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## Approaching the golden ratio with the Fibonacci sequence.

The golden ratio is one of the most difficult numbers to approximate using whole number ratios. The closest approximations are the ratios of fibonacci numbers - and they "converge" very slowly so you have to go a long way up the fibonacci sequence to get a good approximation.

So the golden ratio rhythm "as polyrhythmic as you can get" - as far as you can get from low ratio polyrhythms like 4:3, 5:4 etc. - and also as far from "pure ratio harmony as you can possibly get as well.For the maths see A propery of the golden ratio (in wikipedia). By comparision, polyrhythms based on π are close to 22:7 and the musical pitch interval 22/7 (or 11/7 once reduced to the octave). Or for a very accurate approximation, 335/113 - for the maths see Approximations to PI (wikipedia).

Although it is impossible to play a golden ratio rhythm exactly without a computer to help, you can still get pretty close to it using numbers of beats in the Fibonacci sequence, if you go far enough up the sequence .

This Fibonacci sequence goes: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... - each number is the sum of the previous two. For intance 34 is 21 + 13. The ratio of each number to the previous one approaches the golden ratio - though rather slowly. 144/89 (at 1.61797753) is within 0.0001 of the golden ratio (1.61803399).

So if you play an 8 : 5 polyrhythm that's quite close to the golden ratio. 13 : 8 is closer, 21: 13 closer still, and so on.

So here are some videos to illustrate that.

And here is the video which shows how the beats come close together on successive Fibonacci numbers.

Golden ratio polyrhythm with golden ratio pitch interval Notice how the beats nearly coincide when they reach successive Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,

You can get other "hard to approximate" numbers like the golden ratio with the formula the (a + bφ)/(c + dφ) – where a, b, c, and d are integers such that ad − bc = ±1. So I might explore those in future videos for these pages. See A propery of the golden ratio (in wikipedia)

#### Play all these videos one after another

This plays all these rhythms and also the golden ratio ones.

Practise Tips

It's fun to play along with one of these rhythms, just playing your music with one of the beats, while the others go on in the background of your playing. Probably a good exercise to help develop steady sense of rhythm, and independence.

#### Use these videos as a resource

You can use any of these videos as a resource for your own website or wikis, or make more of them yourself - see Add videos like these to your own site

#### Play these rhythms and animations at any tempo with Bounce Metronome

You can use Bounce Metronome Pro to practise these and many more rhythms at any tempo, including changing tempo. See Harmonic Polyrhythms in the Features list.

Also see Theremin's Rhythmicon.

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