Against Gradience
John J. McCarthy
Date of this version: April 1, 2002
Because thou art lukewarm, neither
hot nor cold, I will vomit thee out of
my mouth.
Revelation 3:16
Abstract
In Optimality Theory, a constraint can assign multiple violation-marks in two ways.
Any constraint can assign several marks if there are several violating structures in the
form under evaluation. Furthermore, some constraints are claimed to be gradient:
they can assign multiple marks even when there is just a single instance of the nonconforming structure.
In this paper, I argue that gradience is not a property of OT constraints. I
examine the full range of gradient constraints that have been proposed and show that
none is necessary, some are insufficient, and some are actually harmful. The
archetypic gradient constraint is ALIGN, and I point out various inadequacies of
gradient ALIGN, arguing instead for a quantized, categorical alternative.
1. Introduction
In Optimality Theory (Prince and Smolensky 1993), a constraint can assign multiple
violation-marks to a candidate. This happens in two situations. First, there can be several places
where the constraint is violated in a single candidate, as when ONSET assigns two marks to the
candidate a.pa.i. Second, some constraints are evaluated gradiently, measuring the extent of a
candidate’s deviance from some norm. The best-known gradient constraints come from the
alignment family (McCarthy and Prince 1993a); for instance, the constraint ALIGN(Ft, Wd, R)
assigns three violation-marks to [(pá.ta).ka.ti.ma], one mark for each syllable that separates the right
foot-edge “)“ from the right word-edge “]“.
Both sources of violation-marks are treated as equivalent in OT, so they are added together
when assessing a candidate. Again, alignment supplies the best-known example. When ALIGN(Ft,
Wd, R) evaluates a word with several unaligned feet, each unaligned foot is a separate locus of
violation, and each locus is assessed gradiently. The violation-marks accumulated from these two
sources are treated homogeneously, as (1) shows.
2
(1) Evaluation by ALIGN(Ft, Wd, R)1
Ft-1
Ft-2
Ft-3
ALIGN(Ft, Wd, R)
a.
[(σ́σ)1 (σ́σ)2 (σ́σ)3 σ]
σσσσσ
σσσ
σ
*********
b.
[(σ́σ)1 (σ́σ)2 σ (σ́σ)3]
σσσσσ
σσσ
Ø
********
c.
[(σ́σ)1 σ (σ́σ)2 (σ́σ)3]
σσσσσ
σσ
Ø
*******
d.
[σ (σ́σ)1 (σ́σ)2 (σ́σ)3]
σσσσ
σσ
Ø
******
e.
[σσσσσ (σ́σ)3]
Ø
Ø
Ø
°
In (1a), for instance, the three feet are misaligned by five, three, and one syllables, respectively. So
this candidate receives eight violation-marks from ALIGN(Ft, Wd, R).
For the purposes of this article, it will be useful to make explicit the ideas that underlie (1).
The hypotheses in (2) state outright the assumptions implicit in Prince and Smolensky (1993).
(2) Multiple Violations in OT
a. Locus hypothesis.
A violation-mark is assigned for each instance or locus of violation in a candidate.
When presented with the right candidate, then, any OT constraint can assign multiple
violation-marks.
b. Gradience hypothesis.2
Some constraints, by virtue of their formulation, assess violations gradiently. These
constraints can assign multiple violation-marks even when there is just a single locus
of violation.
c. Homogeneity hypothesis.
Multiple violations of a constraint from either source are added together in evaluating
a candidate. No distinction is made between multiple violation-marks derived from
the Locus hypothesis and those derived from the Gradience hypothesis.
In my view, the Locus hypothesis is an important insight that should be maintained if possible. It
allows candidates with, say, different numbers of feet to be compared very straightforwardly.
Imaginable alternatives involve a great deal more finagling, such as comparing candidates by
matching the feet in one with the feet in the other. The Gradience hypothesis, though, is less solid.
In this article, I will argue that the Gradience hypothesis is both unnecessary and even problematic.
And, of course, if gradience is eliminated from the theory, then the Homogeneity hypothesis is
superfluous.
1
This approach, which is found in McCarthy and Prince (1993a), ultimately derives from a suggestion made by
Robert Kirchner.
2
The term gradience is not always used in the sense I assume here. For example, Hume (1998) describes LINEARITY
(“no metathesis”) as a gradient constraint because it assigns two violation-marks to the mapping /C1V2C3V4C5/ 6 C1V2C5C3V4.
This is not true gradience, in my view. Rather, it is a case of multiple loci of violation: the input asserts various linear
precedence relations among the segments, and the output contradicts two of those relations, C3 > C5 and V4 > C5.
3
As an alternative to gradience, I propose that categorical constraints may be distinguished
by extent of violation. Instead of gradient ALIGN(Ft, Wd, R), for example, there is a family of
constraints, one for each type of prosodic constituent that can intervene between a foot-edge and a
word-edge.
(3) Quantized ALIGN(Ft, Wd, R)
a. ALIGN-BY-σ(Ft, Wd, R)
No syllable stands between the right-edge of Ft and the right-edge of Wd.
b. ALIGN-BY-FT(Ft, Wd, R)
No foot stands between the right-edge of Ft and the right-edge of Wd.
The candidates [σσσ(σσ)σ], [σσ(σσ)σσ], and [σ(σσ)σσσ] are each assigned one violation-mark by
ALIGN-BY-σ(Ft, Wd, R) — this constraint is categorical, so it treats all three candidates alike,
distinguishing them only from the non-violator [σσσσ(σσ)]. The candidate [σσ(σσ)(σσ)] is assigned
one violation-mark by ALIGN-BY-σ and another by ALIGN-BY-FT(Ft, Wd, R). The candidate
[σ(σσ)(σσ)σ] receives two violation-marks from ALIGN-BY-σ — one for each foot under the Locus
hypothesis, which is not in dispute — and another from ALIGN-BY-FT.
It is apparent from the preceding paragraph that quantized alignment constraints do not make
the same distinctions as gradient alignment. Two examples emphasize the differences. First,
[σσσ(σσ)σ] and [σσ(σσ)σσ] are equally harmonic under quantized alignment but distinct under
gradient alignment. Second, [σσσ(σσ)σ] and [σσ(σσ)(σσ)] violate the same gradient alignment
constraint to different extents, but quantized alignment distinguishes them categorically because only
[σσ(σσ)(σσ)] violates the quantized constraint ALIGN-BY-Ft(Ft, Wd, R). Quantized alignment, then,
is both more and less exact than gradient alignment. A principal goal of this article is to show that
these differences weigh in favor of quantized alignment when the full range of evidence is
considered.
§2 of this paper briefly reviews some of the gradient constraints that have been proposed in
the literature. Except for alignment, all have an obvious and probably necessary reformulation in
categorical terms. Alignment constraints, as I have suggested, can also be reformulated categorically,
and §3 shows that they should be, based on evidence from infixation. §4 then turns to the role of
alignment in stress. Following Kager (2001), I argue that gradient ALIGN(Ft, Wd) constraints
overgenerate and can be replaced by constraints on stress lapses. I also show that other applications
of gradient alignment, such as locating main stress, can be reanalyzed using quantized alignment
constraints. Gradient alignment constraints have also been invoked as a way to produce
autosegmental spreading effects, so this is the topic of §5. As we will see, this use of gradient
alignment is by no means a settled matter, and plausible alternatives exist. Finally, §6 sums up the
results and discusses some of the remaining issues.
Before we begin, there is a final remark to be made about the logic of the argument pursued
here. The Gradience hypothesis is not an essential element of classic OT. (“... [T]he division of
constraints into those which are binary and those which are not, a division which we have adopted
earlier in this section, is not in fact as theoretically fundamental as it may at this point appear.”
(Prince and Smolensky 1993: 81)) If it can be shown that gradient constraints are unnecessary, then
that is sufficient reason to reject them. In this article, I will show that they are indeed unnecessary,
since reasonable non-gradient alternatives can be developed. The argument, however, goes beyond
4
this essential step to show that gradient constraints both underpredict and overpredict the range of
observed behavior in certain situations. The argument, then, is two-pronged: supply an alternative
to gradience and highlight the descriptive superiority of that alternative.
2. Making Gradient Constraints Categorical
This section reviews some of the uses to which gradient constraints have been put in the OT
literature. The section goes on to show how these constraints can — and in some cases must — be
reformulated categorically.
A taxonomy of gradience is useful to help organize the discussion.
(4) Types of Gradience in the OT literature
a. Horizontal gradience.
Assign violation-marks in proportion to distance in the segmental string. Example:
ALIGN(Ft, Wd, R), ALIGN(Prefix, Wd, L).
b. Vertical gradience.
Assign violation-marks in proportion to levels in a hierarchy. Example (paraphrasing
Spaelti 1994): WEAKEDGE . assign a violation-mark for each prosodic category
whose right-periphery is non-empty. E.g., [((dog)σ)Ft]Wd receives three marks.
c. Collective gradience.
Assign violation-marks in proportion to the cardinality of a set. Example (Padgett
1995a): CONSTRAINT(Class) . assign one violation-mark for each member of the
feature-class Class that does not satisfy CONSTRAINT. E.g., aõgMba receives one mark
from ASSIM(Place) and angMba receives two marks.
d. Scalar gradience.
Assign violation-marks in proportion to the length of a linguistic scale. Example
(Prince and Smolensky 1993: 16): HNUC / “A higher sonority nucleus is more
harmonic than one of lower sonority.” I.e., assign a nucleus one violation-mark for
each degree of sonority less than a.
The examples given are typical. Alignment constraints — and perhaps only alignment constraints
— involve horizontal gradience. Vertical gradience is not often encountered, but known cases refer
to the prosodic hierarchy (Selkirk 1980), as in the example cited. Collective gradience is developed
formally in the context of Padgett’s work on feature classes. Scalar gradience appears in Prince and
Smolensky (1993) in the form of the constraints HNUC and PK-PROM.
There is a basic bifurcation between horizontal gradience on the one hand and vertical,
collective, and scalar gradience on the other. Horizontal gradience is responsible for unboundedly
many constraint violations; there is no limit on how many segments or syllables can separate the
constituent edges mentioned in an alignment constraint. Vertical, collective, and scalar gradience,
however, are always limited in extent of violation (setting aside cases where there are multiple loci).
Constraints assessing vertical gradience cannot assign more violation-marks than there are levels in
the hierarchy. Constraints that are collectively gradient cannot assign more violation-marks than
there are members of the set. Constraints that are scalar gradient cannot assign more violation-marks
5
than there are steps in the scale. And since the linguistic hierarchies, sets, or scales referred to in
these constraints are finite, these constraints are always bounded in their assessments.
