomplete neutralisation it has been argued that the phonetic output is not only a blend of the phonological output ( $SR$ ) and the lexical input ( $UR$ ), but also that this blend can be scaled by extralinguistic factors relating to contrastive intent (Port & Crawford Reference Port and Crawford1989; Ernestus & Baayen Reference Ernestus and Harald Baayen2003; Gafos & Benus Reference Gafos and Benus2006). This is an additional factor necessary to adequately account for production. As will be discussed in more detail in §4.2, this is accomplished with the BMP by adding the intent (I) as one of the arguments to production: $A :: L\rightarrow P\rightarrow I\rightarrow \{PR\}$ .

3.3. Currying and uncurrying

This section relates the BMP to earlier theories of phonology in which the outputs of phonology include its inputs (Prince & Smolensky Reference Prince and Smolensky1993; Goldrick Reference Goldrick2000; Revithiadou Reference Revithiadou, Blaho, Bye and Krämer2008; van Oostendorp Reference van Oostendorp2008). This is precisely the claim made in the original formulation of Optimality Theory, where every element of the phonological input representation is contained in the output (Prince & Smolensky Reference Prince and Smolensky1993). Under the feed-forward model, the principle of containment ensures that the phonetic module has access to the lexical form, because it can recover it from the output of the phonology. It follows that if the phonological module obeys the principle of containment then the phonetic module is able to, for example, distinguish between faithful word-final voiceless obstruents and derived ones (van Oostendorp Reference van Oostendorp2008).

Note that the principle of containment is independent of Optimality Theory per se. For instance, it is not difficult to imagine a rule-based theory in which the output of a rule system is a SR presented alongside the UR, which is carried through the derivation. In other words, this principle effectively ensures that the phonological module has something like the type $P'::UR\rightarrow (UR,SR)$ , regardless of whether the phonological module is instantiated by a constraint-based grammar, a rule-based grammar, or some other form of grammar.

Strictly speaking, containment theory, and variants such as turbidity theory (Goldrick Reference Goldrick2000), do not represent the outputs of phonology as a SR paired with an UR. Instead, the output of a word-final devoicing process for the lexical item /gruz/ would be something more like the sequence [(g,g),(r,r),(u,u),(z,s)]. However, our point is that the UR is recoverable from this representation.

What this means from the perspective of the type-functional analysis is that the containment theory of phonology is an uncurried version of the BMP. To explain, consider the fact that since functions in general can return functions, functions with multiple arguments do not need to be given all the arguments at once. If fewer than the totality of arguments is given, then a function is returned.

Consider again addition, which we gave the type: $add ::\mathbb {R}\rightarrow \mathbb {R}\rightarrow \mathbb {R}$ . This can be thought of as the curried version of $add' :: (\mathbb {R},\mathbb {R})\rightarrow \mathbb {R}$ . Whereas $add$ takes two arguments, $add'$ takes a single argument which is a pair of real numbers. It is always possible to convert between a function which takes one input as a pair of arguments and a higher-order function which takes multiple arguments. This conversion is called currying (after Curry Reference Curry, Jon Barwise, Keisler and Kunen1980). Currying itself can be thought of as a higher-order function, which takes an uncurried function like $add'$ and returns the curried version like $add$ . The type signature of currying is thus $curry :: ((A, B) \rightarrow C) \rightarrow (A \rightarrow B \rightarrow C)$ . The argument of the $curry$ function is a function mapping $(a,b)$ pairs to c-type things. The output of the $curry$ function is a function that takes two separate inputs a and b and outputs c. Consequently, $curry(add') = add$ , and thus for all $a,b$ , $add(a,b) = add(a)(b) = a + b$ .

As mentioned, containment theories of phonology essentially have the type $P'::UR\rightarrow (UR,SR)$ . Under the feed-forward model, this output is given to the phonetic module to produce the articulatory representation. Consequently, the phonetic module would have type $A'::(UR,SR) \rightarrow PR$ . This is essentially the uncurried version of the BMP. Currying $A'$ yields a phonetic function of the form in (5):

Since $UR \rightarrow SR$ is the function the phonological module computes, (5) can be rewritten as (6):

Combining (6) and (3) reveals that the BMP can be characterised as shown below:

Generalising over the lexicon again, we get the same type for the BMP:

This shows precisely the relation between containment theories of phonology under the feed-forward model and any theory of phonology computing functions $UR \rightarrow SR$ with the BMP. It also highlights the essential difference between the BMP and the modular feed-forward model: the latter serialises phonology and phonetics while the former does not.

The next two sections discuss two empirical phenomena that have been argued to be problematic for theories of language production based on discrete generative models of phonology: incomplete neutralisation (Port et al. Reference Port, Mitleb and O’Dell1981; Port & O’Dell Reference Port and O’Dell1985) and homophone durational variation (Gahl Reference Gahl2008). We argue that these phenomena are not counterarguments to discrete phonological knowledge under the BMP approach to the phonetics–phonology interface. This is due to the fact that the structure of the interface is itself an analytical assumption that needs to be carefully weighed when discussing the interaction of grammatical and extra-grammatical information in language production.

As many philosophers of science have pointed out, refutation of a given scientific theory is dependent on auxillary assumptions and shared background knowledge (Quine Reference Quine1951; Duhem Reference Duhem1954; Popper Reference Popper1959; Feyerabend Reference Feyerabend, Cohen and Wartofsky1965; Lakatos Reference Lakatos, Lakatos and Musgrave1970). Therefore, phonetic evidence alone does not bear on the nature of phonological knowledge, but rather must be evaluated in tandem with a theory of how phonological knowledge is physically manifested. In other words, phenomena like incomplete neutralisation and variation in homophone duration falsify discrete phonological knowledge only if we assume that the modular feed-forward structure of the interface is a shared assumption (or shared ‘interpretative theory’ in terms of Lakatos Reference Lakatos, Lakatos and Musgrave1970). In this way, our analysis aims to show that arguments for gradient phonological knowledge depend on a certain structure of the production function, but there are alternative ways to structure this function that do not require gradient phonological knowledge to account for the same phonetic facts.

4.1. Background

Final devoicing is probably the best-studied example of a phonological neutralisation process. This is a phenomenon where, at the end of some domain (often syllable or word), an obstruent loses its voicing feature and surfaces as a voiceless segment.Footnote ⁷ It has been attested in a variety of languages including, but not limited to, German (Bloomfield Reference Bloomfield1933), Polish (Gussmann Reference Gussmann2007), Catalan (Wheeler Reference Wheeler2005), Russian (Coats & Harshenin Reference Coats and Harshenin1971) and Turkish (Kopkalli Reference Kopkalli1994). The data in (9) provide an example from German (Dinnsen & Garcia-Zamor Reference Dinnsen and Garcia-Zamor1971).

