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In How does improvisation in Indian ragas differ significantly from jazz melodic improvisation? Phil states:

With a Raga, there is no theme, per se. A Raga is more like a set of rules as to how you progress though the mode than a melody in its own right. Certain notes are designated for emphasis or avoidance, and for being approached or left in a particular manner. With Jazz, an improvisor is left more free to determine how to proceed.

What are the rules of improvisation in Ragas?

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I'll try to be specific and use terms in carnatic music to describe each aspect

  1. Each Raaga provides key phrases which are set forth by the notes which are allowed in the raaga.
    Aarohanam: notes(swaram) that can be played ascending
    Avarohanam: notes that can be played descending.

A list of scales which are common is provided here along with audio:

http://www.ragasurabhi.com/carnatic-music/ragas.html

  1. Signature verses: Signature verses bring out the quality of raaga(bhaava). There are certain notes which are given emphasis even though the aarohanam and avarhonam are the same for some raagas (refer list above)

Along with the notes allowed in the raaga, the style dictates certain notes should not occur together.

  1. Improvisation of a raaga requires a structure which gives its quality. This is often referred to as Manodharama.

Again, there are aspects which are specific to raaga, on how the aalapana(opera style improvisation) should start or end, which notes are emphasised, and which notes are less likely vs more likely.

http://carnaticacademy.weebly.com/uploads/1/2/2/6/12265589/carnatic_music_theory_notes_year_2.pdf

Other than aalapana, Tanam (repetition), and Niraval (melodic transformations of a verse) are popular.

All of these styles provide a framework for a musician to expand, based on her abilities.

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This is a very valid question, but not one that can be answered in a few words or even in a few paragraphs. It requires explaining and demonstrating multiple background concepts before one can begin to answer something this complex.

However, to summarize for the benefit of Stack Exchange users, assuming you already understand how ragas and rhythm function in Indian classical music, there are two major types of improvisation. Free improvisation (without the tabla or a composition) and improvisation around a composition with the tabla.

Free improvisation (alap) is further divided into vistar (exploring the melody of a raga at a gentle pace, showing its various notes in different facets), jod (catchy, lilting patterning of notes as the pace of the performance increases), and taan (mathematical sequences of notes that are performed at double-speed) to bring the alap to a conclusion. The video below demonstrates using Raag Bageshree.

Free improvisation

Traditionally the alap section of a performance forms a large part of it and can go on for up to 40 minutes to an hour or more. It is then followed by improvisation around a composition, called bandish or gat. Here the rules of rhythm must be followed when improvising. A composition is set to a specific rhythm cycle, meaning that each syllable of the composition falls on a particular beat of a particular rhythm cycle. When improvising around the composition, the artist must make sure to come back periodically to the composition at the correct beat. Here is a small demonstration of how that is done around the composition eri aali piya bina in Raag Yaman.

Improvisation around a composition

Again, improvisation around a composition can come at different speeds; and it can use different vocalization techniques in the case of vocal music.

This explanation focuses on Hindustani classical music, but the essential features are somewhat similar in both Carnatic and Hindustani classical music.

Source: The explanations and the links are from my website on Indian classical music, which gives a more comprehensive explanation. The website is not a commercial operation. It is a free educational service with no ads. The videos, like the website, are my own (they are demo videos created for use on the website).

Disclaimer: My singing is good enough to provide a demo, but please note that I am not a professional singer.

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In Indian Classical music, there is no fixed form of music. For improvising upon a raga, you should know the rules that governs that specific raga. For example, in the case of Bhairav Raga, you should know the basic rules that governs the raga. In Indian Classical music, improvisation is termed as sargams or taans, and sometimes even melodies can be improvised. In fact, there is no rule for improvisation in Indian Classical music. The rule of the specific raga governs the ways one can improvise upon a specific raga.

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