Saber El
email: saber.el-arem@polytechnique.edu

Shearing effects on the breathing mechanism of a cracked beam section in bi-axial flexure

Keywords: beam finite element, breathing crack, unilateral contact, shearing effects, convex analysis

 by taking into account the effects of shearing in the constitutive equations of a beam cracked section in bi-axial flexure. The paper describes the derivation of a lumped cracked beam model from the three-dimensional formulation of the general problem of elasticity with unilateral contact conditions on the crack lips. Properties of the potential energy and convex analysis are used to reduce the three-dimensional computations needed for the model identification, and to derive the final form of the elastic energy that determines the nonlinear constitutive equations of the cracked transverse section. We aim to establish a relation of behavior between the applied forces and the resulting displacements field vectors, which is compatible with the beams theory in order to allow the model exploitation for shafts dynamics analysis. The approach has been applied to the case of a cracked beam with a single crack covering the half of its circular cross section.

1 Introduction: state of the art Since the early 1970s when investigations on the vibrational behavior of cracked rotors began, numerous papers on this subject have been published, as a literature survey by [START_REF] Dimarogonas | Vibration of cracked structures: A state of the art review[END_REF] shows. The analysis of the behavior of cracked rotating machinery shafts is a complex structural problem. It requires,

M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS

for a relevant description, a fine and precise modeling of the shaft and cracks in order to allow the identification and calculation of the parameters characterizing their presence. Researchers dealing with the problem of rotating cracked beams recognize its two main features, namely:

• the determination of the local flexibility of the beam cracked section;

• the consideration of the opening-closing phenomenon of the crack during the shaft rotation, commonly called breathing mechanism of the crack, and responsible of the system nonlinear behavior: when the shaft is rotating, then the crack opens and closes according to the stresses developped in the cracked surface. If these stresses are extensive, then the crack opens, resulting in a reduced shaft stiffness. When the stresses are compressive, then the crack remains closed and the shaft has the same stiffness as the non-cracked shaft. Thus, the system stiffness is depending on the cracked section position (Figure 1).

This breathing mechanism depends on the shaft rotation in the case when the static deflection dominates the vibration of the rotating shaft. This is a very common situation in large turbine-generators rotors. During the last three decades, great attention has been paid by several research scientists to the analysis and diagnosis of cracks in rotating machinery. The excellent review papers by [START_REF] Entwistle | Survey of the use of vibration methods in the assessment of component geometry[END_REF], [START_REF] Dimarogonas | Vibration of cracked structures: A state of the art review[END_REF], Wauer [1990a] and [START_REF] Gasch | A survey of the dynamic behavior of a simple rotating shaft with a transverse crack[END_REF] cover many aspects of this area and summarize the most relevant analytical, experimental and numerical works conducted in the last three decades and related to the cracked structures modeling.

There have been different attempts to quantify local effect introduced by cracks. The analysis of the local flexibility of a cracked region of structural element was quantified in the 1950s by Irwin [1957a,b], [START_REF] Bueckner | The propagation of cracks and the energy of elastic deformation[END_REF], [START_REF] Westmann | Stress analysis of cracked rectangular beams[END_REF] by relating it to the Stress Intensity Factors (SIF). Afterwards, the efforts to calculate the SIF for different cracked structures with simple geometry and loading was duplicated [START_REF] Tada | The stress analysis of cracks handbook[END_REF][START_REF] Duong | Mécanique de la rupture fragile[END_REF].

M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS

Most researchers agree with the application of the linear fracture mechanics theory to evaluate the local flexibility introduced by the crack [START_REF] Gross | Stress intensity factors for a single-edge-notch specimen in bending or combined bending and tension by boundary collocation of a stress function[END_REF][START_REF] Anifantis | Identification of peripheral cracks in cylindrical shells[END_REF][START_REF] Dimarogonas | Analytical methods in rotor dynamics[END_REF][START_REF] Dimarogonas | Vibration of cracked structures: A state of the art review[END_REF], Papadopoulos and Dimarogonas, 1987a,b,c, Papadopoulos, 2004]. Obviously, the first work was done in the early 1970s by [START_REF] Dimarogonas | Dynamic response of cracked rotors[END_REF][START_REF] Dimarogonas | Dynamics of cracked shafts[END_REF] and [START_REF] Pafelias | Dynamic response of a cracked rotor[END_REF] at the General Electric Company. The energy release rate approach combines the linear fracture mechanics to rotordynamics theory in order to calculate the compliance caused by a transverse surface crack affecting a rotating shaft. A good review on this method is presented by [START_REF] Papadopoulos | The strain energy release approach for modeling cracks in rotors: A state of the art review[END_REF].

For an elastic structure, the additional displacement u due to the presence of a straight crack of depth a under the generalized loading P is given by the Castigliano theorem

u = ∂ ∂P a 0 G(a)da (1)
G is the energy release rate related to the SIF by the Irwin formula [Irwin, 1957a]. Then, the local flexibility matrix coefficients are obtained by

c ij = ∂ 2 ∂P i ∂P j a 0 G(a) da, 1 ≤ i, j ≤ 6 (2)
Extra diagonal terms of this matrix are responsible for longitudinal and lateral vibrations coupling that could be with great interest when dealing with cracks detection.

In two technical notes of the NASA, Gross andSrawley [1964, 1965] computed the local flexibility corresponding to tension and bending including their coupling terms. This coupling effect was observed by [START_REF] Rice | The part-through surface crack in an elastic plate[END_REF] in their study of cracked elastic plates for stress analysis.

Dimarogonas and his co-workers [START_REF] Dimarogonas | Crack identification of in aircraft structures[END_REF][START_REF] Dimarogonas | Analytical methods in rotor dynamics[END_REF][START_REF] Dimarogonas | A fuzzy logic neutral network structured expert system shell for diagnosis and prognosis-users manual[END_REF][START_REF] Dimarogonas | A general purpose rotor dynamic analysis program-users manual[END_REF], and [START_REF] Anifantis | Identification of peripheral cracks in cylindrical shells[END_REF] introduced the full (6 × 6) flexibility matrix of a cracked section. They noted the presence of extra diagonal terms which indicate the coupling between the longitudinal and lateral vibrations. Papadopoulos and Dimarogonas [1987a,b,c], and [START_REF] Ostachowicz | Coupled torsional and bending vibrations of a rotor with an open crack[END_REF] computed all the (6 × 6) flexibility matrix of a Timoshenko beam cracked section for any loading case. However, there are no results for the SIF for cracks on a cylindrical shaft. Thus, [START_REF] Dimarogonas | Analytical methods in rotor dynamics[END_REF] have developed a procedure which is commonly used in FEM software: the shaft was considered as an assembly of elementary rectangular strips where approximation of the SIF using fracture mechanics results remains possible (Figure 2). integration of the energy release rate on the crack tip by

c ij = ∂ 2 ∂P i ∂P j a 0 b -b G(a, ξ) dξ da (3)
Although it offers the advantage of being easy to insert in a numerical algorithm, this method has some limitations. In fact, some numerical problems were reported by [START_REF] Abraham | Remark on the determination of compliance coefficients at the crack section of a uniform beam with circular cross section[END_REF] when the depth of the crack a exceeds the section radius R. [START_REF] Dimarogonas | Author's reply[END_REF], in his reply, stated that this divergence does not reflect reality; it is due to the assumption of a 2D stress distribution which is not valid near the ends of the crack tip. [START_REF] Papadopoulos | Coupling of bending and torsional vibration of a cracked timoshenko beam[END_REF] suggested to consider equation (3) with :

0.90b ≤ |ξ| ≤ 0.95b
when the crack depth a exceeds the shaft radius R. Wauer [1990b] explored the dynamics of a cracked, distributed parameter rotor component. The proposed model is a rotating Timoshenko shaft which is also flexible in extension and torsion. The stiffness matrix is constructed using the energy release rate approach as described in papers by Papadopoulos and Dimarogonas [1987a,b], and [START_REF] Gudmundson | The dynamical behavior of slender structures with crosssectional cracks[END_REF]. The geometric discontinuity due to the crack is replaced by a load discontinuity. The procedure reduces the problem to equations for one uniform beam with a modified load distribution. The proposed approach is a powerful instrument to obtain approximate results by a relatively small calculation expense. Although the shearing is considered in this approach, the author did not discuss the importance of its effects on the crack breathing mechanism. An original method for deriving a lumped model for a cracked beam section was proposed by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF]. Based on three-dimensional computations, the procedure incorporates more realistic behavior of the cracks than the previous models, namely the unilateral contact conditions on the crack lips and their breathing mechanism under variable loading. The method was derived from three-dimensional formulation of the general problem of elasticity with unilateral contact conditions on the crack lips. The authors established properties of the potential energy of this problem to reduce the amount of computation required for its determination in the case of a beam containing cracks of any shape and number. Convex analysis was also used to derive the final form of the energy that determines the nonlinear constitutive equations of the section of the beam which was incorporated in a FE analysis code. Great attention is paid to the capability of such a model to take into account the real 3D geometry of a crack and to represent, as general as possible, the effects of different load components on its nonlinear behavior.

The experimental validation of the approach is presented by [START_REF] Stoisser | A comprehensive theoretical, numerical and experimental approach for crack detection in power plant rotating machinery[END_REF]. The authors reported that the model reproduces with good accuracy the overall behavior of the shaft line in presence of cracks. The experimental validation allows the use of the model, with confidence and reliability, for the determination of the dynamics of supposed cracked rotors.

Based on the approach of [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF], El [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF]Maitournam [2008, 2007] presented a method of construction of a cracked beam finite element which they used afterwards for the stability analysis of cracked shafts.

The authors distributed the additional energy due to the cracked section on the entire length of the cracked beam finite element. Considerable gain in computing efforts was reached compared to the nodal representation of the cracked section when dealing with the numerical integration of differential equations in structural dynamics.

The cracked structure of Figure 3(a) is subjected to an end moment M 2L = (M x (2L), M y (2L)) at z = 2L. [START_REF] Andrieux | Détermination de la loi de comportement en flexion bi-axée d'une poutre fissurée avec prise en compte du contact sur les fissures[END_REF] has demonstrated certain properties of the problem elastic energy, W * , leading to a considerable gain in the three-dimensional calculus required for the identification of the constitutive equations. In particular, for a linear elastic material, under the small displacements and small deformations assumptions, and in the absence of friction on the crack lips, the energy function could be written by distinguishing the contribution of the cracked section from that of the non-cracked elements, in the form :

W * (M 2L ) = W * (M ) = W * s (M ) + w * (M ) = L EI ||M || 2 (1 + s(Φ)) (4)
where M = (M x , M y ) is the resulting couple of flexural moments at the cracked section, W * s the total elastic energy of non-cracked structure subjected [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] to the flexural moment M and w * the additional elastic energy due to the presence of the cracked section. In order to simulate a rotating load on a fixed beam, the bending load is applied in several aperture angles Φ = atan( My Mx ), with Φ varying over [0, 2π[. E is the Young modulus and I quadratic moment of inertia. In this framework, the nonlinear constitutive equations of the discreet element modeling the cracked section are obtained by differentiating w * with respect to M . We obtain

[θ] =    [θ x ] [θ y ]    = 2L EI    s(Φ) -1 2 s ′ (Φ) 1 2 s ′ (Φ) s(Φ)       M x M y    with s ′ (Φ) = ds(Φ) dΦ (5)
However, for finite element computational codes in rotordynamics, a nonlinear relation of the form [θ] = f (M ) is to be identified. Thus, [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] introduced some properties of the additional deformation energy due to the cracked section, w([θ]). Thus, w could be written as a quadratic function of the rotations jumps as:

w([θ x ], [θ y ]) = EI 4L k(ϕ)||[θ]|| 2 with ϕ = atan( [θ y ] [θ x ] ) (6)
The Légendre-Fenchel transform is used to establish the relation between the two energy functions w * and w, then the stiffness function k is obtained from the compliance function s identified from three-dimensional calculus.

