Not only size matters: limits to the Law of Three Consonants in French phonology

Grammont’s (1914) influential Law of Three Consonants (LTC) states that French schwa is obligatorily pronounced in any CC_C sequence to avoid three-consonant clusters. Later works have shown that schwa presence is also sensitive to the nature of the consonants involved, at least at the word and phrase levels. However the LTC is still generally considered as accurate under its original formulation to describe schwa-zero alternations at the stem level. The goal of the paper is to test whether the LTC should be relaxed even in this context. The paper presents two studies using judgment data to compare the behavior of schwa in derived words (stem-level phonology) and in inflected words (word-level phonology). The results of the two studies show that the nature of consonants involved in the CC_C sequence plays a role at both stem and word levels. Moreover, the same phonotactic asymmetries among consonant clusters are found in both contexts. The data therefore support a weaker version of the stem-level vs. word-level divide than what is usually assumed for French. This conclusion is strengthened by the results of a modeling study showing that a constraint-based grammar with the same phonotactic constraints across stem-and word-level phonologies provides a better fit to the judgment data from Study 1 and Study 2 than a grammar with different phononotactic constraints in the two morphosyntactic domains. The paper also replicates a number of earlier findings on the role of morphosyntactic domains, clash avoidance, and dialectal variation in schwa-zero alternations.


Introduction
Grammont's influential Law of Three Consonants (LTC) states that French schwa is obligatorily pronounced in any CC_C sequence (where C stands for any consonant) to avoid three-consonant clusters (Grammont 1914).In later works, schwa presence has been shown to be sensitive to the nature of the consonants involved in the CC_C sequence, at least at the word and phrase levels.However the LTC is still generally considered as accurate under Grammont's original formulation to describe schwa-zero alternations at the stem level (Dell 1978;Côté 2001).In particular, schwa is generally considered to be obligatorily pronounced in any CC_C sequence within monomorphemic words (e.g.[ʁɡ_ʁ] in marguerite [maʁɡǝʁit] 'daisy') or at the boundary between a stem and a derivational suffix (e.g.[ʁd_ʁ] in garde-rie [ɡaʁd-ə-ʁi] 'Kindergarten'), regardless of the nature of the consonants involved.
The goal of this paper is to test whether the Law of Three Consonants should be relaxed even at the stem level, following Scheer's (1999) insight that at least some speakers might allow for differential treatments of CC_C sequences in this context.This question is not only relevant to French phonology but also more widely to phonological theory.In phonetically based theories of phonology, phonotactic asymmetries ultimately reflect perceptual or articulatory asymmetries (Ohala 1990;1992;Steriade 1997;Flemming 2002;Storme 2019).Phonetic explanations have been put forth to account for the role of the consonantal context in French schwa-zero alternations in particular (Côté 2001).Under the default assumption that segmental properties are largely independent from their morphosyntactic context, these phonetically based analyses predict that the same phonotactic asymmetries should be observed across morphosyntactic domains.If French schwa-zero alternations are sensitive to the nature of surrounding consonants at word and phrase levels but never at the stem level, this is potentially problematic for the hypothesis that phonotactic restrictions are phonetically grounded.
The paper presents two studies using judgment data to test whether only cluster size matters at the stem level or whether the segmental make-up of the cluster is also relevant, as reported for word and phrase levels.The two studies focus on the behavior of schwa-zero alternations at the boundary between stems and derivational suffixes (stem-level phonology) and at the boundary between stems and inflectional suffixes (word-level phonology).Each study controls for different effects that could influence schwa-zero alternations beyond the properties of consonant clusters.
Study 1 controls for dialectal effects, comparing data from a variety that is more prone to delete schwa (Swiss French) and a variety that is less prone to delete schwa (French from France).Study 2 controls for the effect of stem length, comparing schwa realization in monosyllabic stems and in disyllabic stems.The data and code are available on OSF (https://osf.io/5hvxs/).
Section 2 provides some background on schwa-zero alternations in French, with a special focus on the Law of Three Consonants, and introduces the hypotheses to be tested in the paper.
Section 3 presents Study 1. Section 4 presents Study 2. Section 5 implements the two concurrent analyses of stem-level and word-level phonologies (same vs. different phonotactic constraints across levels) as probabilistic constraint-based grammars and compares their fit to the judgment data collected in Study 1 and Study 2, thus providing a theoretically motivated and quantitative assessment of the two analyses.A constraint-based analysis is used because, as noted by Durand & Laks (2000: 32), constraints provide a very intuitive interpretation of the Law of Three Consonants as caused by a general markedness constraint *CCC banning three-consonant clusters.Also several recent theoretical papers have modeled schwa-zero alternations using probabilistic constraint-based grammars (Bayles et al. 2016;Smith & Pater 2020).

The Law of Three Consonants
In French, some morphemes alternate between a form with schwa and a form without schwa.For instance, the noun demande 'request' can be realized with a schwa as [dəmãd] or without schwa as [dmãd].Determining the factors that condition the distribution of schwa-zero alternations has been a central topic in French phonology for more than a century.Table 1 provides a nonexhaustive list of the variables that have been reported to play a role in this alternation along with a non-exhaustive list of the sources that document these effects.This table builds largely but Among these variables, the consonantal context, and in particular the number of consonants surrounding schwa, has received particular attention early on.In his influential treaty on French pronunciation, Grammont (1914: 115-116) states that a preconsonantal schwa is obligatory when preceded by two consonants (CC_C), as illustrated in (1a), but excluded when preceded by a single consonant (C_C), as illustrated in (1b).He calls this generalization the 'loi des trois consonnes' (Law of Three Consonants; LTC) and explains it as a strategy to avoid three-consonant clusters.
In (1a), the schwa form is preferred because it makes it possible to avoid the three-consonant cluster [tdm].In (1b), the schwa-less form is preferred in the absence of three-consonant clusters.

