3.1 Probabilistic selection of ranking

Perhaps the most common approach to optionality rests on the adoption of multiple constraint rankings: each ranking produces one variant, so across the available rankings, the entire range of optionality is produced. The Partial Orders (PO) model of Anttila (1997; 2007) is representative of this kind of theory. It produces variation by positing a partial constraint ranking that is resolved into a complete ranking on each evaluation. This resolution may vary across evaluations yielding different outputs for the same input across tableaux. Anttila illustrates this framework with an analysis of optional vowel coalescence in Finnish. When the second of two adjacent vowels is low, the vowels optionally coalesce:

Coalescence is more common when the first vowel is mid than when it is high. This is captured with the ranking *EA ≫ *IA; in the grammar at large, faithfulness is unranked with respect to these constraints, so three complete rankings are possible, shown in (14). Two trigger coalescence in [ea] sequences, but just one does so with [ia].

Anttila tests this partial ranking (supplemented with handful of additional constraints that capture other properties of the system) against the rate of coalescence in various contexts as revealed by a corpus of Finnish.

Does this analysis produce individual-level variation? One interpretation – perhaps the most plausible one – of the analysis is that the grammar of each speaker of Finnish contains the partial ranking so that all three total rankings in (14) are available to each speaker. This view claims that individual-level variation exists in Finnish because each speaker’s grammar produces both ['suomea] and ['suomee]. Anttila argues for this position but does not show that it is necessary. It is also possible that the variation arises across speakers: we might have no individual-level variation within a non-uniform population so that some speakers invariably produce ['suomea] and others ['suomee]. In this case, we can reinterpret the partial ranking as what we might call a “metaranking”: it characterizes the population of Finnish speakers as a whole – the rankings that they share – but each speaker’s grammar reflects just one of the rankings in (14) with no variation. PO, then, is viable whether or not individual-level variation exists.

The question of non-uniformity in variant frequencies under PO is harder to pin down. Under the assumption that all speakers of a particular linguistic variety have the same ranking, PO predicts uniform frequencies. Because frequencies are directly tied to the total rankings derived from a partial ranking, two speakers who have the same partial ranking are necessarily predicted to produce each variant with the same frequency. But it is easy to imagine that two speakers might have different grammars – non-identical rankings – that produce the same outputs. Since their rankings differ, they may produce variants with different frequencies. For example, by reversing the ranking of *EA and *IA in Anttila’s account of Finnish coalescence, we arrive at a grammar that produces the same outputs but triggers coalescence of [ia] more often than [ea]. Frequencies could also be manipulated across speakers by changing the constraints themselves: replacing *EA with *[–LOW]A, for example, would yield a system in which [ia] is penalized by both active markedness constraints while [ea] is penalized by just one. However, we are not aware of any explicit proposal for accommodating non-uniform frequencies in these ways; the working assumption seems to be that all speakers have the same partial rankings. Which interpretation of PO is the correct one is an empirical question, and in that respect our study can be seen as a means of adjudicating between these options. We return to this issue in section 5.

What is clear, though, is that PO permits a very particular kind of non-uniform frequencies. With variants’ frequencies dictated by the subset of the factorial typology a speaker has access to, any particular set of constraints permits only a discrete set of possible frequencies. To illustrate with the Finnish analysis, if there are no fixed rankings among the three constraints, there are 3! = 6 possible rankings; for any candidate, it will win in n of these rankings, so since each ranking has a $\frac{1}{6}$ chance to emerge, each candidate’s probability is $n·\frac{1}{6}$ . Only multiples of $\frac{1}{6}$ are possible frequencies.

Suppose we impose a ranking between two of the constraints – say, *EA ≫ *IA as in Anttila’s analysis. Half of the factorial typology is now eliminated (the remaining rankings are given in (15)), and each candidate’s probability is $n·\frac{1}{3}$ . This is just a subset of the original multiples of $\frac{1}{6}$ . Imposing a second ranking eliminates either one or two of the rankings in (15). FAITH ≫ *EA eliminates all but the first ranking, and *IA ≫ FAITH eliminates all but the last ranking. *EA ≫ FAITH eliminates just the first ranking, and FAITH ≫ *IA eliminates just the last ranking. If one ranking remains, each candidate’s probability is just n; if two remain, each candidate’s probability is $n·\frac{1}{2}$ . Once again, the available frequencies are a subset of the frequencies made available by the original factorial typology. Given just three constraints, then, at most the frequencies 0, $\frac{1}{6},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{5}{6}$ , and 1 are permitted.

