Hidden Markov Models used for computer music synthesis do not satisfactorily reproduce Indian Carnatic music and also require large training datasets. The essence of Indian Carnatic music is its micro-tonal frequency variations called Gamakas. In this work, we study the flute note properties, features that characterize the Gamakas, and hence attempt to devise a generalized method for synthesizing Carnatic music flute compositions. Our method uses additive sinusoidal synthesis coupled with a stochastic noise model. In time domain, splines are used to model the amplitude envelope to ensure a natural reconstruction. Integrated frequency contours are used for smooth concatenation of notes and modelling of Gamakas and notes. In order to evaluate our synthesis, we use a Mean Opinion Score (MOS) survey to compare our results with the baseline and the original recordings. The MOS of the proposed method is around 3.5 while the baseline is 2.3.