| Description |
This larger work is a one-movement sonata, while at the same time encompassing 4 movements within its structure. As a double-function form, 'Echoes' also employs contrapuntal techniques, bridging the development and recapitulation with a four-voice fugue. The compositional techniques used depict several abstract mathematical concepts.
Tonal stability is increased and decreased throughout, particularly in the recitative sections. This changing of tonal stability depicts the intersecting of dimensions. A being existing in a 2-dimensional world, such as an insect on a flat surface, may occasionally get a glimse of 3 dimensions when a 3-dimensional being or object passes through the second dimension. This glimpse, however, on the part of the 2-dimensional being, is incomplete and does not accurately represent a world of a higher dimension. Similarly, at the intersection of our world with an even higher spatial dimension, we would only catch incomplete glimpses of this higher dimension.
These polytonal and structural elements depict another concept. While a square is 2-dimensional, and a cube is 3-dimensional, it is possible for an object to exist in between 2 and 3 dimensions (by dividing logarithms). This idea of a fractional dimension is explored as thematic material is pulled between tonal centers, but never truly settling. |