In the 1980s, it was discovered that German speakers could discriminate between an underlying voiceless segment and a derived voiceless segment at a rate of 60–70%; further acoustic studies showed that these two types of segments systematically varied along certain acoustic dimensions (Port et al. Reference Port, Mitleb and O’Dell1981; Port & O’Dell Reference Port and O’Dell1985). Acoustically, it was found that the preceding vowel was shorter for underlying voiceless segments, the duration of aspiration noise was longer for underlying voiceless segments, and the amount of voicing into stop closure was longer for underlying voiced segments. These properties make it appear as if the surface form maintained some of the properties of the underlying form. Because the values for the derived voiceless segments were intermediate between a surface voiceless segment derived from underlying voiceless segment and a surface voiced segment in non-coda position, this phenomenon was termed ‘incomplete neutralisation’.

Final devoicing has been extensively studied, and found to be incomplete in many other languages, such as Catalan (Dinnsen & Charles-Luce Reference Dinnsen and Charles-Luce1984), Dutch (Warner et al. Reference Warner, Jongman, Sereno and Kemps2004), Polish (Slowiaczek & Dinnsen Reference Slowiaczek and Dinnsen1985), Russian (Dmitrieva et al. Reference Dmitrieva, Jongman and Sereno2010) and Afrikaans (van Rooy et al. Reference van Rooy, Wissing and Paschall2003). Many other processes, such as coda aspiration in Andalusian Spanish (Gerfen Reference Gerfen2002), French schwa deletion (Fougeron & Steriade Reference Fougeron and Steriade1997) and Japanese monomoraic lengthening (Braver & Kawahara Reference Braver, Kawahara, Albright and Fullwood2016), have also been found to be incomplete. Strycharczuk (Reference Strycharczuk and Aronoff2019) provides a recent review of findings and discusses various hypotheses for the sources of incompleteness.

Returning to final devoicing, Port & Crawford (Reference Port and Crawford1989) find that listeners appear to have control over the level of incompleteness of the neutralisation based on communicative context and how salient a contrast is made. In their experiment, they used five different contexts (based on four sentence conditions) to evaluate how the level of neutralisation changed depending on speakers’ awareness of the task. Conditions 1A and 1B used disguised sentences where the target word was embedded within a sentence. The 1A task involved participants reading the sentence from a written example. The 1B task used the same sentences, but this time participants were read the sentence and asked to repeat it back out loud to the experimenter. Condition 2 used contrastive sentences where both target words were in the same sentence, but clarifying information was included to differentiate the words. Condition 3 also used contrastive sentences, but removed the clarifying information. In Condition 4, the words were in isolation.

They found incomplete neutralisation in every condition when analysing aggregated speaker data. No difference in the amount of incomplete neutralisation was detected between Conditions 1A and 1B in contrast to previous experiments (Jassem & Richter Reference Jassem and Richter1989). In all other cases reported in Port & Crawford (Reference Port and Crawford1989), the level of incompleteness increased when the task highlighted the contrast between the two target words. Condition 2 was more incomplete than Conditions 1A and 1B, and Condition 3 was even more incomplete than Condition 2. This makes sense because Condition 2 highlights the contrast, but includes extra material that can aid in distinguishing between the two words. Therefore, speakers may attempt to highlight the contrast with the amount of ‘voicing’. Condition 3 meanwhile highlights the contrast, but provides no additional information. In this condition, speakers must use the amount of ‘voicing’ to make the contrast is salient. Condition 4 also showed a greater degree of incompleteness than Condition 1A/B and was slightly lower than Condition 2.

These data support the idea that speakers have some level of control over how neutralised a segment is depending on the contrastive condition. The pragmatic conditions therefore influence a speakers intent on maintaining an underlying contrast. In their non-linear dynamic approach to production, Gafos & Benus (Reference Gafos and Benus2006) include a variable called intent to account for this fact. For the remainder of this article, we will also use the term intent as a cover term indicating pragmatic condition/desire to maintain an underlying contrast.

4.2. How the BMP includes intent

§3.2 provided a formal characterisation of the production process. The focus in that section was to show how the BMP conceptualises the phonetic module as taking lexical forms directly alongside the phonology. This means that both UR and SR information is available, something that will be useful in accounting for incomplete neutralisation. That being said, its formulation so far lacks explicit parameters for controlling extra-grammatical factors such as the speaker’s intent. The BMP can be updated to include an intent variable in the input, which will scale the production in some way between the UR and SR targets. We use I for the intent variable, updating the function to: .

In other words, the inputs to the phonetic module reflect multiple factors in production: the lexical form, the phonological instructions, and the pragmatic context. This is a high-level description, and in principle one can see how the phonetic module can account for incomplete neutralisation with this kind of architecture. Nonetheless, it is beneficial to provide one possible concrete instantiation to show how the BMP can simulate the gradient incomplete neutralisation data while maintaining a discrete phonology. The one outlined below is used in the simulations in the remainder of the article.

Recall that the acoustic cues in incompletely neutralised segments are usually in the direction of what might be expected for a phonetic token of the underlying segmental quality. For example, Warner et al. (Reference Warner, Jongman, Sereno and Kemps2004) found that Dutch words containing an underlying voiced stop that was devoiced in word-final position were pronounced with a longer preceding vowel than similar Dutch words contained underlying voiceless stops in the same position. Directionality of incompleteness is therefore essential to any account of incomplete neutralisation. Additionally, Port & Crawford (Reference Port and Crawford1989) showed that the level of incompleteness seems to be scaled according to pragmatic context. Finally, it is a subset of cues that are found to be incomplete when taking the acoustic measurements. Deciding which cues show up as incomplete and why it is only a subset of cues lies beyond the scope of this article. In subsequent discussion, we talk about a single abstract cue along a one-dimensional space for an individual speaker for expositional simplicity and not epistemic commitment.

