This work concerns a problem in modelling people's understanding of music. The problem is cast in the terms of discovering formal rules for transcribing melodies into musical notation, as this might be done by a student in a harmony class, taking musical dictation from a 'deadpan' performance on the keyboard. The score that results reflects important aspects of the structure and interpretation of the piece, which are only implicit in the performance. In Part I it is argued that this paradigm raises questions of general relevance to the study of our perception of music. In carrying out the task of notating a piece, two kinds of problem arise: what are the harmonic relations between the notes, and what are the metric units into which they are grouped? These two problems are considered in isolation from one another. In Part II, algorithms which embody two kinds of rule for the inference of metre are presented . In Part III the harmonic problem is considered. It arises from the fact that the number of keyboard semitones between two notes does not, by itself, identify their harmonic relation, which is what the notation has to express. Among other considerations, the key of the piece is an important characterisation of this identification - but the key is not explicit in the performance, and must itself be inferred. An earlier theory of harmonic relations is further developed into algorithms for assigning key-signatures and notation to melodies. By -the definition of the problem, we are committed to the concern of music belonging to the tradition of Western tonal music, to which the idea of key applies. Most of' the musical examples discussed will be taken from the work of one of its outstanding exponents, J.S. Bach, and in particular we shall be dealing with the fugue subjects of his “Well-Tempered Clavier". Some of the contents of Part II, Section § 2, and of Part III, Section § 3, have already appeared in a paper published in collaboration with H.C. Longuet-Higgins. Part III, Section § 2.1, describes his prior work in formulating the theory of harmonic relations, mentioned above, which forms the foundation of the work described in that section and has been published elsewhere by him.