Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Proposition 2 Let $ºA/left,ºA/right,ºA/conj$ denote the sets of orbits of tone perspectives with respect to the group actions $lef t,right$ and $* cons :º A × ºA --> ºA$ . The monoid isomorphism $?3×4 :º A --> ºA3×4$ induces natural bijections

ºA/left ~= ºA /left = {O × O |O (- º A /left ,O (- º A /left }, ºA/right ~= ºA 3× 4/righ3t×4 = {O 3× O 4|O3 (- º A 3/righ3t ,4O (- º A4 /rig4ht }, ºA/conj ~ ºA 3× 4/conj 3×4 = {O 3× O 4|O3 (- º A 3/conj 3,O 4 (- ºA 4/conj}4. = 3× 4 3×4 3 4 3 3 3 4 4 4

We conclude this paragraph by a series of tables displaying these orbit structures.

Remark 3 In order to display all the tone perspectives, their various orbits and other structures in a suitable and coherent way, we will be using the following $9 × 16$ -table.

PICT

The displayed vertical and horizontal lines in such a TP-Table may be varied in order to group tone perspectives. In the figure above, the lines indicate, that the table is based on a recursive embedding of a standardized $3 × 4$ -table into itself. The outer large $3 × 4$ - frame groups the 144 tone perspectives into 12 small $3 × 4$ - frames according to their multiplication factors, while they are displayed according to their translations inside of each small frame. The standardized $3× 4$ -table is the following:

0 6 9 3 4 10 1 7 8 2 5 11

TP-Table 1 Orbits of tone perspectives under the action of tone symmetries by concatenation from the left: $ºA/left$