Some other ragams

(4) Some other ragams, instead of going up or down simply, can go up or down in a zig zag manner - such as Sa-ri1-ma1-ga2-pa-ni2-dha1-sa etc. That is, you cannot simply go up in scale by merely pressing the keys, but you should spiral to the top of the scale. There are not too many such ragams, but such a meandering structure is called 'Vakram', which literally means crooked. This is an additional constraint imposed on the Ragam, besides the key selection.
(5) In some other instances, it may not be easy to define uniquely the Arohanam and Avarohanam of a ragam. Many Arohanams and Avarohanams (i.e, definitions) can exist for one ragam itself. An example of such a Ragam is Ananda Bhairavi. Of course, those Arohanams and Avarohanams will be close to each other and won't be radically apart. This situation exists especially when we try to extract the Ragam equivalent of folk melodies or alien tunes.
(6) And finally here is a confusing possibility. There can be two Ragams which have identical Arohanams and Avarohanams, but DIFFERENT MICROTONAL ASSOCIATIONS or Gamakams ! The only way to tell these two Ragams apart is to sensitize your ears to the differences to the Gamakams. Of course, you can never possibly play them on the keyboard as two different Ragams!
You can go ahead and create your own ragam by selecting your own five keys (or six or seven) following the above rules and name it after yourself. (But make sure it doesn't already exist !) However, if you created your own pentatonic-pentatonic ragam, you probably did not choose just the first five keys of the octave. You might have distributed the five keys such that they were spread out in the octave instead of being bunched together, just so that your ragam sounded better. In fact, such subjective criteria have given resulted in only a few Ragams being popular.
Mathematically, there are many, many ragams possible. Choosing five, six or seven keys out of possible twelve keys gives rise to a huge number of combinations. Fortunately, many of the possibilities have been deemed 'boring to the ear' by musicians throughout history. Only about six thousand or so ragams have been even cataloged and of these, only about two hundred or so are even used these days. A ragam's popularity can go up and down, depending on people's taste and the existing political climate of the Karnatic music caucus. So, it is really not a tremendous task to learn about fifty or so of the more popular ragams and be good at identifying them, if at least to impress your friends.
It is always possible to break down any song, even the non-Karnatic music songs into its constituent Swarams and define a corresponding Ragam. Even 'Baa baa black sheep' can be broken into a Ragam. Musicians more clever than we are have done such things and created Ragams out of truly Dravidian folk melodies such as 'Aadu Pambey' (the snake song) or Kavadi Sindhu songs like 'Nandavanathil or aandi' and created ragams like Ananda Bhairavi or Kurunji. Sometimes, the ragam corresponding to songs like 'Baa baa black sheep' may not have enormous scope to create a lot of 'characteristic phrases' and thus limiting creating any more songs based on the ragam.

RAGAM

WHAT IS A 'RAGAM'? Now that we have studied the Melakarta scheme inside and out, let us go on to generate the secondary or 'janya' or derived Ragams (the rest of the Ragams, that is) based on some simple guidelines. These are only guidelines and are not hard and fast rules:
(1) A Ragam should use at least five keys in an octave and utmost seven keys in the Arohanam as well as the Avarohanam.
(2) The Arohanam or ascending order of the notes (or Avarohanam or descending order, for that matter) is obtained by simply taking a Melakarta scale and omitting none or one note or two notes. (Remember, the Melakarta scale has seven notes and so we can end up with seven or six or five notes in the derived scale)
For example, let us (yet again !) take Ragam Shankarabharanam. If we omit the keys 'Ma' and 'Ni' and use only the five white keys Sa, Ri, Ga, Pa, Dha then we obtain a famous Ragam called Mohanam. (Hindustani equivalent is Bhoop or Bhopali)
(Usually, the the next octave's Sa is also included for completion and hence the Arohanam will be more correctly given as Sa-ri-ga-pa-dha-Sa. Similarly, the Avarohanam is given by Sa dha pa ga ri sa. You will notice that almost all ragams start with the key Sa. Also, from now on, we will omit saying 'Ri 1' or 'Ri 2' etc. IF THERE IS NO AMBIGUITY AS TO WHICH KEY WE ARE USING.