This bifurcation between the unbounded and the bounded is important because there is an
obvious alternative account of bounded gradience: separate constraints for each level of the hierarchy
or each member of the set. The basic techniques for doing this are introduced in Prince and
Smolensky (1993: 134ff.), where HNUC is deconstructed in this way. As long as there is a bound on
the extent of violation of a gradient constraint, there exists a reformulation in terms of a finite set of
constraints. For example, WEAKEDGE would be restated as a set of constraints demanding an empty
right-periphery of each respective layer of the prosodic hierarchy. As for collective gradience, any
constraint that is gradient over a set of elements can be replaced by individual constraints on each
member of the set. For example, the constraint IDENT(COLOR) is superfluous if there are IDENT
constraints for each of the vowel-color features [back] and [round].3 And HNUC can be replaced by
the constraint hierarchy *PK/OBST >> *PK/NASAL >> ... or by an equivalent set of stringently
formulated constraints (de Lacy 2000, 2002, Prince 1998). In fact, HNUC must be replaced by
individual constraints to account for syllable-structure typology (see Prince and Smolensky (1993:
134ff.) and §3 below).
Unbounded, horizontal gradience, on the other hand, has no translation into a finite set of
categorical constraints. ALIGN(Ft, Wd, R) imposes a harmonic ordering of unlimited depth on words
containing just a single foot: [...(σσ)] ™ [...(σσ)σ] ™ [...(σσ)σσ] ™ [...(σσ)σσσ] ... Categorical
constraints cannot do this; in fact, the purported need to do this is the very reason that gradient
constraints are postulated in OT.
My claim in this article is that the power of unbounded, horizontal gradience is not actually
required in human language, and that a weaker, categorical theory is sufficient.4 The main proposal
is that the actually observed effects of horizontal gradience can be obtained with categorical
constraints that are distinguished by the degree of misalignment that they allow. As in (3), there are
separate categorical constraints for each level of the prosodic hierarchy. These constraints form a
stringency hierarchy in the sense of Prince (1998): violation of (3b) entails violation of (3a). Similar
stringency hierarchies exist for other alignment constraints, such as those regulating prefixes and
suffixes (§3).
The constraints in (3), taken together, make some but not all of the same distinctions as
gradient ALIGN(Ft, Wd, R). To see this, consider in (5) the harmonic ordering imposed by gradient
and categorical ALIGN on all of the possible foot-parsings of a 5-syllable word, assuming that words
contain at least one foot and that degenerate feet are prohibited.
3
If the constraint component CON includes IDENT(round) and IDENT(back), then gradient IDENT(COLOR) is
superfluous because its presence has no visible effect on the resulting factorial typology. Padgett (1995a), following a
suggestion by Alan Prince, looks into eliminating IDENT(round) and IDENT(back) from CON in favor of IDENT(COLOR), but
it has not been shown that this move is possible in this specific case or generally.
4
Compare Zoll (1996), who argues that categorical alignment constraints (COINCIDE) are required in addition to
gradient alignment constraints.
6
(5) Harmonic Ordering of Pentasyllabic Parses by Gradient and Categorical ALIGN(Ft, Wd, R)
Gradient
Categorical
[σσσ(σσ)] ™
[σσσ(σσ)] ™
[σσ(σσ)σ] ™
[σσ(σσ)σ], [σ(σσ)σσ], [(σσ)σσσ] ™
[σ(σσ)σσ], [σ(σσ)(σσ)] ™
[σ(σσ)(σσ)], [(σσ)σ(σσ)], [(σσ)(σσ)σ]
[(σσ)σσσ], [(σσ)σ(σσ)] ™
[(σσ)(σσ)σ]
The ordering imposed by gradient ALIGN was determined by the familiar procedure seen in (1). The
ordering imposed by categorical ALIGN has the perfectly aligned candidate at the top, the candidates
that violate only ALIGN-BY-σ and not ALIGN-BY-FT in the next layer, and the candidates that violate
both ALIGN-BY-σ and ALIGN-BY-FT at the bottom.
As expected, the gradient constraint imposes an ordering of greater depth and even orders
some imperfectly-aligned candidates differently than the categorical constraints (i.e., there isn’t an
order-preserving map between them (see Davey and Priestley 1990: 10)). Furthermore, the
categorical constraints treat the imperfectly-aligned candidates as categorically different in their
alignment performance: [σσ(σσ)σ], [σ(σσ)σσ], and [(σσ)σσσ] obey a constraint that [σ(σσ)(σσ)],
[(σσ)σ(σσ)], and [(σσ)(σσ)σ] violate. The same is not true for the gradient constraint: all of the
imperfect candidates violate the same constraint, ALIGN(Ft, Wd, R), though to different extents.
These observations lead to two empirical questions: is the greater depth imposed by gradient ALIGN
actually required, and in situations where perfect alignment is not possible, do we find categorical
differences among the imperfect candidates? Sections §4 and §5 address the first of these questions,
while §3 addresses the second question, to which we now turn.
3. Alignment and Infixation
The theory of infixation in OT, due to Prince and Smolensky (1991, 1993 33ff.),5 holds that
infixes are imperfect prefixes or suffixes — imperfect because the constraints aligning them
peripherally, ALIGN(Pfx, Wd, L) and ALIGN(Sfx, Wd, R), are crucially dominated and may be
violated. Often, familiar markedness constraints like ONSET or NO-CODA are responsible for nonperipheral placement of an affix.
For example, in Prince and Smolensky’s analysis of Tagalog, infixation of the actor-focus
morpheme -um- is attributed to a constraint hierarchy where NO-CODA crucially dominates gradient
ALIGN(-um-, Wd, L).6 This ranking leads to perfect alignment with vowel-initial words like umabot
5
Applications and extensions of this idea appear in Akinlabi (1996), Buckley (1997), Carlson (1998), de Lacy
(1999), Fulmer (1997), McCarthy (2000), McCarthy and Prince (1993a, 1993b), Noyer (1993), Spaelti (1997), Stemberger
and Bernhardt (1999), and Urbanczyk (1996).
6
The existence of constraints like ALIGN(-um-, Wd, L) is sometimes offered as evidence for the “discovery” that
OT has language-particular constraints. This point is almost jesuitical. ALIGN(Pfx, Wd, L) and ALIGN(Sfx, Wd, R) offer a
universal framework for stating constraints on affix placement. That individual affixes must somehow be identified as prefixes
or suffixes on a language-particular basis comes as no surprise. A real “language-particular constraint,” if any exist, would
presumably have the character of the language-particular rules in other theories: a one-time ad hoc statement with no
typological commitments whatsoever.
7
‘to reach for’ and less-than-perfect alignment with consonant-initial words like sumulat ‘to write’
or grumadwet ‘to graduate’. The tableau in (6) shows how infixation is achieved:
(6) Infixation with Gradient Alignment
/um+gradwet/
NO-CODA
ALIGN(-um-, Word, L)
**
**
a.
L grumadwet
b.
umgradwet
***!
c.
gumradwet
***!
*
d.
gradumwet
**
****!
The gradience of alignment is called on to decide in favor of (6a) grumadwet over (6d) *gradumwet,
since they satisfy NO-CODA equally well. The prefix -um- is infixed no more than is necessary to
optimize performance on NO-CODA. Since grumadwet and *gradumwet tie in their NO-CODA
performance, the better-aligned one wins.
Categorical alignment gives a slightly different picture of this situation. NO-CODA crucially
dominates ALIGN-BY-SEG(-um-, Word, L), which regards grumadwet and *gradumwet as equally
harmonic. The choice between them is made by ALIGN-BY-σ(-um-, Word, L), which favors
grumadwet no matter where it is ranked:
(7) Infixation with Categorical Alignment
/um+gradwet/
NO-CODA
ALIGN-BY-SEG
**
*
a.
L grumadwet
b.
umgradwet
***!
c.
gumradwet
***!
*
d.
gradumwet
**
*
ALIGN-BY-σ
*!
In (7a) grumadwet, the left edge of -um- is not separated by a whole syllable from the left edge of
the word, but in (7d) *gradumwet it is. Clearly, categorical alignment is able to make the necessary
distinctions. (The role of ALIGN-BY-σ in (7) is that of a tie-breaker, a situation that does not support
a ranking argument (McCarthy 2002: 37-8).)
Thus far, we have no empirical basis for choosing one approach or the other. When fuller and
more exact data from Tagalog are considered, however, it turns out that categorical alignment is
superior. This evidence comes from Orgun and Sprouse (1999: 203ff.), though I depart from them
in including initial § in the analysis.7 The data are given in (8).
7
That is, Orgun and Sprouse (1999), like Prince and Smolensky, transcribe ‘to reach for’ as abot and umabot. This
is consistent with Tagalog orthographic practice, but not with the actual pronunciation (Schachter and Otanes 1972: 26). Since
OT constraints evaluate output forms, the initial § in these words cannot properly be disregarded.
8
(8) Infixation in Tagalog
a. C-initial words
sulat
sumulat
§abot
§umabot
b. CC-initial words
gradwet
grumadwet ~ gumradwet
preno
prumeno ~ pumreno
c. m, w-initial words
mahal
*mumahal
walow
*wumalow
d. s + m, w-initial words
smajl
*summajl ~ *smumajl
swiõ
sumwiõ ~ *swumiõ
‘to write’
‘to reach for’
‘to graduate’
‘to brake’
‘to become expensive’
‘to wallow’
‘to smile’
‘to swing’
There are no V-initial words in Tagalog, so this is the complete paradigm. When a word begins with
a single consonant, then -um- is infixed after that consonant, unless the word begins with a labial
sonorant, in which case the verb has no -um- form. (Since verbs are lexically marked to take -umas their actor-focus marker, and since there are other actor-focus markers like ma-, mag-, and maõ-,
it is no great tragedy to be blocked phonologically from having an -um- form. See Schachter and
Otanes (1972: 284ff.).) With cluster-initial roots, -um- is, for at least some speakers, variably infixed
after the first or the second consonant. When the initial cluster contains a labial sonorant, then forms
with ...mum... and ...wum... sequences are again blocked, just as they are in the m- and w-initial
roots. In addition, ...umm... is out by a general prohibition against geminate m (Orgun and Sprouse
1999: 206 fn. 11).
The analysis of these amplified data will proceed de novo, setting aside the previous analysis
sketched in (6, 7). The first order of business is to establish some basic rankings of syllabic
markedness and faithfulness constraints. Since Tagalog permits codas and complex onsets, the
ranking in (9) can be safely assumed.
(9) Tagalog: Some Initial Rankings
DEP-V, MAX-C >> NO-CODA, *COMPLEX-ONS
Since Tagalog syllables also require onsets, we know that ONSET is undominated but, contrary to the
assumption made in previous work that § is epenthesized, we have no actual evidence proving how
ONSET is satisfied with non-compliant inputs. As we will see later, the real story is not quite as
obvious.