Not only size matters, at least at the word and phrase levels
Subsequent works on French schwa have provided a more nuanced view of the LTC.First, the LTC has been found to hold as a gradient rather than a categorical generalization: schwa is not obligatory in CC_C and excluded in C_C but more likely in CC_C overall than in C_C (Bürki et al. 2011;Racine & Andreassen 2012;Côté 2012;Hambye & Simon 2012;Hansen 2012).Second, not only the number but also the nature and order of surrounding consonants has been found to be relevant.C_C sequences with increasing sonority favor the schwa-less form (Bürki et al. 2011).In CC_C, schwa is more likely to be pronounced if the middle consonant is a stop than if it is a fricative (see Côté 2001: 119 and earlier references therein), as illustrated in (2).Also, schwa is more likely to be pronounced if its absence implies that an obstruent-liquid cluster (OL) is not directly followed by a vowel (Dell 1976;1985;Côté 2001), as illustrated in (3).
( This nuanced view of the LTC has been argued to be relevant at the phrase level (e.g. when the three-consonant cluster spans a word boundary, as illustrated in ( 2) and ( 3)) and at the word level (e.g. when the three-consonant cluster spans a boundary between a stem and an inflectional suffix).This latter case is illustrated in (4) with the inflectional future/conditional suffix -r- [ʁ].At the boundary between a stem and this inflectional suffix, schwa is more likely to be pronounced if its absence implies that an obstruent-liquid cluster (OL) is not directly followed by a vowel (Côté 2001: 85).However, at the stem level, the LTC is still generally considered to hold as a categorical generalization in line with Grammont's strict interpretation in (1).Three-consonant sequences (CC_C) at the stem level include sequences that are contained within a single morpheme (e.g.
[ʁg_ʁ] in marguerite 'daisy') or span a boundary between a stem and a derivational suffix (e.g. [ʁd_-ʁ derivational ] in garde-rie 'Kindergarten').In this context, schwa is generally reported to be categorically pronounced and this regardless of the nature and order of surrounding consonants (Dell 1978;Côté 2001: 85, 109;Côté 2012: 258-259; but Scheer 1999: 90 with further discussion in section 2.4).For instance, schwa is reported by Dell and Côté to be obligatory in both OL_C and CO_L at the boundary between stems and derivational suffixes, as illustrated in (5a) and (5b).This contrasts with the differential treatment of OL_C and CO_L at the phrase and word levels illustrated in (3) and (4).
(5) OL_C and CO_L clusters are reported to behave identically at the stem level a. Schwa is reported to be obligatory in OL_-

The hypothesis of strongly distinct phonologies across domains
The common view according to which the nature of consonants involved in CC_C sequences is relevant for schwa-zero alternations in words and phrases but not in stems implies that the phonological grammar may differ quite substantially across morphosyntactic domains.At the word and phrase levels, a set of phonotactic constraints referencing different types of threeconsonant clusters (e.g.*OLC, *COL, etc.) would be active, resulting in different patterns of schwa-zero alternations for different types of three-consonant clusters, as illustrated in (2), (3), and (4).At the stem level, only a single phonotactic constraint banning three-consonant clusters would be active (*CCC), 1 resulting in a single pattern of schwa-zero alternations for all CC_C sequences, as illustrated in (5).
This paper will test the hypothesis of strongly distinct phonologies across morphosyntactic domains for French by focusing on two specific three-consonant sequences (obstruent-liquidconsonant sequences and liquid-obstruent-liquid sequences) and two specific morphosyntactic contexts (derivation and inflection).Obstruent-liquid-consonant sequences (OL_C) and liquidobstruent-liquid sequences (LO_L) were chosen because they behave differently in some morphosyntactic domains (see ( 3) and ( 4)).Derivational suffixes (as illustrated in ( 5)) and inflectional suffixes (as illustrated in (4)) were chosen as examples of stem-level and word-level domains, respectively, because both involve suffixation and therefore form a minimal pair for the stem-level vs. word-level distinction.Derivational suffixes attach to stems to form complex stems, as illustrated in (6a).Inflectional suffixes attach to stems to form words, as illustrated in (6b) (Dell 1978: 7-8).would have different weights at the phrase and word levels but always exactly the same weights at the stem level.
In (7a), P(ə|context) refers to the conditional probability of schwa presence given a particular morphophonological context.According to the hypothesis of strongly distinct phonologies across stem and word levels, this probability varies depending on the nature of the consonants involved in the CC_C sequence at the word level (inflection) but it is constant for all CC_C sequences at the stem level (derivation).
The main features of a phonological grammar that can derive these asymmetries are summarized in (7b).Phonotactic constraints are indexed to specific morphosyntactic domains.
Constraint indexation is one of the ways to derive morpheme-specific phonology (Pater 2007).At the stem level, a single *CCC derivation constraint penalizes any three-consonant cluster that spans a boundary between a stem and a derivational suffix.Hence it penalizes equally OLC clusters (e.g. [ Table 2 shows how a grammar with the properties in (7b) can in principle derive the type of probability distribution hypothesized in (7a) for schwa-zero alternations.Each of the four relevant morphophonological contexts are shown in the first column.The second column shows the schwa and schwa-less variants for each context.Columns 3 to 6 show how each variant is evaluated by the four constraints in the analysis (1 indicates that the candidate in the corresponding row violates the constraint in the corresponding column).In addition to the three constraints in (7b), a faithfulness constraint Dep(ə) penalizing the epenthesis of schwa was added.This analysis assumes that the schwa-less variant is the underlying form and the schwa variant is derived through epenthesis.This is the classic analysis of French schwa at morpheme boundaries (Dell 1985). 2 To materialize the relative importance of constraints, weights (w) are used.The constraint weights in Table 2 were chosen for illustration purposes only.The harmony of a candidate (column 7) corresponds to the weighted sum of its constraint violations, as in Harmonic Grammar (Smolensky & Legendre 2006).Probabilities (last column) are calculated from those harmonies using the MaxEnt framework (Goldwater & Johnson 2003;Hayes & Wilson 2008).
The probabilities in Table 2 follow the predictions in (7a): the probability of using the schwa form (in bold characters) is constant at the stem level (derivation) regardless of the consonants involved in the CC_C sequence whereas this probability varies depending on the nature of the consonants involved at the word level (inflection).Moreover, if the weight of the constraint 2 It would be possible to analyze the schwa variant as underlying and penalize its deletion with a Max(ə) constraint.
However an additional constraint beyond Max(ə) and the three markedness constraints in (7b) would be required to motivate deletion (e.g.*ə), resulting in a slightly more complex analysis (five instead of four constraints).
banning three-consonant clusters at the stem level (derivation) is sufficiently high relative to the weight of the faithfulness constraint penalizing schwa (here the ratio of the corresponding weights is 7/1), then it is possible to derive a near categorical behavior for schwa-zero alternations in this context, with schwa being nearly obligatory (the probability is equal to 1.00 due to rounding).
More variability can be derived at the word level (inflection) if the weights for the relevant constraints are not as far apart (here the ratios for the corresponding weights are 2/1 and 1/1).
The hypothesis of strongly distinct phonologies across stem and word levels seems to be generally assumed in the literature, at least by Dell (1985) and Côté (2001).One exception is Scheer (1999: 90), who reports differential treatments for three-consonant clusters even in stems for some speakers.However he does not provide direct empirical evidence for this claim.
To the author's knowledge, the only study which tested the Law of Three Consonants in stems is Côté (2012): in a corpus study of Quebec French, she found no exception to the LTC under its categorical version (Côté 2012: 258).However, as will be further discussed in section 3.1.1,the corpus used in this study is probably too small to draw strong conclusions regarding the categorical nature of the LTC.

