AMER. ZOOL., 33:289-299 (1993)
Functional Morphology of Mammalian Mastication1
SUSAN W.
HERRING
Department of Orthodontics, SM-46, University of Washington, Seattle, Washington 98195
While chewing is not unique to mammals, it is one of their
most distinctive characteristics. Historically, studies of food processing
in mammals were intended to provide evolutionary insights, but more
progress has been made in understanding mechanistic aspects. Mastication is considered under five headings. (1) Interaction of teeth with food.
Knowledge of comparative dental anatomy and function is advanced in
comparison to understanding of foods and how they are broken down.
(2) Chewing force and its resistance by the skull. The traditional assumption that occlusal force is maximized is not always justified, and experimental results suggest that skull loading is far more dynamic and variable
than had been envisioned from theoretical analyses. (3) How the jaw
moves. The most important masticatory movement is that of the power
stroke, and in most but not all species this is influenced more by the
inclined planes of the teeth and jaw joints than by the musculature. (4)
The role of muscles in producing both force and movement. The most
fundamental distinction among jaw muscles is whether they have a rostral
or caudal direction of pull, as this determines their role in transverse jaw
movements. Reliance on anatomical names tends to obscure functional
similarities and differences among species. (5) Intraoral structures. Because
they are difficult to study, the actions of the tongue and pharynx are still
debated. Even the fundamental question of whether mammals can breathe
and swallow at the same time has not been definitively answered.
SYNOPSIS.
TWO rather different sets of problems are
addressed in studies of mastication. First,
The dictionary defines "mastication" as t h e chewing apparatus is a particularly
the process of grinding or comminuting food intriguing piece of machinery, with its tight
with the teeth in order to prepare it for swal- occlusion and vanable hinges-how does it
lowing and digestion (Stedman's, 1966). w o r k o n a P u r e l y mechanical basis? What
While not unique to mammals (a variety of d o e s Jt m e a n l f a t o o t h 1S s h a r P o r b l u n t ? l f
other vertebrates also do considerable pro- a m u s c l e attaches here or there? These quescessing of food with the teeth), mastication tions a r e a11 t h e m o r e interesting because we
is certainly a mammalian characteristic, humans have our own version of the appaDistinguishing features of the Mammalia r a t u s - Answers emerge from theoretical
which relate to chewing include the occlud- s t u d i e s (**•• computer modeling), in vitro
ing teeth (retained in all but a few very spe- testing (e.g., with photoelastic models) and
cialized groups) and reorganized jaw mus- in viv0 studies of functioning animals, but
culature and jaw joint (Crompton, 1989), a comparative approach is not required, and
the secondary palate (Thomason and Rus- in deed it is often assumed that the results
sell, 1986), and the muscular food-transport Wl11 b e species-invanant. The second set of
apparatus, including changes in the struc- Problems centers around the interpretation
ture of the tongue and pharynx (Smith, of morphological change dunng evolution.
1992)
Not only are teeth the main documentation
of the fossil record in mammals, but also
the diversity of dental apparatus and jaw
1
An invited Review Paper for the Division of Ver- muscles in extant mammals seems to be an
tebrate Morphology.
obvious consequence of adaptive evolution.
289
WHAT IS IT WE WANT TO KNOW ABOUT
MASTICATION?
290
SUSAN W. HERRING
Studies here are obligated to be comparative
and typically involve "classical" functional
morphology, denned here as combining
information on anatomy, mechanics and
behavior in order to understand selective
influences on the animals.
A BRIEF HISTORY
The functional morphology of chewing,
relating as it does to both paleontology and
dentistry, has been a popular subject for
centuries. Of necessity, early studies relied
on the morphology of dried and fossil skulls
and teeth, on cadaver dissections, and to a
certain extent on behavioral observations,
the latter being hampered by the rapidity of
chewing movements (ranging up to 3-5
complete cycles per second in commonly
studied laboratory species). Because of the
nature of the data, interpretations concentrated on comparative biomechanics, with
the lower jaw usually analyzed as a third
class lever with the jaw joint as fulcrum.
