2.1. Fitting rates onto shifted sigmoids at shared abscissas

Tagalog consonant clusters derived by concatenating a prefix that ends with a nasal with a stem that starts with an obstruent undergo nasal substitution: the nasal and the obstruent are coalesced into a single consonant that retains the nasality of the former and the place of articulation of the latter, whereby /maŋ + bigáj/ is realized as [mamigáj] ‘to distribute’ (Zuraw Reference Zuraw2010:421). We focus on two factors that are known to condition the rates of application of this process. One factor is the quality of the stem-initial obstruent, say /b/ versus /k/. Another factor is the identity (perhaps the lexical frequency?) of the nasal-final affix, say /paŋ/ versus /maŋ/.Footnote ¹ HZ observe that the rates of application of nasal substitution to the four resulting underlying concatenations in (1) reported by Zuraw (Reference Zuraw2010) satisfy an intriguing property illustrated in Figures 1a–c.

Figure 1.

The rates of application of nasal substitution in Tagalog to the four underlying concatenations in (1) are fitted onto two shifted sigmoids at shared abscissas.

Figure 1 Long description

The figure consists of three panels labeled a, b, and c. Each panel features a horizontal x-axis with two marked points: abscissa for b and abscissa for k.

Panel a shows a single sigmoid curve labeled sigmoid for m a n g. Two points are plotted on this curve: R m a n g b equals 0.916 above the abscissa for b, and R m a n g k equals 0.993 above the abscissa for k.

Panel b shows the same sigmoid for m a n g curve. It includes the point R m a n g b and R m a n g k on the curve. Additionally, two outlier points are plotted: R p a n g b equals 0.434 located below the curve at the abscissa for b, and R p a n g k equals 0.909 located below the curve at the abscissa for k.

Panel c displays two shifted sigmoid curves. The upper curve is labeled sigmoid for m a n g and the lower curve is labeled sigmoid for p a n g. Vertical dashed lines connect the abscissas to points on both curves. At the abscissa for b, the upper point is R m a n g b and the lower point is R p a n g b. At the abscissa for k, the upper point is R m a n g k and the lower point is R p a n g k.

(1)

The sigmoid function $ \mathbf{S}\left(\mathsf{x}\right)=\frac{1}{1+\exp \left(-\mathsf{x}\right)} $ takes values between 0 and 1 (both excluded) that can therefore be interpreted as application rates. Figure 1a thus plots on the graph of the sigmoid function the rates R(/maŋb/) = 0.916 and R(/maŋk/) = 0.993 of application of nasal substitution to the two underlying concatenations that share the prefix /maŋ/. Next, Figure 1b plots as two points with the same abscissasFootnote ² the rates R(/paŋb/) = 0.434 and R(/paŋk/) = 0.909 of application of nasal substitution to the two underlying concatenations that share the other prefix /paŋ/. HZ discover that the latter two points sit on the graph of some shifted sigmoid, namely the graph of the function S(x + δ) for some shifting constant δ, as verified in Figure 1c. In conclusion, these rates of nasal substitution can be fitted on two shifted sigmoids at shared abscissas. As a result, these four rates can be described through only three parameters: the two shared abscissas and the shifting constant δ. Equivalently, given three of these rates, we can guess the fourth one.

2.2. Independent phonological factors

HZ interpret this surprising property of the rates of application of nasal substitution as the reflex at the level of rates of the fact that the quality of the stem-initial obstruent and the identity of the prefix are independent factors. They formalize this intuition of independence as follows. When modeling the frequencies of nasal substitution within constraint-based phonology, we obviously need some constraints that are sensitive to the quality of the stem-initial obstruent. Furthermore, we also need some constraints that are sensitive to the identity of the prefix, because concatenations with different prefixes but the same stem-initial obstruent have different rates of nasal substitution. Yet, we expect no single constraint to be sensitive to both the quality of the stem-initial obstruent and the identity of the prefix. In conclusion, stems and prefixes are independent factors because no constraint is sensitive to both and therefore no constraint can encode any interaction between them.

To illustrate, the constraint *NC̥ penalizes sequences of a nasal followed by a voiceless obstruent and thus captures the observation that the rate of nasal substitution is higher when the stem-initial obstruent is voiceless. This constraint is only sensitive to the quality of the stem-initial obstruent but crucially insensitive to the identity of the prefix: it assigns the same number of violations to two faithful candidates that feature different prefixes, as long as the stems start with the same obstruent. Conversely, HZ’s faithfulness constraints Uniformity(maŋ) and Uniformity(paŋ) penalize coalescence but only when the coalesced nasal belongs to the prefix /maŋ/ or /paŋ/, respectively. Either constraint is sensitive to the identity of the prefix but crucially insensitive to the quality of the stem-initial obstruent: it assigns the same number of violations to the nasal substitution candidates of two underlying concatenations that feature different stem-initial obstruents, as long as they share the same prefix.Footnote ³