The hypothesis of weakly distinct phonologies across domains
Alternatively, schwa-zero alternations could be gradient and sensitive to the nature of consonants across levels but closer to categorical at the stem level, at least for some speakers (e.g.Scheer 1999).According to this view, the same phonotactic constraints against various types of three-consonant clusters would be active across levels but their effects would be potentially more subtle at the stem level.The predictions of this hypothesis are summarized in (8), with (8a) focusing on the predictions at the level of the data and (8b) on the predictions at the level of the grammar.The grammar in (8b) uses the same phonotactic constraints at the stem and word levels (contrary to the grammar in (7b)).This means that the probability of the schwa form can vary by consonant cluster both at the stem and word levels, as shown in (8a).
But it does not need to vary at both levels if the relevant constraints have the same weights for some speakers.
( The probabilities in Table 3 follow the predictions in (8a): the probability of using the schwa form (in bold characters) varies both at the stem level (derivation) and at the word level (inflection).Moreover, if the weights of the constraint banning OLC clusters (*OLC) and LOL clusters (*LOL) at the stem level are sufficiently high relative to the weight of the faithfulness constraint penalizing schwa (here the ratios of the corresponding weights are 5/1 and 4/1, respectively), then it is possible to derive near categorical behaviors for schwa-zero alternations in these contexts, with schwa being nearly obligatory (probabilities are equal to 0.98 and 0.95, respectively).However the two morphophonological contexts (OL-C derivation , LO-L derivation ) are still predicted to behave slightly differently with respect to schwa-zero alternations (the probabilities of the schwa form are distinct although close; 0.98 vs. 0.95), due to the weights of the two relevant phonotactic constraints being different (5 vs. 4).Moreover, the effects of phonotactic constraints are predicted to be more salient at the word level (inflection) if their weights are not as large relatively to the weight of the faithfulness constraint.Here the ratios for the corresponding weights are 2/1 and 1/1, to be compared to the larger ratios at the stem level (5/1 and 4/1).
A grammar with the property in (8b) can also derive uniform and near-categorical treatments for OL_C and LO_L sequences at the stem level if the two relevant constraints (*OLC derivation and *LOL derivation ) have the same weights and these weights are high (the resulting grammar then behaves like the grammar in Table 2).
In other words, this analysis shows that it is theoretically possible to predict more subtle and closer to categorical effects of phonotactic constraints at the stem level than at the word level while still assuming the same phonotactic constraints in both morphosyntactic domains.
This type of asymmetries in the effect of phonotactic constraints is actually predicted by current models of probabilistic constraint-based grammars such as MaxEnt.This type of models are indeed known to produce floor and ceiling effects, due to the sigmoid relationship they imply between harmony and probability (Zuraw & Hayes 2017: 506-510).Moreover, the same analysis can derive uniform treatments of CC_C sequences at the stem level if the relevant constraints have the same weight for some speakers.
The difference between the hypotheses of strongly vs. weakly distinct phonologies across stem and word levels ultimately hinges on whether some French speakers allow for differential treatments of CC_C sequences at the stem level.Indeed, both hypotheses can derive uniform treatments of CC_C sequences at the stem level.But only the latter hypothesis (weakly distinct phonologies) can derive differential treatments of CC_C sequences at the stem level.

Implications for phonetically based theories of phonotactic constraints
The main goal of this paper is to tease apart the two versions of the stem-level vs. word-level divide presented in sections 2.3 and 2.4 for French.This question also has theoretical implications beyond French.As anticipated in Section 1, phonetically based theories of phonotactics hold that phonotactic asymmetries ultimately reflect perceptual and articulatory asymmetries (Ohala 1990;1992;Steriade 1997;Flemming 2002;Storme 2019).According to these theories, the same phonotactic asymmetries should be reflected across the grammar's morphosyntactic domains if the same phonetic asymmetries among segments hold across these domains.Under the default assumption that segmental properties are largely independent from morphosyntactic context, these theories of phonotactics are more directly compatible with the hypothesis of weakly distinct phonologies across domains in (8).
This point can be illustrated more concretely by considering explanations that have been proposed for asymmetries among three-consonant clusters in French.The phonotactic constraints that drive French schwa-zero alternations in three-consonant sequences have been argued to ultimately have phonetic motivations (Côté 2001: 137-152).For instance, Côté proposed that schwa is more likely to appear after a medial stop (CS_C) than after a medial fricative (CF_C), as illustrated in (2), because stops have weaker perceptual cues than fricatives to signal place of articulation (Wright 2004;Jun 2004) and therefore are more in need of vocalic support.She also proposed that schwa is more likely to occur in OL_C than in CO_L, as illustrated in ( 3) and ( 4), because, in the absence of schwa, OL_C features a local sonority peak (the medial liquid) that does not correspond to a syllable peak (French does not allow syllabic liquids), in violation of the sonority sequencing principle (Clements 1990).This problem is solved if schwa is pronounced in OL_C.By contrast, CO_L already satisfies the sonority sequencing principle in the absence of schwa and therefore schwa presence is not as crucial.In both cases (CF_C vs. CS_C and OL_C vs. CO_L), the explanation refers to phonetically based asymmetries among segments.Because these segmental asymmetries are expected to hold across the grammar's morphosyntactic domains (e.g. a stop should have weaker internal cues than a fricative regardless of morphosyntactic context), then the same phonotactic asymmetries among clusters should be observed across strata, in line with the weak version of the divide between stem-level and word-level phonologies in (8).

Further hypotheses to be tested
In addition to the strong and weak versions of the stem-level vs. word-level divide, four further hypotheses will be tested in this paper.

Effects of morphosyntactic domains on schwa-zero alternations
First, as already discussed above, schwa is reported to be more likely to appear at the stem level (derivation) than at the word level (inflection) overall (Dell 1978;Côté 2001;2012). 3The goal here will be to test whether this hypothesis holds across a range of consonant clusters.The hypothesis is summarized in (9a) at the level of the data and in (9b) at the level of the grammar.
(9) Hypothesis about the effect of stem level vs. word level on schwa-zero alternations a. Data: schwa is more likely to break any given consonant cluster at the stem level (derivation) than at the word level (inflection) P(ə|(C)C_-C derivation ) > P(ə| (C)C_-C inflection ) b. Grammar: a given consonant cluster is more penalized at the stem level (derivation) than at the word level (inflection) If the hypothesis that schwa is less likely at the word level (inflection) than at the stem level (derivation) was confirmed, this would confirm the general claim that schwa becomes less likely as the morphosyntactic domain widens.Evidence for asymmetries between lower and higher morphosyntactic domains is provided by Dell (1977) and Côté (2001) at the phrase level.For instance, schwa is more likely to occur in [st_d] between a noun and its complement within the same noun phrase, as illustrated in (10a), than at the boundary between two phrases, as illustrated in (10b) (Dell 1977: 151).'She puts the list of artists in her pocket.' Côté (2001: 129-132) proposed a prosodic characterization for the kind of asymmetries illustrated in (10a) and (10b), with schwa being less likely at the edge of higher prosodic domains due to strengthening and lengthening effects.Strengthening and lengthening of consonants at the edge of high prosodic domains make the presence of schwa more superfluous for the sake of consonant identification (Côté 2001: 146-151).The prosodic analysis accounts for the asymmetry in (10a) and (10b) under the reasonable assumption that the schwa in (10a) occurs inside a phonological phrase (la liste d'artistes) whereas the schwa in (10b) occurs between two phonological phrases (la liste d'artistes and dans sa poche).However this analysis does not extend to the asymmetry between stem level and word level, as there is no prosodic boundary below the word level in French.If derivational suffixes are found to favor schwa presence more than inflectional suffixes at the stem-suffix boundary, as hypothesized in (9), this means that domain effects cannot be all reduced to prosody and that there are genuine effects of morphosyntactic domains on schwa-zero alternations, as originally formulated by Dell (1977).