Though they cannot compel unfaithfulness to the input because of (9), NO-CODA and
OMPLEX
-ONS play a role in analyzing the grumadwet ~ gumradwet variation. These forms differ
*C
by trading better performance on one constraint for better performance on the other.
9
(10) NO-CODA and *COMPLEX-ONS as Tied Constraints
a. NO-CODA >> *COMPLEX
/um+gradwet/
i.
L grumadwet
ii.
gumradwet
NO-CODA
*COMPLEX-ONS
**
*
***!
b. *COMPLEX >> NO-CODA
/um+gradwet/
i.
grumadwet
ii.
L gumradwet
*COMPLEX-ONS
NO-CODA
*!
**
***
If these constraints are formally tied in the grammar of Tagalog, and if a specific ranking is chosen
at each application of EVAL, then the observed variation can be obtained.8
The role of labial sonorants in blocking -um- affixation is the main focus of Orgun and
Sprouse’s analysis. They propose an OCP-like constraint against sequences of a labial sonorant plus
-um-. Most of the starred forms in (8c, d) violate this constraint: *mumahal, *wumalow, *smumajl,
*swumiõ. They argue that, in order to block -um- affixation entirely with such words, it is necessary
for OCP(labial) to be placed outside EVAL, so it can cause the derivation to crash if EVAL emits
something like *mumahal.
Their argument for this enrichment of OT comes from the impossibility of deeper infixation
to satisfy OCP(labial). Although this constraint rules out *mumahal, it is satisfied by *mahumal.
And since gradient ALIGN(-um-, Word, L) is low-ranked in their analysis, *mahumal’s execrable
alignment should not be a problem. They hint, however, that the post-EVAL application of
OCP(labial) could be avoided “if ALIGN were supplemented with a constraint limiting -um- to the
first syllable” (Orgun and Sprouse 1999: 207), a move they reject on the grounds that “it clearly is
not in the spirit of the alignment approach to infixation” (ibid.). This critique seems apt if gradient
alignment is supplemented with a categorical constraint, but not if it is replaced by a categorical
constraint, as proposed here. I will therefore work through the categorical analysis that they reject.
As background, we first need a way to deal with the problem of ineffability: there is no -umform for words with a certain phonological shape. Within classic OT, some cases of ineffability
involve the null output. Prince and Smolensky (1993: 48ff.) hypothesize that the null output is a
member of every candidate set. In the context of their representational assumptions and the
phenomena they were analyzing, the null output was called the null parse, and it consisted of a
segmental string without prosodic structure. In terms of Correspondence Theory (McCarthy and
Prince 1995, 1999), the null output can be thought of as a candidate whose correspondence relation
8
There is a large body of work applying the idea of partially ordered or tied constraints to problems of phonological
variation (Anttila 1997a, 1997b, Borowsky and Horvath 1997, Hammond 1994, Ito and Mester 1997, Iverson and Lee 1995,
Kiparsky 1993, 1994, Morris 1998, Nagy and Reynolds 1995, Nevin 1998, Noske 1996, Reynolds and Nagy 1994, Ringen
and Heinämäki 1999, Struijke 2000, van Oostendorp 1997) and to variation in versification (Hayes and MacEachern 1998).
10
to the input is undefined. (It is therefore distinct from a candidate where every segment is deleted,
which does have a well-defined correspondence relation.) I will use the symbol “u” to stand for this
candidate.9
No matter what the input, the candidate u is among the candidates emitted by GEN. If GEN
truly has freedom of analysis (McCarthy and Prince 1993b), then it is hard to see how u would
not be a candidate, and if it is a candidate for some input, surely it must be for all. In other words,
any reasonable way of defining GEN is likely to emit the candidate u for free and to include it in
every candidate set.10
As it happens, u is a surprisingly attractive candidate because it is as unmarked as can be.
It vacuously satisfies every markedness constraint in CON. Markedness constraints either militate
against the presence of structure — like NO-CODA — or they require structure, when present, to have
certain properties — like ONSET or many alignment constraints. Since u has no structure whatsoever,
it is never in danger of violating either kind of markedness constraint. Furthermore, because its
input-output correspondence relation is undefined, u vacuously satisfies all faithfulness constraints.
(Faithfulness constraints are defined on correspondence relations; if the correspondence relation of
some candidate is undefined, then the faithfulness constraint cannot possibly be violated.) By
assumption, u violates just one constraint, which Prince and Smolensky dub M-PARSE.
To be specific, the constraint M-PARSE(-um-) is violated by the candidate u whenever the
input contains the morpheme -um-. Since verbs with -um- do sometimes have codas or complex
onsets, we can infer that NO-CODA and *COMPLEX-ONS are dominated by M-PARSE(-um-) (see
(11a)). Furthermore, verbs with -um- are misaligned by one or more segments, so we can also
conclude that ALIGN-BY-SEG(-um-, Wd, L) is also dominated by M-PARSE(-um-) (see (11b)).
(11)
a. M-PARSE(-um-) >> NO-CODA, *COMPLEX-ONS
/um+gradwet/
M-PARSE(-um-)
NO-CODA
i.
L gumradwet
***
ii.
L grumadwet
**
iii.
u
*COMPLEX-ONS
*
*!
b. M-PARSE(-um-) >> ALIGN-BY-SEG(-um-, Wd, L)
/um+sulat/
M-PARSE(-um-)
i.
L sumulat
ii.
u
9
ALIGN-BY-SEG
*
*!
For further development and applications of the null output as a candidate, see Ackema and Neeleman (1998,
2000), Benua (1997), Cohn and McCarthy (1994/1998), Grimshaw and Samek-Lodovici (1995, 1998), Ito, Kitagawa, and
Mester (1996), Kager (2000), McCarthy and Prince (1993b: Chapter 7), and Yip (1998).
10
I am indebted to Jane Grimshaw for this point.
11
These ranking arguments exemplify what Legendre et al. (1998: 257 fn. 9) call a “harmony
threshold” that M-PARSE sets. Because u obeys every constraint except M-PARSE(-um-), no winning
candidate derived from an input with -um- can violate any constraint ranked higher than MPARSE(-um-). Therefore, all constraints that words with -um- are observed to violate must be ranked
below M-PARSE(-um-).
The harmony threshold works to our advantage when it comes to dealing with the effects of
OCP(labial). Because /um+mahal/ maps most harmonically to u, all non-null candidates derived
from this input must violate constraints ranked higher than M-PARSE(-um-). This includes not only
OCP(labial), to rule out *mumahal, but also ALIGN-BY-σ(-um-, Wd, L), to rule out *mahumal.
(12) OCP(labial), ALIGN-BY-σ(-um-, Wd, L) >> M-PARSE(-um-)
/um+mahal/
OCP(labial)
a.
L u
b.
mumahal
c.
mahumal
ALIGN-BY-σ
M-PARSE(-um-)
ALIGN-BY-SEG
*
*!
*
*!
*
This tableau shows a key result. We know from (11) that M-PARSE(-um-) dominates ALIGN-BY-SEG,
since otherwise -um- would never be infixed. To this, (12) adds the information that ALIGN-BY-σ
dominates M-PARSE(-um-). Therefore, M-PARSE(-um-) separates the two quantized, categorical
alignment constraints in the hierarchy. This proves that they must indeed be separate constraints.
This analytic move is not possible in the gradient alignment theory, whence Orgun and
Sprouse’s argument for a post-EVAL check by OCP(labial). Gradient ALIGN is, obviously, a single
constraint, and it must either dominate M-PARSE(-um-) or be dominated by it. Either way, the wrong
result is obtained. If Tagalog is to be analyzed within the strictures of classic input/GEN/EVAL/output
OT, then quantized, categorical alignment constraints are necessary.11
It remains only to clear up a few remaining points about Tagalog before moving on to another
example of the same general type. If deep infixation à la *mahumal is not an option, then why not
skip infixation entirely with such words, opting for *§ummahal or *§umwalow? There is a local
explanation for the ill-formedness of *§ummahal — mm clusters aren’t allowed — but there is no
such explanation for *§umwalow. In fact, we know that -um- words specifically can contain mw
11
Smolensky (1995, 1997) has proposed that constraints can be conjoined with themselves to produce a power
hierarchy of constraints ƒ… C3 >> C2 >> C1„, where Cn is violated if and only if there are at least n distinct instances of Cviolation in the domain of evaluation. Alderete (1997) and Ito and Mester (1998) apply conjunction of a markedness constraint
with itself to the phenomenon of phonological dissimilation, and Legendre, Smolensky, and Wilson (1998) propose selfconjunction as a theory of barriers. One might, then, see it as an alternative approach to the problem of ranking gradient
ALIGN with respect to MPARSE(-um-). The idea would be to set a limit of no more than two segments’ displacement with the
hierarchy ...ALIGN3 >> MPARSE(-um-) >> ALIGN2 >> ALIGN, letting in grumadwet but not *mahumal.
This approach to Tagalog seems fundamentally misconceived. It “rescues” gradient alignment by adding on a theory
of categorical alignment of immense power, able to distinguish two segments from three or three from four. Worse yet, the
intended semantics of ALIGNn — to prohibit misalignment by n or more segments — is not consistent with the semantics of
local self-conjunction in the dissimilation applications. This “theory” of alignment, then, isn’t even well-defined.
12
clusters because of examples like sumwiõ. So *§umwalow must be ruled out for another reason: its
epenthetic initial consonant.
(13) ONSET, DEP-C >> MPARSE(-um-)
/um+walow/
ONSET
a.
L u
b.
umwalow
c.
§umwalow
DEP-C
MPARSE(-um-)
*
*!
*!
Because no surface form of Tagalog violates ONSET, we can safely conclude that ONSET is
undominated. This tableau shows that DEP-C is also high-ranked, crucially dominating
MPARSE(-um-). This forecloses the last way that /um+walow/ could map to a non-null output.
To sum up, we have seen evidence for the following ranking in Tagalog:
(14) Tagalog Ranking Summary
ONSET, OCP(labial), ALIGN-BY-σ, DEP-C, MAX-C, DEP-V
>> MPARSE(-um-)
(no -um- form violates preceding
constraints)
>> NO-CODA, *COMPLEX-ONS, ALIGN-BY-SEG
(-um- forms can have codas and
complex onsets, and -um- can be
infixed.)
On the basis of words beginning with labial sonorants, it has been established that ONSET,
OCP(labial), ALIGN-BY-σ, and DEP-C all dominate MPARSE(-um-). MAX-C and DEP-V are also
unviolated, so they appear in the top ranking stratum as well. The location of MPARSE(-um-) in the
hierarchy sets the harmony threshold: actually occurring -um- words can only violate lower-ranking
constraints. Among those constraints are NO-CODA, *COMPLEX, and ALIGN-BY-SEG.
