Having said all this....

Having said all this, we should also realize that there is nothing inherently scientific or sacred about this 'Equally tempered, twelve key per octave' Western music system, where an arbitrary set point is created at 240 Hz. There are other alternate systems and creative musicians are always experimenting with unconventional systems - in music it is 'cool' to break tradition ! For example, there is no need to have just twelve keys in an octave. In fact, the traditional Indian music system over thousands of years is based on a 22 key per octave system. Even if you chose twelve keys to fill in an octave, there is no reason to tune them in a geometric progression. In other words, you don't have to have an 'Equally tempered scale'. You can locate your frequencies based on some other non-geometric criteria which might 'sound' even better. Such scales in fact, exist and they are called 'Just tempered scales'. In fact, the Indian musical system uses one such scale. And in the final analysis, there is no need to even stick to the concept of octaves when producing music. In short, to produce music, there is no need for a grammar and rules ! If it sounds pleasnt, you are on. However, let us first learn the established grammar and tradition, before we attempt to break them!
Even though Indian musical systems are very different from the traditional Western Music system, we can still get a lot of insight into Indian music using the Equally tempered, twelve keys per octave methodology - essentially because it makes things simple. (Also, the keyboard is probably one of the easiest instruments to play). This has always been a bone of contention between the traditional Indian musicologists and the 'quick and dirty' folks like us. The Indian traditionalist will argue that we are compromising by limiting ourselves to just twelve 'tones' per octave, when tradition, dating back thousands of years, categorically spells out twenty two 'tones' per octave. The twenty two 'sruti' for the middle C octave is given in Table III. By the way, the Indian word for frequency or pitch or 'tone' is 'Sruti. A word of caution though - the term 'Sruti' has several other interpretations and meanings. We will come across some of them later.
Why, some Indian schools of thought even propose that there are infinite frequencies in an octave. The basic reason for such demands for more than twelve 'srutis' per octave is that Indian music, (not just Karnatic music) seems to 'flow' through the frequencies, whereas a Western song seems 'jumpy'. Take for example, 'Baa baa black sheep' and compare it to an Indian song, say, a movie song like 'Roop tera mastaana'. The Indian song seems to involve a lot of vocal acrobatics and nuances and not just go through piano-like jumps. This is the main difference between the Indian and Western music and we will return to this point again and again in this primer.
This is the reason why Indian classical music cannot be played effectively in a twelve key per octave instrument like a piano. Of course, several Western instruments have been 'adapted' with a little modification here and there, to play Indian classical music - violin, mandolin and guitar, for example. Some other instruments have been simply 'used', without modification, such as the harmonium and its latest cousin, the keyboard. Indian purists abhor such blatant use of Western, 'equally tempered' instruments. Expert harmonium player, Rajan Parrikar, points out that 'Just tempered' harmoniums, harmoniums with 22 tones per octave, even over 50 tones per octave etc have been built by various people.
Now let us come back to this basic difference between the Western and the Indian classical music system. We noted that in Indian music it is not enough to produce just twelve or even twenty two 'tones' in an octave. One ought to produce even the intermediate frequencies. These intermediate frequencies, which do not have any keys to produce them, are called 'microtones'. The Indian word for the 'microtone' is 'gamakam'. (of course, 'gamak' in hindi) It is often very difficult to explain this concept clearly and precisely. If the C key produces 240 Hz and the C# key produces 254 Hz what intermediate frequencies are we talking about ? Does Indian music use sounds produced at 247 Hz ? Treatises have been written in India about such microtonal apects of music. Suffice it to say that microtones or gamakams tend to be clustered around the primary key frequency, although this need not always be the case.
Note that if you postulate that an Indian music octave has twenty two or two hundred keys or infinity per octave, then what used to be a 'microtone' in a twelve-key system could now very well be a key. Of course, you can make a piano with such large number of tones per octave. It may be a long piano and you will need a superhuman dexterity to play it. But the positive side of it is that you will be able to play Indian music on it. The bottomline is, the piano produces just twelve frequencies in one octave and that is enough to compose a lot of Western songs. Whereas, to make Indian music, twelve keys are not enough in an octave.
