The rhythmic aspect of Indian classical music uses cycles of a fixed number of beats. Familiar Western analogs are rhythmic accompaniments to 12-bar Blues, which repeat in 12-bar cycle, each played in 4/4 time.

One unique thing I’ve seen in Indian music is the concept of a Tihai. There are many variations, but the most popular variant refers to playing a rhythmic phrase, that, when repeated precisely three times, lands you on the beginning of the next cycle.

For instance, consider the 16-beat Teen Taal:

Dha | Dhin | Dhin | Dha
Dha | Dhin | Dhin | Dha
Dha | Tin  | Tin  | Ta
Ta  | Dhin | Dhin | Dha


And an example Tihai (in bold). My hacky notation can be read as follows:

• Pipe (|) delimits single beats.
• When needed, I use slashes (/) on top to divide the beat into equal parts and then to distribute them among the notes to be played in that beat.
Dha      | Dhin     | Dhin      | Dha
Dha      | Dhin     | Dhin      | Dha
// //      / / / /                // //
/ / / /               // //       / / / /
TiReKeTe | Dha      | DhaDha    | TiReKeTe
----
Dha      | Dhin     | Dhin      | Dha
...


Note how:

• The Tihai starts halfway (after the 8th beat, i.e., on the 9th beat)
• The 3-beat phrase {DhaDha}{TiReKeTe}{Dha} (each {} takes one beat) repeats three times, for a total of 9 beats.
• Since we only have space for 8 beats in this cycle, the last beat starts the next cycle

The first beat of the cycle is known as Sam, and is the highest emotional point of the cycle. When one emphasizes the landing on Sam properly, the Tihai serves to temporarily distort the listener’s rhythmic perception only to resolve it at the Sam.

Here’s me speaking and playing this Tihai on the Tabla:

## Rhythm math

Let:

• $$N$$ be the number of beats in the Tihai phrase. The one above had $$N=3$$.
• $$T$$ be the number of beats in the Taal chosen. e.g., $$T=16$$ for Teen Taal.
• Let $$q$$ be the beat number that should start the Tihai, $$1 \le q \le T$$. In the above example, $$q=9$$, since we started on the 9th beat in the cycle.

If we are to play the $$N$$-beat phrase 3 times and land on the beginning of a rhythmic cycle, we must have:

$3N = 1\mod{T-q+1}$

In the preceding example, we have all the values match up ($$N=3$$, $$q=9$$, $$T=16$$):

$3 \times 3 = 1\mod{\left(16-9+1 = 8\right)}$

Here’s another example of a Tihai which starts on the first beat, and is 11 beats long (first 11 beats underlined, the entire Tihai is emboldened):


// //      / / / /    // / /    / / //
//  //                // //     // //
DhinNa   | Dha      | DhaDhin | Na Dha
// //      //  //                // //
xx Dha   | DhinNa   | Dha     | DhaDha
/ / / /    // / /     / / //    //  //
TiReKiTa | DhaTiRe  | KiTaDha | DhinNa
-----
////       // //      // //      // //
Dha      | DhaDhin  | Na Dha   | xx Dha
//  //                // //     / / / /
DhinNa   | Dha      | DhaDha  | TiReKiTa
// / /     / / //     //  //
DhaTiRe  | KiTaDha  | DhinNa  | Dha
// //      // //      // //       //  //
DhaDhin  | Na Dha   | xx Dha    | DhinNa
-----
Dha  | Dhin ...


Note how this 11-beat phrase, when played 3 times, gives us a piece of 33 beats, which is 2 full 16-beat cycles, and the extra beat that starts the next one. This shows that Tihais need not complete within a cycle – they can span multiple cycles, the only requirement seems to be to land on the Sam after 3 repetitions.

Here, $$N=11$$, $$q=1$$ and $$T=16$$, and we can see that:

$3N = 3 \times 11 = 33$ $= 1 \mod{16}$ $= 1 \mod{16 - 1 + 1}$ $= 1 \mod{T-q+1}$

## Generalization?

I have only ever heard this concept applied to 3x repetitions of fixed phrases (after all, the name Tihai does hint at 3’s). But out of curiosity, what if we want to generalize this to a $$K$$-way repetition? e.g., we could invent a Panchai or something such that a fixed phrase, when repeated 5 times, has to end on the Sam of a rhythmic cycle.

The simple calculations above should generalize to give us:

$K \times N = 1\mod{T-q+1}$

To jam with this idea with $$K=5$$, $$q=1$$ and $$T=16$$, we need an $$N$$ such that:

$5N = 1\mod{16}$
>>> [n for n in range(50) if (5*n) % 16 == 1]
[13, 29, 45]


So there, our first option is a 13-beat phrase. When repeated 5 times, this will be 65 beats, which is 4 cycles of 16 beats and our extra beat to start the next cycle.

Do let me know if such a generalization exists, or if there is maybe some reason why artists choose not to incorporate it. I for one can see that there is something magical about the number 3 – it is an odd number that I, as a listener, seem to be able to keep in my head easily. Maybe 5, 7 or higher odd numbers are just too complex? Maybe they just don’t sound good? I don’t know.