One of my summer projects is to read the entire books I started during last semester and to apply different transformational theories to some pieces.
Steven Rings affirms that transformational theory is a specialist subdiscipline and that very few music scholars pursue a research in this area probably because the mathematical approach has inhibited a broader participation and interest. I don’t think that this is the main reason: the main reason is that math is very simple if it is explained well and it has logical implications. And music theorists may eventually be able to apply mathematical concepts to music, but they are unable to explain in simple words how transformations should work.
Transformational theories try to draw a model for relationships and dynamics in a piece of music, which will be explored from two different perspectives: intervallic and transformational. In both perspectives the elements are considered as members of a group,while the musical entities are members of a set.
The intervallic structure is modeled through the GIS (Generalize Interval System) while the transformational relationships are modeled through a network. David Lewin asserts that the GIS can be converted into a network, Steven Rings thinks that there are GISes that cannot be converted into networks of transformations.