The Size of Regular Diatonic Tunings

The size of a tuning is an important factor if it is to be implemented on an instrument that can produce only a small number of frequencies per doubling, like the piano, which can only produce 12. The size of a tuning is the number of unique values $\ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)} \bmod 1$ takes on. For example, 12TET has size 12 since $\ensuremath{\ensuremath{\underline{\phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)} \bmod 1$ is $(7/12)x\hspace{-1pt}_f\bmod 1$, which takes on only 12 unique values. The size of a tuning is also the number of frequency ratios per doubling that is required to implement it. Most regular tunings are very large (many even infinite!) and therefore difficult or impossible to implement on an instrument with a finite number of frequencies per doubling.

Regular tunings with rational equal to some fraction n/k have size k, assuming n/k is in reduced form. These are the equal temperaments, like 12TET and 19TET ($\ensuremath{\underline{v}}=11/19$). A regular tuning with an irrational has infinite size. Mathematically, this means that $\tau(\mbox{\boldmath$x$})$ is a one-to-one mapping. By Eq. 3.5 and Eq. 3.2,

$$\ensuremath{\ensuremath{\underline{\tau}}_{{}}\!\left(\mbox{... ...ht)} + x_r = x\hspace{-1pt}_f\ensuremath{\underline{v}} + x_r.$$

The function $\tau(\mbox{\boldmath$x$})$ is one-to-one if and only if any change to x results in a different value yielded. Clearly a change to $x\hspace{-1pt}_f$ or x_r alone will result in a different value. In addition, any change to $x\hspace{-1pt}_f$ changes the value of $\tau(\mbox{\boldmath$x$})$ by an irrational amount, so there is no way to cancel this change with a change to x_r since x_r is an integer.

We will sometimes refer to a tuning of size n as an n-tuning.