414 Mr. Farey''s Notation of Musical Intervals 



In Dr. T.'s Table, the specific Gravity of atmospheric Air is 

 assumed to be 1 -0000, and that of Hydrogen is found to be 

 0'0694 ; this last, and 21 others of the numbers expressing the 

 specific Gravities, being repeating decAmoXs ', as an expert Arith- 

 metician would at once perceive, from observing the prime Num- 

 ber 3, to be a multiple, in so many of the Denominators of the 

 Fractions, in the third column of my Table, in the last page; 

 wherein I have omitted these repeating Decimals, and in column 2, 

 set down the Numbers, expressing, how many times the weight 

 of Hydrogen, answers to the weight of each Gas, under equal 

 Bulks and Pressures. The vulgar Fractions in column 3, ex- 

 press the relations which the several specific Gravities, indicated 

 in col. 1, bear to 1 (or y) which is here set against atmospheric 

 or common air. 



The square- roots of the Fractions in col. 3, have furnished 

 the Ratios, from whence the corresponding musical Intervals in 

 col. 4, have been calculated, and expressed in the correct Nota- 

 tion which I had the honour of proposing to musical Calculators^ 

 through your pages, in the year 1807 *. 



For some of your Readers who may not have the Volume at hand, 

 it may be useful here to mention, that the Symbols S, f and m, 

 standingfat the top of their respective ranges of Figures in col. 4, 

 express three small musical Intervals ; their respective magni- 

 tudes being such, that 612^" + \2i + 53m, exactly make an 

 Octave, having the Ratio |; 358:? + 7 f + 31 m make a Fi/tfl 

 (f) ; 197^' + 4 f -'r 17 ni make a major Third ; and 1 1^? + m 

 make a major CowTwa, having the Ratio |-e-, &c. as in the Table 

 referred to: the Interval m being at the same time so exceed- 

 ingly minute, as scarcely to be appreciable by the Ear, in the 

 most extreme case; as may be judged from the fact, of the 

 Fraction expressing the Ratio of the lengths of two Strings cal- 

 culated to yield it, having the first five of its figures alike, in the 

 Numerator and in the Denominator, and a difference of only 3 

 appearing, in the sixth places of these figures; its decimal value 

 with respect to S as an unit is "0078624 ; The middle Interval 

 f, though more considerable in value, so that a difference of 7 

 appears in the fifth figures of its Fraction, and so that its deci- 

 mal value in terms of S is '1496610, is very small, and unim- 

 portant in practical utility, except in as far, as the number of fs 

 in anv expression in the Table, shows, how many of the vulgar 

 Half-notes of Musicians, are contained in the Interval designated : 

 it will for instance, be perceived from inspecting the middle or f 

 range, that phosphuretted Hydrogen is the first Gas which ex- 

 ceeds atmospheric Air in the acuteness of its Sound, by the quan- 

 tity of half a Note ; and in like manner, that Oxygen is the first 

 • See vol. xxvii. Plat* V. p. \40, and vol. xlix, pp. 360 and 3CJ. 



Gas 



