On Musical Temperament. 187 



imd 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 change- 

 able, there is little music which could not be correctly exe- 

 cuted upon it. 



Scholium S. 



In the same general manner, may be found the best system 

 «f 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, 

 &.C. 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 

 intervals except D^ Eb, and G* Ab, — the numbers in Table 

 IV. belonging to C* and Db, E* and F, F« and Gb, &c. must 

 be united, and their sums substituted when they occur, for 

 rt, a, b, &c. in the theorem; while the temperament, for 

 example, of the Illd on C* must not be reckoned 77 as in the 

 complete scale, but 1261 — 77 sharp, since its upper termina- 

 tion has become F, instead of E*. With these variations let 



musical intervals, which practical musicians can seldom possess, there is no pro- 

 bability that it will, ever be extensively adopted. Perhaps, however, four or five 

 sounds, such as D*, E^, Ab, Db, might be added to the common scale of 12 

 intervals by means of his mechanism, with advantage. An instrument thus fur- 

 nished would require the use of pedals but seldom, and would contain chromatic 

 «lll»grces sufficient for the accurate performance of the great mass of organ music. 



16* 