More generally, in a PO analysis using c constraints, each candidate’s frequency is some multiple of $\frac{1}{c!}$ . This predicts that speakers’ output frequencies will be clustered at discrete intervals around the multiples of $\frac{1}{c!}$ . Obviously more gradient-seeming frequencies – and more precise frequency predictions – can be generated by introducing more constraints, but for a theory in which frequencies are supposed to be natural consequences of the constraint ranking, this strategy seems counter to the spirit of the formalism.

In our study we do not have the data to say for sure whether PO’s discrete-intervals prediction is borne out. (Such an evaluation would also require a full PO analysis of French schwa, but we are aware of no such proposal.) PO’s predictions in this regard are quite different from other theories, however, as we will see.

PO is just one of a number of theories that rely on variation within the constraint ranking. Whereas PO exploits a partial ranking, others list the rankings directly (Anttila 2006) or posit floating blocks of constraints that can appear at different points in the ranking on different evaluations (Reynolds 1994; Nagy & Reynolds 1997). With respect to the kinds of variation these theories predict, they all behave similarly to PO because they all rely on the same basic mechanisms to derive variation and output frequencies.

Stochastic OT (S-OT; Boersma 1998; Boersma & Hayes 2001), like PO, attributes variation to the availability of multiple constraint rankings, but it implements that approach via a continuous and stochastic conception of rankings. On each evaluation, noise is added to the constraint ranking so that A ≫ B may become B ≫ A, and variation results. The variation produced by this mechanism is inherently individual-level variation: if two constraints are ranked sufficiently close together so that the added noise makes their ranking variable, there is no way to ensure that for each speaker the ranking settles out the same way on every evaluation. That is, unlike the situation with PO, we see no way of reinterpreting S-OT’s formalism so that it provides variation across but not within speakers. Furthermore, by changing the ranking probabilities for constraints across speakers (without changing the relative dominance relationships), we can produce the same range of variation for different speakers but different variant frequencies.

Serial Variation (SV; Kimper 2011) combines PO with Harmonic Serialism, a theory described by Prince & Smolensky (1993/2004) in which outputs are produced incrementally via multiple passes through the grammar. If a partial ranking is resolved differently on different iterations, an optional process may be triggered at one locus on one iteration but not at a different locus on a subsequent iteration. Since SV allows the ranking to change within a derivation, it necessarily predicts that a single speaker will be able to produce multiple surface forms for a single input. But its mechanism for modeling variant frequencies is the same as PO’s (i.e. one variant is more likely than another if there are more resolutions of the partial ranking that produce it), so its predictions regarding inter-speaker variation are the same as PO’s: it produces non-uniform frequencies if the details of the partial ranking are allowed to vary across speakers.

Finally, Local Constraint Evaluation (LCE; Riggle & Wilson 2005) decomposes each constraint into position-specific instantiations: *SCHWA@1 assigns a violation when position 1 in a candidate contains a schwa, but it assigns no violations for schwas in other positions. Adopting a variable ranking between the parent constraints *SCHWA and MAX means that the position-specific MAX constraints can be freely ranked with the position-specific *SCHWA constraints. While one evaluation might operate under *SCHWA@1, MAX@2 ≫ MAX@1, *SCHWA@2 and thereby delete a schwa in position 1 but not position 2, we could arrive at the opposite ranking on another evaluation, leading to deletion of just the schwa in position 2.

Whether LCE requires individual-level variation is not clear. If position-specific constraints are projected on each evaluation according to the number of positions in the form at hand, then it is difficult to see how, say, the interleaving of the *SCHWA and MAX constraints could be held constant across evaluations (and thus the theory would necessarily predict individual-level variation). But if position-specific constraints are full-fledged members of CON, then the theory reduces to PO in the relevant respects, and it can be reinterpreted as providing a non-uniform population with no individual-level variation if necessary. Under either view, however, non-uniform frequencies result from manipulation of the constraints responsible for variation, just like PO.

