1. Training Phase Modeling Jazz Artist Similarities Mathematically Andres Calderon Jaramillo - Mentor: Dr. Larry Lucas Department of Mathematics and Statistics, University of Central Oklahoma This project attempts to quantitatively model similarities among jazz piano artists by building a relatively simple probabilistic system. We limited our study to monophonic melodies which we assume retain much of the essence of an artist’s style. Our current model makes use of Markov chains to capture the substance and structure of a musical piece. At the initial stage, the system extracts information about attributes such as the transition of pitches, note durations, and phrase lengths. At its later stages, the model uses logistic regression to quantitatively compare a piece by one artist to the style of another artist. INTRODUCTION System Markov Chain Generator Probability Estimator Logistic Regression Influencer Melodies Influenced Melody Duration Chain Pitch Chain Phrase Length Chain Similarity Measurement Inputs Output The diagram above shows a high-level view of our model’s stages: 1.Markov Chain Generator: represents the transitions of pitches, note durations, and phrase lengths as Markov chains. The transition matrices of these chains represent the influencer’s structure. 2.Probability Estimator: for each attribute (pitch, note duration, and phrase length), estimates the probability that the influenced artist’s melody is produced given the Markov chains from the Markov Chain Generator. 3.Logistic Regression: uses the estimates from the Probability Estimator to produce an overall estimate of the probability that the influencer’s structure would have produced the influenced artist’s melody. The Training Phase uses known data about artists’ relationships in order to estimate the coefficients in the regression. PROPOSED MODEL We are currently implementing the model as a computer program. The results of the experiments will be compared to existing information on the relationship among artists in order to measure the performance of our system. EXPERIMENTATION Galovich, S. (2006). Doing mathematics: An introduction to proofs and problem solving. (2nd ed., p. ix). Belmont, CA: Brooks Cole. REFERENCES This project is based on work supported by a grant from the Office of Research & Grants, University of Central Oklahoma In his book, Steven Galovich proposed that “mathematics is the general study (or science) of form, pattern, and structure” (Galovich, 2006). If we accept this definition, it should come as no surprise that music is the object of study of many mathematicians. After all, humans seem to perceive music, not as a purely chaotic process, but as an outcome of human creativity with varying degrees of structure and symmetry. Gaining a deeper understanding of such structures provides a rationale for this project. Our model assumes that a melody is a surface of an artist’s structure. The structure represents the artist’s creativity and patterns of thought that go into creating a piece. The inputs to our model are a set of melodies composed by an influencer artist and a melody composed by an influenced artist. The output is a measurement of the similarity between the artists interpreted as the estimated probability that the influencer would have produced the influenced artist’s melody. BACKGROUND