Seventhly. I then pitch A flat, exactly half way between E and the C next 

 {above. If a monochord be used for this purpose, the length of the wire A flat 

 must be made a geometrical mean proportional between the length of the wire E 

 and the length of the wire C next above. The two sharp thirds produced by 

 this method are peculiarly suited to solemn and to plaintive music. The effect 

 was remarkably striking, in a comparative experiment which I shall relate in the 

 sequel. I was so struck with the peculiar excellency of sharp thirds of this exact 

 value for musical compositions which are either plaintive or majestic, that I have, 

 for the sake of accuracy and methodical discrimination, given them a distinct 

 name. If, from any perfect octave, we deduct one perfect third; and if we then 

 divide the remaining interval into two thirds equally sharp ; each of those two 

 thirds, as well as every other third of that same degree of sharpness, is that 

 sharp third which I shall call a BI-EQUAL THIRD. 



If the pitch of A flat be not determined by means of a monochord, but 

 simplv by the ear, its pitch may be ascertained with great precision, if the tuner 

 pay exact attention to the equality of the beatings of the two successive major 



thirds, E, G sharp* which is the same key as A flat ; and A flat, C. In tuning 



each key throughout the whole instrument, too much attention cannot be paid 

 to the beatings, as that is by far the most accurate way of tuning by the -ear. 

 For, whenever either a third, a fourth, a quint, or an octave is quite perfect, 

 there is, in such case, no beating to be heard. But, on the contrary, whenever 

 either of them is in any degree imperfect, but is not too distant from perfection, 



a beating is always audible. A very slow beating proves that the deviation 



from perfection is not great. A quicker beating shews that the deviation from 

 perfection is more considerable. And, from the equality of the beatings, equal 

 deviations from perfection may be correctly ascertained. 



Eighthly. The pitch of A flat being now determined, I next pitch A flat, 

 E flat, upwards, a perfect quint ; or I tune E flat, A flat, downwards, a perfect 

 fourth. E flat will then be exactly half way between B and the G next above. 



Ninthly. The pitch of A flat being determined, as explained above ; I tune 

 A flat, D flat, upwards, a perfect fourth; or I tune D flat, A flat, downwards, 

 a perfect quint. I then tune the D flat next above, a perfect octave. 



Tenthly. The pitch of D flat being determined, I tune D flat, G flat, up- 

 wards, a perfect fourth ; or I tune G flat, D flat, downwards, a perfect quint. 



Thus, I have already got seven quints quite perfect ; viz. 1. C, G; 2. E, B; 

 3. F, C ; 4. B flat, F ; 5. A flat, E fiat ; 6. D flat, A flat ; 7 . G flat, D flat. 



I have likewise got two quints very nearly perfect, but a little flat, viz. 

 1. B, F sharp, which is the same key as G flat; 2. E flat, B flat. 



Each of those two quints differs from a perfect quint, only one in two 

 thousand six hundred and fifty seven parts and a half nearly ; or only about 

 1 . 1 8*83 1 parts in 3.000.000.000. See the value of those two quints, in page 23. 



II is a fact very worthy of notice, that, in each of those two last mentioned 

 quints, two distinct beatings are to be heard at the same time. The one is very 



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