2.1 Representations: internally layered ternary feet

In line with standard metrical theories, we assume that stress is the manifestation of a foot head (Liberman & Prince Reference Liberman and Prince1977, Selkirk Reference Selkirk1980, Halle & Vergnaud Reference Halle and Vergnaud1987, Hayes Reference Hayes1995).Footnote ² However, rather than adopting the standard assumptions that metrical feet are maximally binary and universally dominated by the prosodic word (2a), we utilise ILT feet in metrical representations. As previously mentioned, ILT feet result from adjoining a weak syllable to a binary foot. More specifically, our working hypothesis will be that foot structure can be recursive (2b), but only minimally so: a single foot layer can be stacked on top of a ‘minimal’ foot by adjunction.

1. (2)

   

Foot recursion involves a departure from the traditional assumption that feet must be exclusively dominated by the prosodic word (ω). However, the idea that a metrical foot might undergo minimal recursion, giving rise to a layered trisyllabic foot (2b), is not new in phonological theory. It was originally proposed by Selkirk (Reference Selkirk1980) and Prince (Reference Prince1980) in their early work on English and Estonian foot structure. Additionally, ILT feet, or fairly similar structures, have been posited in other studies (e.g. Dresher & Lahiri Reference Dresher and Lahiri1991, Rice Reference Rice1992, Kager Reference Kager1994). Such proposals have not met with the same amount of approval as standard metrical theories, but ILT feet have recently experienced a modest revival: they have been invoked for a variety of languages, many of which do not display ternary rhythm (see the references in §1, as well as Caballero Reference Caballero2008 and Kager Reference Kager2012). These studies have shown that minimally recursive feet are supported on empirical grounds that go well beyond the explanation of ternary stress. On the one hand, it has been demonstrated that some phonological processes need to differentiate between two types of foot heads: those that are dominated by one foot projection (cf. (2a)) and those that are dominated by two foot projections (cf. (2b)). This has been argued to be the case in Wargamay and Yidiɲ, two Australian languages with binary stress where only syllables with a double-head status undergo a vowel-lengthening process (see Martínez-Paricio Reference Martínez-Paricio2012, Reference Martínez-Paricio2013 for details). Similarly, it has been proposed that the layered foot provides an optimal framework for capturing reported differences among weak syllables. Note that the two unstressed syllables in an ILT foot have different structures: each weak syllable is immediately dominated by a different foot projection, and hence the phonology may exploit this structural difference in phonological processes. This has been reported to be the case in Chugach, where Low tonal accents only dock onto weak syllables that occupy the adjunct position of an ILT foot. Furthermore, in some ILT configurations, one of the two weak syllables is located at the left edge of the foot, whereas the other is placed in foot-final position, e.g. ((σσ́)Ftσ)Ft, (σ(σ́σ)Ft)Ft. Since the initial position of a foot is prominent (just like the initial position of other prosodic domains), this syllable, even if weak, may be the target of strengthening effects, and behave as phonologically stronger than the other weak syllable in an ILT foot (Bennett Reference Bennett2012, Reference Bennett2013). Such foot-initial strengthening in systems has been documented in English (Davis & Cho Reference Davis and Cho2003, Davis Reference Davis, Downing, Hall and Raffelsiefen2005), Dutch (Kager & Martínez-Paricio Reference Kager and Martínez-Paricio2014), Chugach (Leer Reference Leer1985c, Rice Reference Rice1992) and Huariapano (Bennett Reference Bennett2013), among other languages. In all these systems, only weak syllables that occur in a foot-initial position are targeted by a range of fortition phenomena, e.g. aspiration (English), glottal insertion (Huariapano), fortition of consonants (Chugach) and greater resistance to vowel reduction (Dutch).Footnote ³ Furthermore, beyond these subtle strength distinctions, the ILT foot has been argued to constitute a useful analytical device for restricting the typology of stress-window systems (Kager Reference Kager2012, based on Caballero Reference Caballero2008, Reference Caballero2011).

