# Commuting Groups and the Topos of Triads

@inproceedings{Fiore2011CommutingGA, title={Commuting Groups and the Topos of Triads}, author={Thomas M. Fiore and Thomas Noll}, booktitle={MCM}, year={2011} }

The goal of this article is to clarify the relationship between the topos of triads and the neo-Riemannian PLR-group. To do this, we first develop some theory of generalized interval systems: 1) we prove the well known fact that every pair of dual groups is isomorphic to the left and right regular representations of some group (Cayley's Theorem), 2) given a simply transitive group action, we show how to construct the dual group, and 3) given two dual groups, we show how to easily construct sub…

## 23 Citations

### Morphisms of generalized interval systems and PR-groups

- Mathematics
- 2012

We begin the development of a categorical perspective on the theory of generalized interval systems (GISs). Morphisms of GISs allow the analyst to move between multiple interval systems and connect…

### Voicing Transformations and a Linear Representation of Uniform Triadic Transformations

- Mathematics
- 2016

Motivated by analytical methods in mathematical music theory, we determine the structure of the subgroup J of GL(3,Z12) generated by the three voicing reflections. As applications of our Structure…

### Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality

- MathematicsMCM
- 2013

The dual group to the permutation group acting on n-tuples with distinct entries is constructed, and it is proved that theDual group to permutations adjoined with a group G of invertible affine maps ℤ12 → Ω12 is the internal direct product of the dual to permutation and theDual to G.

### Generalized Inversions and the Construction of Musical Group and Groupoid Actions

- Mathematics
- 2014

Transformational music theory is a recent field in music theory which studies the possible transformations between musical objects, such as chords. In the framework of the theory initiated by David…

### Hexatonic Systems and Dual Groups in Mathematical Music Theory

- Mathematics
- 2016

Motivated by the music-theoretical work of Richard Cohn and David Clampitt on late-nineteenth century harmony, we mathematically prove that the PL-group of a hexatonic cycle is dual (in the sense of…

### Non-Contextual JQZ Transformations

- MathematicsMCM
- 2019

Non-contextual bijections called JQZ transformations that could be used for any kind of chord are introduced that could extend the PLR-group in many situations.

### Using Monoidal Categories in the Transformational Study of Musical Time-Spans and Rhythms

- Mathematics
- 2013

Transformational musical theory has so far mainly focused on the study of groups acting on musical chords, one of the most famous example being the action of the dihedral group D24 on the set of…

### Building generalized neo-Riemannian groups of musical transformations as extensions

- Mathematics
- 2011

Chords in musical harmony can be viewed as objects having shapes (major/minor/etc.) attached to base sets (pitch class sets). The base set and the shape set are usually given the structure of a…

### Non-Commutative Homometry in the Dihedral Groups

- Mathematics
- 2017

The paper deals with the question of homometry in the dihedral groups $D_{n}$ of order $2n$. These groups have the specificity to be non-commutative. It leads to a new approach as compared as the one…

### Towards A Categorical Approach of Transformational Music Theory

- Mathematics
- 2012

Transformational music theory mainly deals with group and group actions on sets, which are usually constituted by chords. For example, neo-Riemannian theory uses the dihedral group D24 to study…

## References

SHOWING 1-10 OF 10 REFERENCES

### The Topos of Triads

- Mathematics
- 2005

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…

### Sheaves in geometry and logic: a first introduction to topos theory

- Mathematics
- 1992

This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various…

### Musical Actions of Dihedral Groups

- ArtAm. Math. Mon.
- 2009

This paper illustrates both geometrically and algebraically how these two actions of the dihedral group of order 24 are dual, used to analyze works of music as diverse as Hindemith and the Beatles.

### THE HEXATONIC SYSTEMS UNDER NEO-RIEMANNIAN THEORY: AN EXPLORATION OF THE MATHEMATICAL ANALYSIS OF MUSIC

- Philosophy
- 2009

Neo-Riemannian theory developed as the mathematical analysis of musical trends dating as far back as the late 19th century. Specifically, the musical trends at hand involved the use of unorthodox…

### Alternative Interpretations of Some Measures from "Parsifal"

- History
- 1998

The analytical potential of the hexatonic organization of the consonant triads was demonstrated in a recent article by Richard Cohn (1996). This was accomplished by providing the consonant triad with…

### Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions

- Art
- 1996

### Neo-Riemannian Operations, Parsimonious Trichords, and Their "Tonnetz" Representations

- Philosophy
- 1997

### Triads and topos theory

- REU Project, University of Chicago
- 2007