In calculating by this formula, it may be previously copied out, the lines being at proper distances, as above, ready to receive the numbers answering to t, w, and L, as they arc determined in the course of the experiment ; 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 su arc both expressed in the same ; and so feet, lines, decimeters, or any 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 w, arc both expressed in the same kind of mea sures It can scarcely be necessary to subjoin, that the two first logarithms arc 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 are 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 un touched for a few minutes, while the above calculations had been made, the following formula might, and may in all similar cases, be used for the necessary calcula tion of the proper, or corrected weight (t'), for such pur pose, viz.
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 10.214 lb. ought to have been used, instead of 10.34 lb. to stretch the wire ; and that the difference of these, or .126 lb. being taken out of the scale-dish, the wire would then sound the proper tenor cliff C of concert pitch.
Then, a round tile 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 flat, 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 beats of the fork and wire, with this comparative 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 vibrations for C', and should be so marked : and if for Mr Loeschitan's proposed enharmonic or perfect piano forte, (see Phil. Mag. xxxix. p. 423.) the forks should be adjusted to 237.032 for C'.
In the tuning of organs and piano-fortes by tables of beats, it will be improper to trust to a tuning-fork, or even to a standard pipe, perhaps, for adjusting C for tempered scales; but the second, third, or fourth, of the methods laid down by Dr Robert Smith, in his Har monics, 2d edit. p. 195, or his fifth method, p. 220. should
sonic one or more of them be repeated, and the pipe slightly altered each time, if necessary, in order to bring it exactly to the proper pitch of C, or of C', according as may be wanted.
On enharmonic organs of Mr Liston's construction, there are a number of concords that are ready tuned ; a comma sharpened or flattened, then, after being re-tuned or examined, may be conveniently used in Dr Smith's second method: so those which arc a schisma, sharpen ed or flattened on such instruments, stand ready for pro ving the truth of the pitch, by counting the beats of such schisma-imperfect concords, and comparing them with a calculation previously made (sec our article BEATS,) of the proper number per 1", or any other pe riod, as Mr Farey has shewn with respect to C E' # ; which fourth should beat 1.0336 times per second. See Phil. Mag. vol. xxxvii. p. 278.
The same gentleman has also derived a ready method of reducing an organ-pipe to 240 vibrations, by those who are furnished with a pocket-watch that beats or ticks 5 in a second, or 10 half-ticks, as those for experi mental purposes always should be made to do, from con sidering the equal-beating system proposed some years ago by Earl Stanhope, (see STANHOPE'S Temperaments,) viz. Tune the notes C, A 5 and C, in the octave below the tenor-cliff C (CC being a true octave) such, that the minor sixth C A 5 may beatfiat 10 in a second, or agree exactly with the half-ticks of the watch, and then try the major third above, A 5 C, and if this also agrees, but beats sharp, exactly with the ticks of the watch, the up per C vibrates just 240 times per I", as it should do. If the third beats faster than the watch, the C's must each be lowered, until, by repeating the operation, they are found to agree ; and the reverse, if it beats slower than the watch. With C, A 5, and f, when C f is a perfect I 1th, C A 5 beats 105, and A Vir !Oil: with C, E, and A 5, when C E is perfect, C A 5 is 10 5, and E is 10 The complete .table of the beats in the octave above the tenor-cliff C (240), in the system above mention ed, spews other whole numbers of beats in a second; they may also be useful, with 10-beat, or with other watch es ; which beats, therefore, it may be right to set down here, viz. with C, E 5, and G, when CG is perfect, or without beats, CE 5 beats 15 flat, and E 5 G beats 15 sharp; with C, E 5, and C, CE 5 is 15 5, and E 5 C is 15 ; with C, E 5, and B, when CB=V+Ill perfect (by help of G between them), CE 5, is 155, and E513 Is 305 ; with C, Al,, and B, (CB as before) CA l, is 205, and A 5B is with C, Al,, and e, when C e is a perfect Xth, CA bp is and e is 405 ; with C, E, D5, and e when CE is a perfect III, and Ee an °eta% e, ED 5 is and Dl, e is 20[7 ; with C, E5, B, and el,, when CB is V+Ill perfect, and CE5 is 15b, EBB is 30b, and Bet? is 30g, then E5 e5 is a true octave ; with C, F, A bp and f, when CF is per fect, FA l, is 20b and Al, f is 20:4, then F and f is a true VIII : and lastly, with C, Al,,B, and a 5, when is 205, A1B is 305, and Ba 5 is 30t1, then CB is V+Ill, and A 5 a 5 is VIII, both perfect ; which many curious definite proportions between the beats (to the pitch 240) arc of far more importance to the tuner, than any thing else belonging to this irregular douzeave, which Mr Farey has calculated, from his lordship's suggestions ; and among them, proportions will be found, either in this or the inferior octaves ; (where these beat only half as fast in succession as we descend,) that will watch the ticks of many watches,