Video pages: Gravity Bounce - New Ones - Time Signatures etc - Polyrhythms - No exact number of beats
3/4 over 4/4 - 4/3 over 4/4 type Polyrhythms - Polyrhythmic paradiddles etc - Swing - Conducting Patterns

Measure With no Exact Number of Beats

See also the fractional beats per measure audio clips

Genres for these rhythms include:: not sure of any particularly, but probably safe to include Contemporary Classical as many composers are interested in borrowing any kind of unusual and interesting rhythms, and you could play them with aid of a special click track for the musicians, or using BM Pro to guide them. Also, microtonal composers (likely to be interested in the golden ratio pitch interval connection), educational and just for fun :-)

I'll add more videos here later on

Golden Ratio Polyrhythm with Golden Ratio Pitch Interval

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You can play rhythms like this endlessly, any tempo, any intervals and many other features- see the Harmonic Fractional Polyrhythm Metronomes page

More about this video

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.

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.

MORE DETAILS

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.

Maths links:

The Golden section ratio: Phi (not too hard if you have a maths background)

An interesting property of the golden ratio (that it is one of the most "difficult" numbers to approximate using ratios).