Phonotactic constraints and learnability: analyzing Dagaare vowel harmony with tier-based strictly local (TSL) grammar

3.3.1 Rounding harmony in verbs: alternations in aspectual suffixes

As stated earlier, the default verb (so-called perfective) form always has a final high vowel and this regardless of what shape the root vowel assumes. This suggests that default suffix or final high front vowel is never sensitive to roundness and backness even if it harmonizes with the root in terms of tongue root feature. This shows that the so-called perfective suffix is default morphology as the vowel remains unchanged in any conceivable environment.

The imperfective suffix always has a nonhigh vowel which always assimilates the preceding root vowel in terms of rounding, backness and tongue root, and height feature. Effectively, when the verb root contains a round back vowel, the vowel of the imperfective suffix is also round back as in (5a) and when the vowel in the verb root is unround and front, the vowel of the suffix assumes identical features, as shown in (5b). In (5c), while agreeing in tongue root feature, a root with low vowel always combines with a suffix with a low vowel.

Based on the data, both tongue root and rounding harmony propagate from the root outward to the suffix or final vowel. Thus, Dagaare harmony can be considered as root-controlled (Clements 1985; Kirchner 1993). With the preceding exposition to the main types of vowel harmony in Dagaare, let us now turn our attention to a proposed analysis.

4.1.1 Strictly local grammars

Strictly Local (SL) grammars are a class of formal languages that serve as the foundation for modeling strictly local dependencies in phonological patterns (Chandlee 2014). Within computational phonology, SL grammar defines phonological well-formedness based on local constraints over contiguous substrings of length k, known as k-grams. A string is considered well-formed with respect to a constraint set C if it does not contain any forbidden k-grams (Graf 2017). In essence, SL grammar characterizes acceptable strings by excluding certain prohibited substrings, making locality the central organizing principle. Formally, a SL grammar consists of the set of k-grams, and G _SL that must not be contained within a well-formed string of the language (Aksënova et al. 2024).

For example, in Samala (a Chumashan language spoken in California’s Santa Ynez Valley), there is a local dissimilation process whereby /sn/, /sl/, and /st/ are realized as [ʃn], [ʃl], and [ʃt], respectively (McMullin 2016). This can be interpreted as a phonotactic restriction that prohibits [s] from being immediately followed by [n], [l], or [t]. Such restrictions are straightforwardly captured using Strictly Local (SL) grammars, which encode well-formedness by banning specific bigrams, that is, sequences of two adjacent segments – in this case, sn, sl, and st.

Another illustration comes from Russian, where obstruent clusters must agree in voicing. Based on the analysis in Aksënova et al. (2024), this voicing agreement constraint is local and can be captured using SL grammars by banning sequences like [+voice] [-voice] or [-voice] [+voice] in obstruent clusters, as illustrated below.

The telic prefix s- is realized as [s] when followed by a voiceless consonant (see 6a), and as [z] when followed by a voiced consonant (see 6b). The relevant alphabet can be defined as ∑ = {s, z, t, b, j, a, i}. The phonotactic grammar in this case prohibits sequences of obstruents with mixed voicing, i.e., adjacent obstruents that disagree in their voicing specifications. This can be formalized as a Strictly Local grammar: G _SL  = ⟨*sz, *zs, *sb, *bs, *zt, *tz, *tb, *bt⟩. This grammar correctly rules out ill-formed sequences where the voicing of the prefix s- does not match that of the following obstruent. For instance, forms where a voiceless [s] precedes a voiced consonant (e.g., *sb, *sz), or where a voiced [z] precedes a voiceless consonant (e.g., *zt, *zs), are predicted to be ungrammatical under this system.

In the context of vowel harmony, SL grammars are used to model patterns where harmonic features such as [±ATR], [±back], or [±round] must match across adjacent or nearby vowels within a word. The core principle of SL grammar is the specification of permitted or prohibited local sequences (typically pairs of segments, or bigrams) in a phonological string. A string is considered well-formed if it does not contain any of the prohibited sequences. SL grammar is especially well-suited for modeling vowel harmony within single morphemes or short stems, where harmony is strictly local. For example, in Dagaare, vowels within roots and suffixes typically agree in [±ATR] (Advanced Tongue Root), as illustrated in (7) below.

