| Description |
This article develops a theory of scale structure based in continuous pitch space. It launches from an investigation of the properties of "modal brightness," a widespread intuition that scales (or modes) with relatively high pitches are qualitatively "bright." The structural features of the diatonic scale (e.g., Myhill's property) mean that its modes demonstrate this phenomenon in a straightforward way, but generalizing brightness to the modes of other scales reveals unforeseen complexity. To explain this behavior, as well as to generalize other scalar properties, the article develops a geometrical model of scalar structures. In this model, the continuous space in which the scales exist (as ordered sets of pitches) is divided by an arrangement of hyperplanes. These hyperplanes characterize small qualitative aspects of a scale's structure, such as whether its first step is larger than its second step. Collectively, the hyperplanes partition the space into discrete regions of qualitatively different scalar structures that the article calls "colors." The article offers computational tools for exploring those spaces, especially for cardinalities 2 through 7, and presents examples of the new insights they offer into the structure of both familiar and novel scales. |
