Fibonacci Rhythm - no bar pattern, highly structured but never repeats - played on harmonics

"What's the best tuning for consonance, beauty, relaxation and peace. I don't want dissonance or tension that needs to resolve.". 

That was Jesse Thom's interesting question recently over at the Xenharmonic Alliance in facebook,

Youtube Video

This rhythmic pattern has no fixed measure size of any description - but it isn't free time either - it's highly structured.

 My idea was to take any very restful unchanging harmony - there are lots of possible choices, including the unison, but one natural choice is the harmonic series - and then use rhythms to maintain the interest.

My first thought was to use polyrhythms for the rhythmic interest, as in the harmonic polyrhythms of Bounce Metronome and the Lambdoma music therapy. But then thought, what about the Fibonacci rhythms? So this is the result.

These fascinating rhythmic patterns never repeat exactly at any time scale - but yet are highly structured. There must be a million different ways to approach her question, this is just one idea :)

These patterns are already there in Tune Smithy and I may add them to Bounce Metronome in the near future. Meanwhile you can do them in Bounce Metronome already, but have to make them partly by hand as I did here.

Here is a longer audio clip:

Fibonacci Rhythm - no bar pattern, highly structured but never repeats - played on harmonics by robertinventor download link: harmonic-fibonacci-fractal-tune (ten minutes) made with Tune Smithy

Here is another version, this time with harp glissandi to help you to hear the separate notes making up the chords.

Fibonacci Rhythm - played on harmonics - with harp glissandi by robertinventor download link: another version, this time with harp glissandi 

I've also added this as a new fractal tune to the latest upload of Tune Smithy today (after download and install, look under Fibonacci Rhythms in the fractal player or fractal composer task for Tune Smithy, and you'll be able to play it endlessly).

Here is how it works

The fastest rhythm plays a pattern of two beats with times in the golden ratio to each other.

All the other parts play exactly the same pattern of beats as the fastest one, only more slowly.

The parts are connected together like this: a large + small beat in a faster part becomes a single large beat in the next, slower, part.Then any left over large beats in the faster part (after pairing up all the short beats with long beats) become short beats in the slower part.


You may find it easier to see how it works with a transcription. The rhythm goes

L S ~ L ~ S L ~ L ~ S L ~ L S ~ L ~ S L ~ L S ~ L ~ L S ~ L ~ S L ~ L S ~ L ~ S L ~ L S ~ L ~ L S ~ L ~ S L ~ L S ~ L ~ S L ~ L ~ S L ~ L S ~ L ~ S L ~ L S ~ L ~ L S ~ L ~ S L ~ L S ~ L ~ S L ~ L ~ S L ~ L S ~ L ~ S L ~ L S ~ L ~ S L

where L and S are the large and small beats, times in the golden ratio to each other. The ~s are like the bar lines, and show where the notes are played in the next slower part

This shows the rhythms of all the parts together so you can see how they relate.

L S | L | S L | L | S L | L S | L | S L | L S | L | L S | L | S L | L S | L | S L | L S | L | L S | L | S L | L S | L | S L | L | S L | L S | L | S L | L S | L | L S | L | S L | L S | L | S L | L | S L | L S | L | S L | L S | L | S L 

There is no fixed measure size

Notice that when a note is played in any of the slower parts, then notes are also played at the same time in all the faster parts. This helps give it the feeling that it is highly structured.

The notes in the slower parts are a bit like bars (of larger and larger sizes in the slower and slower parts) - they give it the feeling of structure - but it is a non repeating structure, with two "bar sizes" no matter which part you choose to use as your bar lines - and the pattern of the "bars" doesn't repeat either, not in any of the parts.

Where does the idea come from?

The original idea of a Fibonacci rhythm of this type is due to David Canright in his article "Fibonacci Gamelan Patterns".

He found that the mathematical ideas behind it have surprising connections with Penrose tiles - those tiles that can tile the plane but can only do so in a non repeating fashion.

In fact the long and short beats of this rhythm correspond to the wide and narrow rhombs of a row of a Penrose tiling. 

David Canright however suggested various musical variations on the construction - you have choices of how you divide each long beat in the slower level into a long + short beat in the next faster part - and here I've done it with his more rhythmically interesting "longest beat near to most deeply reinforced beat".

How the video was made

It's made in Bounce Metronome - but I had to do a lot of the work by hand - copied / pasted the pattern of long and short beats from Tune Smithy for the fastest part - and then skipped beats as necessary in all the slower parts until I got the right pattern. For the harmonies, I just set an interval to play for each part, similarly to the triplet rhythmicon.

How easy (or hard) it is to add as a feature to Bounce Metronome

One easy way to add this to Bounce Metronome would be to do the same thing, just automate the process - so similarly to the triplet rhythmicon, and add a couple of windows to configure the parameters to explore different types of fibonacci gamelan patterns. Won't be able to play them endlessly, just larger and larger finite fragments - but that will let you use the bounce visuals along with the patterns.

This is now done. See  Fibonacci rhythms, sonified pendulum waves, options of interest to instrument tuners, and other new options and bug fixes

I can't think of a natural easy way to do the bounce visuals if you let the patterns play endlessly as you can in Tune Smithy, The problem is that the bounce visuals are based around the width of the window as a time unit, usually the measure. 

The tuning used

The notes are the notes of a harmonic series starting at 5/1 so 5/1, 6/1, 7/1, all the way up to 20/1 (16 parts).

Since it uses the harmonic series, all the notes sound good together.

It's done with a 1/1 of 64 Hz so the lowest pitch note you hear is 320 Hz.

What is the instrument in the clip?

In both the video and the audio clips I used Frank Wen's "Orchestral Harp" sound - it beautifully fits the harmonies I think.

This sound is from the Fluid R3 soundfont - the third release of the Fluid sound font which is a widely available public domain "mega" soundfont made by Frank Wen about ten years ago - for instance I believe it's used to play Midi files in the Denebian distributions of LInux.


Fibonacci Rhythms in Tune Smithy (on-line help from the software).

Fibonacci Gamelan Patterns - online paper by David Canright

The Fibonacci Rabbit Sequence for background about this sequence, and more mathematical, Fibonacci Word (Wikipedia)

You might also be interested in: Connection between the Euclidean Rhythms and the Fibonacci Rhythm (in the Bounce Metronome wish list)

More audio clips

Here is a more complex rhythm, and using the subharmonics (a justly tuned minor chord):

More Complex Fibonacci Gamelan Pattern on Subharmonics by robertinventor download link: more complex fibonacci gamelan

Here is another one, with three beat sizes, and more exotic harmonies:

Fibonaci Rhythm - more exotic harmonies, three beat sizes by robertinventor - Direct download link: more exotic harmonies, three beat size

I made the last two clips by playing a chord in a beta of the FTS Lambdoma (as used for Barbara Hero's Lambdoma music therapy) with the Fibonacci rhythms added in, just as a try out to see how it goes. May well add to FTS Lambdoma and to BM in the near future.