Beyond nasal substitution in Tagalog, Hayes (Reference Hayes2021a,Reference Hayesb, Reference Hayes2022) reviews a number of variable phonological processes in a number of languages: vowel harmony in Hungarian; consonant liaison in French; inversion of final devoicing in Dutch; genitive plurals in Finnish; schwa/zero alternation in French; stress placement in Hupa. For each of these variable processes, Hayes focuses on two factors that condition its rate of application and that are plausibly independent because no reasonable, natural constraint would be sensitive to both factors. He then looks at the rates of application of that variable process to the underlying forms corresponding to the four combinations of two values of those two factors. And he observes that these rates can be fitted on two shifted sigmoids at shared abscissas with good approximation.

This empirical discovery is puzzling. Why do the rates of application of disparate variable phonological processes in unrelated languages all comply with this exotic generalization? What is the proper phonological interpretation of the rate configuration plotted in Figure 1? To answer this question, this paper develops an intuitive and intelligible interpretation of HZ’s generalization in terms of even constraints within the framework of probabilistic harmony-based phonology.

3.1. Probabilistic grammars based on harmony scores

As usual, Gen lists all the phonological mappings that are relevant for the description of the phonological system of interest. Gen(x) denotes the candidate set of an underlying form x, that we assume to be finite to circumvent the subtleties required to define probability masses over infinite sets.Footnote ⁴ A finite number n of constraints C ₁, … , C_n suffice to quantify all the phonologically relevant properties of the mappings listed by Gen. The number of violations assigned by a constraint C_k to the mapping of an underlying form x to a surface realization y from the candidate set Gen(x) is denoted by C_k(x, y).

A harmony function H assigns a strictlyFootnote ⁵ positive harmony score H(x, y) to any underlying form x and any surface form y from the candidate set Gen(x). We assume that this harmony score H(x, y) depends on the mapping (x, y) only through its constraint violations C_k(x, y), as stated in 2b. In other words, H is really a function from n (constraint violation) numbers to positive scores. We then denote by G_H the probabilistic grammar that is based on the constraints C ₁, … , C_n through this harmony function H because G_H realizes an underlying form x as a candidate surface form y with a probability G_H(y | x) that is proportional to the harmony score H(x, y), as stated in (2a).

(2) $$ {G}_H\left(\mathsf{y}|\mathsf{x}\right)\;\overset{(a)}{\propto}\;H\left(\mathsf{x},\mathsf{y}\right)\;\overset{(b)}{=}\;H\left({C}_1\left(\mathsf{x},\mathsf{y}\right),\dots, {C}_n\left(\mathsf{x},\mathsf{y}\right)\right) $$

The proportionality relation (2a) defines the harmony-based probability G_H (y | x) only up to a multiplicative constant Z, namely it says that G_H(y | x) = H(x, y)/Z. Yet, this constant Z is univocally determined as follows. Because of their probabilistic interpretation, the numbers G_H(y | x) must be normalized: their sum over all the surface forms y in the candidate set Gen(x) of the underlying form x must be equal to one. Normalization requires the constant Z to be equal to the sum of the harmony scores of all the surface forms z in the candidate set Gen(x), namely $ Z={\sum}_{z\in Gen\left(\mathsf{x}\right)}H\left(\mathsf{x},z\right) $ . As a result, Z is a constant in the sense that it only depends on the underlying form x but not on the candidate y considered.

The poster child of probabilistic harmony-based phonology is ME. Let me thus recall that a ME grammar G _ME realizes an underlying form x as a candidate surface form y with a probability G _ME(y | x) proportional to the ME harmony score H _ME (x, y), as stated in 3a. This harmony score is obtained by applying the exponential function to the opposite of the sum of the weighted constraint violations, as stated in 3b.Footnote ⁶ The non-negative weight w_k encodes the relative importance of the corresponding constraint C_k. Because of the minus sign inside the exponential (together with the fact that the exponential function is monotonically increasing), the ME grammar G _ME assigns a larger probability to candidates that yield a smaller sum $ {\sum}_k{w}_k{C}_k $ and thus violate the constraints less, taking into account the relative importance of each constraint through its weight.

(3)

3.2. ME grammars predict HZ’s generalization

HZ observe that ME grammars predict their generalization in the following sense. We consider some phonological process that applies variably to four underlying forms x ^TL , x ^TR , x ^BL , and x ^BR . We arrange these underlying forms into a square as in (4), so that the superscripts ‘ t ’ and ‘ b ’ (for top and bottom) specify the row an underlying form sits in while the superscripts ‘ l ’ and ‘ r ’ (for left and right) specify the column. We assume that these underlying forms satisfy two assumptions. The first assumption is that each of them is listed by Gen with only two candidates yes and no, corresponding to application and non-application of the variable phonological process of interest (see footnote 8 for discussion). The second assumption is that each relevant constraint C satisfies either the identities in 5a or those in (5b).

