Henon Map Melody Generator

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The Henon map generates melodic patterns which are usually repetitive with slight variations between repetitions. At fast tempos, a single Henon map sequence will stream into two or three independent lines.

The Henon map depends on two parameters a and b and is defined by the equations:

x_n+1 = 1 + ax_n² + by_n
y_n+1 = x_n

Musical notes are generated from the Henon map with the following equation:
     midikey = floor((x + 1.0)/2.0 * 127.0 + 0.5)
where midikey is a MIDI note number which is a number in the range from 0 to 127. Values of x in the range from -1 to +1 in the Henon map output are converted to the range 0 to 127 with the above equation. The program which generate the melodic sequences can also be given a range of note which are actually played, and notes outside this range are converted to rests. This allows for the constant rhythm of the sequence output to be converted to more complex rhythms.

Parameter Space

black = fixed point ending or goes out of bounds red = 2 point cycle orange = 3 point cycle yellow = 4 point cycle green = 5 point cycle blue = 6 point cycle indigo = 7-11 point cycle violet = 11-20 point cycle white = no repetition (through 10,000 iterations)

Other images:
• zoom onto yellow section where alpha = 0
• changing the starting point to (0.63135448, 0.18940634).
• Zoom into region with chaotic behaviour
• Zoom into region with chaotic behaviour (large)

Further Information

• Henon Map Correlation Dimension, by J.C. Sprott
  
• LISP code for Henon fractal melody generation
  
• C++ source code
  
• Max/MSP C implementation (description).
  
• Improv interactive performance version
  
• Audio filter version