. 



slow, and the other is considerably quicker. Now, as each of those two quints 

 does, as proved by the monochord, approach so very nearly to perfection ; it is 

 evident, that it is the slower beating which is the proper beating to be attended 

 to hi the case of each of those two quints which are so very nearly perfect. 



I have a scientific way of determining the three remaining quints that are 

 also flat, which I shall now explain. They are, 1. G, D ; 2. D, A ; 3. A, E. 



We have already seen that the pitch of G has been determined, as a perfect 

 quint from the first bass C. The pitch of E has likewise been determined, as 

 a perfect third from that same C. The E first octave above from that E, and the 

 E second octave above from that same E, are of course determined likewise. 



Eleventhly and twelfthly. It is now requisite so to pitch the D, and the 

 A, between the G perfect quint from C, and the E second octave from that E 

 which is the perfect third from C, in such a manner, that the interval G, E, may 

 be divided into three equally flat quints, G, D ; D, A ; and A, E. None of 

 those three quints are of such a degree of flatness as to be offensive to the ear ; 

 for, each of those three quints differs from a perfect quint, only one in three 

 hundred and sixty one parts and a half nearly ; or only about 8.298.850 parts 



in 3.000.000.000. See the value of those three quints, in page 23. If a 



monochord be used to determine the pitch of D, and of A ; then, the length of 

 the wire D, and the length of the wire A, must be made two geometrical mean 

 proportionals, between the length of the wire G, and the length of the wire E. 

 But, if a monochord be not used for this purpose, and if the tuner determine the 

 pitch of D, and of A, by the ear; it may be done with great accuracy, if he 

 attend properly to the equality of the beatings of the three successive flat quints, 

 G, D; D, A; and A, E. That fact has been ascertained by repeated trials. 



If the interval G, E, be (as in Kirnberger's method of tuning) divided into 

 one perfect quint, and two equally flat quints ; such, for instance, as the perfect 

 quint G, D, and two equally flat quints, D, A, and A, E ; then, each of those 

 two flat quints, by becoming too flat, is offensive to the ear. I have made that 

 experiment with care ; and the result was what I have just mentioned. And if 

 the same interval G, E, be divided into two perfect quints, and one flat quint ; 

 then, the flat quint, so produced, is still more offensive. This was fully to be 

 expected. 



I shall now give a definition of those three equal and non-offensive quints 

 which result from the scientific division of the interval G, E, above explained, 

 in order clearly to distinguish it from the two exceptionable divisions just de- 

 scribed. That is to say ; if the interval between the perfect quint from a key- 

 note, and the second perfect octave above the perfect third from the same key- 

 note, be divided into three equally flat quints; each of those three equal quints 

 is that which I shall call a TRI-EQUAL QUINT. And the inverse fourth 

 of any tri-equal quint is that which I shall call a TRI-EQUAL FOURTH. 



My NEW mode of TUNING, which I have just minutely explained, is 

 exhibited in a very convenient manner in the following table. 



( 14 ) 



I 



