Section 2: Altered Ratios or Temperament

Abstract

To judge the qualities and effects of sound, it is necessary to attribute to them the ratios mentioned above, but, for practical use, it is completely impossible always to make use of such ratios. If one wanted each progression of a sound to another to be fair, the ratio to the fundamental sound of the absolute pitch would not remain the same; but by assigning to each tone the fair value for the fundamental tone, they are not fair between them. A single example of a very simple succession of six tones: sol, do, fa, re, sol, do would suffice to make this clear. By making the procession of these sounds in the fair ratios and by expressing all these ratios by their least terms, we have:

$$ \begin{array}{lllllllllll} sol\hfill & \hfill & do\hfill & \hfill & fa\hfill & \hfill & re\hfill & \hfill & sol\hfill & \hfill & do\hfill \\ {}\hfill & \left(3:2\right)\hfill & \hfill & \left(3:4\right)\hfill & \hfill & \left(6:5\right)\hfill & \hfill & \left(3:4\right)\hfill & \hfill & \left(3:2\right)\hfill & \hfill \\ {}243\hfill & :\hfill & 162\hfill & :\hfill & 216\hfill & :\hfill & 180\hfill & :\hfill & 250\hfill & :\hfill & 160\hfill \end{array} $$

The sol appears once as 243 and another time as 240, and do appears once as 162 and another time as 160; one will therefore have lowered it by the comma $\frac{81}{80}$. If we wish to repeat this sequence of sounds, or if we want to execute any longer melody with exact ratios, we will be even more in error. If there are several voices that wish to continue their song with fair intervals, each will be incorrect in another way and there will no longer be any harmony. In wishing to carry out the preceding series of sounds in the exact ratios with the fundamental sound, we will have:

$$ \begin{array}{lllllllllll} sol\hfill & \hfill &\ do\hfill & \hfill & fa\hfill & \hfill & re\hfill & \hfill & sol\hfill & \hfill & do\hfill \\ {}\frac{3}{2}\hfill & :\hfill & 1\hfill & :\hfill & \frac{4}{3}\hfill & :\hfill & \frac{9}{8}\kern0.5em \left(\mathrm{or}\ \frac{10}{9}\right)\hfill & :\hfill & \frac{3}{2}\hfill & :\hfill & 1\hfill \end{array} $$

This process will also yield false ratios. If we take do:re as 9:10, the fourth re:sol will not be 3:4, but 20:27, too small by the comma $\frac{81}{80}$ . But if we take do:re as 8:9, the minor third fa:re will not be 6:5, but 32:27, too small by the comma $\frac{81}{80}$. In trying any melody whatsoever in this manner, or if there are other sounds than these that belong to the perfect chord of the fundamental sound and to that of its fifth, we will have false results.