As described by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF], the constitutive equations are finally obtained by differentiating w with respect to [θ] as

M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS    M x M y    = EI 2L    k(ϕ) -1 2 k ′ (ϕ) 1 2 k ′ (ϕ) k(ϕ)       [θ x ] [θ y ]    with k ′ (ϕ) = dk(ϕ) dϕ (7)
The purpose of this paper is to complete the models suggested by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] and El [START_REF] Arem | Loi de comportement en flexion d'une section de poutre fissurée avec prise en compte des effets de cisaillement[END_REF] by considering the shearing effects in the constitutive equations of a cracked beam section in bi-axial flexure. Our main purpose is to establish constitutive relations (nonlinear) between the applied forces and the resulting displacements field, which is compatible with the beams theory in order to allow the model exploitation for shafts dynamics analysis using a beam model.

Description of the current model

The study of the in-plane flexural behavior of a cracked beam by El [START_REF] Arem | Loi de comportement en flexion d'une section de poutre fissurée avec prise en compte des effets de cisaillement[END_REF] and El [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF] showed that for straight line tip cracks the shearing and flexure effects could be separated when aiming to identify the constitutive equations of a cracked transverse section. In fact, the bending moments are only responsible for the rotations discontinuities, and the shear forces for the displacements discontinuities (slips). In this paper, we present the case of the bi-axial flexural behavior.

Our objective is to identify the characteristics of a one-dimensional model (Figure 3 In the one-dimensional (beam) model, the non-cracked parts of the structures are represented by Timoshenko beam elements, and the cracked section by a nodal element (zero length) allowing discontinuities of displacements and rotations. Indeed, this nodal element is composed of two uncoupled nonlinear springs modeling the cracked section flexural and shearing stiffnesses, cf. Figure 4. The unilateral contact between the crack lips is considered.

The following assumptions are considered for the modeling of the structure: • assumption of small displacements and small deformations,

• linear and isotropic elastic material,

• transverse cracks of any shape and in any number,

• unilateral contact without friction between the crack lips,

• the crack is completely closed in the unstressed configuration.

Properties of the energies functions of the problem

Let W * be the total elastic energy of the system. According to what precedes, W * can be put in the form

W * (F ) = W * s (F ) + w * (F ) = W * s (F ) + w * f (F ) + w * c (F ) (8)
where W * s denotes the total elastic energy of the non-cracked structure under the loading F (2L), and w * (F ) the elastic energy due to the presence of the crack. The study of the in-plane flexural behavior showed that the bending and shearing effects can be dissociated. Based on this result, w * is divided into: According to [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF], El [START_REF] Arem | Loi de comportement en flexion d'une section de poutre fissurée avec prise en compte des effets de cisaillement[END_REF] and El [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF], w * is strictly convex and positively homogeneous of degree 2,

• w * f (F ) = w * f (M )

M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS

from which rise the following properties:

Property 1 Functions w * c and w * f are strictly convex.

Property 2 Functions w * c and w * f are positively homogeneous of degree 2 :

∀λ ≥ 0 we have :

     w * f (λM ) = λ 2 w * f (M ) w * c (λT ) = λ 2 w * c (T ) (9)
It should be noticed that an essential hypothesis for obtaining properties 1 and 2 is that the gap between the lips of the crack is zero in the unstressed configuration. For the contribution of the bending moments, the quadratic form of w * f used in the study presented by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF], is considered:

w * f (M ) = L EI ||M || 2 s(Φ) (10)
With Φ = atan( My(L) Mx(L) ) and s(Φ) the additional flexibility due to the presence of the crack and related to the effects of flexural moments at the cracked section of the beam. Energy due to the shearing effects at the cracked section is also quadratic and can be put in the form:

w * c (T ) = L µκS ||T || 2 s c (Φ c ), with Φ c = atan( T y (L) T x (L) ) (11)
µ is the shear modulus and κ the shear correction factor of Timoshenko. Due to property 2, the problem of identification of the function w * c on R 2 is reduced to the identification of the flexibility function s c (Φ c ) on the interval [0, 2π] by considering the particular case of ||T || = 1. This additional flexibility is identified from three-dimensional finite element calculus using Code Aster c . Varé and [START_REF] Varé ; M A N U S C R I P T A C C E P T E D Article | Calcul d'un modèle de rotor fissuré pour le code cadyro[END_REF] presented a simpler form of w * c . They brought the problem back, for certain shapes of cracks, to the identification of three constants by writing:

w * c (T ) = 1 2 T 2 x (L)s x + 1 2 T 2 y (L)s y + s xy T x (L)T y (L) (12)
s x , s y and s xy depend only on the geometry of the crack, they are independent of the parameters of loading and identified using the three following threedimensional finite element calculations cases:

w * c (T x = 1, T y = 0), w * c (T x = 0, T y = 1) and w * c (T x = 1, T y = 1) M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS
This model allowed the obtaining, for certain cracks geometries, of results in excellent agreement with the three-dimensional FE calculations. However, it cannot be applied to any geometry of crack affecting beams or shafts with various cross section forms. Indeed, the shearing forces at the cracked section lead to the opening and closure of the crack according to mode II and/or mode III (modes of fracture). This depends on the orientation of the shearing force defined here by angle Φ c . The contact area between the lips of the crack, and consequently the additional flexibility due to its presence, depends on the direction of the applied shear force.

The present method, based on energy properties, could, without the use of additional simplifying assumptions, be applied to cracks of any shape and number affecting the same beam transverse section.

Constitutive equations of the discrete element with shearing effects consideration

The required relation describing the nodal element (Figure 4(d)) behavior is of the form:

     T M      = f (      [u] [θ]      ) (13)
with

[u] =      [u x ] [u y ]      =      u x (L + ) -u x (L -) u y (L + ) -u y (L -)      and [θ] =      [θ x ] [θ y ]      =      θ x (L + ) -θ x (L -) θ y (L + ) -θ y (L -)     
Consequently, it is the stiffness matrix of the nodal element which would be necessary to identify. Thus, we need to exploit the properties of the additional elastic deformation energy w due to the presence of the cracked section for the displacements discontinuities [u] and the rotations discontinuities

[θ].
Given the assumption of separating the shearing and flexure effects at the cracked section, w could be written in the form:

w([u], [θ]) = w f ([θ]) + w c ([u]) (14)
w f and w c are the elastic deformation energies associated with flexure and shearing, respectively. According to the work by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] and El [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF], w f and w c have the following properties:

M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS
Property 3 Functions w c and w f are strictly convex.

Property 4 Functions w c and w f are positively homogeneous of degree 2 :

∀λ ≥ 0 we have :

     w f (λ[θ]) = λ 2 w f (λ[θ]) w c (λ[u]) = λ 2 w c ([u]) (15)
For w f , the form proposed by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] is considered:

w([θ x ], [θ y ]) = EI 4L k(ϕ)||[θ]|| 2 (16) With ϕ = atan( [θ y ] [θ x ]
) and k the flexural stiffness function of the nodal element.

Function k is deduced from s by using the Légendre-Fenchel transform relating w f to w * f . Following the development steps described in [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF],Varé [2000]) for w f , w c can be put in the form :

w c ([u]) = µκS 4L ||[u]|| 2 k c (ϕ c ) (17)
With ϕ c = atan( [uy] [ux] ) and k c the shearing stiffness function of the nodal element to be identified. Using the Légendre-Fenchel transform we establish the relation between w c to w * c by writing:

w c ([u]) = sup T (T • [u] -w * c (T )) = sup ||T ||=1 λ≥0 (λT • [u] -w * c (λT )) = sup ||T ||=1 λ≥0 (λT • [u] -λ 2 L µκS ||T || 2 s c (Φ c )) (18)
Contrary to the flexural flexibility function s of the nodal element which vanishes for certain couples of flexural moments leading to a total closing of the crack, the linearity noted with respect to the shearing action shows that the flexibility function s c due to shearing is never zero:

∀T = 0, we have [u] = 0
Thus, the required maximum in ( 18) is reached when:

λ =      µκS 2L T •[u] sc(Φc) if T • [u] ≥ 0 0 otherwise (19) M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS
This leads to:

k c (ϕ c ) = sup ||T ||=1 T •[u]≥0 ( (T • [u]) 2 s c (Φ c ) ) (20) Thus, for [u] =      cosϕ c sinϕ c      and T =      cosΦ c sinΦ c     
, we obtain:

k c (ϕ c ) = sup cos(Φc-ϕc)≥0 ( cos 2 (Φ c -ϕ c ) s c (Φ c ) ) (21)
The condition cos(Φ cϕ c ) ≥ 0 implies, for a given ϕ c , that:

Φ c ∈ [ϕ c - π 2 , ϕ c + π 2 ]
The relation ( 21) becomes :

k c (ϕ c ) = sup Φc∈[ϕc-π 2 ,ϕc+ π 2 ] ( cos 2 (Φ c -ϕ c ) s c (Φ c ) ) (22)
The constitutive equations of the cracked section is obtained by differentiating w([u], [θ]) as follows:

             T x = ∂w([u],[θ]) ∂[ux] = ∂wc([u]) ∂[ux] , T y = ∂w([u],[θ]) ∂[uy] = ∂wc([u]) ∂[uy] M x = ∂w([u],[θ]) ∂[θx] = ∂w f ([θ]) ∂[θx] , M y = ∂w([u],[θ]) ∂[θy] = ∂w f ([θ]) ∂[θy] (23)
Finally, the required relation is given by:

          T x T y M x M y           =                  µκS 2L k c (ϕ c ) -µκS 4L k ′ c (ϕ c ) 0 0 µκS 4L k ′ c (ϕ c ) µκS 2L k c (ϕ c ) 0 0 0 0 EI 2L k(ϕ) -EI 4L k ′ (ϕ) 0 0 EI 4L k ′ (ϕ) EI 2L k(ϕ)                            [u x ] [u y ] [θ x ] [θ y ]           (
F = T x (2L) = cos(Φ c ), T y (2L) = sin(Φ c ), M x (2L) = LT y (2L), M y (2L) = -LT x (2L)
which leads, at the cracked section (z = L), to:

(T , M ) = T x (L), T y (L), M x (L), M y (L) = T x (2L), T y (2L), 0, 0 Thus, the additional deformation energy due to the cracked section w * is reduced to w * c . The angle Φ c varies in [0 • , 360 • [ at a rate of a loading case every Since ||T || = 1, and using the relation ( 11), the additional shearing flexibility function is obtained, for all loading cases, by:

s c (Φ c ) = µκS w * c (T ) L
Figure 5 shows the evolution of s c for a straight line front crack with a relative depth a D = 50%. It is noticed in particular that s c does not vanish on [0, 2π[. Also, the strong dependency of s c on the loading direction Φ c is clearly visible. Moreover, we show that the model presented by Varé and [START_REF] Varé ; M A N U S C R I P T A C C E P T E D Article | Calcul d'un modèle de rotor fissuré pour le code cadyro[END_REF] corresponds to particular loading cases since

s x = 2L µκS s c (Φ c = 0), s y = 2L µκS s c (Φ c = π 2 ) and s xy = 2L µκS s c (Φ c = π 4 )
The stiffness function k c is then calculated by using the formula (22), cf. Figure 6(a). Finally, its derivative k ′ c is calculated using the centered differences method, cf. Figure 6(b).

Validation of the approach

Now that we have identified the constitutive equations of the nodal element that represents the cracked transverse section, it is the objective of this section to introduce it in a one-dimensional FE code and see if we get sufficient agreement with three-dimensional representation. For a better validation of the approach, it is certainly more adequat to consider a cracked structure different from the one used in the identification procedure of the nodal element.

Let consider a cylinder of axis (oz), diameter D = 0.5m, quadratic moment of inertia I = πD 4 64 , total length 3m, clamped in z = 0 and subjected at the other end to a couple of forces

T = (T x = cos(α), T y = sin(α)) with α ∈ [0, 2π[
The cylinder contains, at z = 1m, a straight line front crack with a relative depth a D = 50%. Three-dimensional calculations accounting for the unilateral contact without friction between the lips of the crack are carried using the code aster c . The one-dimensional model of the same structure is made of two non-cracked elements of Timoshenko beam type and respective lengths 1m and 2m connected at z = 1m by a nodal element that represents the cracked section ( node 2), cf. Figure 7. Two cases are considered:

• in the first, the effects of shearing at the cracked section are neglected. Only discontinuities of rotations are then allowed at node 2 ([u x ] = [u y ] = 0). In this case we consider the model presented by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF]. • in the second, the nodal element behavior is described by relation (24): shearing effects are considered in this case. ing mechanism are negligible since the [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] model (with no consideration of the shearing effects) and the current model (taking into account the shearing effects) give the same results. The breathing mechanism is, thus, governed by the normal stresses on the crack lips.

