Supervisors info:
Λεώνη Ευαγγελάτου-Δάλλα Καθηγήτρια (Επιβλέπων), Διονύσης Λάππας Καθηγητής, Μιχάλης Δρακόπουλος Καθηγητής
Summary:
The present thesis was created in order to obtain a Master's diploma in Applied
Mathematics, for the Mathematics Department, University of Athens. Its goal is
to explore the relation between Music and Mathematics through the scope of
Fractal Geometry and the Box counting dimension. The material that was used,
was basically books and articles written by musicians that had basic knowledge
of academic mathematics. The guidebook was the Fractals in Music-Introductory
Mathematics in Musical Analysis, 2nd edition, written by Charles Madden.
The thesis is divided in two chapters. The 1st chapter consists of three
topics. The first two are very crucial for the understanding of the content and
they include very useful information about the characteristics of sound and the
musical definitions. The last topic analyzes the relation between Mathematics
and Music through history, focusing mainly in the 16th-17th A.C.
The 2nd chapter is divided in three parts, as well. In the first part (units
2.1, 2.2, 2.3, 2.4) the writer makes an introduction into 1/f^b kinds of noise
as a subject of scientific research. Subsequently, making use of the different
kinds of noise and many other musical compositions, the corelation between the
pitch of one note and the pitch of the next one is being investigated, using
scatter plots (through Excel). In the main body of the thesis (units 2.5, 2.6)
diagrams that match the pitch of a note with the time that is presented in
several compositions are taking place. The purpose is to count the box counting
dimension of each graph, using specially formed grids that are supported
through the Excel chart program. With this procedure every musical composition
has a unique fractal box dimension. The range of the musical species that are
used is quite wide. In the last part of the chapter the scientific article
Fractal Geometry of music is presented. It is written by Andreas and Kenneth
Hsu and it is one of their three articles that aroused the interest of other
academics to make further research. It is also source of inspiration for the
title of the present thesis.
In the appendix the writer presents in details and in strictly mathematical
manner the meaning of the fractal dimension.
Keywords:
Fractal Geometry, Geometry, Music and Mathematics, Fractal Box count dimension
