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• Morse-Thue Sequence
• Logistic Map
• 1/f Noise
• Henon
• Hopalong
• Martin
• Gingerbread man
• Lorenz

Morse-Thue Sequencefile>

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Consider the following sequence of the non-negative integers 0, 1, 2, 3, 4, 5... written in binary notation:

      0, 1, 10, 11, 100, 101...

take the sum of the digits modulo 2 (known as the "digital root") of each number, thus forming the following sequence:

      0, 1, 1, 0, 1, 0, 0...

This is the so-called Morse-Thue Sequence. This sequence can also be generated by iterating the map 0 -> 01 and 1 -> 10. Starting with a single "0" you get:

               0

           0       1

         0   1   1   0

        0 1 1 0 1 0 0 1

Also, each generation can be created appending the complement of the preceding one, as follows:

      0

      0 1

      0 1  1 0

      0 1  1 0   1 0  0 1

Of course, you are not restricted to base 2. You can use any counting system you like. Furthermore, the sequence shows a high degree of self-similarity; for instance, if every even term in the sequence is removed the sequence remains unchanged!:

      0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0



      0   1   1   0   1   0   0   1

Also, removing every second couple keeps the sequence unchanged:

      01 10 10 01 10 01 01 10



      01    10    10    01

How do you make music out of this sequence?. Very simply.
First you select a base in which to count, by which the general counters will be multiplied. Now, this is what happens: The number shown in the Start Counter is multiplied by whatever value you have chosen for the multiplier, then the result is converted to the base you have chosen, for example:

	base 3 multiplier 3 (counter 17) = 3*17 = 51 -> 3 = 1220.

The digits of this number are then added (in base 10):

	1+2+2+0 = 5 

and made the modulo operation with the number of notes in the scale times the number of octaves . Assuming C major (7 notes) and 1 octave, this yields:

	5%(7*1) = 5, 

which is then is added to the starting note note in MIDI values (middle C is No. 60). Since the mapping starts at 0, then the resulting note is an A above middle C.
Some base-multiplier combinations yield an amazing varietyfilefile others are boringly static. The result depends in the right combination of base, multiplier and scale.
Click here for an example of a piece composed with the Morse-Thue algorithm (canons.mid, 25Kb).

Logistic Map

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The Logistic Map is one of most studied non-linear equations in chaotic dynamics . It models very well the behavior of many natural phenomena, such as predator-prey ecosystems, food-population, etc. It is defined as follows:

      X(n+1) = X(n)*k*(1-X(n)),

where k is any real value in the interval [0, 4]. The iteration of this formula gives back values between [0, 1]. For k values less than 3.5 the iteration quickly stabilizes into one value. But at higher values of k the solutions start bifurcating, stabilizing in 2 values then 4, 8, 16... until it reaches total chaos. This is called period doubling. Surprisingly, among this sections of total chaos there are others that have a stable number of values. For a k value of 4, the behavior is totally unpredictable, random. In fact, the power spectrum for this value is indistinguishable from that of "white noise". The logistic map is an example of "deterministic" chaos, that is, for a given value of k, the equation produces the same set of values.
How all this is transformed into notes is a very simple matter: The counters are used as an iteration control. The formula will be iterated until the number shown there. The result (between 0, and 1) is then multiplied by the number of notes in the scale and by the number of octaves . The result of adding this value to the starting note is the final note that will be played. Since the sequence of values is deterministic, for a given value of k you always get the same sequence of values, thus the same melodies.
Click here for an example of a piece composed with the Log Map algorithm (crazy.mid, 20Kb).

1/f Noise

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1/f noise ("one-over-f-noise", occasionally called "flicker noise" or "pink noise") is a type of noise whose power spectra P(f) as a function of the frequency f behaves like: P(f) = 1/f^a, where the exponent a is very close to 1 (hence the name "1/f noise"). Frequencies in "white noise" ( P(f) = 1/f^0 ) are uncorrelated, totally independent from each other, whereas in "brown noise" ( P(f) = 1/f^2 ) they are closely correlated. 1/f noise is somehow middle-ground between white and brown noises.


1/f noise appears in nature all over the places, including traffic flow, radioactive decay, chemical systems, granular flow, ecological systems, human speech...and music!!.

Extensive spectral analysis of "pleasing" pieces of music, such as Bach's Brandenburg Concerti, Vivaldi's four seasons, etc, etc, show the same spectrum as the 1/f noise. The music derived from the 1/f noise is the most pleasing to the human ear: it does not have the unpredictability and randomness of "white noise" (the power spectrum of the Logistic Map for a k value of 4) nor the boringness and predictability of "brown noise" (derived from brownian motion). 1/f noise is simulated with the aid of the following non-linear equation:

     X(n+1) = (X(n)*n)+(sqr(1-n^2)*r),

where n is a real number between [0, 1] and r is any random value. The user has to specify only the n parameter. Instead of using a truly random value for r, the program uses the Logistic Map algorithm with a k value of 4, which generates random but deterministic values. This is necessary so that your melodies are the same for the same set of parameters, otherwise you would never know what will come next.

