Old and New Isometries between Pc sets in the Planet-4D Model

Abstract

The Planet-4D model introduced by Baroin 2011b is a richer model of pitch-class relationships than the standard cyclic model of the clock-face diagram: instead of the distribution of twelve points on a circle in a 2D plane, the Planet-4D model places the twelve pcs on a 4D hypersphere. Beyond the usual T/I group of symmetries, which is still featured in this new musical space, the additional dimensions yield other isometries (most notably now including the M5/M7 operations), which appear here in animated visualizations of musical pieces. The present article elucidates how these isometries are in fact well-known symmetries that do not preserve shape in most previous models with fewer dimensions. The hyper-spherical environment grants each symbol an equivalent physical position, and therefore allows for more symmetries than any 3D model. This article also shows that the model provides a visually intuitive geometric setting of pcs, in accordance with our perception of 3D Euclidean space, where transpositions by major and minor third are elementary rotations.