2.1 Vowel inventory and basic harmony pattern

The Hungarian vowel inventory is given in Table 1.

Phonetic descriptions have noted that [a:] is produced as central or even front, but it patterns phonologically as a back vowel and has been argued to be phonologically classified as such (e.g. Gósy & Siptár 2015). Throughout this paper, I will consider [a:] a back vowel, despite its phonetic implementation. It is worth noting that for all other vowels, the distinction between front and back is synchronically based in the phonetics; the phonetics does have bearing on the phonological patterning, despite the phonetics/phonology mismatch in the behaviour of [a:].

By some researchers, [e:] has been analyzed as being at least sometimes a phonologically low vowel, due to its alternations with [ɛ], [a:], and [ɔ]. For example, Siptár & Törkenczy (2000: 158) class non-alternating [e:] as mid in their feature system, but alternating [e:] emerges as a low vowel in their analysis. However, I follow the phonetic inventory in Siptár & Törkenczy (2000: 52), Hayes & Londe (2006), and Törkenczy (2010), among many others, in categorizing it as mid. Given the height effect among front unrounded vowels in Hungarian harmony, to be discussed in Section 2.2, the distinction between [e:] and [ɛ] in height is relevant to the phonological system. In general, as Törkenczy (2010) notes, the system requires two degrees of backness and three of height.

Notable in this inventory are the gaps where vowels do not have a counterpart that differs only along the front/back dimension. Only non-low rounded vowels have direct harmonic counterparts; the front unrounded vowels and the low vowels ([i], [i:], [e:], [ɛ], [a:], and [ɔ]) do not have counterparts that differ only in the feature [back].¹ Given that short, rounded, back vowels have a three-way height distinction, both [high] and [low] must be active features within the Hungarian phonological system.²

Hungarian has an extensively studied system of front/back harmony, as well as a more limited system of rounding harmony that will not be discussed here (e.g. Vago 1973; Goldsmith 1985; van der Hulst 1985; Ringen 1988; Ringen & Vago 1998; Siptár & Törkenczy 2000; Hayes & Londe 2006; Nevins 2010; Törkenczy 2010; Gafos and Dye 2011; Kimper 2011; Törkenczy et al. 2013; etc.). In general, disregarding the front unrounded vowels, vowels that disagree in the feature [back] are not permitted to co-occur within (native) roots, and suffixes alternate according to the [back] value of the root vowels. Examples are given in (1), with the dative suffix [nɔk]~[nɛk].³

2.2 Neutrality in roots

The front unrounded vowels are generally considered transparent to this harmony; they can co-occur in roots with any other vowel, and are skipped over in determining the harmonic value of the suffix (Siptár & Törkenczy 2000; Törkenczy 2010). Thus, suffixes generally surface as back when they attach to stems that contain a back vowel followed by a (non-low) front unrounded vowel. Examples are shown in (2a–f). While the high front unrounded vowels [i] and [i:] are consistently transparent⁴ (as long as there is only a single one; see below), the mid front unrounded vowel [e:] shows variable behaviour; some back-[e:] stems show consistent transparency (2d–f), while others are able to take either front or back stems (“vacillating”), as in (2g) (Törkenczy et al. 2013). The low front unrounded vowel [ɛ] shows an even greater degree of variable behaviour and is sometimes considered front harmonic (e.g. Ringen 1975; 1978; 1980); while there exist some back-[ɛ] stems that take only back suffixes (2h), most such stems either vacillate (2i) or take only front suffixes (2j) (Törkenczy et al. 2013). This is a morpheme-specific property: different stems have different preferences for front or back suffixes, and the choice is lexically arbitrary (Hayes et al. 2009; Rebrus et al. 2017). Moreover, although a single [i] or [i:] is always transparent, multiple high front unrounded vowels are variably opaque, as shown in (2k).

The distinction between the degree of transparency for high, mid, and low vowels, in terms of how consistently they are transparent, is known as the height effect; it is strongly supported by a large body of data and wug test judgements from native speakers (Hayes & Londe 2006). Essentially, [i(:)] is more transparent than [e:], which is more transparent than [ɛ]; this “transparency scale” is supported by ratios of front suffixation to all suffixation for back-neutral stems calculated from corpus counts (Rebrus & Törkenczy 2016b), with total transparency for [i(:)], a low ratio for [e:], and a high ratio for [ɛ]. The example in (2k), showing that multiple neutral vowels are less transparent, illustrates what is known as the count effect, and also applies to the other front unrounded vowels; again, this effect has been shown to be productive and robust (Hayes & Londe 2006).

While (2) shows that (non-low) front unrounded vowels do not generally trigger harmony when other potential triggers are present, they are “last-resort” triggers, in that roots with only front unrounded vowels typically take front suffixes (e.g. Siptár & Törkenczy 2000).⁵ Examples are shown in (3), with the dative suffix seen in previous examples and the ablative suffix [tø:l]~[to:l]. It is worth noting that within roots, neutral vowels can precede back vowels, as in [bikɔ] ‘bull’; the pattern in (3) occurs solely across morpheme boundaries, in that root-internal harmony for front unrounded vowels is not required even in native vocabulary.

I will consider these examples in (3) to be last-resort triggering, as opposed to a default value, because it is clear from vacillating stems that front unrounded vowels are able to trigger front harmony in some contexts; the same can be thought to apply here.⁷ In new loans and nonce words, front unrounded vowels also always take front suffixes, providing an additional argument for them to behave as triggers.⁸ As I adopt binary features, I also assume for simplicity that both [+back] and [–back] are active in triggering harmony in Hungarian, as has been widely assumed (e.g. Vago 1973; 1976; Farkas & Beddor 1987).

Whether Hungarian has root-internal harmony is a matter of some debate. Many analyses assume that it does not, and that harmony only affects suffixes; in analyses of this type formulated in OT, a high-ranked root faithfulness constraint is assumed (e.g. Ringen & Vago 1998). However, aside from recent loans, Hungarian roots obey similar harmonic restrictions to root-suffix combinations: Törkenczy et al. (2013) note that bisyllabic stems containing both front and back vowels are rare with front rounded vowels (“disharmonic roots”), with all cases being recent loans, but frequent with front unrounded vowels (“mixed roots”), though with [ɛ] most are loanwords. This difference corresponds to the strong versus weak disharmony distinction drawn by Rebrus & Törkenczy (2015). Törkenczy (2010: 8) provides corpus counts supporting this distinction and notes that intuitively for native speakers, disharmonic roots “feel foreign”, while mixed roots do not. In other words, front rounded vowels cannot co-occur with back vowels within roots, while mixed roots containing front unrounded vowels with back vowels are permitted. This fact suggests that, disregarding recent loans, Hungarian roots do show a harmony pattern that should be analyzed, because it works in essentially the same way as the root-to-suffix harmony.

To summarize, Hungarian has a pattern of front/back harmony, in which front unrounded vowels tend to be transparent, but can trigger harmony in certain stems or when there are multiple of them, and behave as last-resort triggers when they are the only vowel in the root. The different front unrounded vowels behave differently with respect to various aspects and degrees of neutrality, with height affecting whether a single neutral vowel is consistently transparent ([i(:)]), usually transparent ([e:]), or usually variable or opaque ([ɛ]). The back vowels and front rounded vowels are generally harmonic, and typically occur only with neutral vowels or vowels in the same harmonic class.