Conclusions

The present approach consists in introducing the effects of shearing in the constitutive equations of a beam cracked transverse section. Three-dimensional calculations were carried out in order to identify the required nonlinear relations. In particular, the 3D nonlinear finite element calculations allowed the breathing mechanism of the crack to be predicted accurately. Based on the properties of the problem energy, the procedure presented in this paper could be applied to cracks of any shape and, moreover, to the case of multiple cracks affecting the same section. The approach was applied to the case of a single straight line front crack covering the half of a cylindrical beam cross section ( a D = 50%). For this form of crack, as illustrated in the Figure 8, the shearing effects on the breathing mechanism of the crack are negligible when compared to those of the flexural moments. The opening and closure of the crack are governed by the normal stress on the crack lips. The relative slip between the crack lips associated with shear forces remains very small compared to the rotations jumps due to the flexural moments at the cracked section (Figure 9).

Fig. 1 .

 1 Fig. 1. Crack breathing mechanism

Fig. 2 .

 2 Fig. 2. Geometry of the shaft cracked section

  Fig.3. The cracked beam model of[START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF]

  (c) ) from accurate three-dimensional FE model (Figure3(a)). The transverse cracked section (Figure3(b)) is replaced by two lumped nonlinear flexural and shearing springs (Figure3(d)) for which we aim to establish the constitutive equations. The three-dimensional FE model considered is that of a cylinder of axis (oz), diameter D = 1m, cross section S, length 2L = 4D, containing, at midspan, a cracked section, cf. Figure4. The structural element, clamped at its end z = 0, is subjected at z = 2L to a couple of shear forces and a couple of flexural moments: (T x (2L), T y (2L), M x (2L), M y (2L)).

Fig. 4 .

 4 Fig. 4. The current cracked beam modeling

  denotes the part of w * (F ) due to the resulting couple of flexural moments M = (M x (L), M y (L)) at the cracked section, and • w * c (F ) = w * c (T ) the part due to the couple of shear forces T = (T x (L), T y (L)).

  24)3.3 Identification of functions s c and k cThree-dimensional finite element calculations were carried out under the Code aster c to identify s c . The cracked structure, cf. Figure4(a), is then subjected, at

Fig. 5 .Fig. 6 .

 56 Fig. 5. Additional shearing flexibility s c , a D = 50%

Fig. 8 .

 8 Fig. 7. Structure Modelling: 3D and beam Finite elements

Figure 8 Fig. 9 .

 89 Figure8shows an excellent agreement between the beam and the threedimensional models. It appears that the effects of shearing on the crack breath-

Acknowledgements The author would like to thank Quoc-Son Nguyen and Habibou Maitournam (LMS, Ecole Polytechnique, France) for very helpful discussions.


Alexis Michaud

Katia Chirkova

Roselle Dobbs

Guillaume Jacques

Liberty Lidz

Martine Mazaudon

Xu Shixuan

THE TONES OF NUMERALS AND NUMERAL-PLUS-CLASSIFIER PHRASES: ON STRUCTURAL SIMILARITIES BETWEEN NAXI, NA AND LAZE *

Keywords: tone, numerals, classifiers, morpho-tonology, Naxi, Na, Mosuo, Laze, Muli Shuitian, Naish languages, phylogeny

Numeral-plus-classifier phrases have relatively complex tone patterns in Naxi, Na (a.k.a. Mosuo) and Laze (a.k.a. Shuitian). These tone patterns have structural similarities across the three languages. Among the numerals from '1' to '10', three pairs emerge: '1' and '2' always have the same tonal behaviour; likewise, '4' and '5' share the same tone patterns, as do '6' and '8'. Even those tone patterns that are irregular in view of the synchronic phonology of the languages at issue are no exception to the structural identity within these three pairs of numerals. These pairs also behave identically in numerals from '11' to '99' and above, e.g. '16' and '18' share the same tone pattern. In view of the paucity of irregular morphology-and indeed of morphological alternations in general-in these languages, these structural properties appear interesting for phylogenetic research. The identical behaviour of these pairs of numerals originates in morphophonological properties that they shared at least as early as the stage preceding the separation of Naxi, Na and Laze, referred to as the Proto-Naish stage. Although no reconstruction can be proposed as yet for these shared properties, it is argued that they provide a hint concerning the phylogenetic closeness of the three languages.

INTRODUCTION

The search for shared innovations between Na, Laze and Naxi

The position of Naxi within Sino-Tibetan is a topical issue in Sino-Tibetan historical linguistics. Naxi was classified as a member of the Loloish group by [START_REF] Shafer | Classification of the Sino-Tibetan languages[END_REF]; however, [START_REF] Bradley | Nahsi and Proto-Burmese-Lolo[END_REF] showed that it does not share the innovations that characterise this group and concluded that Naxi is "certainly not a Loloish language, and probably not a Burmish language either" (p. 6). Some scholars (especially in mainland China) nonetheless maintain the classification of Naxi as a member of the Loloish group (in Chinese: Yí yǔzhī 彝语支); Gai [START_REF] Xingzhi | Nàxīyǔ zài Zàng-Miǎnyǔ zhōng dē dīwèi 纳西语在藏缅语中的地位 (The position of Naxi within Tibeto-Burman)[END_REF] base this renewed claim on the high percentage of phonetically similar words between Naxi and Loloish, but without verifying the regularity of sound correspondences. Naxi has resemblances with neighbouring languages currently classified as Qiangic, such as Shixing (about which see [START_REF] Chirkova | Shǐxīng, a Sino-Tibetan language of South-West China: A grammatical sketch with two appended texts[END_REF] and references therein) and Namuyi (Lama Ziwo 1994; see also [START_REF] Hongkai | Nàxīyǔ zài Zàng-Miǎn yǔzú zhōng de lìshǐ dìwèi 纳 西语在藏缅语族语言中的历史地位 (The position of Naxi among Tibeto-Burman languages)[END_REF][START_REF] Fuquan | Introduction[END_REF].

This issue is related to that of the dialectal boundaries of Naxi. On the basis of a survey conducted in the 1950s, two areas were distinguished: (i) Western Naxi, a relatively homogeneous dialect area subdivided into three "patois" (tǔyǔ 土语): Dayanzhen 大研镇 (Lijiang old town), Lijiang plain 丽江坝, and Baoshan 宝山; and (ii) Eastern Naxi, divided into three "patois": Yongning 永宁, Beiquba 北渠 坝 and Guabie 瓜别 (He Jiren & Jiang Zhuyi 1985: 104-116). The 'Eastern Naxi' domain is actually more diverse than the distinction among three sub-varieties would suggest. The Naxi historians Guo Dalie and He Zhiwu provide a detailed list of subfamilies (zhīxì 支系) of the Naxi, and a dialect map (1999 [1994]: 5-9). The approach chosen here consists in examining language varieties that appear in Guo and He's list, and examining their similarities and differences. Some of these languages may turn out to not be closely related: for instance, the Shixing are mentioned as one of the variegated avatars of the Naxi (Guo Dalie & He Zhiwu 1999[START_REF] Dalie | Nàxīzú Shǐ 纳西族 史 (A history of the Naxi people)[END_REF]: 9), but available linguistic data suggest a relatively large distance between the Shixing language and the Naxi language. Conversely, a possibility to keep in mind is that some language varieties closely related to Naxi may not have been mentioned by these authors.

First-hand data on four language varieties are used here: two "dialects of Western Naxi" (A-sher and Nda-le), and two "dialects of Eastern Naxi": Yongning Na (a.k.a. Mosuo 摩梭) and Laze (a.k.a. Muli Shuitian 木里水田话). I use the language name 'Naxi' for the former, but not for the latter two, choosing to go by their autonyms instead, hence 'Na' and 'Laze'. Concerning Yongning Na, several publications insist on a close relationship with Naxi (see, e.g., a study of syntactic similarities between Yongning Na and Naxi: Jiang [START_REF] Zhuyi | Nàxīyǔ Dōngbù hé Xībù Fāngyán yǔfǎ yìtóng gàikuàng 纳西语东部和西部方言语法异同概况 (Syntactic similarities and differences between Western and Eastern dialects of Naxi)[END_REF]). Concerning Laze, Huang [START_REF] Huang Bufan | Mùlǐ Shuǐtiánhuà gàikuàng 木里水田话概况 (A survey of Muli Shuitian)[END_REF] cautiously concludes that its relationship with neighbouring languages-including Naxi and Loloish-is by no means selfevident and calls for in-depth research.

Naxi, Na and Laze are no exception to the notorious complexity of language names, ethnonyms and toponyms in East and Southeast Asia. Table 1 sets out the various names by which these languages are referred to, as well as the locations where they are spoken. The principle followed in this article is to use romanised autonyms for languages and place names (e.g. 'Laze' for /lɑ˧ze˧/); however, exceptions are made in the case of well-established names, hence the use of 'Naxi', andof 'Yongning Na' (following Lidz 2007, 2010). 1. Some information about the four language varieties under study.

A systematic study of sound correspondences between Naxi, Na and Laze has been conducted [START_REF] Jacques | The reconstruction of Proto-Naish: a preliminary study of the historical phonology of highly eroded Sino-Tibetan languages[END_REF]. The high number of regular correspondences provides strong support for the hypothesis that these three languages are closely related. Still, evidence in favour of a phylogenetic subgroup should ideally comprise shared innovations. Inflectional morphology can provide important evidence for the study of phylogeny-as exemplified, in the Sino-Tibetan domain, by Jackson Sun's comparison of Lavrung and Rgyalrongish languages (Sun 2000a, b; for an overview about morphology in Sino-Tibetan, see [START_REF] Jacques | La morphologie du sino-tibétain[END_REF]. However, there is very little inflectional morphology in Na, Laze and Naxi. Few cases of suppletion have been detected so far; Table 2 shows data on the verbs 'to go' and 'to come', the number '10', and the word for 'year'. Within the broader Sino-Tibetan context, the three subsets in Table 2 have different implications.

(i) The same suppletion for 'year' is also found in Shixing: /kʰu 55 / vs. /bɘ 55 / (data from Huang [START_REF] Bufan | Shǐxīngyǔ 史兴语 (The Shǐxīng language)[END_REF], and in Qiangic languages such as Tangut, Pumi (a.k.a. Prinmi), and Muya (Jacques, manuscript), so that it does not provide evidence of a special closeness between Na, Laze and Naxi.

(ii) Suppletion for 'to go', with a PAST and a NON-PAST, is common in languages of the area. (There also exist imperative forms of 'to go' and 'to come', which will not be discussed here.) Examples include languages where the forms are clearly different from those in Na, Laze and Naxi, such as Lizu [START_REF] Chirkova | Essential characteristics of Lizu, a Qiangic language of Western Sichuan[END_REF], but also languages with comparable forms: Namuyi (a.k.a. Namuzi, Namʑi) and Shixing. In the variety of Namuyi spoken in the county of Muli 木里 (Liangshan, Sichuan), the forms are /bie 13 / for NON-PAST and /hũ 53 / for PAST (Katia Chirkova, p.c.). In Shixing, the forms are /bi 55 / and /hu 53 /, respectively (Huang [START_REF] Bufan | Shǐxīngyǔ 史兴语 (The Shǐxīng language)[END_REF]. If it could be verified that they are cognate, this would provide a (small) piece of evidence for a phylogenetic group including Shixing and Namuyi along with Na, Laze and Naxi (a group referred to as Naic in [START_REF] Jacques | The reconstruction of Proto-Naish: a preliminary study of the historical phonology of highly eroded Sino-Tibetan languages[END_REF], not for a language group containing the latter three languages to the exclusion of others.

(iii) Suppletion for 'to come', which is not found in Laze, will not be discussed further here.