This ranking implies some further results about Tagalog phonology. Infixation is compelled
by ONSET and DEP-C, not NO-CODA (cf. (6, 7)). For example, /um+sulat/ maps to sumulat because
*umsulat and *§umsulat violate ONSET and DEP-C, respectively. The underlying form of §abot must
be /§abot/, as we expect from the lack of §/Ø alternations, and its -um- form §umabot is no different
than any other C-initial root. Of course, under richness of the base, the grammar of Tagalog is also
responsible for disposing of V-initial roots. They must be treated unfaithfully, because Tagalog
words never begin with a vowel, but no active alternations show how they are treated — the
traditional assumption that hypothetical /apak/ becomes §apak is without empirical foundation. If
we assume that /apak/ 6 §apak is indeed the right disposition of V-initial words, then the ranking
ƒONSET >> DEP-C„ must be added to the grammar in (14). Hypothetical /apak/ then would surface
as §apak and would have no -um- form (because DEP-C dominates MPARSE(-um-)). That situation
is undetectable: some surface §-initial verbs take -um- and some don’t anyway, since verbs are
lexically marked to take -um-. The grammar in (14), then, accounts for all of the relevant data.
13
The next example comes from the Austronesian language Nakanai, which is spoken in New
Britain. The data and generalizations are due to Johnston (1980). In this language, as in Tagalog,
infixation shows the effect of categorical alignment: a morphological process is blocked when
misalignment is too severe.
Nakanai disallows codas and allows onsetless syllables freely. Each vowel is said by
Johnston to constitute a syllable on its own, though it is conceivable that what is really meant is some
kind of moraic analysis. Stress falls strictly on the penultimate syllable. Words are minimally
disyllabic in size.
Nakanai forms nominalizations by infixing -il-,12 Tagalog-style, when the word contains
exactly two syllables. With longer words, however, the infix is replaced by the suffix -la.
(15) Nakanai Nominalization
a. Short Words
ilau
tilaga
gilogo
b. Long Words
sagegela
vikuela
vigilemulimulila
‘steering’
‘fear’
‘sympathetic’
‘happiness’
‘fight’
‘story’
The size of the entire word, and not just the size of the root, is decisive. For example, the last
example is based on the disyllabic root gile ‘to sift’.
The difference between short words and longer words is that only short words have initial
stress. The formative -il-, then, is attracted to the main stress of the word, just like the Ulwa
possessive morphemes analyzed in McCarthy and Prince (1990).13 Furthermore, -il- is a prefixal
infix, like Tagalog -um-, showing that ALIGN-BY-SEG(-il-, Wd, L) is crucially dominated. But there
is a limit on how far -il- can be infixed: forms like *sagilege or *vigilemulimiluli are out (cf. Tagalog
*mahumal). Therefore, ALIGN-BY-σ(-il-, Wd, L) is undominated, and so the otherwise identical
ALIGN-BY-SEG and ALIGN-BY-σ constraints must be ranked separately, just like Tagalog.
This difference in ranking of the two categorical alignment constraints is, of course, the
whole point of the Nakanai example. Still, it is necessary to fill in the details of the analysis,
including the -il-/-la allomorphy. Here I take one approach that has been widely adopted in the
literature.14 I assume that -il- and -la are both lexically listed, since there is no regular phonological
12
There are additional alternations in the form of the infix. It is i before l- or r-initial roots. Its vowel is u before u
or o in the next sylllable. And its consonant is r in agreement with an r in the next syllable.
13
The Ulwa possessive morphemes are suffixed to the main-stressed syllable in an iambic stress system: su+kalu
‘his/her dog’, kulukaluk ‘his/her woodpecker’, arakkabus ‘his/her gun’.
14
The idea of lexical entries as sets of allomorphic alternants originated with Hudson (1974) and is adopted by
Hooper (1976). There is a considerable literature applying OT to problems in allomorphy or lexical selection, including
Alcantará (1998), Anttila (1997b), Bresnan (2001, to appear), Burzio (1994, 1997), Drachman, Kager, and Drachman (1997),
Grimshaw (1997), Hargus (1995), Hargus and Tuttle (1997), Kager (1996), Lapointe and Sells (1997), Mascaró (1996),
14
alternation between. Both are supplied in candidates at no cost in faithfulness, leaving the phonology
responsible for choosing the correct allomorph together with the form that contains it.
The allomorph -il- is a formal prefix with its distribution under the control of undominated
ALIGN-BY-σ(-il-, Wd, L). The allomorph -la is a formal suffix, and since it is never infixed, its
distribution is governed by undominated ALIGN-BY-SEG(la, Wd, R). The interesting analytic action
centers around a couple of descriptive problems: -il- is attracted to the stressed syllable, and -la
functions as kind of default, occurring only when -il- is blocked by undominated ALIGN-BY-σ(-il-,
Wd, L).
One way to ensure -la’s default status is to consider its effect on stress. When a word takes
the suffix -la or any other suffix in Nakanai, its stress shifts to the new penult: sagége/sagegéla.
Stress shift like this is a violation of OO-IDENT(stress) or some equivalent output-output faithfulness
constraint (Benua 1997, Pater 1998 and others). Violation of OO-IDENT(stress) can be avoided by
choosing the prefixal allomorph of the nominalizer, and so that is what’s found in tága/tilága. Only
when the -il- allomorph is blocked by undominated ALIGN-BY-σ(-il-, Wd, L) does the -la allomorph
emerge.
(16) ALIGN-BY-σ(-il-, Wd, L) >> OO-IDENT(stress)
/{il, la}+sagege/
a.
L sagegéla
b.
sagilége
ALIGN-BY-σ(-il-, Wd, L)
OO-IDENT(stress)
(cf. sagége)
*
*!
Other candidates like sagégela and salagége violate undominated constraints: foot alignment in the
first case and ALIGN-BY-SEG(-la, Wd, R) in the second.
The other element of the analysis is a constraint attracting -il- to the main-stressed syllable
or head foot, as in Ulwa. In McCarthy and Prince (1993a), a special class of subcategorizational
alignment constraints is proposed to deal with cases like Ulwa, but that does not seem necessary
here. The constraint ALIGN-BY-σ(-il-, Hd, L) is satisfied if no syllable stands between the left edge
of -il- and the left edge of the head of the prosodic word, the main-stress foot. This constraint
dominates OO-IDENT(stress), excluding candidates like *silagége in favor of sagegéla.
(17) ALIGN-BY-σ(-il-, Hd, L) >> OO-IDENT(stress)
/{il, la}+sagege/
a.
L sagegéla
b.
s=ila=gége
ALIGN-BY-σ(-il-, Hd, L)
OO-IDENT(stress)
(cf. sagége)
*
*!
McCarthy and Prince (1993b: Chapter 7), Mester (1994), Perlmutter (1998), Russell (1995), Tranel (1996a, 1996b, 1998),
and Urbanczyk (1999).
15
The left edge of the head foot is marked by the second “=” symbol in (17b), while the left edge of
-il- is marked by the first “=”. Between them stands the syllable la, so ALIGN-BY-σ(-il-, Hd, L) is
violated by this candidate.
The prefix -il- does not fall exactly at the left word edge or foot edge, even in forms like
tilága or iláu. Misalignment by a whole syllable is not possible, as the tableaux (16, 17) show, but
misalignment by a segment is tolerated. This shows that OO-IDENT(stress) dominates the segmentsized versions of the alignment constraints in (16, 17).
(18)
/{il, la}+taga
a.
L t|il|ága
b.
tagála
OO-IDENT(stress)
(cf. tága)
ALIGN-BY-SEG
(-il-, Wd, L)
ALIGN-BY-SEG
(-il-, Hd, L)
*
*
*
This covers the main points of the analysis, summarized in the ranking (19).
(19) Nakanai Ranking Summary
ALIGN-BY-σ(-il-, Wd, L), ALIGN-BY-σ(-il-, Hd, L) >>
OO-IDENT(stress) >>
ALIGN-BY-SEG(-il-, Wd, L), ALIGN-BY-SEG(-il-, Hd, L)
The constraint OO-IDENT(stress) favors -il- over -la, so it sets a kind of threshold: forms with -ilcan violate only the constraints ranked below OO-IDENT(stress). The constraints ranked above OOIDENT(stress) are those that block -il- in favor of -la. These constrants say that the allomorph -il- is
simultaneously attracted to the left word-edge and the left foot-edge. It cannot be displaced from
either of these spots by a syllable or more, so when the word is longer than a single foot, the -ilallomorph fails completely and -la takes its place. But -la has a cost: because it is a suffix in a
language with penultimate stress, it produces a stress alternation. The allomorph -il- avoids this
alternation, and that option is taken when -il- can get close enough to its preferred locus so that it
violates only the low-ranking ALIGN-BY-SEG constraints.
An alternative analysis using gradient alignment is not possible. The problem, as in Tagalog,
is that ALIGN(-il-, Wd, L) and ALIGN(-il-, Hd, L) cannot both dominate and be dominated by OOIDENT(stress). The alignment effect in Nakanai is categorical: some discrepancy is permitted, but
misalignment by a syllable or more is not.
Nakanai, like Tagalog, shows that categorical, quantized alignment constraints are necessary.
Gradient alignment cannot account for either of these languages because gradient alignment cannot
be used to permit modest misalignments while forbidding more severe ones. Gradient alignment
offers much finer distinctions than categorical alignment, but this fineness comes from evaluation
by a single constraint. The separate ranking required in Nakanai and Tagalog, then, is not an option
with gradient alignment.
16
This argument closely parallels the treatment of HNUC in Prince and Smolensky (1993:
139ff.). The original statement of HNUC is a single gradient constraint that judges nuclei by how far
down they are on the sonority hierarchy. The problem with gradient HNUC is that it cannot account
for syllable-structure typology: languages differ systematically in their tolerance for low-sonority
nuclei. For example, English allows any sonorant to be a nucleus, but Spanish allows only vocoids
as nuclei, and Berber allows literally any segment to serve as a nucleus. Gradient HNUC can express
preferences for certain nuclei, but it cannot be used to set a threshold, just as gradient ALIGN cannot
set a threshold for displacement of an affix. Both HNUC and, as I have argued here, gradient ALIGN
must be replaced by categorical constraints whose ranking with respect to other constraints is the
source of typological differences.
4. Alignment and Stress
Another important application of gradient alignment is in the analysis of stress systems.
Gradient alignment does seemingly crucial work in five main situations:
•As shown in (1), gradient alignment reproduces the effect of rule-based directional foot parsing. If PARSE-SYLL dominates ALIGN(Ft, Wd, R), and if no other foot-alignment
constraints are high-ranking, then syllables will be parsed into binary feet from right to left.