Returning to the German final devoicing example in (9), consider a one-dimensional space for some cue c in the set of all cues C that signify the voicing contrast for an individual speaker. Imagine dividing the space in such a way that there is a point where every value equal to or less than that point signifies a [+voice] sound while everything greater than that point signifies a [−voice] sound. Within the [+voice] subsection, there may even be different cue values depending on the position of the voiced sound. For example, an intervocalic voiced obstruent may be further away from a specific cue’s boundary than a word-final voiced obstruent. It is also the case that the [−voice] subsection can be full of different realisations. In the case of final devoicing, a faithful [−voice] sound may have value n in the cue space. Likewise, a [+voice] obstruent in final position may have value m in the cue space.Footnote ⁸

Since the BMP has access to both UR and SR information, the phonetic form is a blend of the phonetic form given the UR and the phonetic form given the SR. This means that the two points m and n provide a theoretical bound on the cue value for the devoiced obstruents in final position. If we assume that the intent variable introduced above controls how much influence the UR or SR has, then the cue c can in theory surface as any value between m and n. Of course, this also depends on the specific implementation of the intent value and scaling process. The next paragraph discusses one way the scaling procedure may be implemented. Figure 2 provides a visualisation of the cue space for the words in (9). Arrows point to possible realisations. Notice that it is only the in alternating case that multiple options exist for a given form.

Figure 2 Hypothetical cue space.

The main idea sketched above is that the phonetic form is some combination of UR and SR influence. How much influence is given to each is controlled by the intent variable. This is the I in the function shown at the beginning of this section. Since intent can be thought of as the percentage that a speaker wants to maintain the underlying form, one way to formalise this notion is as a value in the unit interval $[0,1]$ . Here, the lower bound, 0, represents a speaker with 0% intent to maintain the underlying contrast, while the upper bound, 1, represents a speaker who wants to maintain the underlying contrast 100%. The exact value for cue c is computed by simply taking a weighted sum of $c_{UR}$ and $c_{SR}$ . In Figure 2, $c_{UR}=m$ and $c_{SR}=n$ .

One simple way to combine the two values is to use the intent value directly as a weight. This suggests that the scaling process is linear. Another option is to allow for an exponential scaling process. Since incomplete neutralisation typically results in subtle phonetic differences, a linear weighting might indicate that we would expect to see more intermediate cue values when measuring phonetic forms. Exponential scaling still allows for the UR value to have an effect on the phonetic form, but only in circumstances where there is a high intent value will it result in anything other than subtle variation. The following formula provides an exact formulation of exponential scaling where $\alpha>0$ :

This formula has desirable properties. First, when $I=0$ , there is no effect of the UR on the output, and when $I=1$ there is no effect of SR on the output. While this may seem trivial, it does match the informal explanation of intent. Second, since the scaling weights sum to 1 it is impossible for c to fall outside the bounds set by $c_{UR}$ and $c_{SR}$ . If we assume $c_{UR} < c_{SR}$ , then for any arbitrary values of I, the only way for $c> c_{SR}$ is to have $c_{UR}\times (1-I^\alpha )> c_{SR}\times (1-I^\alpha )$ . But this reduces to $c_{UR}> c_{SR}$ which is a contradiction. This proof works the same way to show how it would not be possible to get a value lower than $c_{UR}$ in this same scenario. Third, because the $\alpha $ parameter is tied to a specific cue, it provides a potential explanation for how only certain cues can be incomplete. Again, we choose not to speculate on why certain cues show up as incomplete while others do not, but do provide this mechanism as a way to include the variation.

When $\alpha =1$ there is a linear effect of the UR on the final output. In this case, the percent influence of the UR is equal to the intent value. As $\alpha $ increases, the influence of the UR becomes less and less for lower intent values. For high values of $\alpha $ , it is only high values of intent that will allow for the UR to have any influence on the output form. This exponential scaling potentially explains why the effects of incomplete neutralisation are subtle, and also that, under extreme circumstances, speakers can produce something very UR-like (see footnote 8).

4.3. Comparison to the dynamical system approach

As discussed above, the approach of Gafos & Benus (Reference Gafos and Benus2006) has been rather influential on our formulation of the BMP. When accounting for final devoicing, they describe a constraint grammar based in nonlinear dynamics that contains separate equations for a markedness constraint (pulling the system towards a voiceless surface form) and a faithfulness constraint (pulling the system towards a voiced underlying form). The two approaches share many aspects: the lexicon and grammar are expressed in terms of functions, extragrammatical information can enter the computation, and there is no direct precedence of phonology over phonetics. Fundamentally, though, these ideas are expressed in two different mathematical frameworks. We use the language of functions and function types as used in programming language theory and other areas of theoretical computer science and discrete math. Gafos & Benus (Reference Gafos and Benus2006) use the language of nonlinear dynamics, which allows them to simultaneously express discrete and continuous aspects of a complex system.

These two approaches make very different philosophical claims about cognition in terms of the symbolic nature of cognitive knowledge. One large advantage to the dynamical systems approach when it comes to phonetics and phonology is the fact that there is no extra translation mechanism needed to turn symbolic phonological knowledge into continuous phonetic substance. Nonetheless, we believe it is instructive to imagine an instantiation of the BMP which draws directly from dynamics of Gafos & Benus (Reference Gafos and Benus2006).

Consider an instantiation of the BMP where the type realisations for the URs, the SRs, and the phonetics representations, are the same (i.e., $UR = SR = PR$ ). In particular, these representations reference specific phonetic cues which are given by differential equations of the form $\dot {x}=f(x)=-dV(x)/dx$ , which describe a time-invariant first-order dynamical system in control of a cue, where $f(x)$ is a force function acting upon the state of the system and $V(x)$ is the related potential. For concreteness, consider the force function $\dot {x}=F(x)=x^{REQ}-x$ with corresponding potential $V(x) = x^2/2 - x^{REQ}x$ , where $x^{REQ}$ is a set of target values $\{-x_0,x_0\}$ which are fixed based on the positive and negative values of some binary feature. If used as the functions for $UR$ and $SR$ in the BMP, they would represent the underlying and surface values of the relevant phonetic cue. The phonetic module in the BMP could then combine them as described in equation (10) to get the final $PR$ form. Ultimately, Gafos & Benus (Reference Gafos and Benus2006) take a different approach to their dynamics. They use a tilted anharmonic oscillator to formalise a markedness force function: $\dot {x} = M(x) = -k + x - x^3$ with corresponding potential $V_M(x)=kx-x^2/2+x^4/4$ . In addition they use a parameter $\theta $ to express contrastive intent within their faithfulness force function as a way to influence the ‘underlying’ form: $\dot {x}=F(x)=\theta (x^{REQ}-x)$ ; and corresponding potential $V_F(x)=\theta x^2/2-\theta x^{REQ}x$ . They then add the two forces together: $\dot {x}= M(x) + F(x)$ .

Our point with this exercise is to emphasise clear parallels between the BMP and the specific approach of Gafos & Benus (Reference Gafos and Benus2006). Where the BMP associates a cue value with a UR, Gafos & Benus have a force equation that places a fixed point at the corresponding lexical/underlying value for voicing (faithfulness). Where the BMP associates a cue value with a SR, they have a force equation that pulls the system towards a point corresponding to the surface value for voicing (markedness). In both cases, these values/equations are summed, but in the case of dynamical systems the scaling controlled by the contrastive intent happens within these equations themselves and not with an external parameter as is the case for the BMP.

