Tuesday, 30 August 2011

Fibonacci's Rabbits - the sound of one mathematician thinking.

Does one always need to know what music is "about" before listening to it?  It's head VS heart. Go with your heart this time and listen first. Connect your device to a decent sound system or enclosed headphones, and click the orange PLAY button:

Alternatively, you could listen to the music at its URL:

If you wish, you can read the score here. It will open in a new window so you can read it while listening:

Many things in Nature, such as cabbages, pine-cones, sunflowers and snail-shells all grow in spiral patterns controlled by the series of numbers known as the Fibonacci Series. The 12th century mathematician Leonardo Fibonacci discovered that exponential growth occurred if the last two numbers of any series were added together (eg, 1+1=2; 1+2=3; 2+3=5; 3+5=8; 5+8=13; 8+13=21, etc). As with Bartok, who always composed with a pine-cone on his desk, these numbers provided the very simple genesis for my short piece "Fibonacci's Rabbits".
When considering the dimension of musical Pitch (assuming one is counting by semitone), the Fibonacci Sequence neatly generates the ratios which create the intervals comprising the major triad. The 'cell' in the first bar-and-a-half of my piece, for instance, outlines the triad of C-major. At this point I choose to bypass the usual debate arguing "Co-incidence or Causality?" ...as that is not my purpose here.
The Fibonacci Series can also be applied to the dimension of Time. Counting by demi-semi quaver (DSQ), my cell exponentially stretches out in time-values as the numbers of DSQ obey the dictates of the series.
Once I had established the mechanical aspects of this cell, however, Rationality was summarily overuled by Art. I proceeded to present the cell in the freely imitative fashion of a Renaissance motet, symbolic of the ordered growth yet highly chaotic interaction of plants in Nature. This image was triggered in the first instance by watching the astonishingly rapid growth of the Climbing Spinach Vine on our balcony in Chiangmai.
The Fibonacci Series can be taken a step further - to suggest Form in music. As the ratios of the numbers spirals from 1+1=2 right through to higher numbers such as 13+21=34, 21+34=55 and so forth, the ratio increasingly accurately converges towards the decimal fraction 0.618, known as the 'Golden Mean', an irrational mathematical constant. View more complete examples here and here. In my example, the number 21 comprises approximately 0.618 of the number 34. Hence in my music, not unlike the music of countless other composers throughout history, I engineered the the climactic point to occur at about six-tenths of the way through - an aesthetically pleasing proportion which defiantly avoids symmetry. Botticelli's 'painting, the Birth of Venus' is a visual demonstration of the principle. He even included a shell for good measure.
I gleefully noted the presence of the G# just after the triad in my cell, which immediately authorized me to write my beloved augmented triads (C-E-G#). The successive entries of the cell in the keys of C, E, and G# (bars A/B/C) resulted in the extended high-pitched augmented triad which materialises in the high notes towards the end of Bar 6:

Similarly, personal choice determined the extended quartal harmonies - traditionally and accoustically  more stable/final than augmented triads - which serve to conclude the piece:
The final gesture includes the only instance of a cell which falls in pitch - symbolic, perhaps, of decay and death  as the necessary counterbalance of growth, the ends of empires, the eternal wheel of life.

Other interesting stuff about the Fibonacci Series:
* The piano keyboard from C to C has 13 keys comprising 8 white keys and 5 black keys, split into groups of 3 and 2.
* In any key, the 'dominant' note is the 5th note of the scale, which is also the 8th note of all 13 notes that comprise the octave. Note that 8:13 = 0.61538, which approximates phi. It's precisely that assymetric wildcard which makes a musician's (and theoretician's) life interesting and happily irrational.
* The spiral layout of cabbage leaves ensures that the smart little plant receives the maximum possible sunlight over the span of a day.

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