Patent Publication Number: US-2020278644-A1

Title: Horological gearing

Description:
This application claims priority of European patent application No. EP19160248.1 filed Mar. 1, 2019, the content of which is hereby incorporated by reference herein in its entirety. 
     The invention concerns a horological gearing. The invention also concerns a horological mechanism comprising a horological gearing of this kind. The invention also concerns a horological movement comprising a horological gearing of this kind. The invention further concerns a timepiece comprising a horological gearing of this kind. The invention finally concerns a method of manufacturing a horological gearing of this kind. 
     In horology, it is known to determine the profile of the teeth of the wheels and of the pinions on the basis of geometric curves such as the cycloid, the epicycloid, the hypocycloid or the involute of a circle. The wheels and the pinions obtained in this way are defined so as to transmit a rotation speed that remains substantially constant during a tooth drive. Tooth profiles having the property of transmitting a substantially constant torque during a tooth drive are also known. 
     Conventional toothing profiles are therefore defined so as to address speed and/or torque objectives. In gearing known from the prior art this results in greater or lesser angular backlash as a function of the axial center distance of the toothed wheels. In the particular case of a wheel kinematically linked to a geartrain of a basic movement (in parallel coupling), which is designed to cause to be displayed, for example, information derived from the time, such backlash may induce risks of shaking of the member displaying the information derived from the time, namely jerky and irregular movement of the member for displaying the information derived from the time. 
     Solutions known from the prior art propose adding friction in a kinematic chain in parallel coupling with the geartrain of the basic movement, so as to generate a resistive torque against the member for displaying the information derived from the time. Solutions of this kind, disclosed for example by the patent applications CH506824 and EP0482443, are however not the optimum in that they may in particular generate a reduction or variation of the amplitude of the oscillations of the balance wheel and therefore degraded chronometric performance. This solution also increases the energy consumption of the movement. 
     Patent application EP2453321 discloses a specific toothing profile having the property of transmitting a substantially constant torque during a tooth drive. A profile of this kind does not enable as great minimization as possible of the angular backlash during a tooth drive of a gearing of this kind. 
     Patent application EP1555584 concerns a backlash-compensation toothed mobile the teeth of which are equipped with at least one elastic element. Although a component of this kind advantageously can be substituted for the friction spring, it nevertheless remains fragile compared to a wheel with rigid teeth. Moreover, the angular backlash is minimized, or even canceled out, by the effect of the compression of the teeth, this compression being able to vary according to the axial center distance variations and thereby affecting the energy consumption of the movement of which said mobile is part. A solution of this kind can therefore be improved on. 
     Patent application EP2053474 concerns a chronograph architecture that has the particular feature of integrating a vertical clutch mobile in the geartrain itself of the basic movement. Thus it is mounted in series between the driving member and the regulating organ of the basic movement, and not in parallel coupling. As a result, the chronograph seconds hand is no longer subjected to random angular movements, independently of any friction spring. An architecture of this kind nevertheless remains highly specific, and requires at least one additional mobile in the geartrain of the basic movement, with the risk of degrading its overall efficiency. 
     Patent application WO2017157764 aims to alleviate the problem of shaking of a display member by way of the gearing in which the teeth of the wheels have the particular feature of having a modulus less than 0.05 mm. This document does not describe any specific profile defined with the objective of minimizing the angular backlash and/or minimizing variation of the angular backlash as a function of the axial center distance. Moreover, such wheels are not very robust and a priori cannot be produced using conventional manufacturing means. 
     The object of the invention is to provide a horological gearing enabling improvement of the horological gearings known from the prior art. In particular, the invention proposes a horological gearing enabling limitation of the gearing backlash, in particular limitation of the sensitivity of the backlash to the gearing axial center distance variations. 
     A horological gearing according to the invention is defined by the following points 1 and 16. 
     1. A gearing for a horological mechanism, comprising a first toothed wheel including first symmetrical teeth and a second toothed wheel including second symmetrical teeth, each of the first and second teeth being conformed so that the primitive backlash ( ) of the gearing is less than 0.3.m, or even less than 0.25.m, or even less than 0.2.m, or even less than 0.15.m, or even less than 0.1.m, or even less than 0.08.m for the nominal axial center distance (e) of the gearing, where m is the modulus of the wheel of which the tooth forms part. 
     16. The gearing obtained by means of the method as defined in either one of the following points 14 or 15. 
     Different embodiments of the gearing are defined by the following points 2 to 10. 
     2. The gearing as defined in point 1, wherein each of the first teeth and of the second teeth is conformed such that the ratio of the variation of maximum angular backlash (jmax) to the variation of the gearing axial center distance (e) is less than 6°/mm or less than 5°/mm or less than 4°/mm or less than 3°/mm for a gearing axial center distance value varying between the nominal gearing axial center distance minus 0.04 mm and the nominal gearing axial center distance plus 0.04 mm. 
     3. The gearing as defined in point 1 or 2, wherein each of the first and second teeth comprises a profile in a plane perpendicular to an axis of the toothed wheel of which the tooth forms part, the profile comprising a functional portion having the shape of a first circular arc, the first circular arc being defined by: 
     a radius (r u ), and 
     a center of coordinates (x u ,y u ) in a direct orthonormal Cartesian system of axes (O 1 ,{right arrow over (e)} x ,{right arrow over (e)} y ;  0   2 ,{right arrow over (e)} x ,{right arrow over (e)} y ) centered on the axis of the toothed wheel of which the tooth forms part, {right arrow over (e)} x  being a unit vector colinear with an axis of symmetry of the tooth, with: 
     
       
         
           
             
               
                 r 
                 u 
               
               = 
               
                 
                   p 
                   
                     i 
                      
                     
                         
                     
                      
                     1 
                   
                 
                  
                 
                   
                     2 
                      
                     π 
                      
                     
                         
                     
                      
                     
                       R 
                       p 
                     
                   
                   z 
                 
               
             