2.3 Patterns of neutrality in suffixes

I turn now to additional complications from suffix alternations and their interaction with neutrality. Törkenczy et al. (2013) discuss the fact that there are both invariant and alternating suffixes containing neutral vowels, but their analysis focuses on paradigm uniformity, not the issue of neutrality. Other literature also notes the presence of alternating suffixes as a factor in the scale of neutrality (e.g. Rebrus & Törkenczy 2016a), but again does not provide any detailed theoretical implementation.

The Hungarian inventory in Table 1 showed that only non-low rounded vowels are paired for [back], but other vowel pairs also alternate harmonically in suffixes. An example is the dative suffix in (1–3), since neither [ɔ] nor [ɛ] is paired. Siptár & Törkenczy (2000), among others, provide a complete list of alternating vowel pairs, shown with examples in Table 2.

Despite the large literature on Hungarian vowel harmony, the suffix alternations between [a:] and [e:], in the penultimate row of Table 2, are rarely explicitly analyzed theoretically. However, this pair is problematic for the widely stated view that neutral vowels are harmonically unpaired, which would suggest that they should be non-alternating (e.g. van der Hulst 2016; see also Törkenczy 2013 on this issue). Under categorical definitions of neutrality, [e:] (unlike [ɛ]) is widely considered a neutral vowel in Hungarian (e.g. Vago 1978; Ringen 1988; Siptár & Törkenczy 2000), but Table 2 shows that it does participate in suffix alternations. Indeed, [a:]~[e:] alternations occur regularly and consistently through Hungarian morphology, in a variety of suffixes. Like for other pairs in alternating suffixes, the front vowel [e:] occurs after stems with front vowels, while the back vowel [a:] occurs following stems with back vowels (Siptár & Törkenczy 2000). Moreover, [e:] is the only possible counterpart for the harmonic vowel [a:] in alternating suffixes (cf. Table 2). Additional examples of the [a:]~[e:] alternation are shown in (4) with four different suffixes, namely the adessive, translative, 3SG definite conditional, and 2SG indefinite conjunctive/imperative; Törkenczy (2010) and Rebrus et al. (2012) note 10 productive suffixes containing this alternation.¹⁰ Note that these are native Hungarian suffixes.

Despite such alternations, there is a distinction between [e:] and [a:] in terms of the number and nature of non-alternating suffixes with these vowels. Siptár & Törkenczy (2000) list twelve non-alternating suffixes with [e:], comparable to the number with [i]. These are common suffixes that are fundamental to the grammatical system of Hungarian, such as the possessive and the causal; implicit in Siptár & Törkenczy’s (2000) discussion of these affixes is that they are native to Hungarian. In contrast, there are only three non-alternating suffixes with [a:], comparable to the four containing [o]; these suffixes are borrowed, mostly Latinate, and their exact semantic content is not always clear from the glosses. Given this and the fact that all three end in [a:l], they may in fact all be the same suffix.¹³ Examples of non-alternating [e:] suffixes are given in (5), and a complete list of the potential non-alternating [a:] suffixes is given in (6).

As mentioned, the status of [a:] as a harmonic vowel and of [e:] as a transparent vowel in Hungarian is widely assumed; this is supported by the data on non-alternating suffixes in (5–6), because [e:] is neutral in a variety of suffixes, while [a:] is not (see Ringen 1978 for argumentation connecting neutrality to non-alternating suffixes in Hungarian). However, the data in (4) clearly illustrate that [a:] and [e:] are a harmonic pair in suffix alternations.

Traditional work has at least implicitly assumed that the underlying vowel in [a:]~[e:] alternations is /a:/, with /e:/ in suffixes surfacing faithfully as a non-alternating neutral vowel. In contrast, in more modern literature on Hungarian, the presence of a neutral vowel in alternating suffixes is considered an aspect of gradient neutrality (e.g. Rebrus & Törkenczy 2016a). As illustrated in Table 2, there are no alternating suffixes with high front unrounded vowels; these vowels occur only in invariant suffixes (Törkenczy 2010; Törkenczy et al. 2013).¹⁵ The data in (4–5) has shown that [e:] is present in both invariant and alternating suffixes. Table 2 also shows that [ɛ] occurs in alternating suffixes; however, it never occurs in invariant suffixes (Törkenczy 2010; Törkenczy et al. 2013). Thus, in this respect too, there is a height effect: high [i] and [i:] are the most neutral, in that they occur only in invariant suffixes; low [ɛ] is the least neutral, in that it occurs only in alternating suffixes; and mid [e:] is intermediate. Under a gradient view of neutrality, this pattern is sensible and consistent with the patterns in Section 2.2, but it is problematic under a categorical view of neutrality, and it needs to be analyzed.

2.4 Summary of patterns

In Hungarian, the distinction between neutral and harmonic vowels is not always clear, and there are multiple distinct aspects and degrees of neutrality; the front unrounded vowels show a variety of behaviours, as shown in Sections 2.2 and 2.3. This has been noted extensively in recent literature on Hungarian. For example, Rebrus & Törkenczy (2016a) note three different notions that are often included as part of neutrality: (i) neutrality as a target (invariant suffixes; see (5)), (ii) neutrality as a trigger with respect to other targets (transparency; see (2)), and (iii) neutrality as the source trigger (anti-harmony; see footnote 5). In Hungarian, these three properties create a scale of neutrality among the front unrounded vowels, which show [i(:)] to be the most neutral, followed by [e:], and finally [ɛ].

In slightly different terms, we can summarize the behaviour of Hungarian vowels based on whether and when they are triggers or targets of harmony (Table 3). As is evident here, there are two primary factors that can affect whether a given vowel is a trigger or target of harmony: the specific vowel quality and the specific morpheme in which the vowel appears (see Rebrus et al. 2012). Additionally, harmony is affected by whether a vowel is in a root or suffix and how many neutral vowels there are.

Despite the discussion of these issues of variability in the literature, theoretical treatments of these issues are lacking. In the present paper, I focus primarily on one aspect of differential neutrality, namely the non-neutrality of [e:] in suffixes. As will be discussed in more detail in the analysis section, the way of understanding harmony targets presented here ties in closely with the view of neutrality as gradient and variable that is dominant in modern literature on Hungarian (e.g. Rebrus et al. 2012; Törkenczy et al. 2013; Rebrus & Törkenczy 2016a).

4.1 The problem

In standard analyses of Hungarian vowel harmony (e.g. Vago 1980a; Ringen & Vago 1998; etc.), [e:] does not undergo vowel harmony because there is no mid, back, unrounded, long vowel to which it could harmonize. For example, Ringen & Vago (1998) propose a constraint *ɨʌ, which requires [+back, –low] vowels to also be round. Ranked above the harmony-enforcing constraint, this general markedness constraint prevents [e:] from undergoing harmony and allows it to be neutral. To my knowledge, there is no discussion in the theoretical literature about why [a:] does undergo harmony, despite also being unpaired.

