One of the most important ragas in Carnatic music, Todi again uses notes that form part of a scale that is used in different genres of music around the world. It is one of the six ragas that are generated through mutual graha bhedam, the other five being Kalyani, Shankarabharanam, Kharaharapriya, Harikamboji and Natabhairavi.
However, the raga Todi is very different from the scale because of the way in which it uses the notes. Its uniqueness comes from the fact that it makes profuse use of gamakas. This brings microtones into the picture and gives the various notes a multitude of shades. Depending on the prayogas and combination of notes, each note takes on a different character, making Todi a raga of infinite variety and range. In fact, Todi is probably the most "microtonal" of all the ragas in Carnatic music.
If you disregard the vivaadi notes (double flats), Todi uses the lower value for each of its notes (except sa and pa, of course). The notes ga and ni are generally used with a lot of oscillation, though the occasional usage of flat notes suddenly gives a different accent to the raga.
Though it is the sustained and exaggerated use of gamakas that gives Todi its unique character, it doesn't seem like the raga was always used this way, especially if one goes by the Sangeeta Sampradaya Pradarshini, written by Subbarama Dikshitar, which arguably contains the most authentic versions of Muthuswami Dikshitar's compositions. Going by the interpretation of the gamakas given in that book, Todi appears to have overwhelmingly used flat or near-flat notes, with the elaborate gamakas being used sparingly and more as an occasional ornamentation to highight the beauty of a phrase. This, however, is not a take that many accept, and the original character of Todi could be at the centre of a never-ending argument of theory!
Getting back to Todi as it exists, different schools use different levels of oscillation in the gamakas. For instance, the GNB school (G N Balasubramaniam, M L Vasanthakumari, Tanjavur S Kalayanaraman, among others), which is reputed for its mastery of Todi, makes calibrated use of oscillation of the notes ga and ni.
Todi is one of the ragas that uses all the forms of gamakas. G N Balasubramaniam's composition, Mama Kuleshwaram, uses all the 10 kinds of gamakas mentioned in texts. The chittaswaras written by Baluswami Dikshitar for Kumara Ettendra's composition, Gajavadana Sammodita Veera, show the various kinds of gamakas that the note ga can use to produce different microtones and effects.
Along with Kalyani, Todi is a sampoorna raga that permits extensive improvisation without using the panchama, and gains a distinct beauty from this panchama-varjya usage. In virtually no other raga is this possible.
Given the importance that Todi has in the order of things, it is no wonder that there are so many important and well known compositions in the raga by virtually every composer of note. It isn't likely that there are people who listen to Carnatic music but don't know Taaye Yashoda, Oothukkadu Venkatasubbier's exceedingly popular song. Also, every note in Todi has been used as a starting point of a composition, proving not just the range of the raga but also the importance of each note to the character of the raga. Tyagaraja has composed two dozen songs in the raga, covering the widest possible range of expression and prayogas in the raga. There is also an interesting composition of M Balamuralikrishna in Todi (Maa Maanini), where almost the whole of the lyrics are in the form of swaraaksharas, where the letters stand for the notes; even through many of the varjya prayogas and leaps over notes that this necessitates, the character of Todi remains steady.
Every student of Carnatic music is taught the adi tala varnam, Eraa Naapai. The ata tala varnam, Kanakaangi, though not as well known, is a grander and much weightier piece.
Some of the important janya ragas of Hanumatodi (its name in the melakarta nomenclature) are Dhanyasi, Asaveri, Hindolam and Punnagavarali.