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Mathematical Modelling of Steps in BharathaNatyam
Mathematical Modelling involves creating a model of a real world system using techniques in Mathematics such as Linear Programming, Differential Equations etc. When the system model has inherent uncertainty, simulation is used in addition to the Mathematical Model to represent either a stationary or a dynamic system (System in Motion).
Adavus in BharathaNatyam (Classical dance art form of south India) represent a set of steps which do not involve expression (nrityam). So Adavus can be studied using Mathematical Models.
Tattu Adavu involves lifting the feet up and down so that one can hear the tapping noise.
The “sollukattu”(tamil word translated into English as Verbal pronunciation of beats) is rendered in varying tempos. There are also repeated movement of the feet in various counts such as 4,6 and 8.
The four verbal beats can be pronounced as tai,ya, tai,hi. If the four verbal beats occur at T(1), T(2), T(3) and T(4) where T(I) is the ith instant of time when the verbal beat is pronounced by the accompanying artiste.
The speed or the tempo is given by T(2) – T(1) T(3) – T(2) and T(4) – T(3). Ideally all these time intervals should be equal. It can be equal if these beats are machine generated. But when an artiste renders these sounds or beats the intervals will not be uniform and will vary randomly. Such variations can be captured using Simulation models.
If the entire step of upwards and downwards movement of the feet one time takes 30 seconds (say) at normal speed. It would take 20 seconds and 10 seconds in the second and the third tempos. For example if tai occurs at 0th instant, ya occurs at the 13.5 seconds, tai is the wait time for 3 seconds and hi occurs at the 30th second, the upward motion of the feet lasts for 13.5 seconds and the downward motion lasts for 13.5 seconds and the wait time lasts 3 seconds. A danseuse and a vocalist cannot render such uniform motion to exactness as demonstrated by the mathematical model and there may be variations.
The dancer’s or the artiste’s movement can be modelled by the position of the torso in space or x,y,z co-ordinates and the relative motion of the Feet, Legs, Upper Hand, Lower Hand, Arms head, neck and eyes with respect to the torso.
For a sequence of Tattu Adavu steps starting at time t = 0 and ending at time t = T the equation of the feet at an instantaneous time t is given by the position of the torso of the dancer and the relative position of the feet with respect to the Torso.
Since Tattu Adavu involves tapping of the feet and movement upwards the resultant motion of say the toes can be modelled using algebra using the following discrete time equations resulting in step functions describing the motion. Differential equations cannot be used as they would represent a system that is continuous.
So writing these equations of the Tattu Adavu as y =0 at t= 0 y = h at t = T/2 and y = 0 at at t = T where T is the time period of a beat and h is the maximum height reached by a foot. This can fixed at 30 cms or can be varied between 25 cms and 50 cms. This is the algebraic model of the 1st Tattu Adavu. In case a model of variation is to be used, then the algebraic model used should be replaced with a simulation model.
The second tattu adavu or the tapping of the feet with two times per beat can be modelled as y =0 at t=0 y = h at t = T/4; y = 0 at t=T/2; y=h at t = 3T/4; y= 0 at t = T.
If the locus of the feet is plotted for more number of points along the time interval then the same equation can be described as y = 0 at t= 0; y = h/10 and t= T/10; y = h/9 at t = T/9 etc.
A dancer with natural motion will not be able to replicate the exact mathematical congruence of the height attained by the moving feet with the respect to the divisions within the time period of the Sollukattu.
If one plots the actual motion of a dancers feet while performing the ‘tattu adavu'(translated in english as tapping of the feet) the resulting equation would be h = 0 at t= 0, y = 0.6h at t= T/2 and h = 1.1h at t = T etc.
These algebraic equations can be used to write computer programs which use graphics to model the motion of a classical dancer’s feet. Hence some aspects of the mechanical steps or adavus can be automatically generated based on using appropriate models to capture the movement of the feet.
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