The fact that [e:] can act as the harmonic counterpart for [a:] raises critical questions about this view of neutrality. Even if the [a:]~[e:] alternation is always underlying /a:/, precisely the same mechanisms that allow [a:] to undergo re-pairing to [e:] should force the same possibility for [e:]. Indeed, if changing the feature [low] is allowed in order for /a:/ to harmonize to [e:], then it is unclear what blocks changes in [low] in so many cases of /e:/ in suffixes surfacing faithfully. The tableaux in Tables 4 and 5 demonstrate this problem, adapting the analysis of Maasai re-pairing from Pulleyblank et al. (1995) and using BACKNESS-HARMONY to stand in for any constraint that enforces harmony. Table 4 derives the correct result, with /a:/ re-pairing to [e:] when it follows a front vowel, but Table 5 incorrectly predicts that /e:/ should consistently surface as [a:] in a back vowel environment. The reverse ranking of BACKNESS-HARMONY and IDENT-IO[low] would predict that /e:/ can surface faithfully in Table 5, but gives an incorrect result in Table 4. Either way, the result is incorrect according to a categorical view, and it is also impossible to derive a gradient view in which /e:/ is variably neutral but /a:/ is not.

Thus, the disparate behaviour of the two unpaired vowels [a:] and [e:] in Hungarian means that we cannot assume that general markedness, alone or in combination with faithfulness to other features, determines whether a vowel is a harmony target. Instead, neutrality must be derived in some other, more vowel-specific way.

4.2 Problems for possible solutions within traditional frameworks

Before considering the target-based view adopted here, I outline two potential solutions that are formulated within more traditional views of harmony, as well as the reasons for rejecting them. First, I consider separating IDENT-IO[–low] from IDENT-IO[+low], and then the possibility that harmony is triggered only by [–back]. Note that neither of these solutions has actually been proposed in the literature to deal with the Hungarian [e:]~[a:] alternation, unsurprisingly given the lack of discussion about this pattern within these kinds of analyses.

4.3 New view of neutrality

Due to the aforementioned problems with solutions in which neutrality is determined by lack of a harmonic counterpart, I argue that the Hungarian patterns require a fundamental change in how we view targets of vowel harmony: it suggests the need to consider the nature of each vowel as a possible harmony target. This view works not only for the [e:]~[a:] asymmetry, but more broadly for the height effect in Hungarian. I propose that the drive to undergo vowel harmony differs by vowel, in a way motivated by cross-linguistic and phonetic properties, and that cases of neutrality do not occur simply because harmony is impossible, but because the vowel-specific drive to undergo harmony is too weak to force unfaithfulness. This notion is gradient, and so will be able to avoid the issues of the categorical views.

In the case of front/back harmony, phonetic and cross-linguistic facts outlined in Section 3 show that vowels that are lower and/or rounded are preferred targets. As such, the drive for lower vowels, such as [a:], to undergo harmony should be stronger than the drive for the non-low, unrounded vowel [e:]. The result, in Hungarian, is that [a:] is consistently a harmonic vowel, while [e:] can be neutral. The additional fact that [a:] is unpaired in the inventory, yet required to be a target, is what forces it to re-pair to [e:]. Moreover, the basic height effect in Hungarian is in direct accordance with the phonetic facts: the higher the vowel, the worse it is as a target of harmony, and therefore the more likely it will be to be neutral.

5.1 Harmonic Grammar (HG)

Harmonic Grammar (HG; Legendre et al. 1990a; b; Smolensky & Legendre 2006; Pater 2009) is a model of phonology that is similar to standard Optimality Theory (OT; Prince & Smolensky 1993) except that constraints are weighted (Potts et al. 2010). In the specific implementation that I adopt here, I follow Potts et al. (2010) in assuming that constraints are negatively formulated, as in standard OT, and weights are positive integers. There are often many options that will work for the weight of each constraint; the specific choice is arbitrary (e.g. Bowman 2013).

Within HG, it has been proposed that violations of a constraint can be multiplied by a scaling factor, which allows properties of a specific violation to influence the degree to which the violation is weighted (e.g. Kimper 2011; see also Coetzee and Kawahara 2013 for scaling factors of a different type). A scaling factor x is a positive constant by which the constraint weight is multiplied; this has the effect of increasing the penalty incurred for a violation if x > 1 and decreasing it if 0 < x < 1. As an example, in this paper, a scaling factor based on target quality will be applied to the harmony constraint. For example, the disharmonic combinations [u…y] and [u…i] both violate a progressive front/back harmony constraint once. However, if the vowels [y] and [i] have target scaling factors of 4 and 2 respectively, then [u…y] will incur twice the penalty for this disharmony than will [u…i]. Further details are reserved for Sections 5.3 and 5.4.

For a given violation, the total penalty assigned by a constraint to a candidate (for a single locus of violation) is –W*S, where W is the weight of the constraint and S is any relevant scaling factors; such penalties are summed across all violations that the candidate incurs for all constraints (adapted from Potts et al. 2010). For example, given a harmony constraint with weight 3 and the target scaling factors defined in the previous paragraph, the harmony violation incurred by [u…y] would receive a penalty of –3*4 = –12, while [u…i] would have one of –3*2 = –6. The total of all penalties for all constraints is called the harmony score, and the candidate with the highest harmony score wins. Since the expression is always negative, the winning candidate will be the one with a harmony score closest to zero (Potts et al. 2010).

5.2 The nature of the harmony-driving constraint

An important theoretical implication of the Hungarian pattern is that the target-sensitive harmony constraint must be negatively formulated, in the sense that disharmony is penalized. While negative constraints are quite common, they are not universal; Kimper (2011) uses a positively formulated harmony constraint, in which harmony is rewarded, and Archangeli & Pulleyblank (2002) propose positive target conditions. However, in Hungarian, the good target [a:] becomes a poor target, [e:], while the reverse does not always occur. If harmonizing to good targets were rewarded, then we would predict the opposite pattern; the poor target [e:] would become the good target [a:] in order to benefit from the reward; /a:/ surfacing unfaithfully as [e:] would be rewarded less and therefore be less likely to occur. Instead, the direction of the Hungarian asymmetry necessitates a penalty for good targets that have not undergone harmony. Non-harmonized [a:] will be heavily penalized, and so [a:] is required to harmonize, whereas non-harmonized [e:] will be penalized less and therefore permitted.

As long as there is a mechanism to distinguish trigger and target, the specific constraint used to enforce harmony is independent of both the principle of target scaling and the perspective that neutrality is caused by the drive to undergo harmony being too weak. I adopt a variation on Pulleyblank’s (2004) infinite distance no-disagreement constraints, defined in (7) and subject to trigger and target scaling as in Sections 5.3 and 5.4.

The boldface indicates that this constraint is oriented towards the second vowel; it is the focus, and therefore the locus of violation, while the first vowel is the context/environment. Violations are counted for each [αBack] preceded by a [–αBack] vowel, and the infinity symbol indicates that such a configuration is a violation regardless of the distance or intervening segments. Only one violation is possible per potential [αBack] target. For example, a sequence such as [u…y₁…y₂] violates (7) twice, once for [y₁] and once for [y₂], both of which are preceded at some distance by a back vowel [u]. The fact that [y₁] intervenes between [u] and [y₂] is irrelevant to the fact that a violation is assigned. On the other hand, the sequence [y₁…y₂…u] violates (7) only once, for the [u] that is preceded by a front vowel.

Since Hungarian harmony is progressive, this constraint counts violations based on the number of vowels that could be targets of harmony but are not. For the purposes of trigger scaling, it is counted from the closest possible potential [–αBack] trigger. Thus, for each pair of vowels violating the constraint, the target scaling factor is based on the vowel for which the violation is counted (the focus), while the trigger is the closest preceding vowel of the opposite [back] value (the context). For example, in [y₁…y₂…u], the one violation has trigger [y₂] and target [u]. The two violations in [u…y₁…y₂] both have trigger [u], and the targets are [y₁] and [y₂] respectively.