2.2 Non-intervention alignment constraints

All constraints used in our analysis conform to McCarthy's (Reference McCarthy2003) alignment format: (i) they are categorical (i.e. they maximally assign one violation mark per locus of violation), and (ii) they qualify as local, in the sense that they do not ‘mention more than two distinct constituents and a relation between them, such as adjacency or shared membership in a superordinate constituent’ (McCarthy Reference McCarthy2003: 80). According to McCarthy's definition, constraints involving three categories can still be local, as long as one of them is a superordinate category that contains the other two. We will see this is relevant for the definition of our constraints. In addition, to avoid complex computational problems that arise when the locus of violation refers to more than one constituent (see §4 for discussion), we follow McCarthy's restriction on the locus of violation: every locus will be a single phonological constituent (Reference McCarthy2003: 77). Finally, to make the assignment of violation marks more explicit and as a means of achieving local foci, we also follow McCarthy's proposal of formulating categorical alignment constraints according to the non-intervention format presented in Ellison (Reference Ellison1994) and Zoll (Reference Zoll1998). This non-intervention schema, which will be implemented in our constraint set, is illustrated in (7), which is adapted from McCarthy (Reference McCarthy2003: 78).

1. (7)

   

Note that Cat1 corresponds to the locus of alignment (or violation), i.e. the designated category that is required to be aligned with the (left or right) edge of a particular domain or superordinate category, Cat3. This locus of violation can be separated from the left/right edge of the superordinate category by a separator category, Cat2. In such a case, the constraint will assign only a single violation mark per locus of violation, no matter how many separators intervene between the locus of alignment and the relevant edge of the domain. To better illustrate the non-intervention format, consider the constraint in (8), which is a concrete example from our set.

1. (8)

   

This non-intervention constraint penalises maximal foot projections (Cat1=locus of violation) that are separated from the left edge of the ω (Cat3=domain, superordinate category) by some foot (Cat2=separator), independently of its minimal/maximal status. As shown in (9), it assigns fewer violations to candidate (b), with two ternary feet, than to (a), with three binary feet. However, the true function of this constraint is to promote candidates with a single foot, like (c).

1. (9)

   

Candidate (c) will be selected when constraints ensuring exhaustive parsing of syllables are low-ranked. We have indicated on the right of the tableau the exact loci that incur a violation. Note that the maximum number of violations which a constraint of this sort can assign is bounded: it equals the number of loci of violation present in a given domain. This boundedness property of our constraints will be argued to be warranted on computational grounds in §4. In the example above, where the locus of alignment is a maximal foot, the number of violation marks a particular candidate incurs will never exceed the number of maximal feet present in a prosodic word.

Note that the non-intervention relation between Cat1 and Cat3 in McCarthy's constraint format cannot always be characterised as ‘local’. Since Cat1 (e.g. an unparsed σ) and Cat3 (e.g. the designated edge of ω) can be at an unbounded distance from one another, the non-intervention constraint must be able to ‘look’ an unbounded distance from the locus of violation (Cat1) to test for the absence of an intervening Cat2 (e.g. a Ft). Locality is considered to be a fundamental principle of grammar (Chomsky Reference Chomsky1965), in the sense that it constrains the expressive formats of rules and constraints in syntax as much as in phonology. Nevertheless, we propose that the relation between Cat1 and Cat3 can be characterised as ‘local’ in a particular way. We assume (following Eisner Reference Eisner1997 and McCarthy Reference McCarthy2003) that edges of Cat3 are accessible to Cat1 and Cat2 by virtue of ‘vertical’ locality. In order to contribute to a more restrictive theory of Con and limit the size of the constraint set, we impose two vertical locality conditions on our non-intervention constraints. These conditions, stated in (10), restrict the types of prosodic elements that can occupy each of the three categories in the non-intervention format. In §3 we will demonstrate that these locality restrictions are typologically motivated.Footnote ⁴

1. (10)

   

Taken together, the conditions in (10) fully determine the values of the separator and the domain category for a given locus. First, the Daughterhood Condition establishes that the locus and separator must be immediate daughters of the superordinate category, thereby establishing the superordinate category and restricting the separator category. For instance, within the ω-domain, where the locus of alignment is the Ft, the separator category can only be a Ftmax or a σ, since only maximal feet and unfooted syllables are directly dominated by ω. Second, the Adjacency Condition requires the separator to be vertically adjacent in the prosodic hierarchy to the superordinate domain category. Hence, in cases where the domain category is ω, the separator will be Ft. In cases where the domain is Ft, the separator will be σ. These conditions greatly restrict the number of possible constraints in our model. For instance, a constraint such as Align-L/R(Ftmax, [σ]Ft, ω), in which the locus of alignment is a Ftmax, the separator category is a footed syllable and the domain or superordinate category is ω, is not possible in our framework, because it does not satisfy the locality conditions in (10). On the one hand, the Daughterhood Condition is violated because the maximal projection of the foot, Ftmax, is an immediate daughter of ω, but the footed syllable is not: it is immediately dominated by Ft, and hence is not a sister of Ftmax. On the other hand, this constraint is also defective according to the Adjacency Condition, because ω and σ are not vertically adjacent categories in the prosodic hierarchy: Ft intervenes between them.