Effects of consonant clusters on schwa-zero alternations
The second hypothesis to be tested concerns the relative markedness of different types of consonants clusters in French.As mentioned before, obstruent-liquid clusters are reported to be more strongly avoided before consonants than before vowels, due to the sonority sequencing principle (Dell 1976;Côté 2001).Moreover, three-consonant clusters are reported to be more strongly avoided than two-consonant clusters (Grammont 1914;Bürki et al. 2011

Effects of stem length on schwa-zero alternations
When testing the various effects mentioned above, the paper will also control for additional effects that could play a role in schwa-zero alternations, in particular the length of stems.Earlier studies have indeed shown that schwa is more likely to break a consonant cluster if schwa can avoid a stress clash, as illustrated in (12a), than in the absence of stress clash, as illustrated in (12b) (Dell 1985;Smith & Pater 2020).In French, the main stress falls on the last syllable of the word.
(12) a. Schwa is more likely to appear between two stressed syllables film russe [ˈfilm(ə)ˈʁys] 'Russian movie' b.Schwa is less likely to appear if only one adjacent syllable bears stress film danois [ˈfilm(ə)daˈnwa] 'Danish movie' French word-initial syllables are usually assumed to bear a secondary stress, corresponding phonetically to a word-initial F0 jump (Vaissière 2002).When a monosyllabic stem bearing word-initial secondary stress is suffixed with a monosyllabic suffix bearing word-final primary stress (e.g.garde-ra [ˌɡaʁd-ˈʁa] 'keep-fut.3sg'),a stress clash is predicted to arise.Schwa can then be epenthesized to avoid this clash (e.g.garde-ra [ˌɡard(ə)-ˈʁa] 'keep-fut.3sg').When the stem is disyllabic (e.g.regarde-ra [ˌʁəɡard-ˈʁa] 'look-fut.3sg'),no stress clash is expected to arise and therefore there is no prosodic motivation for schwa epenthesis.Hence, schwa should be more likely to appear after monosyllabic stems than after disyllabic stems (assuming suffixes are always monosyllabic), as summarized in (13a).At the grammatical level, a phonotactic constraint is needed to penalize stress clash, as summarized in (13b).
(13) Hypothesis about the effect of stem length on schwa-zero alternations a. Data: schwa is more likely to appear after monosyllabic stems than after disyllabic stems P(ə|monosyllabic stem) > P(ə|disyllabic stem) b.Grammar: a phonotactic constraint penalizes two adjacent stress-bearing syllables *Clash

Dialectal effects on schwa-zero alternations
Finally, the paper will also control for dialectal effects on schwa-zero alternations by looking at two French varieties with different rates of schwa realization: French from France and Swiss French.
Earlier studies have reported that French speakers from France are more likely to pronounce schwas than French speakers from Switzerland overall (Racine 2007;2008;Racine 2016).The hypothesis is summarized in (14a) at the level of the data and in (14b) at the grammatical level.3 Study 1 Study 1 tests the hypotheses of strongly vs. weakly distinct phonologies across stem and word levels while controlling for dialectal effects (France vs. Switzerland) on schwa-zero alternations.
The methods are described in section 3.1.The results are presented in section 3.2 and discussed in section 3.3.The data (study1-data.RData) and R code (study1-code.R) are available on OSF.A preliminary version of this study was published in Storme (2021).

Judgment task
The present study uses speakers' metalinguistic judgments as primary data, following a long The design of the present study was inspired by a previous study by Racine (2007;2008) that used metalinguistic judgments to estimate the likelihood of schwa and schwa-less word variants in French (see also Racine & Grosjean 2002: 312-313).More specifically, participants were asked to rate how likely they would be to pronounce schwa variants and schwa-less variants for

Experimental items and fillers
Two variables were manipulated to construct the experimental items: Cluster (with three levels: OL_C, LO_L, C_C) and Morphology (with two levels: derivation, inflection).C_C stands for any two-consonant cluster, LO_L for liquid-obstruent-liquid clusters, and OL_C for obstruent-liquidconsonant cluster.Inflected words all featured the future suffix -r-because this suffix is to the author's knowledge the only consonant-initial inflectional suffix in French.Inflected words were all presented with a subject pronoun preceding them (e.g.je chanterai 'I will sing') to ensure that they were correctly identified as inflected words.Derived words were presented without any additional information (e.g.garderie). 4  Four of the six experimental conditions included 15 words whereas the two remaining ones included 14 words. 5There was therefore a total of 88 experimental items in the study.Table 4 illustrates each condition using items that were featured in the stimulus set.27 filler items were used in addition.The fillers featured schwa in morpheme-internal position.
For each word, the schwa variant was conveyed using the word's graphic form (e.g.garderie).
The graphic form always contains an e corresponding to the schwa phone [ə].The schwa-less variant was conveyed by replacing the e by the apostrophe (e.g.gard'rie).The order of presentation of the experimental items and fillers was randomized.

Participants
21 Swiss French speakers (8 females, 13 males; recruited among students at a Swiss university) and 34 French speakers from France (27 females, 7 males; recruited online via the CNRS's platform RISC) participated in the study online, using the LimeSurvey platform (LimeSurvey 2012).The participants provided their informed consent to participate in the research and agreed to make their data available online.No sensitive information about participants was collected. 4In Study 2, the derived words are presented with a determiner to answer a reviewer's worry that the presence/absence of a clitic before the word might have an effect on the likelihood of the schwa variant. 5This difference is due to an error when typing the stimuli in the online platform.However this error is not problematic because the statistical analysis does not require the same number of observations per condition.

Suffix derivational inflectional
Cluster

Data analyses
While it is common practise to analyze ordinal data such as Likert-scale data as metric variables using linear regression, Liddell & Kruschke (2018) show that this can lead to a number of errors, including false alarms (i.e.detecting an effect that is not real), misses (i.e.failure to detect real effects), and even inversions of effects (i.e. the order of the means according to the metric scale is opposite to the true ordering of the means).One of the major problem with this metric approach is that it assumes that the distance between the response categories is equidistant whereas this might not be necessarily the case.For instance, in the case of a seven-point Likert scale, the distance between 5 and 6 might be treated differently from the distance between 2 and 3 in the participant's mind, even though the two distances are both equal to one on a metric scale.
To avoid these issues, Liddell & Kruschke (2018) recommend to analyze Likert-scale data using ordinal instead of linear regression models.
In this paper, the judgment data were modeled using the ordinal cumulative model (Bürkner & Vuorre 2019: 78-79).The cumulative model assumes that the observed ordinal response variable derives from the categorization of a latent continuous unobserved variable.In the present study, the ordinal variable is the rating of the preference for the schwa or schwa-less variant along the seven-point scale.The latent variable is the participant's underlying opinion about the relative frequency of the two variants.To model this categorization in the case of a seven-point Likert scale, the cumulative model assumes that there are six thresholds which partition the latent variant variable into seven ordered categories (1, 2, …, 6, 7).The model provides estimates both for the different conditions' means along the latent continuous variable and the position of the six thresholds.The reader is referred to Bürkner & Vuorre (2019) for further details.
A Bayesian approach was adopted (rather than a frequentist approach) for inferring the parameters of both the ordinal regression and the probabilistic grammars.This choice was motivated by the fact that Bayesian inference yields outcomes that are intuitive and easy to interpret.In particular, it provides a posterior distribution for all the model's parameters and combinations of parameter values given the data.This makes it very easy to test any hypothesis about the parameter values and about differences between parameter values.Also, Bayesian approaches virtually always converge to accurate values of the parameters (Liddell & Kruschke 2018).

