2.1 Background: bimoraicity requirement

Japanese requires that all prosodic words (ω) have at least two moras (Itô Reference Itô1990, Mester Reference Mester1990, Poser Reference Poser1990, Mori Reference Mori2002, Itô & Mester Reference Ito, Mester, Honma, Okazaki, Tabata and Tanaka2003). This bimoraicity requirement is observed in many word-formation patterns, all of which are based on a bimoraic template, including nickname formation, geisha-client name formation, loanword abbreviation, verbal root reduplication, scheduling compounds and telephone number recitation.

For instance, in the nickname formation pattern, a full name must be truncated to two moras before the suffix -chan can be applied.Footnote ⁶ For example, the five-mora name Wasaburoo can be truncated to two moras, as in (2a), but not one. Similarly, the three-mora name Kotomi can be truncated to either two monomoraic syllables, or a single bimoraic syllable, as in (2b). It cannot, however, be shortened to a single mora.

1. (2)

   

To summarise, a prosodic word must contain at least one foot, and the foot must be binary (at the moraic level in Japanese; McCarthy & Prince Reference McCarthy and Prince1986, Reference McCarthy and Prince1993), as in (3).

1. (3)

   

In spite of this bimoraicity requirement, there are monomoraic nouns in the Japanese lexicon; e.g. [ki] ‘tree’, [i] ‘stomach’ and [e] ‘picture’. When these monomoraic nouns appear in isolation within a prosodic word (e.g. without a case particle), they are produced with vowels of intermediate length – longer than canonical short vowels, but shorter than underlyingly long vowels (Mori Reference Mori2002, Braver & Kawahara Reference Braver2014, Reference Braver and Kawahara2016). Underlyingly bimoraic nouns in the same environment do not show such lengthening. Mori thus concludes that the lengthening is caused by the phonological bimoraic minimality requirement: monomoraic nouns with a case particle within their ω satisfy the bimoraicity requirement (by virtue of the particle's mora), as in (4a), while monomoraic nouns without a particle must gain an additional mora to satisfy this requirement, as in (4b). No lengthening is required for underlyingly bimoraic nouns, as in (4c).

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2.2 Monomoraic lengthening is incompletely neutralising

In spite of the identical surface mora counts in (4b) and (4c) (due to lengthening in the case of (4b)), Braver & Kawahara (Reference Braver and Kawahara2016) found that the vowel durations of lengthened monomoraic nouns were not identical to those of underlyingly long nouns. In that study, twelve native speakers of Japanese were presented with 15 sets of three nouns: (a) a short noun with a case particle (no lengthening), (b) a short noun without a case particle (lengthening expected) and (c) an underlyingly long noun. (5) shows a sample triplet, with nouns in a frame sentence. The target nouns (the first word in each frame sentence) for all three members of the triplet share identical segmental material, differing only in the presence/absence of a case particle and underlying vowel length.

1. (5)

   

Braver & Kawahara (Reference Braver and Kawahara2016) found that lengthened nouns were on average 32.47 ms shorter than underlyingly long nouns, as summarised in Table I. In other words, the short/long vowel contrast is incompletely neutralised: lengthened short vowels are more like long vowels than short vowels in non-neutralising contexts, but they are not identical to long vowels.

Table I

Means, standard deviations and rounded values for vowel duration of nouns (in ms), from Braver & Kawahara (2016) (12 speakers, 15 sets of 3 nouns (= 45 items in total), 7 repetitions).

For ease of explication, in §4 I will focus on the overall means (across speakers and vowel qualities), and will use the rounded values for the target durations of short, lengthened and long vowels given in the rightmost column of Table I.

2.2.1 Bimoraic nouns with and without case particles

It should be noted that the nouns used in the underlyingly long condition in the Braver & Kawahara (Reference Braver and Kawahara2016) study were not followed by a case particle (e.g. kii, rather than kii-mo). An anonymous reviewer points out that in order for the length contrast to be incompletely neutralised, the duration difference between monomoraic nouns with and without a particle (e.g. ki-mo ~ ki, involving degree of lengthening) must be smaller than the duration difference between monomoraic and bimoraic nouns with particles (e.g. ki-mo~kii-mo, with an underlying length contrast). In other words, to be an actual case of incomplete neutralisation, the degree of lengthening in putatively incompletely neutralised lengthened monomoraic nouns must be less than the normal length contrast found in the non-neutralising context of monomoraic vs. bimoraic nouns with particles.

