Golden Ratio Polyrhythm Metronome with Golden Ratio Pitch Interval

see This shows the most polyrhythmic possible rhythm together with the musical interval which is as far from “in tune” as you can get. So far away that it is a very pleasant musical interval to listen to.

You can get Bounce Metronome Pro for Windows to play this and many other rhythms – see

The bouncing numbers help you to see how the beats of the two rhythms nearly coincide when they reach successive Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …So for instance when you get to 21 beats for the blue ball and 34 for the red ball the clicks are close together, and even closer for 34 and 55, etc.


This shows a pattern generated by two beats at the golden ratio to each other. The pitches are also in the golden ratio.

This makes it the most inharmonic possible musical interval and the most polyrhythmic possible rhythm in a certain sense.

In a way it is the most polyrhythmic possible rhythm. First of all, the two rhythms never coincide exactly after the first beat – but any irrational number like PI or E would do that. What is special about this polyrhythm is that the ratio of the two rhythms is hardest to approximate with a pure ratio.

A human player couldn’t play this polyrhythm without assistance from a computer because it continues endlessly without ever repeating the exact same pattern of clicks. In fact there’s a connection betwen this rhythm and the aperiodic Penrose tilings as well.

The pitches are also in the golden ratio – and the interval of a golden ratio is in a certain sense the most inharmonic interval you can have – as far away from “in tune” as you can be in a sense – except that it is so far away it is actually rather pleasant. Pure low numbered ratios of frequencies are the so called “harmonic intervals” – intervals between low numbered frequencies in the harmonic series – which are the intervals that tend to sound most “harmonious”.

The golden ratio is one of the numbers which is hardest to approximate with a pure ratio. The numbers which get closest to it with small number quotients are ratios of successive Fibonacc numbers.

So this means, that after e.g. 8 beats of the blue ball in this video, and 5 beats of the red ball the notes will come closer together than for any earlier beat. Same happens again after 13 and 8, and so on.



Bounce Metronome Pro can play this and various other Harmonic Fractional Polyrhythms based on PI, E, or whatever other intervals you like.

You can play rhythms like this endlessly, any tempo, any intervals and many other features with Bounce Metronome Pro – see the Harmonic Fractional Polyrhythm Metronomes page:

Post time: 09-24-2017