In: Music Education ISBN: 978-1-60876-655-0 Editor: Joao Hermida and Mariana Ferrero © 2009 Nova Science Publishers, Inc. Chapter 6 LINKING PHYSICAL SPACE WITH THE RIEMANN TONNETZ FOR EXPLORATION OF WESTERN TONALITY Reinhold Behringer and John Elliott Leeds Metropolitan University, Leeds, LS6 3QS. ABSTRACT Since about 300 years, Western music is built around the equally-tempered 12-tone scale, which allows transposition into any key. During this time, various efforts have been made by music theorists to create spatial representations of this note space, in which the distance between a pair of notes represents their degree of consonance. The “Tonnetz” by Hugo Riemann (1880) represents the music notes in a plane, as a set of parallel lines created from the circumference of the “circle of fifths”. In this arrangement, each note is surrounded by 6 other notes to which it is related in “consonance”. This Tonnetz paradigm has in recent years received renewed attention, as a tool for automatic analysis of musical tonality structure. There has been, however, no attempt to quantify the actual metric of this Tonnetz, in particular to determine the distance of these lines-of- fifths from each other. In this paper, the Riemannian Tonnetz has been investigated, and numerical parameters of its structure have been determined, based on geometrical constraints and conditions based on musical principles. In particular, the metric has resulted in the following values for the “consonance” of note intervals, which correspond directly to the distance between the notes in the Tonnetz: fifth = 1, fourth = 1, major/minor third = , second = 2, minor second = , minor fifth = 2 x . The remaining interval values can be obtained from these intervals, using octave-transposition. This particular note arrangement has also consequences for the harmony of chords. If the note locations are amended with a circular region in which each tone is sounding, then the overlap of these regions would create chords. If the radius of these note regions is chosen to be 1, then only “consonant” chords will appear: duochords which are “allowed” by the laws of tonality, and trichords which contain only consonant notes. This quantification allows to link the Tonnetz to actual physical space and create a new way of teaching music students about western tonality morphology: a physical area