What we continue to stress in this article is that language production involves the interaction of lexical, phonological, and extragrammatical factors which the modular feedforward model fails to capture. Since this idea can be expressed using different types of mathematical formalisms, we believe that this idea is not a property of the specific mathematical implementation, but rather a property of the high-level architecture (a ‘model’ in our terms). Our simulations below stress this fact by showing that a non-dynamical implementation involving a discrete phonological grammar can also account for the qualitative behaviour of individual language users.

In the remaining parts of this section, we present three case studies to show how our instantiation of the BMP can account for the phonetic facts of incomplete neutralisation in three distinct processes: final devoicing in German, tonal processes in Cantonese, and epenthesis in two dialects of Arabic. These three case studies also highlight the relationship between incomplete neutralisation and near-merger. We show that in all cases, the data can emerge from the same system, therefore providing a unified explanation for these phenomenon, despite previous researchers positing different mechanisms.

Since our simulations do not use dynamical systems, they provide an alternative approach to the interface. However, we are not proposing these simulations in opposition to the dynamical approach. While we believe that the success of our simulations sufficiently captures important qualitative aspects of production, it does not necessarily negate the dynamical systems approach to the interface. Our simulations are introduced with the aim to establish that the BMP is a framework with many possible instantiations. This further helps clarify which level of analysis provides the source of explanation for speaker behaviour in the phenomena we study. In our opinion, it is at the level of the model and not the level of the simulation.

4.4. Final devoicing in German

The intent argument was added to the BMP in order to account for Port & Crawford’s (Reference Hinton and Anderson1989) results from German that show that the level of incompleteness can vary based on pragmatic factors. This section shows how the intent argument and the $\alpha $ parameter can interact to simulate their findings. The simulation below focuses on burst duration, which was the main cue Port & Crawford (Reference Port and Crawford1989) found to be incomplete, and closure duration, which they found to be complete. The exact cues found to incompletely neutralise have varied across studies. For example, there have been conflicting results about whether or not preceding vowel duration is an incomplete cue in German final devoicing. Nicenboim et al. (Reference Nicenboim, Roettger and Vasishth2018) ran a statistical meta-analysis using a Bayesian random-effects regression model and found a main effect that supported vowel duration as a significant cue of incomplete neutralisation. Port & Crawford (Reference Port and Crawford1989), on the other hand, reported preceding vowel duration as being complete in their findings.

Our assumption, based on the results of Port & Crawford (Reference Port and Crawford1989), is that the level of incomplete neutralisation can be dynamically controlled based on pragmatic context. With this in mind, we provide a simulation of their results to highlight the distinction between cues that are complete and cues that are incomplete, while accounting for the pragmatic scaling based on intent. Since the conclusions of Port & Crawford (Reference Port and Crawford1989) and the meta-analysis conflict with respect to vowel duration, we avoid this cue altogether. Ultimately, our simulation results do not depend on the specific cues that neutralise incompletely or not, but rather on the working assumption that cues can vary in this way at all.

Mean values for both closure duration and burst duration for each neutralised final stop pair and each condition are shown in Table 2. These data are taken directly from Port & Crawford (Reference Port and Crawford1989: 265, Table 1).Footnote ⁹ The ratio columns were added by dividing the final /d/ values by the final /t/ values in each condition. Since only the voiceless target is recoverable from the phonetic data in final position, we rely on the ratio to relate surface final /d/ to some hidden underlying target.

Table 2 Data from Port & Crawford (Reference Port and Crawford1989) for neutralised final stops by condition. Ratio indicates the mean value of /d/ divided by the mean value of /t/.

The results are simulated by assuming a single intent value for each pragmatic context, but a different $\alpha $ value for each cue in the scaling function. Abstracting away from specific values, we assume for all cues that a value of 1 is equal to the voiceless target and a value of 0 is equal to the voiced target. Since our focus is on accounting for the different levels of incompleteness, using ratios abstracts away from the condition specific variation. Therefore, the ratios reported in Table 2 can be used to estimate intent values.

Each subfigure in Figure 3 shows the estimated cue values for both burst duration and closure duration. In general, the ratio of closure duration for underlying /d/ segments to underlying /t/ segments was 0.91 or higher for each condition. Since burst duration (represented as $+$ ) was found to vary significantly between derived and faithful surface /t/ segments in the pooled data, but closure duration (represented as $\times $ ) was not, the $\alpha $ parameter was set to 1 for burst duration and 20 for closure duration.

Figure 3 Simulated cue values for Port & Crawford’s (Reference Port and Crawford1989) results. The left plot shows values for /d/-final words; the right plot shows values for /t/-final words. The symbols $+$ and $\times $ represent burst and closure duration, respectively.

The ratios for burst duration varied from 0.29 for condition 3 to 0.88 for condition 4. Intent values were determined by subtracting the burst duration ratios from 1. The resulting plot shows that even with largely varying Intent values, the $\alpha $ parameter can make it so only a single cue shows up as being incomplete.Footnote ¹⁰

From the figures, it is possible to compare both within plots and between plots, resulting in four comparisons. Based on Port & Crawford’s (Reference Port and Crawford1989) data, we expect variation between the two cues for final /d/ and no variation between the two cues for final /t/. We should also expect to see variation between final /d/ and final /t/ for burst duration, but no variation between final /d/ and final /t/ for closure duration. In Figure 3a, the cue values are shown to vary between burst duration ( $+$ ) and closure duration ( $\times $ ), as expected. Cue values close to 1 indicate that the final /d/ that has been neutralised has acoustic properties that are basically similar to the faithful final /t/ segments. The closure duration cue values for /d/ are close to 1, as they are for /t/. For /t/, burst duration is close to 1 as well. Again, this is expected given the data. While it may seem trivial that all of the underlying final /t/ values are right at 1, given that they were the denominator for determining ratios, these values were derived with the same formula that derived the final /d/ values. That is, the same $\alpha $ and same intent values were used, but the formula ensures that final /t/ values are unaltered.

