Gustavo Díaz Jerez | Fractal Sounds, Vol. 1

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Avant Garde: Computer Music Moods: Type: Experimental
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Fractal Sounds, Vol. 1

by Gustavo Díaz Jerez

A musical journey through mathematics. This album features many sonifications (aural representations) of a wide range of mathematical processes, from number theory to fractals. Fascinating for the mind, fascinating for the ear.
Genre: Avant Garde: Computer Music
Release Date: 

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1. Logistic Map
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4:36 $0.99
2. Henon
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3. Gingerbreadman
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1:40 $0.99
4. Gingerbreadman Wave
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5. Martin
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6. Martin Wave
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7. Hopalong
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8. Hopalong Wave
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9. Mira
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10. Lorenz
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11. Lorenz Wave
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12. Rossler
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13. Rossler Wave
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14. Morse-Thue (Base 2)
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15. Morse-Thue (Base 2, 8khz)
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3:35 $0.99
16. Morse-Thue (Base 10)
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17. Morse-Thue Primes
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18. Morse-Thue Primes (8khz)
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19. 3x+1 Wasps
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20. 3x+1 Wind
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21. 3x+1 Steps (Fire)
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22. White Noise
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23. Pink Noise
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24. Brownian Noise
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ABOUT THIS ALBUM


Album Notes
Track 1 -- Logistic Equation
This is a sonification of the logisitic equation bifurcation diagram [X(n+1) = X(n)*k*(1-X(n))], k = [0, 4]. Ten million data points were calculated with k ranging from 3.1 to 4. The data was then read directly as a wave form (each data point being a sample), at a sampling rate of 44.1 KHz. The wave data was then transposed 3 octaves down, in order to hear more clearly the structure. Period doubling starts at k = 3.1 and continues until k = 3.57. It is clearly heard as the addition of several frequencies one octave appart. From the critical value of k=3.57 the signal is most of the time chaotic (noisy), although there are regions of order (settlements) among this sea of chaos. For the final value of k=4, the signal is truly chaotic and identical to white noise.

Track 2 -- Henon
Electronic mapping of the first 20000 iterations of Henon two-dimensional map: [x(n+1) = 1 + y(n) - a*x(n)*x(n); y(n+1) = b*x(n)]. The X coordinate was mapped to an absolute frequency scale ranging from 110 to 880 Hz. The Y coordinate was mapped to a duration scale ranging from 1 to 10 milliseconds. Frequencies are realized as pure sine waves.

Track 3 -- Gingerbread Man
Electronic mapping of the first 20000 iterations of Gingerbread Man two-dimensional map: [X(n+1) = 1 - Y(n) + |X(n)|; Y(n+1) = X(n)]. The X coordinate was mapped to an absolute frequency scale ranging from 110 to 880 Hz. The Y coordinate was mapped to a duration scale ranging from 1 to 10 milliseconds. Frequencies are realized as pure sine waves.

Track 4 -- Gingerbread Man Wave
Sonification of the first 5,300,000 iterations of Gingerbread man attractor equations. Only the X coordinate was used, since the Y coordinate yields identical results. Each data point was then renormalized onto the interval [-1, 1] and read at a sampling rate of 44.1 KHz. The resulting wave was lowered 3 octaves in order to appreciate its structure.

Track 5 -- Martin
Electronic mapping of the first 20000 iterations of Martin two-dimensional map: [x(n+1) = y(n) - sin(x(n)); y(n+1) = a - x(n)]. The X coordinate was mapped to an absolute frequency scale ranging from 110 to 880 Hz. The Y coordinate was mapped to a duration scale ranging from 1 to 10 milliseconds. Frequencies are realized as pure sine waves.

Track 6 -- Martin Wave
Sonification of the first 5,300,000 iterations of Martin attractor equations. Only the X coordinate was used, since the Y coordinate yields identical results. Each data point was then renormalized onto the interval [-1, 1] and read at a sampling rate of 44.1 KHz. The resulting wave was lowered 3 octaves in order to appreciate its structure.

Track 7 -- Hopalong
Electronic mapping of the first 40000 iterations of Hopalong two-dimensional map: [x(n+1) = y(n) - sign(x(n))*sqrt(abs(b*x(n)-c)); y(n+1) = a - x(n)]. The X coordinate was mapped to an absolute frequency scale ranging from 110 to 880 Hz. The Y coordinate was mapped to a duration scale ranging from 1 to 10 milliseconds. Frequencies are realized as pure sine waves.

Track 8 -- Hopalong Wave
Sonification of the first 5,300,000 iterations of Hopalong attractor equations. Only the X coordinate was used, since the Y coordinate yields identical results. Each data point was then renormalized onto the interval [-1, 1] and read at a sampling rate of 44.1 KHz. The resulting wave was lowered 3 octaves in order to appreciate its structure.

Track 9 -- Mira
Electronic mapping of the first 25000 iterations of Mira two-dimensional map. The X coordinate was mapped to an absolute frequency scale ranging from 110 to 880 Hz. The Y coordinate was mapped to a duration scale ranging from 1 to 10 milliseconds. Frequencies are realized as pure sine waves.

