A quick Introduction
Firstly, what is a chord? a chord is a set of notes that are played together at the same instance. The simplest example of a 'chord' is the 'shruthi'. The regular shruthi (usually) has 'sa-pa-Sa'. (_some_ folks use 'sa-ma-Sa' as the shruthi for selected ragas that don't have a pa at all (ex: when singing something in 'abhogi')). While carnatic is already complex enough to take several dozen years to 'master' the art completely it is only complex in terms of being a mono-tonal melody based system (we hardly have multiple notes sounding at the same time other than some very very fundamental fifth chords (ex: sa-r2 is equidistant like pa-d2... so one could play 'sa-pa' and then 'r2-d2'... but thats hardly a chord!). While the 'masses' seem to 'accept' such simple chords, complex chords (where there are more than just 3 or 4 different notes!) don't seem to be well accepted. I also have to say what I believe in... any art form that does not evolve (or have room for natural evolution) will not last long. Carnatic continues to flourish because of the excellent room for creativity and improvisation and change. Some folks have introduced lots of western concepts into the system. Some have tried (albeit with some failure) to introduce "regular" chords into Carnatic. Hindustani concepts ("light music", like bhajans) are very very common these day anyway (I don't know if thats good or bad! I certainly don't enjoy it because I listen to carnatic only for the complexity of its music, I give a damn about any music thats so "simple"!)...
Now there is a key difference between the Carnatic notation and the Western notation. In Carnatic, a 'sa' is at exactly the same 'sa' that the shruthi produces. The 'r1' is a semi-tone (or half-tone) away from the now defined 'sa'. Without defining the 'sa' singing a random note does not make sense in Carnatic at all.
However, in the western system a given note, say C, is always well defined. In the standard way of tuning, the C note is tuned to 440 Hz. All other frequencies that are now 2^n x C is also called C (where n is an integer (negatives included!)). So simply put a note is sound in a said frequency (f) or 2^n x f. So if f=440Hz, C could be 440Hz/880Hz/etc.,. or when n is -1... it could b 1/2 x 440 == 220Hz.
The 'swarasthanams' (for a lack of a better word) in the western system are:
(where any note with the suffix 'b' is called a 'flat'. 'Gb' is called 'G flat'. Its like saying 'B flat' is a semi-tone lesser in frequency than 'B'. The opposite rule is a 'sharp' where 'C#' is read 'C-sharp' and means 'C#' is a semi-tone higher in frequency than a 'C').
Compare this to the (now existing 12 note octave) swarasthanas of Carnatic... they have 12 notes in an octave too! We see something working out... but you just have to remember for now that you cannot take 'C' as 'Sa'. Like with all wise answers... it depends :)
Never compare a scale to a ragam!
The primary confusion arises because of the lack of a shruthi in the western system.
Lets now take a scenario where a singer is trying to sing the notes of Shankarabharanam which when sung 'flat' (without ghamakkams) is similar to the 'major scale'. Shankarabharanam goes (in carnatic terms) like sa-r2-g3-m1-p-d2-n3-S. Here, the 'sa' is still undefined unless we introduce a 'shruthi' into the environment. Ok, lets assume the shruthi is now at '1 kattai'... 1 kattai is the same as singing the 'C' note. After defining the 'sa' and setting it at 'C', shankarabharanam suddenly becomes identical to the western 'C Major scale'.
Now, if the 'shruthi' is now increased to '2 kattai' (== D-A-D (sa-pa-Sa)), then what happens? sa-r2-g3-m1-p-d2-n3-S... its still shankarabharanam! however, the western folks don't recognise it as 'C major scale' anymore! They call it the 'D major scale' (since 'sa' == D)! Now that might bring a new question (confusion): So if I sing a shanarabharanam song in '1 kattai' (== C major scale) today... and later sing the same shankarabharanam song in '2 kattai' (D major scale) tomorrow... will it sound any different? To the average human ear, it wont make a difference! But a very well trained person in the western system can make out the difference!
So then how would one compare a Carnatic ragam to a western scale? The thing to keep in mind is that western scale is absolutely pitched (a C is a C is a C), so a scale is nothing but a ragam at a given "reference" shruthi. So whenever one wants to find out the 'western equivalent' of a carnatic ragam, the shruthi must be taken into account. When you want to a reverse-look-up, only look at the 'relative distance' of the notes in the scale...
So the above table of Western notes, will now have to be re-written as...
Now, if you are singing in 2 kattai, the western notes row in the table will have to go like D,Eb,E,F,Gb,G,Ab,A,Bb,B,C,Cb. You get the idea... :) 'roll over'... play with the chords search to get an idea :)
Given that these are their 'swarasthanams' they have their own 'rules' to construct 'scales' out of this. One such rule is the 'major scale' rule that goes as follows:
1. Pick up a root note (say C, to construct to C major scale)
2. From the above table, start "hopping" from the root note in the following order:
tone, tone, semi-tone, tone, tone, tone, semi-tone.
(where a 'tone' is a two cell jump in the above table... (so if you start at 'C', you get 'D' and not 'Db') and a 'semi-tone' is the next immediate cell)
Lets try to construct the C-Major scale now:
# root node is 'C'
# a full tone jump from 'C' lands on 'D' (Db is skipped)
# another full tone jump from 'D' lands on 'E')
# a semi-tone jump (adjacent cell) from 'E' takes us to 'F' (there is NO Fb!, remember!)
# a tone jump from 'F' takes us to 'G'.. so thus far we have C D E F G...
# a tone jump from 'G' takes us to 'A'
# another tone jump from 'A' takes us to 'B'
# the final semi-tone jump from 'B' takes us to 'C' (the higher sthayi C!)
Now its your exercise to follow the same rule above to construct the D Major scale :)
Given that chords are a native western concept, it might become difficult for the Carnatic person to start translating a given western chord name into the carnatic equivalent. this page exists to ease the problem of having to sit and figure out where is 'C'. To get you started, here are some 'rules'
1. A Chord is defined as just a bunch of notes being played at the same time. Think of three singers, at 1 kattai, each one singing 'sa', 'g3' and 'p' at the same time... that would be the same as C major chord. Of course, It makes a great deal of sense to think of chords in instruments than with vocal. It might be very difficult to co-ordinate three voices and might spoil the purpose of 'free form' music like Carnatic :)
2. Major chords use the formula 1-3-5... ie.,. to construct a C major chord, start counting from the 'root note' of the C major scale and pick up only the 1st, 3rd and 5th notes in the C major scale -- C D E F G A B... so C E G makes a C Major Chord. Similarly D Gb A makes a D major chord.
Hope atleast some of this was useful to you! good luck with the searching for chords!
Edit: The middle 'A' is typically 440 Hz. This article incorrectly refers to 'C' as 440 Hz.