             , 
           
         
       
     
     x u =p i2 r u +R p , y u =−p i3  r u  z: the number of teeth on the wheel of which the tooth forms part; R p , the primitive radius of the wheel R 1  or R 2  concerned; p i1 ,p i2 ,p i3 : parameters determined so that the primitive backlash ( ) of the gearing is less than 0.3.m, or even less than 0.25.m, or even less than 0.2.m, or even less than 0.15.m, or even less than 0.1.m, or even less than 0.08.m for the nominal axial center distance (e) of the gearing, when m is the modulus of the wheel on which the tooth forms part. 
     4. The gearing as defined in one of the preceding points, wherein the profile comprises a tooth head portion having the shape of a second circular arc, the second circular arc being in particular defined by a particular radius (r t ) as follows: 
       r t p i4 (r u +y u ) 
     with: p i4 : a parameter determined so that the primitive backlash ( ) of the gearing is less than 0.3 .m, or even less than 0.25.m, or even less than 0.2.m, or even less than 0.15.m, or even less than 0.1.m, or even less than 0.08.m for the nominal axial center distance (e) of the gearing, when m is the modulus of the wheel on which the tooth forms part. 
     5. The gearing as defined in any one of the preceding points, wherein the parameters (p i1 ) p i2 , p i3 , p i4 ) constitute the coordinates of a first vector ({right arrow over (p)}) satisfying the following condition: 
     ({right arrow over (p)}-{right arrow over (p)}*) t  H ({right arrow over (p)}-{right arrow over (p)}*) ≤1 with: H: a covariance type matrix, and {right arrow over (p)}*: a vector with coordinates (p 1 *,p 2 *,p 3 *,p 4 *). 
     6. The gearing as defined in any one of the preceding points, wherein: 
     the gearing ratio is equal to 1 and/or the first and second wheels have the same number of teeth, and 
     the teeth of the first and second wheels have the same geometry, or wherein: 
     the gearing ratio is equal to 1 and/or the first and second wheels have the same number of teeth, and 
     the teeth of the first and second wheels have different geometries. 
     7. The gearing as defined in any one of the preceding points, wherein the first wheel and/or the second wheel is or are secured to a display member and/or wherein the first wheel and/or the second wheel is or are intended to form part of a geartrain of a horological movement, and in particular is intended to be interleaved, directly or indirectly, between a driving member and a member adjusting a horological movement, and/or wherein the first wheel and/or the second wheel is or are intended to be mounted in parallel coupling with a geartrain of a horological movement. 
     8. The gearing as defined in any one of points 1 to 7, wherein the first and second wheels are rigid. 
     9. The gearing as defined in any one of points 1 to 7, wherein at least one of the first and second wheels comprises an elastic angular-backlash-compensation structure, in particular an elastic structure consisting of at least one cut-out in a wheel. 
     10. The gearing as defined in the preceding point, wherein the angular-backlash-compensation elastic structure is not loaded at the nominal axial center distance. 
     A horological mechanism according to the invention is defined by the following point 11. 
     11. A horological mechanism, in particular a horological mechanism for displaying time information or information derived from time or not time related, comprising a gearing as defined in any one of the preceding points. 
     A horological movement according to the invention is defined by the following point 12. 
     12. A horological movement comprising a gearing as defined in any one of points 1 to 10 and/or a horological mechanism as defined in the preceding point. 
     A timepiece according to the invention is defined by the following point 13. 
     13. A timepiece comprising a gearing as defined in any one of points 1 to 10 and/or a horological mechanism as defined in point 11 and/or a horological movement as defined in point 12. 
     A method according to the invention of producing a gearing is defined by the following point 14. 
     14. A method of manufacturing a gearing as defined in any one of points 1 to 10 and/or a horological mechanism as defined in point 11 and/or a horological movement as defined in point 12 and/or a timepiece as defined in point 13, wherein it comprises: 
     a step of determination of the profiles of the first and second toothings of the first wheel and of the second wheel, 
     a step of shaping the first and second wheels, 
     a step of mounting the first and second toothed wheels so that they mesh with one another. 
     One embodiment of the method of manufacture is defined by the following point 15. 
     15. The method as defined in the preceding point, wherein the determining the profiles of the first and second toothings of the first wheel and of the second wheel comprises, after a step of selection, in particular a step of arbitrary selection, of profiles of the first and second toothings, iteration of the following steps: 
     determination of the performance of the profiles of the toothings; 
     generation of new toothing profiles using genetic algorithms, in particular stochastic optimization algorithms based on mechanisms of natural selection and of genetics and/or algorithms employing evolutionary operators, namely selection and/or crossing and/or mutation. 
     The appended figures represent, by way of example, three embodiments of a timepiece according to the invention. 
       FIGS. 1 and 2  show a gearing known from the prior art. 
       FIG. 3  is a graph representing the angular backlash variations of the gearing from  FIGS. 1 and 2  over one step and for different axial center distance values. 
       FIGS. 4 to 7  show a first embodiment of the timepiece comprising a first gearing embodiment. 
       FIG. 8  is a graph representing the variations of the angular backlash of the first embodiment of the gearing over one step and for different axial center distance values. 
       FIG. 9  shows a second embodiment of a gearing used in a second embodiment of the timepiece. 
       FIG. 10  is a graph representing the variations of the angular backlash of the second embodiment of the gearing over one step and for different axial center distance values. 
       FIG. 11  shows a third embodiment of a gearing used in a second embodiment of the timepiece. 
       FIG. 12  is a graph representing the variations of the angular backlash of the third embodiment of the gearing over one step and for different axial center distance values. 
       FIG. 13  shows a variant gearing. 
    
