Brief Sketch of the Model[1] In the classic 2D circular model of the 12 pcs (the well-known Krenek diagram or clock-face diagram), proximity appears to be restricted to chromatically adjacent points, although other topologies might prove desirable. The Planet-4D model (Baroin 2011b) remedies this restriction by adding two more dimensions to the ambient space, projected here as 3D animations. The 4D model provides a visually intuitive geometric setting of pitch classes, in accordance with our perception of the Euclidean (3D) world, where major-and minor-third transpositions are elementary rotations. The 4D model also allows for all the twelve-tone transformations to be represented as isometries (1) -not only the familiar group of transformations that are generated by transposition and inversion, but also other operations that do not preserve distances under earlier models with fewer dimensions. The aim of this paper is to present the additional symmetries that appear in this model as isometries for Euclidean distance, which is the most intuitive of all. (2) A mathematical description of the model is given in section 1; readers not well versed in mathematics are invited to skip ahead to section 2. Before reading further, however, all readers are invited to look at Video 5 in order to gain a sense of the effectiveness of the model. Section 2 elucidates the elementary motions on the hypersphere, first explaining what is shown in the movies and then moving on to describe the old and new isometries of the Planet-4D model. Twenty-four isometries are identified as the elements of the usual group T/I of transpositions and inversions; twenty-four other isometries are identified with a well-known algebraic group, though one that has never been identified with Euclidean isometries in a music-theoretic setting. In music theory, these additional twenty-four isometries are related by the M5 or M7 operations (multiplication of a pc by 5 or 7, modulo 12), an operation that maps a cycle of fourths or fifths onto the chromatic scale and vice versa. On the 1D circle, the M5/M7 operations do not preserve distance and are thus not isometric, but, surprisingly, in four dimensions they can be represented as isometries. The Appendix collates the proofs of the theorems presented in the main text.[2] In this section we will use the traditional algebraic decomposition of as the starting point for an immersion of the 12 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 th...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.