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How to Compose a Song with the Golden Ratio and the Fibonacci Sequence


The golden ratio defines the harmony of most objects and, thus, is the most perfect expression of beauty.

The section aurea, or golden ratio, is the essence of many artistic works. We can easily find it in architecture, painting and sculpture, which use the pattern to achieve an ideal symmetry. From the leaves that grow on trees to the spirals in pinecones and the geometric formations of snowflakes and the dynamic of black holes and galactic dimensions, the biological configurations of our universe follow this enigmatic algorithm that defines the perfect harmony of most objects. And the most wonderful part of this model is that it can be replicated by the arts. Applying this pattern to musical composition is especially attractive since it is easily done and the result is sublime.

Since the dawn of mankind, the golden ratio has always been close to music: certain theories suggest that, guided by the golden ratio, Pythagoras discovered the resonance of notes on a taught string, and that Plato used this knowledge to create his theory on the Music of the Spheres. If we jump ahead to the 20th century, we find György Ligeti, who dared to compose “Apparitions”, a song divided into sections that were proportional to the golden ratio.

Before we start to create music with the algorithm, it is worth knowing that this measurement is a formula which results in a second number, from which we can set-out: the Fibonacci sequence.

Explained briefly, this succession works when every new number is the result of adding the two previous ones: 1, 1, 2, 3, 5, 8, 13, 21, etc… At the same time, the golden number (also known as Phi, represented by the Greek letter Φ) is a concrete point that we find between the proportions of two segments on a straight line:


A                                                              F                                       B

Where F is the Φ equivalent to “1,6180339885…”, which is also known as the golden number. This is an endless and irrational figure that does not represent a periodic repetition. Based on this basic scheme, we can define what we want to compose. If our song lasts from A to B, then the element F would be the modifier of the track’s rhythm in the following manner: We divide our work into two parts, which will be defined by 61.8% and 38.2%, in accordance with the golden ratio. Afterwards, these will be multiplied by “x”, where x represents the length of the work. Putting it into practice makes it even easier: if your song lasts 4 minutes (240 seconds), then:

240s*0.618Φ = 148.32s or, at 2 minutes with 48 seconds we must intercept the work with a change, a bridge, an arrangement with a different instrument or a new melodic composition. Now, using the Fibonacci sequence, we can also create embellishments and changes in the rhythm of our song to make it all the more attractive, where the sequence 1, 1, 2, 3, 5, 8, 13, 21… will correspond to the minutes or seconds when we will make changes in the tone that implies emphasizing the note that is played at that moment.

Great composers like Beethoven, Mozart and Wagner intentionally changed the rhythm of their sequence. Their compositions were actually very complex, since the numbers they used where not prime, but large ones like 2178309 or 53316291173. However, simply analyzing the score for a piano piece takes us back to the golden ratio again. In a scale we find 8 white keys and 5 black ones, equivalent to musical notes that will be ordered in groups of 2 and 3. This sequence is organized as 2, 3, 5 and 8.

Understood through geometry, we can create several forms using the golden ratio’s straight line: from a simple star-shaped pentagon to infinite hexagonal networks, and they both share the property Φ. Musical notes progress in the same manner: the high and low pitches have the same infinite spiral. We do not have to be mathematicians to understand this.

It has frequently been said that the golden ratio is merely a coincidence, but after understanding its sublime examples in music, we can seriously consider redefining it and trying to see it as a logical and supreme result. Plato used to say that it is impossible to combine two things without a third, there must be a relationship between them that joins them; the best liaison for this relationship is everything.

In sum, the golden ratio is no more than the mathematical translation of an algorithm used by nature, and that stands out because of its hyper-harmonic condition; that is, it is a lesson on aesthetic perfection, courtesy of the natural world.

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