This paper aims to review and to present improvements for the triadic networks explored in 2003 Steven Scott Baker’s thesis called Neo Riemannian Transformations and Prolongational Structures in Wagner’s Parsifal. The improvements appear as a possibility to include the four common-practice triad types – major, minor, diminished and augmented – into a single transformation network.
The well-known Interval Invariance concept is one of the main assumptions of a more orthodox Neo-Riemannian theory that indexes the P, L and R transformations in Brian Hyer’s Tonnetz. On the other hand, the Interval Invariance and Displacement Class concepts allowed Baker to create the “*R” (Fuzzy R) and the “–L” functions that transcends the Hyer’s Tonnetz functions as well as the P1, P2, L1 and L2 functions from Douthett and Steinbach’s Tower Torus. The “*R” function transforms the C into a C+ or a Cm into a B+. The “-L” transforms a C into a C#dim or a Cm into a Cdim.
By including the “-L” function into his innovative Octatonic Propeller Graph, a model that combines three of Douthett and Steinbach’s Octacycles with one of their Hexacycles, Baker enabled the insertion of one diminished triad between each propeller branch. Notwithstanding, the inclusion of the “*R” function generated nothing but a plain graphic version of the Douthett and Steinbach’s Cube Dance.
Based on the combination of “P”, “R”, “-L”, and “*R” functions, we generated two improved graph-models that describe the interaction between the four common-practice triads which we call Generalized Transformation Graphs.