Contrasts were drawn between broad groups
of mammals representing different feeding
behaviors or phyletic lineages (e.g., Maynard Smith and Savage, 1959; Turnbull,
1970). When, in the 1960s, experimental
techniques such as high-speed cinematography, cineradiography, and electromyography (EMG) became generally available,
there was a great scramble to study as many
divergent taxa as possible. A variety of
reviews summarized the state ofthe art after
a dozen or so years of research (Gans et ai,
1978; Hiiemae, 1978; Weijs, 1980), followed by an ASZ symposium organized by
Gorniak in 1983 and published in American
Zoologist in 1985.
Since then, however, the flavor of the field
has changed. Typical studies in the 70s were
functional analyses (EMG and jaw motion)
of various species. Now the emphasis is on
a more fine-grained analysis of the constituent parts of the masticatory system.
Broadly adaptive explanations have yielded
to more mechanistic approaches. I suggest
three reasons for this changing character.
First, little progress was made on the broadly
adaptive explanations. With regard to the
systematic survey, at least one of most kinds
of mammal got looked at but few fundamental generalizations emerged, suggesting
that the focus of the investigations on simple characterization of muscle-bone leverage was incorrect, or at least inappropriate
at a detailed level. For the best treatment
available of these studies, the reader is
referred to an excellent recent review by
Weijs (1993). Second, the methods were
successful in dealing with more mechanistic
questions. The general pattern of food handling in mammals did get clarified, although
of course some mysteries remain. Third, the
new mechanistic information gave rise to
new problems: various arguments and
debates arose in two specific areas: transmission of occlusal force, especially at the
craniomandibular joint (CMJ); and the
nature of the neural control of mastication.
The systematic considerations then paled in
comparison to more specific investigations
of loading and control. This is regrettable;
the study of feeding performance still holds
potential for illuminating mammalian evolution. Eventually, armed with new information about mechanisms, we will have to
return to the comparative arena.
THE ELEMENTS OF MASTICATION
Food vs. teeth: An epic battle
The point of mastication is of course to
break down the food, and the interface where
that occurs is at the teeth; hence I begin here.
It is surprising that the vast literature on
dental evolution in mammals and the use
of dental characters in constructing mammal phylogenies have contributed so little
to functional morphology (Fortelius, 1990).
Available clues in the teeth include crown
form, e.g., the presence of carnassial-type
shearing blades is taken, primarily by analogy to living carnivorans, to indicate meateating (e.g., Van Valkenburgh, 1991), while
brittle, abrasive or tough foods are associated with thick-enameled, bunodont cusps,
increased crown height, and the development of crests, respectively (Janis and Fortelius, 1988). Wear facets and microwear
give information about the direction of the
chewing stroke (Greaves, 1973; Rensberger,
1978, 1986; Gordon, 1984; Teaford and
Byrd, 1989) and features of microwear such
as the proportion of pits to scratches have
also been used to infer diet in both extant
MAMMALIAN MASTICATION
and extinct species (Taylor and Hannam,
1987; Grine and Kay, 1988; Van Valkenburgh et al., 1990; reviewed by Teaford,
1991). Other aspects of tooth structure relevant to functional morphology include the
orientation of enamel prisms (Rensberger
and v. Koenigswald, 1980; Boyde and Fortelius, 1986) and the mineral content of
enamel (Kirkham et al., 1988).
What is mostly missing in these interesting studies is an understanding of the
mechanical properties of foods in relation
to dental structure. Lucas (1982) provided
an influential review of fracture theory,
pointing out the importance of commonly
ignored dental features such as radius of curvature of cusp tips (later used by Freeman
to analyze bat tooth design [1988, 1992])
and lamenting that the theory "exists in a
wilderness" of ignorance about the foods
themselves (p. 161). Little progress has been
made on this front. Kiltie (1982) measured
the compressive strength of various nuts in
an effort to correlate the jaw mechanics and
diets of rain forest peccaries (Tayassuidae).
Wang and Stohler (1990) found in vitro that
the breakage characteristics of common test
foods such as carrot and monkey chow differ
greatly; these in vitro characteristics were
later found to account for about half of the
variation observed in vertical chewing
movements in humans (Wang and Stohler
1991). A number of computational and
empirical investigations of food breakdown
during human chewing have emphasized the
importance of food particle size distribution
and the rate of size reduction during chewing (Lucas and Luke, 1983;Voone/a/., 1986;
van der Glas et al., 1987, 1992). An inverse
correlation of chewed particle size with the
shearing capacity of the molar cusps was
reported by Sheine and Kay (1977) in a study
on two prosimian primates and the tree
shrew Tupaia glis; the ability to grind food
more finely was associated with insectivory,
and the authors proposed that chitin was
the critical food element. In a later study
(1979) Kay and Sheine showed that chitin
particle size was an important criterion of
its digestibility in Galago.