(iv) Suppletion for '10' is attested (in various forms) in several Sino-Tibetan languages, including Tibetan, and borrowings or calques are not unattested (Mazaudon 1985: 127). "The unaspirated initial in 'twenty' is found in nearly all Northern Ngwi languages, and in Central Ngwi languages such as Lisu, Lipo, Lolopo, Sani and so on, but the regular aspirated form is found in other Central Ngwi languages such as Lahu and Jino, and in Southern Ngwi languages" (Bradley 2005: 233). For instance, Nuosu (Liangshan Yi) has /tsʰi˧/ for 'ten' in numerals below 20 and above 29, as against unaspirated /tsi˧/ in the range from 20 to 29 (Li Min & Ma Ming 1983: 108-111). A difference with Naxi, Laze and Na is that, in Loloish (a.k.a. Ngwi), there is a two-way suppletion, not a three-way suppletion: 'ten' has the same form below 20 and above 29. A three-way suppletion is found in Shixing; note, however, that the phonetic forms in Shixing are widely different from those in Na, Laze and Naxi (/qa 55 /, /ʁa 55 / and /qa 33 /, respectively, in the notation of Huang [START_REF] Bufan | Shǐxīngyǔ 史兴语 (The Shǐxīng language)[END_REF], so they may result from an independent development.

To sum up, among the three data sets in Table 2, only the three-way suppletion for '10' could be adduced as potential evidence of a special phylogenetic closeness between Na, Laze and Naxi. The evidence is thin, however: the possibility of borrowing cannot be excluded. "The Thai numbers, for instance, once thought to be derived from the same ancestral forms as the Chinese numbers, and so used to argue for the genetic relationship of these two large language groups, have been shown to have been borrowed from Chinese into Thai at an early date" (Mazaudon 2007, citing Benedict 1942).

Interesting evidence may come-somewhat paradoxically-from the tonal morphology of Naxi, Na and Laze. Some preliminary definitions are useful at this point. The concept of tone sandhi is here restricted to synchronic, productive phonological 'rules' of tone change triggered by adjacent tones; this is a narrower definition than that often given to this concept in the literature (see, e.g., Chen 2000: 25). The phenomena referred to by Bradley 2005 as "irregular tone sandhi" are here labelled as 'irregular tonal alternations'. 'Tonal morphology', a term used interchangeably with 'morphotonology', consists of all the rules that determine the tone patterns of specific syntactic structures, such as determinative compounds, complex predicates, nominal determiners, etc. Guillaume Jacques (p.c.) notes that irregular morphology in Naxi, Na and Laze, as also in Yi and Pumi, largely consists in irregular morphotonology: tone patterns which can be accounted for neither by sandhi rules nor by regular morphotonological rules. However, the differences between the prosodic systems of Naxi, Na and Laze do not allow for a term-to-term comparison of their morphotonology. For instance, the complex tone patterns of compound nouns in Yongning Na, which differ for determinative compounds and copulative compounds [START_REF] Michaud | Yúnnánshěng Lìjiāngshì Yǒngníng qūyù Mósuōhuà zhōng bīnlín xiāoshī de shēngdiào yǔ 摩 梭 话 中 濒 消 位 (Endangered tonal and phonemic oppositions in the variety of Mosuo/Na spoken in Yongning, Lijiang, Yunnan)[END_REF], have no equivalent in Laze and Naxi. Likewise, the peculiar tone patterns of reduplication in Naxi appear to reflect an evolution that is specific to this language (Michaud & Vaissière 2007a, b).

There is nonetheless at least one set of tonal data that lends itself to comparison across the three languages: the tone patterns of numeral-plusclassifier phrases. These patterns, though different in the three languages, share structural similarities. Three pairs of numerals emerge: '1' and '2' always have the same tonal behaviour; likewise, '4' and '5' share the same tone patterns, as do '6' and '8'. Sections 2 to 4 set out the facts in Naxi, Laze and Na, going from the simplest system (Naxi) to the most complex (Na). Section 5 provides a general discussion, arguing that the identical behaviour of these pairs of numerals originates in morpho-phonological properties that they shared at least as early as the stage preceding the separation of Naxi, Na and Laze, referred to as the Proto-Naish stage. Although no reconstruction can be proposed as yet for these properties, they appear to support the hypothesis that Na, Laze and Naxi belong in the same subgroup of Sino-Tibetan.

Reflections on the potential time depth of tone sandhi and irregular tonal patterns

The notion that tonal irregularities can go back a long way may require some clarification. It is well established that the diachronic evolution of tone systems can proceed at a rapid rate; close dialects can have widely different tone systems; witness the great diversity of prosodic systems across dialects of Japanese or Chinese. Contact between dialects is apt to add extra complexity to tonal correspondences (a detailed example in the Japanese domain is provided by [START_REF] Kubozono | Tonal change in language contact: Evidence from Kagoshima Japanese[END_REF]; this greatly lessens the chances of recovering hints about the distant past by the comparative method. Naxi, Na and Laze are no exception: their prosodic systems have important structural differences. Naxi is close to the 'omnisyllabic' type (one tone per syllable), with only a few phenomena of tonal change (Michaud 2006b, Michaud & He Xueguang 2007a); at the other extreme, Yongning Na has a complex tonal system involving a paradigm of four tone patterns for monosyllabic verbs and five for monosyllabic nouns, as well as ubiquitous morphotonological phenomena [START_REF] Michaud | Phonemic and tonal analysis of Yongning Na[END_REF]. The diachronic evolution of the prosodic systems of Naxi, Na and Laze has remained little explored since Bradley's hypotheses on the development of the High tone of Naxi (1975) and Matisoff's study of the evolution of stopped syllables in " Burmo-Naxi-Lolo" (1991). [START_REF] Jacques | The reconstruction of Proto-Naish: a preliminary study of the historical phonology of highly eroded Sino-Tibetan languages[END_REF] deliberately focus on vowels and consonants, provisionally leaving tone aside. In this situation, adducing tone patterns as phylogenetic evidence may appear as a misguided venture.

However, there is convincing evidence from other languages showing that tonal alternations and irregularities may have great historical depth, sometimes going back to segmental phenomena that predate the development of tones. Evidence to this effect is found in Vietnamese (Austroasiatic) and in Hmông-Miên, as well as in Sino-Tibetan. In Vietnamese, the word for 'ten' has two tonal realisations: mười (IPA: /mɯɤj/, tone A2) in the numbers from 10 to 19; and mươi (same segmental phonemes: /mɯɤj/, with a different tone: A1) in the numbers from 20 to 99. There exist a few other pairs:

-'many' is nhiều, with tone A2; 'how much' is bao nhiêu, where the syllable nhiêu has tone A1 -ngồi 'to sit' (A2); ngôi 'to enthrone' (A1) -vòi '(elephant's) trunk' (A2); voi 'elephant' (A1)

An important fact about Vietnamese is that it does not have any tone sandhi phenomena. What tonal alternations it has are due to the historical influence of a preceding syllable which came to function as a prefix and finally coalesced with the main syllable. Michel Ferlus (p.c.) hypothesises that the phrase which evolved into the present-day structure for numerals from 20 to 90 (hai mươi '20', ba mươi '30', etc.) contained a prefixed syllable before the root for '10', with the meaning 'a set of 10': '20' was 'two sets of 10', etc. The only trace which remains of this prefix is the changed tone of the syllable mươi: the prefix must have had an unvoiced initial, leading to the treatment of the word for '... set of 10' as an unvoiced-initial word at the stage where the voiced and unvoiced series of initials merged [START_REF] Haudricourt | De l'origine des tons en vietnamien[END_REF][START_REF] Haudricourt | Bipartition et tripartition des systèmes de tons dans quelques langues d'Extrême-Orient[END_REF]. The word therefore developed a high-series tone (A1), whereas the unprefixed word for 'ten' went into the low series (hence its tone A2 and not A1) due to its voiced initial. (The reconstruction and its time frame remain to be worked out in detail.) Likewise, 'elephant' can be analysed as deriving from 'trunk' ('trunk-bearer'). The tonal alternations found in Hmông-Miên languages also go back a long way (Ratliff 1992, passim). Within Sino-Tibetan, the history of Tibetan provides clear examples of the effects of historical prefixes and final consonants in tonogenesis [START_REF] Sun | Variegated tonal developments in Tibetan[END_REF]. Although further discussion of this topic falls outside the scope of this paper, it appeared useful to recall these illustrations of the fact that tonal morphology can "furnish invaluable clues to earlier stages of the language" (Matisoff 1973: 28).

The syntax of the noun phrase: some observations that hold for Naxi, Laze and Na

As a final introductory note, here are some remarks about the syntax of noun phrases which include a numeral+classifier phrase in Naxi, Laze and Na. The basic structure is noun+numeral+classifier. The structure of the noun phrase is as in (1), where NP=Noun Phrase, N=Noun, DET=Determiner, NUM=Numeral, and CL=Classifier.

(1) NP N DET NUM CL Bradley's generalisations about Loloish languages are also valid for Naxi, Laze and Na: "The numeral plus classifier construction follows the head of the NP. It is closely-bound and cannot have anything inserted between the final numeral and the following classifier" (Bradley 2005: 224). That the determiner makes up a unit within the noun phrase is evidenced by the tonal interaction between the numeral and the classifier, studied at length in the present article. There is no tonal interaction between the noun and the determiner: the tone pattern of the determiner is fully independent from that of the noun. Lastly, concerning the inventory of classifiers and their semantic extension, the reader is referred to Thomas Pinson's Naxi-Chinese-English glossary (Pinson 1998: 245-251), which contains some 90 classifiers, and to Lidz (2006: 8-14) for a description of about 50 classifiers in use in Yongning Na. The following sections focus exclusively on the tone patterns of numeral-plus-classifier phrases.

NUMERAL-PLUS-CLASSIFIER PHRASES IN NAXI

The Naxi facts are presented first because they are simpler than those in Laze (Section 3) and Na (Section 4). There are three basic lexical tones in Naxi: H(igh), M(id) and L(ow); the three tonal categories of classifiers correspond transparently to these three categories.

A-sher Naxi

A-sher (/ɑ˧ʂə˞˩/) is the Naxi name of a village located to the northeast of the city of Lijiang (see Michaud 2006a for a map and detailed coordinates). The tonal patterns of numeral-plus-classifier phrases are shown in Table 3. 1 The phrases 1 The phonetic transcription used here is narrower than a strictly phonemic analysis. In particular, in Naxi, Na and Laze, [ɲ] is an allophone of /ŋ/ before high front vowels, and [ɿ] and [ʅ] are allophones of /ɯ/ after fricatives and affricates ([ɿ] after dentals and [ʅ] after retroflexes). A narrow notation appeared more adequate to convey a feel for the pronunciation of the items at issue. For phonemic analyses of Naxi, the reader is referred to He Jiren & Jiang [START_REF] Jiren | Nàxīyǔ jiǎnzhì 纳西语简志 (A presentation of the Naxi language)[END_REF]Michailovsky &Michaud 2006. Concerning the phonemic analysis of Yongning Na, one correction is to be made to Michaud 2008: [e] and [i] were considered as allophones; I have now realised that they contrast after coronal fricatives and affricates. The suppletive forms of '10' in Table 2 are a case in point: the /e/ and /i/ are contrastive in /tsʰe/ and /tsʰi/; in turn, these contrast with /ɯ/, as in /tsʰɯ˧˥/ 'goat'. A phonemic analysis of Laze is set out in Michaud and Jacques (to appear).

˧ ˧ ˥ ˧ ˧ ˧ ˩ ˧ ˥ ˧ ˧ ˧ ˩ 2 ɲi˧ ɲi˧pʰe˥ ɲi˧ly˧ ɲi˧kʰɯ˩ ˧ ˧ ˥ ˧ ˧ ˧ ˩ ˧ ˥ ˧ ˧ ˧ ˩ 3 sɿ˩ sɿ˥pʰe˧ sɿ˥ly˧ sɿ˧kʰɯ˩ ˩ ˩ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˧ ˩ ˧ ˩ 4 lu˧ lu˥pʰe˧ lu˥ly˧ lu˧kʰɯ˩ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˧ ˩ ˧ ˩ 5 wɑ˧ wɑ˥pʰe˧ wɑ˥ly˧ wɑ˧kʰɯ˩ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˧ ˩ ˧ ˩ 6 ʈʂʰwɑ˥ ʈʂʰwɑ˥pʰe˧ ʈʂʰwɑ˥ly˧ ʈʂʰwɑ˥kʰɯ˩ ˥ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˩ ˥ ˩ 7 ʂə˞˧ ʂə˞˧pʰe˥ ʂə˞˧ly˧ ʂə˞˧kʰɯ˩ ˧ ˧ ˥ ˧ ˧ ˧ ˩ ˧ ˥ ˧ ˧ ˧ ˩ 8 ho˥ ho˥pʰe˧ ho˥ly˧ ho˥kʰɯ˩ ˥ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˩ ˥ ˩ 9 ŋgv̩ ˧ ŋgv̩ ˧pʰe˥ ŋgv̩ ˧ly˧ ŋgv̩ ˧kʰɯ˩ ˧ ˧ ˥ ˧ ˧ ˧ ˩ ˧ ˥ ˧ ˧ ˧ ˩ 10 tsʰe˩ tsʰe˩pʰe˥ tsʰe˩ly˧ tsʰe˧kʰɯ˩ ˩ ˩ ˥ ˩ ˧ ˧ ˩ ˩ ˥ ˩ ˧ ˧ ˩ numeral in isolation Tone of classifier H M L H M L
The tones of numeral-plus-classifier phrases in Naxi, Na and Laze 9 with irregular tone patterns are set in bold type. In order to facilitate reading, the tone patterns are recapitulated on the right-hand side of the table, with identical cell shadings for combinations with identical tone patterns. The data collection method is the same for all the language varieties described in this article: it consists of direct elicitation via Mandarin Chinese and verification in a set of transcribed narratives. For ease of reference across publications, the identity of the language consultants is indicated by means of a unified code: M for male, F for female, followed by a number.