Track 10 -- Lorenz
Electronic mapping of the first 20000 iterations of Lorenz three-dimensional chaotic attractor: [x(n+1) = x(n) + (-a*x(n)*dt) + (a*y(n)*dt); y(n+1) = y(n) + ( b*x(n)*dt) - (y(n)*dt) - (z(n)*x(n)*dt); z(n+1) = z(n) + (-c*z(n)*dt) + (x(n)*y(n)*dt)]. The X coordinate was mapped to an absolute frequency scale ranging from 110 to 880 Hz. The Y coordinate was mapped to a duration scale ranging from 1 to 10 milliseconds. The Z coordinate was mapped to amplitude, ranging from .25 to 1. The intertwining orbits of this attractor are clearly audible as ascending and descending glissandi. Frequencies are realized as pure sine waves.

Track 11 -- Lorenz Wave
Sonification of the first 5,300,000 iterations of Lorenz attractor equations, for each coordinte X, Y, and Z. Each data point was then renormalized onto the interval [-1, 1] and read at a sampling rate of 44.1 KHz. Coordinte X was assigned to the right channel, Y coordinate to the left. The Z coordinate was mixed to both channels. The result was lowered 3 octaves in order to appreciate its structure.

Track 12 -- Rossler
Electronic mapping of the first 30000 iterations of Rossler three-dimensional chaotic attractor: [x(n+1) = x(n) - y(n)*dt - z(n)*dt; y(n+1) = y(n) + x(n)*dt + a*y(n)*dt; z(n+1) = z(n) + b*dt + x(n)*z(n)*dt - c*z(n)*dt]. The coordinates X, Y, and Z were mapped to three different "voices," representing each dimension. In voice 1, coordinate X was mapped to an absolute frequency scale ranging from 110 to 880 Hz, coordinate Y to a continous duration scale ranging from 1 to 10 milliseconds, and coordinate Z to an amplitude .25 to 1. In voice 2, Y was mapped to frequency, X to duration, and Z to amplitude. Finally, in voice 3, Z was mapped to frequency, Y to duration, and X to amplitude. The chaotic orbits of this attractor are clearly audible as ascending and descending glissandi. Frequencies are realized as pure sine waves.

Track 13 -- Rossler Wave
Sonification of the first 2,650,000 iterations of Rossler attractor equations, for each coordinte X, Y, and Z. Each data point was then renormalized onto the interval [-1, 1] and read at a sampling rate of 44.1 KHz. Coordinte X was assigned to the right channel, Y coordinate to the left. The Z coordinate was mixed to both channels. The result was lowered 3 octaves in order to appreciate its structure.

Track 14 -- Morse-Thue Numbers (Base 2)
Sonification of the natural number sequence. The first 1000 multiples of the first 10000 natural numbers were calculated in sequence, yielding ten million data points: 1,2,3,4,5,6,...1000,2,4,6,8,10,12,14,...,2000,3,6,9,12,15,18,21,...,3000,4,8,12,16,20,24,28,...,5000,... All numbers were then expressed in binary base (base 2): 1,10,11,100,101,110,...,10,100,110,1000,1010,...,11,110,1001,1100,1111,... Finally, the 1s in each number were added, yielding the final sequence: 1,1,2,1,2,2,...,1,1,2,1,2,...,2,2,2,2,4,... This ten-million data set was then normalized to the interval [-1,1] and interpreted as a wave form, at 44.1 KHz sampling rate. The result shows a huge amount of structure. The intensity increases slowly and steadily in dB. There is a constant pedal tone with a frequency of about 43 Hz. There is a signal that increases in frequency continously. There is a periodic pattern of "pitches" in the very high frequency range. At the same time there are ascending and descending "glissandi" throughout, which seem to fall all the way for numbers that are a power of 2.

Track 15 -- Morse-Thue Numbers (Base 2) (8 Khz)
The same as 'Morse-Thue Numbers (Base 2)', but the data was read at a sampling rate of 8 KHz and then resampled at 44.1 KHz. Only the first 1760 natural numbers were used. This is a slower and transposed version of 'Morse-Thue Numbers (Base 2)'. At 8 KHz everything sounds 5.5125 times slower and almost 2.5 octaves lower. The 43 Hz pedal tone is no longer audible. The ever-ascending frequency as well as the up and down glissandi are audible, although at a much more slower pace. The periodic patterns of "pitches" can now be clearly distinguished and appear to be formed by several distinct frequencies, being really "chords", each of then representing the 1000 multiples of each number.