This simulation shows that the interaction of the intent and the alpha parameters captures the aggregate behaviours observed by Port & Crawford (Reference Port and Crawford1989), where burst duration was incompletely neutralised and varied according to pragmatic context, while closure duration did not. A reviewer points out that closer inspection of individual behaviour in Port & Crawford (Reference Port and Crawford1989) shows that there was variation across individuals in the manifestation of cues in relation to incompleteness as well as interpretation of pragmatic context. Our simulation could be modified to simulate this kind of population-level behaviour in different ways. One way would include a probability distribution over the I and $\alpha $ parameters from equation (10). While this may better capture group behaviours, we don’t believe that would provide any further insight into what we find important: the capacity of the individual speakers. The simulations therefore are deterministic under the assumption that a given speaker, with specific intent values, and specific alpha values would act in a certain way. Likewise, we could add an error term to account for noise in the system not captured by the simulation, but this again would not change the interpretation of the qualitative behaviour the simulation exhibits.

The overall structure of the BMP allows for lexical influence on phonetic form. It also accounts for incomplete neutralisation while maintaining a singular phonological devoicing rule, contra Port & Crawford (Reference Port and Crawford1989), who claim that their data refutes such a possibility. They write, ‘One can apparently only write accurate rules for German devoicing by making them speaker-dependent and by employing a very large set of articulatory features to capture the detailed dynamic differences between the speakers’ implementation of the contrast’ (Port & Crawford Reference Port and Crawford1989: 280). This interpretation follows from conceptualising the phonetics–phonology interface in terms of the modular feed-forward model, but it does not follow from conceptualising it in terms of the BMP. This is because the BMP is able to capture ‘dynamic differences between the speakers’ implementation of the contrast’ by recognising multiple simultaneous factors influencing phonetic production. One factor is the lexical form, and another can be a discrete phonology with a singular devoicing process. Port & Crawford (Reference Port and Crawford1989) show that pragmatic context is a necessary ingredient, which is formalised in the BMP as intent. Individual speakers’ implementation of contrast does not need to be encoded in the phonological grammar, because with the BMP speakers have access to the contrast outside of the phonological module. This highlights the roles that both competence and performance play in the production process (cf. Chomsky Reference Chomsky1965). In both cases, there is knowledge that is being used during implementation: lexical knowledge, discrete phonological knowledge and a continuous representation of contrastive intent. It follows that under the BMP a continuous phonetic output does not require a continuous phonological grammar.

4.5. Tonal near-merger in Cantonese

The similarity between incomplete neutralisation and near-merger has been well documented (Ramer Reference Ramer1996; Winter & Röttger Reference Winter and Röttger2011; Yu Reference Yu, Oostendorp, Ewen, Rice and Hume2011; Braver Reference Braver2019). While the term ‘incomplete neutralisation’ emerged from the phonetic and phonological literature, the term ‘near-merger’ originated in sociolinguistics. Near-merger can be traced back to Labov et al. (Reference Labov, Yaeger and Steiner1972) and their work on New York City English. Words like source and sauce were reported to be identical by speakers, but then consistently produced with slightly different phonetic forms. The term ‘near-merger’ is therefore usually used when two classes of sounds are perceived as being of the same category, but produced with subtle variation.

One aspect of Port et al.’s (Reference Port, Mitleb and O’Dell1981) argument for incomplete neutralisation was that listeners could correctly guess the specific word at an above-chance level, highlighting the perceptibility of the contrast. This suggests that the primary difference between incomplete neutralisation and near-merger is whether the difference is perceptible. There is also the synchronic versus diachronic distinction. Near-merger has been used by sociolinguists to explain sound change, while incomplete neutralisation is often related to the active production process.

Alternations also help distinguish the two. In the source–sauce example, there is no alternation driving the neutralisation, but incomplete neutralisation is dependent on there being an alternation. Regardless of whether or not these two phenomena are one and the same, we believe that certain cases of near-merger can be explained with the same mechanisms we have developed for incomplete neutralisation using the BMP.

Tonal near-merger in Cantonese as discussed by Yu (Reference Yu2007) is one such case. Unlike the source–sauce example, it involves morphological alternations called pinjam. These alternations involve a non-high-level tone turning into a mid-rising tone.

The derived mid-rising tones of these pinjam words were compared with lexical mid-rising tones in lexical near-minimal pairs. The f0 value at the onset of the tone, the inflection point, and peak of the rise were all found to be higher for the pinjam words. Furthermore, a follow-up study on this phenomenon showed that listeners were unable to tell the two types of mid-rising tones apart, thus giving it its near-merger status.

On first glance, this seems to make the opposite prediction of what might be expected given the UR/SR scaling account we have developed so far. The derived pinjam 35 tones should be lower than the lexical mid-rising tones since they (potentially) correspond with a a non-high-level tone. A closer look shows that the phonological analysis involves an underlying floating high tone: pɔnɡ ${}^{22}$ (55) $\rightarrow $ pɔnɡ ${}^{35}$ ‘a scale’ where parentheses indicate a floating tone. In this case, it may be interpreted that the reason that the pinjam mid-rising tone has higher f0 values than the lexically specified mid-rising tones is due to the inclusion of an underlying high tone.

Yu (Reference Yu2007) explains the data using an exemplar model with further support coming from contracted syllables (sandhi). The morphemes /tsɔ/ and /tɐk˺/ both surface with a mid-rising tone in contracted syllables:

What makes it interesting is that /tsɔ/ has an underlying mid-rising tone while /tɐk˺/ has an underlying high tone. The f0 value at all three points was found to be higher for the mid-rising tone derived from the underlying high tone than for the mid-rising tone that was underlyingly mid-rising. In the BMP, this is exactly what would be expected. That is, a surface mid-rising tone that was derived from an underlying high tone should have its f0 values raised, given a non-zero intent value. Despite the exemplar interpretation, Yu (Reference Yu2007: 207) recognises this fact and writes, ‘Thus, the extra-high f0 of the [derived mid-rising tone] can be interpreted as the retention of the tonal profile of an underlying [high] tone.’

Figure 4 shows simulated data for the sandhi process. Our goal with this simulation is to show the qualitative ‘in between-ness’ of the derived mid-rising tone at all three measured points. We take the same approach as in §4.2, where we abstract to a [0,1] cue space. In this example, 1 corresponds to a high tone (5) and 0 corresponds to a low tone (1). Using an $\alpha $ value of 2 and Intent value of 0.4, the values for three types of mappings are shown: a faithful mapping of the high tone (/55/ $\rightarrow $ [55]), a faithful mapping of a mid-rising tone (/35/ $\rightarrow $ [35]), and an alternation where an underlying high tone turns into a mid-rising tone (/55/ $\rightarrow $ [35]). Shapes indicate surface tone: squares are mid-rising and circles are high. Color indicates underlying tone: white is mid-rising and black is high. The derived mid-rising tone is therefore represented by a black square.

Figure 4 BMP prediction for Cantonese tonal merger: Simulated cue values for Yu’s (Reference McCarthy2007) tone-sandhi data.

