/sci/ - Science & Math

The four interpretations of a major or minor scale, starting with the singletons, up to the tetradic interpretation for jazz harmony. We have successive refinement morphisms, and associated maps between nerves (see section 14.4 for this association). Every interpretation uniquely maps to the ‘uninterpreted’ local scale composition shown to the right. Observe that the tetradic interpretation has a nerve which is a full torus, in fact a union of tetrahedra, the intersection configuration around any fixed tone. The harmonic strip of the triadic interpretation is embedded in the tetradic torus as a strip which is ‘entwined’ along the torus’ interior circle. So the absence of orientation of the harmonic strip vanishes on the tetradic jazz harmony.