    
     A first embodiment of a timepiece  300  is described hereinafter with reference to  FIGS. 4 to 8 . The timepiece  300  is, for example, a watch, in particular a wristwatch. 
     The timepiece comprises one embodiment of a horological movement  200 . The horological movement may be of the electronic or mechanical, in particular automatic, type. 
     The horological movement comprises one embodiment of a horological mechanism  100 . This mechanism may be a mechanism for displaying time information or information derived from time, or displaying time information or information derived from the time of day, or displaying information of a function that is not time related. In particular, the mechanism may be mechanically linked to the geartrain of the movement, in parallel coupling with that geartrain. For example, the display mechanism may comprise a minute wheel, a chronograph module, a countdown module, a display train of a chronograph module or of a countdown module, or again a display system comprising a rack meshing with a toothed wheel mechanically connected to a display hand. Alternatively, the display mechanism may, for example, comprise a mechanism for displaying information from an altimeter or from a depth meter. 
     The mechanism  100  comprises first embodiment E 1  of the horological gearing. 
     The gearing E 1  comprises a first toothed wheel R 1  including symmetrical first teeth dl and a second toothed wheel R 2  including symmetrical second teeth d 2 . 
     The first toothed wheel is mounted to be mobile in rotation about a first axis A 1 . The second toothed wheel is mounted to be mobile in rotation about a second axis A 2 . The first and second toothed wheels are, for example, mounted on a common frame. The first and second axes are preferably parallel or substantially parallel. The axial center distance e between the first and second axes is such that the first and second wheels mesh with one another. 
     Each of the first and second teeth is conformed and/or arranged so that the primitive backlash   of the gearing is less than 0.3.m, or even less than 0.25.m, or even less than 0.2.m, or even less than 0.15.m, or even less than 0.1.m, or even less than 0.08.m for a nominal axial center distance e, where m is the modulus of the wheel of which the tooth forms part. 
     The considerations of the above three paragraphs are preferably equally valid for a second embodiment E 2  of a gearing comprising a first toothed wheel R 1 ′ and a second toothed wheel R 2 ′ and for a third embodiment E 3  of a gearing comprising a first toothed wheel R 1 ″ and a second toothed wheel R 2 ″. These other two embodiments are referred to later. 
     In other words, the gearing El has tooth or toothing profiles P 1 , P 2  of a driving wheel and of a driven wheel having the property of minimizing as much as possible the angular backlash during a tooth drive whilst being only very slightly sensitive to axial center distance variations. The properties in terms of efficiency and wear of a gearing of this kind may also advantageously be optimized. 
     By “angular backlash j” is meant, for a given axial center distance e, the angular movement with which a first wheel R 10  can turn freely relative to a second wheel R 20  immobilized in a specified orientation, or in a given position. By way of explanatory illustration,  FIGS. 1 and 2  show wheels R 10  and R 20  of this kind of a gearing E 0 , in which the profiles of the teeth are defined in accordance with the standard NIHS 20-25 known from the prior art. The backlash j is angle α that may be expressed in degrees (or in radians) as represented in  FIGS. 1 and 2 . In one particular case, the backlash j may in particular be defined as a function of the primitive backlash J. 
     By “primitive backlash J” is meant, for a nominal axial center distance e, a maximum arc length l along which can travel a point on a primitive circle (R p2  in  FIG. 2 ) of the first wheel R 10  relative to the second wheel R 20  immobilized in a specific orientation, as defined by the standard ISO 1122-1:1998. Thus the backlash J is a length that may in particular be expressed in millimeters. The backlash J may also be expressed as a function of the pitch p or of the modulus m of the wheels R 10 , R 20  forming part of the gearing E 0 . 
     The angular backlash j can vary as a function of the respective orientations of the first and second wheels R 10  and R 20 . Accordingly, over a tooth drive, the backlash j is caused to evolve. For a give gearing, the greatest angular backlash j is denoted j max*  at the nominal axial center distance e of said gearing, the angular backlash j max  determined on one or the other of the wheels R 10 , R 20  is expressed as follows: 
     
       
         
           
             
               j 
               max 
             
             = 
             
               
                 J 
                 
                   1 
                   , 
                   2 
                 
               
               
                 R 
                 
                   
                     P 
                      
                     
                         
                     
                      
                     1 
                   
                   , 
                   2 
                 
               
             
           
         
       
     
     with j 1,2  expressed in radians, and R p1,2  the respective primitive radii of the wheels R 10 , R 20 . 
     The backlash j may be defined by one or more pairs of teeth according to the profile chosen for the teeth of each of the wheels forming part of the gearing. 
     An angular backlash j is necessary for the wheels R 10  and R 20  to mesh. Nevertheless, too large an angular backlash j risks degrading the quality of the transmission of the movement from the driving wheel to the driven wheel, which could be reflected in vibrations or jerking of the driven wheel. 
     Moreover, an increase in the axial center distance because of manufacturing and assembly tolerances runs the risk of inducing an increase in the angular backlash j. It is therefore necessary to reduce as much as possible the angular backlash j at the nominal axial center distance of a gearing. However, excessive minimization of the angular backlash j at the nominal axial center distance could induce risks of jamming of the gearing at the minimum axial center distance because of the manufacturing and assembly tolerances. 
     It proves that conventional horological profiles do not satisfy the definition of gearing in which the wheels have as constant as possible an angular backlash whatever the axial center distance. To be more specific, conventional clock profiles do not satisfy the definition of a gearing in which the wheels have the smallest possible angular backlash that is as constant as possible whatever the axial center distance. 
     The gearings E 1 , E 2 , E 3  that are the subject matter of the present document are configured so that the toothing profiles of a driving wheel and of a driven wheel minimize as much as possible the angular backlash during a tooth drive, whilst being only very slightly sensitive to the axial center distance variations induced by industrial manufacturing processes and the means for positioning the means for guiding these wheels, for example for a axial center distance varying over a range [e−20 μm, e+60 μm], e being the nominal axial center distance. 
     The toothing geometries of the gearings E 1 , E 2 , E 3  according to the invention typically allow the use of a gearing in which the wheels have a primitive backlash ( ) less than 0.3.m, or even less than 0.25.m, or even less than 0.2.m, or even less than 0.15.m, or even less than 0.1.m, or even less than 0.08.m for the nominal axial center distance e, with: 
     
       
         
           
             m 
             = 
             
               
                 2 
                  
                 e 
               
               
                 
                   z 
                   1 
                 
                 + 
                 
                   z 
                   2 
                 
               
             
           
         
       
     