There are two reasons to believe that the degree of lengthening (69.99 ms; see Table I) is indeed less than the underlying length contrast. First, Mori (Reference Mori2002) shows that bimoraic nouns hardly differ in duration when uttered with vs. without particles – those without case particles are only longer by 5–6 ms (4–5%). Even if we assume that particle-less kii is 10% longer than kii-mo, we should expect kii-mo to be approximately 143 ms. Assuming this generous estimate, the underlying length contrast is 88.15 ms – a great deal larger than the degree of lengthening in monomoraic nouns.

Second, an earlier version of the Braver & Kawahara (Reference Braver and Kawahara2016) study included a condition with bimoraic nouns with particles (e.g. kii-mo; Braver & Kawahara Reference Braver and Kawahara2014). In this first experiment, such bimoraic nouns were on average 26.55 ms longer than lengthened particle-less monomoraic nouns (e.g. ki-mo ~ ki) – much smaller than the underlying contrast between monomoraic nouns with particles and bimoraic nouns with particles (e.g. ki-mo ~ kii-mo) of 72.20 ms. These results are summarised in Table II.

Table II

Means and standard deviations for vowel duration of nouns (in ms), from Braver & Kawahara (2014) (7 speakers, 11 sets of 3 nouns (= 33 items in total), 10 repetitions).

2.2.2 Lengthened nouns are phonologically bimoraic

As discussed in the introduction, monomoraic noun lengthening is perhaps a more compelling case of truly phonological neutralisation than final devoicing. There are two pieces of evidence to support this claim: (i) lengthened monomoraic nouns are allowed in phonological contexts which require bimoraic minimality, and (ii) lengthened monomoraic nouns can carry a pitch accent.

Japanese pitch accent is realised as H*L – a high tone on the accented mora, followed by a low tone realised on the following mora. Since the tone-bearing unit in Japanese is the mora (Haraguchi Reference Haraguchi1977, McCawley Reference McCawley and Hyman1977), any word which manifests both the H* and L tones of the Japanese pitch accent must be minimally bimoraic. This property is exemplified in the pitch tracks in Fig. 1 for the sentence sho’/sho'u (mo) dokusenshita yo ‘book/chapter (also) monopolised disc’. (Following Japanese transliteration conventions, accent is marked as <’> between the moras expressing H and L tones, and the long vowel in sho'u is represented as ou, but pronounced [oː].)

Figure 1

Pitch tracks for Japanese nouns in a carrier sentence (male speaker reading sho'/sho'u (mo) dokusenshita yo ‘book/chapter (also) monopolised DISC’): (a) underlyingly long; (b) short with particle; (c) lengthened short noun (no particle).

In the pitch track for sho'u dokusenshita yo in Fig. 1a, the start of the long vowel [oː] is realised as a high tone, descending to a low tone by the end of the second mora. Similarly, in sho’ mo dokusenshita yo in Fig. 1b, the high tone is realised on the first (and only) mora of sho, while the descent to a low tone extends beyond the word boundary and into the mora of mo. If lengthened monomoraic nouns can show pitch accent (and are therefore phonologically two moras), we should expect the fall from H* to L in sho’ dokusenshita to spread across the two moras of the lengthened o – and this is precisely what occurs (Fig. 1c).

In addition to the pitch-accent data, there is additional evidence that the particle-less context described above does indeed require two moras. If a monomoraic noun without a particle is followed by a glottal stop or a pause, these additions can provide the relevant ω with the requisite mora (Mori Reference Mori2002). Mori argues that this possibility is allowed because the additional glottal stop or pause takes up a similar amount of time as a second mora would.

3.1 Previous accounts

Perhaps the earliest theoretical account of incomplete neutralisation is due to Anderson (Reference Anderson1975), who argues that some phonetic rules can precede phonological ones. Without using the term incomplete neutralisation, Anderson points out that flapping in American English, where /t/ and /d/ become /ɾ/ in certain prosodic contexts, leads to a gradient distinction in the duration of the preceding vowel: vowels preceding /d/-flaps are slightly longer than vowels preceding /t/-flaps. He further notes that a standard rule-based analysis of this phenomenon, as in (6), cannot account for the data.

1. (6)

   

Under the analysis in (6), vowels lengthen first, followed by /t/ and /d/ becoming flaps. This ordering – which is necessary to ensure that vowels are lengthened only before /d/-flaps and not /t/-flaps – contradicts the modular feed-forward model, since the gradient (and hence phonetic) vowel-lengthening rule precedes the (phonological) flapping rule. In order to resolve this conflict, Anderson proposes that (some) phonetic rules may, in fact, precede (some) phonological rules.

