The Virahāṅka–Fibonacci numbers
Each number is the sum of the two before it.
Counting rhythms again
The number of rhythms with 1, 2, 3, 4, 5, 6 beats is 1, 2, 3, 5, 8, 13. You found that each count is the count for one beat fewer plus the count for two beats fewer.
How many rhythms do you think there are with 7 beats?
What this lesson covers
The idea
The Virahāṅka sequence 1, 2, 3, 5, 8, 13, 21, 34, … gives the number of rhythms having 1, 2, 3, … beats; each number in it is the sum of the two previous numbers.
Counting rhythms again
The number of rhythms with 1, 2, 3, 4, 5, 6 beats is 1, 2, 3, 5, 8, 13. You found that each count is the count for one beat fewer plus the count for two beats fewer.
How many rhythms do you think there are with 7 beats?
- 18
- 21
- 26
Continue the sequence
Here are the first five numbers. For each next length, pick the number of rhythms. The two numbers used are lit up. Go all the way to 10 beats.
A famous sequence
The sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … gives the number of rhythms having 1, 2, 3, … beats. Each number is the sum of the two numbers before it.
The Prakrit scholar Virahāṅka wrote this rule around the year 700 CE. In the West the same numbers are called Fibonacci numbers, after a mathematician who wrote about them in 1202 CE. That is about 500 years later.
The Virahāṅka sequence 1, 2, 3, 5, 8, 13, 21, 34, … gives the number of rhythms having 1, 2, 3, … beats; each number in it is the sum of the two previous numbers.
- Beats | Rhythms
- 5 | 8
- 6 | 13
- 7 | 21
- 8 | 34
- 9 | 55
- 10 | 89
Notes
The Virahāṅka sequence 1, 2, 3, 5, 8, 13, 21, 34, … gives the number of rhythms having 1, 2, 3, … beats; each number in it is the sum of the two previous numbers.
Check yourself
How many rhythms are there with 9 beats?
Answer: 55
8 beats: 34 rhythms. 7 beats: 21 rhythms. 34 + 21 = 55.
The Virahāṅka numbers go …, 34, 55, 89, … What comes after 89?
Answer: 144
55 + 89 = 144.
A daisy usually has a Virahāṅka number of petals. Which number of petals fits?
A part of the sequence is …, 21, ?, 55, … What number is missing?
Answer: 34
Each number is the sum of the two before it, so 21 + ? = 55. The missing number is 55 − 21 = 34.
- 24. 24 is not in the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, … It is between 21 and 34.
- 30. 30 is not in the sequence. It is between 21 and 34.
- 34 — correct. Yes! 13 + 21 = 34, so 34 is in the sequence.
- 35. 35 is not in the sequence. After 21 and 34 comes 55.