Mathematics in the Indian Rhythm System
Abstract. In this note, I explain the tricky concept of "tihai" with a mathematical approach to the concept.
Introduction. The tala (also known as the "taal" or the "tal") which literally means a "clap" in Sanskrit is a 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 اصول" in the theory of Ottoman/Turkish music and "Igha ایقاع" (also spelled as Iqa') in Arabic and ancient Persian music. The "sam" is the starting point or the first beat (in Hindi "matra") of an Indian rhythm (tala) from which the cycle begins and upon which it ends.
The "tihai" is the figure at the end of an elaborated or fixed composition, consisting of three equal phrases, which begins on any beat in the given "tala" and after being played three times, ends on the "sam". Note that the length of a rhythmic phrase is the number of beats that a rhythmic phrase contains.
Different kinds of tihais. Essentially, there are two kinds of tihais.
A tihai is said to be "dumdar" if there is a time interval between the first and the second phrase and also between the second and the third phrase of the tihai. Note that the word "dumdar دُمدار" is a Persian word meaning "with a tail".
A tihai is said to be "bedum" if there is no time interval between the first and the second phrase and also between the second and the third phrase of the tihai. Let me add that the word "bedum بیدُم" is a Persian word that means "without a tail".
Now, if we let "d" be the length of a tala T that we are supposed to play and "p" the length of a phrase that is repeated three times to make the tihai related to the tala T, and if we start our tihai from the first beat, we must have the equation
where "n" is the number of cycles that the supposed tala T 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 "Tintal" which is a rhythm cycle in d=16 beats we have
Since "t" is an integer number and
we obtain that the fraction (n+1)/3 needs to be a positive integer. It is clear that the first suitable "n" is 2 which gives t=11.
Conclusion. If we repeat a phrase with a length of 11 beats, it starts from the beginning "sam" and ends with a suitable "sam" of Tintal which is in 16 beats:
3 x 11 = 33 = 2 x 16 + 1.
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!).
Also, see my note on Persian mathematics.