In my master thesis I start with the history of mathematics in music. First, I introduce the harmonic relationships that were already known by the Ancient Greeks.
Then, I describe Fibonacci numbers, Fibonacci sequence and the golden ratio and then I present the use of Fibonacci sequence with a piece of music. The golden ratio was also used in violin construction by Stradivarius. After that, I define transpositions and inversions and present their usage with the Bach's fugue.
Then, I connect mathematics with dancing. First, I mention the contra dancing for which the best mathematical model is the dihedral group, and later, I continue with folk dances which can be mathematically described by using permutations, fixed points and compatible sequences.
Next, I continue with the description of dihedral groups and pitch classes. Then, I define the group acting on sets and major and minor triads.
After that, I continue with PLR-transformations and precisely define their effects and usage in graphs, torus and in Ludwig Van Beethoven's Ninth Simphony. Then I prove that the groups T/I and PLR are dual.
Finally, I name some examples of usage of tone-network and PLR-transformations.
|