In the simulation, the faithful mappings are unaffected by the $\alpha $ and intent values, and the values for the alternation mapping are an interpolation between these two extremes. This shows once again that this instantiation of the BMP captures important qualitative aspects of this tonal phenomenon.

A reviewer points out that our simulation fails to capture the size of the difference at different points. We agree that this is a shortcoming of the specific implementational choices. Nonetheless, our goal was to simulate the in between-ness and not the exact magnitude. One potential fix would be to vary the cue value for [55] at each point. Currently, the size of the difference is based on the size of the difference between the [35] target and the [55] target. It seems reasonable to say that the [55] peak is the true maximum (1) value on the cue dimension, and the onset and inflection point values are lower. Therefore, if we make the [55] peak relatively high enough, the difference at the peak will always be greatest. Since there is no data provided by Yu (Reference Yu2007) on the phonetic properties of the [55] tone, we leave this for future work. We stress once again that this specific implementational choice does not actually impact any claims about the model structure (i.e., the type of information available and the way it combines).

Yu (Reference Yu2007) also found that the mid-rising tone derived from an underlying high tone in the contracted syllables had higher f0 values than the pinjam mid-rising tone (also derived from an underlying high tone). The exemplar model explains this data with an averaging effect. An alternative explanation is that the act of syllable contraction highlights the underlying form more directly than pinjam, and therefore speakers are more likely to have a higher intent value, thus pulling the final phonetic form towards the underlying high tone values.

This section shows that while near-merger and incomplete neutralisation have been described as two separate phenomena, they can, in certain cases, emerge from the same basic system. The BMP only relies on a correspondence between underlying and surface forms which is anticipated through the phonological mapping. Any phonological change, whether it be morphologically driven or otherwise, will predict the same type of phonetic effects in this model. The phonetic distribution of any segment should therefore be bounded between what we would expect given the underlying form and the actual surface form.

4.6. Epenthesis in Arabic

Lebanese Arabic speakers epenthesise an [i] vowel to break up word final CC clusters. Gouskova & Hall (Reference Gouskova, Hall and Parker2009) performed an acoustic study that had speakers pronounce words with underlying forms /CVCC/ and /CVCiC/. Words of the first form are pronounced the same as the second form due to the epenthesis process. In both cases, the last vowel is an [i]. Measurements of the acoustic properties of these vowels found that the epenthetic [i] showed statistically significant differences in duration and occasionally F2 frequency when compared to [i] tokens that were present in the underlying form. Notably, the authors write, ‘epenthesis introduces something less than an [i]: the vowel is backer and shorter, all properties that would make this vowel closer to [ɨ] or [ə] – and, arguably, to zero’ (emphasis original). While they use Optimality Theory with Candidate Chains (OT-CC; McCarthy Reference McCarthy2007) to explain these findings, the fact that the acoustic properties of the epenthetic vowel are more similar to zero is expected given the BMP.

Since the BMP relies on a segmental correspondence between underlying and surface forms, the correspondent of an epenthesised segment is arguably zero. The spatial cues for a zero segment may be the neutral articulatory values for the speaker/language, but the durational cue would be zero. This means that phonologically epenthetic vowels would range from 0 ms when Intent is 1 to the target duration for an [i] vowel when Intent is 0. If the level of Intent is between 0 and 1, then the duration of the epenthetic vowel will always be closer to zero, which is exactly what Gouskova & Hall (Reference Gouskova, Hall and Parker2009) find.

Hall (Reference Hall2013) follows up on Gouskova & Hall’s (Reference Gouskova, Hall and Parker2009) work with a larger number of speakers. In the original study, it was found that the level of incompleteness varied from person to person and this finding was strengthened in the follow up study, most notably in relation to formant values. In fact, no difference in duration was found between the lexical (61 ms) and epenthetic (60 ms) vowels at the group level.Footnote ¹¹ Hall (Reference Hall2013) hypothesises that this may be a result of the faster speech rates used in data collection for this study than those used in data collection in Gouskova & Hall (Reference Gouskova, Hall and Parker2009). For this reason, our simulation focuses on the formant values.

When comparing the mean value of epenthetic versus lexical [i], Hall (Reference Hall2013) groups speakers into two categories: dramatic difference and non-dramatic difference. She further claims that the non-dramatic difference ranges from speakers with a small difference to those with no difference at all. We therefore use three groups in our simulation: dramatic difference, small difference and no difference. Notably, the dramatic-difference speakers all have a higher/fronter lexical [i] compared to the other speakers. We can take this into account in a simulation by having the dramatic speaker have a different surface [i] target than the other two types of speakers. Figure 5 shows the simulated F1 and F2 values for each type of speaker. The starting point of the arrow is the lexical [i] values and the end point of the arrow is the epenthetic [i] values.

Figure 5 Simulated formant values for lexical and epenthetic [i] vowels based on Hall (Reference Hall2013) for a dramatic-difference speaker, a small-difference speaker and a no-difference speaker.

This paragraph lists the parameters used to determine the values in the scaling simulation. For the dramatic-difference speaker, lexical [i] was assigned the F1 $\times $ F2 vector (400, 2200); the other two speakers were each assigned the vector (450, 2000), to indicate a more central vowel. Some noise was added to the second two speakers’ vectors to provide visual separation in the plot. This is because otherwise the lines on which each arrow sat would be overlapping. Since there was more movement along the F2 dimension in Hall’s (2013) data, the F2 cue was determined with an $\alpha $ value of 2 while F1 was determined with an $\alpha $ value of 2.4 (since a higher $\alpha $ leads to less incompleteness). Finally, Intent levels were set to 0.5, 0.3 and 0.15 for the dramatic-difference, small-difference and no-difference speakers. This is not the only way to simulate the different type of speakers. For example, it is possible to have a single Intent value and instead have the $\alpha $ levels for different cues vary across speakers. There is not enough empirical data to choose between simulation strategies here. Therefore, we again emphasise that this simulation is only one way to instantiate the BMP.

Another dimension that can affect the simulation results is the spatial parameters of the underlying zero form. This also varies drastically based on what choices are made in regard to PRs. If PRs are acoustic targets, then a zero morpheme would have to have some type of acoustic target even if its duration is also 0. One plausible set of values is that corresponding to the default/neutral segment in the language (Archangeli Reference Archangeli1984; Broselow Reference Broselow1984; Pulleyblank Reference Pulleyblank1988; McCarthy & Prince Reference McCarthy and Prince1994). In the simulation above, we chose a neutral vowel (schwa) as the basis for the F1 and F2 targets, but this is ultimately an implementation choice rather than an architectural choice. Our main point continues to be about the latter, but by being explicit we can investigate consequences of the former. Ultimately, it may make more sense to think about zero morphemes in terms of articulation. A durationless target may still have spatial targets, but they can be thought of as the neutral position of the articulators – which would also lead to the vowel being more central.

