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

Author:

Amiot Emmanuel1,Baroin Gilles2

Affiliation:

1. Perpignan University

2. Université de Toulouse, France

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.

Publisher

Society for Music Theory

Subject

Music

Reference13 articles.

1. Baixas, Guillaume. 2012. “Etude des isométries du Model-Planet, un modèle de représentation des notes de musique.” Master’s thesis, Montpellier 2 University.

2. Baroin, Gilles. 2011a. “The Planet-4D Model: An Original Hypersymmetric Music Space Based on Graph Theory.” InMCM 2011: Proceedings of the 3rd International Conference on Mathematics and Computation in Music, 326–29. Springer.

3. Baroin, Gilles. 2011b. “Applications de la théorie des graphes des objets musicaux. Modélisations, visualisations en hyperespace.” PhD diss., University of Toulouse le Mirail.

4. Cohn, Richard. 1991. “Properties and Generability of Transpositionally Invariant Sets.”Journal of Music Theory35 (1–2): 1–32.

5. Cohn, Richard. 2012.Audacious Euphony: Chromatic Harmony and the Triad's Second Nature. Oxford University Press.

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