Tonal Composition in Multidimensional Virtual Realms

Abstract

The increasing prevalence of touchscreen mobile devices and multiplayer online worlds allows for a new theory of tonal composition in which voice-leading graphs are mapped onto the two- and three-dimensional physical spaces suggested by virtual realms. A single musical piece composed in this manner can be heard in countless different ways based on each listener's path of movement within the virtual realm. Multidimensional harmonic progressions involve temporal sequences not of chords but of mappings, of which the simplest example is any bijection. However, the palette of possible mappings grows richer as the order and dimensionality of voice-leading graphs increase relative to the virtual realm, granting more options for the former to be reduced and "flattened" onto the latter. For example, graphs of equal cardinality may intersect and have their voice leadings combined to include additional chords such as diminished triads (which lie outside the hexatonic cube) and major sevenths (which lie outside the octatonic tesseract). The additional temporal dimension in the notation of such progressions underscores the same issues that arise with the addition of each new spatial dimension. This fluid interchangeability between time and space may help point the way for expansion of this theory into extra dimensions in both voice-leading graphs and virtual realms.