In the original study, Gouskova & Hall (Reference Gouskova, Hall and Parker2009) claim that the phenomenon at hand is a case of incomplete neutralisation, but Hall (Reference Hall2013) suggests that what is going on is more likely to be near-merger. Regardless of what it should be called, there is some type of intermediary effect between an underlying form and a surface form, and this is what the BMP predicts by having access to the lexicon, the phonological grammar and the pragmatic context in which utterances are being made. The BMP is agnostic to perception, and therefore the perceptibility of a given token plays no role in the synchronic phonetic realisation. This is what allows for a unified explanation of German final devoicing, Cantonese tonal merger and Lebanese Arabic epenthesis.

5.1. Background

Up to this point, we have discussed scenarios where various lexical items have identical surface forms but phonologically distinct underlying forms. In these cases, the variation between underlying and surface forms allows for interpolation between the two. We now turn our attention to a different scenario: homophony. It has been reported that many homophonic pairs have subtle phonetic differences, most notably along the temporal dimension (Walsh & Parker Reference Walsh and Parker1983; Losiewicz Reference Losiewicz1995; Gahl Reference Gahl2008; Lohmann Reference Lohmann2018a,Reference Lohmannb). Like neutralised pairs, homophones share the same surface phonological form, but unlike neutralised pairs there is no guarantee that they have diverging underlying forms. Nonetheless, the architecture of the BMP offers an explanation for the phonetic variation of homophones.

Frequency has long been known to play a role in the phonetic realisation of phonological units (Fosler-Lussier & Morgan Reference Fosler-Lussier and Morgan1998; Bybee Reference Bybee2001; Jurafsky et al. Reference Jurafsky, Bell, Gregory, Raymond, Bybee and Hopper2001; Bell et al. Reference Bell, Brenier, Gregory, Girand and Jurafsky2009). Leslau (Reference Leslau1969) reports that the Arab grammarians were attuned to this phenomenon as they noted that more frequent words become ‘weaker’. Another dimension that can play a role in this phenomenon is part of speech. For example, words like road (N) and rode (V) have been found to vary in their pronunciation (Bell et al. Reference Bell, Brenier, Gregory, Girand and Jurafsky2009). Gahl (Reference Gahl2008) looked at non–function word homophone pairs such as time (N) and thyme (N) and found that there was a difference in duration that correlated with frequency of the lemma. This clearly implicates lexical frequencies in production. Based on these findings, Gahl (Reference Gahl2008) rejects discrete, symbolic lexical representations and instead argues for an exemplar-based organisation of the grammar.

5.2. Adding frequency to the BMP

In the same way that Intent is an input to the phonetic function in §4.2, frequency information is yet another input. Frequency is represented as a function F, and the phonetic function is updated accordingly: . In other words, phonetic implementation is a function that takes in the lexicon, the phonology, an intent variable and a frequency function. The frequency function we envision has the type $F :: L \rightarrow \mathbb {R}$ . Since the lexicon L is a set, the frequency function maps each item in the lexicon to a number that corresponds to its frequency. Again, the inclusion of the input form of lexical items vis-à-vis the lexicon is what allows us to account for phonetic variation. Furthermore, it is important that the same phonological form does not entail the same lexical item, since they are distinguished by syntactic and semantic information in the lexicon.

Another way to think about this is through the analogy of a computer’s memory system. Each lexical item would be represented in memory as a unique bit string. The memory system does not care about the content of what it is storing; it just has different values stored at different bit addresses. The lexicon can be thought of in this same way. Under this type of architecture, the frequency information for a given lexical item is determined by a function rather than stored directly in the lexical entry. We see this as a way to encode the difference between knowledge of language and knowledge about language. The former refers to grammatical knowledge, while the latter refers to language use. Based on the studies discussed in the previous section, it is clear that both are necessary for the production process.

Before continuing further, we introduce a function $\pi :: (UR\ |\ SR) \rightarrow PR$ that converts objects of the type $UR$ or $SR$ into a PR. Here, we assume this is a tuple of ordered cue parameter vectors. These may be articulatory or acoustic cues as long as they contain both spatial and temporal information. Given $\pi $ , formula (10) discussed in the previous section for the implementation of the intent scaling would now be (13):

Recall that L contains URs and $P(L)$ returns SRs. So this is just the intent scaling over all cues for all phonemes of a given lexical item that is being produced. Here, $\tau $ can be thought of as determining the overall target value with type $\tau :: L \rightarrow P \rightarrow I \rightarrow \{PR\}$ . It therefore provides a foundation that other factors can slightly alter. With that idea in mind, consider a duration scaling factor $\delta :: \mathbb {R}\rightarrow [0,1]^n$ . Specifically, $\delta $ maps frequencies to the unit interval. These functions $\pi $ , $\tau $ and $\delta $ can be considered sub-programs within the larger phonetic function .

In order to complete our description of this process, we also need to explain how the various input elements interact. In this simulation, we propose that the target value output by the $\tau $ function is multiplied by the output of the $\delta $ function to provide a frequency-scaled phonetic output. Following the assumption that the PR is a vector of parameters, the $\delta $ function outputs a vector rather than a scalar. In this way, frequency effects can occur under the architecture of the BMP without needing to be encoded directly in the lexicon or the phonological grammar. Instead, they are just one more factor that influences production alongside the lexicon, the phonological grammar and pragmatic intent.

Our particular implementation is inspired by Pierrehumbert’s (Reference Pierrehumbert, Gussenhoven and Warner2002) simulation of leniting bias. She defines the production of a given token x as $x = x_{\textit {target}} + \varepsilon + \lambda $ , where $x_{\textit {target}}$ is the specific phonetic target that has been computed based on an exemplar model, $\varepsilon $ is some random error, and $\lambda $ is the leniting bias. This is motivated because leniting bias is closely related to duration (Priva & Gleason Reference Priva and Gleason2020), and duration is related to frequency. For our implementation, the equivalent of $x_{\textit {target}}$ is the output of $\tau (L,P,I)$ , the equivalent of $\lambda $ is the output of $\delta (F(L))$ , and instead of adding the bias term to the target, our implementation multiplies them.