We will assume that the locus category can be instantiated by different projections of a recursive foot, by feet with specific prominence and branchingness properties (head feet, non-branching feet) and by unfooted syllables or syllables with foot-head status. On the basis of this, we derive the sets of constraints in (11) and (18) below, all of which satisfy the two vertical locality conditions. The constraints are arranged in two classes, determined by the superordinate category, ω or Ft. In (11) we present the class that determines the location and number of feet, as well as the location of unfooted syllables within a prosodic word. Later, in (18), we present the class of constraints which regulates the location of foot heads and foot dependents. We use square brackets with a subindex to represent immediate domination in our constraint set. Since in (11) the superordinate category is the prosodic word, all loci are indicated with […]ω. In (11a) the indication that the maximal foot is immediately dominated by the prosodic word ([Ftmax]ω) is redundant, but in other cases, like (11b), where the locus category is [Ftmin]ω, the […]ω notation becomes crucial: it indicates that the constraint will only affect minimal feet that are immediately dominated by ω (i.e. non-recursive feet). This notation is equally crucial in (11f), where the locus is stated as [σ]ω, which means that the constraint only targets syllables that are immediately dominated by ω, i.e. unfooted syllables.

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A complete factorial typology of these constraints will be presented in §3. Here we concentrate on illustrating the main functions of some of these constraints and their roles in predicting ternary and binary patterns. Since the first pair of constraints, Align-L/Rtmax, was exemplified in (8) and (9), we turn to the effects of Align-L/Rmin in (11b). The locus of violation in this pair of constraints is a minimal foot that is simultaneously maximal, i.e. a non-recursive foot. The constraints have two functions. First, they draw non-recursive feet toward the left/right edge of their containing ω. Second, they share the economising function of the other alignment constraint: whereas Align-L/Rmax minimises the number of maximal feet in general, Align-L/Rmin minimises the number of maximal feet that are minimal, maximising the emergence of recursive feet. As such, they promote ternarity. The two functions are illustrated in the tableau in (12). Given a candidate with eight syllables, Align-L/Rmin favours the candidate with two ILT feet (a), rather than alternative parsings with four minimal binary feet (b) or two ILT feet and a binary foot (c), (d). (We abstract away from headedness parameters for the moment, so that the head of all candidates in (12) is in foot-initial position.)

1. (12)

   

These constraints alone cannot be responsible for selecting the ternary pattern in (13a). Note that candidates with a single recursive foot (b) or a single non-recursive foot (c) satisfy the two constraints equally well, as with Align-L/Rmax in (8).

1. (13)

   

To derive the ternary stress pattern in (12a) and (13a), Align-L/Rmin must crucially dominate the constraint that minimises the number of unparsed syllables and chains feet towards the left edge of the prosodic word. This is Chain-L, the left version of the constraint in (11f), which promotes exhaustivity while chaining any unfooted syllables to the left edge of ω and feet to the opposite edge. Hence, for a pattern like (12a), Align-L/Rmin and Chain-R must be undominated, so that all unfooted syllables are chained to the right edge of ω. When these three constraints dominate Chain-L, a system like Cayuvava, with extreme ternarity, emerges, as in (14).

1. (14)

   

Candidate (f), with unfooted syllables chained at the right edge of the prosodic word, is eliminated because it violates Chain-R. Candidates (d) and (e) are also ruled out because they incur various violations of the Align-L/Rmin constraints. Finally, candidates (a)–(c) all respect high-ranked Chain-R and Align-L/Rmin, but violate Chain-L. Since (a) has only two unfooted syllables at the right edge of ω, it incurs fewer violations of Chain-L than (b) and (c), and is thus optimal. The ranking in (14) thus promotes ternarity, while allowing a small amount of underparsing, i.e. of two syllables. Observe that Chain-L and Chain-R together eliminate the need for Parse-σ, by minimising the number of unparsed syllables. When Chain-L and Chain-R are both undominated, candidates with exhaustive parsing of syllables will surface as optimal, even if exhaustivity is achieved by means of a binary foot. This pattern is illustrated in (15a), and corresponds to languages like Estonian. These systems promote ternarity, but prefer building a binary foot to leaving two syllables unparsed.

