The Golden Ratio & The Fibonacci Sequence
The combination of studying music whilst having a deep interest in mindset, spirituality, and ‘the universe’, has left me with a huge fascination over everything to do with frequency and vibration. So, I have decided to start a new blog series exploring just that… and this is the first installment!
Now, the golden ratio doesn’t entirely relate to frequency and vibration, yet at the same time it isn’t completely unrelated. It is its own thing that can be applied to and discovered in all kinds of things throughout our world and beyond. As it is a huge topic within the arts and also within spirituality, I felt it would be an interesting starting point, laying the foundation for other blog posts in the future that will specifically deal with frequency and vibration. With that being said, let’s get into it!
The Golden Ratio
Written as the Greek letter phi or Φ, the golden section is a ratio between two numbers that equals approximately 1.618 (1:1.618). Seen as a divine geometric proportion, it is believed to create the most beautiful shapes, appearing in so many forms within nature that it has been described as the universal constant of design. Artists and architects throughout history have applied it to their work, resulting in some of the most famous buildings and art work we admire today.
The Formula
In simple terms, the golden ratio appears when a line is split into two sections and the longer section (a) is divided by the shorter section (b). This is then equal to the sum of the entire length of the line (a+b) divided by the longer section (a), equalling 1.618, approximately.
The Fibonacci Sequence & Golden Spiral
Closely related to the golden section, is the Fibonacci sequence, created by a talented Italian mathematician from the Middle Ages called Leonardo Bonacci, later known as Fibonacci. Essentially, it is a sequence of numbers starting from 0, in which each number is created by adding together the previous two numbers. The ratio between each number and the previous number gradually approaches a value of 1.618 or phi.
0, 1 (0 + 1), 1 (1 + 1), 2 (2 + 1 ), 3 (3 + 2), 5 (5 + 3), 8 (8+5), 13 (13 +8), 21 (21 + 13), 34...
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181 …
It is between the numbers 55 and 89, that the ratio of 1.618 is first reached —> 89 ÷ 55 = 1.618181818181818.
The Fibonacci sequence is used to create the proportions of the Golden Spiral, an iconic diagram used to illustrate the Golden Section. It is formed using square shapes that follow the Fibonacci sequence. For example, the first number is 1, so the height and width of the first square will have a value of 1. Then, within each square an arch is drawn from one corner to the other, building the distinguished spiral shape. This pattern fits within rectangles that have been split at the point of the Golden Ratio.
Take a look at the diagrams below:
The Golden Section in Music
Not only does the Golden Section and Fibonacci sequence permeate art, architecture, and the natural world that surrounds us, but also it is present in music. In fact, it is integral to the foundation of Western harmony and musical scales.
Some examples include:
On a piano, there are 13 notes in an octave, with 8 white keys and 5 black keys.
There are 8 notes in a scale, of which the 1st, 3rd, 5th, and 8th notes create a basic chord.
The 5th note in a scale (8 notes) is known as the dominant note. This same note is the 8th note in the octave, when counting all the black and white notes together.
When numbering the notes in a scale (8 notes), the Fibonacci sequence can be seen on the 1st, 2nd, 3rd, 5th, and 8th notes, as expected. When numbering the notes in an octave (13 notes), numbers from the Fibonacci sequence, although slightly shifted, land on the same keys. The ratio between the ‘scale’ numbers and the ‘octave’ numbers align with the Fibonacci ratios- 1:1, 2:3, 3:5, 5:8, and 8:13.
Aside from harmony, the golden section can appear in the structure of a musical work. For example, if a piece has its climax around 61.8% (almost 2/3s) into the composition, it is often seen to have the most impact on the listeners.
There are many other ways the Golden Ratio and Fibonacci sequence appear and are utilised in music, and I will likely explore it in more depth in future posts. However, for now, in fear of writing far too much, I will leave it there.
Criticisms
The golden ratio isn’t without its criticism, with many questioning whether it always results in beauty and whether it can be successfully applied to absolutely everything in our world.
I would imagine that this would depend on how it is implemented and what outcome is desired. For, when we apply this to man-made things, it can be nothing more than a tool we can use to achieve beauty, if we know how. But, it cannot save nor be the solution to something that just isn’t beautiful to begin with.
There is a possibility that an ugly building could use the golden section, because maths is simply maths. If we are the creators, it is through our own eyes and our own judgement that we can see and manipulate our work to achieve beauty. Although this ratio or equation could perhaps elevate our work and enhance its beauty, I don’t believe it can do it for us. We must take some responsibility and recognise our own creative role on this planet, which often requires certain choices to be made…
Sources:
Classic FM: What is the Fibonacci Sequence – and why is it the secret to musical greatness?
Canva: What is the golden ratio?
The Golden Number: Music & The Fibonacci Sequence