(4)

(5)

a.

b.

The ‘horizontal’ identities in (5a) say that the constraint C is insensitive to the column dimension because it does not distinguish between two underlying forms x ^TL versus x ^TR or x ^BL versus x ^BR that belong to the same row and thus only differ for the column. Analogously, the ‘vertical’ identities in (5b) say that the constraint C is insensitive to the row dimension because it does not distinguish between two underlying forms x ^TL versus x ^BL or x ^TR versus x ^BR that belong to the same column and thus only differ for the row.Footnote ⁷ Since no constraint is sensitive to both column and row dimensions, no constraint can encode an interaction between these two dimensions, that are therefore independent. Under this independence assumption, HZ show that the ME grammar corresponding to any constraint weights predicts that the four underlying forms in (4) undergo the variable process considered (namely are realized as the candidate yes) with probabilities that can be fitted onto two shifted sigmoids at shared abscissas, as in Figure 2a.

Figure 2.

The ME grammar G _me in (a) and the probabilistic grammar G_H corresponding to an arbitrary harmony function H in (b) predict that the variable phonological process of interest applies to the four underlying forms in 4 with probabilities that can be fitted onto two shifted sigmoids at shared abscissas.

Figure 2 Long description

A two-panel figure showing phonological process probabilities.

Panel a, labeled lowercase a, displays the M E grammar G sub M E. The horizontal x-axis has two marked points: abscissa for L on the left and abscissa for R on the right. Two sigmoid curves rise from left to right. The upper curve is labeled sigmoid for T and the lower curve is labeled sigmoid for B. Vertical dotted lines connect the abscissas to points on the curves. On the upper curve, the left point is G sub M E open parenthesis Y E S vertical bar x super T L close parenthesis and the right point is G sub M E open parenthesis Y E S vertical bar x super T R close parenthesis. On the lower curve, the left point is G sub M E open parenthesis Y E S vertical bar x super B L close parenthesis and the right point is G sub M E open parenthesis Y E S vertical bar x super B R close parenthesis.

Panel b, labeled lowercase b, displays the probabilistic grammar G sub H. The layout is identical to panel a, but the labels use G sub H instead of G sub M E. The curves and points are positioned at the same relative coordinates as in panel a, showing the shared abscissas for L and R across both grammar types.

To illustrate, we revisit the four underlying forms in (1). In this case, the column dimension corresponds to the stem-initial obstruents (/b/ versus /k/) and the row dimension to the prefixes (/paŋ/ versus /maŋ/). Each of these four underlying forms is listed by Gen with only two candidate surface realizations, corresponding to application and non-application of nasal substitution. Finally, the constraint set consists of two types of constraints. Either constraints such as Uniformity(paŋ) that satisfy the ‘horizontal’ identities in (5a) because they are insensitive to the column dimension corresponding to the stem-initial obstruents, or constraints such as *NC̥ that satisfy the ‘vertical’ identities in (5b) because they are insensitive to the row dimension corresponding to the prefixes. The ME grammar corresponding to any constraint weights predicts that the four underlying forms in (1) undergo nasal substitution with probabilities that can be fitted onto two shifted sigmoids at shared abscissas.

3.3. Harmony-based grammars that predict HZ’s generalization

We can now easily move beyond the special case of ME grammars to arbitrary harmony-based grammars. We say that an arbitrary harmony function H predicts HZ’s generalization provided it satisfies the following condition. Choose four arbitrary underlying forms (4) with two candidates yes and no each, corresponding to application and non-application of the process of interest. Choose an arbitrary set of constraints that each satisfy the horizontal or vertical identities in (5). Then, the resulting harmony-based grammar G_H predicts that the four underlying forms in (4) undergo the variable process considered (namely are realized as the candidate yes) with probabilities that can be fitted onto two shifted sigmoids at shared abscissas, as in Figure 2b.Footnote ⁸

To understand HZ’s generalization within the framework of harmony-based phonology means to understand which phonological property is both necessary and sufficient in order for a harmony function H to predict HZ’s generalization in the sense just made explicit. The rest of this paper shows that this property turns out to be remarkably simple and intelligible: it is the simple and intuitive property that a harmony function H is oblivious to those constraints that intuitively should not matter because they are even, as formalized in the next Section.

4.1. Underlying forms that only differ for the even constraints

We consider two underlying forms x and $ \hat{\mathsf{x}} $ that satisfy the following two assumptions. The first assumption is that they are listed by Gen with the same (finite) number m of (possibly different) candidates, as stated in (6). Two candidates y _i and $ {\hat{\mathsf{y}}}_i $ that bear the same index are called corresponding candidates, capturing the intuition that they are obtained from the two underlying forms x and $ \hat{\mathsf{x}} $ by applying the same phonological processes. The second assumption is that each relevant constraint C satisfies either the ‘vertical’ identities in (7a) or the “horizontal” identities in (7b).