1. (15)

   

To account for languages that also display exhaustive parsings of syllables, but invoke binary footing instead of achieving exhaustivity by means of building ternary feet, we need to introduce the constraint in (11c), which regulates the position of non-minimal feet, Align-L/Rnon-min. As can be seen in the tableaux in (16), the role of this constraint is to minimise ternary feet. When either the left or right edge version of this constraint is undominated, and ranked together with Chain-L/R above the Align-L/Rtmin constraints, a system with binary feet in even-parity forms arises, as in (16a.i). However, in odd-parity forms, exhaustivity is ensured by the presence of one ternary foot, as in (16b.ii).

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Tableaux (14)–(16) show that our constraint set predicts that there is no absolute boundary between binary and ternary systems, but that there is indeed a typological rhythmic continuum, as anticipated in §1. In §3, we will present evidence for the predicted continuum from language examples in the context of a full factorial typology. First, however, we discuss the two remaining pairs of constraints needed to account for the typology of rhythmic stress, Align-L/Runary (11d) and Align-L/Rmain (11e), which are necessary to minimise the number of unary feet in languages and select the primary stress in a word. Both constraints pull unary feet and main stress towards the edges of words.

The function of Align-L/Rmain corresponds to the old EndRule-L/R, and for this reason, there is no need for illustration. Tableau (17) shows the principal effects of Align-L/Runary: they minimise the number of unary feet in ω and, when present, they chain the unary feet to a particular ω-edge.

1. (17)

   

The second class of constraints, given in (18), has the foot as the superordinate category.

1. (18)

   

The first two constraints correspond to the traditional foot-type constraints: Trochee favours feet beginning with a strong syllable, and Iamb feet beginning with a weak syllable. The formulation of these constraints does not follow the non-intervention format in (7): they do not specify a separator category. Instead, these alignment constraints favour the presence of a head at the left edge of a foot (in the case of Trochee) or its absence (in the case of Iamb). This is for typological reasons, to avoid unary-only languages. Note that a unary foot (σ́) starts and ends with a strong syllable, while not containing any separators. Hence, if Trochee and Iamb were formulated as mirror-image non-intervention constraints, such that Trochee banned feet whose head was separated from the left edge of the foot by an intervening syllable, and Iamb banned feet whose head was separated from the right edge of the foot by an intervening syllable, a unary foot would simultaneously respect both constraints. This problem with mirror versions of the foot-form constraints is not specific to our framework, and has been noted by previous authors, who have solved it in a similar asymmetrical way, assuming all unary feet to incur a violation of Iamb (e.g. Tesar & Smolensky Reference Tesar and Smolensky2000). Finally, we propose the foot-form constraints in (18c) to determine the location of the adjunct in an ILT foot: the left version (Trocheenon-min) favours ILT feet with a right adjunct ((σσ)σ), whereas the right version (Iambnon-min) prefers feet with a left adjunct (σ(σσ)).

The interaction of these constraints will be crucial in selecting the particular shape of any ILT foot. To illustrate the effects of the constraints in (18), in (19) we provide a list of the possible feet in our model and their respective violations of these constraints.

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3.1 Method

The factorial typology of the constraint set was calculated using OTSoft (Hayes et al. Reference Hayes, Tesar and Zuraw2003). Candidates and their violations were generated using a computer script. The constraint set consisted of the twelve ω-domain constraints in (11) and the four Ft-domain constraints in (18). With 16 constraints, the number of logically possible rankings equals 16! (2·1×1013).

Candidates to be evaluated were all the logically possible metrifications of input strings ranging in length from two to eight syllables. Gen was limited to metrifications containing all logically possible combinations of the seven feet in (6) (ternary, binary and unary) plus unparsed syllables. We also excluded two types of candidates. First, candidates whose primary stress foot was in a non-leftmost or non-rightmost position were excluded. These candidates are harmonically bounded under the current constraint set. Second, all candidates satisfied culminativity: each contained one and only one primary stress foot. The total number of candidates equalled 10,612 (distributed over different word lengths).

