Evaluations of Fifth Tunings

We have derived the just tuning of P5 and M3 in the previous section. Since we have decided to evaluate tunings on the basis of how close they come to the just versions of these intervals, we need to be able to derive this information from a fifth tuning $\phi(x\hspace{-1pt}_f)$. Let us first consider how to do this for P5. There can be as many tunings of P5 as there are pitch classes: for example, the P5 from C3 to G3 could be different from that from G3 to D3, and so on. (Tunings of P5 from register to register are always the same; for example the P5 from C3 to G3 is always the same as that from C4 to G4.) We shall develop a function, $\Phi(x\hspace{-1pt}_f)$, that describes the tuning of the P5 from a pitch with class $x\hspace{-1pt}_f$ to one with class $x\hspace{-1pt}_f+1$ in terms of its deviation from just. For a fifth tuning $\phi(x\hspace{-1pt}_f)$, 

$$\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(x\hsp... ...\hspace{-1pt}_f\right)}\right) - \ensuremath{\underline{3/2}}.$$ (3)

(Recall that we use underlining to indicate the base-2 logarithm.) So is how far CN to GN is from just, is how far GN to DN is from just, is the same thing for DN to AN, and so on. The function $\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)}$ is similar to Lindley and Turner-Smith's $\mbox{tem}(q,\mbox{V})$ [14, 58].

Let us now consider how to evaluate a fifth tuning in terms of its tunings of M3. Again, there can be as many tunings of M3 as there are pitch classes: for example, the M3 from C3 to E3 could be different from that from G3 to B3, and so on. We shall develop a function, $\Theta(x\hspace{-1pt}_f)$, that describes the tuning of the M3 from a pitch with class $x\hspace{-1pt}_f$ to one with class $x\hspace{-1pt}_f+4$ in terms of its deviation from just. We begin by observing that for any pitch x, the tuning of the M3 from x to $\mbox{\boldmath$x$}+\mbox{M3}$ is
$$\begin{align*} \ensuremath{\ensuremath{\underline{\tau}}_{{}}\!\left(\mbox{\bold... ...emath{\underline{\phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)} - 2. &&\end{align*}$$
This motivates the definition
$$\begin{align*} \ensuremath{\ensuremath{\underline{\Theta}}_{{}}\!\left(x\hspace{... ...}_{{}}\!\left(x\hspace{-1pt}_f\right)} - \ensuremath{\underline{5}}.\end{align*}$$
So is how far CN to EN is from just, is how far GN to BN is from just, is the same thing for DN to FN, and so on. The function $\ensuremath{\ensuremath{\underline{\Theta}}_{{}}\!\left(x\hspace{-1pt}_f\right)}$ is similar to Lindley and Turner-Smith's $\mbox{tem}(q,\mbox{III})$ [14, 24].

We will now show how $\Theta(x\hspace{-1pt}_f)$ can be expressed in terms of $\Phi(x\hspace{-1pt}_f)$. First, we will express $\ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\left(x\hspace{-1pt}_f+4\right)}-\ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)}$ in terms of $\Phi(x\hspace{-1pt}_f)$ as follows:

$$\begin{align*} \begin{split} \ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\l... ...+3}\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(i\right)}}.\end{align*}$$
Now we can proceed to express $\Theta(x\hspace{-1pt}_f)$ in terms of $\Phi(x\hspace{-1pt}_f)$ as

 $$\begin{align} \ensuremath{\ensuremath{\underline{\Theta}}_{{}}\!\left(x\hspace{-... ...{\underline{\Phi}}_{{}}\!\left(i\right)}}}\mbox{ where } C_S = 81/80.\end{align}$$
The constant C_S is an important frequency ratio called the syntonic comma, or the comma of Didymus. It is the difference between four just P5 and a just M3 plus 2 doublings. We will now proceed to discuss some specific diatonic tunings, using $\Phi(x\hspace{-1pt}_f)$ and $\Theta(x\hspace{-1pt}_f)$ as our criteria for evaluation.