The constraint in (7) is equivalent to the two separate constraints *[–Back]∞[+Back] and *[+Back]∞[–Back]. This approach therefore differs from that of Pulleyblank (2004) in that the constraint is consistently oriented not to a specific value of the feature [back], but rather to the second vowel in a disharmonic sequence. This choice offers a conceptual advantage for a case of harmony like Hungarian, which is arguably progressive and triggered by either feature value, because it offers an easy way of identifying the trigger and target for each violation. In comparison, other types of harmony constraints, such as AGREE[back] (e.g. Baković 2000), do not have a natural way of referring to the potential triggers and targets of harmony.

In addition to providing an inherent definition of trigger and target, this type of constraint has a more natural way of dealing with transparency than other options. Spreading-type constraints like ALIGN[back] or SPREAD[back] generally require significant representational complexity, such as line crossing, gapped configurations, or output underspecification, to deal with transparency (e.g. Ringen & Vago 1998; Walker 1998; 2012 Jurgec 2011; Ní Chiosáin & Padgett 2001). Further, constraints like AGREE[back] do not have a natural way of referring to non-adjacent vowels, and they naturally prefer opacity to transparency in order to reduce the number of disharmonic transitions (Baković & Wilson 2000). In contrast, a target-oriented constraint captures the generalization that all vowels should be targets of harmony when a potential trigger is present; no additional representational complexity is required, and transparency is derived because whether other potential targets intervene is irrelevant to whether the constraint is violated.¹⁶ See Pulleyblank (2004) for additional argumentation for this style of constraint. Some of the constraints used by Hayes & Londe (2006) in dealing with Hungarian are of a similar type, though formulated differently.

It is worth noting that the main cases of harmony that Pulleyblank (2004) discusses are dominant-recessive systems, and the constraints are oriented towards the recessive (target) feature value, such as ATR in Yoruba. Thus, although Pulleyblank (2004) does not discuss targets, the constraint here is consistent with the insight of his constraints, despite the distinct implementation. Nonetheless, in the standard OT framework adopted by Pulleyblank (2004), without weights or scaling, the constraint in (7) could not predict the Hungarian transparency facts; both Back-Front-Back and Back-Front-Front sequences violate *[–αBack]∞[αBack] twice, and so this constraint without scaling could not distinguish these candidates. The next two subsections deal with how this constraint will be scaled in the current framework to allow for transparency.

5.3 Kimper’s (2011) trigger strength

In order to explain asymmetries in triggering behaviour in vowel harmony, Kimper (2011) introduces a trigger strength scaling factor for his harmony constraint. As described in Section 5.1, this scaling is a constant for a given vowel, and the weight of a violation of the harmony constraint is multiplied by this scaling value to determine the penalty (or, in Kimper’s framework, reward) of a particular trigger configuration. Thus, it has the effect of increasing the likelihood of harmony when the potential trigger is perceptually impoverished for the harmony feature, which is a characteristic of good harmony triggers (Kimper 2011). Trigger strength is therefore vowel-specific, but cross-linguistic and phonetically motivated, exactly parallel to the concept of target-specific harmony advocated for here.

Kimper (2011) employs a more complex formal model than the one necessary here, but the scaling factor that he develops for trigger strength forms the basis for the current approach to vowel-specific targeting. A simplified version of Kimper’s (2011) trigger scaling factor, quoted from Bowman (2013: 4), is given in (8).

This constraint refers to rewards, since Kimper (2011) and Bowman (2013) use a harmony constraint that is formulated positively, meaning that harmonic configurations are rewarded. As discussed in Section 5.2, such a constraint is impossible here. However, the trigger strength scaling factor can easily be reformulated for a negative constraint, as in (9).¹⁷

This scaling factor has the effect of increasing the penalty incurred for a disharmonic trigger/target pair when the trigger is perceptually impoverished. The result is that not harmonizing is worse with a good trigger than with a bad trigger. For example, in a front/back harmony system, if [y] is more perceptually impoverished for [back] than [i] (i.e. [y] is a better trigger), then the trigger strength scaling for [y] should be greater than or equal to that of [i]. This scaling factor is then applied to the weight of the harmony constraint in the way described in Section 5.1. As such, a disharmonic sequence in which [y] is the potential trigger will receive a greater penalty than one in which [i] is the potential trigger. In order to capture the fact that Hungarian neutral vowels are transparent instead of opaque, I adopt the trigger scaling factor in (9) in addition to the target one introduced in Section 5.4. (See Kimper 2011 and Bowman 2013 on the application of trigger strength to Hungarian.)

Kimper (2011) also proposes a distance scaling factor, which is particularly relevant for the count effect in Hungarian; he suggests that the reward of harmony decreases as the distance between the trigger and target increases. Since I formulate the harmony constraint negatively here, in this case, the penalty will decrease. A definition of the distance scaling factor, based on the simplified version in Bowman (2013), is given in (10).

In Hungarian, this factor will have the effect of penalizing disharmony less at a greater distance, which will create the count effect; with multiple neutral vowels, the back vowel in the root is further from the suffix target, and so a front suffix will be penalized less than it would with only a single neutral vowel.

5.4 Target quality

As argued in Sections 2 through 4, whether a vowel undergoes harmony or not is sensitive to its quality as a potential target. This observation is similar to Kimper’s (2011) discussion about triggers, and so I formalize it by adapting Kimper’s (2011) trigger strength scaling factor in (8–9) for use with targets. The statement of this scaling factor is given in (11).

Parallel to the one for trigger strength, this scaling factor has the effect of increasing the penalty for disharmonic pairs when the potential target is a good harmony target. In this definition, I remain agnostic on the precise factors that determine target quality; an in-depth analysis of the underlying mechanisms behind target quality is left for future research. However, as discussed in Section 3, it is clear from cross-linguistic patterns that low and round vowels are better targets of front/back harmony than non-low, unrounded vowels, and there is significant phonetic motivation for this asymmetry. As such, for front/back harmony, the target quality scaling factor of vowels that are low and/or round should be greater than or equal to that of non-low, unrounded vowels. This scaling is implemented within the model as described in Section 5.1.

5.5 Markedness conflation

Both trigger and target scaling factors are hypothesized to be universal, but only in the sense that they are motivated by cross-linguistic phonetic facts and are options across languages. A language may choose to treat all vowels equally in terms of triggering harmony and/or being targets; in such cases, the harmony system would be symmetrical. However, if a language chooses to scale for trigger strength and/or target quality, there are limited options for the relative scaling factors across vowels, and these options are built into the definition through the phonetic motivations. For example, since target quality in front/back harmony is motivated by articulatory and perceptual properties of height and rounding with respect to backness, languages can pattern like Hungarian, with only vowels that are round and/or low undergoing harmony, but reverse Hungarian is predicted to be impossible. Specifically, for front/back harmony systems the target scaling factor for non-low, unrounded vowels must be less than or equal to that of vowels that are low and/or rounded. The option for equality allows for symmetry, but if there is asymmetry, it must be in the direction of less participation of non-low, unrounded vowels.

