Colloquium on Mathemathical Music Theory
نویسندگان
چکیده
The article studies the topos Sets of actions of an 8-element monoid T on sets. It is called the triadic topos as T is isomorphic to the monoid of affine transformations of the twelve tone system Z12, leaving a given major or minor triad invariant. The subobject classifier Ω of this topos and its Lawvere-TierneyTopologies j are calculated. We explore the characteristic functions of the restricted T-action on triads as subobjects of the corresponding full affine T-action on Z12. The ’upgrade’ operations on such triad-actions which are induced by the Lawvere-Tierney-Topologies relate the triads to actions on larger tone sets, such as the modal mixture, hexatonic, and octatonic systems. Further we calculate the minimal j-dense chords for three interesting cases: jP, jR, jL, which are related to Riemannian and Neo-Riemannian concepts.
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