More recently, van Oostendorp (Reference Oostendorp2008) has argued that incomplete neutralisation in German final devoicing can be accounted for by ensuring that devoiced (underlyingly voiced) and underlyingly voiceless segments have distinct phonological representations. Under this analysis, based on Turbidity Theory (Goldrick Reference Goldrick2000), segments can stand in two possible relations with a given feature: (i) the projection relationship, which is an ‘abstract, structural relationship’, and (ii) the pronunciation relationship, which ‘describes the output realisation of structure’ (van Oostendorp Reference Oostendorp2008: 1368). The turbidity model analyses incomplete neutralisation in coda devoicing by providing different structures for underlyingly voiceless and voiced segments: in the phonological output, devoiced segments maintain a projection relation (but not a pronunciation relation) with a [voice] feature, but underlyingly voiceless segments are not related by projection or pronunciation to [voice]. Since underlyingly voiceless and devoiced segments are thus distinct in the phonological output, the phonetic module can differentiate between them, allowing longer vowels to precede devoiced segments, for example.

Another proposal, due to Gouskova & Hall (Reference Gouskova, Hall and Parker2009), capitalises on the idea of a segment's derivational history. They present a study showing that epenthetic vowels in Lebanese Arabic are either shorter, backer or both shorter and backer than their lexical counterparts. In order to model this phenomenon in a phonological grammar, they argue that the phonetics must have access to an intermediate stage of phonological derivations. Following the assumptions of Optimality Theory with Candidate Chains (OT-CC; McCarthy Reference McCarthy2007: ch. 3), they take a candidate to consist of a derivational chain from the phonological input to the phonological surface form, with gradual, incremental steps along the way. Under this approach, the epenthetic vowel [i] goes through a number of steps – starting as null, then gradually becoming more and more like a full vowel, forming a chain like the one in (7) (Gouskova & Hall's (8)).

1. (7)

   

If the phonetics can access the entire chain, they argue, rather than just the last step in the chain, epenthetic vowels in Lebanese Arabic might be realised as one of the other steps in the chain – explaining why some speakers produce the epenthetic vowel more like [ɨ] or [ə], rather than [i]. Underlying /i/, however, has no such chain of changes, and therefore must surface as canonical [i].

A different class of analysis, which I draw on in the model presented below, concerns paradigm uniformity among morphologically related forms (e.g. Benua Reference Benua1997, Steriade Reference Steriade, Broe and Pierrehumbert2000, Yu Reference Yu2007). The effects of paradigm uniformity can be observed when a given allomorph avoids a language-general pattern in favour of similarity to some morphological neighbour. Steriade (Reference Steriade, Broe and Pierrehumbert2000) argues, for example, that words within a morphological paradigm share certain phonological and phonetic properties. She describes the (optional) process of schwa deletion in French, which renders forms such as (8a) (with surface schwa) as (8b) (schwa deleted).

1. (8)

   

Crucially, the underlined [ʁ] in (8b) (with schwa deletion) is not phonetically identical to the one in (8c), where there was no underlying schwa. The [ʁ] in (8b) surfaces with ‘qualities that would only be appropriate if the schwa was still present’ (Steriade Reference Steriade, Broe and Pierrehumbert2000: 327; cf. Rialland Reference Rialland, Wetzels and Sezer1986, Fougeron & Steriade Reference Fougeron and Steriade1997). On Steriade's phonetic analogy analysis, forms that have undergone schwa deletion as in (8b) are influenced by forms like (8a), with surface schwa, thus accounting for their similarity.

An assumption of many theories of incomplete neutralisation is that speakers have relatively fine-grained control over phonetic implementation. In the model proposed by Yu (Reference Yu, Goldsmith, Riggle and Yu2011), the phonology has just such control over phonetic implementation of contrast. Following Kingston & Diehl (Reference Kingston and Diehl1994: 420, n.2), an allophone is simply ‘any phonetic variant of a distinctive feature specification or arrangement of such specifications that occurs in a particular context’. That is to say, contrastive features can vary considerably in their realisation, depending on their context. For example, [+voice] in English may be realised with closure voicing intervocalically, but as voiceless unaspirated word-initially. Kingston & Diehl (Reference Kingston and Diehl1994) argue that speakers can therefore choose between the various methods to articulate a feature such as [+voice]. Yu (Reference Yu, Goldsmith, Riggle and Yu2011) claims that subphonemic differences, such as incomplete neutralisation and near-merger, should be seen in a similar light. In incomplete neutralisation of final devoicing, the non-final voiced ‘allophone’ and the word-final voiceless ‘allophone’ are distinct, even if their phonetic cues are so impoverished as to ‘escape detection by traditional methods of linguistic data collection … Nonetheless, the contrast is maintained from the perspective of the native speaker, albeit covertly’ (Yu Reference Yu, Goldsmith, Riggle and Yu2011: 311).

