1. Carnatic - the south Indian classical music. Traditionally Purandara Dasa is supposed to be the originator, from around the 13th century. This came up in the Vijayanagar empire. But there are older composers like Vyasa Raya whose compositions are still popular.(1)
2. Hindustani - the north Indian classical music, origins attributed to Amir Khusro, around 12-13 century.(2)
There are older musical systems, for example, Sopana system in Kerala. (3)
Moreover, instruments like the Veena (the Indian lute) are much older, as can be attested from Gupta era paintings (6th century A.D.), so it is possible that a lot of the history is now lost. (4)
The ragas are arranged in a combinatorial manner, roughly, out of 12 notes, how many harmonious combinations can be made? (5)
Related: combinatorics was used in prosody, in addition to music: One of the earliest occurrenc of the Pascal's triangle is in Indian prosody. (6)
(1) https://en.wikipedia.org/wiki/Carnatic_music
(2) https://en.wikipedia.org/wiki/Hindustani_classical_music
(3) https://en.wikipedia.org/wiki/Sopana_Sangeetham
(4) https://en.wikipedia.org/wiki/Veena
(5) https://cs.uwaterloo.ca/journals/JIS/VOL5/Balasubramanian/ba...
(6) https://en.wikipedia.org/wiki/Pascal%27s_triangle#History
https://en.wikipedia.org/wiki/Sangeet_Natak
https://www.youtube.com/watch?v=kdlLqzs9LKk
There is an interesting form of performance called Jugalbandi where different artists perform duet (mostly without rehearsal).
https://en.wikipedia.org/wiki/Jugalbandi
Another rare video - https://www.youtube.com/watch?v=lk60ObnbIOk
Edit: The use of this mugam by Artemiev is mentioned in the wikipedia page https://ru.wikipedia.org/wiki/%D0%91%D0%B0%D1%8F%D1%82%D1%8B...
P.S: Amir Khusrao's role in development of Qawwali, Tarana and other forms of Islamic Sufi music making sounds plausible. Anything beyond that is venturing into speculation.
In India "educated" or "learned" people will be intimately familiar with many of "raags/ragas" which can be thought of as "tunes" (but they are deeper than that). These ragas are usually meant to be played at different times of day/year or to evoke certain moods. Players will improvise these raags, interpreting not only their "version" of the song, but custom jamming a version specifically for that place and time and audience. Some of these raags have a very long history. I think when people talk about "classical" Indian music this is what they mean. Temporary it's deeply historical music that is constantly evolving.
I'm generalizing and glossing over so much to be brief, and much of this varies by region and tradition, but that's a gist.
This is on contrast to folk music, which focuses less on theory.
In India, which just as in the west, development of the theory was aided by the patronage of the wealthy/elite.
Also as in the west the theoretical system serves the purposes of preservation, reproducibility, and most of all, pedagogy.
This music is also not written down (intentionally, I might add), so while roughly the same forms have been practiced for 100s of years, most of what is remembered and played is from the past 150 years or so. You can call this era classical, if you will.
[0] - https://www.goodreads.com/book/show/20320205-a-southern-musi...