proportionals between the length of the wire C, and the length of the wire C 

 next above. Those three equally sharp thirds, of this last mentioned value, are 

 the very same as the sharp thirds which result from that mode of tuning which 

 is commonly termed the equal temperament. Not one of those three 

 equally sharp thirds has either of those two striking characters which are to be 

 found in an instrument that is tuned in the most advantageous manner. For, 

 not one of those three equally sharp thirds has either the beauty of a third 

 which is perfect, nor the peculiar atid solemn character of that other third 

 which I have already denominated a bi-equal third. This is, therefore, another 

 inherent defect belonging to that method which is called the equal tem- 

 perament ; inasmuch as that defect necessarily extends to all the twelve keys, 

 in that ill contrived mode of tuning. 



I can now explain to the reader in what manner the NEW temperament 

 which I have discovered affects the twelve thirds, in a keyed instrument which 

 has exactly twelve t keys in each septave. 



Two of those twelve thirds are quite perfect. Those two are, 1. C, E; 

 2. G, B. 



Six of the remaining ten thirds are sharper than perfect ; and each of them 

 is of the exact value of a bi-equal third. Those six are, 1. E, G sharp, which 

 is the same key as A flat ; 2. A flat, C ; 3. B, D sharp, which is the same key 

 as E flat ; 4. E flat, G ; 5. G flat, B flat ; 6. D flat, F. 



The remaining four thirds are likewise sharp, but less so than the preceding 

 six ; and each of them is, in respect to sharpness, intermediate between the other 

 two classes above specified ; namely, intermediate between a perfect third and a 

 bi-equal third. See page 23. 



Of those four last mentioned sharp thirds, two are nearer to a bi-equal third 

 tha.i to a perfect third. Those two are, 1. A, C sharp, which is the same key 

 as D flat ; 2. B flat, D. See page 23. 



t The usual number of twelve keys seems to be pointed out to us from the natural 



number of twelve musical intervals. There are three reasons for considering' that division 



of the septave as a natural one. 



First. If we start from any key, (such, for example, as from C,) then, twelve quinls, 

 duly tempered according to my natural and scientific temperament explained above, will 

 bring us again to a key of the same denomination. Therefore, the number of twelve succes- 

 sive quinls naturally leads to the number of twelve keys of twelve distinct denominations. 



Secondly. There are four natural sets or columns of major thirds, and each of those four 

 • i respectively contains three series of major thirds, as was fully explained above. See 

 pagM i and 8. This makes up the same number of twelve keys. 



Thirdly. The like number of twelve kri/s is found also in the following manner; namely, 

 !.;. i -iMi-iilering the three natural sets or columns of minor thirds; for, each of those thief, ttta 

 respectively contains four scries of minor thirds, making likewise twelve keys in all, viz. 



Ffeli let, 1. (', Kllai, 3. R-flat, Gflat; 3. G flat, A; 4. \, C. 

 grand ft. l. 1) Hat, Ej 2. B, Gr; 3. O, Bflat; 4. Bflat, L) flat. 

 Tlurd set. .. I>, I', '.'. P, A Hal; 3. Aflat, 1$ ; 4. B, D. 



Vol. 25. N'8.100. S''/)M806. U ( 1" ) 



