Music Composer, music recognition software. Composer recognizes polyphonic music from a microphone or other Wave input or file and converts it into Midi sequences. You can sing, whistle or play guitar, piano, flute and so on with your microphone, Composer will automatically recognize and score your music and create standard Midi sequences. No Midi keyboard or musical experience required.
AKoff Guitar Assistant designed to assist in visual tuning a guitar with microphone or an electrical guitar connected to PCs sound card. It analyzes in real time a stream of audio signals from WAVE input of your sound card and calculates the main frequency. This frequency is shown by a pointer on graphic guitar signature stamp (frets), comparing it to the proper frequency. Tuning a string now simply means centering the pointer on the appropriate graphic fret.
The Mandelbrot set can be defined as the set of points, 
, in the complex plain for which the infinite series 
, converges to a finite value.
In the images shown, the real axis is the horizontal (x) axis and the imaginary axis is the vertical (y) axis. There is no reason to go farther from the origin than -2 or 2 on the real axis or -2i or 2i on the imaginary axis, since any point that far from the origin will quickly diverge for the series shown above. The dark pixels represent starting points, , in the complex plane that did not diverge. My test for convergence is whether or not for each starting point, , the series, , reached an absolute value of 2 or greater after 100 iterations of the series shown above. You can also get beautiful pictures by shading differently or coloring differently depending on how long it took to diverge, or by the absolute value reached by a certain number of iterations or both.
The closer you look at the border between what is in the set and what diverges, the more structure you find.
When creating these images it is convenient to use a simple x-y plane and assume a certain number of pixels in your image, say 400 by 400. You choose a center for your image somewhere in the plane, (x′, y′), somewhere less than 2 from the origin, and scale your pixels to represent numbers in the complex plane represented by c=(x-x′)/(scale)+i(y-y′)/(scale) and for each pixel start at z=0+0i and iterate until you reach above 2 or get tired of iterating. You then choose to color the pixel as being in the set or not (can even be just black or white) or shade according to the quickness of divergence or maximum value of z² reached for those you decide are in the set.
Note that you don't actually have to have the computer do complex math for you, since you are only finding one new (x,y) value from one old value each time and then testing whether or not x² + y² > 2. and new x value is just the old x value squared minus the old y value squared plus the pixel x value while each new y value is just 2 times the old x times the old y plus the pixel y. (It is kind of fun to work this out and show that it is that simple).
Benöit Mandelbrot published a paper in 1980 (after decades of work in complex analysis and complex dynamics) discussing the parameter space of quadratic polynomials. Adrien Douady and John H. Hubbard did detailed mathematical studies of the set establishing many of the fundamental properties and they are responsible for naming the set in honor of Mandelbrot. Popularization of the set came with the publications of images by Heinz-Otto Peitgen and Peter Richter. See "The Beauty of Fractals" 1986.
If you happen to have Matlab, you can right click here and choose to save the m-file and right click here to save the *.fig file that I wrote in Matlab to create images of the mandelbrot set. I think that as long as these two files are in the same directory, you can direct matlab to that directory and run the command, "Mandelbrot" to use this simple application.
 Get this wonderful home study course on human anatomy in 3-D on your computer before the price goes up!. |
$18M in Affiliate Secrets!