Below are actually just some parts of my thesis that I
translated into english (the paragraphs marked with a * are
unfinished or missing). I mean this merely to be an introduction
into the subject (for current academic research I recommend the
links section); I did not focus on an
extremely strict mathematical notation (this made it possible to
write most of it in plain text format to maximize compatibility),
and I also left out some of the proofs. (The original, written in
german, includes both strict notation and proofs; at the moment,
however, it is not presentable because some of the formatting got
lost in the conversion from Mac to DOS format.)
If you want just a quick overview, I recommend reading the abstract, the introduction and the application examples part.
- Introduction of the approach and the mathematical notation used (ZxZ, Z12 x Zn, affine transformations).
- Classification, part 1
- Notes: non-reversible transformations, basic properties of ZxZ and Z12 x Zn, classification of motifs with 2 elements.
- Classification, part 2
Classification of motifs with 2 elements: Invertible expansion
transformations, non-invertible transformations, distribution of
Invariants of the isomorphy class of a motif: (1) Recursive classification, (2) Volume.
Classification of motifs in (Z12)^2 with 3 elements.
- Classification, part 3
- List of the isomorphy classes of motifs in (Z12)^2 with 3 elements.
- Mathematics, part 1
- Some basic properties of finite abelian groups and their automorphisms. Example: Z7 x Zn.
- Mathematics, part 2
- Invariants (3): Range. Application to motifs with 2 elements. Splitting into p-components: the transformation.
- Classification, part 4
Towards the classification of motifs with 4 elements. Application of recursive classification, volume and range.
Invariants (4): Refining the subset classification.
- Mathematics, part 3
Module-theoretic approach for the complete classification of arbitrary motifs.
Complete classification of arbitrary motifs in (Z12)^2.
- Classification, part 5
- List of the isomorphy classes of motifs in (Z12)^2 with 4 elements.
- *Algorithmical stuff
*Algorithm to solve linear equations in modular arithmetics.
*Algorithm for complete classification of arbitrary motifs in (Z12)^2.
- Application examples
- Melody analysis: class statistics. Algorithmic composition.
- Classification of motifs in different measures (added 2003-04-05)
- Some investigations I did after my diploma, about how isomorphy classification is affected by the measure a motif is noted in. In german.
- Classification in Z10 x Z10 and Z5 x Z5 (added 2005-07-08)
- The same procudure as in my thesis applied to basic spaces based on the number five, including a composition of mine in Z5 x Z5.