Fifth Tunings

Previously, we decided to model diatonic tunings as register-zero transposable tunings. If diatonic pitches are represented as FRV, a diatonic register-zero transposable tuning is a fifth tuning, $\phi(x\hspace{-1pt}_f)$, that maps the pitch class $x\hspace{-1pt}_f$ to a frequency ratio. It is called a fifth tuning because the class of an FRV is its ``fifths'' component. Without loss of generality, we assume that  

$$\ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\left(0\right)}=0$$ (1)

in order to simplify calculations involving $\phi(x\hspace{-1pt}_f)$. (Recall that we use underlining to indicate the base-2 logarithm.) From the definition of a register-doubling tuning, the relationship between a diatonic transposable tuning $\tau(\mbox{\boldmath$x$})$ and its underlying fifth tuning $\phi(x\hspace{-1pt}_f)$ is  

$$\ensuremath{\ensuremath{\underline{\tau}}_{{}}\!\left(\mbox... ...remath{\begin{bmatrix} x\hspace{-1pt}_f \\ x_r \end{bmatrix}}.$$ (2)

At this point we have completed the development of the formal framework in which we will examine diatonic tunings. Before we discuss actual diatonic tunings, though, we need to have some reference or basis for what a ``good'' diatonic tuning would be. Our formal framework makes no value judgments along these lines and therefore gives us no guidance. As will be explained in the next section, our basis for such value judgments will be the beatless tunings of M3 and P5.