3.2 Probabilistic selection of output

The second kind of theory produces optionality by introducing some way of making multiple outputs available under a single ranking. For example, the Rank-Ordered Model of Eval (ROE; Coetzee 2004; 2006) posits that all candidates that survive to a designated point in an evaluation are possible outputs. Those that are ruled out before this cut-off line are eliminated from consideration as normal. This theory produces multiple outputs with a single constraint ranking, and since there is no way to designate one of those candidates as the winner across all evaluations for a particular input and speaker, it necessarily predicts that a given speaker will have access to multiple surface forms. ROE is intentionally vague about frequency predictions, claiming only that more harmonic outputs should be more frequent than less harmonic ones. More specific frequency predictions are left to other factors (e.g. sociolinguistic considerations). This leaves plenty of room for these other factors to trigger different variant frequencies for different speakers.

Markedness Suppression (MS; Kaplan 2011) produces variation by allowing Eval to discard violation marks at random for markedness constraints that receive a special designation on a language-particular basis. Kaplan applies this framework to variation in French schwa. The analysis designates *SCHWA as suppressible, meaning that violations it assigns can be discarded; this is indicated by ʘ*SCHWA. The tableau in (16) shows how this framework produces the variation seen in the phrase envie de te le demander ‘feel like asking you’ (see (10) and (11) in section 2), using the LTC-based grammaticality judg-ments for simplicity. Each of the indicated winners is a possible output if its violations of ʘ*SCHWA are removed,⁹ but candidate (i) cannot win because it is eliminated by higher-ranking constraints against the [dtldm] cluster.

This theory models only individual-level variation. There is no mechanism by which we can always eliminate the ʘ*SCHWA violations for only, say, [ãvidətlədəmãde] for one speaker and [ãvidtələdmãde] for another. Variation arises within a single ranking by manipulating violation marks in a way that cannot be held constant for a single speaker.

On the other hand, MS allows variants to be produced at different rates across speakers. The likelihood of suppression for any violation mark is specified by a probability p; under p = .5, e.g., each violation mark has a 50% chance of suppression. By specifying different values of p for each speaker, we can produce different output frequencies across speakers.

Finally, Maximum Entropy grammars (MaxEnt; Goldwater & Johnson 2003; Jäger & Rosenbach 2006; Jäger 2007) assign a probability to each candidate based on its performance on the constraint weighting: more harmonic candidates have a greater probability. (See especially Jesney 2007 for a study of MaxEnt’s applicability to optionality.) Under MaxEnt, violations are interpreted numerically, as in Harmonic Grammar (Legendre et al. 1990; Smolensky & Legendre 2006), whereby the number of violations of a constraint is multiplied by that constraint’s weight so that candidates that violate higher-weighted constraints receive greater penalties. Since every candidate in MaxEnt has some non-zero probability of being chosen as the output, the theory produces individual-level variation. Non-uniform frequencies can result from manipulating constraint weights so that candidates’ probabilities vary across speakers.

3.3 Summary

The theoretical landscape surveyed here is summarized in (17). Clearly, the presence or absence of individual-level variation and non-uniform populations/frequencies in French is important for more than what it tells us about French specifically. Determining whether individual speakers show variation in the realization of schwa is an important step in assessing the viability of theories that require this sort of variation. If variation in French schwa (or Finnish coalescence, for that matter) arises only across speakers, these frameworks are unsuitable. Even worse, if all optional processes behave similarly, with variation arising only across speakers, these theories model a nonexistent phenomenon. Likewise for non-uniform variant frequencies: some theories predict such non-uniformity, and others do so only under certain interpretations (interpretations that have not been explicitly developed in the literature, moreover), so determining whether or not frequencies are uniform across speakers can provide empirical support for or against particular theories.

It is worth noting that a number of these theories – SV, LCE, and MS, especially – were conceived to address what Riggle & Wilson (2005) call local optionality, a particular kind of optionality that PO and related theories are seen as incapable of producing (but see Kaplan to appear for an argument that PO can produce local optionality under the right circumstances). Under local optionality, when a form contains multiple loci of variation, those loci behave independently of each other. For example, in envie de te le demander, there is no requirement that the optional schwa deletion/insertion process apply in the same way to all four loci. This is unexpected under PO: to take all four schwas as underlying for purposes of illustration, a variable ranking between MAX and *SCHWA produces either no deletion (MAX ≫ *SCHWA) or the most deletion allowed by the LTC or other phonotactic constraints (*SCHWA ≫ MAX). Mechanics introduced by other theories, such as LCE’s position-specific constraints, are designed to grant OT more flexibility.

