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Mathematical Modelling of Steps in BharathaNatyam
Mathematical modeling involves creating a model of a real-world system using mathematical techniques 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 a stationary or dynamic system. (System in motion).
Adavus in BharathaNatyam (South Indian classical dance art form) represents a set of steps that do not involve expression (nrityam). Thus, Adavus can be studied using mathematical models.
Tattu Adavu involves lifting the feet up and down so that the sound of the tapping can be heard.
The “sollukattu” (Tamil word translated into English as verbal pronunciation of the beats) is rendered at varying tempos. There are also repeated movements of the feet in various counts such as 4, 6 and 8.
The four verb tenses can be pronounced as tai,ya, tai,hi. If the four verb tenses occur at T(1), T(2), T(3) and T(4) where T(I) is the ith instant of the time when the verbal tense is pronounced by the accompanying artist.
The speed or tempo is given by T(2) – T(1) T(3) – T(2) and T(4) – T(3). Ideally, all of these time intervals should be equal. It may be equal if these beats are generated by a machine. But when an artist renders these sounds or beats, the intervals will not be uniform and will vary randomly. These variations can be captured using simulation models.
If the entire step of moving the feet up and down once takes 30 seconds (say) at normal speed. It would take 20 seconds and 10 seconds in the second and third tempos. For example if tai occurs at 0th instant, ya occurs at 13.5 seconds, tai is the waiting time for 3 seconds and hi occurs at 30th second, the upstroke of the feet lasts 13.5 seconds and the downward movement lasts 13.5 seconds and the waiting time lasts 3 seconds. A dancer and a singer cannot render such a uniform movement with exactness as demonstrated by the mathematical model and there may be variations.
The movement of the dancer or performer can be modeled by the position of the torso in space or the x, y, z coordinates and the relative movement of the feet, legs, upper hand, lower hand , head, neck and arm eyes. at the torso.
For a sequence of Tattu Adavu steps beginning 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 dancer’s torso and the relative position of the feet with respect to in the Torso.
Since Tattu Adavu involves the tapping of the feet and upward motion, the resulting motion of the toes, for example, can be modeled using algebra using the following discrete time equations, yielding step functions describing movement. Differential equations cannot be used because they would represent a continuous system.
So write these Tattu Adavu equations as y=0 to t=0 y=h to t=T/2 and y=0 to t=T where T is the time period of a beat and h is the maximum pitch reached by one foot. This can be fixed at 30 cm or vary between 25 cm and 50 cm. This is the algebraic model of the 1st Tattu Adavu. In the case where a variation model must be used, the algebraic model used must be replaced by a simulation model.
The second adavu tattu or foot tapping with twice per beat can be modeled as y=0 to t=0 y=h to t=T/4; y=0 to t=T/2; y=h to t=3T/4; y= 0 to t = T.
If the locus of the feet is plotted for a larger number of points along the time interval, the same equation can be described as y = 0 to t = 0; y = h/10 and t = T/10; y = h/9 to t = T/9 etc.
A dancer with natural movement will not be able to reproduce the exact mathematical congruence of the height reached by the moving feet with respect to the divisions in the time period of the Sollukattu.
If one plots the actual movement of a dancer’s feet when performing ‘tattu adavu’ (translated into English as tapping the feet), the resulting equation would be h=0 to 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 that use graphics to model the movement of a ballet dancer’s feet. Therefore, some aspects of mechanical steps or adavus can be generated automatically using appropriate models to capture foot movement.
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