     where z 1  and z 2  are the respective numbers of teeth of the wheels R 1 , R 1 ′, R 1 ′″ and R 2 , R 2 ′, R 2 ″ forming part of the gearing concerned, namely the number of teeth of each of the wheels R 1 , R 1 ′, R 1 ′″ and R 2 , R 2 ′, R 2 ″. 
     By way of example, the horological profile defined in accordance with the standard NIHS 20-25 enables the use of a gearing in which the wheels have a greater primitive backlash. 
     The geometries of the toothing profiles according to the invention also enable minimization of the variation of the angular backlash as a function of the axial center distance variation. 
     The modulus m is preferably greater than or equal to 0.05 mm so as to employ a robust gearing able in particular to transmit a predetermined minimum torque. 
     The tooth or toothing “profile” of a wheel may be defined by the intersection of the tooth or toothing surfaces of the teeth with a plane P perpendicular to the rotation axis of the wheel. 
     To achieve the aim of minimization of the primitive backlash and minimization of the variation of the angular backlash, the tooth or toothing profiles may have the particular feature of including at least one functional portion PF the shape or the profile of which is a particular circular arc. Based on the work of the applicant, it has been found, in fact, that a gearing with wheels at least one of which has teeth each of which has a functional portion defined by a circular arc of this kind has an angular backlash that can be minimized and rendered substantially constant as a function of the axial center distance of said gearing. 
     By “functional portion PF” is meant a zone of the profile of a tooth that is conformed to enable minimization of the angular backlash and that is designed to cooperate, at least in part, through contact so as to participate in the meshing of a gearing. 
     With the aim of simplification of the definition of each of the profiles of the teeth of the wheels R 1 , R 1 ′, R 1 ″ and R 2 , R 2 ′, R 2 ″, the wheels R 1 ′, R 1 ″ could more simply be referenced R 1  and the wheels R 2 ′, R 2 ″ more simply referenced R 2 . 
     In the embodiment from  FIGS. 4 to 7 , in the embodiment from  FIG. 9  and in the embodiment from  FIG. 11 , the toothing profiles of the wheels R 1  and R 2  are preferably defined so that a first functional portion PF 1  of the first wheel R 1 , characterized by a first circular arc, cooperates through contact with a second functional portion PF 2  of the second wheel R 2 , characterized by a second circular arc, as represented in  FIG. 5 . The first and second functional portions PF 1 , PF 2  can in particular cooperate through contact on the line of the centers passing through the respective centers O 1 , O 2 , of the wheels R 1 , R 2 . The functional portions PF 1  and PF 2  are preferably characterized by circular arcs having the same radius of curvature r u  for a gearing of 1:1 ratio. 
     The circular arc of radius 7 1 , and with center C 1 , C 2  at coordinates (x u ,y u ) may preferably be determined as follows in a direct orthonormal Cartesian system of axes defined by a triplet (O 1 , {right arrow over (e)} x ,{right arrow over (e)} y ) or (O 2 ,{right arrow over (e)} x ,{right arrow over (e)} y ), as represented in  FIG. 6 . O 1  or O 2  coincides with the rotation axis A 1  or A 2  of the wheel R 1  or R 2  concerned, the axis A 1  or A 2  being represented in  FIG. 4 . {right arrow over (e)} x  is a unit vector colinear with the axis of symmetry S 1 , S 2  of a tooth d 1 , d 2  having said functional portion PF 1  or PF 2 . {right arrow over (e)} y  is a unit vector perpendicular to {right arrow over (e)} x . 
     
       
         
           
             
               r 
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                   1 
                 
               
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                   2 
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                     R 
                     p 
                   
                 
                 z 
               
             
           
         
       
       
         
           
             
               an 
                
               d 
             
              
             
               : 
             
           
         
       
       
         
           
             
               x 
               u 
             
             = 
             
               
                 
                   p 
                   
                     i 
                      
                     
                         
                     
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                     2 
                   
                 
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                   r 
                   u 
                 
               
               + 
               
                 R 
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               y 
               u 
             
             = 
             
               
                 - 
                 
                   p 
                   
                     i 
                      
                     
                         
                     
                      
                     3 
                   
                 
               
                
               
                 r 
                 u 
               
             
           
         
       
     
     with: z, the number of teeth of the wheel R 1  or R 2  concerned; R p : the primitive radius of the wheel R 1  or R 2  concerned, with: 
     
       
         
           
             
               
                 R 
                 p 
               
               = 
               
                 e 
                 2 
               
             
             , 
           
         
       
     
     wnere e is the nominal axial center distance of the gearing of which the wheel R 1  or R 2  forms part in the case of a gearing with ratio 1:1; p i2 ,p i2 ,p i3  are parameters determined on the basis of optimization algorithms so that the primitive backlash   of the gearing of which the wheel R i , forms part, namely the wheel R 1  or R 2 , is less than 0.3.m, or even less than 0.25.m, or even less than 0.2.m, or even less than 0.15.m, or even less than 0.1.m, or even less than 0.08.m for a nominal axial center distance e. 
     The length of the functional portion PF 1 , PF 2  is preferably between m and 4.m inclusive. A functional portion PF 1 , PF 2  of this kind may be associated with at least one other portion such as a tooth head portion PT or a tooth root portion PP so as to define a complete tooth profile. The portions PP, PF, PT preferably define an unbroken continuous function between the various portions. This is reflected in a toothing surface having no edge. The portions PP, PF and PT may all consist of circular arcs tangential to one another. 
     The tooth head portion PT may advantageously also be defined by a circular arc the radius r t  of which may be expressed as follows in the Cartesian system of axes (O 1 ,{right arrow over (e)} x ,{right arrow over (e)} y ) or (O 2 ,{right arrow over (e)} x ,{right arrow over (e)} y ): 
       r t =P i4 (r u +y u ) 
     with p i4 , a parameter determined on the basis of optimization algorithms so that the gearing is able to meet the aforementioned condition, that is to say the primitive backlash   of the gearing of which the wheel R i  forms part, namely the wheel R 1  or R 2 , is less than 0.3.m, or even less than 0.25.m, or even less than 0.2.m, or even less than 0.15.m, or even less than 0.1.m, or even less than 0.08.m for a given nominal axial center distance e. 
     The resulting outside or total radius R e  of the wheel R 1  or R 2  may then be expressed in the following manner: 
       R e =x u +√{square root over ((r u −r 1 ) 2 −y u   2 )}+r t  
 
