And those four, together with THE QUINT WOLF, make the five 

 wolves which I have mentioned above. And it was from my having observed 

 these five distinct wolves that I was led to find out that superior mode 

 of tuning keyed instruments which I am now going to describe. 



If, in any musical instrument which has exactly twelve fixed keys or twelve 

 fixed tones in each septave, the octaves be tuned perfect; then, the quints 

 cannot all be tuned perfect, as the table of octaves and of quints, 

 in page 6, clearly shews. And if, in any such instrument, the quints be tuned 

 perfect; then, the octaves cannot be tuned perfect. 



Now, such is the construction of the human ear, that we can bear to hear 

 a much greater deviation from perfection in the quints than we can bear to hear 

 in the octaves. And we can bear a still greater deviation from perfection in the 

 major thirds than we can bear either in the octaves or in the quints. Musicians 

 and tuners have therefore agreed not to tune all the quints perfect, as that would 

 make the octaves intolerable. 



Some tuners, however, in order to assist the quints, have very improperly 

 proposed to tune the octaves a little imperfect. The objections to this method 

 are obvious. For, if we sharpen the octaves to assist the quints, it injures the 

 thirds ; and if we flatten the octaves to assist the thirds, it injures the quints. 



Besides, however small the deviation from perfection may be in a single 

 octave, it will become very sensible in two or three ; and, in the extent of six 

 or seven octaves, such a deviation will become very offensive. 



It is necessary, therefore, that all the octaves should be tuned perfect. 

 Now, from what has been already said, it is evident that this object can be 

 obtained in one way only ; and that is, by tuning some one, or more, of the 

 twelve quints above mentioned flatter than perfect. 



In order for all the octaves to be tuned perfect, it is likewise necessary 

 that some one, or more, of the three successive major thirds, in each of the 

 four natural columns above mentioned, must be tuned sharper than perfect. 

 As for example, in the C column, some one, or more, of the three successive 

 major thirds, viz. C, E ; — E, G sharp, which is the same key as A flat ; — and 

 A flat, C, must be tuned sharper than perfect; for, otherwise, CC, which is 

 produced by means of those three successive major thirds, could not be (as it 

 ought to be) a perfect octave ; because three successive perfect thirds do not 

 make up a perfect octave. Therefore, in tuning any musical instrument which 

 has exactly twelve fixed keys or twelve fixed tones in each septave, the problem 

 does, of necessity, resolve itself into this ; namely, to ascertain which one, or 

 more, of the three successive major thirds, in each of the above mentioned four 

 natural columns respectively, ought to be tuned sharper than perfect, and in what 

 proportion each is to be so tuned ; and also, which one, or more, of the twelve 

 successive quints ought to be tuned flatter than perfect, and in what proportion. 



What I have just stated will enable the reader easily to understand what 

 musical TEMPERAMENT is. This term signifies the precise adjustment of 



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