While the formalization differs, this approach parallels previous work on similar hierarchies, such as de Lacy’s (2004) concept of markedness conflation. The target concept here is similar in that it reflects universal properties of which vowels are better targets, and the “ranking” of possible targets can be conflated. For example, considering only height, since lower vowels are better targets of front/back harmony, we predict the following possibilities: no vowels participate, only low vowels participate (neutrality of mid and high vowels), low and mid vowels participate (neutrality of high vowels), or all vowels participate (no neutrality). Languages where front/back harmony does not fit within these categories, such as with neutrality of low vowels to front/back harmony but participation of high and mid vowels, are not expected. These predictions are consistent with the cross-linguistic height generalizations in front/back harmony.

Formally, this universal hierarchy is implemented here through the phonetic motivations behind the target scaling factor in the definition, and the possibility of conflation comes from the option for multiple categories to have the same scaling factor. Thus, the target scaling values by height for front/back harmony (given consistent rounding) must be such that low ≥ mid ≥ high; the different patterns are derived by the possibility of equality. Recall that the target scaling factor itself is defined in terms of phonetic correlates of target quality, so that this hierarchy is not simply stipulated.

Similar to de Lacy’s (2004) markedness conflation constraints, this approach does not require a stipulated universal ranking, because the scaling factor is defined to be motivated by universal, cross-linguistic properties. It remains for future research to determine whether the scaling factors described here and the stringency hierarchy of constraints discussed by de Lacy (2004) are notational variants or whether they in fact make distinct predictions.

5.6 Morpheme-specificity

As discussed in Section 2.4, whether a particular vowel is a trigger or target in Hungarian depends not only on vowel quality, but also whether it is in a stem or suffix, as well as the specific morpheme. The former property is simple to account for in terms of root faithfulness (Beckman 1998), since in general in Hungarian, there are stricter harmony requirements on suffixes than on roots (e.g. [ɛ] is required to harmonize in a suffix, but not in a root).

Morpheme-specific behaviour is more complicated, in particular given its interaction with vowel quality; variation is lexically determined only for [e:] and [ɛ] (Rebrus et al. 2012). In particular, under the more modern assumption that [a:]~[e:] suffixes are not all underlyingly /a:/, we need to be able to capture the fact that [e:], unlike all other vowels, can be a target in some but not all suffixes. Further, both [e:] and [ɛ], unlike other vowels, can be triggers in some but not all roots in which they follow back vowels. For example, (2d) showed a back-[e:] root that takes only back suffixes, [ka:ve:-nɔk], while (2g) showed a similar root that can take either back or front suffixes, [ɔste:k-nɔk/nɛk]. Similarly, (2i) showed a back-[ɛ] root that can take either back or front suffixes, [ma:gnɛʃ-nɔk/nɛk], while (2j) showed one that takes only front suffixes, [ko:dɛks-nɛk]. In other words, whether a given vowel is a trigger or target depends not only on its quality, but also on the morpheme it occurs in. The choice of a given morpheme is arbitrary, except that it is limited to the “zones of variation” (Hayes et al. 2009: 835) that are possible for morphemes of its type.

It would be conceivable to think of Hungarian harmony as scaled entirely by morpheme rather than by vowel quality, with each root morpheme specified to take certain types of suffixes. A full analysis does require a morpheme-specific component, in order to account for the fact that a given morpheme patterns consistently in the system, in a way that can be distinct from other morphemes of the same phonological shape (e.g. the different back-[ɛ] roots in (2)). However, allocating all of the phonological work to morpheme-specific factors would be highly stipulative, given that the set of options for the behaviour of a given morpheme is predictable based on vowel quality in Hungarian. Incorporating factors of trigger and target vowel quality into the analysis reduces this stipulation, by limiting the set of possibilities for morpheme-specific behaviour to the existing phonologically-defined trends, in a way that is typologically and phonetically motivated. Thus, an analysis that incorporates both morpheme- and vowel-specific factors is preferable to an account that deals with only one of these.

The morpheme-specific behaviour seen in Hungarian is of a very predictable type: a vowel may exceptionally have the normal behaviour of a vowel that is one step better as a trigger or target, in the scale (from low to high) of [i(:)] – [e:] – [ɛ] – harmonic vowels. Specifically, [i(:)] and the harmonic vowels are not variable, the former because the normal behaviour of [e:], like that of [i(:)], is neither triggering nor being a target of harmony, and the latter because there is no category above. In contrast, [e:] can, in some morphemes, show the normal triggering and targeting behaviour of [ɛ], in that it variably triggers harmony (vacillation) in some roots (e.g. (2g)) and can be a target of harmony in some suffixes; both of these are the normal behaviour of [ɛ]. Similarly, in some morphemes, [ɛ] can show the normal behaviour of the fully harmonic vowels, in consistently triggering harmony (e.g. (2j)). Note that [ɛ] already shows the typical targeting behaviour of the other vowels, since it does not occur in invariant suffixes.

Due to these generalizations about the possible morpheme-specific behaviour, I will encode morpheme-specific behaviour into the trigger and target scaling factors, with vowels exceptionally able to take the trigger and target scaling of a vowel in the category above. It remains an open question why there appears to be this one-step limit in morpheme-specific behaviour (e.g. why [i] cannot behave like [ɛ]), but this issue is beyond the scope of the present paper. I suggest the possibility that a one-step change can be consistent with the phonetic properties of a given vowel, which motivate its normal behaviour in relation to how the language sets its scaling factors, but a larger change cannot be.

6.1 Constraints, weights, and scaling

As discussed in Section 5.2, I adopt a target-oriented harmony constraint that reflects that Hungarian harmony is progressive and triggered by either feature value. This constraint is repeated in (12). One violation of this constraint is counted for each [αBack] vowel preceded by a [–αBack] vowel at any distance.¹⁸

In addition to this harmony constraint, we require general markedness constraints to prevent vowels that are unattested in Hungarian from surfacing, as well as a faithfulness constraint against changes to [low]. These constraints are defined in (13).

While the harmony constraint is in conflict with faithfulness to the feature [back], the general constraint IDENT-IO[back] is not active within the Hungarian system, and so I omit it from the analysis.²⁰ Instead, I will assume that there is an active positional faithfulness constraint (e.g. Beckman 1998), specifically a highly weighted constraint to preserve the [back] specification of the word-initial vowel. This constraint will not be illustrated in the tableaux for space reasons, but will ensure that the initial vowel is the trigger of harmony.²¹ Since Hungarian also has rounding harmony, I assume the same about IDENT-IO[round] as about IDENT-IO[back].

Many previous analyses of Hungarian harmony (e.g. Ringen & Vago 1998) ignore root-internal harmony, assuming a highly ranked root faithfulness constraint. In reality, a high ranking of such a constraint is only necessary for loanwords (see footnote 21 on loanword treatment). As discussed in Section 2, disharmonic roots consisting of non-neutral vowels are recent loans; within native Hungarian roots, non-neutral vowels must agree in backness, while front unrounded vowels may both precede and follow either front or back vowels. This pattern is in contrast to the behaviour of neutral-back sequences across morpheme boundaries, because neutral vowels trigger last-resort harmony (see (3)). I suggest that these facts can be accounted for using root-specific faithfulness to [back], defined as in (14).

Weighted correctly with respect to harmony, this constraint will allow non-neutral vowels to harmonize within a root, but require root-internal disharmony involving neutral vowels to surface faithfully. Given that the constraint is root-specific, it will not affect last-resort triggering in which suffix vowels harmonize to neutral root vowels.

