The Tonnetz Invented by Leonard Euler, popularized during the XXth century by fellow Mathemusicians, this is the usual decomposition of the 12 tones scale into 4×3 = 3×4.
The Tonnetz Invented by Leonard Euler, popularized during the XXth century by fellow Mathemusicians, this is the usual decomposition of the 12 tones scale into 4×3 = 3×4. This is known as the Sylow decomposition of the cyclic group Z_12= Z_3 ⨂ Z_4 Each note having therefore two coordinates, we associate it with a two dimensional (Shape and Color) graphical symbol. That is equivalent to labelling the graph with a Gaussian integer, z=a+ib,a∈[0,1],a∈[0,3] ,whose real part is the position along Z_3 , the imaginary one , along Z_4 [5]. Musically speaking, one axis progresses by major 3rd (3 ½ tones), the other by minor 3rd (2 tones).As it shall be, but unfortunately not feasible IRL, the Tonnetz is designed on an endless plane, we simulate its boundlessness in VR with the help of hidden teleportations and camera effects. The user can therefore explore this infinite space, observe the organization, and travel according to his imagination. The Tonnetz is made of tones, each note symbol has an associated sound object. Since the Tonnetz of Chords (Triads = perfect Maj or Min), is the dual space of the Tonnetz of Notes [19][11], by adjusting the hearing range in the software, the user will be able to hear multiple tones at once. If situated i.e., at the barycenter of [do,mi,sol], he will perceive the three notes at the same time : a CMaj chord.
