Mathematical Modeling of Steps in BharathaNatyam

May 23, 2021 0 Comments

Mathematical modeling 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, the 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). So Adavus can be studied using mathematical models.

Tattu Adavu involves lifting the feet up and down so that one can hear the noise of the blows.

The “sollukattu” (Tamil word translated into English as verbal pronunciation of rhythms) translates into different rhythms. There are also repeated foot movements at 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 in T (1), T (2), T (3) and T (4) where T (I) is the i-th instant of time when the accompanying artist pronounces the verbal beat.

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 the same. It can be the same if these rhythms are generated by a machine. But when an artist plays these sounds or rhythms, the intervals will not be uniform and will vary randomly. These variations can be captured using simulation models.

If the whole step of the up and down movement of the feet once takes 30 seconds (say) at normal speed. It would take 20 seconds and 10 seconds in the second and third time. For example, if tai occurs at instant 0, it already occurs at 13.5 seconds, tai is the waiting time of 3 seconds and hi occurs at second 30, the upward movement of the feet lasts 13.5 seconds and the downward movement lasts 13.5 seconds. and the time-out lasts 3 seconds. A dancer and a vocalist cannot represent a uniform movement with the accuracy that the mathematical model demonstrates and there may be variations.

The movement of the dancer or artist 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, arms, head, neck and eyes with respect. to 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 instantaneous time t is given by the position of the dancer’s torso and the relative position of the feet with respect to the torso.

Since Tattu Adavu involves foot tapping and upward motion, the resulting motion of, say, the toes can be modeled using algebra using the following discrete time equations that result in step functions that describe motion . Differential equations cannot be used as they would represent a continuous system.

So, write these equations of the Tattu Adavu as y = 0 at t = 0 y = h at t = T / 2 and y = 0 at t = T where T is the time period of one beat and h is the maximum height reached. by one foot. This can be set at 30 cm or it can be varied between 25 cm and 50 cm. This is the algebraic model of the first Tattu Adavu. In case a variation model is to be used, the algebraic model used should be replaced by a simulation model.

The second adavu tattu or the tapping of the feet twice per beat can be modeled as y = 0 at t = 0 and = 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 a greater number of points over 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 movement will not be able to replicate the exact mathematical congruence of the height reached by the moving feet with respect to the divisions within the Sollukattu time period.

If the actual movement of a dancer’s feet is plotted while performing the ‘tattu adavu’ (translated as tapping of the feet), the resulting equation would be h = 0 at t = 0, y = 0.6h at t = T / 2 and h = 1.1 h at t = T, etc.

These algebraic equations can be used to write computer programs that use graphics to model the movement of a classical dancer’s feet. Therefore, some aspects of mechanical steps or adavus can be generated automatically based on the use of appropriate models to capture the movement of the feet.

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