To start with, we should note that a tihai is made up of three identical phrases played one after the other to reach the sam (1st beat of the tala cycle) or in some cases to reach the matra from where the mukhda or the gat starts.
Why is tihai made up of three phrases? Number 3 seems to have a very significant and effective reason. Apparently no satisfying explanation of why this magical no. 3 is frequently used to describe important entities. There are many examples relating to number 3 (a) Brahma, Vishnu, Mahesh (b) akash, prithvi, patal etc. I was just wondering whether the logical number of 3 could be based on 1 + 2 on which (according to some) entire mathematics is based.
The role of a tihai is to fill the gap of matras between ending of a taan and the distance from that point to the sam, in an artistic manner. Thus a tihai is an adjunct to a taan and not the main focus of the overall melodic expression. In other words, a taan is not played as an adjunct to a tihai but other way round.
Whether it is musically more satisfying if a tihai is composed on the spur of moment and not fixed in advance, is an interesting question. However to improvise in such a manner, the performer has to be capable enough to do so. No doubt, prefixed tihais becomes mechanical and hence are not as satisfying.
Tihai in gat
Apart from the role of tihai in relation to taan, it has been noted that some earlier drut sitar gats also occasionally used tihai as an ending part of the gat. It is very important to note that such tihais are not introduced in the sthayi portion of the gat but only in the antara section. It stands to reason therefore that when a tihai is used after execution of taan at the end of taan – as an adjunct, it could be argued that the use of tihai in drut instrumental gat is also woven into a fag end of the antara portion. In my long career, I have never come across a gat having a tihai in the sthayi portion.
Chakradhar tihai
This type of tihai are seldom used by contemporary sitar players. I know a few of chakradhar tihais which U. Vilayat Khan used to play as a part of gat development – especially in taan portion. However, these tihais were not an adjunct to a taan but had a complete melodic structure of their own.