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Mathematical Games: White and brown music, fractal curves and one-over-f fluctuations. Gardner, Martin. Scientific American, 1978.
Garnder spends a great deal of time philosophizing over why fractal curves are interesting and important. When he finishes discussing Plato and program music he begins describing white and brown music, as well as 1/f fluctuations. He describes an algorithm which generates pseudo 1/f fluctuations (which Kirby borrows with only minor alterations and is described below in the summary of his paper). This algorithm comes from Voss, who notes that music generated from the pseudo 1/f algorithm is indistinguishable from real 1/f music.
Evolving electroacoustic music: the application of genetic algorithms to time-domain waveforms. Magnus, Cristyn.
The paper discusses fairly typical approach to genetic algorithms. Magnus focuses on wave forms. She defines gene segments to be the space between zero crossings of the wave form so that “sexual reproduction” and “mutation” techniques can be applied meaningfully. She also chooses a “target” waveform to act as her fitness function. Of course, using a specific target function with a genetic algorithm will cause the sample wave forms to converge to a local minimum where they all approximate the target reasonably well. To solve this problem, Magnus focuses on the parallels to biology. She divides the waveforms of her “population” into separate “locations” where each location has a different target function. However, the individual waveforms are individually allowed to move from location to location and “interbreed” with the waveforms at the new location. Her end result, is to concatenate all the waveforms created during the process into one sound file.
Compositional Chaos and Musical Pleasure. Kirby, Wayne J.
Kirby described different types of fractional noise and discussed their aesthetic value. He finished by describing algorithms he used for generating fractional noise.
First he discribed the differences between different types of fractional noise: White noise (1/f^0) is purely random and has no corollation to the past. Brown noise (1/f^2) on the other hand, has a high correlation to the past and produces variations like a random walk. The last type, 1/f noise is in between the complete randomness of white noise and the high correlation of brown noise.
He then discussed several studies showing that people tend to find 1/f noise the most pleasing of the three types of fractional noise. He also mentioned that while analyzing the power density spectrum of a particular Beethoven score and he found that it displayed 1/f characteristics. He hypothesized that other scores which people found pleasing would also share these characteristics.
Lastly, He discusses Gardener's algorithms and his implementations of fractional music generation. He used a similar approach to Gardner for generating 1/f noise using random number generation and binary numbers.
The process is (roughly) as follows:
- Decide on “how many” random number generators (Garder used 3, Kirby used 5). - Choose how many notes you want in your tune. - List these numbers in binary, starting with zero. The number of digits must be the same as the number of generators chosen in step (1). - “Assign” a generator to each digit. - Use each generator to generate a random number and add them together. Then use a predetermined system to assign the sum to a note frequency. - For each subsequent note, examine the binary number. For each digit that is a one, take the corresponding number generator and generate a new number to replace the old one in the sum. Now assign the new resulting sum to a new note frequency.
Paper summer 1: Alpern, Adam. Techniques for Algorithmic Composition of Music. 1995.
Alpern discusses three approaches to composition.
The first is accomplished using his program Contour, which he wrote in Common Lisp. Contour does a very basic musical analysis of a piece, and then expands the piece based on that analysis. The analysis consists of labeling if notes are rising or falling to determine the “contour” (rising or falling) of a phrase or piece. At this point, Contour uses each note of the original melody as a starting point of a new melody which must be characterized by the same contour as the original. Lastly, Contour appends all the new melody fragments together. Contour works well in combination with using 1/f noise to generate the new melodies. On its own, 1/f noise tends to be to wandering and random to be pleasing, but applying Alpern's Contour program gives more structure to the result.
Alpern also explores using Nonlinear Dynamical Systems and Chaos in composition. He specifically focused on Henon Maps and Logistic Map. To use these equations in composition, they are evaluated iteratively, by calculating a solution and using that solution as input values for the next iteration. Using chaos to compose produces interesting results because as the equations are evaluated iteratively the solutions will be similar but not the same due to the properties of attractors.
The last approach Alpern considers is genetic programming. Alpern identifies the weakest point of genetic composition is the “Computational Critic” (the fitness test). Alpern's solution is to use several independent tests which critique different aspects of melodies and rate them by comparing them to a “case-base” of melodies imputed by the user. The Critic tests “Tonal-novelty-balance,” “Rythmic-novelty-balance,” “Tonal-response-balance,” “Skip-balance,” and “Rhythmic-Coherence.”