Thus these theories grow out of the effort to construct a formalism that provides all and only the possible outputs in optional phenomena, and much of the literature cited above with respect to theses theories is concerned with measuring formalisms against this metric. That is clearly important work, and our goal here is to add to that strand of research other criteria by which theories of optionality can be evaluated: do they place variation in the right place (individuals versus populations), and do they provide the right level of flexibility with respect to output frequencies?

In what follows, we present the results of a corpus study aimed at assessing the nature of the variation of French schwa by searching for evidence of individual-level variation. But it is not sufficient to simply show that a single speaker produces schwa in some utterances but not others. Many factors have been argued to affect schwa’s realization, and if schwa is truly optional (in some circumstances), these factors must be weeded out. For example, as we’ve seen, certain phonological contexts require schwa and others forbid it; lumping these contexts together to show that schwa appears only some of the time would be misleading. Discourse type affects schwa’s appearance (Hansen 1994), so it would be equally dangerous to show that schwa is variable in a corpus that includes a variety of discourse types: perhaps all the tokens with schwa are from formal utterances and all the tokens without it are from informal ones. In this case, schwa is not necessarily variable; instead, we’ve uncovered evidence for two different grammars, one which requires schwa and another which bans it. Fortunately, the PFC corpus allows us to control for these factors, at least to a degree. The phonological context for each token is provided by the corpus, and the corpus is composed of a handful of discourse contexts according to which tokens may be segregated.

4.1 Overview of the pfc corpus

An investigation of schwa’s variation must carefully navigate phonological, discourse, and dialectal factors. As we’ve seen, this vowel’s realization may be conditioned by these considerations, so simply determining that [ə] is realized in a subset of a particular speaker’s tokens in which it could appear does not show that schwa is in fact optional for that speaker: what appears to be variation may actually be consistent realization in one context and consistent omission in another. Likewise, in comparing frequencies of schwa omission across speakers, it is necessary to control for dialect: non-uniform frequencies between speakers may reflect different dialects, not different rates of schwa omission within a single dialect.

The PFC corpus (http://www.projet-pfc.net/; Durand et al. 2002; 2009a) allows a degree of control over all of these factors. It is an audio corpus of French, with accompanying transcriptions; in a portion of the corpus, potential and actual sites for schwa are coded for many of the factors discussed above. It is this subset of the corpus that we analyze in this paper.

The corpus represents speakers from Francophone areas around the world and includes detailed demographic information about each speaker. To our knowledge, the extant literature on schwa’s variability is most robust for Parisian and Canadian varieties of French (though many other varieties have been studied; see especially Durand et al. 2009b and Gess et al. 2012). For this reason, we restricted our analysis to the 47 speakers identified as residents of either Île de France (the region including Paris) or Canada. Beyond place of residence, the corpus has fields for potentially relevant demographic information for each speaker, such as age and level of education. Unfortunately, not all information is supplied for each speaker, and these additional demographic categories are provided for too few speakers and tokens to allow us to draw reliable conclusions about them. Consequently, we limit our analysis of individual-level factors to place of residence.

The corpus provides a total of 23828 tokens of potential schwas – positions in a word in which a schwa can appear if the word or morpheme surfaces in the proper context – for the Île de France and Canadian speakers; of these, 22% were produced with [ə], 78% were produced without [ə], and 117 are coded as unclear. These ‘unclear’ productions were excluded from further analysis.

Our choice of contexts to investigate is constrained by the potential for matching contexts coded in the corpus with contexts described in the literature. In practice, this means we are able to probe just the contexts identified in section 2. Each token is coded for the following factors which we included in our analysis.

• Discourse type (Discourse): read text, guided conversation, and free conversation. All three were included in our analyses. The corpus also includes some readings of word lists, but tokens of this type were not available for the speakers we analyzed.

