Atonal music theory / An introduction to atonal music theory through abstract algebra.

Analyzing post-tonal or atonal music is a far-cry from classical musical analysis of functional harmony. If you've studied classical theory, the familiar jargon (tonic chords, dominant chords, tonicization, modulation, etc.) is done away with. In its stead we are left with numbers, sets, and transformations; entities that are deeply rooted in mathematics. So before discussing anything about music, it's more than appropriate to familiarize ourselves with this math that will be in the background of all post-tonal analysis.

Numerical Matrix Analysis / Class Notes.

Atomic Density Research / Notes for research project into atomic density of algebraic congruence monoids.

Preliminary Algebra Notes

For a non-empty set $S$ and associative binary operation $\ast$, the association $M=(S,\ast)$ is called a monoid if $S$ is closed under the operation $\ast$, and there exists an identity element $e\in S$ such that for all $a\in S$, $a\ast e=e=e\ast a$.

A monoid is called commutative/abelian if its operation is commutative.

A submonoid is a subset of the elements of a monoid that in themselves form a monoid under the same operation.

Numerical Matrix Analysis / Class Notes.

Linear algebra / Some linear algebra notes.

Linear systems

A linear equation is simply an equation involving variables/indeterminates with highest power one (or zero). For example, the equation $x=4$ is linear while $x^2=4$ is not linear but quadratic.

Differential Equations / Class Notes.

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Differential equations are useful for modelling problems.

Abstract algebra / Some abstract algbera notes.