291
and Gans, 1976; Thexton et al., 1980; Fish
and Mendel, 1982; Lucas et al., 1986; Plesh
et al., 1986; Byrd, 1988). Suffice it to say
here that the effects are usually substantial
(rabbits being an exception [Morimoto et
al., 1985]), but differ in the various studies,
perhaps because of the different species
examined (goats, cats, tree shrews, humans
and rats in the studies cited above) and/or
because of the different (and uncharacterized) foods used. Most workers probably
agree that these effects result from sensory
inputs (particularly from periodontal receptors) modifying the motor program of mastication rather than from any direct
mechanical feature of the food. Huang et al.
(personal communication) found that dental anesthesia diminished, but did not abolish, the effect of food consistency on pig
mastication. However, the residual effect
could still have had a sensory basis, since
muscle spindles were not anesthetized.
Bones and biomechanics
The traditional mechanical analysis of the
jaws centers on the estimation of occlusal
force, specifically the vertical or jaw-closing
component, using a static analysis (e.g.,
Turnbull, 1970, and references therein).
Experimental approaches have also emphasized static vertical force because of the ease
with which it can be measured by transducers placed between clenched teeth (e.g.,
Daunton, 1977; Robins, 1977;Dechowand
Carlson, 1986; human literature reviewed
by Gibbs and Lundeen, 1982), in contrast
to the difficulty of achieving dynamic and/
or three-dimensional measurements. Perhaps because so many of the available data
pertain to static vertical force, many analyses assume, implicitly or explicitly, that (all
other things being equal) occlusal force normal to the teeth should be maximized. While
not an unreasonable assumption for anyone
who has ever labored over a substantial hunk
of peanut brittle, it may not always be the
case that masticatory success depends on
vertical force. For example, the grinding
strategy used by many ungulates, rodents
and other species requires crests on the lower
One area that has received considerable teeth to shear across crests on the upper
attention is how various foods affect the rate teeth, a task comparable to a snow shovel
and movements of chewing (e.g., De Vree scraping along an uneven sidewalk; while
292
SUSAN W. HERRING
the shovel requires enough vertical force to
keep it in contact with the sidewalk, success
in snow removal depends on its forward
momentum. Furthermore, both modeling
(Koolstra et ai, 1988) and measurement
(Southard et ai, 1990) have demonstrated
non-vertical components to occlusal force.
If, for the purpose of argument, we accept
the assumption that occlusal force is maximized, then several aspects of morphology
need to be considered. Relevant here is the
mechanical arrangement of the bones, muscles and teeth in terms of struts, links and
levers. The traditional analysis, performed
in lateral view, provoked a long and entertaining controversy over whether the mammalian jaw should be modeled as a third
class lever (which implies that the craniomandibular joint is reactively loaded) or as
a "link" (with essentially no loading at the
joint [Gingerich, 1971; Roberts and Tattersall, 1974]). The latter view was in part
inspired by the "inefficiency" of wasteful
reactive loading. This battle (in my opinion)
has been won by the lever supporters, buttressed both by the theoretical analyses and
empirical evidence (Picq et al., 1987 and
references cited). Available EMG evidence
suggests that it is unlikely that loading at
the CMJ is limited or even controlled
(Osborn and Baragar, 1985; Throckmorton
et ai, 1990). Evidently it is more important
for mammal jaws to be effective than to be
efficient. However, there are probably some
exceptions to the general rule of CMJ loading. During the origin of mammals, the
primitive lever mechanism appears to have
been de-emphasized and later redeveloped
with different jaw bones (Crompton and
Hylander, 1986). Among recent mammals,
the reorientation of the masseter muscle in
many rodents suggests that the "link" model
may apply for mastication, although not for
incision (see Weijs and Dantuma, 1975, for
some actual calculations). Further evidence
for relative unloading of the jaw joint in
rodents comes from the small size and free
mobility of the joint structures. At the other
end of the spectrum, similar arguments have
been made for elephants (Maglio, 1973).