The vowels involved in the harmony pattern are either directly adjacent or separated by a single intervening consonant. Crucially, they must agree in their tongue root feature – either [ATR] or [RTR]. The relevant vowel alphabet is ∑ = {i, e, u, o, ɪ, ɛ, ʊ, ɔ}, and the SL grammar rules out combinations where [ATR] and [RTR] vowels co-occur locally. Specifically, the following bigrams are prohibited: G _SL  = ⟨*iɛ, *ɛi, *ɪe, *eɪ, *uɔ, *ɔu, *ʊo, *oʊ⟩. These sequences represent ill-formed vowel pairings that violate tongue root harmony and are correctly excluded by the SL bigram constraints.

Now let us look at another harmonic process. In Lokaa (as discussed in Akinlabi 2009; Aksënova et al. 2024), non-high vowels must harmonize in ATR value with the preceding non-high vowel, while high vowels and consonants are transparent to harmony. This pattern results in both well-formed and ill-formed surface forms, with agreeing segments that may not be adjacent to each other. The following examples illustrate this harmonic agreement, highlighting instances where agreement is achieved across intervening segments.

The strings in (8a) and (8b) need 5-grams to capture the pattern as there are three intervening elements between the two non-high vowels, [e] and [o]. In (8d), there are five segments intervening between [ɛ] and [a] and because there is no upper bound on the amount of material which separate the two non-high vowels agreeing in terms of [ATR] feature, no SL grammar, which depends on a fixed locality window, can adequately account for this pattern. This limitation highlights the need for Tier-Based Strictly Local (TSL) grammar, which can handle long-distance dependencies by projecting only harmony-relevant segments onto a separate tier. In short, while SL grammars are effective for modeling strictly local patterns, they fail to capture non-local vowel harmony, especially when intervening consonants or syllables must be ignored. This necessitates the use of more expressive models, such as TSL, to accurately represent these long-distance phonological processes.

4.1.2 Tier-based strictly local grammars

Tier-based Strictly Local (TSL) grammars (Aksënova et al. 2024; Heinz 2011, 2018; Heinz et al. 2011) enable the modeling of non-local dependencies by projecting specific input elements onto a tier, creating locality among otherwise distant segments and ruling out forbidden strings. TSL grammar consists of a tier alphabet, T, and a set of k-grams, G _TSL , that must not appear in a well-formed tier representation of a string. This approach allows TSL grammar to block forbidden substrings over the tier, even when the segments are not contiguous in the linear string. For example, the harmony pattern in Lokaa in (8) shows non-high vowels harmonizing across intervening segments. Although this pattern is impossible to capture with purely SL grammar, TSL enables projection of these non-adjacent segments onto a tier, where ill-formed combinations can be effectively ruled out. Thus, Table 1 below shows the active constraints on tier-adjacent non-high vowels. The ATR pattern in the surface forms is obtained by blocking the combinations of non-high vowels that do not agree in [tense] specifications.

There are many languages in which segments agree in more than one single feature. For instance, in Turkish, vowels agree in backness and rounding (see, Kaun 1995, 2004); in Bukusu, one agreement pattern involves vowels and another involves liquids (Odden 1994); in Dagaare verbs, the imperfective suffix agrees in ATR and rounding with the verb root (Angsongna 2023). Such languages with more than one harmony pattern may require more than one tier and these tiers may be related in some ways. Figure 2 illustrates the four logically possible relations between tier 1 and tier 2 (Aksënova et al. 2024): (i) T1 and T2 can be identical as in double harmony involving only vowels; the set of blockers and undergoers is the same for both harmony processes (same tier) (ii) T1 might share no common elements with T2 – disjoint tiers. This relation involves independent vowel harmony and consonant harmony processes (iii) embedded relation in which T1 is a subset of T2 or vice versa. This is the case of consonant harmony such as sibilant harmony (iv) partially overlapping relation in which T1 and T2 share some elements, but neither is a subset of the other.

Figure 2:

Possible tier relations.

Cases where T1 is the only tier that partially overlaps T2 appear to be typologically unattested. Meanwhile, out of the four logical possibilities, the partially overlapping case is reported to be the most common.

TSL approach is used to capture different harmonic patterns. A study of double vowel harmony patterns shows they require a single tier, although more than one feature is involved. For sibilant harmonies however, two tiers are needed. Languages with both vowel and consonant harmonies also require two tiers though the two tiers do not share any common element. That is, such cases involve disjoint tiers.

With double vowel harmonies, the initial assumption might be that each harmonic feature requires a separate tier. However, data reveals that both harmonies can coexist on the same tier. This does not imply that undergoers and blockers are identical for both harmonies; each segment participating in one harmony remains relevant to the other. Typically, one harmony affects all vowels, while the other spreads only among tier-adjacent vowels. In some instances, both features spread simultaneously, while in others, all vowels act as undergoers for both harmonies, with a subset further propagating both features. Essentially, TSL grammar formalizes a particular constraint within a language phonological structure. In the following sections, I analyze the multiple vowel harmonies observed in Dagaare using TSL grammar as the framework.

