Taxonomy of Clifford Cl3,0 subgroups: Choir and band groups
Abstract
We list the subgroups of the basis set of Cl3,0 and classify them according to three criteria for construction of universal Clifford algebras: (1) each generator squares to +1 or -1, (2) the generators within the group anticommute, and (3) the order of the resulting group is 2n+1, where n is the number of nontrivial generators. Obedient groups we call choirs; disobedient groups, bands. We classify choirs by modes and bands by rhythms, based on canonical equality. Each band generator has a transposition (number of other generators it commutes with). The band's transposition signature is the band's chord. The sum of transpositions divided by twice the number of generator pair combinations is the band's beat. The band's order deviation is the band's disorder. For n less than or equal 3, we show that the Cl3,0 basis set has 21 non-isomorphic subgroups consisting of 9 choirs and 12 bands.