Within the mainstream of the lever model
of jaw mechanics, the major recent development has been the extension of the anal-
ysis to three dimensions and the resulting
conclusion that the balancing (i.e., the side
without the food) jaw joint typically bears
greater reaction forces than the working (the
side with the food) joint (e.g., Greaves, 1978;
Smith, 1978; Korioth and Hannam, 1990;
reviewed by Hylander, 1992). Another
interesting perspective on CMJ loading was
developed by Bramble (1978), who considered the interaction between muscle action
lines and a second fulcrum at the bite point.
The forces of chewing are not only resisted
by the teeth and the CMJ, of course. The
bones themselves distort during function
and move relative to each other at the
sutures. Knowledge in this area is expanding
rapidly with the advent of finite element
computer analyses (e.g., de Jongh et al.,
1989; Korioth and Hannam, 1990; Hart et
ai, 1992) and in vivo measurements of bone
strain using foil strain gages bonded to bones
or across cranial sutures (e.g., Weijs and de
Jongh, 1977; Hylander, 1979; Hylander et
ai, 1991; Herring and Mucci, 1991). The
strain studies address arguments about the
meaning of mandibular form (e.g., Daegling, 1989; Demes et ai, 1984), the significance of the mandibular symphysis (Scapino, 1981; Greaves, 1988), and the
transmission of forces across the cranium
(Greaves, 1985). At this point it seems reasonable to say that bone strain in vivo is a
far more dynamic and variable parameter
than had been envisioned by theoretical
workers. For example, the typical loading
in the zygomatic suture of the pig is compressive in part of the structure but tensile
elsewhere, but this pattern is reversed whenever the opposite side masseter muscle is
unopposed (Herring and Mucci, 1991).
Moving the jaw along inclined planes
In defiance of static analysis, mastication
is a dynamic process. The CMJ is a moving
as well as a load-bearing element, and the
muscles effect the chewing cycle in addition
to providing occlusal force. The variety of
chewing cycles exhibited by various mammals has been discussed elsewhere (Hiiemae, 1978; Weijs, 1993). The critical part
of the cycle is the power stroke, during which
the mandibular teeth move past the maxillary teeth, presumably exerting forces on
MAMMALIAN MASTICATION
the food. In extant mammals (but not multituberculates, see Krause, 1982), the mandibular teeth are directed medially and rostrally during the power stroke. The
movement is accomplished by rotation
around the long axis of the working side
dentary bone (if the symphyseal joint is
patent as in primitive mammals [Crompton
and Hiiemae, 1970; Dotsch, 1982]), by
mediolateral translations at the CMJ
(especially in carnivorans, [Scapino, 1965]),
and/or by anteroposterior condylar movements, either a caudal translation of the balancing condyle (ungulates and primates,
including man) and/or a rostral translation
of the working condyle(s) (primarily in
rodents, e.g., Byrd, 1981; Offermans and De
Vree, 1990). The absolute excursions made
by the occluding cheek teeth may be as small
as a few millimeters (for example, the lateromedial shift in cats [Gorniak and Gans,
1980]) or as large as several centimeters
(large rodents and ungulates). Here I would
like to consider the determinants of the
power stroke, i.e., why the mandible takes
the precise pathway which is observed in
each species. There are basically only three
possibilities.
First, the mandible may be physically
constrained, for example by interlocking
canine teeth (Herring, 1972, but see Kay et
al, 1986, for a different finding) or by preand post-glenoid processes surrounding the
condyle (e.g., in carnivorans [reviewed by
Dessem and Druzinsky, 1992]). Such constraints are actually somewhat unusual, most
mammal jaws being relatively mobile, even
sloppy. In any case physical constraints can
only establish the absolute limits of the
power stroke excursion, not influence the
pathway within those limits. Second, the
mandible may slide along inclined planes
formed either by the teeth or by the CMJ.
The inclined planes formed by carnassial
teeth or by inwardly (rabbits and ungulates)
or outwardly (caviid rodents) sloped grinding surfaces clearly guarantee a repeatable
power stroke even given variation in the
direction of the muscle force provided
(Becht, 1953). A strongly inclined sliding
joint surface, such as is found in humans,
can serve a similar purpose, although less
precisely. Inclined planes are arguably the
293
main determinant of the power stroke.
However, some omnivorous species have
such low-cusped molars that any planes
formed are small relative to the irregularities of the food (e.g., pigs, some primates
and some rodents) that a third mechanism
is needed, namely precise control of the
power stroke by the muscles.
