Plots and sounds from roots of polynomials
The plots of the roots of some sets of polynomials make for interesting viewing. I've been sonify these plots, using each root as a sound event, with time the horizontal axis and pitch (frequency) the vertical axis. This method creates a sound whose spectrogram looks like the plot.
The real line corresponds to 2500 hz, and each has a range of 100 hz to 4900 hz. Each sound event (grain) has length 0.01 seconds.
All have the same timescale, so the larger the range of real parts of the roots is, the longer the whole sound. So some sounds are longer than others.
The sounds are palindromic, so the second half is a reversal of the first half.
Some results are below. Click the plots to embiggen.
Plot of all roots of all Littlewood polynomials (i.e., polynomials with all coefficients either 1 or -1) of degree 16.
Sonification of the above. Frequency calculated using (2500-100)/1.42(Im z)+2500 for each root, z.
Plot of all roots of all polynomials of degree 16, with terms of the form ±(m+1)zm.
Sonification of the above. Frequency calculated using (2500-100)/1.31(Im z)+2500 for each root, z.
Plot of all roots of all polynomials of degree 16 with terms of the form ±(m+1)2zm
Sonification of the above. Frequency calculated using (2500-100)/1.22(Im z)+2500 for each root, z.
Plot of all roots of all polynomials of degree 16 with terms of the form ± cos m zm
Sonification of the above. Frequency calculated using (2500-100)/1.26(Im z)+2500 for each root, z.
Plot of all roots of all polynomials of degree 16 with terms of the form ± 1/(m+1) zm.
Sonification of the above. Frequency calculated using (2500-100)/1.53(Im z)+2500 for each root, z.
Plot of all roots of all polynomials of degree 16 with terms of the form ± 1/(m+1)2 zm.
Sonification of the above. Frequency calculated using (2500-100)/1.68(Im z)+2500 for each root, z.
Plot of all roots of all polynomials of degree 16 with terms of the form ± m! zm.
Sonification of the above. Frequency calculated using (2500-100)/0.15 (Im z)+2500 for each root, z. This a short sound piece since the roots are all quite small compared to the roots of the polynomials above.
Plot of all roots of all polynomials of degree 16 with terms of the form ±(9-|8-m|)zm (i.e., plus or minuses applied to each term of the polynomial with coefficient vector (1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1)).
Sonification of the above. Frequency calculated using (2500-100)/2.1 (Im z)+2500 for each root, z.
Plot of all roots of all polynomials of degree 16 with terms of the form ±(1+|8-m|)zm (i.e., plus or minuses applied to each term of the polynomial with coefficient vector (9,8,7,6,5,4,3,2,1,2,3,4,5,6,7,8,9)).
Sonification of the above. Frequency calculated using (2500-100)/1.3 (Im z)+2500 for each root, z.