The constraint weights and scaling factors are set as in (15). There are many choices that would work, and the specific numbers are arbitrary, as long as certain conditions are satisfied; throughout the analysis, I will argue for these specific conditions on weighting that are required for Hungarian. Note that the scaling factors in (15b–d) are specific to the particular constraint *[–αBack]∞[αBack], and are not applied to the grammar as a whole; this follows Kimper (2011) and Bowman (2013).

As discussed in Section 5.6, note that both trigger and target scaling can vary morpheme-specifically, in that a morpheme may select for its vowel to behave like the vowel one category above it. This will be discussed further in Section 6.4.

6.2 Basic harmony facts

The present analysis retains the ability to capture the basic facts of Hungarian harmony that are the focus of other analyses. Tables 8, 9, 10, 11, 12, 13 illustrate. Following Vago (1980b), among others, I assume that the dative suffix underlyingly has a front vowel; this assumption is not critical to the analysis, due to the low (zero) hypothesized weight of the general IDENT-IO[back].

Table 8 shows that suffix [ɛ] is required to harmonize following non-neutral roots; the same will apply for front rounded and back vowels. Disharmony violates *[–αBack]∞[αBack], while the harmonic form does not violate any of the constraints due to the absence of IDENT-IO[back] in the analysis (i.e. its very low hypothesized weight). This tableau does not demonstrate any weighting of the included constraints.

Table 9 shows that front rounded and back vowels are required to harmonize within a root; disharmony violates *[–αBack]∞[αBack], while the harmonic form in this case violates IDENT-IO[back]_ROOT. This tableau demonstrates the need for the weighting condition in (16). (See footnote 21 on the distinct analysis required for loanwords, in which disharmony is allowed.)

Table 10 demonstrates that front unrounded vowels are able to be transparent. In candidate (a), the neutral vowel [e:] is opaque, with the suffix surfacing as front. This candidate has two violations of *[–αBack]∞[αBack]: one for the potential target [e:], with trigger [a:], and the other for the potential target [ɛ], again with trigger [a:]. The first violation is scaled by 16(a:)*1(e:) = 16. The second is scaled by 16(a:)*8(ɛ)*1/2(distance) = 64, since there is one syllable intervening between them. The total score for candidate (a) is therefore –80. Candidate (b), with transparency, similarly has two violations of *[–αBack]∞[αBack]: one is again for the potential target [e:], with trigger [a:], but the other is for the potential target [ɔ] with trigger [e:]. Again, the first violation is scaled by 16, but this time, the second is scaled by 2(e:)*16(ɔ) = 32. There is no distance scaling here, since the vowels are in adjacent syllables. The total for candidate (b) is thus –48. A fully harmonic candidate, changing the /e:/ to [a:], violates both root faithfulness to [back] and faithfulness to low, and therefore receives a score of –220. Thus, candidate (b) wins, since –48 is the highest score; transparency (b) is preferred over opacity (a) because [e:] is a poor trigger, and over full harmony (c) due to root faithfulness. The required weighting conditions demonstrated in this tableau are given in (17); note that here is where it is necessary to maintain Kimper’s (2011) trigger strength scaling in order to derive transparency rather than opacity.

Table 11 illustrates that neutral vowels are last-resort triggers of suffix harmony, even in cases where harmony involves an additional faithfulness violation. Candidate (a) is faithful, but violates harmony, with scaling of 2 for trigger [i:] and 16 for target [a:]. The total is therefore –32. In contrast, candidate (b) harmonizes, violating only IDENT-IO[low], which gives a score of –20. The harmonic candidate is the winner. The same result will also hold for suffixes that do not violate IDENT-IO[low]. The weighting condition from this tableau is given in (18).

Importantly, the last-resort triggering in Table 11 happens only across a suffix boundary; root-internal neutral-back sequences surface faithfully due to IDENT-IO[back]_ROOT. Table 12 illustrates; the faithful candidate wins because harmony violates IDENT-IO[back]_ROOT. (19) gives the required weighting conditions. This result will also hold for any combination of back vowels with front unrounded vowels (i.e. any of these vowels, in any order), because of the high weight of IDENT-IO[back]_ROOT relative to the trigger and target scaling factors of front unrounded vowels. Specifically, they highest trigger or target scaling of a front unrounded vowel is 8; because 8*16 = 128 < 200, even root-internal disharmony involving [ɛ] can surface faithfully. This is in contrast to Table 9, where root faithfulness is not maintained with harmonic vowels; in that case, the harmony constraint was scaled such that it outweighed root faithfulness. This result is consistent with the generalization that non-neutral vowels in native Hungarian roots are from the same harmonic class, whereas front unrounded vowels can co-occur with either class.

6.3 [a:]~[e:] alternation

Given the assumptions and constraints outlined in Section 6.1, this analysis can also accurately capture the asymmetric re-pairing behaviour of Hungarian [a:] and [e:], in which [a:] is required to re-pair but [e:] is not. Table 13 shows that /a:/ re-pairs to surface as [e:] when it follows a front rounded vowel. Direct harmony in [back], as in candidate (c), violates the highly weighted constraint *[æ:]. The decision is then between re-pairing, in (b), and faithfulness, in (a). Re-pairing violates IDENT-IO[low], with a weight of –20. The faithful candidate is disharmonic, therefore violates *[–αBack]∞[αBack]. Trigger and target scaling are each 16, giving a total of 1*(–1*16*16) = –256 as the score for candidate (a). Since –20 > –256, candidate (b) wins, and /a:/ is required to re-pair to [e:]. To ensure that harmony occurs in this case, we require the weighting conditions stated in (20). Note that /a:/ will not re-pair to [i:] or [y:] because such candidates are harmonically bounded by the one in which /a:/ re-pairs to [e:]: all of these candidates violate IDENT-IO[low], but re-pairing to a high vowel also violates IDENT-IO[high]. I therefore omit candidates in which /a:/ re-pairs to a high vowel, as well as the constraint IDENT-IO[high], from the tableaux for presentation reasons. The generalization here is that /a:/ re-pairs to the front vowel that requires the fewest feature changes, and so changing [high] in addition to [low] will be worse than changing only [low].

In contrast, Table 14 demonstrates that /e:/ is not required to re-pair when it follows a back vowel. Similarly to Table 13, direct harmony in [back], as in candidate (c), violates the highly weighted constraint *[ɤ:], and the decision is then between re-pairing, in (b), and faithfulness, in (a). Again, re-pairing violates IDENT-IO[low], with a weight of –20, while the faithful candidate is disharmonic and therefore violates *[–αBack]∞[αBack]. However, since the potential target is [e:], target scaling is only 1. Thus, the total score for candidate (a) is 1*(–1*16*1) = –16. Since –16 > –20, candidate (a) wins in this case, meaning that /e:/ is not required to re-pair to [a:]. This tableau demonstrates the need for the weighting conditions given in (21). Critically, comparing the second condition in (21) to the second condition in (20) shows the need for scaling(target [e:]) < scaling(target [a:]), illustrating the need for the target quality concept argued for here.

These examples show that this analysis correctly predicts the asymmetry between [a:] and [e:] in Hungarian, by appealing to the fact that [a:] is a better target of front/back harmony than [e:] is. Disharmony involving the potential target [a:] is therefore penalized to a greater extent than disharmony involving the potential target [e:]; the result is that harmony outweighs faithfulness for /a:/, but the reverse is true for /e:/. Notably, unlike an analysis that uses separate faithfulness constraints for [+low] and [–low], the current approach does not predict the possibility of reverse Hungarian. Indeed, due to the phonetic basis and markedness conflation inherent in the definition of target scaling, the target scaling factor for [e:] must be less than or equal to that of [a:]. As such, these vowels must either pattern like they do in Hungarian or behave symmetrically.

