44 Temperaments of different musical Systems. 



Corollary \o. The minor Lim ma of Dr. Smith, p. 223, 



or the value of a sharp or a flat, is, 2 -f- f -f- 



5u—7t . . _ fc . 58"149()5— 7r 

 — m: or, without f*, we have • • t- 



5 0]36u-7t 



. m . 



u i 

 As is easily deduced from my theorems above, by 

 the process in page 158, of the Harmonics. As is 

 also the following, 



46$-f-5r 



Corollary 1 1 . The major Limma of Dr. Smith is 2 



4u + 5t . - . 46-1496j + 5r_, 

 _j_ f j m . or, without i, we have : 2 



1 • - : w ' • •' ^ 



40130M + 5* 

 -f m. 



Corollary 12. The wzeaft Tbra* of regularly tempered sy- 

 . J04*— 2r _ 9^—2/ 

 stems, is -2 -f 2r -| m; or, without 



104'2992^2r Q'0272v-2t 



f we have 2 -f m. 



s u 



By multiplying these general Corollaries, I should, per- 

 haps, exceed your limits. I must therefore content mvself 

 for the present with the following, as particular applications 

 of the theorems and corollaries above, viz. 



Scholium ] . If a douzeave he required, in which the fifths 



should be perfect, we have by theorem 1, \ =0; 



which condition will be answered, if r and / each ^= 

 and s and u each = + 1 5 which values substituted in 



the fifth wolr 2 -\ -m, gives — 12 2 



s u ' & 



— m as it ought to do, by cor. i. 



. . lls-4r u—At 

 Also, m theorem 3, ■ 2 4- m gives 112 



-f m or c, a major comma, as the sharp temperament 

 of each major third in this system ; also in theorem 6. 

 l]5__3r ' u— 3t 

 2 -| m gives 1 12 +m or c, as the sharp 



* By an examination of plate V in your 28th volume, and of a more ex- 

 tensive table of Intervals, it appears, that the number of 2s, always exceeds 

 the number ofm's in ratios between those of 12*5 : 1 and 10*: 1 (those of 

 Sand g). But the major comma and its aliquot parts, most frequently 

 occurring; in temperaments, I have adopted its ratio of 11 1 1, and thus find 

 f = -H062 + *Oi:56m ; from which equation, the latter ones in cor. 10, 1 1 

 aud 12 are obtained. tem- 