4.1.3 Tongue root harmony in TSL

Tongue root harmony in Dagaare as described in previous sections affects all vowels. As a result, TSL grammar then allows the projection of dependent items on the tier and bans all ill-formed combinations over the tier. There are two harmonizing features in Dagaare that are either plus or minus and therefore all vowels within a simple word have a choice of the following two possibilities: [ATR] – [ATR] and [RTR] – [RTR]. For each of these combinations, there is an example below which shows that suffixes must agree with the root in these parameters.

The data above and harmony requirements of the language give rise to the following well-formedness constraints as adopted from Pulleyblank (2002).

The constraints prohibit vowel sequences where the tongue root features differ. Table 2 shows the active constraints on the tier adjacent to the tier alphabet. The constraints illustrate the illicit featural combination. The tier alphabet includes all vowels in the language. In addition, there are corresponding ill-formed segmental bigrams which make up the forbidden substrings in the TSL grammar H _ATR . Any bigram in which the vowels disagree in tongue root feature must be banned.

Figure 3 below illustrates the analysis above. The subfigure in (a) shows the well-formed word tíɡé ‘feasts’ while the one in (b) shows the illicit word.

Figure 3:

Tongue root harmony in Dagaare.

The ATR vowels [i, e] are projected on the tier and their combinations i.e. is not among the sequences that are required to be ruled out and therefore the word tíɡé is well-formed in (a). The other subfigure in (b) shows ill-formedness with the word *tíɡɛ́, where the two adjacent vowels í and ɛ disagree in their tongue root specifications. Accordingly, when i and ɛ are projected on a tier, we see that the bigram *iɛ is prohibited by the grammar and so the word *tiɡɛ must be ruled out.

It is important to note that the notion of directionality – such as root/stem control – does not directly factor into the implementation of TSL grammars. Rather, TSL focuses on the sequencing of relevant segments (e.g., vowels), without reference to morphological structure. In this framework, directionality is not explicitly encoded but instead emerges from the nature of the prohibited featural combinations specified in the grammar.

Overall, TSL grammars are particularly well-suited for capturing the phonotactic patterns of vowel sequences in simplex words, especially in languages like Dagaare, where vowels within a word must harmonize in a specific harmonic feature (e.g., tongue root) specification. However, this effectiveness diminishes when applied to morphologically complex structures such as compounds, which present significant challenges for the TSL framework due to its lack of morphological sensitivity. I return to this issue in Section 4.2.

The following section, however, shifts focus to how TSL can be extended to model multiple types of vowel harmony – specifically those involving tongue root, rounding, backness, and height features.

4.1.4 Multiple vowel harmonies in TSL (ATR, rounding, backness, and height)

As previously stated, with TSL grammar, a harmonic process selects a set of elements (undergoers) and establishes an agreement relation among them with respect to a particular feature. These elements are projected on the tier alphabet T (which contains all the vowels of the language) in order to achieve locality. In the previous section, using TSL grammar, I give an account of tongue root harmony, which is the main harmonic pattern exhibited in nouns. In this section, I present a formalization of the harmony patterns involving the imperfective form of the verb. The imperfective form in Dagaare, unlike nouns and the default verb forms, exhibits cases where vowels agree in more than one harmonic feature – tongue root, rounding, backness and height. This multiple feature agreement is commonly observed in many languages. For instance, in Turkish, vowels agree in rounding and backness; in Imdlawn Tashlhyt sibiliants agree in voicing and anteriority and in Bukusu one agreement pattern involves vowels and another involves liquids. These are cases often described as double harmonies. These patterns raise questions as to whether more than one tier is required to account for double harmonies or whether there is a restriction on the tier alphabet for patterns that exhibit double harmonies.

Considering a single tier is required to account for the single harmonic pattern ATR, it is likely to easily assume that each feature of a double or multiple harmony needs its own tier. That is not the case though, because a typological study (Aksënova et al. 2024; Burness et al. 2021) reports that double vowel harmony requires just a single tier despite the presence of more than one feature. All harmonies fit on the same tier. It should be noted that the undergoers and blockers are not the same for both or all harmonies but the elements taking part in one harmony are relevant for the other because sometimes the double or multiple harmonic features are transmitted simultaneously.