• Word size and position in the word (Word): monosyllabic word, first syllable of a polysyllabic word, interior (i.e. neither initial nor final) syllable of a polysyllabic word, and last syllable of a polysyllabic word. An additional code for ‘metathesis’ indicates situations in which a schwa precedes a consonant it would normally follow; there were 5 such tokens in our dataset, which we excluded from further analysis. For the LTC-based analysis of VC__C, we also excluded the interior syllable of a polysyllabic word and last syllable of a polysyllabic word, positions which do not show variation in the dialects under consideration (see section 4.3). For the CC__C context, we examined only tokens from monosyllabic words or the first syllable of a polysyllabic word. The other contexts were excluded to avoid influence from morphology (see the discussion above (7)) and prosodic boundaries (see (8)). For our investigation of clitics, we limited our study to the monosyllabic words: all of the monosyllabic words containing schwa are clitics, and all clitics containing schwa are monosyllabic.

Furthermore, each token is coded for the following additional factors. We used these to select the tokens to be analyzed.

• Left-hand context (Left): V(#)C__, C(#)C__, start of an intonation group, uncertain schwa to the left, and simplified consonant group. These last two codes indicate contexts in which the schwa was preceded by another potential schwa whose presence was unclear to the coder or by a cluster in which not all of the consonants were realized; we excluded these 500 tokens from our analysis to ensure a homogenous dataset. We also excluded the 8 tokens where Left was coded as the beginning of an intonational phrase but Word was coded as an interior or final syllable of a polysyllabic word, as the interpretation of this context was unclear. For our LTC-based analysis and our analysis of clitics, we included only the V(#)C__tokens. For the CC__C context, we used only the C(#)C__tokens. In all three cases, tokens at the start of an intonational group were excluded to ensure a more homogenous dataset and avoid the complications introduced by prosodic boundaries (see (8)).

• Right-hand context (Right): __V, __C, strong intonational boundary or end of utterance, and weak tonic boundary. We used only the __C tokens in our analyses: schwa is generally prohibited prevocalically (Dell 1980; Côté 2001), and we sought to exclude the effects of prosodic boundaries from our analysis; furthermore, in this case there were insufficient tokens that carried codes for boundaries. We also omitted 1 token coded as ‘NA’ for Right.

To summarize, we identified tokens for our three contexts by combining the factors listed above in appropriate ways. For the LTC-based analysis of VC__C, we used schwas in monosyllabic words and in initial syllables of polysyllabic words that were preceded by VC and followed by C. For the CC__C context, we used schwas in monosyllabic words and the first syllable of a polysyllabic word where the schwa was preceded by C#C and followed by C.¹⁰ For clitics, we used only monosyllabic words with a schwa preceded by V#C and followed by C. In all VC__ contexts, there is the possibility that the initial V is itself a schwa; likewise, in CC__ contexts, it is possible that schwa has been omitted between the two consonants (i.e. the CC sequence alternates with CəC). We did not exclude any such tokens from our study because the code for Left indicates whether a vowel (including schwa) is present and therefore whether the token in question belongs in the VC__ category or the CC__ category. Neither the LTC nor Côté’s analysis treats VC__ where V = [ə] differently from VC__ where V is another vowel, so we also chose not to distinguish them. Likewise for CC__ in which a schwa between the consonants is possible and CC__ where it is not.

With certain tokens excluded as described in the foregoing discussion, the final dataset contained 23197 tokens, and speakers contributed a median of 453 tokens each. The smallest number of tokens contributed by any speaker was 117, and the largest number was 956.

4.2 Analysis

In what follows, we analyze the occurrence of [ə] with respect to the contexts discussed above. For each phonological context, we ask two questions. The first question is whether there is in fact individual-level variation in the use of [ə]. In each case, the answer is yes: most or all of the speakers in the corpus produce [ə] some, but not all, of the time in a given phonological context.

The second question we consider is whether all speakers produce [ə] in that context with the same frequency. To address this question, we adopt the simplifying assumption that each speaker s_i has a fixed probability p_i of producing [ə] in that context. The question, then, is whether it is likely that all 47 speakers have the same probability p – in other words, whether all speakers have the same grammar (for that context). Since the dataset represents a random sample of speakers’ productions, not every speaker will be observed to produce [ə] at a rate of exactly p_i. Thus, our question is whether the observed variation in speakers’ rates of [ə] production is consistent with the hypothesis that all speakers have the same probability of producing [ə] in a given utterance.

To test this question, we fit a series of mixed-effects logistic models predicting the occurrence of [ə] in each phonological context. Every model predicts the outcome Schwa (1 for yes, 0 for no) from the fixed effect Word (i.e. word size and position) and the random effect Discourse.¹¹ Thus, our base model for each context is a random-intercept model with the structure

where α_j[i] is the baseline probability of realizing [ə] in discourse type j, β is the effect of Word, and ɛ_i is the error term.

