BibTeX
@MISC{Noll05thetopos,
author = {Thomas Noll},
title = {The Topos of Triads},
year = {2005}
}
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Abstract
The article studies the topos Sets^T 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-Tierney-Topologies 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.
Keyphrases
8-element monoid article study subobject classifier modal mixture tone set twelve tone system z12 neo-riemannian concept characteristic function upgrade operation octatonic system corresponding full affine t-action affine transformation topos set interesting case triadic topos minimal j-dense chord minor triad invariant lawvere-tierney-topologies relate