The imperfective form in Dagaare involves multiple harmonies (tongue root, rounding, backness and height). All harmonic features in these patterns are simultaneously transmitted. As shown in the following data (11), within an imperfective verb, all vowels agree in the above features – the imperfective suffix always agrees in terms of the above features with the root.

These examples in (11) and the requirements for vowel harmony in Dagaare leads to the constraint, agree (f). This constraint is further expanded into the following three-way set.

Similar to the tongue root harmony discussed above, the four harmonizing features for the imperfectives can be plus or minus. Therefore, all vowels within an imperfective verb have a choice among the following possibilities: [±ATR], [±round], [±back], [±low]. Table 3 below presents the TSL grammar for Dagaare imperfective verbs in various harmonic components. The tier alphabet T incudes all the vowels in Dagaare. H _ATR rules out sequence of vowels that disagree in tongue root feature; H _round prohibits co-occurrence of vowels that do not agree in rounding feature; H _back bans vowels in a sequence that do not share same feature for backness and finally H _low rules out vowels in a domain that do not share same feature low. In effect, only one TSL grammar is necessary – its tier alphabet T, and G _TSL  = H _ATR ∪ H_round ∪ H _low . That is, this grammar is basically a three-way conjoined constraint of AGREE (F) banning all V-V sequences that disagree in one or more of these three features.

Only one TSL grammar is necessary to capture these patterns. This grammar operates within the tier alphabet T and the grammar G _TSL is a combination of H _ATR, H _round, and H _low which contain the sets of forbidden bigrams. Figure 4 presents a graphic representation of the analysis above. In each subfigure (a–c), the well-formed word appears on the left, with its corresponding illicit form on the right.

Figure 4:

ATR, rounding/backness, and height harmony in Dagaare.

The subfigures in (a) represent ATR harmony (though it exhibits backness/rounding as well). The left subfigure represents the well-formed word, wùló. The two vowels are projected on a tier, and nothing is blocked because the combination uo is acceptable as they both agree in ATR features. The right figure in (a) represents a violation of tongue root harmony where *uɔ is ruled out by the H _ATR part of the grammar. The left subfigure of (b) is a graphic representation of rounding/back harmony where nothing is blocked while the subfigure on the right illustrates disagreement in round/back harmony wherefor the bigram *ue is avoided by the H _round of the grammar. Finally, the subfigures in (c) demonstrate height harmony in which aa is not blocked since there is agreement is the feature [low]. The figure on the right however is illicit because the combination of the low and mid vowels *aɛ is ruled out by the H _low component of the grammar. In summary, the harmonic patterns in the imperfective involve cases where the undergoers for one harmonic spreading is the same for the others, hence only a single TSL grammar is required to capture these patterns.

Overall, Dagaare exhibits multiple harmony processes that are effectively captured within a single Tier-Based Strictly Local (TSL) grammar framework. This unified TSL grammar is essential because each vowel participating in one harmony is relevant to the others; the harmonic processes are interdependent rather than isolated. TSL grammar serves as a formal mechanism for encoding specific phonotactic constraints that govern permissible and impermissible surface sequences within a language. It accomplishes this through either individual constraints or a set of constraints acting in conjunction, but it does not account for underlying phonological alternations that arise when roots, suffixes, and clitics are combined. Thus, TSL grammar is known for its efficiency in modeling surface phonotactics directly, without relying on underlying forms, a feature that distinguishes it from many generative phonological models such as Optimality Theory (OT). It focuses exclusively on surface-level segmental sequences, allowing it to characterize observable harmonic patterns with precision.

Crucially, the TSL account of Dagaare multiple harmonies – specifically tongue root, rounding, backness, and height – is comparable to the double vowel harmony systems observed in languages such as Kirgiz, Yakut, and Buryat, as discussed by Aksënova et al. (2024). In all these cases, the harmonies target the same set of segments (i.e., vowels), allowing a single TSL grammar to capture all the harmony patterns simultaneously. This makes Dagaare typologically similar to these languages in terms of its representational simplicity under the TSL framework.

However, Dagaare differs from languages like Imdlawn Tashlhiyt (Berber), which exhibits sibilant harmony involving both anteriority and voicing, and Kikongo (Bantu), which displays vowel harmony (in height) alongside nasal harmony. In these cases, the harmonies affect distinct sets of segments (e.g., sibilants and nasals) and thus cannot be captured within a single tier. As a result, each harmony pattern in these languages requires a separate TSL grammar, with distinct tiers for each type of harmony. This contrasts with Dagaare, where a single tier suffices for modeling multiple vowel harmonies, highlighting the representational efficiency of its harmonic system under the TSL framework.