For each context, our primary question is whether all speakers have the same probability of producing [ə]. To show that the population is indeed non-uniform, we must establish two things: (1) that there are significant differences among speakers in their frequency of [ə], and (2) that these differences are not just dialect-level differences.

To explore the first question, we compare our baseline model to a model with an additional random intercept for Speaker. If the two models do not differ significantly in how well they fit the data, as measured by an analysis of variance (ANOVA), then we can conclude that speakers do not differ in their overall probability of producing [ə]. But if the model with a random effect of Speaker is significantly better than the one without, then we can conclude that speakers do in fact differ.

To explore the second question, we use the factor City as a crude approximation of dialect-level variation, and treat the factors City and Speaker as competitors. We build a superset model that includes both City and Speaker as random intercepts; if this model is significantly better than a model with City only – but not significantly better than a model with Speaker only – then we can conclude that City does not explain any variation in the data beyond what is explained by Speaker, and therefore that the variation we see is truly between individual speakers.

Because we build a separate statistical model for each phonological context, we are unable to say anything about more general effects of the phonological or morphosyntactic features that define these contexts. We adopt this procedure because we are interested, not in the factors that influence the realization of [ə] by themselves, but rather in what the variability of [ə] looks like once known sources of non-optionality have been factored out. For example, all three subsets of the data include clitics; however, we are unable to say anything about how clitics in general affect the realization of [ə] (or how they interact with other factors). Such a question is beside the point: we are not in fact interested in clitics per se; we want to zero in on environments that are known in the literature to foster variation, and the environments we analyze all happen to include clitics.

4.3 The Law of Three Consonants

In this section we present analyses that assume that the Law of Three Consonants accurately captures the environments that lead to schwa’s variability. In particular, we follow Gess et al.’s (2012) implementation of the LTC: “a schwa is usually present after two consonants, while it may not be realized after a single one” (Gess et al. 2012: 5). According to this view, schwa is required (or at least strongly favored) in the context CC__C, whereas it is optional in the context VC__C. Thus in this section we include data like that in (2) (e.g. j’arriverai demain ‘I shall arrive tomorrow’ [ӡarivrɛdəmɛ̃] ∼ [ӡarivrɛdmɛ̃]) but not data like that in (3) (e.g. une fenêtre ‘a window’ [ynfənɛtr] ∼ [ynfnɛtr]) because only in the former is schwa predicted by the LTC to be optional.

Gess et al.’s simple statement of the LTC requires modifications for particular dialects. For this reason we focus on the speakers from Île de France and Quebec from the corpus and present separate analyses of each dialect. We take as our guides for these dialects Hansen (2012) and Côté (2012), who describe the behavior of schwa in Parisian and Quebec French, respectively, using the PFC corpus.

4.3.1 Parisian French

Hansen (2012) finds that while the LTC is generally a good guide to Parisian speakers’ behavior, certain contexts require special mention. First, she finds that [ə] almost never appears in word-medial VC__C tokens (i.e. schwa does not appear after a single consonant in non-word-initial, non-word-final syllables), and that it is extremely rare in word-final syllables in this context. Our findings in the PFC corpus for the Île de France region are similar: only 4.71% (19 out of 403 tokens) of word-medial tokens, and 1.11% (53 out of 4790 tokens) of word-final tokens, realize [ə] in this context. Thus, we omit word-medial and -final syllables from further analysis because they do not appear to foster robust variation.

The final dataset comprises 2359 tokens, of which 1125 (47.7%) are realized with [ə]. As shown in Figure 1, there is clear individual-level variation: in free and guided conversation, the vast majority of speakers neither invariably produce nor omit [ə]. (In read speech, unsurprisingly, [ə] is much more frequent, and many speakers approach categorical behavior.)

Figure 1

Histograms of individual Île de France speakers’ frequency of word-initial and clitic [ə], broken down by discourse type.

