A Tonnetz Model for pentachords

This article deals with the construction of surfaces that are suitable for representing pentachords or 5-pitch segments that are in the same $T/I$ class. It is a generalization of the well known \"Ottingen-Riemann torus for triads of neo-Riemannian theories. Two pentachords are near if they differ by a particular set of contextual inversions and the whole contextual group of inversions produces a Tiling (Tessellation) by pentagons on the surfaces. A description of the surfaces as coverings of a particular Tiling is given in the twelve-tone enharmonic scale case.Comment: 27 pages, 12 figure

Reconciling the statistics of spectral reflectance and colour

The spectral reflectance function of a surface specifies the fraction of the illumination reflected by it at each wavelength. Jointly with the illumination spectral density, this function determines the apparent colour of the surface. Models for the distribution of spectral reflectance functions in the natural environment are considered. The realism of the models is assessed in terms of the individual reflectance functions they generate, and in terms of the overall distribution of colours which they give rise to. Both realism assessments are made in comparison to empirical datasets. Previously described models (PCA- and fourier-based) of reflectance function statistics are evaluated, as are improved versions; and also a novel model, which synthesizes reflectance functions as a sum of sigmoid functions. Key model features for realism are identified. The new sigmoid-sum model is shown to be the most realistic, generating reflectance functions that are hard to distinguish from real ones, and accounting for the majority of colours found in natural images with the exception of an abundance of vegetation green and sky blue