In this demo I demonstrate a particular computer-based formalisation of musical notation. I apply this representation to historical musical works written in non-standard temperaments, with a view to accurate playback while retaining a completely flexible choice of temperament.
While almost all modern applications assume the use of equal temperament with twelve identical semitones (so-called 12-TET), compositions from the Renaissance and Baroque periods make no such assumptions. Many use some form of meantone, such as the quarter-comma meantone; or even more exotic equal temperaments, such as a division of the octave into nineteen semitones (19-TET). Many keyboards of the era are “enharmonic”, i.e. they split their accidentals [1,5]. It is assumed that highly chromatic works (e.g. Girolamo Frescobaldi, Michelangelo Rossi [7,8]) were written with such instruments in mind. Any meantone temperament or any equal temperament with >12 semitones requires these “split” notes, so that for example F sharp is played with a different key to G flat. While more general keyboards continue to be developed (sometimes called “isomorphic” keyboards), a historically-accurate realisation of a given composition still requires a painstaking manual choice of notes. To remedy this situation I demonstrate a program that parses musical notation and plays back the result with a fully general choice of temperament.
All of the most commonly-used tuning systems, the so-called “regular” temperaments , have the mathematical structure of a free Abelian group. This can be thought of as a lattice, reminiscent of Euler’s Tonnetz , in either two dimensions (in the case of syntonic or meantone temperaments) or one dimension (in the case of equal temperaments).
I argue on purely syntactical grounds that the notation system for notes and intervals lends itself to this lattice structure, even before a particular temperament has been chosen. When consistently represented this way, the set of notes and intervals is closed under the operations of interval addition/subtraction (this is not the case for most commonly-used computer music systems, such as Euterpea, music21, Sibelius etc.).
Concretely, each interval is represented in terms of two basis intervals, which can be any two linear-independent intervals. One choice of basis (among infinitely many) is the augmented unison (A1) and the diminished second (d2); intuitively the d2-part of any interval corresponds to its “number”, and the A1-part to its augmentation/diminution.
The task of translating the notated music into specific frequencies of sound is then considerably simplified: for syntonic temperaments, any two intervals can be assigned arbitrary frequency ratios – though typically the octave is always fixed to a ratio of 2 – and the ratios of all the other intervals then follow automatically, from some simple two-dimensional linear algebra; and, for equal temperaments, a projection onto a one-dimensional subspace is performed, with one interval (the “comma”) mapping to a unison, with different octave-divisions corresponding to different choices of comma (e.g. 12-TET follows from using the diminished 2nd as the comma).
Because my focus is on music from the C16th and C17th, the representation of larger musical structures (phrases, entire compositions, etc.) is geared towards polyphony: each phrase is a linked list, and each piece of music consists of a branching “rose tree” data structure, with phrases as the leaves. This simplifies the implementation of the demonstration program  (written in Haskell), which parses input from Lilypond files (using Parsec), performs the transformation to the appropriate tuning system, and produces audio output (using Csound-expression).
I will give an audio demonstration  of music written in traditional notation, but intended to be played using certain exotic temperaments: in particular, the piece ‘Seigneur Dieu ta pitié’ by Guillaume de Costeley (C16th), written for a keyboard with 19 keys per octave.
References and links