It is very difficult to say about the root of Bengali music, but one thing is sure that Bengali music has lots of variations.
Here is given an overview of different kinds of Bengali music:

Vocal - Kheyal, different ragas.

Instrumental - Sitar, Sarod, Flute, Tabla, Ektara
 
Folks Songs
Baul - these are the songs sung by a specific group of people known as Bauls.
Kirtan - religious songs that are mostly sung in chorus.
Bhatiali - these songs are more commonly sung by the boatmen. S.D.Burman experimented a lot with these tunes and used them in several popular modern songs.
Majlishi Classical Songs - thungri, toppa
Devotional Songs - Ramprasadi, ShyamaSangeet, Bhaktigeeti
Rabindra Sangeet - songs written and tuned by Rabindranath Tagore
General Modern Bengali Songs- Film and non-film songs
Ganasangeet- these songs convey a social message and are usually sung in chorus.
Classical Music