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Forum for Tune Smithy, Bounce Metronome and other software from Robert Inventor
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Author Topic: Golden Ratio Polyrhythm Metronome with Golden Ratio Pitch Interval  (Read 9545 times)
Robert Walker
Full Member
Posts: 165

« on: May 16, 2010, 02:28:19 PM »

Golden Ratio Polyrhythm Metronome with Golden Ratio Pitch Interval

Here is a video of Bounce Metronome Pro playing the golden ratio preset for the Harmonic Fractional Polyrhythm Metronomes page.

It's the most polyrhythmic possible rhythm and played using the musical interval which is as far away from "in tune" as you can get. It is so far from in tune that it is a rather pleasant musical interval to listen to.

One way of putting it is that this is the most unsynchronised two players can get with both playing a steady beat - and as out of tune (in a rather specialised sense of the word) as they can possibly get as well.

I've done it with numbered beats so you can 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.

See: Phi and the Fibonacci numbers

 A property of the golden ratio (wikipedia)

See also the Harmonic Fractional Polyrhythm Metronomes page

« Last Edit: May 16, 2010, 02:34:34 PM by Robert Walker » Logged
Robert Walker
Full Member
Posts: 165

« Reply #1 on: May 16, 2010, 02:47:25 PM »

Oh and the instrument I used is the "Slightly noisy plucked Bessel" of Bounce Metronome Pro's Waveform Player.

Did it like that since it means it can play the desired golden ratio pitch interval very precisely so you can get a good idea of what it sounds like.

More information from the help for Bounce Metronome Pro:

The interval is 833 cents which is (a slightly sharp minor sixth which is) the inversion of a major third just a bit flat (i.e. closer to a minor third) at 367 cents instead of 386 cents for a pure 5/4 (or 8/5). Which makes sense since the inversion of the major third is 8/5 which is the ratio of two successive Fibonacci numbers.

This page: Nature, The Golden Ratio,and Fibonacci too ...

has a nice demo of how the golden ratio is as far as you can get from any simple fraction, and how this explains the way sunflowers form using the golden ratio and fibonacci number related spirals.

The various types of major, neutral and minor thirds and their inversions are generally regarded as pleasant to modern ears - well for most - some musicians so train their ears so that the only intervals that sound acceptable to them are the modern tempered twelve equal ones, and if they later take an interest in microtonal music it can take a while to undo the effect of that training.

Also more generally, this is culturally dependent - for instance in Western Europe in medieval times these intervals including the major and minor thirds were all regarded as dissonances needing to be resolved - and only the perfect fifth, perfect fourth and octave were considered to be consonances - all music had to resolve to perfect fifths or fourths.

This is also timbre dependent, which intervals sound consonant or dissonant can depend on the instrument you play it on - some intervals sound better on some instruments and others on others. Just to take a few examples, the frequency sepctrum of bell timbres have inharmonic partials, clarinets have only odd harmonis in their frequency spectrum, and pianos have detuned higher partials because of the high tension in the strings. You can also custom build new textures with the frequency spectrum designed to make almost any interval sound consonant (this approach was pioneered particularly by Bill Sethares).

« Last Edit: May 17, 2010, 01:39:42 AM by Robert Walker » Logged
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