     The parameters p i1 ,p i2 ,p i3 ,p i4  are preferably determined on the basis of genetic algorithms that are stochastic optimization algorithms based on the mechanisms of natural selection and genetics. On the basis of a population of initial potential profiles chosen arbitrarily, their relative performance in terms in particular of angular backlash are evaluated. On the basis of this performance, a new population of potential profiles is created using evolutionally operators, namely selection, crossing and mutation. These latter operations are iterated until a satisfactory solution appears. 
     Accordingly, based on this method, it is possible to determine parameters: (p 11 ,p 12 ,p 13 ,p 14 ), (p 21 ,p 22 ,p 23 ,p 24 ) respectively characterizing portions PF 1 , PT 1 , PF 2 , PT 2  of the wheels R 1 , R 2  able to satisfy angular backlash objectives for given axial center distance variation of the wheels R 1 , R 2  and a given number of teeth z 1 , z 2  of the wheels R 1 , R 2 . 
     The tooth root portion PP may be constructed in continuity with the profile. The junction between the portion PF and the portion PP preferably defines an inflection point PI. 
     In the case of a gearing with ratio 1:1 equipped with wheels R 1  and R 2  provided with the same number of teeth (z 1 =z 2 ), the respective teeth d 1 , d 2  of the wheels R 1  and R 2  may be identical or have identical geometries. 
     In particular, these respective teeth may have functional portions PF 1 , PF 2  that are identical, namely: (p 11 ,p 12 ,p 13 ,p 14 )≡(p 21 ,p 22 ,p 23 ,p 24 ). 
     To be more specific, these respective teeth may have portions PF 1 , PT 1 , PF 2 , PT 2  that are identical, namely: (p 11 ,p 12 ,p 13 ,p 14 )≡(p 21 ,p 22 ,p 23 ,p 24 ). 
     Alternatively, in the case of a gearing with ratio 1:1, the respective teeth of said wheels R 1 , R 2  may be different. In particular, these respective teeth may have different functional portions PF 1 , PF 2 . In fact, given that the profile of said teeth is not constructed on the basis of a modulus m, it is possible to define the functional portions PF 1 , PF 2 , and to be more specific the portions PF 1 , PT 1 , PF 2 , PT 2 , best satisfying angular backlash objectives. 
     Accordingly, in this case: (p 11 ,p 12 ,p 13 )≢(p 21 ,p 22 ,p 23 ) and, to be more specific: (p 11 ,p 12 ,p 13 ,p 14 )≢(p 21 ,p 22 ,p 23 ,p 24 ) 
     The specific parameters (p i1 ,p i2 ,p i3 , p i4 ) may constitute the coordinates of a specific vector {right arrow over (p)} satisfying a predefined condition for the primitive backlash   of a gearing employing a wheel R 1  or R 2 . 
     The applicant&#39;s research moreover shows that it is possible to determine a set of vectors {right arrow over (p)} centered on a vector {right arrow over (p)}* satisfying a specific condition on the primitive backlash   of a given gearing. 
     For example, the set of vectors {right arrow over (p)} satisfying the condition: J&lt;0.3.m must satisfy the following condition: ({right arrow over (p)}-{right arrow over (p)}*) t H({right arrow over (p)}-{right arrow over (p)}*)≤1 where H is a covariance type matrix and {right arrow over (p)}* a vector with coordinates: (p 1 *, p 2 *, p 3 *, p 4 *). 
     The wheels R 1  and R 2  may advantageously be rigid. In other words, they preferably do not include an elastic structure enabling compensation of the angular backlash, either on the teeth or on the arms connecting the teeth to the respective hub of the wheels. 
     Alternatively, whatever the embodiment or the variant, and as represented in  FIG. 13 , at least one wheel R 1 , R 2  may have at least one elastic structure  81 , such that the profiles PF 1 , PF 2  can be defined so as to cancel out the angular backlash over a predefined axial center distance range. The cancellation of the angular backlash over a given axial center distance range enables minimum prestressing of the elastic structure whatever the axial center distance, and therefore minimization of the energy consumed by the gearing. 
     At the nominal axial center distance e, the at least one elastic structure is preferably not actuated or prestressed. This elastic structure is preferably implemented by cut-outs  82  formed on some or all of the teeth of the wheel R 1  and/or of the wheel R 2 . 
     The wheels R 1 , R 2  may be produced by machining, in particular by cutting. Alternatively, they may be obtained by micromachining processes such as etching, photolithography or additive manufacturing techniques. These latter techniques have the advantage of very faithfully reproducing the circular arc characterizing the functional portion PF and, more widely, the continuous function characterizing the portions PP, PF and PT. The wheel R 1  or R 2  may advantageously be made of nickel or a nickel-phosphorus alloy, of silicon, of glass, or of ceramic. 
     Of course, a first wheel R 1  may be obtained by a first method of manufacture while the second wheel R 2  may be obtained by a second method of manufacture. Of course, a first wheel R 1  may be manufactured in a first material while the second wheel R 2  may be manufactured in a second material. 
     A first wheel R 1  or R 2  may form part of a geartrain  92  of the horological movement  200 . More particularly, a first wheel R 1  or R 2  may be interleaved, directly or indirectly, between a driving member  91  and a regulating member  93  of the horological movement. Moreover, a second wheel R 1  or R 2  may be mounted in parallel coupling with said geartrain  92  of the horological movement. By way of example,  FIG. 4  more particularly shows a wheel R 1  forming part of a geartrain  92  and a wheel R 2  mounted in parallel coupling with said geartrain. 
     The wheel R 1  or R 2  may be secured to a display member O, in particular a member for displaying a time indication or an indication derived from the time such as a second or a fraction of a second. This display member preferably comprises a hand. Alternatively, the display member may comprise a disk. By “secured to” is preferably meant “fixed to”. However, other mechanical connections may be envisaged. By way of example,  FIG. 4  shows the display member O taking the form of a hand that is secured to the wheel R 2  mounted in parallel coupling with the geartrain  92 . 
     By way of example, the angular backlash of the gearing E 0  consisting of two identical wheels R 10 , R 20  each having 70 teeth defined on the basis of the standard NIHS 20-25 and the modulus m of which is 0.0726 mm, as represented in  FIGS. 1 and 2 , can in particular be minimized by way of genetic algorithms that enable identification of the parameters characterizing optimized profile portions and modification of the profiles of the teeth. 
       FIG. 3  shows a graph representing the angular backlash of the gearing E 0  over an angular pitch p and for different axial center distances. Note that the maximum angular backlash j max  at the nominal axial center distance e is of the order of 0.4°, which corresponds to a primitive backlash J of the order of 0.018 mm, i.e. approximately 0.3.m. Moreover, the angular backlash can vary by a maximum amplitude of the order of 0.65° for a axial center distance varying over a range [e−40 μm, e+40 μm]. In such an example, the sensitivity of the maximum angular backlash as a function of the gearing axial center distance is about 8.3°/mm when considering a gearing axial center distance range of 0.08 mm, the range being centered on the nominal gearing axial center distance (Δjmax/Δe=8.3°/mm). 
       FIGS. 4 to 7  show the gearing E 1  optimized from the point of view of the angular backlash relative to E 0  and in which the respective teeth of the wheels R 1 , R 2  are identical, with a profile that is characterized by a vector {right arrow over (p)} respecting the following condition: 
     
       
         
           
             
               
                 
                   ( 
                   
                     
                       p 
                       → 
                     
                     - 
                     
                       
                         p 
                         → 
                       
                       * 
                     
                   
                   ) 
                 
                 t 
               
                
               
                 H 
                  
                 
                   ( 
                   
                     
                       p 
                       → 
                     
                     - 
                     
                       
                         p 
                         → 
                       
                       * 
                     
                   
                   ) 
                 
               
             
             ≤ 
             1 
           
         
       
       
         
           
             with 
              
             
               : 
             
           
         
       
       
         
           
             
               
                 p 
                 → 
               
               * 
             
             = 
             
               ( 
               
                 
                   
                     
                       
                         0 
                         . 
                         3 
                       
                        
                       0 
                        
                       9 
                        
                       8 
                        
                       8 
                     
                   
                 
                 
                   
                     
                       
                         0 
                         . 
                         1 
                       
                        
                       6 
                        
                       5 
                        
                       0 
                        
                       5 
                     
                   
                 
