# When to start a tihai

20 Jan 2022

In Indian classical music, a tihai is a rhythmic phrase that repeats 3 times, usually with a gap/pause of some fixed duration between each phrase, and ends (often but not always) on the first beat of the next cycle. The problem then, at least for those of us who have not been exposed to tihais long enough, is determining when to start the first note of the tihai.

Consider playing 3 iterations of 4 over a 16-beat cycle ending just before beat one of the next bar. If you start on 1 you would’ve finished it too early, so where do you start? One way of figuring this out is to determine the length of the tihai, which in this case is . Counting backwards from 0 to 12 starting on beat one of the next bar takes you to beat 5, so in effect beat 5 is where this tihai begins.

which translates as follows:

Let’s do this again except now we’ll add a gap of 1, sometimes expressed as . Where do we start now? On beat 3, or the 3rd semiquaver of the first beat.

Now with a gap of two:

With a gap of 3:

And wih a gap of 4:

So there you go. Now to automate the process I use the following function, which takes 3 arguments, the length of the cycle, the number of beats, and the length of the gap.

```(defun tihai (length beats gap)
"Find the start of a tihai."
(let ((n (+ (- (+ length 1)
(+ (* beats 3) (* gap 2)))
length)))
(if (> n length)
(- n length)
n)))
;; (tihai 16 5 0) => 2
```

The function is based broadly on the following equation, where l is the length of the cycle, b is the number of beats, and g is the length of the gap.

Conversely, to create a tihai that starts on beat N of a given cycle, we’d have to calculate the number of beats left until the next downbeat and divide that by 3. Then use the remainder (if there is one) to determine the gap. For example, in a 16-beat cycle, a basic structure of a tihai that starts on beat 1 would be 3 phrases of `5` beats with a gap of `1`, namely .

This is all too technical and I can’t imagine how traditional musicians conceive tihais intuitively, but at least now I can come up with my own tihais. By the way, I cannot claim any of this is correct.