A final model of incomplete neutralisation makes use of ‘phonetic traces’. In their study of tongue-twister errors, Goldrick & Blumstein (Reference Goldrick and Blumstein2006) show that ‘traces’ of an intended target can have an effect on pronunciation. For example, in the mistaken use of [g] rather than [k], VOT is longer than in intended pronunciations of [g] – the mistaken production of [g] shows a trace of the intended [k] target. This phenomenon is modelled in a system with ‘cascading activation’ – information flows unidirectionally from the phonological module to the phonetic module; however, earlier stages in this process can generate and transmit multiple representations downstream. In the case of mistaken [g] for [k], representations of both [g] and [k] are activated at the phonological level and transmitted to the phonetics. Goldrick & Blumstein (Reference Goldrick and Blumstein2006) argue that this sort of analysis can be applied to incomplete neutralisation: in German final devoicing, a devoiced [d̥] is pronounced in a ‘t-like’ fashion, due to partial activation of the underlying /t/ being sent to the phonetic module.

3.2 Weighted phonetic constraints and paradigm uniformity

The model presented in this paper relies on two independently motivated theoretical mechanisms: weighted phonetic constraints (Legendre et al. Reference Legendre, Miyata and Smolensky1990, Zsiga Reference Zsiga2000, Flemming Reference Flemming2001) and paradigm uniformity (Benua Reference Benua1997, Steriade Reference Steriade, Broe and Pierrehumbert2000). These are discussed in the following sections.

3.3 Choosing a morphologically related base

There are four major approaches to choosing the base to which a form must be faithful: morphological complexity, orthography, frequency and informativeness. In the Japanese case, the base to which a monomoraic noun must be faithful is the form with a particle (e.g. ki mo), not the bare form (e.g. ki) – this ensures that when monomoraic nouns lengthen, they are prevented from becoming as long as underlyingly long nouns. This section briefly describes these approaches, and argues that informativeness most unambiguously selects the form with particles as the base in monomoraic lengthening.

3.3.2 Frequency

Another approach to basehood relies on frequency: the most frequent member of a morphological paradigm serves as the base (Mańczak Reference Mańczak1958). To examine whether this approach is appropriate in the case of Japanese monomoraic noun lengthening, the NINJAL Parsed Corpus of Modern Japanese (NPCMJ) was searched for nouns and their immediately following morpheme.Footnote ⁸ Of the 6550 nouns in the corpus, 5786 (88.34%) were immediately followed by a particle, while the remaining 764 nouns were followed by some other part of speech, such as another adjective, a noun or number (i.e. not a particle). The five most frequent particles are shown in Table III.

Table III

Frequency of most common particles in the NPCMJ corpus. Percentages indicate frequency among all nouns (n=6550).

Given that the caseless nouns are less frequent than those followed by o, no, ni or ga, the frequency approach to basehood predicts, given the data above, that the base for Japanese nouns should be a noun with a case particle – the desired outcome. Putting aside for the moment which particular case particle attaches to the base, the base of a monomoraic noun will always contain sufficient material to meet the bimoraicity requirement, by virtue of the case particle.

One question raised by this analysis, pointed out by an anonymous reviewer, is whether base frequency is computed as the cumulative frequency of all suffixed forms vs. unsuffixed forms, or whether one particular suffixed form must be more frequent than unsuffixed forms. In the analysis presented above, any one of o, no, ni or ga is more frequent in the corpus data than unsuffixed forms, and could therefore be a licit base.

It should be noted, however, that the omission of a case particle is more frequent in informal speech than in writing. Since the NPCMJ is based on written Japanese, it is possible that the relative frequency of dropped particles is higher in spoken Japanese than in the corpus. If unsuffixed nouns were even slightly more frequent in speech than in the corpus, unsuffixed nouns could become relatively more frequent than any one of the suffixed forms. If this were the case, the frequency analysis would, in order to accurately represent the facts, necessarily have to take the cumulative frequency of all suffixed forms into account, rather than that of any one suffixed form. In this event, the vowel duration of the base could be computed as, for example, the mean vowel duration across all suffixed forms.