Above 99, the higher-level units are: 100; 1,000; 10,000 (the equivalent of Chinese wàn 万); and 10,000 times 10,000 (one hundred million, the equivalent of Chinese yì 亿). The last two digits are expressed the same way as in the range between 1 and 99.

The facts in Table 3 are identical with those reported in linguistic descriptions of Naxi (such as the data on the Qinglong 青龙 dialect-present-day Changshui 长水-reported by He Jiren et al. 1985: 12-14) and with the examples found in the editions of traditional Naxi tales and rituals, such as He [START_REF] Kaixiang | Nàxī Dōngbā gǔjí yìzhù 纳西东巴古籍译注(一) (An annotated translation of Naxi Dongba classical texts[END_REF]. The only tonal difference that I noticed concerns '1' and '2'. In the book coauthored by He Jiren and Jiang [START_REF] Jiren | Nàxīyǔ jiǎnzhì 纳西语简志 (A presentation of the Naxi language)[END_REF], they are transcribed with Mid tone on page 59 (/ɖɯ˧/, /ɲi˧/), but with Low tone on page 170 (/ɖɯ˩/, /ɲi˩/). This is a dialectal difference: '1' and '2' are pronounced in the Mid tone in some dialects, including A-sher, but with Low tone in other dialects, including Qinglong 青龙 (present-day Changshui 长水; see, e.g., He Zhiwu 1981: 145) and Yangxi 漾西, He Jiren's native dialect (we made recordings in 2002). Both of these dialects are spoken in neighbouring areas in the Lijiang plain, i.e. at the heart of the Naxi area. This observation will be taken up again in the general discussion (Section 5).

Most of the tone sequences observed in numeral-plus-classifier phrases correspond to a simple sequence of the tone of the numeral and that of the classifier. Those that do not (set in bold type in the table) concern the numerals '3', '4', '5', '6', '8' and '10'. If these cases resulted from a synchronic rule of tone sandhi, the same output would be expected from each of the M-tone numerals '1', '2', '4', '5', '7' and '9'. This prediction is not borne out (e.g., with M-tone classifier: /ɖɯ˧ly˧/ 'one...', with MM tone pattern, but /lu˥ly˧/ 'four...', with HM). Similarly, the outputs of the two L-tone numerals '3' and '10' differ from each other.

Two options are open in the analysis of these facts: either the patterns are irregular, and learnt as such by the speakers; or the inventory of lexical tones must be extended: if '4' and '5' pattern differently from '1' and '2', it could indicate that they have different lexical tones, and that the contrast between the two tonal categories at issue is neutralised in isolation. Such neutralisations are indeed observed in Yongning Na, a closely related language discussed in Section 4. In the case of Naxi, however, postulating additional lexical tones is hardly a satisfactory solution: one would then need to postulate five tones, two of them neutralising to L in isolation (those of the words for '3' and '10'), and two others neutralising to M in isolation (those of {4, 5} on the one hand, and those of {1, 2, 7, 9} on the other).

There is no evidence anywhere else in the language for these categories, however, whereas there do exist a few other irregular tonal patterns in the lexicon. 2 The 2 For instance, /nv̩ ˧hə˞ ˩/ 'mung bean' can be transparently analysed as composed of /nv̩ ˩/ 'bean' and /hə˞ ˩/ 'green'; its tone pattern is synchronically opaque: in the present state of the language, compounding is not accompanied by any tonal change (Michaud & Vaissière 2007a: 34). The tones of numeral-plus-classifier phrases in Naxi, Na and Laze interpretation chosen here is that some numeral-plus-classifier phrases have irregular tone patterns. The patterns in Table 3 that do not coincide with the simple sequence of the lexical tones of their constituting elements are taken to be irregular patterns, inherited from earlier historical stages. This situation is areally common; detailed examples in Loloish are analysed by [START_REF] Bradley | Why do numerals show 'irregular' correspondence patterns in Tibeto-Burman? Some Southeastern Tibeto-Burman examples[END_REF].

The irregular patterns of Naxi are complex, in that both the numeral and the classifier can undergo tone change: in /ho˥pʰe˧/ ('8' plus H-tone classifier), the tone of the classifier is different from its realisation in isolation; in /sɿ˥ly˧/ ('3' plus M-tone classifier), the tone of the classifier is as expected, whereas that of the numeral is modified; finally, in /wɑ˥pʰe˧/ ('5' plus H-tone classifier), both differ from their realisation in isolation.

From Table 3, it appears that there are five sets of numerals: '3' and '10' are singletons, each with its own tonal behaviour, whereas '4' and '5' pattern together, as do two other sets: {6, 8} and {1, 2, 7, 9}. Interestingly, these sets also behave identically in numerals from '11' to '99' and above: thus, '16' and '18' share the same tone pattern, as do '60 ' and '80', '106' and '108'; '14' and '15', etc. These observations will be central to the general discussion.

Nda-le Naxi

Nda-le (/ndɑ˧le˩/) is the Naxi name of a village located southeast of the city of Lijiang (see, again, Michaud 2006a for a map and detailed coordinates). The Naxi dialect of Nda-le is close to that of A-sher; the comparison of these dialects allows for a fine-grained investigation into dialectal diversity. There are only three tonal categories of classifiers in Nda-le, which coincide exactly with those of A-sher. On the other hand, the tonal patterns of numeral-plus-classifier phrases are not identical: see Table 4. As in Tables 3 and5, the irregular cases are indicated in bold: in these cases, the tone pattern of the determiner is not simply the concatenation of the tones of the numeral and the classifier.

Again, the question arises whether these cases result from a sandhi rule taking as its input the tones of the numeral and of the classifier. Nda-le Naxi does have one sandhi rule which comes into play for some of the forms in Table 4; since this relatively complex phenomenon is not central to the present issue, it is set out in the Appendix. From the vantage point of the present research, an important observation is that tone sandhi does not account for all the data: to take an example, '3' and '10' have the same tone (L) but do not pattern together in every case. In association with a H-tone classifier, 'ten' yields LH (/tsʰe˩pʰe˥/) whereas 'three' yields /sɿ˧pʰe˥/. There is therefore more to the data in Table 4 than the application of sandhi rules.

The irregular patterns concern the numerals '3', '4', '5', '6', '8' and '10', as in A-sher Naxi (Section 2.1). Again as in A-sher Naxi, the numerals of Nda-le Naxi can be classified into five sets on the basis of their tonal behaviour: '3' and '10' are singletons, each with its own tonal behaviour, whereas '4' and '5' pattern together, as do two other sets: {6, 8} and {1, 2, 7, 9}. Another similarity with A-sher is that these sets also behave identically in numerals from '11' to '99' and above: thus, '16' and '18' share the same tone pattern, as do '60' and '80', etc.

As a final note about Naxi dialects, the language variety spoken in Fv-kho (Naxi: /fv̩ ˥kʰo˧/; Chinese: 峰 科 Fèngkē; see Michaud 2006a for detailed coordinates) also partakes in the structural pattern studied here; the pairs that pattern together are {1, 2}, {4, 5}, and {6, 8}.

NUMERAL-PLUS-CLASSIFIER PHRASES IN LAZE

Laze is an as yet unclassified language spoken in the county of Muli 木里 (Liangshan 凉山, Sichuan); the reader is referred back to Table 1 for information about the various names that have been given to this language. It was first studied by Huang Bufan in the 1980s (Huang Bufan 2009). Its tonal system, while somewhat richer than that of Naxi, remains relatively straightforward (see [START_REF] Michaud | Mùlǐ shuǐtiánhuà shēngdiào xìtǒng yánjiū 木里水田话声 调系统研究 (The prosodic system of Muli Shuitian/Laze)[END_REF]. There are four categories of monosyllabic nouns: H(igh), M(id), L(ow), and M-to-H, and all classifiers likewise have one of these four tones. However, a further distinction must be made among H-tone classifiers on the basis of their tonal behaviour in numeral+CL phrases: the classifiers 'year' /kʰv̩ ˥/ and 'basket' /kʰɤ˥/ have different behaviours when associating with a numeral, as shown in Table 5a, where a 'year' type and a 'basket' type are distinguished among H-tone classifiers. Note that no citation forms are provided for Laze numerals, because they cannot be said in isolation. Unlike in Naxi, the numerals in the range from '11' to '19' do not behave simply as in the range from '1' to '9'. For reasons of space, numbers above '10' are not presented here for all types of classifiers. However, it appeared useful to present data from '11' to '20' for the 'year' and 'basket' types of H-tone classifiers, because these data constitute additional evidence that these two types are indeed distinct: see Table 5b. Numerals could not be elicited in isolation, the consultants insisting on providing a complete determiner, with a classifier-as is done when counting items. For want of more direct evidence, one may reason as follows. Tone patterns in which the determiner receives a tone that differs from its lexical tone will be considered as irregular; they are therefore set in bold in Table 5a. The lexical tone of numerals can then be extracted from the remaining forms, considered as regular. This yields: L tone for '6' and '8'; M tone for '1', '2', '4', '5', '7' and '9'; and H tone for '3' and '10' (unless '3', which is thoroughly irregular, is analysed as having a MH tone). In turn, this allows for a distinction between two sets of irregular patterns:

(i) those that result from two processes that are also observed elsewhere in the language: M+L>H+H, and L+M>L+L. In the process of lexicalisation, compounds with an M+L pattern are observed to change to H+H, and compounds with an L+M pattern to L+L, e.g. /wɤ˩/ 'pig' and /dze˧/ 'food' (with M tone) yielding /wɤ˩dze˩/ 'pig feed, swill' [START_REF] Michaud | Mùlǐ shuǐtiánhuà shēngdiào xìtǒng yánjiū 木里水田话声 调系统研究 (The prosodic system of Muli Shuitian/Laze)[END_REF].

(ii) those that involve the numerals '3' and '10', for which no parallel has been found elsewhere in the language.

The former can be interpreted as relatively recent tonal changes, the latter as relatively ancient irregularities. To sum up the observations made about Laze, (i) unaccountable irregularities concern the numerals '3' and '10', and (ii) numerals can be classified into five sets on the basis of their tonal behaviour: '3' and '10' are singletons, each with its own tonal behaviour, whereas '7' and '9' pattern together, as do two other sets: {6, 8} and {1, 2, 4, 5}. As in Naxi, these sets also behave identically in numerals from '11' to '99' and above: thus, '17' and '19' share the same tone pattern, as do '70' and '90', etc. These facts will be taken up in the general discussion.