Let us talk some more about microtones or gamakams. The vocal gliding and rolling in Indian music, (Remember Kishore Kumar's yodelling ?) whether it sounds good or not, are again examples of microtone usage. In fact, the microtones add variety to the Indian classical music - an extra dimension. From movie songs to folk music to classical music, the very heart of Indian music is this 'continuous flow' or 'gliding through a continuum of frequencies' or gamakam or microtonal excursions. Thus it is often said that Indian music is 'melody-based'. Since microtones are so important in Karnatic and Hindustani music and very few instruments can produce all the frequencies in an octave, the best enunciation of Indian classical music is in vocal singing. Many instruments like the violin, Gottuvadhyam (called Chitra Veena these days) and even the simple bamboo flute can produce a lot of gamakams, of course.
Just to drag the concept a little farther - some of you who have grown up in India may have developed a taste for Indian music, be it movie songs or highly classical songs. When you were exposed to Western rock and roll music you may have been drawn toward the numbers rich in gamakams (The Beatles, Simon and Garfunkel et al have produced many such pieces) rather than the rhythm oriented heavy metal numbers.
In the same breath, people say that the Western music is 'harmony-based', which brings out yet another difference between the two systems. 'Harmony' is produced when several instruments play different melodies or pieces simultaneously like in an orchestra. Harmony is also produced when more than one tone is produced at the same time. In the Western Music, 'harmony' is an important element.
Orchestration and 'harmony' are absent in Indian classical music. People have tried out orchestration of Indian classical music time and time again with limited success. (and there have been probably as many Western attempts to compose and play 'Indian style' melody based classical music) There is not much of a market for such 'Fusion' music, except perhaps among the fringe elements of the immigrant Indian community :-) Even if there is a 'Jugal bandhi' - a standard fare in Hindustani classical music where two instruments (or even two vocalists) are featured together, the musicians usually follow the same melodic pattern one after another with minor variation rather than play different melodies simultaneously.
There are a number of other differences as well between the two systems of music. Indian classical music, for example, does not use what are called chords, or pressing more than one key simultaneously. Chords are a major aspect of Western music and producing harmony via chords is a natural consequence of the Equally tempered (geometric series) arrangement of the keys. If keys were arranged in a Just tempered sequence, pressing more than one key at a given time might produce an unpleasant sound pattern resulting in what is called 'Besur' (in Hindustani music) or 'Abaswaram' (in Karnatic music). By the way, one more advantage of Equal temperment of pianos and keyboards is that it makes it easier to 'tune' them, (they go out of tune every once in a while and need to be tuned periodically) since each key is harmonically related to the other keys. In case of Just tempered arrangement, since the key ratio between adjacent keys is not a constant, most keys will have to be tuned individually.
Also absent is 'polyphony' - where several instruments (melodies) come and go asynchronously instead of at periodically predictable times. Also, the Western scales are standardized. The middle C octave ranges from 240 to 480 Hz. There is no reason to stick to these frequencies. In fact, in Indian music, you have the freedom to choose the frequency range of the octave from anywhere to twice anywhere. You can start at 230 Hz, if you wish. Also, in Western classical music, most musicians have music notated on sheets of paper and 'read' it when performing. Indian music is always played by 'ear'.
Just to summarize, the essential differences between Indian classical music system and the Western music are (a) the Western keyboard is 'Equally tempered' whereas the Indian keyboard ideally should be 'Just tempered' (b) Only twelve keys per octave are used in the West, whereas to play Indian music one needs to produce several intermediate microtones, not represented by a conventional keyboard - This is the most major difference (c) Harmony, chords, polyphony etc are absent in Indian classical music (d) In Indian music, there is no need to standardize an octave to begin at 240 Hz.