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

Figure 13:

Brahms: Piano Concerto No. 1, second movement.

Generally, a group of the form $Z2 wO Sn$ , where there are $n$ copies of the fiber $Z2$ , and the control group $Sn$ acts as a permutation group on those $n$ copies, is called the hyperoctahedral group of degree $n$ . When $n$ is 2, the hyperoctahedral group is $Z2 wO S2$ , which is actually the dihedral group of order 8 (the standard symmetry group of the square). As illustrated in the above example, the top two levels of the 12/8 signature comprise this hyperoctahedral group. In the general case, therefore, the time signature 12/8 is the wreath-subappendment of $Z3$ to the hyperoctahedral group, thus:

Z3 wO Z2Ow Z2.

Now let us give a theory of simultaneous division. For example, some bars can have double and triple division occurring simultaneously:

Simultaneous Division. Simultaneous division of an interval by different numbers $D1$ , $D2$ , ... $Dn$ , will be given by wreath sub-appendment by the direct product $ZD1 × ZD2 × ...× ZDn$ .

This can be illustrated with the second movement of Brahms 1st Piano Concerto,