1 ɖɯ˧ʈʂae ɖɯ˥gv̩ ˥ ɖɯ˧ɲi˧ ɖɯ˧kʰv̩ ˥ ɖɯ˧kʰɤ˥ ˧ ˥ ˥ ˧ ˧ ˧ ˥ ˧ ˥ ˧ ˥ ˥ ˧ ˧ ˧ ˥ ˧ ˥ 2 ɲi˧ʈʂae ɲi˥gv̩ ˥ ɲi˧ɲi˧ ɲi˧kʰv̩ ˥ ɲi˧kʰɤ˥ ˧ ˥ ˥ ˧ ˧ ˧ ˥ ˧ ˥ ˧ ˥ ˥ ˧ ˧ ˧ ˥ ˧ ˥ 3 su˥ʈʂae˩ su˥gv̩ ˧ su˥ɲi˥ su˥kʰv̩ ˩ su˥kʰɤ˩ ˥ ˩ ˥ ˧ ˥ ˥ ˥ ˩ ˥ ˩ ˥ ˩ ˥ ˧ ˥ ˥ ˥ ˩ ˥ ˩ 4 ʐv̩ ˧ʈʂae ʐv̩ ˥gv̩ ˥ ʐv̩ ˧ɲi˧ ʐv̩ ˧kʰv̩ ˥ ʐv̩ ˧kʰɤ˥ ˧ ˥ ˥ ˧ ˧ ˧ ˥ ˧ ˥ ˧ ˥ ˥ ˧ ˧ ˧ ˥ ˧ ˥ 5 ŋwɤ˧ʈʂae ŋwɤ˥gv̩ ˥ ŋwɤ˧ɲi˧ ŋwɤ˧kʰv̩ ˥ ŋwɤ˧kʰɤ˥ ˧ ˥ ˥ ˧ ˧ ˧ ˥ ˧ ˥ ˧ ˥ ˥ ˧ ˧ ˧ ˥ ˧ ˥ 6 qʰɔ˩ʈʂae qʰɔ˩gv̩ ˩ qʰɔ˩ɲi˩ qʰɔ˩kʰv̩ ˥ qʰɔ˩kʰɤ˥ ˩ ˩ ˩ ˩ ˩ ˩ ˥ ˩ ˥ ˩ ˩ ˩ ˩ ˩ ˩ ˥ ˩ ˥ 7 ʂʅ˧ʈʂae ʂʅ˥gv̩ ˥ ʂʅ˧ɲi˧ ʂʅ˥kʰv̩ ˥ ʂʅ˧kʰɤ˥ ˧ ˥ ˥ ˧ ˧ ˥ ˥ ˧ ˥ ˧ ˥ ˥ ˧ ˧ ˥ ˥ ˧ ˥ 8 ɕi˩ʈʂae ɕi˩gv̩ ˩ ɕi˩ɲi˩ ɕi˩kʰv̩ ˥ ɕi˩kʰɤ˥ ˩ ˩ ˩ ˩ ˩ ˩ ˥ ˩ ˥ ˩ ˩ ˩ ˩ ˩ ˩ ˥ ˩ ˥ 9 gv̩ ˧ʈʂae gv̩ ˥gv̩ ˥ gv̩ ˧ɲi˧ gv̩ ˥kʰv̩ ˥ gv̩ ˧kʰɤ˥ ˧ ˥ ˥ ˧ ˧ ˥ ˥ ˧ ˥ ˧ ˥ ˥ ˧ ˧ ˥ ˥ ˧ ˥ 10 tsʰi˥ʈʂae tsʰi˥gv̩ ˧ tsʰi˥ɲi˥ tsʰi˥kʰv̩ ˩ tsʰi˥kʰɤ˧ ˥ ˥ ˥ ˧ ˥ ˥ ˥ ˩ ˥ ˧ ˥ ˧ ˥ ˥ ˥ ˩ ˥ ˧

NUMERAL-PLUS-CLASSIFIER PHRASES IN YONGNING NA

In the lexical data provided by He Jiren et al. (1985: 135-172), as also in the language synopsis by Yang [START_REF] Zhenhong | Mósuōhuà gàikuàng 摩梭话概况 (An overview of the Mosuo language)[END_REF][START_REF] Zhenhong | An overview of the Mosuo language[END_REF], four tones are reported for Yongning Na: High, Mid, Low, and Rising. The prosodic system of Yongning Na is actually of greater complexity, but several of the tonal contrasts are neutralised when words are spoken in isolation. A similar situation is found in Pumi (see [START_REF] Ding | A typological study of tonal systems of Japanese and Prinmi: Towards a definition of pitch-accent languages[END_REF][START_REF] Jacques | Tonal alternations in the Pumi verbal system[END_REF]). In the variety analysed here, on the surface, each syllable has one of three level tones (High, Mid or Low) or one of two contours (Low-to-Mid or Mid-to-High); there are strong arguments in favour of the analysis of contours as sequences of two levels. Five tonal classes of monosyllabic nouns have been brought out, eleven classes of disyllabic nouns, The tones of numeral-plus-classifier phrases in Naxi, Na and Laze and four classes of monosyllabic verbs [START_REF] Michaud | Phonemic and tonal analysis of Yongning Na[END_REF]. If a classifier is "a type of limited noun that occurs only after numerals (...), and whose selection is determined by a preceding (overt or implicit) noun" (Matisoff 1973: 88), one may expect a degree of tonal complexity for classifiers that is comparable to that of nouns-and hence a considerably more complex picture than in Naxi (Section 2), which only has three basic tones. The tonal complexity of numeral-plus-classifier phrases clearly bears out this prediction, witness the six categories brought out in Table 6. The form for '1' followed by the classifier for humans is provided in brackets because a different classifier is used with this numeral; no tone pattern is indicated in the right hand-side of the table for this combination.

˥ ˥ ˩ ˥ ˥ ˧ ˥ ˥ ˩ ˥ ˥ ˧ 12 tsʰi˥ɲi˥kʰv̩ ˩ tsʰi˥ɲi˥kʰɤ˧ ˥ ˥ ˩ ˥ ˥ ˧ ˥ ˥ ˩ ˥ ˥ ˧ 13 tsʰi˥su˥kʰv̩ ˩ tsʰi˥su˥kʰɤ˧ ˥ ˥ ˩ ˥ ˥ ˧ ˥ ˥ ˩ ˥ ˥ ˧ 14 tsʰi˥ʐv̩ ˥kʰv̩ ˩ tsʰi˥ʐv̩ ˥kʰɤ˧ ˥ ˥ ˩ ˥ ˥ ˧ ˥ ˥ ˩ ˥ ˥ ˧ 15 tsʰi˥ŋwɤ˥kʰv̩ ˩ tsʰi˥ŋwɤ˥kʰɤ˧ ˥ ˥ ˩ ˥ ˥ ˧ ˥ ˥ ˩ ˥ ˥ ˧ 16 tsʰi˥qʰɔ˩kʰv̩ ˥ tsʰi˥qʰɔ˩kʰɤ˥ ˥ ˩ ˥ ˥ ˩ ˥ ˥ ˩ ˥ ˥ ˩ ˥ 17 tsʰi˥ʂʅ˥kʰv̩ ˩ tsʰi˥ʂʅ˥kʰɤ˧ ˥ ˥ ˩ ˥ ˥ ˧ ˥ ˥ ˩ ˥ ˥ ˧ 18 tsʰi˥ɕi˩kʰv̩ ˥ tsʰi˥ɕi˩kʰɤ˥ ˥ ˩ ˥ ˥ ˩ ˥ ˥ ˩ ˥ ˥ ˩ ˥ 19 tsʰi˥gv̩ ˥kʰv̩ ˩ tsʰi˥gv̩ ˥kʰɤ˧ ˥ ˥ ˩ ˥ ˥ ˧ ˥ ˥ ˩ ˥ ˥ ˧ 20 ɲi˧tsi˧kʰv̩ ˥ ɲi˧tsi˧kʰɤ˥ ˧ ˧ ˥ ˧ ˧ ˥ ˧ ˧ ˥ ˧ ˧ ˥

Tone of classifier H

The data presented here are from a female speaker (F4) born in 1950; in the speech of younger speakers, some of these contrasts are becoming lost, bringing the tone system closer to an 'omnisyllabic' type (one tone per syllable) as in Chinese, the dominant language. Observations on this topic are proposed by [START_REF] Michaud | Yúnnánshěng Lìjiāngshì Yǒngníng qūyù Mósuōhuà zhōng bīnlín xiāoshī de shēngdiào yǔ 摩 梭 话 中 濒 消 位 (Endangered tonal and phonemic oppositions in the variety of Mosuo/Na spoken in Yongning, Lijiang, Yunnan)[END_REF].

Taking into account the range of numerals from 1 to 30, I have found no less than eleven different categories of classifiers. For instance, /kʰwɤ˥/ 'piece of...', which has the same behaviour as /ɲi˥/ 'day' up to '13', has different tone patterns for '14 ', '15', '16', '18', '19', '22', '25', '29' and '30', requiring the recognition of these two classifiers as members of distinct categories. This system may appear staggeringly complex; however, numeral-plus-classifier phrases are extremely frequent in discourse, a factor which is known to favour the preservation of irregular morphology. The system of Yongning Na would warrant a detailed study of the occasional hesitations or 'mistakes' of speakers in the tone patterns of such phrases, in the spirit of Henri Frei's Grammar of Mistakes [START_REF] Frei | La grammaire des fautes : introduction à la linguistique fonctionnelle, assimilation et différenciation, brièveté et invariabilité, expressivité[END_REF]. It is not clear at this stage which of the forms can be considered as irregular: in Naxi and Laze, the 'regular', 'default' mode of association of a numeral and a determiner is that each surfaces with its lexical tone, whereas the Na data in Table 6 cannot be accounted for by a set of simple rules. For the sake of the present argument, let us simply note that three pairs of numerals stand out: '1' and '2' always have the same tonal behaviour; likewise, '4' and '5' share the same tone patterns, as do '6' and '8'. As in Naxi and Laze, these sets also behave identically in numerals from '11' to '99' and above: thus, '14' and '15' share the same tone pattern, etc.

GENERAL DISCUSSION AND CONCLUSION

The mathematical structure of the system found in Laze, Na and Naxi, a decimal system with higher-level units of 10,000 and 100,000,000 as in Chinese, is widespread in the area, unlike the more unusual counting systems analysed by Mazaudon 2002Mazaudon , 2007. . It is therefore useless for phylogenetic classification. As for the facts presented in Sections 2-4, they can be interpreted as separate innovations, in light of the lack of surface similarity between Naxi, Laze and Na; 3 and '10' are stable (though not in collocations for '20'), but no classifiers are; and in the Na data, no numerals or classifiers are stable.

Table 6. Tonal patterns in Yongning Na for [ɲi] 'day', [kʰv̩ ] 'year', [nɑ] (CL for knives), [kv̩ ] (CL for humans),

[kʰɯ] (CL for elongated objects) and [mae] 'monetary unit'. Consultant: F4.