The above comments on chewing movements have some general validity for mammals, but the reader should remember that
there exist some truly oddball species (primarily non-chewing) for which these comments are quite irrelevant. For example,
echidnas apparently do not depress or elevate the jaw sensu stricto, but manage to
open and close the mouth by axial rotation
of the curved dentaries (Murray, 1981).
Muscles: Force or movement?
Numerous studies on the anatomy of
mammalian jaw muscles have been published, notably Turnbull's (1970) heroic
review. Evolutionary trends in jaw muscles
include: (1) changes in orientation, often as
a byproduct of evolutionary or ontogenetic
modifications of the skull (Herring, 1985a,
b)\ (2) alterations of internal architecture,
for example, the warthog (Phacochoerus)
masseter has aponeuroses oriented perpendicular, rather than parallel, to the muscle
surface, a change which may be related to
packing problems in very pinnate muscles
(Herring, 1980); (3) changes in absolute or
relative size of the various muscles; and (4)
losses and fusions of muscle subdivisions,
e.g., loss of the superficial portion of the
temporalis in rabbits. No examples of complete loss are known to me—the rabbit
retains a substantial deep temporalis.
Aspects of muscle anatomy with particular relevance to chewing include the directions of pull and the forces produced. I have
recently reviewed issues related to the architectural complexity of jaw-closing muscles
(Herring, 1992) and their physiological
properties (Herring, 1993), and Weijs (1993)
has reviewed muscle contraction patterns;
hence these subjects will be given short shrift
here. Instead, I will highlight some problems that in my opinion need emphasis.
With regard to direction of pull, one of the
worst obstacles to progress has been our reli-
294
SUSAN W. HERRING
ance on the anatomical names of muscles.
The simple use of the term "superficial masseter" implies that such a structure exists as
a functional entity across taxa. Unfortunately, jaw-closing muscles are not only heterogeneous internally but are also linked to
each other externally, and so their homologies, while undoubted, are imprecise.
Worse, even if superficial masseters are
homologous in different mammals, this does
not imply any functional similarity. The
action line of the superficial masseter relative to the toothrow varies from about 20°
(Rattus) to about 90° (Equus) (Turnbull,
1970). Nor has any phylogenetically conserved central motor program been observed
for activation of specific named muscles.
Rather, muscles seemed to be coordinated
according to whether they pull rostrally or
caudally on the mandible. The medial
excursion of the power stroke is produced
by a force couple of protrusors on the working side and retrusors on the balancing side
(a patterned referred to as diagonal by Herring [1985*] and as triplets by Weijs [ 1993]).
Even jaw-opening muscles such as the
digastric work in such couples (Weijs et ai,
1989). For functional purposes, it might be
better to designate muscles as vectors rather
than as named parts.
For traditional static analyses, muscle
anatomy (in particular the physiological
cross section) is used to estimate muscle
force. Current models incorporate information on force-velocity and force-length
relations (reviewed by Weijs and van
Ruijven, 1990) and are reasonably successful at duplicating actual masticatory strains.
Gans and De Vree (1987) present some general considerations on the placement of sarcomeres with respect to both force and
excursion. Muscle forces clearly depend as
much on sarcomere stretch and speed of
contraction as they do on muscle size.
Therefore, it is interesting to note that the
different chewing movements observed in
various mammals have consequences for
these parameters. Consider the power stroke,
a variably medial and rostral movement of
the nearly closed working-side dentary. In
ungulates and primates, this is accomplished by rotating the mandible around a
vertical axis in the general vicinity of the
working side CMJ. The muscles on the
working side therefore shorten relatively little. Because force and velocity are inversely
related, the working side muscles will produce high force. In rodents, however, rostral
translation of the jaw exceeds rotation
(Offermans and De Vree, 1990), so that the
working side muscles must undergo substantial excursions. The rodent condition
appears to emphasize movement at the
expense of force.