Turning to our second question – whether speakers have different probabilities of producing [ə] – we find that a model with a random effect of Speaker improves significantly over a baseline model without it (χ²(3, 4) = 49.3, p = 2.18e – 12). A model with a random effect of City does not improve on the baseline model (χ²(3, 4) = 2.34e – 10, p = 1); unsurprisingly, a model with both Speaker and City improves on the model with City only (χ²(4, 5) = 49.3, p = 2.18e – 12), but not on the model with Speaker only (χ²(4, 5) = 0, p = 1). We conclude that in this context, the population is genuinely non-uniform in terms of variant frequencies, and this non-uniformity cannot be reduced to dialect-level variation.

Table 1 shows the fixed effects of the final model. The effect of Word is marginally significant; the positive sign of the estimate for this factor shows that [ə] is somewhat more likely to appear in the first syllable of a polysyllabic word than it is in the baseline condition (monosyllabic words, which are all clitics). This is consistent with claims in the existing literature (Côté 2001; Hansen 2012).

Table 1

Fixed effects of the final model of VC__C schwa in Île de France.

	Estimate	St. Error	z-value	Pr(>|z|)
(Intercept)	0.11	0.88	0.12	0.90
Word: 1st σ of polysyllabic	0.26	0.13	1.94	0.05

4.3.2 Quebec French

Côté’s (2012) study shows that Quebec French is roughly consistent with the LTC. Like Hansen, Côté finds that [ə] does not appear word-medially in the context VC__C or word-finally, a claim also supported by our data: among speakers from Quebec, [ə] is realized in 1.89% (2 out of 106 tokens) of word-medial tokens, and 1.52% (15 out of 986 tokens) of word-final tokens. With so little variation in these positions, we excluded them from our analysis. In monosyllabic words and in word-initial syllables, Côté reports variation in both VC__C and CC__C. Variation in the latter context is inconsistent with the LTC, so we excluded it, too. (These tokens are analyzed in the following section.) We are left with monosyllables and the initial syllable of polysyllabic words in the context VC__C.

As shown in Figure 2, the final dataset clearly shows non-categorical behavior by individual speakers, particularly in guided conversation and read speech. However, we find no evidence for non-uniform frequencies across speakers in Quebec French: a model with a random effect of Speaker does not significantly improve on a model without it (χ²(3, 4) = 2.4, p = 0.121). (Because the Quebec data includes speakers from only one city, we are not able to test for a random effect of City.) However, it is possible that this null result is due to inadequate power: we have only 9 speakers from Quebec, as compared to the 27 speakers from Île de France.

Figure 2

Histograms of individual Quebecois speakers’ frequency of word-initial and clitic [ə], broken down by discourse type.

4.4 Schwa in CC__C: Côté (2001)

As discussed above, the context CC__C is a much more nuanced environment under the analysis of Côté (2001) than it is under the LTC. This is not a homogenous environment for Côté: sometimes schwa is obligatory, and some-times it is optional. The summary table of the factors influencing schwa under Côté’s approach is repeated below.

In this section we focus on the CC__C tokens (like une fenêtre ‘a window’ [ynfənɛtr] ∼ [ynfnɛtr]) that were excluded in the previous section, and we do not include any VC__C tokens. To narrow the focus to those CC__C environments that Côté identifies as exhibiting optional schwa, we consider here only CC__C involving schwa in the first syllable of a polysyllabic word or in a clitic. This eliminates contexts involving derivational suffixes, which Côté claims render schwa obligatory (see (7), repeated below).

Also, while the PFC corpus provides information concerning the presence of prosodic boundaries, there are not enough such tokens in our dataset to support an analysis that includes boundary information; we therefore exclude tokens coded as involving prosodic boundaries from the analysis presented here.

Finally, to deal with the segmental contexts that preclude variation, the first author (a near-native speaker of Canadian French) transcribed the consonants in all 1232 CC__C tokens for six of the eleven Center-of-Paris speakers and all 9 Quebecois speakers. (To ensure that we had a sufficient number of tokens to analyze, we selected the six Parisian speakers with the largest numbers of tokens.) To check for reliability, another near-native speaker (not affiliated with this study) transcribed a random sample of 50 of these tokens; agreement between the two coders was 92%.

Recall that Côté argues that schwa’s absence cannot yield a triconsonantal cluster in which the middle consonant is more sonorous than both of its neighbors.¹² Relevant data are repeated in (21).

