VOL. XC.] PHILOSOPHICAL TRANSACTIONS. 02S 



necessary in rapid transitions, as in a shake ; and then probably because, by the 

 contraction of the capacity of the trachea, an increase of the pressure of the breath 

 can be more rapidly effected this way, than by the action of the abdominal muscles 

 alone. The reflection of the sound thus produced from the various parts of the 

 cavity of the mouth and nostrils, mixing at various intervals with the portions of 

 the vibrations directly proceeding from the larynx, must, according to the temporary 

 form of the parts, variously affect the laws of the motion of the air in each vibra- 

 tion, or, according to Euler's expression, the equation of the curve conceived to 

 correspond with this motion, and thus produce the various characters of the vowels 

 and semi-vowels. The principal sounding board seems to be the bony palate : the 

 nose, except in nasal letters, affords but little resonance ; for the nasal passage may 

 be closed, by applying the finger to the soft palate, without much altering the 

 sound of vowels not nasal. A good ear may distinctly observe, especially in a loud 

 bass voice, besides the fundamental note, at least 4 harmonic sounds, in the order 

 of the natural numbers ; and the more reedy the tone of the voice, the more easily 

 they are heard. Faint as they are, their origin is by no means easy to be explained. 

 This observation is precisely confirmed, in a late dissertation of M. Knecht, pub- 

 lished in the musical newspaper of Leipsic. Perhaps by a close attention to the 

 harmonics entering into the constitution of various sounds, more may be done in 

 their analysis than could otherwise be expected. 



l6. Of the Temperament of Musical Intervals. — It would have been extremely 

 convenient for practical musicians, and would have saved many warm controversies 

 among theoretical ones, if 3 times the ratio of 4 to 5, or 4 times that of 5 to 6, 

 had been equal to the ratio of 1 to 2. As it happens to be otherwise, it has been 

 much disputed in what intervals the imperfection should be placed. The Aristoxe- 

 nians and Pythagoreans were in some sense the beginners of the controversy. 

 Sauveur has given very comprehensive tables of a great number of systems of 

 temperament; and his own now ranks among the many that are rejected. Dr. 

 Smith has written a large and obscure volume, which, for every purpose but for 

 the use of an impracticable instrument, leaves the whole subject precisely where it 

 found it. Kirnberger, Marpurg, and other German writers, have disputed with 

 great bitterness, almost every one for a particular method of tuning. It is not 

 with any confidence of success, that one more attempt is made, which rests its 

 chief claim to preference, on the similarity of its theory to the actual practice of 

 the best instrument-makers. However we estimate the degree of imperfection of 

 1 tempered concords of the same nature, it will appear, that the manner of dividing 

 the temperament between them does not materially alter its aggregate sum; for 

 instance, the imperfection of a comma in a major-third, occasions it to beat very 

 nearly twice as fast as that of half a comma. If indeed the imperfection were 

 great, it might affect an interval so materially as to destroy its character; as, in 

 some methods of temperament, a minor 3d diminished by 1 commas approaches 

 more nearly to the ratio 6 to 7> than to 5 to 6; but, with this limitation, the sum 



