xMid = .3515
yPoint = .42073
WindowSize = .001
MaxIteration = 255
xResolution = 100

These parameters define a line of 100 points that mark the upper sixth of image seen in Figure 2. To generate a single line of values instead of a plane, some changes were made in the basic program. The single yPoint value on the y-axis is held constant (and so also is the LamdaImag variable). The return statement is changed to a print statement that returns a Csound I-statement for each point calculated on the line.

In order to use the xCounter as a time counter for sound events the n-value for each previous x-point is stored in the variable PreviousN. Now n-counts can be tested so that a new I-statement is created only when a different value is returned from the process. In other words a new sound event occurs only when there is a change in the number returned from the process. A basic time increment is defined such that each pass through the xCounter loop adds that increment to the clock that marks the start times for each event in the mapping. The variable xResolution determines how many time samples (or events) will be calculated in the mapping (there are 100 events calculated in first example). The total length from zero to the start time of the last event will be TimeIncrement * xResolution.

Each number returned from the process will now be mapped into a pitch instead of a color. The variable PitchCount determines how many different pitches will be available. For example a four octave C major scale, 7 notes per octave times 4, equals 28 different pitches. (The pitch range will be specified in the Csound score file.) The basic program for creating a sonic map of the Mandelbrot set is then as follows:

define xMid = .3515
define yPoint = .42073
define WindowSize = .001
define MaxIteration = 255

define TimeIncrement = .2
define PitchCount = 28
define xResolution = 100

xStart = xMid-.5*WindowSize

xIncrement = WindowSize/xResolution
PreviousN = 1000
StartTime = 0.0
LamdaImag = yPoint

for (xCounter=0; xCounter < xResolution; ++xCounter){
    LamdaReal=xStart + xIncrement*xCounter
    x = 0.0
    y = 0.0

    for (n=0; n < MaxInteration; ++n){
        xNew = x^2 - y^2 + LamdaReal
        yNew = 2*x*y + LamdaImag
        x=xNew
        y=yNex

        if ((x^2+y^2) >= 4) break;
    }
    if (n==PreviousN) {
      StartTime += TimeIncrement
    } else {
      printf("i1 %7.2f 12.0 75 %d\n", StartTime, n%PitchCount)
        PreviousN = n
    }
}

For the examples in this report a simple Csound instrument is used, based on the pluck instrument. The orchestra is shown as follows:

; mandel.orc
sr = 44100
kr = 441
ksmps = 100
nchnls = 1

instr 1
   ipitchtable = 1  ; pitch table in score      
   ipitchndx = p5   ; p5=pitch index from table
   iamp = ampdb(p4)
    
   ipch table  ipitchndx, ipitchtable
   kenv expseg  1.0, 1.0, 1.0, 11.5, .0001
   asig pluck iamp*kenv,cpspch(ipch),cpspch(ipch),0,1

    out asig
endin

An I-statement in the Csound score file for this orchestra has the format

;p1      p2          p3              p4                   p5
i1   starttime   duration   amplitude(in decibels)   pitchindex

A pitch table that defines the pitch domain for the sonic map is setup at the beginning of the score using GEN 2. The values generated from the Mandelbrot set program function as index values into this table. The PitchCount variable in the program is equal to the number of elements in the table. The score for the illustrated program is seen here.

; mandel.sco

; f1 is a pitch table defining a four octave C major 
; scale C2-B5.
f1 0 32 -2 6.00 6.02 6.04 6.05 6.07 6.09 6.11
           7.00 7.02 7.04 7.05 7.07 7.09 7.11
           8.00 8.02 8.04 8.05 8.07 8.09 8.11
           9.00 9.02 9.04 9.05 9.07 9.09 9.11

; ins start   dur  ampdb(p4)  pitchndx(p5)
  i1    0.0  12.0   75             3
  i1    1.6  12.0   75             4
  i1    3.4  12.0   75             5
  i1    4.2  12.0   75             6
  i1    4.4  12.0   75             7
  i1    4.6  12.0   75             9
  i1    4.8  12.0   75            10
  i1    5.0  12.0   75             5
         .     .     .             .
         .     .     .             .
         .     .     .             .
  i1   18.6  12.0   75            11
  i1   19.2  12.0   75            10
  i1   19.8  12.0   75             9
e

One hundred I-statements were generated from the Mandelbrot set program (example 1).