Wikipedia
Vadi, in both Hindustani classical music and Carnatic music, is the tonic (root) swara (musical note) of a given raga (musical scale). "Vadi is the most sonant or most important note of a Raga." It does not refer to the most played note but it rather refers to a note of special significance. It is usually the swara which is repeated the greatest number of times, and often it is the swara on which the singer can pause for a significant time. Vadi swara in a raga is like a king in a kingdom. Specialty of any raga depends on vadi swara and because of this, the vadi swara is also called the Jeeva swara or the Ansha swara. A good artist uses vadi swara in different ways like singing vadi swara again and again, starting a raga with vadi swara, to end a raga with vadi swara, singing vadi swara many times in important places with different swaras or sometime singing vadi swara for a longer time in one breath.
Vadi swara is also helpful to identify the appropriate time for singing or playing a raga. If the vadi swara of a raga is from the purvanga part of the saptak i.e. “Sa Re Ga Ma”, then it is called purvanga vadi raga and usually the time for singing or playing purvanga vadi ragas is from 12 pm to 12 am. For example, ragas like Bhimpalasi, Pilu, Purvi, Marwa, Yaman, Bhoopal, and Bageshree etc. have purvanga vadi swara and so the time for singing and playing these ragas is between 12 pm to 12 am.
In the same way, if the vadi swara of a raga is from the utranga part of the saptak i.e. “Pa Dha Ni” then it is classed as a utranga vadi raga and the time for singing or playing utranga vadi ragas is from 12 am to 12 pm. For example ragas like Bhairav, Bhairavi, Bilawal, Kalingada, Sohini, and Asavari etc. have utranga vadi swara and so the time for singing or playing these ragas is between 12 am to 12 pm.
Vadi swara, along with the Samvadi swara of a raga, usually brings out the uniqueness of the raga and its bhava (mood) and rasa (emotion).