How Composers use Fibonacci Numbers & Golden Ratio | Composing with Fibonacci



I'm going to talk about music Fibonacci and the golden min but first I want to talk about rabbits his little puzzle for you proposed by Leonardo Bonacci in his 1202 book Lieber Tabuchi the book of the abacus Bonacci puzzle goes like this if you put a pair of rabbits in a field and they give birth to a new pair of rabbits every month and it took each new pair a month to reach adulthood how many rabbits would you end up with after a year counting month by month you get the following sequence starting with one pair of rabbits after a month they produce a pair of baby rabbits you book two pairs at the end of the second month the original pair produces two more babies so you end up with three pairs in total after three months the original rabbit couple produce another new pair of babies whilst the oldest baby rabbits have now reached adulthood and given birth to their own first babies giving five pairs after four months you've got three adult pairs and two baby pairs after five months you've got 500 pairs and three baby pairs after six months eight adult pairs and five baby pairs and so on like this 13 21 34 55 89 144 so the answer at the end of one year is a lot of rabbits now maybe this number sequence looks familiar what Leonardo Byrne actually better known by his nickname Fibonacci had uncovered with the help of his rabbits is the so called Fibonacci sequence you'll notice that each new term in the sequence is produced by adding together the two previous terms one plus two equals three two plus three plus five three plus five is eight and so on giving the famous sequence of Fibonacci numbers and it's not just rabbits these numbers seem to appear everywhere in nature in the number of petals on a plant in the way a tree produces branches in the world of a pine cone in the growth of seashells even in the flight paths of birds of prey it's amazing to think that some of the most beautiful aspects of nature come through the application of the mathematics in the Fibonacci series so this connection between mathematics and beauty has led musicologists to question whether music that most mathematical of arts could also have Fibonacci numbers hidden within it now there all sorts of quite dubious theories you can find online about this about how there are 13 chromatic notes in the octave eight white notes and five black notes or how various Fibonacci ratios do generate musical intervals we recognize but for me the most interesting area is how the proportions of the series have effect whether consciously or not the way a piece of music is composed and structured there's a great example it's short enough to include in its entirety it's the first of Chopin's preludes opus 28 in C major the piece is 34 bars long with a climax at bar 21 the first major chromatic event is bar 13 and the opening statement lasts 8 bars so 8 13 21 34 a perfect Fibonacci sequence have a listen and judge for yourself whether these constitute audible events another plausible example is the first movement of Mozart's sonata in C major K 279 which has 100 measures and whose main sections come from far 62 and 38 very close to Fibonacci proportions now no one's suggesting that Mozart or sharp unconsciously worked with Hitachi nimbus which weren't widely known about until the 1870s I think the most convincing aspect of the shoppin is the placement of the climax of the piece which happens at a place known as the golden section also known as the golden ratio or the golden mean the golden section is the division that cuts a fixed length in such a way that the shorter portion bears the same ratio to the longer portion as the longer portion bears to the whole length the actual ratio is about 1.618 often written as the Greek letter fee fee is intimately connected with Fibonacci and you can see this in the maths if you look at the ratio between successive numbers in the Fibonacci sequence the higher up you go the closer and closer they get this golden ratio unlike the Fibonacci series the golden ratio has been known about and studied since ancient Greek times it represents a point on a line that's neither 1/3 nor half and some argue that when a work of art or a building incorporates the ratio it looks better it looks more in proportion Leonardo da Vinci certainly knew about – illustrating a book called the de vino' property on the divine proportion by Luca Pacioli published in 1509 and there's some evidence that some of his paintings are structured around the golden section although this is all hotly debated in architecture the Great Mosque of Cairo one in Tunisia built in 670 ad seems to have a consistent use of golden section proportions throughout finding such comparable examples in the structures of music is more challenging at least before the 20th century and of course there's the major problem that music operates in time so you can only tell that you're at the golden section of a piece in retrospect once you've heard the whole length of the piece but it's certainly interesting to consider how quite a lot of pieces do have a major climax around this point in their structure Fibonacci's work really came to a wider public knowledge from the mid 1870s onwards when the French mathematician Eduardo Luca named and studied them and from where they filtered into the minds of French Cimber list artists and then on into the work of composers like Debussy and Ravel which is why we start seeing a sudden plethora of Fibonacci style structures in pieces from this time onwards musicologists like Roy Hart and ernõ lenva have analyzed pieces by Debussy Ravel and Bartok and found evidence that they're structured using Fibonacci sequences how it claims for example that the formal boundaries of Debussy la Mer correspond exactly to the golden section and he delves deep into the piece uncovering Fibonacci numbers and golden sections were every year for example the 55 bar long introduction to dialogue Devon a la Mer can be broken down into five sections of twenty one eight eight five and fifteen bars in length it's not really known though how much these composers knew of these ideas or applied them to their work perhaps the most suggestive piece of evidence that Debussy really did have mathematical considerations in mind when composing at least some of his pieces comes in an enigmatic letter to his publisher shackled around written in 1903 about the corrective proofs of his piano keys Jardin de 3 in which he writes you'll see that there's a bar missing my mistake as it's not in the manuscript however it's necessary as regards number the divine from de bercy's time onwards composers increasingly started consciously composing pieces using golden section and Fibonacci numbers cranek no no and Stockhausen are all known to have used them but whether they're anything more than a curiosity I'm not totally convinced too often these studies seem to rely on picking out pretty arbitrary points in the music to justify the measurement of a particular number but despite that it's hard to escape the feeling that there must be some kind of connection between these numbers and music no one can deny that these numbers exist or that they occupy a central place in our experience of beauty in the natural world so why not in music as well thanks very much for watching I hope that was deaky enough for you and I'd love to hear what you think please put your comments below and if you'd like to see more videos like this please do like and subscribe see you next time it standing at the golden section you

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