3.2 Results

There were 316 different output patterns, all with distinct bracketings. This represents a huge reduction in the number of logically possible rankings, by a factor 1·5×10−11. For conciseness, we collapsed the 316 patterns by abstracting over the following properties: (a) primary/secondary stress; (b) left/right mirror image; (c) foot headedness. The collapsed forms for the three foot types are given in Table II.

Table II

Collapsed patterns.

We will refer to the collapsed patterns as BTU-patterns. Abstracting in this way allows for a more condensed and insightful discussion of the factorial typology than presenting all the individual patterns, since headedness at the level of Ftmin and Ftnon-min and differences between primary and secondary stresses never interact with foot distribution. For the BTU-patterns presented below as illustrations, we chose to have binary feet (σ́σ) and (σσ́) always agree in headedness with the heads of ILT feet that co-occur in the same pattern.Footnote ⁵

The factorial typology contained a total of 22 BTU-patterns, classified into nine categories on a rhythmic continuum. None of the potentially optimal candidates in the tableaux submitted to OTSoft had tied violation profiles. This was checked by running the same input file in OTWorkplace (Prince et al. Reference Prince, Tesar and Merchant2015). It was established that the total number of predicted languages was the same as the number generated in OTSoft. The factorial typology of 316 languages was also independently replicated from a newly generated input file. This replication was run in OTWorkplace on an automatically generated enlarged set of 27,526 candidates, rather than 10,612, which included all candidates with primary stress on a medial (non-peripheral) foot, all of which turned out to be harmonically bounded.

We will now discuss the 22 BTU-patterns, subdivided into nine categories. For each BTU-pattern, we will present the ranking that is responsible for generating it, and an example showing data from a language that instantiates it, with the foot boundaries hypothesised. References for metrical systems will be taken mainly from Gordon (Reference Gordon2002), and occasionally from other typological studies (Elenbaas & Kager Reference Elenbaas and Kager1999, Kager Reference Kager2005, Reference Kager2012).

3.3 The factorial typology

3.4 Discussion

The factorial typology contains 316 patterns, grouped into nine rhythmic categories, giving a total of 22 BTU-patterns. All nine rhythmic categories, except possibly one (category I), contain at least one attested stress pattern. Our discussion will focus on two criteria: undergeneration and overgeneration. We will then turn to the binary-to-ternary rhythmic continuum.

4.1 Minimal recursion

After the presentation of the empirical force and typological coverage of our theory of ILT feet, there is a major conceptual issue that still needs to be addressed: why are recursive feet different from other recursive prosodic structures? In other words, why is recursion at the foot level minimal, if prosodic recursion at other layers of the prosodic hierarchy can in principle be unbounded (Wagner Reference Wagner2005, van der Hulst Reference Hulst and van der Hulst2010)?

We believe that the answer to this question lies precisely in the different nature of supra-foot prosodic categories (e.g. the prosodic word, the phonological phrase, etc.) and lower prosodic categories such as the foot. As traditionally acknowledged in prosodic hierarchy theory (e.g. Selkirk Reference Selkirk1980, Reference Selkirk and Fretheim1981, Nespor & Vogel Reference Nespor, Vogel, van der Hulst and Smith1982, Reference Nespor and Vogel1986), prosodic constituents above the foot are generally defined in relation to their corresponding morphosyntactic categories. That is, the prosodic word is defined in relation to the morphosyntactic word, the phonological phrase in relation to the syntactic phrase, etc. (Ito & Mester Reference Ito and Mester2013: 21 and references therein). Since syntactic structure in principle displays unlimited layers of recursion, it is not unexpected that prosodic categories above the foot display unlimited layers of recursion too (e.g. Ladd Reference Ladd2008, Wagner Reference Wagner2010, Ito & Mester Reference Ito and Mester2013). The foot is different in that it is not defined by its relation with morphosyntax, but according to phonological properties such as rhythm, sonority and relative prominence. A foot expresses a rhythmic relation between strong and weak nodes in the prosodic word. Consequently, foot recursion is in principle unmotivated, and has therefore often been completely banned from Gen (Ito & Mester Reference Ito and Mester2007).