•Gradient ALIGN(Ft, Wd) is also used to obtain non-iterative footing. Example (1e) shows
that ALIGN(Ft, Wd, R) can be satisfied best by placing a single foot at the right edge, and this
will be optimal if PARSE-SYLL is low-ranked. In reduplicative emergence of the unmarked,
gradient ALIGN(Ft, Wd) is used to force reduplicants to contain no more than a single foot
(McCarthy and Prince 1994).
•The locus of main stress in both rhythmic and prominence-driven stress systems is
determined by gradient evaluation of ALIGN(Hd, Wd). Constraints of this type pick out, say,
the rightmost foot or the rightmost heavy syllable as the strongest in the word.
•In languages like Latin or Macedonian that deploy a single foot at the right edge modulo
final-syllable extrametricality, the constraint ALIGN(Wd, Ft, R) must be evaluated gradiently
under domination by NON-FINALITY(Ft).
•In languages with “foot extrametricality”, main stress is forced onto the penultimate foot by
gradient ALIGN(Hd, Wd, R).
I may have missed one or two sporadic applications of gradient alignment in stress, but these are the
main effects to investigate. The following subsections address each of these bulleted points in turn.
The metrical alignment constraints I assume in the analyses below are essentially the same
as those in McCarthy and Prince (1993a), but with gradience eliminated and quantized violation
added, as proposed above in §2. Specifically, the following constraints, in their left and right
versions, are assumed:
17
(20) Quantized, Categorical Metrical Alignment Constraints
a. ALIGN(Ft, Wd, Edge) — every foot is aligned at the edge of some word.
ALIGN-BY-σ(Ft, Wd, Edge)
ALIGN-BY-FT(Ft, Wd, Edge)
b. Align(Wd, Ft, Edge) — every word has a foot at its edge
ALIGN-BY-σ(Wd, Ft, Edge)
[ALIGN-BY-FT(Wd, Ft, Edge)]
c. Align(Hd, Wd, Edge) — the head foot is at the edge of the word.
ALIGN-BY-σ(Hd, Wd, Edge)
ALIGN-BY-FT(Hd, Wd, Edge)
ALIGN-BY-SEG and ALIGN-BY-µ metrical constraints are in principle possible, but usually their
violations will be the same as those assigned by ALIGN-BY-σ. They may turn out to be useful,
however, in situations involving consonant or mora extrametricality. The constraint ALIGN-BYFT(Wd, Ft, Edge) is bracketed because it has no practical value, since every candidate that contains
at least one foot vacuously satisfies it.
4.1 Directional Foot-Parsing
As shown in §1, gradient alignment constraints of the ALIGN(Ft, Wd) variety simulate the
effects of a directionally iterative rule of foot-parsing in rule-based metrical phonology (Hayes 1980,
1995, Prince 1983). Recently, though, Kager (2001) has argued that constraints of this type yield an
overly rich typology. Through permuted ranking ALIGN(Ft, Wd) allows free choice of parsing
direction independent of foot type. This is not correct: iambic systems are probably never right-toleft (Kager 1993, McCarthy and Prince 1993b). Furthermore, Kager claims that bidirectional
systems, with a single foot at one end and iteration from the other end, never iterate from the main
stress toward the secondary. That is, while there are languages like Polish, which parses
heptasyllables like [(1σσ)(1σσ)σ(0σσ)], there are no solid cases of languages with the parse
[(1σσ)σ(1σσ)(0σσ)] (though see fn. 15). Free permutation of gradient ALIGN(Ft, Wd) predicts that
these non-existent patterns should occur.
Kager proposes to eliminate the ALIGN(Ft, Wd) constraints entirely — a fortiori eliminating
a prime example of gradient evaluation. In their place, he substitutes an enriched theory of the
constraint *LAPSE (Elenbaas and Kager 1999, Green and Kenstowicz 1995, Hung 1994, Nespor and
Vogel 1989, Selkirk 1984). A lapse is a sequence of unstressed syllables. Though lapses are marked
in general, Kager proposes that they are less marked in two positions, word-finally and adjacent to
the main-stressed syllable, and more marked in another position, word-initially. The responsible
constraints are these:
18
(21) Lapse Constraints in Kager (2001) (definitions paraphrased)
a. *LAPSE
Assign one violation-mark for pair of adjacent unstressed syllables σ
-σ
-.
b. LAPSE-AT-END
Assign one violation-mark for each pair of adjacent unstressed syllables that is not
word-final (i.e., not ...σ
-σ
- ]).
c. LAPSE-AT-PEAK
Assign one violation-mark for each pair of adjacent unstressed syllables that is not
adjacent to the main-stressed syllable (i.e., not ...0σσ
-σ
- ... or ...σ
-σ
- 0σ...).
d. *INITIAL-LAPSE
Assign one violation-mark for each pair of adjacent unstressed syllables that is wordinitial (i.e., [σ
-σ
- ...).
The constraint *LAPSE militates against stress lapses generally, but the constraints LAPSE-AT-END
and LAPSE-AT-PEAK can license lapses in those specific environments. Conversely, *INITIAL-LAPSE
disfavors lapses word-initially. Certain directionality effects can be obtained from the interaction of
these categorical constraints, with a better fit to observation than the gradient ALIGN(Ft, Wd)
constraints. In fact, the only alignment constraints required are ALIGN(Wd, Ft, L) and ALIGN(Wd,
Ft, R), which demand a single foot initially and finally.
Table 1 shows how the effects of directionality are obtained. This table is an unranked
tableau. It shows how candidates fare on the various constraints without considering the constraints’
ranking. This organization allows quick inferences about which candidates are harmonically
bounded. (A candidate is harmonically bounded if it incurs a proper subset of a competitor’s marks.)
The table uses 7-syllable words, since directionality is usually visible only in words containing an
odd number of syllables or moras, and 7 syllables are need to show the full range of observed
patterns. Exhaustive parsing is assumed up to degeneracy — that is, FT-BIN dominates PARSE-SYLL.
(On non-exhaustive parsing, see §4.2.) The candidates are grouped according to the position of main
stress (indicated by the numeral 1) and whether feet are iambic or trochaic. Each group, such as
(a–d), contains a set of candidates that compete with one another, holding main-stress location and
foot-type constant. In effect, the candidates in each group compete in directionality only.
[Put Table 1. about here.]
The shaded rows in table 1 contain candidates that are harmonically bounded by other
candidates in the same group. For example, (b) is harmonically bounded by (c), because (b) has a
proper superset of (c)’s violation-marks. Rows that are not greyed-out are predicted to be possible
in this theory under some ranking(s) of the given constraints.
The constraints included in this table are those in (21) plus the quantized, categorical
constraints ALIGN-BY-σ(Wd, Ft, Edge) and ALIGN-BY-σ(Ft, Wd, Edge). This is a departure from
Kager in two respects: the alignment constraints are quantized, though his are not; and Kager
assumes that there are no ALIGN(Ft, Wd) constraints. I include categorical ALIGN(Ft, Wd) in the
table to show that its presence does not alter the results. (Comparison of the last four columns of the
table shows that, when all candidates have the same number of feet, categorical ALIGN-BY-σ(Wd,
Ft) and ALIGN-BY-σ(Ft, Wd) track one another exactly.)
19
Kager shows that all of the non-harmonically-bounded trochaic systems are attested: (a) is
Pintupi, (c) is Garawa, (d) is Wargamay, (e) is Warao, (g) is Piro, and (h) is Cairene Arabic
(substituting moras for syllables). The harmonically bounded pattern in (b) has not been reported;
there are a few reports of (f) in the literature, but all are subject to other interpretations.15 If indeed
(b) and (f) are impossible, then this must count as evidence against gradient ALIGN(Ft, Wd) and in
favor of the categorical constraints operating in Table 1. The problem with gradient alignment is that
it easily produces unattested (b) and its symmetric counterpart (f) — for instance, the ranking to get
(b) with gradient constraints is ALIGN(Wd, Ft, L) >> ALIGN(Wd, Ft, R) >> ALIGN(Ft, Wd, L).
Attestation of the non-harmonically-bounded iambic systems is less complete. Pattern (i) is
Araucanian and (p) is Creek (again, moraic rather than syllabic). Neither (k) nor (o) has been
observed. Still, this is real progress over the model based on gradient ALIGN(Ft, Wd). It predicts not
only (k) and (o), but also the remaining patterns (j), (l), (m), and (n), all of which are unattested.
Kager also examines stress systems that allow degenerate feet and therefore have no lapses.
In Murinbata, for example, 7-syllable words have the trochaic stress pattern (10)(20)(20)(2), while
in Weri they have the iambic pattern (2)(02)(02)(01). This too looks at first glance like a
directionality effect. With gradient alignment, one would say that ALIGN(Ft, Wd, L) is active in
Murinbata, as the following tableau shows:
(22) Gradient ALIGN(Ft, Wd, L) >> ALIGN(Ft, Wd, R) in Murinbata
ALIGN(Ft, Wd, L)
ALIGN(Ft, Wd, R)
9 *’s
12 *’s
a.
L (2)(02)(02)(01)
b.
(02)(2)(02)(01)
10 *’s !
11 *’s
c.
(02)(02)(2)(01)
11 *’s !
10 *’s
d.
(02)(02)(01)(2)
12 *’s !
9 *’s
This tableau reveals a curious property of gradient ALIGN(Ft, Wd) that was first noted by Crowhurst
and Hewitt (1995). In systems that disallow degenerate feet, the effect of right-to-left footing is
obtained from high-ranking ALIGN(Ft, Wd, R), as shown in (1). But in systems that permit
degenerate feet, right-to-left footing requires high-ranking ALIGN(Ft, Wd, L). The directional sense
of gradient alignment is oddly reversed, depending on which of FT-BIN or PARSE-SYLL is ranked
higher.
Apart from this formal peculiarity of gradient alignment, there is a typological problem as
well. The left-to-right stress pattern in (22d) is attainable by simply permuting the gradient
constraints, yet it does not seem to exist. This typological skew leads to Kager’s further proposal that
constraints on clash, rather than gradient alignment, are responsible for apparent directionality effects
in stress systems with degenerate feet. Of all the candidates in (22), only (22a) avoids clash
completely. It therefore satisfies *CLASH better than its competitors. This constraint and its allies are
15
The strongest apparent counterexample is Indonesian (Cohn 1989, Cohn and McCarthy 1994/1998). Kager
observes that the crucial forms are Dutch loans and may simply be reproducing stress from the original language.
20
sufficient, Kager argues, to account for the full range of observed directionality effects in these
systems.
It seems clear that a theory of directional foot-parsing based on the distribution of lapses and
clashes is superior on typological grounds to the gradient ALIGN(Ft, Wd) constraints. These
typological results for stress systems in Kager’s work converge with the results developed here on
other grounds, such as abstract considerations of the nature of multiple violation in OT and the
properties of infixation. Convergence of results coming from such different directions is a bit
unusual and is perhaps confirmation that the approach advocated here is on the right track.
