Algebraic music theory rests on the observations that the set I of intervals forms a Z-module (or free abelian group) under concatenation and that the set of pitches forms an affine space (or torsor) over this module. A change of key is simply a shift of origin.

Within this framework a melody is a walk in the affine space subject to welldefined local constraints (conjunct motion, leap compensation, tendency-tone resolution, diatonic membership, contour, range, etc.). A chord is expressed, via Chasles’ identity, as a vector sum of intervals; the same language yields a clean description of closed and open voicings, pitch-equivalent fingerings, and the affine subspaces that correspond to whole-tone scales, augmented triads, diminished-seventh chords and tritones.

The paper develop these ideas in elementary terms, supply the necessary multiplication and inversion tables, give explicit fretboard realisations of the principal intervals and dyads, and illustrate the theory with a detailed analysis of the traditional melody and harmony of Greensleeves. An appendix collects interval-stack constructions for the most common triads, seventh chords and ninth chords.

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This article is intended as a minimal introduction to some basic musical ideas which fall within the intersection of mathematics and music. We discuss the science underlying the construction of Pythagorean and other tunings, giving mathematically clear definitions of the necessary musical terms. Careful distinction is made between the frequential and perceptual measures of pitch. The admissible changes of scale for each of these measures are derived. It is shown that commutative diagrams can provide a useful tool for the construction and analysis of scales. All needed musical notions are explained.

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