Given that there is recent non-stress evidence for ILT feet (see again §1 and §2 and references therein), as well as stress evidence (there are languages with ternary rhythm and mixed binary–ternary rhythm, but not with quaternary rhythm), this paper has investigated the possibility of minimally relaxing the ban on recursion at the foot level, exploring the typological consequences of a framework with minimally recursive feet. More specifically, in our model only Ftmin can participate in the adjunction operation in Gen, giving rise to ILT feet.

A thorough investigation of the representational and typological predictions made by a theory allowing feet with more than one layer of recursion would require another full-length study. Although we have not found compelling linguistic evidence for multiple layers of recursion at the foot level (not even in languages with one stress per ω), and even though there is no apparent syntactic motivation for recursion at this level of the prosodic hierarchy, it still needs to be explored (i) whether it is desirable to allow multiple layers of foot recursion, and (ii) whether there is in fact any compelling distributional evidence for positing such structures. If there is evidence that Gen can produce such types of multiply recursive feet, our constraint set would need to be reconsidered, since our current non-intervention constraints would favour multiply recursive feet, e.g. [((((Ft)σ)σ)σ)σ)]ω, over flatter multiply footed structures, e.g. [(Ft)(Ft)(Ft)]ω, and over structures with a single foot plus a sequence of unparsed syllables, e.g. [(Ft)σσσσ]ω, under certain circumstances. This is because our non-intervention constraints specifically target multiply footed structures and sequences of unparsed syllables, but not multiply recursive feet.

In an earlier stage of our research, we explored a version of Gen which also allowed Ftnon-min to participate in the adjunction operation, i.e. we assumed that feet could display more than one layer of recursion. However, the factorial typologies in this approach (comparing our current constraint set with others containing no restrictions on the levels of recursion allowed by Gen) were unable to predict half of the attested systems. For these reasons, and given that it is uncertain which metrical systems could possibly benefit from being analysed with multiply recursive feet, we have followed the lead of authors like van der Hulst (Reference Hulst and van der Hulst2010) and Ito & Mester (Reference Ito and Mester2007, Reference Ito and Mester2013), who suggest that the absolute upper limit on recursion in the foot reflects its intrinsic relation with rhythmic well-formedness, whereas supra-foot prosodic categories, which display a closer relation with syntax, may involve various layers of recursion.Footnote ¹⁰

4.2 Unbalanced recursion

Prosodic recursion can take one of two forms: (i) balanced, where a higher node dominates two (full) categories of the same type (41a), and (ii) unbalanced, where a category of a lower level is adjoined to a higher-level category (41b) (van der Hulst Reference Hulst and van der Hulst2010: 322, Vigário Reference Vigário2010: 491). This latter type of structure has been proposed for clitic adjunction in various languages, and matches the type of structure currently proposed for ILT feet, where a syllable is adjoined to a foot to form a superordinate foot, e.g. ((Ft)σ)Ft.

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Given that balanced prosodic structures have been proposed for other prosodic categories (e.g. a prosodic word dominating two prosodic words (Ito & Mester Reference Ito, Mester, Grijzenhout and Kabak2009: 96a), or an intonational phrase dominating two intonational phrases (Myrberg Reference Myrberg2013)), one might wonder whether it is appropriate to ban balanced feet (e.g. ((σσ)Ft(σσ)Ft)Ft) from Gen. Some authors have posited balanced feet (referred to as ‘cola’) to account for the existence of tertiary stress in languages like Hungarian (Hammond Reference Hammond1987, Green Reference Green1997) and Passamaquoddy (Stowell Reference Stowell1979). However, later research has questioned the empirical status of tertiary stress in these languages, weakening the arguments for cola (e.g. LeSourd Reference LeSourd1993, Siptár & Törkenczy Reference Siptár and Törkenczy2000).

In our study, we have also banned balanced feet from Gen. That is, we have assumed that Gen does not generate any non-minimal feet such as those in (42) below. Once again, we believe that this difference between recursive feet and recursive supra-foot prosodic categories, which may be balanced, emerges from the close relation between foot-level prominence and rhythm. At higher levels of the prosodic hierarchy, where syntactic structure is mapped onto prosodic structure, prosodic categories are not defined by their phonetic properties, but by their particular relation with (morpho-)syntax. In syntax, a category can be self-embedded in a category of the same type, and hence a head can contain more than one head. In contrast, the rhythmic nature of the metrical foot strictly links up the notion of ‘head/stress’ and ‘foot’ in a one-to-one fashion: a foot can have at most one head. In the same way that binary feet with two heads (e.g. (σhσh)) have generally been excluded from Gen (cf. Bye Reference Bye1996, Kager Reference Kager, Nespor and Smith1996), we assume that recursive feet that dominate two feet are also excluded. Note that a balanced foot implies that a non-minimal foot contains more than one foot head at the syllable level. One of the predictions of a language with balanced feet of this sort is that multiple levels of stress can arise, as illustrated in (42), where tertiary stress is represented as σ̆.