4.1 Duration targets for monomoraic and bimoraic nouns

I assume here that, for any given segment, there is some language-specific target duration. For the case of Japanese, I propose two cover targets that govern the duration of all monomoraic and bimoraic segments: all segments which bear one mora in the output have a target duration represented as TargetDur(μ), and all segments which bear two moras in the output have a target duration represented as TargetDur(μμ). Using the rounded values from Braver & Kawahara (Reference Braver and Kawahara2016), shown in Table I, unlengthened monomoraic short vowels have TargetDur(μ) = 50 ms and underlyingly bimoraic long vowels have TargetDur(μμ) = 150 ms. In non-neutralising contexts, both of these targets are generally met.

Along the lines of Flemming's (Reference Flemming2001) C-duration and σ-duration constraints, I assume that candidates are under pressure to conform to language-specific duration targets. This pressure is codified in two constraints of the family Dur(x)=TargetDur(x). Dur(μ)=TargetDur(μ), defined in (13), penalises monomoraic candidates which fail to match their target duration. In a parallel fashion, Dur(μμ)=TargetDur(μμ) penalises bimoraic candidates which do not meet their target duration.Footnote ¹²

1. (13)

   

To see Dur(μ)=TargetDur(μ) in action, consider the tableau in (14) (which assumes a constraint weight of 1).

1. (14)

   

The Dur(μ)=TargetDur(μ) column shows the cost of each candidate for that constraint; the parentheses show the calculation used to reach that cost. For example, in candidate (14a), the cost associated with Dur(μ)=TargetDur(μ) is calculated as in (15), where the candidate's duration (Dur(cand)) is 30 ms and the target duration for a single mora (as per the rounded data) is 50 ms.

1. (15)

   

As can be seen in (14), Dur(μ)=TargetDur(μ) penalises short vowels whose duration differs from the target (50 ms); as candidates’ durations diverge from this target, the cost increases.

Dur(μμ)=TargetDur(μμ) works nearly identically, with two moras instead of one. As can be seen in the tableau in (16), candidates with surface-bimoraic vowels (both underlyingly long vowels as in (16a) and lengthened short vowels as in (b)) decrease in cost for Dur(μμ)=TargetDur(μμ) as they approach TargetDur(μμ) (150 ms).

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In order to prevent candidates which fail to epenthesise a second mora, and thus vacuously satisfy Dur(μμ)=TargetDur(μμ), the tableau in (16) also contains two additional constraints not yet discussed: FtBin(μ) prefers feet to be minimally bimoraic (with a cost of 1 per monomoraic foot), and Dep(μ) assesses a cost of 1 per epenthesised mora. When FtBin(μ) is weighted above Dep(μ) (here w _FtBin(μ) = 2, w _Dep(μ) = 1), candidates like (16b.iv) cannot win by failing to epenthesise and thus vacuously fulfilling Dur(μμ)=TargetDur(μμ). In all future tableaux and cost tabulations, I exclude monomoraic candidates which fail to lengthen, on the assumption that a weighting in which FtBin(μ) outweighs Dep(μ) prevents such candidates from being viable optima.

4.2 Paradigm uniformity in lengthened monomoraic nouns

Given the constraints Dur(μ)=TargetDur(μ) and Dur(μμ)=TargetDur(μμ), it is clear how unlengthened short vowels and underlyingly long vowels are under pressure to reach their duration targets. We are left, however, with vowels in lengthened short nouns, which, with an average duration of 125 ms, meet neither the target for short nouns (50 ms) nor the target for long nouns (150 ms). Because lengthened short nouns are surface-bimoraic, they are subject to the constraint Dur(μμ)=TargetDur(μμ); note, however, that they do not comply completely with this constraint. I claim that the distinction between underlyingly long nouns on the one hand and lengthened short nouns on the other is caused by pressure on lengthened short nouns to maintain a degree of faithfulness to canonical short vowels – a pressure that underlyingly long vowels do not face. To see how this works, we will first examine a single example based on the rounded data in Table I, and then proceed to a more formal definition.

Consider the three utterances in (17), which contain (a) unlengthened short vowels, (b) lengthened short vowels and (c) underlyingly long vowels. Vowel durations of the underlined words are given in parentheses.Footnote ¹³

1. (17)

   

The vowel in (17b) has two surface moras (due to lengthening), and is thus pressured by Dur(μμ)=TargetDur(μμ) to reach a target of 150 ms. It does not, however, reach this target. This failure is due to pressure from a competing constraint, which requires faithfulness to a morphologically related base form (i.e. paradigm uniformity).