                 
                   
                     
                       
                         0 
                         . 
                         1 
                       
                        
                       7 
                        
                       0 
                        
                       5 
                        
                       7 
                     
                   
                 
                 
                   
                     
                       0 
                        
                       .46491 
                     
                   
                 
               
               ) 
             
           
         
       
       
         
           and 
         
       
       
         
           
             H 
             = 
             
               ( 
               
                 
                   
                     
                       8 
                        
                       4 
                        
                       1 
                     
                   
                   
                     
                       
                         5 
                         . 
                         1 
                       
                        
                       5 
                     
                   
                   
                     
                       
                         - 
                         4 
                       
                        
                       2 
                        
                       7 
                     
                   
                   
                     23 
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                       3 
                        
                       
                         5 
                         . 
                         4 
                       
                     
                   
                   
                     2.69 
                   
                   
                     
                       - 
                       12.8 
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                       2 
                        
                       2 
                        
                       3 
                     
                   
                   
                     
                       
                         - 
                         1 
                       
                        
                       
                         7 
                         . 
                         7 
                       
                     
                   
                 
                 
                   
                     
                       sym 
                       . 
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     13.9 
                   
                 
               
               ) 
             
           
         
       
     
       FIG. 8  shows a graph representing the angular backlash   of the gearing E 1  over an angular pitch P and for different axial center distances e, for a given vector {right arrow over (p)} respecting the aforementioned condition. Note that the maximum angular backlash j max  at the nominal axial center distance e is of the order of 0.08°, i.e. approximately five times less than the backlash j max  of the gearing E 1  from  FIGS. 1 and 2 . Accordingly, for this particular vector {right arrow over (p)},   is of the order of 0.6.m. Moreover, in this particular case, the angular backlash can vary with a maximum amplitude of the order of 0.18° for an axial center distance varying over a range [e−40 μm, e+40 μm], i.e. an angular backlash variation close to four times less than referred to above. In such an example, the sensitivity of the maximum angular backlash as a function of the gearing axial center distance is about 1.7°/mm when considering a gearing axial center distance range of 0.08 mm, the range being centered on the nominal gearing axial center distance (Δjmax/Δe=1.7° /mm). 
     The exercise has also been carried out for a gearing consisting of two identical wheels R 1 , R 2  each having 60 teeth, with a profile that is characterized by a vector {right arrow over (p)} respecting the condition: ({right arrow over (p)}-{right arrow over (p)}*) t H({right arrow over (p)}-{right arrow over (p)}*)≤1 
     The gains observed are equally surprising, with   of the order of 0.6 m with a vector {right arrow over (p)} giving the best result in terms of minimization of the angular backlash. 
     In this precise case: 
     
       
         
           
             
               
                 p 
                 → 
               
               * 
             
             = 
             
               ( 
               
                 
                   
                     0.30046 
                   
                 
                 
                   
                     0.20695 
                   
                 
                 
                   
                     0.14800 
                   
                 
                 
                   
                     0.64031 
                   
                 
               
               ) 
             
           
         
       
       
         
           
             H 
             = 
             
               ( 
               
                 
                   
                     3467 
                   
                   
                     
                       - 
                       204 
                     
                   
                   
                     
                       - 
                       1282 
                     
                   
                   
                     
                       - 
                       91 
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     96 
                   
                   
                     95 
                   
                   
                     
                       - 
                       12.4 
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     496 
                   
                   
                     
                       - 
                       49.6 
                     
                   
                 
                 
                   
                     
                       sym 
                       . 
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     48.5 
                   
                 
               
               ) 
             
           
         
       
     
       FIG. 9  shows a gearing E 2  in accordance with a second embodiment optimized from the point of view of the angular backlash relative to the gearing E 0 , in which the respective teeth of the wheels R 1 ′, R 2 ′ are different, with the teeth of the wheel R 1 ′ wider than those of the wheel R 2 ′. Thus: 
     
       
         
           
             
               p 
               → 
             
             = 
             
               ( 
               
                 
                   
                     
                       p 
                       
                         1 
                          
                         1 
                       
                     
                   
                 
                 
                   
                     
                       p 
                       
                         1 
                          
                         2 
                       
                     
                   
                 
                 
                   
                     
                       p 
                       
                         1 
                          
                         3 
                       
                     
                   
                 
                 
                   
                     
                       p 
                       
                         1 
                          
                         4 
                       
                     
                   
                 
                 
                   
                     
                       p 
                       
                         2 
                          
                         1 
                       
                     
                   
                 
                 
                   
                     
                       p 
                       
                         2 
                          
                         2 
                       
                     
                   
                 
                 
                   
                     
                       p 
                       
                         2 
                          
                         3 
                       
                     
                   
                 
                 
                   
                     
                       p 
                       
                         2 
                          
                         4 
                       
                     
                   
                 
               
               ) 
             
           
         
       
       
         
           
             
               with 
                
               
                 : 
               
                
               
                   
               
                
               
                 
                   ( 
                   
                     
                       p 
                       → 
                     
                     - 
                     
                       
                         p 
                         → 
                       
                       * 
                     
                   
                   ) 
                 
                 t 
               
                
               
                 H 
                  
                 
                   ( 
                   
                     
                       p 
                       → 
                     
                     - 
                     
                       
                         p 
                         → 
                       
                       * 
                     
                   
                   ) 
                 
               
             
             ≤ 
             
               1 
               . 
             
           
         
       
     
     In this specific case: 
     
       
         
           
             
                 
             
              
             
               
                 
                   p 
                   → 
                 
                 * 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         0 
                          
                         .30513 
                       
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                           1 
                         
                          
                         4 
                          
                         7 
                          
                         4 
                          
                         2 
                       
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                         
                          
                         1 
                          
                         9 
                          
                         0 
                          
                         6 
                          
                         9 
                       
                     
                   
                   
                     
                       0.56802 
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                           3 
                         
                          
                         0 
                          
                         9 
                          
                         6 
                          
                         7 
                       
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                           1 
                         
                          
                         4 
                          
                         3 
                          
                         1 
                          
                         1 
                       
                     
                   
                   
                     
                       
                         
                           
                             ( 
                             ) 
                           
                           . 
                           2 
                         
                          
                         1 
                          
                         4 
                          
                         4 
                          
                         4 
                       
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                           5 
                         
                          
                         6 
                          
                         5 
                          
                         4 
                          
                         5 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             H 
             = 
             
               ( 
               
                 
                   
                     317 
                   
                   
                     74.1 
                   
                   
                     
                       - 
                       134 
                     
                   
                   