H MH M M L M H MH M M L M 1 ɖɯ ɖɯ˧ɲi˥ ɖɯ˧kʰv̩ ɖɯ˧nɑ˧ (ɖɯ˧v̩ ˧) ɖɯ˧kʰɯ˩ ɖɯ˧mae˩ ˧ ˥ ˧ ˧ ˧ ˧ ˥ ˧ ˧ ˧ ˧ ˩ ˧ ˩ ˧ ˩ ˧ ˩ 2 ŋi ɲi˧ɲi˥ ɲi˧kʰv̩ ɲi˧nɑ˧ ɲi˧kv̩ ɲi˧kʰɯ˩ ɲi˧mae˩ ˧ ˥ ˧ ˧ ˧ ˧ ˧ ˥ ˧ ˧ ˧ ˧ ˧ ˩ ˧ ˩ ˧ ˩ ˧ ˩ 3 so˧ so˩ɲi so˩kʰv̩ so˧nɑ˧ so˩kv̩ so˧kʰɯ˧ so˩mae ˧ ˩ ˩ ˧ ˧ ˩ ˧ ˧ ˩ ˩ ˩ ˧ ˧ ˩ ˧ ˧ ˩ 4 ʐv̩ ˧ ʐv̩ ˧ɲi˧ ʐv̩ ˧kʰv̩ ˩ ʐv̩ ˩nɑ ʐv̩ ˩kv̩ ʐv̩ ˧kʰɯ˥ ʐv̩ ˧mae˥ ˧ ˧ ˧ ˧ ˧ ˧ ˩ ˧ ˩ ˩ ˩ ˧ ˥ ˧ ˥ ˩ ˩ ˧ ˥ ˧ ˥ 5 ŋwɤ˧ ŋwɤ˧ɲi˧ ŋwɤ˧kʰv̩ ˩ ŋwɤ˩nɑ ŋwɤ˩kv̩ ŋwɤ˧kʰɯ˥ ŋwɤ˧mae˥ ˧ ˧ ˧ ˧ ˧ ˧ ˩ ˧ ˩ ˩ ˩ ˧ ˥ ˧ ˥ ˩ ˩ ˧ ˥ ˧ ˥ 6 qʰv̩ qʰv̩ ˧ɲi˥ qʰv̩ ˧kʰv̩ ˥ qʰv̩ ˧nɑ˥ qʰv̩ ˧kv̩ ˥ qʰv̩ ˧kʰɯ˥ qʰv̩ ˧mae˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ 7 ʂʅ˧ ʂʅ˧ɲi˧ ʂʅ˧kʰv̩ ʂʅ˧nɑ˧ ʂʅ˧kv̩ ʂʅ˧kʰɯ˩ ʂʅ˧mae˩ ˧ ˧ ˧ ˧ ˧ ˧ ˧ ˧ ˧ ˧ ˧ ˧ ˧ ˧ ˩ ˧ ˩ ˧ ˩ ˧ ˩ 8 hũ hũ˧ɲi˥ hũ˧kʰv̩ ˥ hũ˧nɑ˥ hũ˧kv̩ ˥ hũ˧kʰɯ˥ hũ˧mae˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ ˧ ˥ 9 gv̩ ˧ gv̩ ˧ɲi˧ gv̩ ˧kʰv̩ ˩ gv̩ ˩nɑ gv̩ ˩kv̩ gv̩ ˧kʰɯ˩ gv̩ ˧mae˥ ˧ ˧ ˧ ˧ ˧ ˧ ˩ ˧ ˩ ˩ ˩ ˩ ˩ ˧ ˩ ˧ ˩ ˧ ˥ ˧ ˥ 10 tsʰe˧ tsʰe˩ɲi tsʰe˩kʰv̩ tsʰe˧nɑ˧ tsʰe˩kv̩ tsʰe˧kʰɯ˧ tsʰe˩mae ˧ ˩ ˩ ˧ ˧ ˩ ˧ ˧ ˩ ˩ ˩ ˧ ˧ ˩ ˧ ˧ ˩ numeral in isolation

Tone of classifier

The tones of numeral-plus-classifier phrases in Naxi, Na and Laze they potentially encapsulate phylogenetic evidence nonetheless, since they bring out similar parallels between pairs of numerals. Table 7 recapitulates groupings among numerals in Na, Laze and Naxi; anticipating the argument about the relatedness of these three languages, the table also proposes hypotheses about their common ancestor, Proto-Naish. Proto-Naish is defined as the (hypothetical) common ancestor of the present-day languages described in Chinese scholarship as "dialects of Naxi", and which will be referred to as Naish languages. Naish is proposed as a name for the entire group because, as pointed out by Yang [START_REF] Fuquan | Introduction[END_REF], the syllable 'Na' appears in the autonyms of several of the communities at issue, including 'Na' and 'Naxi'. The second syllable of 'Naxi' means 'human being'; as for the name 'Laze', it apparently originates in a place name (Guo Dalie & He Zhiwu 1999[START_REF] Dalie | Nàxīzú Shǐ 纳西族 史 (A history of the Naxi people)[END_REF]: 6-7); the /lɑ˧/ in 'Laze' cannot be cognate with 'Na', since Laze has a contrast between /n/ and /l/ and the latter regularly corresponds to /l/ in Na and Naxi. Table 7 abstracts away from the tone patterns themselves to bring out the cases of identical tonal behaviour of two or more numerals (in the numeral system itself and in numeral-plus-classifier phrases). These cases are indicated by the use of the same shading in the cells at issue. The cells corresponding to numerals which are not members of larger subsets are left in white. The table brings out the identical behaviour of the pairs of numerals {1, 2}, {4, 5} and {6, 8} in Naxi, Laze and Na (though the patterns for {1, 2} and {4, 5} merge in Laze). Given the considerable differences in the surface forms between the three languages, it can safely be asserted that this similarity is not due to borrowing.

Before interpreting these data as a confirmation of the initial hypothesis of a close relationship between Naxi, Laze and Na, the status of the data in Table 7 needs to be examined carefully. This is clearly a case of retention of a common structure. The facts in Table 7 are synchronically opaque: the identical behaviour of these pairs of numerals dates back to morpho-phonological properties that they shared at least as early as the Proto-Naish stage, and for which no reconstruction can be proposed as yet. The similar behaviour of numerals could be put down to 'affix-runs' of the type brought out by Michailovsky 2003 in Kiranti; but not all the numerals that pattern together in the Naish languages are contiguous, witness the pair {6, 8}.

The shared morpho-phonological properties of these sets of numerals must have endured until the stage when the classifiers developed, since they left traces on the tone patterns of numeral-plus-classifier phrases; and the development and elaboration of classifiers is a relatively recent development in Tibeto-Burman (Bradley 2005: 224). It would be paradoxical to suggest that the entire structure in Table 7 reflects a language state of great antiquity: to my knowledge, this structure has not been reported so far in Burmic, Qiangic or other Sino-Tibetan language groups. In Sino-Tibetan, well-attested affinities are observed between '2' and '7' (which probably both had a final -s at early stages of the ancestor language: see Matisoff 1997), but these do not pattern together in Naxi, Na and Laze. Examples of languages of the area that do not partake in the structure set out in Table 7 include Nuosu (Loloish), Shixing, Namuyi, and Pumi (Qiangic). For want of detailed evidence on numeral-plus-classifier phrases on each language, the comparison below is based on the behaviour of numerals, comparing the forms in the interval [1: 10] with those in the numerals from '1' to '10' in higher numerals.

(i) In Nuosu, numerals fall into the following tonal subgroups: {1, 2, 7}, {3, 4, 5, 9, 10}, and {6, 8}; however, '6' and '8' differ in that the former is not subject to tonal alternations (Li Min et al. 1983: 108). The 'Naish pattern', if it may be so called, is not found in Lisu or Lahu either (data from [START_REF] Bradley | Why do numerals show 'irregular' correspondence patterns in Tibeto-Burman? Some Southeastern Tibeto-Burman examples[END_REF].

(ii) In Shixing, the numerals '1' and '2' do appear to pattern together, but not the numerals '4' and '5', nor '6' and '8': the tone patterns of '16' and '18' differ from each other, as do those of '60' and '80' (data from Huang Bufan et al. 1991).

(iii) In Namuyi, '1' and '4-10' all have L tone, whereas '2' and '3' have M tone. The tones of numerals do not change in higher-level combinations, from '11' to '99', except that there is a special form for '10' in the range from 20 to 39: /hɔ˧/, as opposed to /hɔ˩/ in all the other cases. No tonal change takes place when the classifier /la˧/ is added after numerals from 1 to 100. (Data from Libu Lakhi, recorded at Academia Sinica in 2008; see also Libu Lakhi et al. 2010.) As a last observation on languages of the area, let us mention that Pumi also lacks the 'Naish pattern' among numerals (data from Lu Shaozun 2001).

Thus, the grouping of {1, 2}, {4, 5} and {6, 8} can be proposed as one of the criteria that can be used to determine whether a language belongs in the Naish language group or not. This criterion should of course be used in combination with as many others as possible, including regular sound correspondences, common lexical innovations, and cases of suppletion such as those reported in Table 2. This criterion offers additional evidence for the set of hypotheses recapitulated in Jacques et al. 2011: (i) as argued by [START_REF] Bradley | Nahsi and Proto-Burmese-Lolo[END_REF], Naish and Loloish are separate subgroups; (ii) Shixing is not part of the Naish group proper, pace Guo Dalie & He Zhiwu (1999[START_REF] Dalie | Nàxīzú Shǐ 纳西族 史 (A history of the Naxi people)[END_REF]: 9); and (iii) Namuyi, despite numerous surface similarities with Na and Naxi, is not part of Naish either. This leads to further hypotheses, such as that the striking phonetic closeness of the forms of numerals from '1' to '9' in Na and Namuyi is due to language contact. Needless to say, systematic comparison of Shixing and Namuyi with Naxi, Na and Laze is required before any solid conclusion on issues of cognacy vs. areal convergence can be reached.

Does Naxi stand apart within the Naish language group?

Data from A-sher Naxi and Nda-le Naxi suggest the existence of a Naxi innovation whereby the pair of numerals {1, 2} came to have the same tonal behaviour as '7' and '9', resulting in the set of four similar numerals {1, 2, 7, 9} observed in A-sher and Nda-le. However, this phenomenon is not attested in all dialects of Naxi. In Section 2.1, it was mentioned that the linguists He Jiren and He Zhiwu, who are native speakers of the dialects of Yangxi and Changshui, respectively, describe '1' and '2' as having Low tone. This observation takes on special significance in view of the general picture in Table 7. The realisation of '1' and '2' with Low tone, which sets them apart from '7' and '9', is an archaic feature, reminiscent of the state of affairs in Laze and Na. The evolution whereby the pair of numerals {1, 2} came to have the same tonal behaviour as {7, 9} in several Naxi dialects must therefore be more recent than the separation of Naxi from Na and Laze. The identity of the tone patterns of {1, 2} and {4, 5} in Laze must likewise be supposed to have taken place after this language variety began its separate evolution.

Perspectives for reconstruction

The reconstruction of a Proto-Naish system in which the pairs {1, 2}, {4, 5} and {6, 8} patterned together raises new issues concerning the phonetic nature of the Proto-Naish system and the evolution leading up to the present-day Naish languages. At this stage, the phonological properties conditioning the similar behaviour of the above pairs are not clear. Table 8 presents the segmental correspondences for numerals from 1 to 10, and a preliminary reconstruction (the method and results of this reconstruction are set out in [START_REF] Jacques | The reconstruction of Proto-Naish: a preliminary study of the historical phonology of highly eroded Sino-Tibetan languages[END_REF] The history of numerals in Sino-Tibetan is notoriously complex. Quite a few of the items in Table 8 raise special difficulties: they do not fit in with the regular correspondences observed elsewhere, explaining why reconstructions are only proposed for three of the ten items ('2', '5' and '9'). This may point to morphological renovation. The forms for '6' cannot be straightforwardly reduced to a single etymon; the correspondence between Naxi /wɑ/, Na /v̩ / and Laze /ɔ/ is not observed elsewhere. A prefix may have coalesced with the numeral; there may have existed two different forms at the stage of Proto-Naish, with different uses. A relationship with the widespread Sino-Tibetan etymon for '6', reflected by Tibetan drug, is likely; the Na and Laze forms can be reconstructed back to *qʰU, which can be related to drug, as final stops are lost without trace in *-uC rhymes; the disappearance without trace of a medial *-r-is also postulated in the historical phonology of the Naish languages. As for the Naxi form, it could come from the same root, but with a dental prefix instead of a uvular one; the rhyme remains unaccountable at present. The vowel correspondence for '10' is likewise isolated.

Looking beyond Naish, '6' and '8' are the only two numerals that have a final stop in conservative languages such as Tibetan (drug and brgyad) and Japhug Rgyalrong (/kɯtʂɤɣ/ and /kɯrcat/). The similar tonal behaviour of '6' and '8' in Naish might therefore constitute an indirect trace of final stops, which otherwise left few traces in the historical phonology of Naish [START_REF] Jacques | The reconstruction of Proto-Naish: a preliminary study of the historical phonology of highly eroded Sino-Tibetan languages[END_REF]. Further documentation on related dialects and languages will hopefully allow for a fullfledged reconstruction of the phonetic substance of the numerals and classifiers of Proto-Naish; this will shed some light on the historical origin of the observed patterns.

APPENDIX: THE REVERSAL OF H+M TONE SEQUENCES ON DISYLLABLES TO M+H IN NDA-LE NAXI

The presence of tone sandhi in Nda-le Naxi was mentioned in Section 2.2: this language variety has a tone sandhi rule which affects some of the phrases shown in Table 4. The present Appendix sets out this phenomenon. A peculiarity of Ndale Naxi vis-à-vis neighbouring Naxi dialects (including A-sher, as well as the dialect of the old town of Lijiang) is the systematic reversal of H+M patterns to M+H on disyllables-said differently, the neutralisation of H+M and M+H patterns to M+H, as illustrated in Table 9. This phenomenon will be referred to below as "reversal of H+M to M+H". Only two exceptions have been observed so far. The first is /ʈʂʰʅ˥kv̩ ˧/ 'that place' (distal deixis): the preservation of its H+M pattern may be due in part to the iconic value of the H tone on the distal deictic, and in part to the avoidance of homophony with /ʈʂʰʅ˧kv̩ ˥/, which means 'this one' (proximal deictic plus classifier used for one human individual). The second is a set of terms of address calqued from Chinese: /ə˥lø˧/ for elderly men (from Chinese ālǎo 阿老), /ə˥njae˧/ for women older than oneself aged about 40 to 50 (āniáng 阿娘), /ə˥nae˧/ for 'grandmother, old woman' (ānǎi 阿奶). These are part of a specific set of terms of address for people older than oneself.