(6)

(7)

a.

b.

The ‘vertical’ identities in (7a) say that the constraint C assigns the same number of violations to any two corresponding candidates y _i and $ {\hat{\mathsf{y}}}_i $ of the underlying forms x and $ \hat{\mathsf{x}} $ . In other words, C does not distinguish between these two underlying forms. The ‘horizontal’ identities in (7b) say that the constraint C assigns the same number of violations to all the candidates of the underlying form x and it also assigns the same number of violations to all the candidates of the other underlying form $ \hat{\mathsf{x}} $ . In other words, C is even relative to both underlying forms (C yields an e in all ERCs; Prince Reference Prince2002). In conclusion, these two underlying forms x and $ \hat{\mathsf{x}} $ only differ for constraints that are intuitively irrelevant because they are even.

To illustrate, we suppose that Gen lists the underlying forms /rat/ and /rut/ with the faithful candidates [rat] and [rut] together with the candidates [ra] and [ru] obtained through coda deletion. Furthermore, we consider the constraints C ₁ = NoCoda, C ₂ = Max, and C ₃ = NoRound, described in 8. NoCoda and Max do not distinguish between the two underlying forms: they assign the same number of violations to the two corresponding deletion candidates as well as to the two corresponding faithful candidates, as required by the ‘vertical’ identities (7a). Furthermore, NoRound is even relative to both underlying forms: it assigns the same number of violations to the two candidates of the underlying form /rat/ (both candidates satisfy the constraint) as well as to the two candidates of the underlying form /rut/ (both candidates violate it), as required by the ‘horizontal’ identities (7b). We conclude that the underlying forms /rat/ and /rut/ only differ for the constraint NoRound that is intuitively irrelevant because it is even.Footnote ⁹

(8)

4.2. OT and HG categorical grammars are oblivious to the even constraints

Let us consider the categorical HG grammar corresponding to some weights w ₁, w ₂, and w ₃ of the constraints C ₁ = NoCoda, C ₂ = Max, and C ₃ = NoRound, respectively. How does this grammar behave on the underlying forms /rat/ and /rut/ that only differ for the even constraint NoRound? The HG harmony H _hg of a mapping is simply the sum of its weighted constraint violations. The difference between the HG harmony of the deletion candidate [ra] minus the HG harmony of the faithful candidate [rat] can therefore be made explicit as in (9a). Since the constraint NoRound is even between these two candidates, the two quantities () and () are identical and therefore cancel out, yielding (9b). Since the remaining constraints NoCoda and Max do not distinguish between the two underlying forms /rat/ and /rut/, we can replace the former underlying form and its candidates with the latter underlying form and its corresponding candidates, yielding (9c). Finally, the identity (9d) holds by reasoning as in (9a–b).

(9)

This reasoning in (9) shows that the difference between the HG harmony of the deletion candidate minus the HG harmony of the faithful candidate is the same for the two underlying forms /rat/ and /rut/. It follows that our categorical HG grammar cannot distinguish between these two underlying forms /rat/ and /rut/ that only differ for the even constraint NoRound: our HG grammar deletes the coda of /rat/ if and only if it deletes the coda of /rut/ as well. The reasoning in (9) obviously extends to the general case. We conclude that no HG grammar can distinguish between two underlying forms x and $ \hat{\mathsf{x}} $ that only differ for even constraints in the sense of Subsection 4.1: every HG grammar realizes the underlying form x as a surface candidate y _i if and only if it realizes the other underlying form $ \hat{\mathsf{x}} $ as the corresponding candidate $ {\hat{\mathsf{y}}}_i $ . We summarize this conclusion by saying that HG grammars are oblivious to the even constraints. This conclusion extends to categorical OT grammars, as they are a special case of HG grammars.

4.3. ME grammars are oblivious to the even constraints

We now turn from categorical to probabilistic phonology. Let us consider the probabilistic ME grammar G _ME corresponding to some weights w ₁, w ₂, and w ₃ of the constraints NoCoda, Max, and NoRound, respectively. How does this grammar behave on the underlying forms /rat/ and /rut/ that only differ for the even constraint NoRound? We start by computing the logarithm of the ratio between the ME harmony of the faithful candidate [rat] divided by the ME harmony of the deletion candidate [ra]. Step (10a) holds because the logarithm of a ratio is the difference of the logarithms. Step (10b) holds because the ME harmony H _ME recalled in 3b is simply the exponential of the opposite of the HG harmony H _HG. Step (10c) holds because the logarithm and the exponential functions invert each other: whenever we feed some number x to the exponential and then feed the result exp x to the logarithm, we get back the number x we started from, because x = log(exp x). Step (10d) holds because the reasoning (9) has shown that the difference between the HG harmony scores of the faithful and deletion candidates is the same for the two underlying forms /rat/ and /rut/. Finally, step (10e) holds by reasoning as in (10a–c).