4.2 Non-iterative Footing
Gradient ALIGN(Ft, Wd) has also been used, when ranked above PARSE-SYLL (or *LAPSE),
to obtain non-iterative footing. As shown in (1e), ALIGN(Ft, Wd, R) is perfectly satisfied only in a
candidate with a single foot at the right edge. The candidates with fuller footing (1a–d) all violate
ALIGN(Ft, Wd, R), since it is impossible to have more than one foot properly aligned at the right
edge. Similarly, ALIGN(Ft, Wd, L) and PARSE-SYLL together determine the shape of the reduplicant
in languages with minimal-word reduplication (McCarthy and Prince 1994, 1999). By dominating
the base-reduplicant faithfulness constraint MAX-BR, gradient ALIGN(Ft, Wd, L) and PARSE-SYLL
rule out all candidates where the reduplicant is bigger than a single foot.
In non-iterative footing and minimal-word reduplication, gradient ALIGN(Ft, Wd) is being
used as a way to penalize all feet except one. But gradience is a rather big club to brandish at the
small problem of limiting words to a single foot. The quantized, categorical constraints ALIGN-BYFt(Ft, Wd, R) and ALIGN-BY-Ft(Ft, Wd, L) are also impossible to satisfy in forms containing more
than one foot. Any foot that has another foot separating it from the word-edge incurs a single
violation-mark from each of these constraints. The following tableau shows how non-iterative
footing is obtained in a language with penult stress:
(23) Non-Iterative Footing from ALIGN-BY-Ft(Ft, Wd, R) >> PARSE-SYLL
ALIGN-BY-Ft
(Ft, Wd, R)
PARSE-SYLL
a.
L [σσσσσ (σ́σ)]
*****
b.
[σσσ (σ́σ) (σ́σ)]
*
***
c.
[σ (σ́σ) (σ́σ) (σ́σ)]
**
*
Candidates (23b, c) receive one mark for each unaligned foot. Observe that there is no gradience here
— candidate (23c) receives two alignment marks because it has two unaligned feet, in conformity
with the Locus hypothesis.
4.3 Main Stress
Gradient alignment has been used as a way of assigning main stress to the rightmost or
leftmost foot. In rhythmic stress systems like those exemplified in table 1, main stress usually falls
on the leftmost or rightmost foot, which sometimes does not fall in absolute word-initial or word-
21
final position. In prominence-driven stress systems, main stress falls on the leftmost or rightmost
heavy syllable, with no limit on how far it can be displaced from the word edge. Standardly, minimal
violation of gradient ALIGN(Hd, Wd, Edge) is the source of these leftmostness and rightmostness
effects (McCarthy and Prince 1993b, Prince and Smolensky 1993).
Quantized, categorical alignment constraints can produce the same effects. The constraints
ALIGN-BY-FT(Hd, Wd, Left) and ALIGN-BY-FT(Hd, Wd, Right) are violated if some other foot stands
between the head foot of the prosodic word and the left or right word-edge. For example, ALIGN-BYFT(Hd, Wd, Left) correctly favors [(10)(20)(20)0] over alternative placements of the main stress:
(24) ALIGN-BY-FT(Hd, Wd, Left) >> ALIGN-BY-FT(Hd, Wd, Right)
ALIGN-BY-FT
(Hd, Wd, Left)
a.
L [(10)(20)(20)0]
b.
[(20)(10)(20)0]
c.
[(20)(20)(10)0]
ALIGN-BY-FT
(Hd, Wd, Right)
*
*
*
*
Reversing the ranking correctly favors the last candidate, [(20)(20)(10)0]. The middle candidate,
[(20)(10)(20)0], is harmonically bounded, as it should be. (On foot extrametricality, see §4.5.)
These results also extend to prominence-driven stress systems, once the proper role of footstructure is recognized. Bakovic (1998) has shown that the problematic default-to-opposite/defaultto-same distinction in prominence-driven stress can be obtained from gradient ALIGN(Ft, Wd)
interacting with other constraints. Here, I will argue that the main elements of Bakovic’s carry over
to the theory proposed here, with gradient alignment replaced by categorical, quantized constraints.
Bakovic’s main idea is that prominence-driven stress systems, which had been attributed to
unbounded feet in the past (Halle and Vergnaud 1978, Hayes 1980), actually involve binary feet.
Unlike rhythmic stress systems, though, feet are rather sparse in prominence-driven stress: they parse
all the heavy syllables and a pair of light syllables at the default edge. As in rhythmic stress systems,
the first or last foot is singled out for main stress. Schematically, the stress patterns are like these:
(25) Prominence-driven Stress
a. Default to opposite: Rightmost heavy, else leftmost.
(Ll)(H)l(H)ll
(Ll)ll
b. Default to same: Leftmost heavy, else leftmost
ll(H)l(H)ll
(Ll)ll
22
The letters h and l stand for heavy and light syllables, respectively. Capitalization marks foot heads,
and underlining shows the position of main stress. In the default-to-opposite system, the last foot in
the word takes the main stress; in the default-to-same system, the first foot takes the main stress.16
As in rhythmic stress systems, the location of main stress is determined by the constraint
ALIGN-BY-FT(Hd, Wd, Edge). In the rightmost-else-leftmost system (25a), the head foot is never
separated by any other feet from the right word-edge, so ALIGN-BY-FT(Hd, Wd, R) is undominated
(see (26)). In the leftmost-else-leftmost system (25b), the head foot is never separated by any other
feet from the left word-edge, so ALIGN-BY-FT(Hd, Wd, L) is undominated (see (27)). Since, by
assumption, every heavy syllable projects a foot, the constraint WSP (“if heavy, then a foot-head”)
is also undominated in systems of both types. And since the default-to-opposite system (25a) has an
initial foot in all words, ALIGN-BY-σ(Wd, Ft, L) must also be undominated in this system.
The tableau in (26) shows all of the constraints that are visibly active over a wide range of
candidates in the default-to-opposite system.
(26) Default-to-Opposite Stress
ALIGN-BY-Ft
(Hd, Wd, R)
a.
L (Ll)(H)l(H)ll
b.
(Ll)(H)l(H)ll
c.
(Ll)(H)l(H)(Ll)
d.
(Ll)(H)l(H)ll
e.
(Ll)hlhll
f.
ll(H)l(H)ll
g.
ll(H)l(H)ll
h.
L (Ll)ll
i.
(Ll)(Ll)
j.
ll(Ll)
ALIGN-BY-σ
(Wd, Ft, L)
WSP
*!
ALIGN-BY-σ
(Hd, Wd, L)
ALIGN-BY-σ
(Ft, Wd, L)
*
**
*
**
*
***!
*!
**
**!
*!
*!
*
**
*!
*
**
*!
*!
*
*
*!
For reasons of space, it is not possible to show all the constraints, but in general it is safe to assume
that a constraint aligning something at one edge crucially dominates its mirror-image counterpart at
the other edge. The demand that the head foot be the rightmost foot in the word — that is, ALIGN-BYFT(Hd, Wd, R) — rules out many of the candidates. The only lively competitors that remain are
16
Bakovic’s analysis, which I am mostly adopting here, presupposes a difference in foot-structure between defaultto-opposite and default-to-same stress systems: in the former (25a) but not the latter (25b), a pair of light syllables at the
default edge is always footed. This would seem to predict a difference in the presence of secondary stress, but a caution is
in order. Reports and non-reports of secondary stress in prominence-driven stress systems sometimes lead to confident
pronouncements about what the facts are, but this confidence is unwarranted. It’s hard enough to hear secondary stress in
rhythmic systems, unless there is vowel reduction. Given the notorious difficulty of hearing even primary stress in
prominence-driven systems, no conclusions about theories should hinge on details of secondary stress. Furthermore, it seems
absurd to conclude that a language has no secondary stress just because none is mentioned in a published description.
23
those like (26b, g, e, i, j) that have different foot-parsings than the intended output. Of these, (26g)
and (26j) are excluded because they do not have a foot at the left word-edge, a violation of
undominated ALIGN-BY-σ(Wd, Ft, L), while (26e) fatally violates undominated WSP. The remaining
failed candidates, (26c) and (26i), posit an additional foot to the right of the intended main stress.
In (26i), low-ranking ALIGN-BY-σ(Hd, Wd, L) emerges to force placement of the main stress on a
word-initial foot — provided that no foot follows it. And in (26c), ALIGN-BY-σ(Ft, Wd, L) (or its
ALIGN-BY-FT counterpart) has the same foot-economizing effect shown in the immediately preceding
section. The problem with (26c) is that its final foot is not demanded by any of the top-ranked
constraints. Since this foot is not aligned with the left edge, ALIGN-BY-σ(Ft, Wd, L) penalizes it.
The next tableau shows the constraints active over the same candidate set in the default-tosame system (25b).
(27) Default-to-Same Stress
ALIGN-BY-FT
(Hd, Wd, L)
WSP
ALIGN-BY-Ft
(Ft, Wd, L)
a.
(Ll)(H)l(H)ll
*!
**
b.
(Ll)(H)l(H)ll
*!
**
c.
(Ll)(H)l(H)(Ll)
*!
***
d.
(Ll)(H)l(H)ll
e.
(Ll)hlhll
f.
L ll(H)l(H)ll
g.
ll(H)l(H)ll
h.
L (Ll)ll
i.
(Ll)(Ll)
j.
ll(Ll)
ALIGN-BY-σ
(Wd, Ft, L)
**!
**!
*
*
*!
*
*
*!
*
*!
The candidates where the head foot is not the leftmost foot (27a, b, c, g, i) are eliminated by ALIGNWd, R). As in the previous tableau, WSP rules out candidate (27e), because of its
unfooted heavy syllables. WSP crucially dominates ALIGN-BY-FT(Ft, Wd, L), which again has a
foot-economizing effect — the only feet that are occur are those that are demanded by WSP or by
the immutable requirement that every word contain at least one foot (i.e., culminativity or
headedness (Hayes 1995, McCarthy and Prince 1993a, Selkirk 1980)). And ALIGN-BY-FT(Ft, Wd,
L) dominates ALIGN-BY-σ(Wd, Ft, L): the requirement that every word start with a foot is satisfied
only when there are no other feet in the word.
BY-FT(Hd,
These two tableaux show that the main elements of Bakovic’s analysis of prominence-driven
stress can be obtained without gradient constraints. The linchpin of the reanalysis is the constraint
ALIGN-BY-FT(Hd, Wd, Edge). This constraint is indifferent to the distance between the head foot and
the word-edge, as long as no other feet intervene. It therefore selects the leftmost/rightmost foot as
the optimal location for main stress, but it does so with categorical rather than gradient evaluation.
