To being with, scales in classical western music and ragas in classical Indian music are not the same. You cannot just say, this raga in Carnatic or Hindustani system of Indian music is the equivalent of this scale in Western music system. Why?
Absolute Pitch vs Relative positioning of Notes
Every note in a Scale of the Western Music has absolute pitch. But that is not the case in Indian music. A given note of western music can have different pitches in Indian music.
In other words, the note Sa of a Raga can start not only at C of a western scale, but can be any note like D, E, F etc of the western scale. The remaining notes of the raga will be relatively placed to the note Sa.
So the swara sa of MayaMalavagowla in Carnatic Music can start at C, D, D# or whatever note of western music. The remaining notes of the raga will be placed accordingly. So for the same Raga of Indian music, two different singers may sing complete different notes of the western music, because they may have different comfort zones for the starting pitch sa.
Western Music = Fixed Logarithmic Scale
In Western Music, the difference between the frequency of pairs of adjacent notes always has the same ratio which is in a logarithmic scale. The ratio is 12th root of 2, and hence an octave has 12 keys in Western Music.
This ratio is fixed, and all western musical instruments like Piano, Keyboard are tuned to this twelve-tone equal temperament. The note A in the octave is fixed (usually at 440 Hz) and all other notes are fixed at some multiple of semitones near this frequencey. For example A# will be 440+one semitone ie 440 x 12th root of 2.
So, notes within an octave have an irrational number ratio between them in Western Music.
Indian Music = Whole Number Ratios of any fundamental frequency
In Indian Music on the other hand, every note of a raga is a whole number ratio of a fundamental frequency of your choice. So the notes form a Harmonic Series. No note is fixed at a frequency, unlike western music.
You can select any frequencey as a fundamental frequency. All notes of the raga are placed relative to the fundamental frequency, as a whole number ratio. So if fundamental frequency is 60 Hz, you can define sa,ri as say 2:3 which means sa will be 120 Hz (2 times the fundamental) and ga will be 180 hz(3 times the fundamental).
In other words, even if you start singing MayaMalavagowla at C of Western Music, other swaras of the raga still may not match the corresponding notes in western music, because your fundamentaly frequency can vary.
12 Semitones of an Octave vs 22 Shruthis in an Octave
As we saw earlier, the number of semitones in an Octave is 12 and this is fixed. But because the Indian Raga pitch is a Harmonic Series of a fundamental frequency, theoretically you can have any number of pitches within an octave in Indian music. The Natya Shastra, the ancient Indian text related to performing arts, defines 22 Shrutis in an Octave.
Classical Indian musical instruments like Veena, can be plucked to play in all these Shrutis. But you cannot play them all in a western instrument like say Piano.
This is the reason they say that a normal western Piano or Keyboard cannot play an Indian musical tune with complete justification. There simply are not enough number of keys in an octave of a Piano to play Indian music. The traditional Indian Harmonium can play them, because you can vary the pitch as you want.
Western Scale is its Notes, but Raga is not just its notes
A Western Scale is the set of notes used to play that scale. A western scale can be identified just by looking at its notes.
But an Indian Raga is not just a collection of notes. It is the relative positioning of the notes and each raga has its own rules about how those notes can or should be played! Two different ragas can be played with same notes, and still be different, because of their rules!
The order of the notes, the importance of certain notes, the Shruti inflections played in a Raga all matter and make the Ragas different from one another. So, a given scale in Western Music will be mapped to more than one Raga even if we keep the Sa fixed at C of the Octave!
In other words, same raga can be sung in two different western scales, and at the same time two different ragas can be sung in the same western scale!
Human Brain recognizes Harmonic Series in Sound
The Human brain does not use a fixed absolute pitch. Music is perceived in terms of relative ratio with respect to the fundamental frequency. And Indian Music uses this fact to create numerous possible variations. If two people sing the same Song using two different fundamental pitches, the the brain will recognize it to be the same tune.
Brain hears the Missing Fundamental
This becomes so fascinating when you understand the fact that, when a music or sound containing harmonic overtones is missing any sound in its fundamental frequencey, the human brain will still hear it!
So if you are hearing three overtones at 300 Hz, 450 Hz and 600 Hz, then note that the fundamental frequency is 150 Hz because all other frequencies are a whole number multiple of 150. The sound itself does not contain any noise at 150 Hz though. But even then, the human ear will brain will hear an additional frequency of 150 Hz. This is called the Missing Fundamental, implying the fundamental sound is not present, but the human brain still hears it!
This aspect is used by Musical Software, loud speakers etc to imply a missing fundamental which they do not produce! So, even if a speaker cannot produce a sound at low frequency, it can still make you feel listening to that sound by generating higher frequency sounds that are a whole number multiple of the low frequency Sound!
Western Scale is just one of the numerous ways of playing that Indian Raga
So, we can conclude that there is no simple one to one mapping between Indian Ragas and Western Scales. And also bear in mind, that the number of ragas runs from Hundreds to Thousands, and you simply do not have that number of scales in Western Music!
So, we can at the most map an Indian raag to a western scale keeping in mind that the scale contains notes which is one of the numerous ways in which that raga can be played, provided the sa and pa of the raga are selected so that they match the C and G of the western octave, and the raga rules are followed in playing the notes of the scale.
See a list of Western Scales equivalent of Classical Indian Ragas provided the above mentioned conditions in playing the raga are met.