Truncated Tunings

To realize a piece using a regular tuning, a large number of frequencies are often required, either because is irrational or because has a large denominator, where anything greater than 12 is considered large. One way to limit the size of a regular tuning is to truncate it. A truncated tuning is the same as a regular tuning for a range of 12 or more pitch classes, but tunes all other pitch classes to be enharmonic to ones inside this range. Two pitch classes $x\hspace{-1pt}_f$ and $y\hspace{-1pt}_f$ are enharmonic if and only if $\ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)} ... ...suremath{\ensuremath{\underline{\phi}}_{{}}\!\left(y\hspace{-1pt}_f\right)} + k$ for some integer k. In other words, two pitch classes are enharmonic if and only if they are tuned to a multiple of a doubling apart.

The most common form of enharmony tunes pitch classes separated by 12 P5 so that they are separated by 7 doublings. For example, G (8) and A (-4) can be tuned enharmonically such that $\ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\left(8\right)} = \ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\left(-4\right)} + 7$. This would make the pitches GN and AN be tuned enharmonically for all N.

Truncated tunings of size 12 always take the same form, since all pitch classes separated by 12 must be tuned enharmonically. A truncated 12-tuning derived from regular tuning can be expressed as

$$\ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\left(x\hspa... ...-1pt}_f+c) \bmod 12\right)} + 7((x\hspace{-1pt}_f+c) \div 12).$$

The transposition of c simply allows the regular range to be shifted in phase. For example, c=0 makes the range 0...11, c=1 makes the range $-1\ldots10$, and so on. Since we consider diatonic tunings to be transposable anyway, c can be dropped, forming the simpler equation  

$$\ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\left(x\hsp... ...(x\hspace{-1pt}_f\bmod 12\right)} + 7(x\hspace{-1pt}_f\div 12).$$ (5)

The general structure of truncated 12-tunings can be thought of as follows. They consist of chains of 12 pitch classes tuned according to the regular tuning they were derived from. These chains are connected by P5 tuned to a wolf fifth characteristic to the truncated tuning. For example, if `-' indicates v in the tuning they were derived from and `$\sim$' indicates a wolf fifth, truncated 12-tunings can be pictured as follows:

...-A-E-B-F $\sim$ C-G-D-A-E-B-F-C-G-D-A-E $\sim$ B-F-C-...

By transposition, the ``phase'' of the chains can be aligned arbitrarily. For instance, the chains below are also possible.

...-A-E $\sim$ B-F-C-G-D-A-E-B-F-C-G-D $\sim$ A-E-B-F-C-...

Let us calculate $\Phi(x\hspace{-1pt}_f)$ for truncated 12-tunings. If is the difference between the tuning of the wolf fifth and a just fifth,

$$\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(x\hspa... ...ath{\underline{\Phi_R}} & \mbox{otherwise.} \end{array}\right.$$

We can calculate as
 $$\begin{align} \ensuremath{\underline{\Phi_W}} &= \ensuremath{\ensuremath{\underl... ...} \ensuremath{\underline{C_P}}=12(\ensuremath{\underline{3/2}})-7. &&\end{align}$$
The constant C_P is an important frequency ratio called the Pythagorean comma. It is the difference between twelve just P5 and 7 doublings. Now that we have calculated , we can see that

$$\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(x\hspa... ...ath{\underline{\Phi_R}} & \mbox{otherwise.} \end{array}\right.$$

For example, a truncated Pythagorean 12-tuning ($\ensuremath{\underline{\Phi_R}}=0$)has

$$\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(x\hspa... ...pt}_f\bmod 12 = 0 \\ 0 & \mbox{otherwise.} \end{array}\right.$$

Now let us calculate for truncated 12-tunings. We know that it will take on two values: some value for thirds whose classes span the wolf fifth, and otherwise. In other words, it will be of the form

$$\ensuremath{\ensuremath{\underline{\Theta}}_{{}}\!\left(x\hs... ...h{\underline{\Theta_R}} & \mbox{otherwise.} \end{array}\right.$$

We can calculate as
$$\begin{align*} \ensuremath{\underline{\Theta_W}} &= \ensuremath{\ensuremath{\und... ... \ensuremath{\underline{C_P}} - 8\ensuremath{\underline{\Phi_R}}. &&\end{align*}$$
So, for truncated 12-tunings,

$$\ensuremath{\ensuremath{\underline{\Theta}}_{{}}\!\left(x\hs... ...mbox{otherwise (by \eqref{Theta_regular}).} \end{array}\right.$$

For example, truncated Pythagorean 12-tuning ($\ensuremath{\underline{\Phi_R}}=0$) has

$$\ensuremath{\ensuremath{\underline{\Theta}}_{{}}\!\left(x\hs... ...remath{\underline{C_S}} & \mbox{otherwise.} \end{array}\right.$$

Truncated tunings of size greater than 12 are quite interesting. Historically, truncated 13- and 14-tunings were actually used on keyboard instruments, typically through the use of a split key for G/A and/or D/E [14, 138-9]. Some modern meantone organs continue to have this feature. For example, the Brombaugh organ in Duke University Chapel, completed in 1997, has a single key for G/A and D/E but the tuning of these keys can be toggled between the two pitches through the use of a foot pedal [4].

The structure of a truncated 14-tuning, in terms of its fifth relations, appears below.

...-G-D $\sim$ A-E-B-F-C-G-D-A-E-B-F-C-G-D $\sim$ A-E-...

The size of the wolf fifth ($\sim$) is the same as for a truncated 12-tuning derived from the same underlying regular tuning. Unlike truncated 12-tunings, larger truncated tunings have some ambiguity with respect to how enharmonics are assigned. For example, should C be tuned the same as D or E? The same issue arises with pitch classes like F. This is a fairly esoteric theoretical issue, though, since very little (if any) music uses a range of pitch classes so extreme as to make these questions important.