54 



C O N C E R T. 



names of the inventors of the systems, &c. are adapted 

 to this pitch of the tenor-ctifF C, and will only apply to 



instruments of that pitch. 



In .our article Acoustics, vol. i. p. 115, we have 



.given a simple and useful theorem, that may also be 



thus expressed, viz. N = / 96 - 5 L lJ, for finding the 



number of complete vibrations mad e by any note in 

 one second of time ; winch we shall here illustrate, by 

 the calculation of an example at length, by logarithms, 

 so arranged, as to serve as & formula for all similar ope- 

 rations, of an experiment that we once made on a C 

 tuning-fork, or rather, on the vibrations of a brass- 

 wire tuned unison therewith, the wire taken from a 

 coil, of which (58.4 inches in length weighed .007.3 lb. 

 avoirdupois) 1 inch weighed .0001251b. (w) One end, 

 of a sufficient length, of this wire was fixed to the 

 top peg of a vertical monochord, or Sonometer, 

 hanging over a pretty sharp steel edge below the 

 peg, and to the other end of this wire a sort of 

 scale-dish, for receiving weights (decimals of the lb. 

 of 7000 grains) was attached ; and a 10 lb. weight be- 

 ing placed in the scale-dish, another moveable steel 

 bridge on the monochord was set, and fixed so, that 

 when, by tilting the monochord a little, or, rather, 

 bringing it nearer to a vertical position than it before 

 hung, by means of a strong hinge by which it was at- 

 tached at the top to its frame and stand, this lower 

 bridge cut off or limited such a part of the wire be- 

 tween it and the upper bridge, as gave a sound, when 

 struck, something lower than the fork under experi- 

 ment. Other smaller weights were then added in the 

 scale-dish, until the sound of this portion of the wire, 

 11.7 inches long (l) between the bridges, exactly 

 agreed with the fork. The weights in the scale-dish, 

 the weight of that dish and its apparatus, and the 

 weight of the remaining length of the wire over and 

 above 1 1 .7 inches, being calculated, as above, and add- 

 ed together, gave 10.34 lb., extremely near (<), for the 

 stretching weights ; whence we have 



Constant logarithm (of 96.5 



inches) = 1.9845273 



Logarithm of the stretching 



weight (t), =10.34 lb. . . . =1.0145205 



2*9990478 

 Logarithm of weight of an mch 



of the wire («>), =.000125 lb. =6.0969100 



2)6.9021378 



3.4510689 

 Logarithm of thelength of vibra- 

 ting wire (l), =17-7 inches = 1.0681859 



Logarithm of the complete, or 



double vibrations, in 1" (n) . = 2.828830 =241.48 



Whence it appeared, that this tenor-cliff C fork gave 

 241.48, or very near 241| vibrations per 1". 



In calculating by this formula, it may be previously 

 sopied out, the lines being at proper distances, as above, 

 ready to receive the numbers answering to t, w, and L, 

 as they are determined in the course of the experi- 

 ment ; and it may not be amiss to observe respecting 

 these, that grains, ounces, grammes, or any other weights, 

 and their decimals, may be used, instead of pounds, 

 provided t and w are both expressed in the same ; and 



so feet, lines, decimetersromnj other measures, may be 

 used instead of inches, provided the constant length, 

 (96.5 inches, and its logarithm,) and the unit length 

 of wire, as to the weight vo, are both expressed in the 

 same kind of measures. It can scarcely be necessary 

 to subjoin, that the two first logarithms are added, the 

 next subtracted from their sum, and the remainder 

 halved, and from this the next is deducted, and the 

 numbers answering to the remainder is taken from the 

 Tables, as at the end. If it had been wished to reduce 

 this fork to the proper standard of 240 vibrations, the 

 apparatus remaining untouched for a few minutes, 

 while the above calculations had been made. The fol- 

 lowing formula might, and may in all similar cases, be 

 used for the necessary calculation of the proper, or cor- 

 rected weight (£'), for such purpose, viz. 



Constant logarithm (of 240*) . . . 4.7604225 

 Logarithm of the former weight 



(t) = 10.541b 1.0145205 



5.7749430 



Logarithm of the former vibrations 



(n) =241.48 2.3S28SI2 



2 



4.7657624 



Logarithm of the weight (t'), for 



240 vibrations per 1" 1.0091806: 



10.214 



Goncert 

 fitch. 



Where the two first logarithms are added, the double 

 of the third is deducted from their sum, and the number 

 sought in the tables, that answers to this remainder. 

 Whence it appears that 10.2 i 4 lb. ought to have been 

 vised, instead of 10.34 lb. to stretch the wire; and that 

 the difference of these, or ,1261b. being taken out of 

 the scale-dish, the wire would then sound the proper 

 tenor-cliff C of concert pitch. 



Then, a round file being provided, and the fork fixed 

 in a vice, by slightly enlarging the size or depth of the 

 bottom of the opening of the prongs of the fork, and 

 repeatedly trying its sound with the wire, it might at 

 length be brought to yield the proper number of 240 

 complete vibrations. If the fork had proved too fiat, 

 its pitch might be raised, by filing a small portion off 

 the top of one or both of the prongs, so as to shorten 

 them. 



The above method, if repeated with care, and with 

 different lengths of vibrating wire> and of weights, 

 would very exactly adjust a tuning-fork, as exact, at 

 least, as a unison could be judged of or adjusted by the 

 ear, between the fork and the wire ; and if, instead of 

 trusting to this, a third sound from a wire or fork, a 

 very little different from the unison intended to be ad- 

 justed, be provided, and the teats of the fork and wire, 

 with this comparative sound, be made equal in a second 

 or any other period, as Mr W. Nicholson recommends 

 in his Phil. Jour. 8vo. i. p. 320, every desirable degree 

 of accuracy may be obtained by this method, in the 

 pitch of a 240 fork, as such might be called and 

 marked. 



If a fork or a pipe or major comma higher were want- 

 ed for tuning organs of Mr Liston's construction, as Mr 

 Farey has shewn to be necessary, in the Phil. Mag. 

 vol. xxxix. p. 420, the same must be adjusted to 243 vi- 

 brations for C, and should be so marked : and if for 

 Mr Loeschman's proposed enharmonic or perfect piano 