While the data we model only involves temporal cues, our implementation would equally apply to spectral cues as well. This raises the question of whether frequency information can also influence non-temporal cues. The answer appears to be yes. In a recent review of phonetic reduction, Clopper & Turnbull (Reference Clopper, Turnbull, Cangemi, Clayards, Niebuhr, Schuppler and Zellers2018) discuss how various factors such as frequency affect both spectral and temporal cues. The primary spectral cue that has been investigated in relation to frequency is the F1 $\times $ F2 vowel space, which has been shown to be more contracted for more frequent words (Munson & Solomon Reference Munson and Solomon2004). Crucially, Munson & Solomon (Reference Munson and Solomon2004) found vowels in low-frequency words to be longer than vowels in high-frequency words, but found no statistically significant interaction between duration and vowel-space expansion. Therefore, a simulation that accounts for both spectral and temporal cues would necessarily have to tease apart the influence of duration from the influence of frequency. This, however, has no impact on the architecture of the BMP, since it already claims that both those types of information are available during the production process.

5.3. Homophone duration variation in English

In this section, we present a simulation that shows how the functions described in the previous section may be implemented using frequency data from CELEX (Baayen et al. Reference Baayen, Piepenbrock and Gulikers1996) and duration data from the Switchboard corpus (Godfrey et al. Reference Godfrey, Holliman and McDaniel1992). We gathered this data following the methodology presented by Gahl (Reference Gahl2008: 479–480), including using the time-aligned orthographic transcript originally created by Deshmukh et al. (Reference Deshmukh, Ganapathiraju, Gleeson, Hamaker and Picone1998). Figure 6 shows the mean duration and log frequency of 17 homophonous pairs. Each point represents a word in the corpus and is connected to its homophone by a dashed line. While there is an overall negative correlation between duration and log frequency in the plotted pairs, it is not the case that every individual pair showed a negative relationship.Footnote ¹²

Figure 6 Average duration and log frequency for 17 homophonous pairs. These data come from the Switchboard corpus (Godfrey et al. Reference Godfrey, Holliman and McDaniel1992). Thin dashed lines connect all pairs. The thick grey line is the output of a linear model ( $\mathcal {I}$ ) of these points, showing a general negative correlation.

This simulation uses a linear model to predict the effect of frequency on duration. In the remainder of this section, it will be referred to as $\mathcal {S}$ to reinforce the difference between the abstract model and this specific implementation. $\mathcal {S}$ ’s outcome variable (y) is duration (in ms) and has two predictor variables: log frequency and phonological form. This results in a single slope based on log frequency and varying intercepts based on phonological form, and can be directly related to the functions for determining duration-influenced phonetic output. (14) shows the structure of $\mathcal {S}$ in full.

These parameters can be broken down to show how they relate to the functions above. Under the operating assumption that duration scales linearly with frequency, the underlying target value, which corresponds to the function $\tau (L,P,I)$ , will be equal to the equation in (14) with $\beta _1\times LogFreq(x)$ removed. In other words, the intercept for each phonological form is the hypothesised target value.Footnote ¹³

To relate $\mathcal {S}$ to the duration-scaling function $\delta (F(L))$ above, it is necessary to do some rearranging of terms. In its current form, $\mathcal {S}$ is similar to Pierrehumbert’s (Reference Pierrehumbert, Gussenhoven and Warner2002) approach. For the sake of exposition, replace $\beta _0 + \sum _{i=1}^{|L|}\beta _i \times [l_i = x]$ with a constant k and remove the error term. The formula then becomes $y = \beta _1\times LogFreq(x) + k$ . Basic algebra derives an equivalent form: $y = k \times (1 + \frac {\beta _1\times LogFreq(x)}{k})$ . Since $\beta _1$ , the slope coefficient, is negative and $LogFreq(x)$ is guaranteed to be non-negative, the value of $1 + \frac {\beta _1\times LogFreq(x)}{k}$ is guaranteed to be less than 1. As long as $\beta _1\times LogFreq(x)$ is less than or equal to k, the value of $1 + \frac {\beta _1\times LogFreq(x)}{k}$ is also guaranteed to be greater than or equal to 0. Under these conditions, this works as a scaling factor in exactly the way necessary to implement the effect of frequency with the functions described above. The function $\delta (F(L))$ above is therefore instantiated as (15).

Figure 7 shows the individual duration values for nine randomly selected homophonous pairs as well as the output of $\mathcal {S}$ for each phonological form. $\mathcal {S}$ has a significant effect for frequency ( $\beta = -5.761$ , $t=-3.561$ , $p < 0.001$ ). To illustrate how this works, consider the pair bye $\sim $ buy. $\mathcal {S}$ predicts an intercept of 293.852 for this phonological form and therefore provides the equation $\hat {y} = 293.852 - 5.761\times LogFreq(x)$ . This can now be translated into the form $PR_{dur} = \tau (L,P,I) \cdot \delta (F(L))$ . The term $\tau (L,P,I)$ equals 293.852. For the form bye, LogFreq is equal to 5.30, making $\delta (F(L))$ equal to $(1 + \frac {-5.761\cdot 5.3}{293.852}) = (1 + \frac {-30.5333}{293.852}) = (1\, {-}\, 0.1039071) = 0.896$ . Using the same method, $\delta (F(L))$ for buy is $0.835$ . These values therefore predict that the frequency-influenced duration value for bye should be $293.852\cdot 0.896\approx 263$ . The mean duration for all tokens of bye in the data set is 261 ms. The frequency influenced duration value for buy is $293.852\cdot 0.835\approx 245$ . The mean duration for all tokens of buy in the data set is 246 ms.

Figure 7 Frequency and duration information for individual tokens of 9 randomly selected homophonous pairs. Each plot represents a single pair. The solid black lines are the predicted linear relationship for that phonological form. Dashed lines indicate 95% confidence intervals.

Success on an individual pair does not tell the entire story. To begin with, word frequency is not the only factor that affects duration. Second, the previous paragraph pairs the predicted value with the mean value for a given lexical item. Visual inspection of Figure 7 clearly shows that the data for each lexical item is quite spread. This suggests that the error term in $\mathcal {S}$ can be directly thought of as the aspects of production other than frequency that influence duration for a given production. Therefore, specific results of $\mathcal {S}$ presented here should be interpreted conservatively.

Rather than focussing on perfect prediction, the goal here was to show how the architecture of the BMP can be used to simulate this type of frequency and duration data. The assumptions being made in this simulation are: 1) the phonology maps discrete inputs to discrete outputs; 2) there are multiple inputs to the phonetic module: the target lexical item, the phonological map, the intent value and frequency information; 3) the lexical item, phonological map and intent are used to produce a PR; 4) this representation is further scaled based on frequency information for individual lexical items. Consequently, adopting an exemplar model or gradient phonology is not necessary to account for the types of duration effects that Gahl (Reference Gahl2008) and others have documented.