Tonnetze and Tori for the 19-, 31-, and 53-Tone Equal Temperaments

The Tonnetz is a graphical representation of consonant intervals and chords, together with their corresponding parsimonious transformations, which are the basis of neo-Riemannian theory. It has proved to be useful for analyzing some chromatic passages of nineteenth century, as well as for composition. Furthermore, it has led to several generalizations and variants, which have been applied to different musical styles. In most cases, it implements the ordinary 12-tone equal temperament or 12-TET, which has a finite number of notes and allows to transform it into a torus. However, regarding consonances, they are only exactly matched in just intonation, which has infinite notes. Therefore, other finite tuning systems were developed to better approximate it, among them 19-, 31-, and 53-TET, which in turn gave rise to related new instruments. In this paper, the Tonnetze and tori for these tuning systems are developed, which are intended to be helpful when using them.