The same reversal of H+M to M+H takes place in reduplication: H-tone adjectives reduplicate to M+H in Nda-le, and to H+M in A-sher and other neighbouring dialects (about reduplication in Naxi, see Michaud and Vaissière 2007a; a shortened version of this research was published in English as Michaud and Vaissière 2007b).

In light of the existence of a reversal of H+M to M+H in Nda-le, the identical surface patterns of the phrases /ʈʂʰwɑ˧ly˥/ and /ʈʂʰwɑ˧pʰe˥/ ('6' + M-tone CL and H-tone CL, respectively) can be analysed as resulting from different sequences of processes. The combination /ʈʂʰwɑ˥+ly˧/ yields /ʈʂʰwɑ˥ly˧/; this determiner surfaces as such in A-sher Naxi, whereas it is reversed to /ʈʂʰwɑ˧ly˥/ in Nda-le Naxi, following the regularity described above. As for the combination /ʈʂʰwɑ˥/+/pʰe˥/, for reasons that remain unknown it yields /ʈʂʰwɑ˥pʰe˧/ (H+M) in Naxi dialects of the Lijiang area (including A-sher); for Nda-le, the same underlying H+M pattern can be hypothesised to be present: it gets modified to M+H by the systematic process (specific to Nda-le Naxi) of reversal of H+M to M+H. The same argument can be made for the phrases made of {4, 5, 8} followed by a M-tone classifier-and even for '3': their surface tone pattern, M+H, can be derived by application of the H+M reversal rule to the patterns of A-sher Naxi (Table 3), which suggests that they may be underlyingly identical.

  The tonal patterns of numeral-plus-classifier phrases in Nda-le Naxi, using the same classifiers as in Table3. Consultant: M10.

Table

		Place spoken	Language	Autonym	Chinese	Alternative
	Name in local	Present-day Chinese name	name used here		name	names
	language hw̃ɑ˥ dʑy˥ ɑ˧ʂə˞ ˩	Mùlǐ xiàn Xiàngjiǎo xiāng 木里县项脚乡 Lìjiāng gǔchéngqū Wénhuà cūn 丽江古城	Laze A-sher Naxi	/lɑ˧ze˧/	Lārè 拉热 Mùlǐ Shuǐtián 木 里水田话 AS Naxi, A Ser Naxi
	ndɑ˧le˩	区文化村 Lìjiāng gǔchéngqū Guìfēng Dàlái cūn 丽江	Nda-le Naxi	/nɑ˩hi˧/	Nàxī 纳西	NL Naxi, Nda Le Naxi
		古城区贵峰大来村
	ɬi˧di˩	Lìjiāng Nínglàng xiàn Yǒngníng xiāng 丽江宁	Yongning Na	/nɑ˩˧/	Nà 纳	Mósuō 摩梭话
		蒗县永宁乡

Table 2 .

 2 Cases of suppletion in Naxi, Na and Laze.

	Cases of suppletion	Na	Laze Naxi
	'year' except in time ordinals 'year' in time ordinals: 'this year, next year, two years ago', etc. 'to go', non-past 'to go', past 'to come', non-past 'to come', past '10' in the range [10..19] '10' in the range [20..29] '10' in the range [30..39]	kʰv̩ ˧˥ i˥ bi˧ hɯ˩ ju˧ tsʰɿ˩ tsʰe˧ tsi˧ tsʰi˧	kʰv̩ ˧˥ vie˧ bi˧ hɯ˧ i˧ i˧ tsʰi˥ tsi˧ tsʰi˧	kʰv̩ ˥ be˧ bɯ˧ hɯ˧ ɭɯ˧ tsʰɿ˩ tsʰe˩ tsae˩ tsʰae˩

Table 3 .

 3 The tonal patterns of numeral-plus-classifier phrases in A-sher Naxi, using [pʰe˥] (CL for flat objects),[ly˧] (CL for round objects) and [kʰɯ˩] (CL for elongated objects) as examples. Consultant: M4.

	1
	ɖɯ˧
	ɖɯ˧pʰe˥
	ɖɯ˧ly˧
	ɖɯ˧kʰɯ˩

Table 5a .

 5a Tonal patterns in Laze for [ʈʂae] 'section', [gv̩ ] (generic CL),

[ɲi] 'day', [kʰv̩ ] 'year

' and [kʰɤ]

'basket

'. Consultant: F7.

Table 5b .

 5b Behaviour of the 'year' and 'basket' types of H-tone classifiers of Laze in numeral-plus-classifier phrases from eleven to twenty.

	11
	tsʰi˥ɖɯ˥kʰv̩ ˩
	tsʰi˥ɖɯ˥kʰɤ˧

Table 7 .

 7 Groupings among numerals from '1' to '10' on the basis of their tonal patterns in association with classifiers in Na, Laze and Naxi, and hypotheses about Proto-Naish.

	1

Table 8 .

 8 ). Segmental correspondences between the numerals of Naxi, Na and Laze, and preliminary reconstructions. (Italics are used for forms presumed not to be cognate.)

	A-sher	Nda-le	Na	Laze	Preliminary
	Naxi	Naxi			reconstructions
	1 2 3 4 5 6 7 8 9 10 tsʰe ɖɯ ɲi sɿ lu wɑ ʈʂʰwɑ ʂə˞ ho ŋgv̩	ɳɖɯ ɲi sɿ ɳɖu wɑ ʈʂʰwɑ ʂə˞ hø ŋgv̩ tsʰe	ɖɯ ɲi so ʐv̩ ŋwɤ qʰv̩ ʂʅ hũ gv̩ tsʰe	ɖɯ ɲi su ʐv̩ ŋwɤ qʰɔ ʂʅ ɕi gv̩ tsʰi	*rd-(rhyme: uncertain) *ŋi *s-(+nasal rhyme *ũ?) *rl-(rhyme: uncertain) *ŋwa *qʰ-(+rhyme *U?) *sr-(+nasal rhyme?) (uncertain) *ŋgu *tsʰ-(rhyme: uncertain)

Table 9 .

 9 Some correspondences illustrating the neutralisation of H+M and M+H patterns inNda-le Naxi. (A Chinese gloss is added for disambiguation where necessary.)

	Meaning	Nda-le Naxi	A-sher Naxi	Correspondence
	mist, fog forehead heart lid (of pot/pan) sickle middle, centre nettle mortar, 臼 rubbish, refuse brains beard spittle, mucus beggar lean meat cloth, fabric man's waistband earring hoe morning hors-d'oeuvre, 酒菜	ki˧sae˥ tø˧mu˥ nv̩ ˧me˥ ku˧pu˥ sɿ˧kv̩ ˥ ly˧ŋgv̩ ˥ le˧pʰy˥ ʈʂu˧mu˥ ɳɖə˞ ˧mə˞ ˥ kv̩ ˧fv̩ ˥ mv̩ ˧tsɿ˥ ʈʂə˞ ˧pʰe˥ hɑ˧me˥ ʂʅ˧nɑ˥ tʰø˧pv̩ ˥ mbu˧kɯ˥ he˧kʰv̩ ˥ tsʰø˧kv̩ ˥ mv̩ ˧sɿ˥ ʐʅ˧ʂə˞ ˥	ki˥sae˧ to˥mu˧ nv̩ ˥me˧ ku˥pu˧ ʂv̩ ˥kv̩ ˧ ly˥gv̩ ˧ le˥pʰy˧ ʈʂu˥mu˧ ndae˧mae˥ kv̩ ˧fv̩ ˥ mv̩ ˧tsɿ˥ ʈʂʷə˞ ˧pʰe˥ hɑ˧me˥ ʂʅ˧nɑ˥ tʰo˧pv̩ ˥ bɯ˧kɯ˥ he˧kʰv̩ ˥ tsʰo˧kv̩ ˥ mv̩ ˧sɿ˥ ʐʅ˧ʂʷə˞ ˥	M+H: H+M M+H: H+M M+H: H+M M+H: H+M M+H: H+M M+H: H+M M+H: H+M M+H: H+M M+H: M+H M+H: M+H M+H: M+H M+H: M+H M+H: M+H M+H: M+H M+H: M+H M+H: M+H M+H: M+H M+H: M+H M+H: M+H M+H: M+H

In A-sher Naxi the numerals '1', '2', '6', '7', '8' and '9' have stable tones: there is no sandhi in numeral-plus-classifier determiners. In Nda-le Naxi the stable numerals are '1', '2', '4', '5', '7', '9' and '10', and the stable classifiers are the high and low tone ones; in Laze '3', '6', '8'

research project "What defines Qiang-ness? Towards a phylogenetic assessment of the Southern Qiangic languages of Muli" (ANR-07-JCJC-0063).

Volume 1: Typological studies in word and sentence prosody, 323-351. Berlin/New York: Mouton de Gruyter. Lama Ziwo 拉 玛 兹 偓 . 1994. Nàmùyīyǔ zhīshǔ yánjiū 纳 木 依 语 支 属 研 究 (Research on the classification of the Namuyi language). Minzu Yuwen 民族语 文 (Minority Languages of China) 1.50-60. Li Min 李民 & Ma Ming 马明. 1983. Liángshān Yíyǔ Yǔyīn Gàilùn 凉山彝语语 音概论 (Introduction to the phonetics of Liangshan Yi). Chengdu: Sichuan Minzu Chubanshe. Libu Lakhi, Tsering Bum & Kevin Stuart. 2010. China's Namuyi Tibetans: Life, language, and folklore, vol. 1. Asian Highlands Perspectives 2A (online publication). Lidz, Liberty. 2006. A synopsis of Yongning Na (Mosuo) [long version of handout]. Paper presented at the 39th International Conference on Sino-Tibetan Languages and Linguistics, University of Washington, Seattle, 2006. Lidz, Liberty. 2007. Evidentiality in Yongning Na (Mosuo). LTBA 30(2).45-87. Lidz, Liberty. 2010. A descriptive grammar of Yongning Na (Mosuo). PhD dissertation, Austin: University of Texas. Lu Shaozun 陆绍尊. 2001. Pumiyu fangyan yanjiu 普米语方言研究 (Pumi dialectology). Beijing: Minzu Chubanshe. Matisoff, James A. 1973. The grammar of Lahu. Berkeley: University of California Publications. Matisoff, James A. 1991. Jiburish revisited: tonal splits and heterogenesis in Burmo-Naxi-Lolo checked syllables. Acta Orientalia 52.91-114. Matisoff, James A. 1997. Sino-Tibetan numeral systems: Prefixes, protoforms and problems. Canberra: Pacific Linguistics, series B-114. Mazaudon, Martine. 1985. Dzongkha number systems. In R. Suriya et al. (eds.), Southeast Asian linguistic studies presented to André-G. Haudricourt, 124-157. Bangkok: Mahidol University. Mazaudon, Martine. 2002. Les principes de construction du nombre dans les langues tibéto-birmanes. Mémoires de la Société de Linguistique de Paris, Nouvelle série, 12.91-119. Mazaudon, Martine. 2007. Number building in Tibeto-Burman languages. Paper presented at the North-East India Languages Symposium (NEILS2), Gauhati (Assam, India). Michailovsky, Boyd. 2003. Suffix-runs and counters in Kiranti time-ordinals. In David Bradley, Randy LaPolla, Boyd Michailovsky & Graham Thurgood (eds.), Language variation: Papers on variation and change in the Sinosphere and the Indosphere in honour of James A. Matisoff, 237-251. Canberra: Pacific Linguistics. Michailovsky, Boyd & Alexis Michaud. 2006. Syllabic inventory of a Western Naxi dialect, and correspondence with Joseph F. Rock's transcriptions. Cahiers de linguistique -Asie Orientale 35(1).3-21.