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The primary-stressed syllable is at the same time the head of a minimal foot, the head of a balanced foot and the head of ω. The secondary-stressed syllable is the head of a minimal foot, which is also maximal. Finally, the tertiary-stressed syllable is the head of a minimal foot, which is located in the dependent position of the balanced foot (or colon). It surfaces with stress, but this stress is weaker than the others, due to its dependent status. Self-embedding more than one foot in a recursive foot has various implications with respect to prominence, multiplying the levels of stress, but also the types of weak syllables. Since we do not know of any system or phonological phenomena that could benefit from these structures, and we believe that at lower levels of the prosodic hierarchy, headedness is a definitional property (i.e. the presence of one syllable head or one foot head entails the presence of exactly one syllable/foot), we have not incorporated balanced feet into our analysis.

4.3 Categorical vs. gradient alignment

Our theory of non-intervention categorical alignment constraints offers a new way of accounting for rhythmic stress distributions, which have previously been analysed by means of gradient alignment. Here we focus on some formal properties of categorical and gradient alignment, and on some differences in typological predictions.

In a formal sense, categorical and gradient constraints differ in the maximal number of violation marks they assign to a locus of violation (McCarthy Reference McCarthy2003: 79). Gradient alignment constraints are known to cause a non-linear ‘quadratic’ relationship between the number of elements in a domain evaluated by a constraint and the number of violation marks it assigns (Bíró Reference Bíró2003; similar arguments are put forward by Riggle Reference Riggle2004 and Heinz et al. Reference Heinz, Kobele and Riggle2005). In contrast, the maximum number of violation marks assigned by a non-intervention categorical alignment constraint is a linear function of the number of loci of violation per form.Footnote ¹¹

A potential concern about non-linearity is related to computational tractability. Several proposals have been made to implement OT grammars as finite state transducers (FSTs) (e.g. Ellison Reference Ellison1994, Frank & Satta Reference Frank and Satta1998, Karttunen Reference Karttunen, Karttunen and Oflazer1998, Gerdemann & van Noord Reference Gerdemann, van Noord, Eisner, Karttunen and Thériault2000). Yet it is known that quadratic constraints cannot be implemented as FSTs (e.g. Eisner Reference Eisner1997, Frank & Satta Reference Frank and Satta1998, Bíró Reference Bíró2003). However, as pointed out by an anonymous reviewer, ‘interpretability by a finite state grammar is neither necessary nor always sufficient for computational tractability under all forms’. In sum, quadratic constraints have more computational power than linear ones, yet this only leads to formal concerns if we adopt assumptions on the implementation of OT grammars that are not universally accepted.

A second set of arguments raised against gradient alignment constraints concerns incorrect typological predictions, such as ‘midpoint pathology’ (Table XIIc; see Eisner Reference Eisner1997, Hyde Reference Hyde2012) and unattested distinctions in directionality among metrical systems (e.g. Kager Reference Kager2001, McCarthy Reference McCarthy2003). Yet, as we argued in §3, criteria for pathology and insights into effects on learnability are currently insufficiently developed to base a decisive argument on.

Empirically, it might be asked whether gradient alignment is crucially needed to model metrical patterns. Among the critical cases which have been examined (McCarthy Reference McCarthy2003: 116ff, Hyde Reference Hyde2012: 809ff), one stands out. In languages such as Maithili, Hindi, Paumari and Banawá, the primary stress foot cannot be in word-final position, so that the primary stress is retracted to the penultimate foot. This n−1 pattern, described using foot extrametricality in rule-based theory (Hayes Reference Hayes1995: 77), can be analysed in gradient alignment theory by ranking Non-finality(Ftmain) above Rightmost(Ftmain), pulling the primary stress onto the penultimate foot. It may seem that categorical alignment has no account, yet this is not the case. The pattern is analysable by ranking Non-finality(Ftmain) above a non-intervention constraint Align-L([Ftmax]ω, [Ftmain]ω, ω): ‘For every maximal foot [Ftmax]ω assign a violation mark if some main foot stress [Ftmain]ω intervenes between Ftmax and the left edge of its containing ω’, as in (43).