                     
                       - 
                       0.771 
                     
                   
                   
                     207 
                   
                   
                     
                       - 
                       73.7 
                     
                   
                   
                     
                       - 
                       126 
                     
                   
                   
                     4.38 
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     52.9 
                   
                   
                     
                       - 
                       26.9 
                     
                   
                   
                     0.596 
                   
                   
                     17.9 
                   
                   
                     
                       - 
                       44 
                     
                   
                   
                     
                       - 
                       17.8 
                     
                   
                   
                     1.45 
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     61.8 
                   
                   
                     
                       - 
                       0.163 
                     
                   
                   
                     
                       - 
                       94 
                     
                   
                   
                     27.1 
                   
                   
                     55.3 
                   
                   
                     
                       - 
                       1.43 
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     1.06 
                   
                   
                     
                       - 
                       1.06 
                     
                   
                   
                     
                       - 
                       0.789 
                     
                   
                   
                     0.79 
                   
                   
                     
                       - 
                       0.194 
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     175 
                   
                   
                     
                       - 
                       23.2 
                     
                   
                   
                     
                       - 
                       95 
                     
                   
                   
                     2.84 
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     41.3 
                   
                   
                     20 
                   
                   
                     
                       - 
                       1.46 
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     57.6 
                   
                   
                     
                       - 
                       1.74 
                     
                   
                 
                 
                   
                     
                       sym 
                       . 
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     0.99 
                   
                 
               
               ) 
             
           
         
       
     
       FIG. 10  shows a graph representing the angular backlash of the gearing E 2  over an angular pitch P and for different axial center distances  , for a given vector {right arrow over (p)}. Note that the maximum angular backlash j max  at the nominal axial center distance is of the order of 0.05°, i.e. approximately eight times less than the maximum angular backlash j max  defined by the gearing from  FIGS. 1 and 2 . Accordingly,   is of the order of 0.04.m. Moreover, in this particular case, the angular backlash can vary with a maximum amplitude of the order of 0.1° for a axial center distance varying over the range [e−30 μm, e+30 μm]. In such an example, the sensitivity of the maximum angular backlash as a function of the gearing axial center distance is about 1.1°/mm when considering a gearing axial center distance range of 0.06 mm, the range being centered on the nominal gearing axial center distance (Δjmax/Δe=1.1°/mm). 
     By way of further example, the angular backlash of a gearing E 30  that is not shown consisting of a pinion R 10 ″ having 33 teeth defined on the basis of a Treybal profile known from the prior art and the modulus m of which is 0.0602 mm, driving a wheel R 20 ″ having 110 teeth defined on the basis of the Treybal profile and the modulus In of which is 0.0602 mm, can in particular be minimized by means of genetic algorithms. This kind of third embodiment E 3  of the gearing is represented in  FIG. 11 . The profiles of the teeth of the wheels R 1 ″ and R 2 ″ of the gearing E 3  according to the third embodiment can be characterized by a vector {right arrow over (p)} respecting the condition ({right arrow over (p)}-{right arrow over (p)}*) t H({right arrow over (p)}-{right arrow over (p)}*)≤1, with: 
     
       
         
           
             
                 
             
              
             
               
                 p 
                 → 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         p 
                         
                           1 
                            
                           1 
                         
                       
                     
                   
                   
                     
                       
                         p 
                         
                           1 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       
                         p 
                         
                           1 
                            
                           3 
                         
                       
                     
                   
                   
                     
                       
                         p 
                         
                           1 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       
                         p 
                         
                           2 
                            
                           1 
                         
                       
                     
                   
                   
                     
                       
                         p 
                         
                           2 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       
                         p 
                         
                           2 
                            
                           3 
                         
                       
                     
                   
                   
                     
                       
                         p 
                         
                           2 
                            
                           4 
                         
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   p 
                   → 
                 
                 * 
               
               = 
               
                 ( 
                 
                   
                     
                       0.47511 
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                           1 
                         
                          
                         8 
                          
                         8 
                          
                         4 
                          
                         3 
                       
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                         
                          
                         3 
                          
                         0 
                          
                         1 
                          
                         9 
                          
                         2 
                       
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                           4 
                         
                          
                         0 
                          
                         5 
                          
                         7 
                          
                         4 
                       
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                           3 
                         
                          
                         3 
                          
                         0 
                          
                         4 
                          
                         1 
                       
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                           1 
                         
                          
                         4 
                          
                         1 
                          
                         5 
                          
                         8 
                       
                     
                   
                   
                     
                       
                         
                           0 
                           . 
                           4 
                         
                          
                         8 
                          
                         2 
                          
                         3 
                          
                         2 
                       
                     
                   
                   
                     
                       0.43577 
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
                 
             
              
             and 
           
         
       
       
         
           
             H 
             = 
             
               
                 ( 
                 
                   
                     
                       
                         7 
                          
                         
                           2 
                           . 
                           2 
                         
                       
                     
                     
                       
                         1 
                          
                         
                           0 
                           . 
                           1 
                         
                       
                     
                     
                       
                         - 
                         46.5 
                       
                     
                     
                       
                         
                           1 
                           . 
                           7 
                         
                          
                         9 
                       
                     
                     
                       
                         4 
                          
                         
                           6 
                           . 
                           2 
                         
                       
                     
                     
                       
                         - 
                         
                           2 
                           . 
                           8 
                         
                       
                     
                     
                       
                         
                           - 
                           2 
                         
                          
                         
                           9 
                           . 
                           6 
                         
                       
                     
                     
                       
                         
                           0 
                           . 
                           4 
                         
                          
                         5 
                          
                         9 
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                         1 
                          
                         
                           9 
                           . 
                           9 
                         
                       
                     
                     
                       
                         - 
                         6.14 
                       
                     
                     
                       
                         
                           - 
                           
                             0 
                             . 
                             0 
                           
                         
                          
                         9 
                       
                     
                     
                       
                         
                           2 
                           . 
                           0 
                         
                          
                         5 
                       
                     
                     
                       
                         
                           - 
                           
                             2 
                             . 
                             4 
                           
                         
                          
                         9 
                       
                     
                     
                       
                         
                           - 
                           
                             3 
                             . 
                             6 
                           
                         
                          
                         8 
                       
                     
                     
                       
                         
                           1 
                           . 
                           4 
                         
                          
                         8 
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         3 
                          
                         
                           4 
                           . 
                           3 
                         
                       
                     
                     
                       
                         
                           - 
                           
                             0 
                             . 
                             9 
                           
                         
                          
                         7 
                       
                     
                     
                       
                         