(10) $$ {\displaystyle \begin{array}{l}\begin{array}{lll}\log \frac{H_{\mathrm{ME}}\left(/\mathrm{rat}/,\left[\mathrm{ra}\mathrm{t}\right]\right)}{H_{\mathrm{ME}}\left(/\mathrm{rat}/,\left[\mathrm{ra}\right]\right)}\;& \overset{\left(\mathrm{a}\right)}{=}& \log {H}_{\mathrm{ME}}\left(/\mathrm{rat}/,\left[\mathrm{ra}\mathrm{t}\right]\right)\hbox{--} \log {H}_{\mathrm{ME}}\left(/\mathrm{rat}/,\left[\mathrm{ra}\right]\right)\\ {}& \overset{\left(\mathrm{b}\right)}{=}& \log\;\exp\;\left(\hbox{--} {H}_{HG}\left(/\mathrm{rat}/,\left[\mathrm{ra}\mathrm{t}\right]\right)\right)\hbox{--} \log\;\exp\;\left(\hbox{--} {H}_{HG}\left(/\mathrm{rat}/,\left[\mathrm{ra}\right]\right)\right)\\ {}& \overset{\left(\mathrm{c}\right)}{=}& {H}_{HG}\left(/\mathrm{rat}/,\left[\mathrm{ra}\right]\right)\hbox{--} {H}_{HG}\left(/\mathrm{rat}/\left[\mathrm{ra}\mathrm{t}\right]\right)\\ {}& \overset{\left(\mathrm{d}\right)}{=}& {H}_{HG}\left(/\mathrm{rut}/,\left[\mathrm{ru}\right]\right)\hbox{--} {H}_{HG}\left(/\mathrm{rut}/,\left[\mathrm{ru}\mathrm{t}\right]\right)\\ {}& \overset{\left(\mathrm{e}\right)}{=}& \log \frac{H_{\mathrm{ME}}\left(/\mathrm{rut}/\left[\mathrm{ru}\mathrm{t}\right]\right)}{H_{\mathrm{ME}}\left(/\mathrm{rut}/,\left[\mathrm{ru}\right]\right)}\end{array}\\ {}\hskip0.72em \end{array}} $$

This reasoning in (10) shows that, since the differences between HG harmony scores are the same for the two underlying forms /rat/ and /rut/, the (logarithms of the)Footnote ¹⁰ ratios of ME harmony scores are the same as well. This observation makes intuitive sense: since ME is a probabilistic version of categorical HG, it makes sense that formal HG properties have ME counterparts. We now use this observation to compare the ME probabilities of deletion of the underlying codas of /rat/ and /rut/. Step (11a) holds because of the definition of harmony-based grammars from Subsection 3.1. Step (11b) holds because the normalization constant Z is equal to the sum of the ME harmony scores of the two candidates [ra] and [rat]. Step (11c) holds by dividing both the numerator and the denominator by the ME harmony score of the deletion mapping (/rat/, [ra]) (which is licit, because ME harmony scores are different from zero). Step (11d) holds because the two harmony ratios in the denominators have been shown to coincide in (10). Finally, step (11e) holds by reasoning as in steps (11a–c).

(11) $$ {\displaystyle \begin{array}{lll}{G}_{\mathrm{ME}}\left(\left[\mathrm{ra}\right]\;|/\mathrm{rat}/\right)& \overset{(a)}{=}\frac{H_{\mathrm{ME}}\left(/\mathrm{rat}/,\left[\mathrm{ra}\right]\right)}{Z}& \overset{(b)}{=}\frac{H_{\mathrm{ME}}\left(/\mathrm{rat}/,\left[\mathrm{ra}\right]\right)}{H_{\mathrm{ME}}\left(/\mathrm{rat}/,\left[\mathrm{ra}\right]\right)+{H}_{\mathrm{ME}}\left(/\mathrm{rat}/,\left[\mathrm{ra}\mathrm{t}\right]\right)}\\ {}& \overset{(c)}{=}\frac{1}{1+\frac{H_{\mathrm{ME}}\left(/\mathrm{rat}/,\left[\mathrm{ra}\mathrm{t}\right]\right)}{H_{\mathrm{ME}}\left(/\mathrm{rat}/,\left[\mathrm{ra}\right]\right)}}& \overset{(d)}{=}\frac{1}{1+\frac{H_{\mathrm{ME}}\left(/\mathrm{rut}/,\left[\mathrm{ru}\mathrm{t}\right]\right)}{H_{\mathrm{ME}}\left(/\mathrm{rut}/,\left[\mathrm{ru}\right]\right)}}\\ {}& \overset{(e)}{=}{G}_{\mathrm{ME}}\left(\left[\mathrm{ru}\right]\;|/\mathrm{rut}/\right)& \end{array}} $$