Description of analysis
A Bayesian hierarchical ordinal cumulative regression was fit to the seven-point Likert-scale data as a function of dummy-coded factors Morphology (reference level 'derivation'), Cluster (reference level OL_C), and Origin (reference level 'France') and all their interactions, using Stan (Carpenter et al. 2017) and the brms package (Bürkner 2017) in R (R Core Team 2020).
The model included the maximal random effect structure justified by the study's design (Barr et al. 2013), allowing the effects and their interactions to vary by participant (Morphology, Cluster) and by word (Origin). 6The probit link function was used in order to apply a cumulative model assuming the latent variable to be normally distributed (Bürkner & Vuorre 2019: 84).The default priors of the brms package were used.Equal variances were assumed for the unobserved variables that underlie the observed ordinal variable.
Four sampling chains with 4,000 iterations with a warm-up period of 2,000 iterations for each chain were run, resulting in a total of 8000 samples.To avoid initialization at too small or too large values, initial values for the MCMC sampler were set to zero. 7 For all relevant parameters, their mean and 95% credibility interval (CI) according to the model's posterior distribution are reported.In the analysis, the parameters concern the latent unobserved continuous variable corresponding to participants' opinion about the likelihood of schwa absence.Due to the way the Likert-scale was set up, greater values correspond to a greater likelihood of schwa deletion (according to the participants).For testing hypotheses about the difference Δ between two conditions, Franke & Roettger (2019)'s recommendations were followed.The posterior probability that this difference is larger than zero (Δ > 0) is reported.
If this probability is close to 1 and furthermore zero is outside of the posterior 95% CI for Δ, compelling evidence is considered to be provided for the hypothesis that posits the existence of a difference between the relevant conditions.

Description of results
Figure 2 shows the posterior distribution (mean and 95% CI) of each response category (1, 2, …, 6, 7) for all cells in the factorial design.This posterior distribution was calculated using Equation 5 in Bürkner & Vuorre (2019: 79).This equation expresses the probability of each response category k as a function of the predictors, their corresponding regression coefficients, and the thresholds τ k and τ k-1 inferred along the latent continuous variable.
Controlling for individual differences in the treatment of three-consonant sequences in derived words.As discussed in section 2.4, only some speakers might be driving the general asymmetry between OL_C and LO_L in derived words (stem level).Differences between clusters in inflected words.Participants were also found to rate schwa absence as more likely in LO_L than in OL_C in inflection and there is compelling evidence for this difference for participants from both France ((μ French, LO_L, inf -μ French, OL_C, inf ) = 1.37,CI = [0.92,1.78], P(Δ > 0) = 1) and Switzerland ((μ Swiss, LO_L, inf -μ Swiss, CO_L, inf ) = 1.11,CI = [0.63,1.57], P(Δ > 0) = 1).Participants were found to rate schwa absence as more likely in C_C than in LO_L in inflected words and there is compelling evidence for this difference for participants from both 8 Notation () is a shorthand for the expectation (mean) of the posterior distribution of interest.It can be concluded that there is sufficient evidence to support the hypothesis that the number and nature of surrounding consonants are revelant in both derived and inflected words, with OL_C being judged overall as more likely to feature schwa than LO_L and LO_L more likely to feature schwa than C_C in both derived and inflected words.However there is individual variation at the stem level, as some speakers do not treat OL_C and LO_L differently in this context.

Discussion
The results of Study 1 are summarized in Table 5 for the six morphophonological contexts, with > indicating a greater estimated likelihood of the schwa variant.
The results of Study 1 support the hypothesis that phonologies are weakly distinct across stem and word levels (see section 2.4) against the hypothesis that they differ strongly (see section 2.3).At the stem level (derivation), not only the number but also the nature of surrounding consonants matters for schwa-zero alternations, at least for some speakers from both France and Switzerland.This means that Grammont's Law of Three Consonants should be relaxed not only at the word level but also at the stem level.At the grammatical level, the fact that some speakers show sensitivity to the nature of consonants involved in the CC_C sequence in derived words is problematic for the view that only a single *CCC constraint is active at the stem level (see the modeling study in section 5 for further confirmation).By contrast, the absence of effect for other speakers is not particularly problematic for the view that different phonotactic constraints (e.g.*OLC, *LOL) are active at the stem level: indeed, the relevant constraints could have the same (or very similar) weights for these speakers.
Furthermore, the relative markedness of clusters was found to be the same in both derived and inflected words, with OLC being more strongly avoided than LOL and LOL being more strongly avoided than CC.This is in line with the hypotheses presented in section 2.6.2.This is also consistent with phonetically based theories of phonotactic constraints that hold that asymmetries between phonotactic constraints ultimately derive from perceptual/articulatory or sonority asymmetries and therefore predict that these phonotactic asymmetries should be consistent across domains.
Three-consonant clusters were found to be more strongly avoided in derived than in inflected words, in line with the hypothesis that phonotactic restrictions are stronger in lower than in higher morphosyntactic domains (see section 2.6.1).This is consistent with the claim that there are genuine effects of morphosyntactic domains on schwa-zero alternations below the word level (Dell 1977).
However, this result did not extend to two-consonant clusters: two-consonant clusters were unexpectedly found to be more strongly avoided in inflected than in derived words, and for both speakers from France and Switzerland.This result is unexpected in two ways.The literature on French indeed does not report any asymmetry between inflection and derivation in schwa likelihood for two-consonant clusters (e.g.Côté 2001: 85).Moreover, if any asymmetry was to be observed, one would expect it to go in the opposite direction, with derivation favoring schwa presence more than inflection.Indeed, this is what has been observed for three-consonant clusters in the present study and this is also what is expected under the general hypothesis that schwa is more likely at the boundary of lower morphosyntactic domains (Dell 1977;Côté 2001).A follow-up analysis was run to test whether this could be due to uncontrolled differences in sonority in the stimuli (sonority-increasing C_C sequences favor schwa deletion).But the same results were found.
One possibility to explore in further research would be that derived words are more likely to be analyzed as morphologically simple than inflected words, and even more when the cluster at the stem-suffix boundary is simple (C_C).Derived words often have less transparent semantics than inflected words, making their morphological structure less salient (Haspelmath & Sims 2010).The presence of a simple consonant cluster (C_C) at the stem-suffix boundary could make the morpheme boundary even less salient, therefore favoring even further a non-decompositional analysis of derived words (Hay 2003).As a result, morpheme-boundary schwas could end up being reanalyzed as being morpheme-internal in (some) derived words.Contrary to morphemeboundary schwas, morpheme-internal schwas have to be analyzed as underlying, because their distribution is not entirely predictable (e.g.pelouse [p(ə)luz] 'lawn' vs. plus [p(*ə)lys] 'more').
If schwas are sufficiently unlikely in derived words with C_C sequences at the stem-suffix boundary, speakers could then tend to reanalyze these words as featuring a morpheme-internal CC sequence (without any underlying schwa).If a faithfulness constraint specifically penalizes schwa epenthesis in morpheme-internal consonant clusters (e.g.contiguity) and has a larger weight than the general constraint penalizing schwa epenthesis, then schwa could end up being more likely in inflected words with C_C at the stem-suffix boundary than in derived words reanalyzed as featuring a morpheme-internal CC sequence, despite clusters being generally more marked at the stem level than at the word level.Exploring the detailed predictions of this account (in particular whether derived words with C_C are more likely to be analyzed as morphologically simple than inflected words) is left for further research.
The finding that Swiss French speakers are systematically more likely to accept the schwaless variant than French speakers from France is in line with the conclusions reached by Racine (2007;2008) based on judgment data and by Racine et al. (2016) (among others) based on production data.Despite these systematic differences, participants from France and Switzerland behaved remarkably similarly with respect to the hypotheses studied in this paper, except for the treatment of OL_C in derived vs. inflected words (see Table 5).This similarity between the two groups of participants is line with Racine (2007)'s observation that speakers from France and Switzerland differ in their baseline rates of schwa production but otherwise follow the same general principles for schwa-zero alternations.

