On Musical Temperament. 187 
and at the same time allowing both the unisons to be used 
together—either by an apparatus for slightly increasing the 
tension of the strings, or by one which shall intercept the 
vibrations of such a part of the string, at its extremity, as shall 
elevate its tone, by the diesis of the system of temperament 
adopted. Were only 4 degrees to the octave, furnishing the 
instrument with 5 sharps and 4 flats, thus rendered changeable, 
there is little music which could not be correctly executed 
Upon it. 
Scholium S. 
In the same general manner, may be found the best system 
of intervals, for a scale confined to a less number of degrees 
than that of the complete Enharmonic scale. In such an 
investigation, the numbers in Table IV. expressing the fre- 
quency of all such adjacent degrees as have but one sound in 
the given scale, must be united ; and the temperaments m, n, 
¢. of the theorem, when belonging to concords whose ter- 
minating degrees are united to those adjacent, must be taken, 
not what they were in the complete scale, but what they 
become, considering them as terminated by the substituted 
adjacent degree. 
If, for example, the best temperaments were required for a 
scale of 15 degrees to the octave, such as is that of some 
European organs, or in other words, having no Enharmonic 
1ntervals except D# Eb, and G# Ab,—the numbers in Table 
ly, belonging to C# and Db, EF and F, F# and Gb, &c. must 
€ united, and their sums substituted when they occur, for 
% a’, b, &c. in the theorem; while the temperament, for 
€xample, of the [IId on C# must not be reckoned 77, as in the 
“omplete scale, but 1261—77 sharp, since its upper termina- 
ton has become F, instead of E#. With these variations let 
and would ¢ 
ance of 