This reasoning in (11) shows that our ME grammar G _ME cannot distinguish between the two underlying forms /rat/ and /rut/ that only differ for the even constraint NoRound: G _ME deletes the underlying coda of /rat/ with the same probability it deletes the underlying coda of /rut/, namely G _ME([ra] | /rat/) = G _ME([ru] | /rut/). The reasoning in (10–11) obviously extends to the general case. We conclude that no ME grammar can distinguish between two underlying forms x and $ \hat{\mathsf{x}} $ that only differ for even constraints in the sense of Subsection 4.1: every ME grammar realizes the underlying form x as a surface candidate y _i with the same probability it realizes the other underlying form $ \hat{\mathsf{x}} $ as the corresponding candidate $ {\hat{\mathsf{y}}}_i $ , namely G _ME(y _i | x) = G _ME( $ {\hat{\mathsf{y}}}_i $ | $ \hat{\mathsf{x}} $ ). In other words, ME grammars are oblivious to the even constraints.

4.4. Harmony-based grammars that are oblivious to the even constraints

We can now easily move beyond the special case of ME grammars to arbitrary harmony-based grammars from Subsection 3.1. We say that an arbitrary harmony function H is oblivious to the even constraints provided, no matter how we choose two underlying forms x and $ \hat{\mathsf{x}} $ that only differ for even constraints in the sense of Subsection 4.1, the resulting harmony-based grammar G_H predicts the same probability for their corresponding candidates y _i and $ {\hat{\mathsf{y}}}_i $ , namely G_H (y _i | x) = G_H ( $ {\hat{\mathsf{y}}}_i $ | $ \hat{\mathsf{x}} $ ). We now have two general formal properties of harmony functions and their grammars: the property of predicting HZ’s generalization, formulated in Subsection 3.3; and the property of being oblivious to the even constraints, formulated here. The former property is empirically well supported by HZ’s work, but hard to interpret. The latter property instead admits a straightforward interpretation. The main result of this paper is that these two properties are equivalent, as shown in the next Section.

5.1. Main result: HZ’s generalization and even constraints

Let us take stock. Section 2 has reviewed HZ’s generalization that the rates of application of variable phonological processes governed by independent factors fit onto shifted sigmoids at shared abscissas. How can we make intuitive sense of this apparently exotic generalization? Can we relate it to some simple intuition? To this end, Section 4 has turned to the simple intuition that a constraint that is even among the candidates of an underlying form does not matter: grammars are oblivious to the even constraints when choosing (categorically or probabilistically) the surface realization of that underlying form. The main result of this paper, boxed below, is that HZ’s exotic generalization is equivalent to this simple intuition that the even constraints do not matter, at least within the framework of probabilisticFootnote ¹¹ harmony-based phonology reviewed in Section 3.

The restriction to harmony-based phonology is crucial for this boxed result to hold. To illustrate, probabilistic SOT and NHG grammars are obviously constraint-based: they are defined solely in terms of constraint violations. Yet, they are not harmony-based in the sense of Subsection 3.1: de la Fuente et al. (Reference de la Fuente, Jeong, Anttila, Magri, Avelino, Balihaxi, Colvin, Czarnecki, Joo, Wang, Zobarlar, Jardine and McCollum2025) show how to construct candidate and constraint sets such that SOT and NHG probabilities are not proportional to the harmony scores of the constraint violation profiles, no matter how these harmony scores are defined. And indeed the boxed result fails for SOT and NHG. On the one hand, HZ observe that SOT massively fails at their generalization and that NHG as well admits counterexamples. On the other hand, SOT and NHG grammars are oblivious to the even constraints in the sense of Subsection 4.4. In fact, the SOT or NHG probability of a candidate is equal to the probability of sampling a categorical OT or HG grammar that indeed chooses that candidate surface realization. And categorical OT and HG grammars are themselves oblivious to the even constraints, as verified in Subsection 4.2.

The rest of this Section explains intuitively why the result boxed above holds, with the more technical steps of the reasoning relegated to the Appendix. The reasoning has the shape in (12). To start, Subsection 5.2 reviews from Magri (Reference Magri2025) the notion of harmony scores that factorize into products of rescaled constraint violations and therefore encode a particularly simple mode of constraint interaction. Then, Subsection 5.3 recalls from Magri Reference Magri2025 that a harmony function H predicts HZ’s generalization in the sense of Subsection 3.3 if and only if (this is a complete characterization) H is factorizable, as stated in (12a). Finally, Subsections 5.4–5.5 (together with the Appendix) show that a harmony function H is oblivious to the even constraints in the sense of Subsection 4.4 if and only if (again, this is a complete characterization) H is factorizable, as stated in (12b). The result boxed above that a harmony function predicts HZ’s generalization if and only if it is oblivious to the even constraints then follows straightforwardly.