Study 2
Study 2 further tests the hypotheses of strongly vs. weakly distinct phonologies across stem and word levels, specifically controlling for an effect that was not controlled for in Study 1, namely the effect of stem length on schwa-zero alternations.Because French speakers from Switzerland and France were found to behave similarly with respect to the Law of Three Consonants in Study 1, Study 2 focuses on a single variety (Swiss French).The methods are described in section 4.1.

Judgment task
The same judgment task was used as in Study 1.The reader is referred to section 3.1.1for further details.

Experimental items and fillers
Three variables were manipulated to construct the experimental items: Cluster (with two levels: OL_C, LO_L), Morphology (with two levels: derivation, inflection), and Stem Length (with two levels: monosyllabic stem, disyllabic stem).C_C clusters were not considered in this follow-up study because they are not directly relevant to the paper's main research question.Inflected words were all presented with a subject pronoun preceding them (e.g.je chanterai 'I will sing') to ensure that they were correctly identified as inflected words.Whereas derived words were presented without additional information in Study 1, they were preceded by a determiner in Study 2 (e.g.la garderie).This change was implemented to address a reviewer's concern that the fact that inflected and derived words were presented differently in Study 1 could have affected the results.
Each of the eight experimental conditions included 10 words, for a total of 80 experimental items.Table 6 illustrates each condition using items that were featured in the stimulus set.55 filler items were used in addition.The fillers featured words with schwa in morpheme-internal position as well as inflected and derived words with C_C sequences.
As in Study 1, the schwa variant was conveyed using the word's graphic form (e.g.la garderie).
The schwa-less variant was conveyed by replacing the e by the apostrophe (e.g.la gard'rie).The order of presentation of the experimental items and fillers was randomized.

Data analyses
The judgment data were modeled using the ordinal cumulative model (Bürkner & Vuorre 2019: 78-79), as in Study 1.The reader is referred to section 3.1.4for details.

Description of analysis
A Bayesian hierarchical ordinal cumulative regression was fit to the seven-point Likert-scale data as a function of dummy-coded factors Morphology (reference level 'derivation'), Cluster (reference level OL_C), and Stem Length (reference level 'monosyllabic') and all their interactions, using Stan (Carpenter et al. 2017) and the brms package (Bürkner 2017) in R (R Core Team 2020).The model included the maximal random effect structure justified by the study's design (Barr et al. 2013), allowing the effects and their interactions to vary by participant (Morphology, Cluster, Stem Length) and by word (random intercept by word).An alternative model with word frequency as an additional variable was fit to the data, but this variable was not found to have a significant effect on schwa-zero alternations (β = 0.10, 95%

Description of results
Figure 4 shows the posterior distribution (mean and 95% CI) of each response category (1, 2, …, 6, 7) for all cells in the factorial design.
Controlling for individual differences in the treatment of three-consonant sequences in derived words.
As discussed in section 2.4 and as found in Study 1, only some speakers might be driving the general asymmetry between OL_C and LO_L in derived words (stem level).It can be concluded that there is sufficient evidence to support the hypothesis that the number and nature of surrounding consonants are revelant in both derived and inflected words, with OL_C being judged overall as more likely to feature schwa than LO_L in both derived and inflected words.However there is individual variation at the stem level, as some speakers do not treat OL_C and LO_L differently in this context, in particular in monosyllabic stems.

Discussion
The results of Study 2 are summarized in Table 7 for the eight morphophonological contexts, with > indicating a greater estimated likelihood of the schwa variant.
Study 2 mainly replicates the results of Study 1.The results indeed support the hypothesis that phonologies are weakly distinct across stem and word levels (see section 2.4) against the hypothesis that they differ strongly (see section 2.3).At the stem level (derivation), not only the number but also the nature of surrounding consonants matters for schwa-zero alternations, at least for some speakers.This further supports the claim that Grammont's Law of Three Consonants should be relaxed not only at the word level but also at the stem level.
Furthermore, the relative markedness of clusters was found to be the same in both derived and inflected words, with OLC being more strongly avoided than LOL, in line with the hypotheses presented in section 2.6.2 and with the results of Study 1. LOL clusters were found to be more strongly avoided in derived than in inflected words, in line with the hypothesis that phonotactic restrictions are stronger in lower than in higher morphosyntactic domains (see section 2.6.1) and with the results of Study 1.This is consistent with the claim that there are genuine effects of morphosyntactic domains on schwa-zero alternations below the word level (Dell 1977).No effect was observed for OLC clusters though.This might be interpreted as a ceiling effect, as the likelihood of schwa presence was high in this context.
The finding that monosyllabic stems are more likely to feature a schwa at the stem-suffix boundary than disyllabic stems is also consistent with the results of earlier studies showing an avoidance of adjacent prosodically prominent syllables in French (see section 2.6.3).

