Math day 5:
Fibonacci Sequence and Music
In western music theory, we know an octave including 13 notes:
C, C#, D, D#, E, F, F#, G, G#, A, A#,B, C (1,1#,2,2#,3,4,4#,5,5#,6,6#,7,i)
Is there anything special about numbers 13?
We notice an octive including 5 sharps notes (C#,D#,F#,G#, A#), and 8 regular notes (C,D,E,F,G,A,B,C).
Now if we go one more step further look at the key distribution on a piano:
The 5 sharp notes are in groups of 2 and 3.
Now, if we re-think about the numbers, we have 2+3=5, 5+8=13.
so we have a sequence of number: 2,3,5,8,13, which is a segment of Fibonacci sequences.
Fibonacci Sequence Definition:
Personally, I think it is very interesting how the Fibonacci Sequence emerged in Music Theory.
Where is the Fibonacci Sequence come from?
this was a story my previous student told me, there were 2 bunnies on an island, then they started a family, the family members formed Fibonacci sequence ….
Now, given a random positive nature number, can we tell it is a Fibonacci Number very quick?
Personally, I could not tell if the number is >100. but, our friend Python can!
A little Fibonacci number Yes or No Python game:
import math def isPerfectSquare(x): i=int(math.sqrt(x)) return (x==i*i) def isFibnoacci(n): if (isPerfectSquare(5*n*n+4) or isPerfectSquare(5*n*n-4)): print (n, "is a Fibonacci Number") else: print(n,"is not a Fibonacci Number" )
Now we can run our Game Test:
isFibnoacci(4) 4 is not a Fibonacci Number isFibnoacci(168) 168 is not a Fibonacci Number isFibnoacci(1000) 1000 is not a Fibonacci Number isFibnoacci(89) 89 is a Fibonacci Number
Note: The Keypoint is every Fibonacci number is in the form of 5n²+4 or 5n²-4.
The Fibonacci number is so fascinating in music, art and science not only because of its practical use but also it leads to the Golden Ratio!