Though only 7 symbols are used, actually most musical systems use 12 notes in an octave. The additional 5 notes can be easily seen on a keyboard as they are black. See a picture of a keyboard covering 2 octaves with Carnatic music symbols on the keys for an Aadhaara Sruthi of 1 kattai or C. If you use Google Chrome or Mozilla Firefox browser you can click on a key to hear the note. The five notes ri, ga, ma, da and ni have 2 variations each (black and white). The variations are often informally mentioned in teaching (as 'small ri' or 'big ri' etc.).
On a fretted instrument like the Veena there are 12 frets in an octave. See a picture and listen to the notes of a Veena by clicking as described (use Google Chrome or Mozilla Firefox browser for audio). The audio is based on a tuning for 3 kattai as Aadhaara Sruthi (commonly used for Veena). The symbols ri1, ri2 etc. have been used based on the informal system (also used in the Varnam Book by A.S.Panchapakesa Iyer). Other systems are also used leading to ambiguity. Only the descriptive names for the notes such as 'Suddha Rishabham' etc. are unambiguous and are based on a more elaborate 72 Melam system described later.
The possibility of choice of two variants each for the notes ri,ga,ma,da and ni would imply 32 selections (25 = 32). It is the choice of 7 notes out of the 12 that enables creation of melodies with widely varying moods.
Though a Carnatic musician can choose his Aadhaara Sruthi at any pitch, it is the practice to make it coincide with the pitches of the keys of the Harmonium/Keyboard to help accompaniments to use appropriate instruments for that sruthi. In the 'kattai' system, the white key corresponding to C (usually the first key) is taken as 1 kattai and each subsequent white key is given the next number. The black keys are given half numbers - the black key between 1 kattai and 2 kattai is 1.5 kattai. (There is no 3.5 kattai or 7.5 kattai!). See a picture of keyboard with Kattai names marked. With the advent of Electronic Thamboora and Sruthi Box the corresponding Western music symbols are also often used.