24
The key to eliminating gradience, in this case as in others, is to quantize alignment violations as
proposed in §2.
4.4 Syllable Extrametricality
The combination of non-iterative footing and final extrametricality supplies another situation
where gradient ALIGN(Wd, Ft, R) or, equivalently, gradient ALIGN(Ft, Wd, R) has been invoked. If
a language has the stress pattern [σσσ(0σσ)σ] or its moraic equivalent, the standard approach has been
to posit the ranking ƒNON-FINALITY(FT) >> ALIGN(Ft, Wd, R) >> PARSE-SYLL„ — the final syllable
is unfooted, and, since every word must contain at least one foot, the sole foot is aligned as far to the
right as possible.
The problem for the categorical theory is that the intended output [000(10)0] ties with its
competitors *[00(10)00] and *[(10)0000] on the various right-edge ALIGN-BY-σ constraints: ALIGNBY-σ(Wd, Ft, R), ALIGN-BY-σ(Ft, Wd, R), and ALIGN-BY-σ(Hd, Wd, R). All three candidates have
an unaligned head foot and no other feet, so all incur exactly one violation-mark from each of these
constraints. Nor are the *LAPSE constraints of §4.1 particularly useful in addressing this problem;
unsurprisingly, constraints that enforce rhythm are not going to be of much help in a fundamentally
arrhythmic, one-stress-per-word language.
This problem is a good deal less serious than it seems. The case that there are any languages
with a [000(10)0] stress pattern is not strong. The mora-counting version of this pattern is usually
ascribed to Latin, but the only reason for doing so is that grammar books never say anything about
Latin secondary stress. In fact, there is good evidence that Latin stress is actually iterative,
conforming to the right-to-left pattern shown in row (a) of Table 1: [(20)(10)0]. The second syllable
of words like pudi:citiam ‘chastity (acc. sg.)’ or vere:bamini ‘you (pl.) were afraid’, although
underlyingly long, is observed to scan as short: pudicitiam, verebamini. This makes sense if the pair
of pretonic syllables pudi and vere is formed into a single bimoraic foot, as argued by Mester (1994).
Latin stress is iterative, then, which means that Latin is not a sound example of the problematic
[000(10)0] pattern. Of course, the many languages that have inherited or borrowed the Latin stress
system — Spanish, Italian, English, German, Dutch, and others — languages that can actually be
observed, all seem to have secondary stresses, so they too must be iterative.
If indeed the [000(10)0] stress pattern is a spurious result of overinterpreting the lack of a
reported secondary stress (also see fn. 16), then it poses no problem for categorical alignment.
Moreover, since Kager’s argument, summarized in §4.1, makes a pretty good case that NONFINALITY(FT) is not necessary for iterative stress systems, the need for NON-FINALITY(FT) needs to
be re-examined. This constraint may simply be superfluous, with its genuine effects being obtained
by judicious ranking of LAPSE-AT-END, as in row (a) of table 1.
4.5 Foot Extrametricality
Hayes (1995) introduces and makes heavy use of foot extrametricality. The idea is that a
word-final foot is skipped over, so main stress is assigned to the penultimate foot. An example is a
left-to-right iambic stress system with main stress at the right, but not on a word-final syllable:
compare odd-parity [(02)(01)0] with even-parity [(02)(01)+(02),]
In a theory with gradient alignment constraints, this pattern can be obtained with the ranking
ƒALIGN(Ft, Wd, L), NON-FINALITY(Hd) >> ALIGN(Hd, Wd, R)„ — i.e., the head is as far to the right
25
as possible, given left-to-right foot-parsing and the antagonism between main stress and word-final
position. Without categorical alignment, there’s an obvious problem: the candidates [(02)(01)(02)]
and *[(01)(02)(02)] are tied in their performance on ALIGN-BY-FT(Hd, Wd, R). The unresolved tie
then devolves onto low-ranking ALIGN-BY-FT(Hd, Wd, L), which wrongly favors main stress on the
first foot. Hayes (1995) cites three examples of this type: Negev Bedouin Arabic, Delaware, and
Cayuga. All have left-to-right iambic feet with the final foot extrametrical, yielding results like
Cayuga (tewa)(katá)(weR nye§) ‘I’m moving about’. Categorical alignment cannot produce this result.
Since this analysis cannot be reconstructed with categorical alignment, it’s necessary to
consider alternatives. Suppose, instead of [(02)(01)(02)], that the actual winning candidate is
[(02)(01)00]. It fully satisfies ALIGN-BY-FT(Hd, Wd, R), handily beating *[(01)(02)00]. It is,
moreover, readily attainable if NON-FINALITY(Hd) dominates *LAPSE. The constraint LAPSE-AT-END
is also crucial.
(28) Iambic “Foot Extrametricality” Pattern
ALIGN-BY-FT
(Hd, Wd, R)
a.
L [(02)(01)00]
b.
[(01)(02)00]
c.
[(02)(02)(01)]
d.
[(02)0(01)0]
e.
[(02)(01)(02)]
NON-FINALITY(Hd)
LAPSE-AT-END
*LAPSE
*
*!
*
*!
*!
*
*!
Together, ALIGN-BY-FT and NON-FINALITY(Hd) force there to be a stress lapse. With LAPSE-AT-END
anywhere in the hierarchy, this lapse must fall on the last two syllables. Since the last two syllables
are adjacent to both the peak and the end, there is no better place for the forced lapse to go.
Hayes also describes a trochaic version of this analysis for Palestinian Arabic and Hindi. The
idea is that preantepenultimate stress can be obtained with trochaic feet and foot extrametricality,
as in [(Ll)+(Ll),]. The sort of analysis given in (28) won’t carry over to this trochaic case:
*[(20)0(10)0] is actually better, lapse-wise, than [(20)(10)00].17 Fortunately, neither of these
examples is probative. Arabic word- and syllable structure is such that the crucial penta- or
hexasyllables examples never occur, nor are any cited for Hindi. In any case, there are serious
problems establishing what the Hindi facts really are (Hayes 1995, Ohala 1977).18
The other main application of foot extrametricality in Hayes (1995) is the “extrametricality
in clash” phenomenon. A trochaic language with foot extrametricality in clash is Manam: main stress
17
To be specific, *[(20)0(10)0] violates *LAPSE once, but [(20)(10)00] violates it twice, because there are two
overlapping “00" sequences, and it also violates LAPSE-AT-END, because of the first “00" sequence.
18
Hayes (1995: 131) cites “Egyptian Radio Arabic” as a case of optional foot extrametricality. This is to account
for the variation between mùškíla and múškila ‘problem’ or between kàtabáhu and kátabàhu ‘he wrote it’. Egyptian Radio
Arabic is, however, not a natural language; it is the result of radio announcers pronouncing Classical Arabic words under the
influence of diverse stress systems. This is particularly apparent from the mùškíla/múškila variation: the first pronunciation
is typical of Cairo, and the second is typical of the rest of the Arab world. Grammatical analysis of this variation does not
seem appropriate.
26
falls on the antepenult only in words of the form [(H) +(Ll),]. The idea is that the final foot is
extrametrical just in case there is a clash between antepenult and penult. There is also an iambic
version of extrametricality in clash, which Hayes finds in several Austronesian languages. In
Javanese and Malay, stress falls on the penult unless it contains schwa, in which case stress falls on
the ultima. Under the assumption that all and only syllables with c are light, extrametricality in clash
is needed to get penult stress when neither of the last two syllables contains c: [(H) +(H),] vs.
*[(H)(H)].
Foot extrametricality in clash requires a considerable enrichment of the underlying theory of
extrametricality in both rule-based phonology and OT. The problem is that the rules or constraints
responsible for extrametricality are usually context-free — the only context they require is the word
boundary, and that comes for free from the Peripherality Condition (Harris 1983). Assignment of
extrametricality only in clash is a big leap in expressive power, and this ought to encourage
skepticism. In fact, alternative analyses exist.
Manam has been reanalyzed without foot extrametricality by Buckley (1998). Analogous to
(28), the idea is that the final (LL) foot is absent completely, rather than extrametrical; its absence
is forced by *CLASH. As for the Austronesian cases, they are surely spurious. Closely related
Indonesian has virtually the same stress pattern, but is clearly trochaic (Cohn 1989, Cohn and
McCarthy 1994/1998). Cohn and McCarthy show that the final stress in words like kccíl ‘small’ is
not evidence that the basic stress pattern is iambic, but rather that FT-FORM(TROCHAIC) is crucially
dominated by a well-motivated constraint against stressed c. Penult stress in words like bicára
‘speak’ is not evidence of foot extrametricality, but rather of the normal trochaic foot structure of
the language: [bi(cára)].
In summary, there are no solid examples of foot extrametricality, where main stress is
assigned to the penultimate foot. To be specific, there is no evidence of stress being assigned to the
penult foot in preference to another foot further to the left. Were such evidence to be forthcoming,
it would surely challenge the theory proposed here, because stress on a non-peripheral foot is
harmonically bounded.
(29) [F F F] Harmonically Bounded With Categorical Alignment
ALIGN-BY-FT
(Hd, Wd, R)
a.
L [F F F]
b.
L [F F F]
c.
[F F F]
ALIGN-BY-FT
(Hd, Wd, L)
NON-FINALITY(Hd)
*
*
*
*
*
Form (29a) harmonically bounds (29c) — with categorical alignment, only the leftmost and
rightmost feet are possible hosts of main stress. This result, which appears to be empirically correct,
is quite emphatically not true of gradient alignment.
27
(30) [F F F] Not Harmonically Bounded With Gradient Alignment
ALIGN-BY-FT
(Hd, Wd, R)
a.
L [F F F]
b.
L [F F F]
c.
[F F F]
ALIGN-BY-FT
(Hd, Wd, L)
NON-FINALITY(Hd)
**
*
**
*
*
Under the ranking ƒNON-FINALITY(Hd) >> ALIGN-BY-FT(Hd, Wd, R) >> ALIGN-BY-FT(Hd, Wd, L)„,
gradient alignment can yield the non-occurring pattern [F F F] — it is not harmonically bounded.
5. Alignment and Autosegmental Spreading
In Kirchner (1993) and much subsequent work, gradient alignment constraints are used to
obtain the effect of autosegmental spreading. For example, in Turkish the feature [round] spreads
rightward until it reaches the end of the word (dost-un-uz ‘your friend’) or encounters a non-high
vowel (dost-un-uz-dan ‘from your friend’). In the latter case, gradient ALIGN([round], Wd, R) is
minimally violated under domination by *[+rnd, –hi] (which is itself dominated by a positional
faithfulness constraint that is protective of the initial syllable).
(31) Gradient ALIGN([round], Wd, R) in Turkish
/dost-In-Iz-dAn/
*[+rnd, –hi]
ALIGN([round],
Wd, R)
*
4 *’s
a.
