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I know this question is out of the blue, but please, if anyone knows what is the name of the sitar composition Pandit Ravi Shankar is playing while teaching George Harrison in this clip then it would really be helpful.

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This is the raag Todi in Hindustani (north Indian classical) music. It takes the following notes in the ascent and descent:

Aroh: Sa, Komal Re, Komal Ga, Tivra Ma, Komal Dha, Shuddha Ni, Sa
Avaroh: Sa, Shuddha Ni, Komal Dha, Pa, Tivra Ma, Komal Ga, Komal Re, Sa

The closest equivalent raga in Carnatic (south Indian classical) music is the ragam Shubhapantuvarali. The slight difference is that Shubhapantuvarali allows Pa to occur in the ascent as well:

Arohanam: Sa, Shuddha Ri, Saadhaarana Ga, Prati Ma, Pa, Shuddha Dha, Kaakali Ni, Sa
Avarohanam: Sa, Kaakali Ni, Shuddha Dha, Pa, Prati Ma, Saadhaarana Ga, Shuddha Ri, Sa

Just for completeness, the seven notes in Indian classical music are called Sa, Re/Ri, Ga, Ma, Pa, Dha, Ni and can be thought of as the analogues to the movable do solfège in Western classical music.

Here is a short but excellent rendition of Raag Todi by Pandit Ravi Shankar, with Ustad Alla Rakha accompanying on the tabla, for you to compare:

Provided to YouTube by NAXOS of America

Raga Todi - Rupak Tal (7 beats) · Ravi Shankar

Alla Rakha: Live at Monterey (1967)

℗ 1993 Delos

Released on: 1993-01-01

Artist: Alla Rakha
Artist: Ravi Shankar
Composer: Miyan Tansen

Auto-generated by YouTube.

Pandit Ravi Shankar describes the raag briefly at the start of this video, which I've transcribed below:

I will start with one of our most beautiful morning raga known as Todi. Raga Todi. This raga uses the flattened second, the flattened third and the flattened sixth, and also uses the sharpened or the augmented fourth. Its principal mood is that of... little sad and also devotion.

In the clip in the question, it appears that Pandit Ravi Shankar isn't playing or teaching any particular composition, but rather teaching George Harrison the raga (and the sitar) by singing and playing phrases of the raga and having him repeat them. Extemporaneous explorations of ragas are the norm in Indian classical music. For instance, the video I linked above begins with a solo by Pandit Ravi Shankar which is just such an exploration, known as "aalaap" or "aalaapanai".

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    Thank you so much for this beautiful answer!! Feb 16 at 9:23
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    @PhysicsEnthusiast Namaskār! :) If you are interested, then this to inform that there is a Chatroom here in Music Fans SE , which is solely dedicated to Indian Music.Feel free to drop by in... :)
    – Malavika
    Feb 16 at 19:50

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