                           - 
                           3 
                         
                          
                         
                           0 
                           . 
                           9 
                         
                       
                     
                     
                       
                         
                           2 
                           . 
                           1 
                         
                          
                         5 
                       
                     
                     
                       
                         1 
                          
                         
                           9 
                           . 
                           5 
                         
                       
                     
                     
                       
                         
                           - 
                           
                             0 
                             . 
                             3 
                           
                         
                          
                         8 
                          
                         8 
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
                           1 
                           . 
                           2 
                         
                          
                         3 
                       
                     
                     
                       0.581 
                     
                     
                       
                         - 
                         1.08 
                       
                     
                     
                       
                         - 
                         0.656 
                       
                     
                     
                       
                         - 
                         0.11 
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         3 
                          
                         
                           5 
                           . 
                           1 
                         
                       
                     
                     
                       
                         
                           1 
                           . 
                           4 
                         
                          
                         4 
                       
                     
                     
                       
                         - 
                         19.8 
                       
                     
                     
                       
                         
                           0 
                           . 
                           3 
                         
                          
                         3 
                          
                         3 
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
                           6 
                           . 
                           2 
                         
                          
                         6 
                       
                     
                     
                       
                         
                           0 
                           . 
                           7 
                         
                          
                         1 
                          
                         1 
                       
                     
                     
                       0.321 
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         1 
                          
                         
                           4 
                           . 
                           2 
                         
                       
                     
                     
                       
                         
                           - 
                           
                             0 
                             . 
                             1 
                           
                         
                          
                         5 
                          
                         4 
                       
                     
                   
                   
                     
                       
                         sym 
                         . 
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
                           1 
                           . 
                           0 
                         
                          
                         9 
                       
                     
                   
                 
                 ) 
               
               . 
             
           
         
       
     
       FIG. 12  represents the maximum angular backlash j max  of the gearings E 30  and E 3  over an angular pitch P in accordance with a variation x of the nominal axial center distance   of said gearings over a range [e−40 μm, e+60 μm]. In such an example of the gearing E 3 , the sensitivity of the maximum angular backlash as a function of the gearing axial center distance is about 1.7°/mm when considering a gearing axial center distance range of 0.06 mm, the range being centered on the nominal gearing axial center distance (Δjmax/Δe=1.7°/mm). Note that the maximum angular backlash j max  of the gearing E 3 , at the nominal axial center distance, is of the order of 0.2°, i.e. approximately 2.4 times less than the maximum angular backlash j max  defined by the gearing E 30 . Over the given axial center distance range, the variation of the maximum angular backlash j max  of the gearing E 3  is moreover of the order of four times less than that induced by the gearing E 30 . Note furthermore that with a gearing of this kind the variation in the amplitude at the balance wheel is reduced. 
     The gearings described above therefore have teeth profiles conformed so as to minimize the angular backlash of a horological gearing, in particular of a gearing comprising a wheel mounted in parallel coupling with a geartrain. Moreover, the sensitivity of the variation of the backlash as a function of the gearing axial center distance is limited. 
     As already seen, the invention concerns a method of manufacturing the gearings E 1 ; E 2 ; E 3  and/or the horological mechanism  100  and/or the horological movement  200  and/or the timepiece  300 . The method comprises: 
     a step of determination of the profiles of the first and second toothings of the first wheel R 1 , R 1 ′, R 1 ″ and of the second wheel R 2 ; R 2 ′; R 2 ″, 
     a step of shaping the first and second wheels, 
     a step of mounting the first and second toothed wheels so that they mesh with one another. 
     The step of determination of the profiles of the first and second toothings of the first wheel R 1  and of the second wheel R 2  preferably comprises, after a step of selection, in particular a step of arbitrary selection, of profiles of the first and second toothings, iteration of the following steps: 
     determination of the performance of the profiles of the toothings; 
     generation of new toothing profiles using genetic algorithms, in particular stochastic optimization algorithms based on mechanisms of natural selection and of genetics and/or algorithms employing evolutionary operators, namely selection and/or crossing and/or mutation. 
     In this document, by “symmetrical tooth” is meant that there exists a straight line segment passing through the rotation axis of the wheel and constituting an axis of symmetry of the profile of the tooth. 
     By “toothed wheel” is meant any wheel having a toothing. This definition includes the pinions. This toothing may extend over 360° or over a particular angular range. Accordingly, this definition also includes any rack with a toothed sector. This definition preferably also includes any rack. By “gearing” is meant any assembly comprising toothed wheels of this kind. 
     By “method of manufacture” of the gearing is meant a method leading to the definition of the profiles of the first and second toothings of the wheels forming part of a gearing of this kind, and shaping or manufacturing each of those wheels. 
     As previously mentioned, the angular backlash of the gearings according the invention are preferably very slightly sensitive to axial center distance variations. Preferably, in the various embodiments disclosed above or according to the invention, each of the first teeth and of the second teeth is conformed such that the ratio of the variation of maximum angular backlash jmax to the variation of the gearing axial center distance is less than 6°/mm or less than 5°/mm or less than 4°/mm or less than 3°/mm, for a gearing axial center distance value varying between the nominal gearing axial center distance minus 0.04 mm and the nominal gearing axial center distance plus 0.04 mm. Preferably, the upper limits of the ratio mentioned above apply globally to the whole above mentioned range (nominal gearing axial center distance minus 0.04 mm to nominal gearing axial center distance plus 0.04 mm). Preferably, the upper limits of the ratio mentioned above apply locally on the whole above mentioned range (nominal gearing axial center distance minus 0.04 mm to nominal gearing axial center distance plus 0.04 mm) too, i.e. the derivative with respect to the gearing axial center distance of the maximal angular backlash is less than 6°/mm or less than 5°/mm or less than 4°/mm or less than 3°/mm on the whole above mentioned range. 
     In other words, each of the first teeth and of the second teeth is conformed such that the ratio Δjmax/Δe is less than 6°/mm or less than 5°/mm or less than 4°/mm or less than 3°/mm, Δjmax being the maximal angular backlash of the gearing and Ae being the gearing axial center distance variation and Δe being less or equal to 0.08 mm and Ae being centered on the nominal gearing axial center distance. 
     Preferably: 
     the first toothed wheel includes first symmetrical teeth d 1 , first teeth based on cycloid curves and/or epicycloid curves and/or hypocycloid curves and/or involutes of a circle being excluded, and/or 
     the second toothed wheel includes second symmetrical teeth d 2 , second teeth based on cycloid curves and/or epicycloid curves and/or hypocycloid curves and/or involutes of a circle being excluded.