1. (43)

   

Note that the current set only includes constraints whose separator category is an unspecified category, e.g. Ft. Separator-specific non-intervention constraints are not included, since we have not considered non-finality effects (see §4.4). Interestingly, (43) shows that a categorical non-intervention constraint may produce gradient effects while still being linear (not quadratic), as its locus of violation is Ft. In sum, there seems to be no compelling evidence that gradient alignment is crucially needed.

Our approach to circumventing gradient alignment is not the first on record. Previous attempts have been based on the ‘rhythmic licensing’ framework of Kager (Reference Kager2001, Reference Kager2005). This framework sets up constraints to license rhythmic configurations such as (extended) lapses and clashes near specific landmarks, such as word edges and stress peaks. Typological and computational studies exploring this approach include McCarthy (Reference McCarthy2003), Heinz et al. (2005) and Houghton (Reference Houghton2006, Reference Houghton2008). A full review of this framework is beyond the scope of this paper, but we note here some criticisms that have been raised against it, such as the problem of stressing sequences of light syllables occurring between heavy syllables (Alber Reference Alber2005), pathological patterns such as ‘lapse-licensing’ (Heinz et al. Reference Heinz, Kobele and Riggle2005), the complex locus of violation of lapse-licensing constraints (Buckley Reference Buckley2009) and the inability to deal with distance-counting patterns (Hyde Reference Hyde2012). None of these problems carry over to the present framework, however.

Since no decisive formal or typological arguments can be made against gradient alignment that might favour categorical alignment, we conclude that the two theories seem to be equally able to account for gradient metrical distributions. As pointed out by Alan Prince (personal communication), having two full-fledged theories of alignment instead of only one allows us to better evaluate their insights and shortcomings.

4.4 Non-finality

This study has not considered effects of constraints that enforce non-finality in the factorial typology. In order to motivate this, we must separate out the three main types of non-finality effects. It will become clear that in two out of three cases our account is nearly identical to standard accounts, while in a third case, which we consider first, we have offered a reanalysis.

The first non-finality effect is syllable unparsability at edges, the device which leads to locating a foot one syllable away from the edge (Hayes Reference Hayes1982, Prince & Smolensky Reference Prince and Smolensky1993). At the right edge this leads to parsings such as …(σ́σ)σ]ω, needed for antepenultimate stress in Macedonian, Latin, etc., and at the left edge to parsings such as ω[σ(σσ́)…, required for third-syllable stress in Kashaya, Azkoita Basque, etc. ILT feet generally alleviate the need for non-finality in such cases, reanalysing them as …((σ́σ)σ)]ω and ω[(σ(σσ́))… respectively (see §4). (See Kager Reference Kager2012 for discussion of the ILT foot typology in trisyllabic window systems at the left and right word edges.) In summary, in an ILT framework, syllable unparsability at edges is not needed, and is clearly undesirable, because ILT feet in combination with unparsability incorrectly predict stress on the fourth syllable from the edge. Residual cases of unparsability fall out from the interaction of syllable unstressability at edges with foot-form constraints.

The second type is foot extrametricality (Hayes Reference Hayes1995, McCarthy Reference McCarthy2003, Hyde Reference Hyde2012), which leads to the desired effect of removing the primary stress of the final foot, and relocating it on the rightmost non-final foot. Such cases were discussed in §4.3, where it was shown that our theory accounts in principle for such effects by the interaction of Non-finality(Ftmain) – the constraint also used in standard accounts – with the separator-specific non-intervention constraint Align-L([Ftmax]ω, [Ftmain]ω, ω).

The third non-finality effect is syllable unstressability at edges, associated mainly with patterns with iterative binary iambs with a foot-prominence reversal (to a binary trochee) on the final foot, e.g. (σσ́)(σσ́)(σσ́)(σ́σ). The constraint responsible bans stress (but not foot structure) from final syllables (see Gordon Reference Gordon2002, for example). Such patterns occur in languages such as Hopi and Ashininca, and in ‘restricted trisyllabic window’ systems in which primary stress can be antepenultimate or penultimate, but not final. Our analysis is identical to the standard analysis (see Kager Reference Kager2012 for discussion of window systems).