If we could see inside
As in most scientific fields, research in
mammalian mastication is dominated by
technique; we study those aspects which are
amenable to investigation. This explains
why so much work has been done on the
mandible and its muscles. Events inside the
mouth, such as food transport and swallowing, are actually more critical for feeding,
but unfortunately, they are technically difficult to study. The soft intraoral structures
leave little evidence of their presence on the
skeleton, and the hyolaryngeal skeleton itself
is rarely preserved even in museum preparations, not to mention fossils. Further, the
muscles of the tongue and pharynx are
numerous but small and deep; many are not
easily accessible even surgically. Thus, even
though the tongue-and-throat apparatus has
many interesting anatomical variations
(Howes, 1896; Linton, 1905; Livingston,
1956; Iwasaki et al, 1987), very little is
known about how these relate to function,
with the exception of the total reorganization of the apparatus in ant- and termiteeating mammals (Doran, 1975).
Studies on the tongue outside the mouth
have included Abd-el-Malek's (1955) wellknown study on edentulous humans,
Schonholzer's (1958-9) intriguing comparative observations on drinking in zoo animals, and invasive studies of muscle action
in anesthetized animals (Bennett and
Hutchinson, 1946). Electromyography of
extrinsic tongue muscles verifies the role of
these muscles in producing gross tongue
movements (reviewed by Lowe, 1981,
1990). However, the complex intertwining
of the intrinsic fibers has thus far baffled
functional interpretation of electrical activity (except for the obvious fact that these
295
MAMMALIAN MASTICATION
muscles are active during feeding); thus a
test of Kier and Smith's (1985) theoretical
treatment of tongue-like organs is not yet
possible.
Of course, it is possible to see inside to
some degree. Cineradiography and videofluoroscopy have been used on animals with
radioopaque markers implanted in the
tongue and/or hyoid bone (review by Hiiemae and Crompton, 1985; more recent
studies by De Gueldre and De Vree, 1984;
Franks et al., 1985; Anapol, 1988; German
et al., 1989; German and Franks, 1991).
These studies have revealed a cyclic movement of the hyoid body during chewing that
varies with species and that differs strikingly
from the swallowing movement. How these
movements of the hyoid body relate to the
stylohyoid chain of elements is not known,
but is of great interest, because the stylohyoid chain is extremely diverse in various
mammals. Markers in the tongue show that
contractions are regionally asynchronous
and asymmetrical, with twisting movements occurring during the closing and
power strokes (German et al., 1989; Cortopassi and Muhl, 1990).
The activity of pharyngeal muscles in
swallowing has been studied primarily in
anesthetized animals (reviewed by Bosma,
1957; Miller, 1982; Smith, 1992). Since
swallowing is a reflex, these observations
are assumed to reflect awake swallowing,
except for the absence of a bolus. A longstanding controversy over the coordination
of breathing and feeding still rages. In adult
humans, breathing stops during swallowing,
and indeed is made impossible by the folding down of the epiglottis over the entrance
to the larynx (Ardran and Kemp, 1952;
Ekberg, 1983). However, based on the resting posture of the epiglottis relative to the
soft palate, Negus (1929) postulated that in
many if not most other mammals, swallowing and breathing could take place simultaneously, the food bolus passing sideways
around the epiglottis. This claim, extended
to human infants (Laitman and Crelin, 1980)
as well as all non-primate mammals (Cave,
1967), has become a major ingredient in
some scenarios of the evolution of speech
(Lieberman, 1991). Experimental observations on whether breathing stops and/or the
epiglottis folds down during swallowing have
not resolved the controversy. Laitman et al.
(1977) reported that the epiglottis, marked
with a radioopaque clip, remained upright
during radiographically observed swallowing in monkeys, although a "momentary
separation" from the soft palate occurred.
However, in pigs, which have a particularly
long intranarial epiglottis and should therefore be extreme examples of simultaneous
breathing and swallowing, Herring and Scapino (1973) saw epiglottic movement.
Biewener et al. (1985) found that breathing
in dogs was disrupted not only during swallowing, but even during chewing (which is
not the case in pigs), and epiglottic closure
during swallowing in dogs (Suzuki and
Nomura, 1973) has been inferred from
muscle activity patterns. Clearly, more direct
evidence is necessary, but at the moment,
the claim that the human pharynx functions
differently from that of all other mammals
should be regarded as highly suspect.
ACKNOWLEDGMENTS
My work has been supported by PHS
grants, currently DE 08513, for which I
thank NIH. I also thank Raymond Fink for
discussions about the epiglottis, Wim Weijs
for allowing access to his manuscript before
publication, and Dave Wright for helpful
comments on the manuscript.
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