Grammatical modeling
The analysis of the judgment data was also supplemented with a linguistic analysis using probabilistic constraint-based grammars.In this framework, the likelihood of schwa presence/absence in the different experimental conditions can be directly interpreted in terms of constraint weights.This makes it possible to interpret the judgment data in terms of the relative strengths of phonotactic constraints against consonant clusters.In this section, the judgment data were aggregated across participants of a given French variety and across words.The constraint weights of all four grammars were inferred using a Bayesian binomial regression implemented in rjags (Plummer 2016).To help with model convergence, one of the weights was set to a constant value of 1 (the weight of *CC der in Study 1, the weight of Dep(ə) in Study 2).Following Goldwater & Johnson (2003), a Gaussian prior with mean equal to zero was chosen for all other constraint weights.Informally, this prior specifies that zero is the default weight for constraints (which means that the constraint has no effect on the output).The variance of the Gaussian prior was set to 1,000.Three MCMC chains were used with 100,000 samples and a thinning interval of 10 (which means that every 10th value in the chain was kept in the final MCMC sample while all other values were discarded).The first 5,000 samples of each chain were used for burn-in (which means they were also discarded).Convergence of the chains on the posterior distribution was assessed using the Gelman-Rubin statistic: it was very close to 1 for all parameters, 10 indicating that the samples were representative of the posterior distribution (Kruschke 2015: 181).The effective sample size for each constraint weight estimated by the model was superior to 10,000, indicating that the MCMC samples were large enough for stable and accurate numerical estimates of the posterior distributions (Kruschke 2015: 184).For model comparison, the deviance information criterion (DIC; Gelman et al. 2013: 172-173) was used.

Results for Study 1
The posterior distributions for the constraint weights are shown in Table 8a and 8b for the grammar that distinguishes three-consonant clusters in derived words and for the grammar that does not, respectively.Figure 6 shows the frequencies that each grammar predicts for the schwa variant in the 12 contexts (6 morphophonological contexts × 2 varieties) against the frequencies attested in Study 1.The grammar with the same phonotactic constraints across word and stem levels (Table 8a, Figure 6a) was found to have a smaller deviation information criterion (Δ = -18.02)than the grammar with distinct phonotactic constraints in the two levels (Table 8b, Figure 6b), indicating that the increase in goodness of fit is worth the added complexity in the grammar.In other words, the data provide evidence for constraints referencing the nature of consonants in CCC clusters even at the stem level.found in section 3.2.2 was replicated at the grammatical level: the weight of *CC inf is larger than the weight of *CC der , meaning that phonotactic constraints referring to two-consonant clusters are more strongly enforced at the word level (see section 3.3 for a potential explanation for this unexpected effect).Finally, as expected (see section 2.6.4),French speakers from Switzerland were found to weigh Dep(ə) higher than French speakers from France.

Results for Study 2
The posterior distributions for the constraint weights are shown in Table 9a and 9b for the grammar that distinguishes three-consonant clusters in derived words and for the grammar that does not, respectively.Figure 7 shows the frequencies that each grammar predicts for the schwa  variant in the 8 contexts (2 morphological contexts × 2 stem lengths × 2 three-consonant clusters) against the frequencies attested in Study 2. The grammar with the same phonotactic constraints across word and stem levels (Table 9a, Figure 7a) was found to have a smaller deviation information criterion (Δ = -77.36)than the grammar with distinct phonotactic constraints in the two levels (Table 9b, Figure 7b), indicating that the increase in goodness of fit is worth the added complexity in the grammar.In other words, the data provide evidence for constraints referencing the nature of consonants in CCC clusters even at the stem level.
Moreover, most of the grammatical hypotheses presented in section 2.6 are supported by the results of the modeling study.*OLC was found to have a greater weight than *LOL, as shown in Table 9a.As expected under the hypothesis that phonotactic constraints are more strongly enforced at the stem level than at the word level (see section 2.6.1),*OLC and *LOL were found to have greater weights in derived than in inflected words.Finally, *Clash was found to have a non-zero weight, meaning that the language does avoid sequences of stressed syllables, as expected (see section 2.6.4). (

Conclusion
Grammont's influential Law of Three Consonants (LTC) states that schwa is obligatorily pronounced in CC_C sequences in French to avoid three-consonant clusters.Although the LTC has been shown to depend on the nature and order of consonants in CC_C at the word and phrase levels, Grammont's categorical formulation is still generally considered as accurate to describe schwa-zero alternations at the stem level.The judgment data collected in the two studies presented in this paper support the hypothesis that not only the number but also the nature of surrounding consonants matters for schwa-zero alternations at the stem level (in derived words), at least for some speakers.This means that Grammont's Law of Three Consonants should be relaxed not only for word and phrase levels but also at the stem level.Furthermore, the same phonotactic asymmetries were found across levels.This is compatible with theories of phonotactics that hold that phonotactic asymmetries are not arbitrary but rooted in extragrammatical factors such as perception, articulatory effort or sonority.
The results also replicate some earlier findings about schwa-zero alternations.In particular, obstruent-liquid-consonant clusters were found to be more strongly avoided than liquid-obstruentliquid clusters, in line with findings that French follows the sonority sequencing principle.Schwa was found to be more likely to break a three-consonant cluster than a two-consonant cluster, in line with earlier findings that cluster size matters in schwa-zero alternations.Schwa was found to be more likely to be pronounced in monosyllabic stems than in disyllabic stems, in line with previous findings on clash avoidance in French.Schwa variants of words were found to be generally more acceptable by French speakers from France than from Switzerland, in line with earlier findings on the difference between the two varieties.Finally, schwa was generally found to be more likely to be pronounced in derived words (at the stem level) than in inflected words (at the word level), in line with the hypothesis that lower morphosyntactic domains favor schwa presence.This result is particularly interesting because it means that there are genuine morphosyntactic effects on schwazero alternations below the word level and therefore that asymmetries among domains cannot be all reduced to prosodic effects.One exception to the generalization that derivation favors schwa presence more than inflection was the case of C_C sequences at the stem-suffix boundary, where the opposite effect was observed (schwa was more likely in inflected than in derived words).This unexpected effect was hypothesized to be due to the greater morphological decomposability of inflected words, with derived words with simple consonant clusters at the stem-suffix boundary being more likely to be reanalyzed as monomorphemic than inflected words.This hypothesis should be explored further in future work.
Finally, the modeling study showed that it is possible to get a very good match to the judgment data using the Maxent framework for constraint-based grammars.This is line with previous research showing that this framework is well adapted to deal with phonological variability in general (e.g.Zuraw & Hayes 2017;Smith & Pater 2020).
as a stem-formation process: [[ɡarde] stem -rie] stem 'Kindergarten' b.Inflection as a word-formation process: [[ɡarde] stem -ra] word 'keep-fut.3sg'Thepredictions of the hypothesis of strongly distinct phonologies across domains are summarized in (7) for the two relevant consonant sequences (OL_C and LO_L) and the two relevant morphological contexts (derivation and inflection).The predictions are summarized in (7a) at the level of the data (probability distribution of schwa-zero alternations) and in (7b) at the level of the grammar (constraint set in each stratum).(7)Thehypothesis of strongly distinct phonologies across stem and word levels a. Data: schwa-zero alternations are sensitive to the nature of consonants at the word level (inflection) but not at the stem level (derivation) P(ə|OL_-C inflection ) ≠ P(ə|LO_-L inflection ) P(ə|OL_-C derivation ) = P(ə|LO_-L derivation ) b. Grammar: phonotactic constraints differ at the stem level (derivation) and at the word level (inflection) {*OLC inflection , *LOL inflection } {*CCC derivation } 1 Or equivalently phonotactic constraints referencing different types of three-consonant clusters (e.g.*OLC, *COL, etc.) is more likely between a noun (N) and a prepositional phrase (PP) Elle met [la D list([ə]) N d'artistes PP ] NP dans sa poche.'She puts the list of artists in her pocket.'b.Schwa is less likely between a noun phrase (NP) and a prepositional phrase (PP) Elle [met [la liste d'artist([ə])] NP [dans sa poche] PP ] VP .
(14)   Hypothesis about dialectal effects on schwa-zero alternations a. Data: schwa is more likely to appear in French from France than in Swiss French P(ə|France) > P(ə|Switzerland) b.Grammar: the faithfulness constraint penalizing schwa epenthesis has a larger weight in Swiss French than in French from France w(Dep(ə) Switzerland ) > w(Dep(ə) France ) tradition in linguistics(Schütze & Sprouse 2013;Schütze 2016;Myers 2017) and in the study of French schwa(Dell 1985;Côté 2001;Racine & Grosjean 2002;Racine 2007;Smith & Pater 2020).However, because many recent studies on French schwa are based on speech corpora instead of judgments (e.g.Bürki et al. 2011;Racine & Andreassen 2012;Côté 2012;Eychenne 2019), the use of judgment data may require some justification.First, corpus data suffer from a problem of data sparsity that is expected to be particularly acute in the present case.Indeed, the hypotheses tested in this study bear on words that feature very specific morphological and phonological properties, namely inflected and derived words with OL_C and LO_L clusters at stem-suffix boundaries.Corpora would probably need to be very large in order to feature enough occurrences of the relevant words.For instance, the corpus of Laurentian French used in Côté (2012) contain 2,530 contexts for schwa but only three instances of CC_C at the boundary between a stem and the inflectional future/ conditional suffix -r-(Côté 2012: 258).Moreover, speakers might behave differently with respect to the treatment of CC_C sequences at the stem level, as discussed in section 2.4.Controlling for this type of individual variation requires having multiple occurrences of the relevant morphophonological contexts by speaker.The corpora usually used for phonological research in French (like the corpus Phonologie du Français Contemporain; PFC;Durand et al. 2009) are not large enough to allow for this level of granularity.Finally, to the author's knowledge, available corpora of French speech do not provide the morphological information necessary to readily test the hypotheses that are central in the present study.In particular, the PFC corpus does not provide information about whether a word-internal schwa is morpheme-internal or at a morpheme boundary (seeRacine et al. 2016 for a description of the variables that were coded for schwa in PFC).
set of 115 words.The task was slightly different from the task used byRacine.In the present study, the task corresponds to a judgment of relative frequency whereas participants in Racine's study were asked to rate the absolute frequency of each variant independently.A judgment of relative frequency was used because it makes it possible to directly obtain the information most relevant for the research question of interest, namely the estimated relative frequency of the two variants.In Racine's work, an extra-step is needed to calculate the relative frequency of the two variants from their individual frequencies (seeRacine 2007: 127).FollowingRacine (2007), the judgments were elicited using a seven-point Likert scale, with 1 indicating a categorical preference for the schwa variant (e.g.garderie), 7 indicating a categorical preference for the schwa-less variant (e.g.gard'rie), and 4 indicating no preference for either form.An example is shown in Figure1.

