Well Temperaments

Well-temperaments, like truncated tunings, are limited in size, and, in addition, do not have unpleasant wolf intervals. As with many topics in tuning theory, there is much misunderstanding surrounding the notion of a well temperament. Most people are familiar with the term though exposure to Bach's masterpiece, Das Wohltemperierte Klavier (The Well-Tempered Klavier). Unfortunately, most people assume that ``wohltemperierte'' is just an antiquated German word for ``equal-tempered.'' This is not the case, since even during Bach's time there was a specific word for ``equal-tempered'' (gleichschwebend) and so it seems likely that if he meant ``equal-tempered'' he would have entitled the piece accordingly [19], [14, 59-71,195].

A well temperament is any tuning satisfying the following two criteria. First, that it be a tuning of size 12. A tuning of size twelve tunes a range of 12 pitch classes and tunes all other pitch classes to be enharmonic to ones inside this range. Mathematically, this means that a tuning of size 12 obeys the rule

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

Thus 12TET is a well temperament, but not the only well temperament. It is unique in that it is the only regular well temperament. All other well temperaments are similar to 12TET, but they impart a slightly different character to music in different keys since they are irregular.

Well temperaments can specified by tabulating values of $\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)}$ over a range of 12 values of $x\hspace{-1pt}_f$ . Instead of doublings, it is useful to express $\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)}$ in Pythagorean commas. This is a convenient unit because of a special property of tunings of size 12 which is that for all i₀,

$\begin{displaymath} \sum_{i=i_0}^{i_0+11} \ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(i\right)} = -\ensuremath{\underline{C_P}} \end{displaymath}$

Several well temperaments appear in Table 3.3, taken from [14, 62]. The abbreviations identifying the temperaments are as follows: W = Werckmeister 1681, Y1 = Young 1800, V = Vallotti circa 1750, L = Lambert 1774, Y2 = Young 1800, N = Neidhardt 1724, ET = 12TET. Although we have chosen to model diatonic tunings as transposable, historically, well temperaments were not thought of as transposable. This is why Y1 and V are listed as separate tunings even though they are just transpositions of each other.

Table 3.3: Some well temperaments (in units of $\ensuremath{\underline{C_P}}$ )
$x\hspace{-1pt}_f$

$\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)}$ -2 -1 0 1 2 3 4 5 6 7 8 9

W 0 0 $-4^{\mbox{-}1}$ $-4^{\mbox{-}1}$ $-4^{\mbox{-}1}$ 0 0 $-4^{\mbox{-}1}$ 0 0 0 0

Y1 0 0 $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ 0 0 0 0

V 0 $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ 0 0 0 0 0

L 0 $-7^{\mbox{-}1}$ $-7^{\mbox{-}1}$ $-7^{\mbox{-}1}$ $-7^{\mbox{-}1}$ $-7^{\mbox{-}1}$ $-7^{\mbox{-}1}$ $-7^{\mbox{-}1}$ 0 0 0 0

Y2 $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ 0 0 0 0

N $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-6^{\mbox{-}1}$ $-12^{\mbox{-}1}$ 0 $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ 0 0

ET $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$ $-12^{\mbox{-}1}$

Figure 3.11 shows these well temperaments in terms of $\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)}$ and $\ensuremath{\ensuremath{\underline{\Theta}}_{{}}\!\left(x\hspace{-1pt}_f\right)}$ . (The Valotti temperament is excluded because of its similarity to the first Young temperament.) The points on the graphs in Figure 3.11 are shaped differently in order to indicate what pitch class they belong to. Figure 3.10 illustrates this mapping of shape to pitch class. This shaping, reminiscent of a clock hand, is designed so that multiple points may lie on top of each other and still be distinguished.

						$x\hspace{-1pt}_f$
$\ensuremath{\ensuremath{\underline{\Phi}}_{{}}\!\left(x\hspace{-1pt}_f\right)}$	-2	-1	0	1	2	3	4	5	6	7	8	9
W	0	0	$-4^{\mbox{-}1}$	$-4^{\mbox{-}1}$	$-4^{\mbox{-}1}$	0	0	$-4^{\mbox{-}1}$	0	0	0	0
Y1	0	0	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	0	0	0	0
V	0	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	0	0	0	0	0
L	0	$-7^{\mbox{-}1}$	$-7^{\mbox{-}1}$	$-7^{\mbox{-}1}$	$-7^{\mbox{-}1}$	$-7^{\mbox{-}1}$	$-7^{\mbox{-}1}$	$-7^{\mbox{-}1}$	0	0	0	0
Y2	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	0	0	0	0
N	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-6^{\mbox{-}1}$	$-12^{\mbox{-}1}$	0	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	0	0
ET	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$	$-12^{\mbox{-}1}$