We discarded all 60 tokens with this property, of which 59 were realized with schwa, suggesting that schwa is indeed obligatory (or nearly so) in this environment. Furthermore, Côté claims that schwa’s absence cannot yield a triconsonantal cluster in which the middle consonant is a stop unless one of the following conditions holds: (a) the schwa appears in a clitic (i.e., our monosyllabic words; (22a)), (b) the following consonant is a continuant (22b), or (c) the preceding consonant is a glide (22c), a category which according to Côté includes preconsonantal /r/.

We excluded 10 tokens on these grounds. (Of these tokens, 2 lacked the supposedly obligatory schwa; although we lack sufficient data for analysis, it is interesting to note that in both cases schwa was absent from the initial syllable of the word petite “small (fem.)”.)

For these analyses, we accounted for the variability in the realization of /r/ in French (see section 2 as well as Côté 2001 and references therein) by treating prevocalic /r/ as an obstruent and /r/ in other positions as a glide.

As shown in the histograms in Figure 3, there is indeed individual-level variation in the realization of schwa in the remaining tokens, at least in the Free and Guided Conversation discourse types, confirming Côté’s claim that schwa is optional in this subset of CC__C tokens.

Figure 3

Proportion schwa use in the context CC__C by speaker.

Turning now to the question of whether the population of speakers is uniform, we find that adding a random effect of Speaker significantly improves the baseline model (χ²(3, 4) = 9.31, p = 0.00227). The random effect of City does not improve the baseline model (χ²(3, 4) = 0, p = 1); moreover, a model with both Speaker and City improves over a model with just City (χ²(4, 5) = 0, p = 0.00227), but not over a model with just Speaker (χ²(4, 5) = 9.31, p = 1). Thus, there are in fact differences among speakers in how often they produced schwa in the context CC__C, and these differences cannot be attributed to dialect-level variation. Fixed effects of the final model are shown in Table 2.

To summarize, once tokens of CC__C that possess consonantal properties that preclude schwa’s omission are excluded, we find that schwa is indeed optional in this environment. Not only does variation within speakers exist (outside of read speech, at least), but we find variation across speakers.

That is, speakers do not behave homogenously with respect to the rate of schwa’s appearance in this context, and these differences cannot be attributed to dialect-level variation.

4.5 Clitics

In this section we probe schwa’s behavior in clitics more thoroughly. While consonantal properties potentially interfere with schwa’s optionality in CC__C, Côté (2001) states that schwa in clitics is always optional with just one preceding consonant, an environment in which the LTC also predicts variation. Her examples, including those in (12) above, are limited to cases in which the clitic is not utterance-initial but is preceded by another (vowel-final) word; utterance-initial clitics do not behave quite so uniformly (Malécot 1976). We therefore restrict ourselves to the V#C__C environment. The data includes all of these tokens from Canada (and not just Quebec, unlike the previous analyses) and Île de France, so while there is overlap between this context and the tokens we used for the LTC-based analysis from section 4.3, the current tokens are not a subset of the data from that section.

As shown in the histograms in Figure 4, speakers exhibit a wide range of frequency of [ə] in clitics in this environment; there are a few speakers with categorical behavior at either end of the scale (i.e., speakers who never or always produce [ə] in this context in at least some discourse type), and many more speakers who fall somewhere in the middle. Because the data is broken down by discourse type, we see that there is substantial variation even within a single discourse context; thus, the observed variation in the use of [ə] in clitics is not an artefact of pooling data across categorical behavior in a variety of discourse types.

Figure 4

Proportion schwa use in clitics by speaker.

Our baseline models for the clitic dataset do not include a fixed effect for Word, since all tokens have the same value for this factor (monosyllabic words). A random effect of Speaker significantly improves on a model without it (χ²(2, 3) = 60.7, p = 6.56e – 15); thus, there are real differences among speakers in the frequency of [ə] in this environment.

A model with a random effect of City is superior to the model with neither random effect (χ²(2, 3) = 40.2, p = 2.32e – 10). The model with both random effects is superior to a model with only a random effect of Speaker (χ²(3, 3) = 7.18, p = 0.00736), and also superior to a model with only a random effect of City (χ²(3, 4) = 27.7, p = 1.4e – 07). Thus, for [ə] in clitics, we find evidence of non-uniformity at both the individual level and the dialect level. A population-level measure of the frequency of [ə] in clitics, even if drawn from a single dialect area, would therefore fail to capture the true range of grammars possessed by French speakers. Fixed effects of the final model are shown in Table 3.