6.4 Variability

The behaviour in Table 14 is not the only possibility for how [e:] will behave in suffixes. As noted in Section 2, [e:] has intermediate neutrality along this dimension; the recent literature seems to assume that the underlying form of affixes that alternate between [a:] and [e:] is not necessarily /a:/, but rather that /e:/ is not fully neutral and could be required to harmonize in suffixes. This is possible under the current analysis due to the option for morpheme-specific target scaling factors. As discussed in Section 5.6, Hungarian morphemes may exceptionally select for their vowels to take the scaling trigger and/or target factors of a vowel that is one step better as a trigger/target. If we have such a suffix with [e:], it will be required to harmonize, as shown in Table 15, where the subscript “ɛ” indicates an [e:] that morpheme-specifically behaves like [ɛ], in the way justified in Section 5.6.

In other words, while rounded vowels and back vowels are required to harmonize, [e:] may or may not: it is of intermediate neutrality with respect to being a target in a suffix. This possibility does not exist for [i(:)], because the next category up is [e:], which has the same target scaling as [i(:)]. It also does not exist for [ɛ] or other vowels, which already have target scaling factors sufficient to ensure that harmony in suffixes must always occur. Thus, [i(:)] is consistently invariant, [e:] is variable, and [ɛ] and other vowels are consistently targets, reflecting the suffix height effect illustrated in Table 3 in Section 2.

I turn now to variability in triggering, which occurs in distinct ways for both [e:] and [ɛ]: the former can be transparent or variable, while the latter is typically variable or opaque. An example of transparent [e:] was given in Table 10; for it to be variable, it must select for the morpheme-specific trigger scaling of [ɛ]. This is shown in Table 16, where the subscript “ɛ” indicates that the vowel morpheme-specifically behaves like [ɛ]. In such a case, the front and back variants of the suffix receive identical harmonic scores, meaning that either one could surface. Note that this result holds regardless of whether we assume that [e:] is also morpheme-specifically a better target in this case.

Given that this trigger scaling factor is the usual one for [ɛ], the analysis predicts that back-[ɛ] roots will typically vacillate. Moreover, [ɛ] can morpheme-specifically have the trigger strength of a fully harmonic vowel; in such a case, the suffix will be consistently front, with [ɛ] opaque, similarly to the case of harmony with non-neutral vowels in Table 8.²² In contrast, [i(:)] can never behave as vacillating or opaque, because its only morpheme-specific option is to have the trigger scaling of [e:], which generally behaves as transparent. Thus, the analysis is able to derive the height effect in transparency, in which high vowels are consistently transparent, mid vowels are sometimes variable, and low vowels are generally variable or opaque. This is due to the fact that morpheme-specificity in Hungarian is limited to a vowel behaving as “one category up”, which I hypothesize could relate to the phonetic motivations for trigger and target; a vowel cannot morpheme-specifically take on the behaviour of one too distinct from its own phonetic properties.

Finally, multiple neutral vowels can also show vacillation, taking both front and back versions of the suffix. This is a simple effect of distance scaling, as shown in Table 17.

Note that this view of vacillation is a simplification; a more nuanced analysis, which is beyond the scope of the present paper, would convert harmonic scores to percentages, because the Hungarian vacillation data is not as simple as each possible suffix form surfacing half the time. This issue will be discussed further in Section 6.5. Moreover, there is a role for paradigm uniformity in the behaviour of vacillating roots; generalizations of other aspects of paradigm uniformity in Hungarian harmony have previously been made by Törkenczy et al. (2013) and Rebrus et al. (2017). In this case, paradigm uniformity is required to ensure that vacillating roots behave the same way across multiple types of suffix vowels; specifically, we might otherwise predict suffixes with alternants containing [ɛ] or [e:] to behave differently than suffixes with fully harmonic vowels, because the target scaling factors are different. Again, a full implementation is beyond the scope of the present paper. The critical aspect of the current analysis is that paradigm uniformity and morpheme-specificity do not need to bear the entire burden of explanation. Instead, the explanation here is shared by vowel-specific factors, which are less arbitrary because they are motivated by cross-linguistic and phonetic evidence.

To summarize, this section and those preceding it have demonstrated that the target-focused framework developed in Section 5 maintains the ability to capture the crucial facts about Hungarian harmony that have been previously analyzed, while also extending to the gradient nature of neutrality in the language. The specific numbers given here work, but what is crucial is that all of the relative weighting requirements are satisfied, specifically in terms of the relative ranges of trigger and target scaling factors for the different vowels.

6.5 Further directions for Hungarian

While it captures much of the gradient variability in Hungarian, this analysis still abstracts away from some of the more problematic aspects of Hungarian harmony. In this section, I consider how the analysis could be extended to additional data for which a full treatment is beyond the scope of this paper.

First, as mentioned in Section 2, Hungarian has some anti-harmonic roots, which contain only front unrounded vowels, yet take back suffixes. An adequate analysis of these roots should explain why they all contain only front unrounded vowels, and why the majority of them contain [i(:)]. This patterning exactly parallels the other aspects of neutrality (invariant suffixes and transparency), and so should receive a similar account. One possibility in the framework here is that all alternating Hungarian suffixes are by default [+back], and that there is additional variability in the trigger strength of [i(:)], and to a lesser extent [e:]. If the trigger strength of these vowels can, in specific morphemes, be low enough to not trigger harmony at all, then default suffix vowels should occur; if the default is [+back], then anti-harmony is derived. The assumption of a default [+back] for suffixes changes a number of aspects of the above analysis, in particular the generalizations related to [e:] sometimes undergoing harmony in suffixes, but it is worth exploring as a future direction for explaining anti-harmony.

A further property that this analysis could be extended to deal with is stem/suffix interactions. Research has shown that of the vacillating stems that can take either front or back suffixes, stems vary in terms of their preference for front or back, and this further varies by the specific suffix attached to the stem. For example, Törkenczy (2013) notes that the root [fotɛl] ‘armchair’ more often takes the back version of the illative suffix [bɔ]/[bɛ], while the root [kontsɛrt] ‘concert’ takes only the front version of this suffix. With the plural [ok]/[ɛk], [fotɛl] instead takes the front version more often, while the root [hɔvɛr] ‘friend’ almost always takes the back version. In other words, the preferred choice for vacillating cases depends on both the stem and suffix.²³ Since the present analysis deals with morpheme-specific properties, a more nuanced version of it could be derived, in which there is some correlation between the harmonic scores of candidates and their percentage frequency, and this could depend on both the specific stem and suffix. This direction ties in with the general comment about the nuances of vacillating roots as discussed in Section 6.4. Moreover, it could reduce the need for an appeal to paradigm uniformity to explain the similarity in behaviour of vacillating roots across suffixes, regardless of whether the suffix alternants contain [e:], [ɛ], or only harmonic vowels.

