Mapping Carnatic ragas to Western scales is a complex task because the two systems have significant differences in terms of pitch organization, microtonal nuances, and ornamentations. However, it is possible to find rough correspondences between some Carnatic ragas and Western scales for reference. Keep in mind that these mappings are approximate, and the nuances of each raga may not fully align with Western scales. Here's an example of mapping the Carnatic raga Natabhairavi to the Western natural minor scale:
Carnatic Raga Natabhairavi:
Sa (Tonic): C (for example, if we choose C as the tonic)
Ri (Major Second): D (two half steps above the tonic)
Ga (Minor Third): E♭ (three half steps above the tonic)
Ma (Perfect Fourth): F (five half steps above the tonic)
Pa (Perfect Fifth): G (seven half steps above the tonic)
Da (Minor Sixth): A♭ (eight half steps above the tonic)
Ni (Minor Seventh): B♭ (ten half steps above the tonic)
Western Natural Minor Scale (Aeolian Mode):
Major Second: B
Minor Third: C
Perfect Fourth: D
Perfect Fifth: E
Minor Sixth: F
Minor Seventh: G
In the mapping above, we've chosen C as the tonic for the Carnatic raga Natabhairavi to make it easier to compare with the Western natural minor scale. However, it's important to note that Natabhairavi can have different starting notes (tonics) in different performances or compositions within the Carnatic tradition.
Also, it's crucial to recognize that Carnatic ragas are not restricted to the fixed pitches of Western scales. They often include microtonal variations, ornamentations (gamakas), and specific patterns that give each raga its unique identity. Western scales, in contrast, use equal temperament, where pitch intervals are standardized and evenly spaced.
Therefore, while you can make approximate mappings between some Carnatic ragas and Western scales, these mappings are more for reference and may not capture the full essence of the raga. Carnatic music relies heavily on improvisation and nuanced expressions, which go beyond the scope of Western scales.
Mapping ragas to Western scales can be a helpful starting point for cross-cultural musical collaborations or for Western musicians looking to incorporate elements of Carnatic music into their compositions. However, to fully understand and perform Carnatic ragas, it's essential to study them within the context of the Carnatic tradition, including their unique ornamentations, tonal nuances, and emotional expressions.
Here is a table I put together -- which serves as a good starting point to atleast visualize the notes to begin with.