Track 16 -- Morse-Thue Numbers (Base 10)
The same as 'Morse-Thue Numbers (Base 2)', but instead of converting the numbers in the sequence to base 2, they were kept in base 10: 1,2,3,4,5,6,...1000,2,4,6,8,10,12,14,...,2000,3,6,9,12,15,18,21,...,3000,4,8,12,16,20,24,28,...,5000,.... The digits of each number in the sequence were added, yielding the final sequence: 1,2,3,4,5,6,...1,2,4,6,8,1,3,5,...,2,3,6,9,3,6,9,3,...,3,4,8,3,7,2,6,10,...,5,....etc. The same process as in 'Morse-Thue Numbers (Base 2)' was then applied. There are, naturally, many similarities with 'Morse-Thue Numbers (Base 2)': the low pedal tone, the glissandi, the periodic pattern, and the ever-ascending signal are present. However, their periodicity and frequency are not the same. In general, glissandi go up to higher frequencies. The ever-ascending signal starts also at a higher frequency. The periodic pattern of "pitches", on the other hand, seems to be in a lower frequency range.

Track 17 -- Morse-Thue Primes
The same as 'Morse-Thue Numbers (Base 2)', but instead of the natural number sequence, the prime number series was used. The first 1000 multiples of the first 10000 primes were used and applied the same operation as in 'Morse-Thue Numbers (Base 2). The aural result is similar to that of 'Morse-Thue Numbers'. However, careful listening reveals this sonification is much more unpredictable. The patterns of "pitches", which are periodic in 'Morse-Thue Numbers', seem almost random here. Ascending and descending glissandi do not seem to follow any periodic pattern either. Both the 43 Hz pedal tone and the ever-ascending-in-frequency signal are present here.

Track 18 -- Morse-Thue Primes (8 KHz)
The same as 'Morse-Thue Primes', but the data was read at a sampling rate of 8 KHz and then resampled at 44.1 KHz. Only the first 2000 primes were used. This is a slower and transposed version of 'Morse-Thue Primes'. At 8 KHz everything sounds 5.5125 times slower and almost 2.5 octaves lower. The 43 Hz pedal tone is not audible. The ever-ascending frequency as well as the up and down glissandi are audible, although at a much more slower pace. The patterns of "pitches" can now be clearly distinguished and appear to be formed by several distinct frequencies, being really "chords", each of then representing the 1000 multiples of each prime.

Track 19 -- 3x+1 Numbers (Wasps)
Sonification of the '3x+1' problem, also known as the Collatz problem, or hailstone numbers. The controlling function is x = {3x+1 if x = 1 mod 2; x/2 if x = 0 mod 2}, that is, 3x+1 if x is odd; x/2 if x is even. Starting with any number x, the function is iterated until the number 1 is reached, yielding an infinite loop 4,2,1,4,2,1...It is conjectured that all numbers eventually 'fall' to 1, thus entering the 4,2,1 loop. Proving it seems to be extremely difficult. The series of the first 40000 natural numbers were calculated, yielding a four million data set. If any number in the series is greater than 49, its digits are added until the result is no greater than 49. The data set was then normalized in the interval [-1, 1] and read directly at 44.1 KHz. The resulting wave is extremely complex, having a large amount of structure in all frequencies. It resembles the sound one would hear in the heart of a wasp's nest!

Track 20 -- 3x+1 Wind
Electronic mapping of the '3x+1' problem, using the number of steps required for any number x to hit 1 [S(x)]. The following mapping scheme was applied: S(x) was computed for x = [2, 100]. Starting from a fundamental frequency of 300 Hz, the following operation is performed: if S(x) is greater than S(x-1) the frequency is incremented |S(x)-S(x-1)| times in intervals of 2 Hz, with a duration of 45.35 milliseconds per interval. If S(x) is less or equal than S(x-1) the frequency is decremented |S(x)-S(x-1)| times in intervals of 2 Hz, every 45.35 milliseconds. Frequencies are realized as pure sine tones. The result yields ascending and descending 'glissandi' of varying durations, resembling the wind, which is an aural representation of the apparently chaotic behavior of the process.

Track 21 -- 3x+1 Steps (fire)
Another sonification of the '3x+1' problem, but instead of using the iteration values of x, the number of steps (iterations) required to hit 1 were used. The step values of the first 4 million natural numbers were calculated. If any number in the series is greater than 49, its digits are added until the result is no greater than 49. The data set was then normalized in the interval [-1, 1] and read directly as a 44.1 KHz wave file. The result resembles very much the sound of cracking wood in a fire.

Track 22 -- White Noise
White noise created by a 661,500 numerical data set of random values in the interval [-1, 1]. Frequencies are absolutely uncorrelated. White noise has an even distribution of power (in dB). There is as much power between 100 and 200 Hz than between 10000 and 10100 Hz.

Track 23 -- Pink Noise
1/f noise (a.k.a. pink noise) created by a 661,500 numerical data set in the interval [-1, 1] using Voss' algorithm. 1/f noise has an even distribution of power if measured in a logarithmic scale, that is, there is the same amount of power between 100 and 200 Hz than between 10000 and 20000 Hz. The power decreases about 3 dB per octave.

Track 24 -- Brownian Noise
Brownian noise created by a 661,500 numerical data set of random but correlated values in the interval [-1, 1]. Frequencies are highly correlated, resembling a random-walk in three dimensions. The power of Brownian noise decreases about 6 dB per octave.


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