Fibonacci's Private Fantasy in C major

Oh no, I've dabbled again with those addictive Fibonacci numbers. Get yourself some good wraparound headphones or connect to a decent sound system, crank up the volume, and click the orange play button:


Like my earlier piece Fibonacci's Rabbits, its form and content are based on a musical expression of the number series, in terms of pitch (where the unit is the semitone) and also in terms of time. In the first example below, the unit of time is the 32nd note, and the pitches each rise by the corresponding number of semitones:

Technically, the Fibonacci series can be understood to begin with zero  (viz, 0+1=1, 1+1=2, 1+2=3, 2+3=5, etc), so by rights I ought to have started my music with a very brief silence. But as silence theoretically precedes the beginning of every piece of music (except in supermarkets, of course), I decided that this was just an academic w*nk and no-one except me would notice its shocking absence.

As with 'Fibonacci's Rabbits', the climax of the piece comes at the golden mean - as you'd expect - and I'm sure you'll detect the sudden aggressive change in mood just after the half-way mark at 1' 15". It effectively cleaves the piece into two unequal sections (AB) the B being shorter than the A in the proportion of the golden mean (0.618). To account for this shorter time-span, the underlying bass figure is consequently slightly speeded up in the B section, but still retains its Fibonacci ratios, of course. This B section, besides being announced by the aggressively louder instrumentation, is tempered and made somehow more familiar and 'gentle' by use of the diatonic C-major scale in the upper instruments (the numerical 1st, 2nd, 3rd, 5th, 8th, 13th, and 21st notes of the scale, ie, c d e g c' a'' g'''. This generates an entirely different harmonic feel by comparison with the A section, even though the original Fibonacci-derived bass continues to the end, complete with its naughty G-sharp, and finds itself at odds with the major scale environment. Parallel universes, multiple existences.

Overall, the music is clearly C-centric, ie pivoting around the pitch-class C as the epicentre of its sound universe. Given the pitches in the Natural Overtone Series (closely related to the Fibonacci series), the triad of C-major therefore cannot help but be prominent. But there is that unrepentant G-sharp to spice things up a little and introduce some delightful (and badly-needed) irrationality. Several bars before the climax, there is a somewhat 'brittle and quiet' sounding region where the harmony swings vaguely towards G, the V chord, although it is deliberately ambiguous with that  G-sharp lurking in the bass. This is reminiscent of the old classical 'Development' section, and is even introduced by a subtle V/V applied dominant.  So in fact, the piece is extremely traditional in its overall tripartite XYX form, redolent of the eighteenth-century Enlightenment. Nothing new to see here, Mr Mozart, move along now. But hey, what the hell... listen anyway. You might even like the noise it makes.

Confused? You should be. The form of the piece is deliberately ambiguous, with a 2-section format superimposed over a 3-section format (Note that 2 and 3 themselves are Fibonacci numbers). The 2-part AB aspect derives from the golden mean of Nature, as revealed by the great mathemagician Fibonacci,  and the 3-part XYX aspect derives from Nurture, the artifice of human creation. (Some might argue that the 3-part form should be labelled XYZ, given that the last part is so considerably less chromatic than the first. Whatever, my thesis is still valid:- whatever you want to call them, there are still 3 sections. Making art ambiguous is not yet illegal in Australia - (well, at least not until Snot Morrison's inevitable coup dumping Malcolm Turnbull).

Instrumentation: a variety of synthesizers and handbells, as per Sibelius 7.2 software. This time there is no score - maybe until someone asks for one.

Mrs Fibonacci Baked a Pie.

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:


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

http://soundcloud.com/peter-gore-symes/fiibonaccis-rabbits-the-sound

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

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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.