Additudes!: Fibonacci/Lucas Mod Cycles

set in the Order-21 Perfect Square

for Csound 'pluck'

[24:21]

Six centuries apart, Leonardo of Pisa (aka "Fibonacci") and Edouard Lucas discovered a pair (one each) of additive integer series whose generative rules might be combined as follows:

Each next term sums the current term and its immediate predecessor.  Term #1 is always 1. Term #0 is either one less than that (0, in the Fibonacci) or one more (2, in the Lucas).

These series, either forwards by addition as here or reversed via subtraction, continue of course indefinitely.  As my intent to harness them musically called for finite sources, I settled on the modulo function to provide these, their values and occurrence frequencies to be mapped respectively to pitch and duration.  The question then became, which moduli -- more specifically, what context (preferably organic!) for their choice?  This was answered on a separate formal basis as I encountered the Order-21 Perfect Square.

A square is "perfect" if tiled entirely by unique-sized squares.  While only a few instances of such a figure have been discovered, it is known that its tiles must number no fewer than 21, and that 21 permit just one selection of sizes in exactly one configuration.  As with the others, its area sums the tile areas, and its side sums the tile sides through any horizontal or vertical cross section. This 21's framing side measures 112 units, its tile sides 2 4 6 7 8 9 15 16 17 18 19 24 25 27 29 33 35 37 42 and 50.

I have taken the figure to define Additudes! ' outer structure, mapping its dimensions to sound parameter ranges -- the frame vertically to a pitch span of 6 octaves, horizontally to a time span of 60 seconds.  In this context each tile square becomes the corresponding frame for one of the Fibonacci/Lucas cycles described, with that tile's dimension determining the modulus itself.

There are 21 movements, each named as a series of flags indicating chord membership by density (R to L: 1, 2, 4, >4):

01. 0 0 0 0           08. 0 0 1 1           15. 0 1 1 1
02. -  -  -  -           09. 0 1 0 1           16. 1 0 1 1
03. 0 0 0 1           10. 1 0 0 1           17. 1 1 0 1
04. 0 0 1 0           11. 0 1 1 0           18. 1 1 1 0
05. 0 1 0 0           12. 1 0 1 0           19. -  -  -  -
06. 1 0 0 0           13. 1 1 0 0           20. 1 1 1 1
07. -  -  -  -           14. -  -  -  -           21. E p  i  l

Access: CD-brochure.  For per-movement audio at IMSLP click here,
then on Performances: Synthesized and 20 more, for links A1-A21.