Fibonacci Sequence and The Golden Ratio

In Music

Teagan LombardoMath 371

Fibonacci Sequence

1,1,2,3,5,8,13,21,34,55,89,…

Each number is the sum of the previous two numbers, starting at 1,1, or in more modern use, 0,1.

The Golden Ratio

• The golden ratio is represented by Φ (phi)• Φ = 1 + √5 = 1.6180339887………

2

• The reciprocal of phi is represented by φ (small phi)

• φ = 0.6180339887……..

Relation Between Fibonacci Numbers and Phi

• If you take a Fibonacci number and divide it by the previous Fibonacci number, you get phi (approximately)!

• Ex) 8/5= 1.6 ; 55/34=1.6176…….• If you take a Fibonacci number and divide it by

the next Fibonacci number, you get small phi (approximately)!

• Ex) 3/5=0.6 ; 21/34= 0.6176…….

Fibonacci Sequence In Music

• The Fibonacci sequence is seen in music various ways

• One way is with the keys of a piano:• There are 8 white keys in an octave when you

play a C major scale. 8 is a Fibonacci number!

1 2 3 4 5 6 7 8

• There are 5 black keys in an octave! • 5 is a Fibonacci number! • The 5 black keys are divided into a group of 2

and a group of 3- two more Fibonacci numbers!

1 2 3 4 5

• With the 8 white keys and 5 black keys, there are 13 keys total in the octave.

• 13 is a Fibonacci number

1 3 5 6 8 10 12 13

2 4 7 9 11

• It is said that composers use the golden ratio to choose the “climax” or bridge of the song

• Also, Fibonacci numbers are used to mark important measures in the music.

The Golden Ratio in Music

• http://www.youtube.com/watch?v=uv7rpcU29Nk

• (1:45)

Formula By Gary Ewer

• Convert length of song to seconds• [Length of song x 0.618…(small phi)]• 60• Take the result of that and multiply the

decimal part by 60 to get the exact point in the song

Skyscraper- Demi Lovato

• Song is 3:42• 3:42-> 60x3 = 180 + 42 = 222 • 222 x 0.618… = 137.196 = 2.2866• 60 • 0.2866 x 60 = 17.196• So the bridge of the song is at 2:17196

Wrecking Ball- Miley Cyrus

• Song is 3:43• 3:43-> 60x3=180 + 43= 223• 223 x 0.618….= 137.814 = 2.2969• 60• 0.2969 x 60 = 17.814• So the bridge of the song is at about 2:18

We Are Young- Fun.

• Song is 4:12• 4:12-> 252• 252 x 0.618 = 155.736 = 2.5956 600.5956 x 60 = 35.737So the bridge of the song is at about 2:36

Can You Feel The Love Tonight- The Lion King

187 x 0.618 = 115.566 = 1.9261 600.9261 x 60 = 55.566

So the bridge is at about 1:56.

Thinking of You- Katy Perry

• 4:06=246• 246 x 0.618 = 152.028 = 2.5338 600.5338 x 60 = 32.028So the bridge is at 2:32

• When most composers use Fibonacci numbers and the golden ratio in their compositions, these were not the basis of their musical style.

• Casey Mongoven of Germany designed a very complicated style of music in which the basis was Fibonacci numbers and the golden ratio.

Bar 5

Bar 55

• As you could see, the vocals come in on measure 5, a Fibonacci number.

• On bar 55 (a Fibonacci number), the vocals come back in after a long instrumental, which is a high point in the song.

• This long instrumental is 8 measures long (a Fibonacci number)

Exam Question

• What are two ways that Fibonacci numbers or the Golden Ratio are found in music?

• In the keys on a piano (8 white keys, 5 black keys, 13 total in an octave)

• Fibonacci numbers marking important measures in a song

• Reciprocal of phi used to determine where the bridge of a song is.

Referenceshttp://www.youtube.com/watch?v=uv7rpcU29Nk

http://www.goldennumber.net/music/

Mongoven, Casey. (2010). A Style of Music Characterized By Fibonacci Numbers and the Golden Ratio.

Leonard, Hal. Contemporary Rock: Keyboard Play-Along Volume 4.

https://garyewer.wordpress.com/2010/05/27/songwriting-and-the-golden-mean/