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\title{TJHSST Senior Research Project\\Computer Music Analysis\\Computer Systems Lab, 2007-2008}
\author{Josiah Boning}
\date{April 4, 2008}
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\begin{abstract}
Although music is one of the most universal aspects of human culture, it is very 
difficult to define. Most definitions of music have been dependent on attributes 
such as rhythm, melody, and harmony, which are extremely subjective, so the 
ability to identify music has been limited to humans. This project aims to 
better define ``music'' by applying machine learning techniques to music 
analysis and recognition, allowing computers to autonomously identify whether a 
given audio sample is musical in nature.
\par
{\bf Keywords:} music analysis, machine learning
\end{abstract}


\section{Introduction}

This project aims to unify machine learning and signal processing techniques in 
a program that will learn to distinguish between musical and non-musical audio 
recordings. The end product will be a program that will learn to identify audio 
samples of musical nature. The program will be ``trained'' on a number of audio 
samples typically agreed upon as ``music'' as well as a number of non-musical 
samples.  After sufficient training, the program will be able to distinguish 
between musical and non-musical audio recordings.


\section{Background}

Computers have already been used to perform analysis of music. In 1999, 
Bigerelle and Iost determined that different genres of music could be 
distinguished by fractal dimension, and in 2004, Basili et al. showed that 
machine learning techniques could successfully indentify musical 
genres\cite{bigerelle}\cite{basili}. Other research has attempted to deconstruct 
music in terms of rhythmic and melodic patterns, and even looked at writing 
software to generate music conforming to such patterns\cite{leach}.  However, as 
Bigerelle and Iost point out, each instrument has a different sound quality, and 
composers write music with these timbral differences in mind.  Simply analyzing 
the notes on sheet music precludes the use of these differences in the analysis.  
Audio recordings, in contrast, allow analysis of exactly what the composer 
intended his audience to hear.


\section{Program Design}

\subsection{Spectral Decomposition}
The program performs a Fourier transform on the audio data to obtain a frequency 
spectrum of the audio file. This data will then be processed in the neural 
network. The program uses the open source FFTW library to perform the transform.

\subsection{Fractal Dimension}
The program calculates the fractal dimension of the input audio data using two 
methods, the Variation method and the ANAM method.

The Variation method:
\begin{equation}
\Delta = \lim_{\tau \to 0} \left( 2 - \frac{\log \left( \frac{1}{b-a} \int_a^b \left| {max(f(t))}_{ \left| x-t \right| < \tau} - {min(f(t))}_{ \left| x-t \right| < \tau} \right| dx \right) }{\log \tau} \right)
\end{equation}

The ANAM method:
\begin{equation}
\Delta = \lim_{\tau \to 0} \left( 2 - \frac{\log \left( \frac{1}{b-a} \int_{x=a}^{x=b} \left[ \frac{1}{\tau^2} \int_{t_1=0}^{\tau} \int_{t_2=0}^{\tau} \left| f(x+t_1) - f(x-t_2) \right|^\alpha \right]^{1/\alpha} dx \right) }{\log \tau} \right)
\end{equation}

\subsubsection{Accuracy}
Because the Variation and ANAM methods are being applied to discrete audio data, the accuracy of the numerical integration is limited by the sample rate. To work around this difficulty, cubic splines will be used to interpolate between audio data sample points and increase the accuracy of the calculations.

\subsection{Machine Learning}
The program uses a feedforward perceptron topology with a single hidden layer. The perceptron will be trained on audio samples that clearly are or are not music. The network will probably use a Back Propagation learning algorithm.

\subsection{Program Architecture}
The program is written in C. The source is organized into files by subject: the input functions, fourier transformation bindings, fractal dimension calculations, and neural network functions each have their own file. The main file is simply a driver that calls the functions in the other files.

\section{Conclusions}

The program can currently read audio data from WAV files. It can perform spectral analysis to determine the frequency makeup of the audio. It can also perfom fractal dimension calculations, although the numeric integration methods are currently somewhat inaccurate. The integration methods will be improved; instead of using Riemann sums, the program will model small sections of the data with cubic splines and perform the integrations mathematically. The program also currently creates rudimentary neural networks, but does not have a training algorithm in place.

\begin{thebibliography}{99}
\bibitem{basili} Basili, Roberto, Alfredo Serafini, and Armando Stellato. 2004.
	"Classification of Musical Genre: A Machine Learning Approach."
	Presented at the 5th International Conference on Music Information Retrieval.
\bibitem{bigerelle} Bigerelle, M., and A. Iost. 2000.
	"Fractal Dimension and Classification of Music."
	{\it Chaos, Solitons \& Fractals.} 11(14):2179-92.
\bibitem{leach} Leach, Jeremy, and John Fitch. 1995.
	"Nature, Music, and Algorithmic Composition."
	{\it Computer Music Journal.} 19(2):22-23.
\end{thebibliography}

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