Short and long beats

The idea

Each way of writing n as a sum of 1s and 2s (an n-beat rhythm of short and long syllables) starts with 1+ or 2+, so the count for n is the count for n–1 plus the count for n–2.

Poets and beats

In Indian poetry, a short syllable takes 1 beat and a long syllable takes 2 beats. A poet wants to fill exactly 3 beats.

In how many different ways can 3 beats be filled?

• 2 ways
• 3 ways
• 4 ways

Build the rhythms

Tap Short and Long to fill the beats. Every new rhythm goes in a column, by its first syllable. Then use the book’s method for 5 and 6 beats.

Start with 1 + or 2 +

Every rhythm of *n* beats starts with a short (1 +) or a long (2 +). After a 1 +, the rest is any rhythm of n − 1 beats. After a 2 +, the rest is any rhythm of n − 2 beats.

Each way of writing *n* as a sum of 1s and 2s starts with 1 + or 2 +, so the count for *n* is the count for n − 1 plus the count for n − 2.

• Beats | Rhythms | How
• 3 | 3 | 2 + 1
• 4 | 5 | 3 + 2
• 5 | 8 | 5 + 3
• 6 | 13 | 8 + 5

Notes

Each way of writing *n* as a sum of 1s and 2s starts with 1 + or 2 +, so the count for *n* is the count for n − 1 plus the count for n − 2.

Check yourself

To count the 7-beat rhythms we add the numbers of…

How many 6-beat rhythms start with a short syllable (1 +)?

Answer: 8

After a short syllable 5 beats remain, and there are 8 rhythms with 5 beats. So 8 of the 6-beat rhythms start with a short syllable.

A 5-beat rhythm starts with a long syllable (2 beats). In how many ways can the remaining beats be filled?

Answer: 3

After a long syllable 5 − 2 = 3 beats remain. There are 3 rhythms with 3 beats (1 + 1 + 1, 1 + 2, 2 + 1).

There are 8 rhythms with 5 beats and 13 with 6 beats. How many rhythms have 7 beats?

Answer: 21

A 1 + leaves 6 beats (13 ways). A 2 + leaves 5 beats (8 ways). 13 + 8 = 21.

• 6-beat rhythms and 1. The short syllable is only the first step. After it, all the rhythms of the other 6 beats are possible, not just 1.
• 6-beat rhythms and 6-beat rhythms. After a long syllable only 5 beats remain, not 6.
• 6-beat rhythms and 4-beat rhythms. After a long syllable (2 beats) of 7, 5 beats remain, not 4.
• 6-beat rhythms and 5-beat rhythms — correct. Yes! A 1 + in front leaves 6 beats and a 2 + in front leaves 5 beats.