1. Fibonacci Sequence and The Golden Ratio In Music Teagan Lombardo Math 371

2. 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.

3. 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……..

4. 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…….

5. 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 2345 6 78

6. 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! 123 45

7. With the 8 white keys and 5 black keys, there are 13 keys total in the octave. 13 is a Fibonacci number 135 6 81012 13 2 47911

8. 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.

9. The Golden Ratio in Music http://www.youtube.com/watch?v=uv7rpcU2 9Nk http://www.youtube.com/watch?v=uv7rpcU2 9Nk (1:45)

10. 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

11. 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

12. 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

13. We Are Young- Fun. Song is 4:12 4:12-> 252 252 x 0.618 = 155.736 = 2.5956 60 0.5956 x 60 = 35.737 So the bridge of the song is at about 2:36

14. Can You Feel The Love Tonight- The Lion King 187 x 0.618 = 115.566 = 1.9261 60 0.9261 x 60 = 55.566 So the bridge is at about 1:56.

15. Thinking of You- Katy Perry 4:06=246 246 x 0.618 = 152.028 = 2.5338 60 0.5338 x 60 = 32.028 So the bridge is at 2:32

16. 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.

17. Bar 5

18. Bar 55

19. 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)

20. 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.

21. References http://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/