As per the discussion in §3.3, we will take the base to which monomoraic nouns must be faithful to be a form which contains the noun and a following case particle. I formalise this faithfulness to a morphologically related base in the form of an output–output constraint, OO-Id(dur), which enforces faithfulness between a given segment in a candidate and the corresponding segment in the candidate's base, as in (18).

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To see this more concretely, let us apply OO-Id(dur) to the lengthened [ki Ø]_ω in (17b). As stated above, the base to which (17b) must be faithful has a vowel duration of 50 ms. OO-Id(dur) therefore pressures [ki Ø]_ω to attain a vowel duration of 50 ms.

The effect of OO-Id(dur) on lengthened short nouns can be seen in the tableau in (19): the cost of this constraint increases as vowel durations move away from 50 ms.

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4.3 Interaction of OO-Id(dur) and Dur(µµ)=TargetDur(µµ)

We consider now the tension between paradigm uniformity and the canonical duration of a segment – in other words, between OO-Id(dur) and Dur(μμ)=TargetDur(μμ). It should be noted that Dur(μ)=TargetDur(μ) is not relevant for assessing lengthened short vowel inputs, because it only penalises mora-bearing units with a single mora on the surface. Since lengthened short candidates by definition have two moras on the surface, Dur(μ)=TargetDur(μ) does not apply, and will not be shown in the tableaux that follow. Similarly, as discussed above, the relative weighting of FtBin(μ) over Dep(μ) prevents lengthened nouns from surfacing with only one mora; the costs of these two constraints will be left out of the calculations below, to simplify the exposition.

The key tension between Dur(μμ)=TargetDur(μμ) and OO-Id(dur) can be expressed in terms of their role in setting target durations for lengthened short candidates. Dur(μμ)=TargetDur(μμ) is least costly for lengthened short candidates whose durations approach TargetDur(μμ) (150 ms), while OO-Id(dur) is least costly for those candidates whose durations approach TargetDur(μ) (50 ms), since their base has only one mora. The relative strength of these constraints – as reflected in their weighting – determines how the conflict between these two pressures is resolved. As the weight of Dur(μμ)=TargetDur(μμ) increases relative to that of OO-Id(dur), the candidates with the lowest cost are those that approach 150 ms. Conversely, as the weight of OO-Id(dur) increases relative to that of Dur(μμ)=TargetDur(μμ), the candidates with the lowest cost are those that approach 50 ms.

The degree to which a candidate satisfies the compromise between Dur(μμ)=TargetDur(μμ) and OO-Id(dur) can be assessed by its total cost – the sum of the costs of all constraints for that candidate. The candidate which best satisfies the compromise (and is therefore the winner) is the one which has the lowest overall cost. Given our two constraints, the total cost of a candidate is as in (20).

1. (20)

   

As shown in Table IV, with the weighting <w _Id = 1, w _{μ μ} = 3> (and with the same assumptions about durational targets as above), the total cost of a candidate decreases as the duration of lengthened vowels approaches 125 ms – the minimum cost is attained by candidates of 125 ms (our desired winner).

Table IV

Costs for given lengthened short vowel durations, where w _Id=1, w _μμ=3, TargetDur(μ)=50 ms and TargetDur(μμ)=150 ms.

Varying the weights of these two constraints results in a different minimum to the cost function, and a different degree of neutralisation, as might be expected to be found in other languages, as shown in Table V. As the weight of w _Id increases relative to that of w _{μ μ}, the duration of the lengthened vowels is predicted to decrease (and, vice versa, to increase). This principle can also be seen in the graph in Fig. 2, in which the x- and y-axes represent the weights of OO-Id(dur) and Dur(μμ)=TargetDur(μμ) (w _Id = x; w _{μ μ} = y; 0 < x, y ≥ 5), and the z-axis represents the predicted duration of lengthened vowels: the WPU model can generate languages anywhere along the plotted surface.

Figure 2

Predicted lengthened vowel duration, given weights for w _Id and w _μμ (TargetDur(μ)=50 ms; TargetDur(μμ)=150 ms).

Table V

Predicted lengthened short vowel durations (in ms) for sample weightings of OO-Id(Dur) (w _Id) and Dur(μμ)=TargetDur(μμ) (w _μμ), where TargetDur(_μ)=50 ms and TargetDur(μμ)=150 ms.