Figure 3
shows how each individual participant in Study 1 treats OL_C and LO_L in derived words.The results confirm the hypothesis of individual variation at the stem level: some participants are more likely to delete schwa in LO_L than in OL_C (participants on the left side of the figure) whereas some participants do not treat the two sequences differently (participants on the right side of the figure).

Figure 2 :
Figure 2: Posterior distribution (mean and 95% CI) of each of the seven response categories as a function of Morphology, Cluster, and Origin.

Figure 3 :
Figure 3: Posterior distribution (mean and 95% CI) of the difference between LO_L and OL_C in derived words (stem level) by participant.A positive value means that schwa deletion is more likely in LO_L than in OL_C.

40
Swiss French speakers (22 females, 18 males; recruited among students at a Swiss university) participated in the study online, using the LimeSurvey platform (LimeSurvey 2012).The participants provided their informed consent to participate in the research and agreed to make their data available online.No sensitive information about participants was collected.

Figure 5
shows how each individual participant in Study 2 treats OL_C and LO_L in derived words with monosyllabic and disyllabic stems.The results confirm the hypothesis of individual variation at the stem level.Some participants are more likely to delete schwa in LO_L than in OL_C whereas some participants do not treat the two sequences differently.Generally, participants are more likely to treat the two clusters differently with disyllabic stems than with monosyllabic stems.There is also one outlier who is more likely to delete schwa in OL_C than in LO_L (Participant 33).Differences between clusters in inflected words.Participants were also found to rate schwa absence as more likely in LO_L than in OL_C in inflection and there is compelling evidence for this difference both in monosyllabic stems ((μ monosyll, LO_L, inf -μ monosyll, OL_C, inf ) = 1.29,CI = [0.88,1.73], P(Δ > 0) = 1) and in disyllabic stems ((μ disyll, LO_L, inf -μ disyll, CO_L, inf ) = 1.34,CI = [0.90,1.82], P(Δ > 0) = 1).

Figure 4 :
Figure 4: Posterior distribution (mean and 95% CI) of each of the seven response categories as a function of Morphology, Cluster, and Stem Length.

Figure 5 :
Figure 5: Posterior distribution (mean and 95% CI) of the difference between LO_L and OL_C in derived words (stem level) by participant and stem length (monosyllabic, disyllabic).A positive value means that schwa deletion is more likely in LO_L than in OL_C.

Table 2 :
Illustration of the analysis assuming strongly distinct phonologies at the stem level (derivation) and at the word level (inflection).O = obstruent, L = liquid, C = consonant.

Table 3 :
Illustration of the analysis assuming weakly distinct phonologies at the stem level (derivation)and at the word level (inflection).O = obstruent, L = liquid, C = consonant.
;Smith & Pater 2020).In other words, this means that schwa should be more likely to occur in OL_C than in LO_L and in LO_L than in C_C.The predictions of this hypothesis on cluster markedness are summarized in (11a) at the level of the data and in (11b) at the level of the grammar.
(11)Hypothesis about the effect of cluster type on schwa-zero alternations a. Data: schwa is more likely to appear in OL_C than in LO_L, and in LO_L than in C_C P(ə|OL_C) > P(ə|LO_L) > P(ə|C_C) b.Grammar: OLC clusters are phonotactically more marked than LOL clusters and LOL clusters are more marked than two-consonant clusters w(*OLC) > w(*LOL) > w(*CC)

Table 5 :
Summary of Study 1 results: probability of the schwa variant as a function of Cluster, Morphology, and Origin (France vs. Switzerland).

Table 7 :
Summary of Study 2 results: probability of the schwa variant as a function of Cluster, Morphology, and Stem Length.

Table 8 :
Posterior distribution of the constraint weights (mean and 95% CI) in the two grammars.

Table 9 :
Posterior distribution of the constraint weights (mean and 95% CI) in the two grammars.