Recent literature on Hungarian has noted that invariant and alternating suffixes with [e:] behave differently with respect to vacillating roots: the invariant ones (e.g. [e:] in Table 14) are transparent, in that the root+suffix combination remains vacillating, while the alternating ones (e.g. [na:l]~[ne:l] in Tables 13 and 15) are opaque, in that the root+suffix combination can only take front suffixes (e.g. Törkenczy et al. 2013). As Törkenczy et al. (2013) note, part of the explanation can be due to paradigm uniformity. In the present analysis, another factor to consider is the correlation between morpheme-specific trigger and target scaling. It is clear in Hungarian that a correlation between trigger and target behaviour exists in the general case: the height effect for triggering follows the exact same pattern as the height effect for being targeted. Since this analysis treats morpheme-specific variation as bumping a vowel into the behaviour of a vowel one level better on the trigger or target scale, we predict the possibility that in cases where [e:] behaves like [ɛ] as a target, it could also behave like [ɛ] as a trigger. As such, alternating [e:] should behave differently than invariant [e:], as the former is morpheme-specifically a better trigger than the latter. This possible correlation between trigger and target factors, including in morpheme-specific cases, should be explored further in additional research.

A question arising from this analysis is why the trigger and target scaling can change morpheme-specifically. It is clear that they must, with limited options given the type of morpheme, because the variable patterning in Hungarian shows morpheme-specific properties in addition to vowel-specific ones. A direction to explore here is the effect of loans: vacillating back-[e:] roots tend to be loans, as do many of the roots in which [ɛ] co-occurs with a back vowel, regardless of their behaviour. A potential hypothesis is that loanwords come into the lexicon with trigger and target scaling that may not match the typical pattern of native words, but that a degree of closeness to the phonetically motivated behaviour of native words is required (i.e. the one category limit). This idea could also potentially be extended to “disharmonic” roots, in which back vowels co-occur with front rounded vowels. As noted, such roots are always recent loans, and perhaps the factor that prevents root-internal harmony from applying is a difference in the scaling factors for these vowels. Thus, examining the connection between loans and (dis)harmonic behaviour in Hungarian, in relation to the issue of “one category up” variability, is a worthwhile future direction.

7.1 Target conditions in other languages

Hungarian [e:] is not alone in the fact that it is generally neutral to harmony despite appearing to have the option to participate through pairing with [a:]. As noted in Section 1, vowels that are able to undergo harmony do not always do so; moreover, there is a degree of cross-linguistic consistency in the choice between neutrality and participation, regardless of whether a vowel is paired. Mayak is one such case: the low vowels [a] and [ʌ] are paired for [ATR] and participate in tongue root harmony in limited contexts, but they are generally neutral (Andersen 1999). Moreover, low vowels in tongue root harmony systems are often unpaired and neutral, as in languages like Yoruba, which has an opaque [a] but no [+ATR] low vowel (Archangeli & Pulleyblank 1994). Archangeli & Pulleyblank (1994) note that this kind of consistency is in fact a cross-linguistic pattern: paired vowels that are idiosyncratically neutral, as is the case in Mayak, tend to be the same types of vowels that are unpaired and neutral in other languages.

Any theory in which neutrality is based on the impossibility of undergoing harmony will have difficulty explaining cases like Mayak. More importantly, they miss the generalization that which vowels are neutral tends to be the same cross-linguistically regardless of whether the neutral vowels are paired or unpaired. This consistency is clear evidence that vowel-specific, harmony-specific factors beyond inventory pairing (and the associated general markedness) affect whether a vowel is a target for harmony or is neutral, contra many claims in the literature.

The analysis presented here can also be extended to cases not commonly considered as neutrality, namely harmony that targets only certain vowels. Previous analyses of vowel harmony have also noted that harmony may be subject to target conditions. An example is the constraint HI/ATR “if [+high], then [+ATR]”, which Archangeli & Pulleyblank (2002) use to account for high vowels being preferred targets in Kinande ATR harmony. This mechanism for incorporating features of targets differs from the current proposal in that it directly references the harmony-independent notion of feature compatibility, rather than the quality of a vowel as a potential harmony target. These notions are clearly related, as feature compatibility affects the phonetic and cross-linguistic factors that determine target quality. Though the details of the present approach are novel, the idea that harmony may be subject to target conditions is not, and target-specific harmony is necessary outside of Hungarian.

It is worth noting that the approach of Archangeli & Pulleyblank (2002) differs from the one here, for instance in the case of low vowels in front/back harmony; neither LO/BACK nor LO/FRONT would capture that low vowels are good targets, the former because it cannot explain why the low back vowel [a:] undergoes fronting in front harmony contexts, and the latter because it is unmotivated both cross-linguistically and in Hungarian, in which low vowels tend to be back, in terms of the number of such vowels in the phonologically defined inventory. ²⁴ In contrast, in the present approach, the phonetic motivation for low vowels to be good targets for front/back harmony is built into the theory’s prediction of their behaviour.

Overall, target conditions are necessary beyond Hungarian, and the present approach has the ability to extend to other languages and types of harmony. With the cross-linguistic concept of target quality, this analysis accounts for generalizations missed by a featural compatibility approach to neutrality, and can capture more complex cases like paired neutral vowels.

7.2 Trigger/target interactions

An additional implication concerns the interactions between triggers and targets, considered in this subsection. Given the use of scaling for both triggers and targets, the present analysis predicts that trigger and target should be able to interact in such a way that harmony depends on both of them. In other words, we should find cases in which harmony occurs only with both a strong trigger and a quality target, or in which disharmony is permitted only with a weak trigger and a poor target.

This prediction is borne out in rounding harmony in Kirgiz and Altai. Kaun (1995; 2004) shows that in rounding harmony, good triggers are vowels that are non-high and/or front, while good targets are those that are high.²⁵ In Kirgiz and Altai, rounding harmony is obligatory except when the potential trigger is high and back and the potential trigger is low; only in such cases is disharmony permitted (Korn 1969). Table 18 summarizes, with examples from Korn (1969).

The only trigger/target combination where disharmony is permitted is precisely the one that is predicted: high, back vowels are weak triggers, while low vowels are poor targets. Thus, Kirgiz and Altai manifest a trigger/target interaction predicted and easily accounted for by the present framework.

Nonetheless, the more common trigger/target relationship in rounding harmony is that trigger and target must agree in height, which is known as parasitic harmony and occurs for example in Yokuts (Kaun 1995). On the surface, this pattern would be problematic under the present analysis; as Kaun (1995) notes, good triggers of rounding harmony are non-high, while good targets are high, which means that height agreement involves either a bad trigger or a bad target. The approach here, without modifications, therefore predicts that if harmony occurs in cases of height agreement, then it should also occur when both the trigger and target are good.

However, this problem only occurs under the assumption that what it means to be a good trigger is independent of what it means to be a good target, and vice versa. Particularly in rounding harmony, this assumption is false; as Kaun (1995) argues, rounding is realized distinctly on high versus non-high vowels, meaning that height agreement allows for uniformity in the phonetic realization of [round]. Based on this argument, the relationship between trigger and target scaling factors should be dependent, which would permit an account of parasitic harmony. Formally, this could be implemented with either a single scaling factor for each trigger/target pair or with the addition of a scaling factor for the interaction effect.

To summarize, the prediction that trigger strength and target quality can interact is borne out in Kirghiz and Altai rounding harmony. Parasitic rounding harmony, which appears at first glance problematic for this approach, is in fact possible to analyze; it requires the addition of an interaction factor, which is motivated by the fact that triggers and targets are not independent.