As this temperament of Mr Broadwood's of which we arc treating, or some other, which perhaps by chance, and without any fixed principie, his tuners practise, has obtained considerable celebrity in London, and being also the first that has occurred to be described in our work, we trust that we shall be excused by our more learned readers, for setting down the whole of the opera tions necessary for obtaining the vibrations and the beats of this system; as an example of the rules that we intend to submit, for enabling those to understand and perform all the necessary calculations, who are acquainted only with common decimal arithmetic, the use of the alge braic signs -1-, x, ÷, and =, (for addition, subtrac tion, multiplication, division, and equality,) and the use of the common Tables of logarithms, (oF which Callot's stereotype are the best,) than which nothing is more easy than to acquire a knowledge and facility in their use ; and to which we are the more induced, from there being no works extant, to which we can refer, for familiar explanations or examples of the calculations necessary in considering musical temperaments.
By a reference to Plate XXX., in Vol. II., and article APOTOME, where it is explained, it will be seen that the reciprocal logarithm, or recip. log. of S, or the major semitone, is .0280287,2. This, divided by 40, or removing the decimal point one place to the left hand, and dividing by 4, we get .0007007,2, the recip. log. of the flat temperament of the fifth, in Mr Broadwood's system, = 1.429724.4Z : and, from the same Plate, we get .1760912,6, (not .17669, Etc. as there engraved by mistake,) the recip. log. of V, or the fifth ; the difference of which two last numbers is .1753905,4 = the recip. log. of the tempered fifth, to be added, wherever, accord ing to the preceding directions, the tuning of it is up wards, and subtracted wherever the same is downwards, as in columns of the following table ; in which the VIII =.3010300,0, is added when an octave is directed to be tuned upwards, and subtracted when the same is to be tuned downwards. It is right here also to explain, that the logarithm of the vibrations of the note A, at the beginning, and in the middle of the first column of the table, has been assumed by previous trial, or work ing backwards, such, that the note C may have a log. of 2.3802112,4, answering to the number 240 of vibra tions, which is understood to be the present CONCERT Pitch, (see that article,) and to which the pitch of the instrument to be tuned, must be carefully adapted, ac cording to the rules that will there be given, (see also Dr R. Smith's Harmonics, prop. xviii.) otherwise the beats here calculated will not apply.
The first column in the above Table or process, had better be calculated through, as above directed, and written wide, before proceeding to the second, and let the resulting log. of C.738 be deducted from that of E b,
which, in the present case, will give .1779140,6 for this bearing or resulting fifth, from which, taking the perfect fifth .1760912,6, we get .0018228,0, the rccip. log. of the quint wolf or sharp and fifth in Mr Broadwood's system, =3.719106Z; and by reference to Air Farey's 15th corol lary in the Philosophical Magazine, vol. xxxvi. p. 374, or to our article TEMPERAMENT, we find that 11 x temp. of V — d, ought to give this same Vth wolf; or, 11 ; which differ ing only 2 in the eighth place of logarithms, shews that all the several operations in this column have been cor rectly performed; otherwise they must have been gone over again and corrected. We next proceed carefully to take out the numbers in the logarithmic Tables, an swering to the several notes marked by the letters in front of the first column, and place them opposite in co lumn two, after the sign = ; the next operation is, to halve all these numbers where an octave has been tuned downwards, as from A, B, FW and Gar, which arc to be placed below (and opposite to their respective logs,) in the first half or the Table, and to double all those where an octave has been tuned upwards, as from G, C, and Bb, to be placed also below, in the lower half of the ta ble, as the letters placed after the second column indi cate.
We now turn to the new and correct theorem for cal culating BEATS by our 4th method, and multiply the least number of vibrations by '3, and the greatest num ber by 2, (the terms of the ratio, i) in each correspond ing pair of vibrations of the fif ha, and place the products below and above in the intervals in column 2 ; by which means two numbers nearly alike will come together, ready for subtracting to obtain the beats, that are set opposite in column 4, to each of these pairs of products ; by which process, all the trouble and risk of mistakes in transcribing numbers are avoided, and the whole opera tion may be preserved for future use or revision. The products for Cok and lib, at the two extremities of the parts of the Table, may easily be deducted to obtain the heats, where they stand, and without transcribing. Me thods so very simple and easy as these, of obtaining the beats of the fifths, (and of all the other concords by the same theorem that has been referred to,) to the utmost degree of exactitude, will, we hope, stimulate many to apply them in the calculations on other systems, who have been deterred by the very operose method hitherto known and recommended for the purpose. In practice, the index and decimal point of the logarithm, in column I, ni.1- very well be dispensed with. (g)