As with the Logistic Map , the counters are used as an iteration control. Try using values of n that are not to much to the extremes (0 and 1), for the music gets "brownish". Best values are between .6 and .85. Keep in mind that this is only an approximation of the real thing. Simulating 1/f noise is a complex matter, well beyond the scope of this program.
Click here for an example of a piece composed with the 1/f noise algorithm (autumn.mid, 7Kb).

Henon

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Michel Henon was an astronomer at Nice observatory in southern France. He became interested in fractals by investigating the orbits of astronomical objects. The strange attractor most often linked with Henon's name comes not from a differential equation, but from the world of discrete mathematics - difference equations. The Henon map is an example of a very simple dynamic system that exhibits strange behavior. The orbit traces out its characteristic banana shape , but on close inspection, the shape is made up of thicker and thinner parts. Upon magnification, the thicker bands resolve to still other thick and thin components. And so it goes forever! The equations that generate this strange pattern perform the mathematical equivalent of repeated stretching and folding, over and over again:

     X(n+1) = 1 + Y(n) - a*X(n)^2

     Y(n+1) = b*X(n)

Two parameters a and b. Default values are a=1.4, b=0.3. As with other two dimensional maps, the length of the vector is used to compute the note to be played, which depends on your choice of octaves, scale, etc.
Click here for an example of a piece composed with the Henon algorithm (The Subway.mid, 8Kb).

Hopalong

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This fractal is attributed to Barry Martin of Aston University in Birmingham, England. Hopalong is an "orbit" type fractal like Lorenz. The image is obtained by iterating this formula after setting X(0) = Y(0) = 0:

     X(n+1) = Y(n) - sign(X(n))*sqrt(|sin(a)*X(n)-cos(a)|)

     Y(n+1) = a - X(n)

The only parameter is the angle a, measured in radians (-2pi to 2pi). The function sign() returns 1 if the argument is positive, -1 if argument is negative or 0 if the argument is 0. This fractal continues to develop in surprising ways after many iterations. Notes are computed from this fractal by iterating the equation and taking the vector length of the (X,Y) values, from which the pitch is finally computed.
Click here for an example of a piece composed with the Hopalong algorithm.

Martin

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This simple 2-dimentional map is also attributed to Barry Martin . Although graphically not as interesting as other fractal types, it generates very interesting melodies. It is defined by the following formulas:

     X(n+1) = Y(n) - sin(X(n))

     Y(n+1) = a - X(n)

Only one parameter: the angle a measured in radians (-2pi to 2pi). Try values near pi. As with other 2-dimensional maps, notes are computed from the value generated by the length of the vector for the (X,Y) values.

Gingerbread man

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This chaotic map in two dimensions was proposed and studied by R. Devaney . It is chaotic in certain regions and stable in others; for example, (1,1) is a fixed point. All other points within the hexagonal region whose vertices are (0,0), (1,0), (2,1), (2,2), (1,2) and (0,1) are periodic with period 6. This contrasts with a point just outside this region: it leads to a chaotic orbit. There is a stable hexagonal region forming the belly and five others forming the legs, arms and head of the gingerbread man. These points are, of course, points of stable orbits.

The gingerbread man fractal is defined by the following formulas:

	X(n+1) = 1 - Y(n) + |X(n)|

	Y(n+1) = X(n)

FractMus' counters are used as an iteration control. The user has to input the initial values of X and Y. Values can range from -20 to 20. After every iteration, in order to get a unique value, the vector length is computed (the square root of the sum of the values squared). From this value, depending on the choice of scale, octaves, modulation, etc, the actual note is computed.
Click here for an example of a piece composed with the Gingerbread man algorithm.

Lorenz

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This algorithm does not use Lorenz' world-wide famous three-dimensional chaotic map, defined as follows:

     dx/dt = -a + ay

     dy/dt = -xz + bx - y

     dz/dt = xy - cz

Instead it utilizes a more humble-looking (although equally interesting in its output) chaotic equation proposed by Lorenz in 1976 as part of his research in climatology. The equation is defined as follows:

     X(n+1) = a(3X(n) - 4X(n)^3)

One parameter: value of a. Range is [0, 1]. Range with best results is [.65, 1]. Parameters under .5 produce monotonic results, but you may want to use them anyway depending on the situation.
Click here for an example of a piece composed with the Lorenz algorithm (jamaica.mid, 12Kb).