Ternary parallelogram scales are MOS substitution: Difference between revisions

Since the two non-axial steps differ by ''n'''''w''', to identify the two we wrap ℤ² with the vector ''n'''''w''', identifying lattice points separated by ''n'''''w'''. This makes the space ℤ × ℤ/''n''ℤ. Now the image of <math>\pi(\mathcal{I}_w)</math> under this wrap is of the form [0 : ''m''] × ℤ/''n''ℤ. Note that ''m'' is also a generator of ℤ/''n''ℤ, establishing that the period of the template word is ''m''. Now do another wrap, identifying ''m'' with 0, and and we're left with [0 : ''m''] in ℤ. We're now in period-equivalent pitch-class space with a generator chain. Thus we have a MOS with period ''m''.