L dostunuzdan
b.
dostunuzdon
**!
*
c.
dost‚n‚zdan
*
9 *’s
d.
dostun‚zdan
*
6 *’s
The featural alignment constraint must be gradient in order to distinguish (31a) from (31c, d) — all
are imperfectly aligned, but (31a) is the best of this bad lot. (In assigning violation-marks, I’ve
assumed that all segments are counted by gradient ALIGN. The results are similar if only vowels are
counted.)
Tone systems present similar phenomena. For example, in the southeastern dialects of Shona,
high tone spreads rightward up to the penult, falling short of the final syllable, which is presumed
to be extrametrical (Myers 1997: 157ff.): /ku-mú-ereng-er-a/ 6 kumúéréngéra ‘to read to him/her’.
Gradient ALIGN(H, Wd, R) is better satisfied by kumúéréngéra than by alternatives like
*kumúéréngera. The same thing can be observed in cases where the OCP blocks H from spreading
onto a syllable immediately followed by another H: /á-ri ku-téng-a/ 6 árí kuténga, *árí kúténga ‘he
is buying’ (cf. /á-ri ku-gar-a/ 6 áríkúgára ‘he is staying’).
The attractiveness of gradient alignment lies in its similarity to the familiar: it faithfully
reproduces the effects of directional spreading rules in rule-based autosegmental phonology. This
familiarity is not actually a virtue, however; we need to ask whether there are non-gradient
28
approaches to the same phenomena, even if they involve a more radical departure from spreading
rules.19
The alternatives exploit the Locus hypothesis in lieu of gradience. In essence, they say that
dostunuzdan is more harmonic than *dostun‚zdan because the latter has more unassimilated vowels,
not because the feature [+round] is further from the right word-edge. One approach, due to Myers
(1997: 861-3), sees high-tone spreading as motivated by the constraint SPECIFY(T), defined as “a
syllable must be associated with a tone”. This constraint presupposes that Shona opposes high-toned
syllables with toneless ones; the requirement that every syllable have a tone is, in effect, a
requirement that H spread, particularly since SPECIFY(T) crucially dominates DEP(H) (so tones
cannot be filled in by epenthesizing H). Since SPECIFY(T) is violated once by each toneless syllable,
it charges multiple violations under the locus hypothesis, yielding spreading without gradience.
Another, more radical approach says simply that every segment must be associated with the
spreading feature or tone (Padgett 1995b).
(32) SPREAD(F)
If any segment is associated with F, then every segment is associated with F. Assign one
violation-mark for each segment that is not associated with F.
SPREAD is like Myers’s SPECIFY, but without the representational assumptions. In fact, SPREAD is
the most stringent pro-spreading constraint imaginable: it settles for nothing less than total hegemony
over all segments in the candidate.20
SPREAD(round) makes the right decisions in (31). It correctly distinguishes (31a) from (31c,
d), assigning fewer violation-marks to (31a), and it does so simply by adding up the loci of violation,
without gradient evaluation. In Shona, SPREAD(H) correctly favors kumúéréngéra over
*kumúéréngera, since the former has fewer loci of violation than the latter.
SPREAD is not a complete theory of autosegmental spreading, but then neither is ALIGN.
Greater satisfaction of SPREAD can in principle be achieved by non-local agreement in the
harmonizing feature, so e.g. *kumúerengéra is more harmonic, SPREAD-wise, than faithful
kumúerengera. And perfect satisfaction of ALIGN can be obtained by flop, delinking the feature or
tone and reassociating it at the designated edge, as in *kumuerengerá. So the SPREAD and ALIGN
models each require some auxiliary apparatus to achieve real spreading effects. In the case of
SPREAD, what’s needed is the assumption that feature spreading is strictly local (Gafos 1996, 1998,
McCarthy 1994, Ní Chiosáin and Padgett 2001, Walker 1998 and others). In the case of ALIGN,
what’s needed is some additional constraint preventing the feature or tone from deassociating from
its original segmental sponsor.
19
Another alternative to gradient ALIGN is AGREE (Bakovic 2000, Lombardi 1999, 2001). The constraint
AGREE(round), defined as “adjacent vowels must have identical values for [round]”, cannot make the crucial distinction
between (31a) and (31c, d). The problem is that all three candidates violated this constraint once.
20
SPREAD might seem to echo Kaun’s (1995: 98) EXTEND(round), defined as “the autosegment [+round] must be
associated to all available vocalic positions within a word”. Kaun never explains what she means by “available”, and her
actual practice in tableaux seems inconsistent with her definition: one violation-mark is assigned for each instance of [+round]
that has not spread. This is obviously very much unlike (32).
29
In fact, the only significant difference between categorical SPREAD and gradient ALIGN is that
the latter, but not the former, incorporates a directional element into its formulation. SPREAD is
directionless, so any extension of a feature’s domain is an improvement. But ALIGN constraints are
inherently directional, so they are only satisfied by spreading toward the designated edge. If SPREAD
is to supplant ALIGN as a theory of vowel harmony and tone spreading, then something must be said
about how directionality effects are obtained.
Directionality of spreading is a controversial matter, quite independently of the gradience
issue that is my focus here. Bakovic (2000), Beckman (1997, 1998), Kaun (1995), and Lombardi
(1999), among others, have argued on the basis of harmony processes that directionality effects are
illusory. In their view, the direction of spreading is not stipulated in some rule or constraint; rather,
it follows from the privileged status of the segment initiating the spreading, which is typically initial,
in a root or stem, or both. What looks like directional spreading is then a side effect of positional
licensing (Beckman, Kaun) or cyclic derivation (Bakovic).21
The case for directional spreading of tone seems more compelling, but still, questions arise.
For example, the Bantu languages seem to have a predilection for rightward high-tone spreading
(see, e.g., Bickmore 1996, Myers 1986). This could be arbitrarily stipulated in an analysis with
gradient ALIGN(H, Wd, R), but one wonders whether the direction of speading is a descriptive
artifact of the typical Bantu penult stress, which tends to attract tone, or of prefixing inflection. And
even if direction truly is arbitrary, with no connection to other facts about the language, it can be
obtained without recourse to gradient alignment. For example, Myers (1997: 868) introduces a
faithfulness constraint, ANCHOR-L, that is defined as follows:
(33) ANCHOR-L
Assign a violation if and only if:
(a) there is an output syllable SN that has an input correspondent S,
(b) both S and SN bear tone, and
(c) either S or SN is the leftmost syllable associated with its tone, and its
correspondent syllable is not the leftmost syllable associated with its tone.
This constraint is violated whenever tone spreads leftward. Ranked above SPREAD(H), it will block
spreading to the left while allowing spreading to proceed unimpeded to the right. As desired, no
gradient, directional alignment constraint is needed.
Needless to say, this brief review of alignment in harmony and tone spreading is not
sufficient to do justice to these broad and highly productive areas of research. Nonetheless, it has
been possible to show some plausible alternatives to gradient alignment as the source of
autosegmental spreading. The thesis of this paper — that gradient constraint evaluation does not
exist — finds support from these considerations.
6. Conclusion
In this article, I have argued against the gradience hypothesis, the premise that some OT
constraints can assign multiple violation-marks to a single instance of a marked structure. Two main
types of gradience were identified, bounded and unbounded. Bounded gradience is met with
21
In fact, dominant/recessive harmony is clearly non-directional. Spreading is initiated by specific feature values,
rather than initial syllables, roots, or stems. See Bakovic (2000) and Bakovic and Wilson (2000) for discussion and analysis.
30
sporadically in the OT literature, in certain constraints on hierarchies, scales, and classes. Bounded
gradience is unnecessary; any boundedly gradient constraint can be replaced by a set of categorical
constraints (§2). Unbounded gradience is in all likelihood limited to alignment constraints,
particularly constraints on affix placement, which have been important in analyzing infixation, stress,
and autosegmental spreading. I have argued that gradient alignment both undergenerates and
overgenerates: it cannot account for infixation phenomena in Tagalog and Nakanai and it yields a
stress typology that is too rich.
In place of gradient alignment, I proposed a set of categorical alignment constraints where
extent of violation is quantized by the prosodic hierarchy. For example, ALIGN-BY-FT(Cat1, Cat2,
Edge) is violated whenever a metrical foot stands between Edge of Cat1 and Edge of Cat2. These
categorical constraints appear to be sufficient to account for observed alignment effects in infixation,
foot-parsing, and main-stress assignment. The last bastion of gradient alignment, autosegmental
spreading, was also discussed, and it was shown that plausible alternatives exist there too.
The conclusion I draw, then, is that there are no gradient constraints. The Gradience and
Homogeneity hypotheses have no place in OT. No constraint can assign more than one violationmark for each locus of violation, and multiple violation-marks always indicate multiple loci of
violation. This improvement in the theory, I have argued throughout, is not merely aesthetic but also
empirical: gradience both over- and undergenerates when confronted with the full range of observed
phonological behavior.
Acknowledgements
For their comments, criticisms, and suggestions, I am grateful to all the participants in the UMass
phonology seminar, Spring, 2002: Della Chambless, Paul de Lacy, Maria Gouskova, Jin-Hyung Kim,
Steve Parker, Joe Pater, Lisa Selkirk, Taka Shinya, Monica Sieh, Melissa Svendsen, Anne-Michelle
Tessier, Ellen Woolford, and Hosuk Yoon.
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Table 1.
*LAPSE
LAPSE-AT-END
LAPSE-AT-PEAK
*INITIAL-LAPSE
ALIGN-BY-σ
(Wd, Ft, L)
ALIGN-BY-σ
(Wd, Ft, R)
ALIGN-BY-σ
(Ft, Wd, L)
ALIGN-BY-σ
(Ft, Wd, R)
*
**
***
**
**
**
**
*
***
**
*
***
**
**
**
**
**
*
**
***
*
**
***
**
**
**
**
Trochaic,
Main-Stress Left
a.
(10)(20)(20)0
*
*
b.
(10)(20)0(20)
*
*
c.
(10)0(20)(20)
*
*
d.
0(10)(20)(20)
*
Trochaic,
Main-Stress Right
e.
0(20)(20)(10)
f.
(20)0(20)(10)
*
*
g.
(20)(20)0(10)
*
*
h.
(20)(20)(10)0
*
*
Iambic,
Main-Stress Left
i.
(01)(02)(02)0
j.
(01)(02)0(02)
*
*
k.
(01)0(02)(02)
*
*
l.
0(01)(02)(02)
*
*
*
*
*
***
**
*
*
***
**
**
**
**
**
**
***
Iambic,
Main-Stress Right
m.
0(02)(02)(01)
*
*
*
n.
(02)0(02)(01)
*
*
*
o.
(02)(02)0(01)
*
*
p.
(02)(02)(01)0
*