40 Theorems on Musical Temperament. 



the temperaments proper to the six several concords re- 

 spectively. 



In regular douzeaves, none of these wolves occur, in 

 any of the six concords, upon any of the four middlemost 

 key-notes, viz. G, D, A or E respectively. 



C, F, bB and bE have no wolves in their major concords 

 (i e. the III, V or VI upon them, re- 

 spectively.) 



B, *F, *C and *G have no wolves in their minor con- 

 cords (i. e. 3d, 4th, or 6th.) 



Whence it follows (see Dr. Smith's Harmonics, Plate 

 XIX, p. 162, 2d edit.) that in major Keys, modulation can 

 be made from C by *s, through the keys G, D, A and E 

 without false concords or wolves; but if we proceed further 

 by *s, B has a false III, *F has its III and VI false, *C has 

 its III and VI false, and *G has its V, III and VI false, 

 which last chord Mr. Smyth calls the wolf, by way of 

 eminence, p. 450 of last volume. 



And in major kevs, modulation can also be made from 

 C by bs, through F, bB, and bE, and no major wolves occur, 

 (yet bE has a false 4th) ; but on proceeding further to bA 

 (or *G) its V, III and VI are false, as above. 



In minor Keys, modulation can be made from A by *s 

 through E, B, * F, *C, and *G, without any minor wolf (yet 

 *G has a fa'se Vth),but on proceeding further to *D (or bE), 

 its 4th, 3d and 6th are false. 



And in minor keys, modulation can also be made from 

 A by bs through D and G, without anv false concords or 

 wolves ; but on continuing thus to modulate, C has a false 

 0th, F has its 3d and-6th false, bB its 3d and 6th false, and 

 bE (or *D) has its 4th, 3d, and 6th false, as above. 



The six following Theorems, express in terms, of the 



r t 



fractions and — (either proper or improper) of the small 



intervals Schisma and most Minute* or 2 and m, all the 

 temperaments and wolves of the 72 concords, in any re- 

 gular douzeave; and whence, such temperaments can 

 rrad.ly be calculated for any proposed system, or the va- 

 rio s properties and relations of its intervals can be dis- 

 covered and computed : and by means of other theorems, 

 the beats can be calculated from such temperaments, in 

 terms of £ and m. In the article beats in the " Edin- 



• See vol. xxviii. p. 142, and engraved table, plate V. 



burgh 



