# Tihai in Indian music

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 // // / / / / // //DhaDha | TiReKeTe | Dha | DhaDha / / / / // // / / / / 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**):

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

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 *Tihai*s 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.