Transposable Tunings

A transposable tuning is a relative tuning of a transposable pitch set. Mathematically, a transposable pitch set is one that can be made to form an Abelian group under addition. Musically, this means that the pitch set must have the following two properties. The first is that the interval from any pitch x₁ to any other pitch x₂, x₂-x₁, must be well-defined. The second is that the transposition of any pitch x by any interval y, x+y, must be a well-defined pitch. For example, the diatonic pitch set (all pitches representable in traditional Western notation) is transposable.

The thing that is special about a relative tuning of a transposable pitch set is that it isn't just one tuning: it can be thought of as summarizing an entire class of tunings. Mathematically, a transposable tuning $\tau(x)$ can be thought of as summarizing an entire class of tunings of the form $\tau_{y}(x) = \tau(x+y)$. For example, some diatonic tunings are playable only in keys closely related to C major. We will refer to such a tuning as being ``defined about C.'' Because diatonic pitches are transposable, this special relationship to C is totally incidental. For theoretical purposes, it is still the same tuning even if it is transposed to be defined about B, or any other pitch. For example, if $\tau(x)$ is the tuning defined about C, its transposed version about B is simply $\tau_{\mbox{\tiny M2}}(x) = \tau(x+\mbox{M2})$.