(12)

5.2. Factorizable harmonies

A harmony function H from Subsection 3.1 effectively takes n (constraint violation) numbers and returns positive harmony scores. Magri (Reference Magri2025) focuses on the special case where the harmony score H(x, y) of the mapping consisting of some underlying and candidate surface forms x and y can be computed as in (13). First, we compute the number of violations C_k(x, y) assigned by each constraint C_k to this mapping. Then, we rescale this number of violations by feeding it to some corresponding unary function h _k. Finally, we take the product of these rescaled numbers of violations h_k(C_k(x, y)) over all constraints.

(13) $$ H\left(\mathsf{x},\mathsf{y}\right)=\prod \limits_{k=1}^n{h}_k\left({C}_k\left(\mathsf{x},\mathsf{y}\right)\right)={h}_1\left({C}_1\left(\mathsf{x},\mathsf{y}\right)\right)\cdotp \dots \cdotp {h}_n\left({C}_n\left(\mathsf{x},\mathsf{y}\right)\right) $$

In other words, this condition (13) says that the harmony score is simply the product of the constraint violations. The only leeway is that, before taking these products, we are allowed to rescale the number of violations assigned by each constraint C_k through the corresponding unary function h_k. Equivalently, the n-ary function H factorizes into the product $ H={\prod}_k{h}_k $ of the unary functions h_k, each attuned only to the corresponding constraint C_k. Footnote ¹² As a result, these factorizable harmonies encode a particularly simple, restricted mode of constraint interaction.

Subsection 3.1 has reviewed ME grammars as the poster child of probabilistic harmony-based phonology. Now we observe that ME harmonies are actually the poster child of factorizable harmonies. In fact, the reasoning in (14) shows that the ME harmony factorizes into a product (13) of unary functions h_k that are defined through the position h_k (x) = exp(–w_kx). Step (14a) recalls the definition (3b) of ME harmony scores as the exponential of the opposite of the sum of the weighted constraint violations. Step (14b) holds because the exponential of a sum is the product of the exponentials. Finally, step (14c) holds because of the choice of the unary functions h_k.

(14) $$ {H}_{\mathrm{ME}}\left(\mathsf{x},\mathsf{y}\right)\;\overset{(a)}{=}\;\exp \left(-\sum \limits_{k=1}^n{w}_k{C}_k\left(\mathsf{x},\mathsf{y}\right)\right)\;\overset{(b)}{=}\;\prod \limits_{k=1}^n\exp \left(-{w}_k{C}_k\left(\mathsf{x},\mathsf{y}\right)\right)\;\overset{(c)}{=}\;\prod \limits_{k=1}^n{h}_k\left({C}_k\left(\mathsf{x},\mathsf{y}\right)\right) $$

5.3. Factorizable harmonies and HZ’s generalization

Subsection 3.2 recalled HZ’s observation that ME harmonies predict HZ’s generalization. HZ’s reasoning extends straightforwardly from ME harmonies to any harmony that factorizes in the sense of (13). In other words, factorizability entails HZ’s generalization. The main result of Magri Reference Magri2025 is that the reverse holds as well: whenever a harmony function predicts HZ’s generalization, it can be shown to factorize. In conclusion, a harmony function H predicts HZ’s generalization in the sense of Subsection 3.3 if and only if (this is a complete characterization) it factorizes into a product (13) of unary functions that attend each to a single constraint and thus encodes a simple mode of constraint interaction, as stated in (12a).

This equivalence between factorizability and HZ’s generalization can be understood intuitively as follows. We focus on a variable phonological process that depends on two phonological dimensions. HZ’s generalization starts from the assumption that this process can be described through constraints that are simple, in the sense that each constraint is only sensitive to one of the two relevant phonological dimensions whereby no constraint can capture any interaction between those two dimensions. Under this assumption, HZ’s generalization predicts that the pattern of rates of application of the process considered ought to be simple as well, in the sense that the rates of application to four underlying forms that cross-product the two relevant phonological dimensions can be fitted onto two shifted sigmoids at shared abscissas. Thus, they depend on only three parameters and are therefore so tightly connected that we can guess any of the rates from the others. Intuitively, in order for a simple constraint set to always predict a simple pattern of rates, we need a mode of constraint interaction that is simple. The result just quoted says that this intuition is correct when simple constraint interactions are formalized through harmonies that factorize into products of rescaled numbers of constraint violations.

5.4. Factorizable harmonies are oblivious to the even constraints

Subsection 4.4 observed that ME harmonies are oblivious to the even constraints. That reasoning extends straightforwardly from ME harmonies to any harmony that factorizes in the sense of (13). To illustrate, we revisit the example of the underlying forms /rat/ and /rut/ from Subsection 4.1 that only differ for the even constraint NoRound. We consider some harmony function H for the n = 3 constraints C ₁ = NoCoda, C ₂ = Max, and C ₃ = NoRound that factorizes into the product (15) of three unary functions h ₁, h ₂, and h ₃, each attuned only to the corresponding constraint. In other words, the harmony score of a mapping is simply the product of the numbers of violations assigned to that mapping by the three constraints considered. The only leeway is that, before taking these products, we are allowed to rescale the constraints through the corresponding unary functions h ₁, h ₂, and h ₃.

