After I have tuned, in the manner ahove particularly specified, those 

 septaves which are about the middle of the instrument ; I then tune all the 

 octaves perfect, both in the upper and in the lower septaves. If it be a piano- 

 forte, or any other instrument which has two or more strings to each key ; then, 

 great attention should be paid to the tuning of those unisons perfect. 



When the instrument is tuned, according to the rules above mentioned; the 

 octaves should be tried as follows. Strike four Cs at once all through the 

 instrument. Then four Ds, &c. successively. Try the short keys all through 

 the instrument, in like manner. And if the instrument will not stand this test, 

 we may be quite certain that it is not in tune. 



It will not be necessary for me to go regularly through the twelve fourths ; 

 as they always follow their corresponding quints. Thus, a perfect quint, such as 

 the perfect quint, for example, from B flat, namely, B flat, F, will give, for the 

 inverse key F, a perfect fourth, viz. F, B flat next above. And a flat quint, 

 such as the flat quint from D, viz. D, A, will give, for the inverse key A, a sharp 

 fourth, namely, A, D next above. And the degree of sharpness of the sharp 

 fourth will always, of necessity, correspond with the degree of flatness of the flat 

 quint, wherever a flat quint occurs. My mode of tuning does not produce a 

 single offensive quint ; neither does it produce a single offensive fourth. 



Having accurately explained how the new temperament which I have 

 discovered regulates the twelve quints, and also the twelve fourths ; I will now 

 shew in what manner it affects the twelve thirds. But, before I can make myself 

 clearly understood upon this interesting and curious part of the subject, I must 

 first state a few things, by way of preliminary observations. 



I have established the following facts, from experiments made with the 

 greatest attention, by means of a very excellent monochord, and also by means 

 of a grand piano-forte, built by one of the best piano-forte makers in England. 



I divided the perfect octave C C, between the first bass C and the middle 

 C, into three successive thirds, in the three following ways, viz. 



First. I divided that perfect octave CC, into two perfect thirds, namely, 

 C, E ; and E, G sharp, which is the same key as A flat ; and into one third 

 sharper than a perfect third, namely, A flat, C next above. This sharper third, 

 so produced, was, as I expected, very offensive. 



Secondly. I divided that same perfect octave C C, into one perfect third, 

 C, E ; and into two equally sharp thirds, viz. E, G sharp, which is the same key j 

 88 A flat ; and A flat, C next above. The length of the wire A flat, in the ij 

 monochord, was made a geometrical mean proportional between the length of 

 the wire E, and the length of the wire C next above. This last mentioned 

 division of the octave C C is excellent ; for, the two sharp thirds thus produced 

 were bi-equal thirds, as was particularly explained above. See page 13. 



Thirdly. I then divided the same perfect octave CC. into three thirds, 

 all equally sharper than perfect. This equal division was effected, by the 

 wires E, and A flat, in the monochord, having been made two geometrical mean 



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