The goal of the present work is to improve the analysis and synthesis of
musical audio signals by the application of tonal pitch spaces.
The first part written by Gabriel Gatzsche consists of the Chapters 2 to 6.
It discusses the mathematic-geometrical description of tonality on several
hierarchical levels based on Fred Lerdahl’s Tonal Pitch Space, David
Gatzsche’s Cadence Circle and Elaine Chew’s Spiral Array (calculation of
geometric centroids within tonal pitch spaces). Using two formulas, the
symmetry model generator formula and the SYM operator, it is possible 1.)
to describe the emergence of the most important levels of western tonality
out of an array of fifths and 2.) to generate several key related models
which are centered to the corresponding symmetry tone. With that steps it
becomes possible to link several existing pitch spaces into a unified
framework called symmetry model. To enable also the analysis of real music
signals based on pitch spaces the centroid vector within the circular pitch
space is introduced. This feature vector is a low dimensional
representation of important tonal properties of musical audio signals. Such
properties are functional relationships, the mode, tension and relaxation
or harmonic ambiguities. Furthermore the pitch class - pitch height space
is introduced. This space assigns geometric positions to different octaves
of a given pitch class such that ”well sounding” chords can be created by
choosing a simple shaped region of the space. By transforming (rotating,
translating, scaling etc.) such a region also well sounding chord
transitions are generated. This leads to the development of a new musical
instrument, called HarmonyPad. The HarmonyPad allows a musician to create
music by interacting with pitch spaces directly. 
Within the second part of the dissertation consisting of the Chapters 7 to
12 Markus Mehnert investigates the applicability of the symmetry model to
concrete problems of music information retrieval (MIR) particularly chord
and key recognition. The state of the art in the field of key recognition
focuses on the estimation of major and minor keys. Within that work a new
symmetry model based algorithm is presented which exceeds the results of
current algorithms clearly. Additionally a new approach is proposed which
extends key recognition to the estimation of the most often used six church
modes. The latter represent the character of a musical piece in a better
way then the standard modes ”major” and ”minor” do. Furthermore a new
benchmark is introduced which allows the comparison of the current approach
with future algorithms. A new machine learning algorithm (HMM/KNN) is
proposed. The new algorithm combines the approaches Hidden Markov Models
and k Nearest Neighbours. In the field of chord recognition the new
approach achieves better results then all of the previous algorithms. It is
shown that the symmetry model feature vector leads to significant better
chord recognition results then the chroma vector which represents the state
of the art.