reduced to two thirds of its length, the sound produced by the two thirds will 

 be that of a perfect quint. Now, the number of vibrations of a wire of uniform 

 thickness, which has any given degree of tension, will be inversely as the length 

 of the wire. That is to say, that a wire, for example, of thirty inches long, will 

 vibrate twice, during the time that an equal wire, of twenty inches long, under 

 the same tension, will vibrate three times. If the longer wire be tuned to C, the 

 shorter wire will be the G perfect quint. It follows from this, that the third, 

 sixth, and ninth vibrations, &c. of the G wire so tuned, will respectively keep 

 pace with the second, fourth, and sixth vibrations, &c. of the C wire above 

 mentioned. And it is from that very circumstance that concords are produced. 

 But, let us suppose, that the length of the G wire, instead of being precisely 

 two thirds of the length of the C wire, be either something more, or something 

 less ; then, the third, sixth, and ninth vibrations, &c. of the G wire, will no 

 longer respectively keep pace with the second, fourth, and sixth vibrations, &c. 

 of the C wire. Therefore, instead of concords, discords will be heard. For. 

 true concords can only be obtained by means of exact geometrical proportions 



Ibetween the vibrations produced. There are various classes of discords. 



ISome of them are offensive; others are not. Some of them are characteristic; 

 others are not so. How to limit the extent of the discordancies, and how to 

 distribute the non-offensive discords in such a manner that the various keys, 

 linstead of being injured, shall on the contrary be improved, will be very fully 

 explained hereafter. But, to have, in any instrument, nothing but discords, is 

 abominable ; and that is always and necessarily the case, whenever that mode of 

 tuning which is denominated the equal temperament is adopted. 



By the scheme of equal temperament, not only every third is sharp, 

 but is equally sharp ; every fourth is sharp, and is also equilly sharp ; and every 

 quint is flat, and is equally flat. Therefore, not only every major key is rendered 

 imperfect, but is rendered similarly and equally imperfect. This of course destroys 

 the difference of character which ought to exist, in a well tuned instrument, 

 between the different major keys. And the minor keys are liable to the same 

 dfffct, for a similar reason. Thus it is that dull monotony is substituted for 

 pleasing and orderly variety. And modulation from key to key loses, in great 

 measure, the very object of modulation, which is to relieve the ear, and to cause 

 us to return to the original key with an increased pleasure, which arises from the 

 systematic variety of the different keys through which we have successively passed. 



Some tuners, who prefer an UNEQUAL TEMPERAMENT, throw the whole 

 of the quint wolf into the key of E flat. Others, divide it between the two 

 keys of A flat, and D flat ; or between some other two keys. Those persons 



rally say, that they throw THE wolf (as they inaccurately call it) into a 

 single key, or into two keys, according to their respective method of tuning. 

 The absurdity of this assertion must now be apparent to the reader. For, since 

 Ithere is not only in the quints a quint wolf, but there are likewise in the 

 major thirds FOUR DISTINCT wolves, each of which four is by nature confined 



( IX ) 



