Mathematics in Indian Rhythm System
A Short Note on the Mathematical Approach to the Concept of Tihai in Indian Rhythm System
By Dr. Peyman Nasehpour
This is a short note for a mathematical approach to the concept of "tihai" in Indian rhythm system (tala).
Let us remind the following definitions:
Tala, Taal or Tal (Sanskrit tala, literally a "clap"), is the term used in Indian classical music for the rhythmic pattern of any composition and for the entire subject of rhythm similar to the notion of "Usul" in the theory of Ottoman/Turkish music and "Igha'" (also spelled as Iqa') in Arabic and ancient Persian music.
"Sam" is the starting point or the first beat (matra) of an Indian rhythm (tala) from which the cycle begins and upon which it ends.
"Tihai" is the figure at the end of an elaborated or fixed composition, which begins on any beat in the given tala and after being played three times, ends on "sam".
Length of a rhythmic phrase: The length of a rhythmic phrase is the number of the beats that a rhythmic phrase contains.
The length of a tihai for a rhythm cycle, while starting from sam and ending on sam, can be obtained from solving an equation explained as follows:
Let "d" be the length of a tala X and "t" the length of a phrase repeated three times to make the tihai related to the tala X. Therefore if we start our tihai from the first beat, we must have the following equation:
3t=nd+1, where "n" is the number of cycles that the supposed rhythm cycle is played.
This means that we need to find a suitable "n" in such a way that it gives us a positive integer solution for "t". For example, in the case of tintaal, d=16. Therefore we have that t=(16n+1)/3. As we know "t" is an integer number. But 16n+1=15n+n+1. This shows that if we want "t" to be a positive integer, the fraction (n+1)/3 must be a positive integer. It is clear that first suitable n is 2. Hence, d=11.
Conclusion. If we repeat a phrase that its length is in 11 beats, then it starts from the sam and ends to a sam of a tala, which it is in 16 beats.
Remark. A tihai for a 7-beat rhythm cycle can be a phrase in 5 beats and a tihai for a 5-beat rhythm cycle can be in 7 beats (prove it!).