The very first entry of this blog was devoted to how Pythagoras created the Western musical scale through the use of mathematical ratios. It is not surprising, therefore, that the most famous of all ratios–Euclid’s golden ratio–would find its way into the music. While interest in the golden ratio was common in the ancient world, artists renewed their interest in it through a sequence of numbers created by mathematician Leonardo Pisano: The Fibonacci Sequence. In the Fibonacci Sequence, each number is the result of the sum of the previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. As the Fibonacci Sequence increases, the closer it gets to the golden ratio of 1.618.
When the Fibonacci Sequence is drawn, it looks like this:
Leonardo da Vinci, for example, used the Fibonacci Sequence in many of his works. For example:
Now, let’s layer the Fibonacci Sequence over it:
And that’s not the only famous Leonardo image based on Fibonacci:
Leonardo was hardly alone. Consider these famous images. All Fibonnaci inspired.
Just as the Fibonacci Sequence helps artists to create pleasing proportions in their paintings, the golden ratio helps composers create music that is intrinsically pleasing to the ear. While he wasn’t the first to do so, it is worth noting young Mozart was nearly as infatuated with math as he was with music. In his first piano sonata, Mozart based the overall structure of the first movement on the golden ratio. It is written in classic sonata form (exposition, development, and recapitulation). The exposition is 38 bars; the development and recapitulation are 62 bars combined.
Wolfgang Amadeus Mozart, Piano Sonata No. 1 in C Major, K. 279: I. Allegro
Erik Satie, composing a century later, used the golden ratio for the beat counts in each movement of his Trois sonneries de la Rose+Croix. Later scholarship found contemporary correspondence with Claude Debussy in which the composers had discussed utilizing the golden ratio in their music. This video, which annoates the score, is a great introduction to Satie’s remarkable music and his use of the golden ratio.
Erik Satie, Trois sonneries de la Rose+Croix:
Interest in the golden ratio transcends eras and genres of music. For example, the prog rock band Genesis incorporates the golden ratio into The Firth of Fifth, which features solos 55, 34, and 13 bars–all golden numbers. Unsurprisingly, The Firth of Fifth is often rated as the band’s best song.
Genesis, The Firth of Fifth:
Other bands have taken the golden ratio and Fibonacci Sequence to extremes. Dream Theater structured their eighth studio album, Octavarium, around the Fibonacci Sequence–even the title, which alludes to the number 8, has 5 syllables. Fives and eights feature all over the album and booklet, with the key being shown on page 5 of the booklet by dominoes showing a five and and 8. Radiohead is also famous for using the golden ratio to organize their music, most notably on the album In Rainbows. That album is 42 minutes, 34 seconds long. If you divide that by the golden ratio (1.618), you find yourself at 2 minutes and 49 seconds into Reckoner, where the background vocalist can be heard singing “In rainbows.”
Lest this entry go too far down the progressive rabbit hole, I’ll also mention a song written by a really talented NYC kid that some of you might have heard of. Although she was accepted into The Juilliard School to study piano, she decided instead to try her hand at pop music. That worked out pretty well for her. And in this song from 2016, the dramatic key change comes at the 111 second out of 179 seconds–bang on 1.618. Yeah, that’s right–Lady Gaga bringing math and classical music theory to the masses.
Lady Gaga, Perfect Illusion