(15)

We now compute the ratio between the harmony scores of the faithful and deletion candidates [rat] and [ra] through the reasoning in (16), which is analogous to reasoning in (9), above. Step (16a) holds because H has the shape (15). Step (16b) holds because the constraint NoRound is even between the two candidates [rat] and [ra], whereby the two quantities () and () are identical and their ratio is therefore equal to one. Step (16c) holds because the remaining constraints NoCoda and Max do not distinguish between the two underlying forms /rat/ and /rut/, whereby the former underlying form and its candidates can be replaced with the latter underlying form and its corresponding candidates. Finally, step (16d) holds by reasoning as in steps (16a–b).

(16)

This reasoning in (16) shows that the ratios between the harmony scores of the faithful and deletion candidates are the same for the two underlying forms /rat/ and /rut/. By reasoning as in (11), we conclude that the grammar G_H based on this factorizable harmony H cannot distinguish between these two underlying forms /rat/ and /rut/ that only differ for the even constraint NoRound: G _h deletes the underlying coda of /rat/ with the same probability it deletes the underlying coda of /rut/, namely G_H ([ra] | /rat/) = G_H([ru] | /rut/). The reasoning in (16) obviously extends to the general case. We conclude that no grammar G_H based on a factorizable harmony H can distinguish between two underlying forms x and $ \hat{\mathsf{x}} $ that only differ for even constraints in the sense of Subsection 4.1: G_H realizes the underlying form x as a surface candidate y _i with the same probability it realizes the other underlying form $ \hat{\mathsf{x}} $ as the corresponding candidate $ {\hat{\mathsf{y}}}_i $ , namely G_H (y _i | x) = G_H ( $ {\hat{\mathsf{y}}}_i $ | $ \hat{\mathsf{x}} $ ). In other words, probabilistic grammars based on factorizable harmonies are oblivious to the even constraints in the sense of Subsection 4.4.

5.5. Harmonies oblivious to the even constraints are factorizable

The reverse holds as well. Suppose that a harmony H is oblivious to the even constraints: no matter the candidates and the constraints considered, whenever two underlying forms x and $ \hat{\mathsf{x}} $ only differ for even constraints in the sense of Subsection 4.1, the grammar G_H based on this harmony H realizes the underlying form x as a surface candidate y _i with the same probability it realizes the other underlying form $ \hat{\mathsf{x}} $ as the corresponding candidate $ {\hat{\mathsf{y}}}_i $ , namely G_H (y _i | x) = G_H( $ {\hat{\mathsf{y}}}_i $ | $ \hat{\mathsf{x}} $ ). A more involved reasoning relegated to the Appendix shows that this harmony H then factorizes into the product (13) of unary functions h_k each attuned only to the corresponding constraint C_k. In conclusion, a harmony function is oblivious to the even constraints if and only if (this is a complete characterization) it is factorizable, as stated in (12a).

To understand this result intuitively, we reason once again on the underlying forms /rat/ and /rut/ from Subsection 4.1. Which difference between these two underlying forms would justify deleting their codas with different probabilities? Within constraint-based phonology, the answer must be some difference in constraint violations. Let us thus consider each constraint in turn. The constraints NoCoda and Max do not distinguish between the two underlying forms /rat/ and /rut/ because they assign the same number of violations to their faithful candidates [rat] and [rut] as well as to their deletion candidates [ra] and [ru]. As a result, in and by themselves, these constraints NoCoda and Max justify no difference between the probabilities with which deletion applies to the underlying codas of /rat/ versus /rut/.

The remaining constraint NoRound is even relative to both underlying forms /rat/ and /rut/ because it assigns the same number of violations to the candidates [rat] and [ra] of the former as well as to the candidates [rut] and [ru] of the latter. Thus, NoRound has nothing to say on how the candidates [rat] and [ra] split up the probability mass between the two of them nor on how the candidates [rut] and [ru] split it up. As a result, in and by itself, NoRound justifies no difference between the probabilities with which deletion applies to the two underlying codas of /rat/ versus /rut/.

The only relevant difference between these two underlying forms /rat/ and /rut/ is that the faithful candidate [rut] of the latter violates both markedness constraints NoCoda and NoRound, while the faithful candidate [rat] of the former violates only NoCoda. Yet, this property is invisible to a harmony H that factorizes into the product (15) of three unary functions h ₁, h ₂, and h ₃, each attuned to only one constraint. In fact, H makes decisions by narrowly taking into account only one constraint at a time, ignoring all other constraints. In conclusion, this factorizable harmony H cannot distinguish between the two underlying forms /rat/ and /rut/ because it cannot see the only relevant difference between them, namely that faithful realization of one but not of the other violates both markedness constraints.