BROADWOOD'S Temperament of the Musical Scale. For several years past, Mr. James Broadwood, a piano forte maker in London, has been celebrated for the excellence of his instruments, as to perfection of workmanship and tone ; and he has been supposed, also, to employ the best set of practical tuners for attending to the tuning of the instgruments of his customers at stated periods : in pursuance, therefore, of the notice we gave at the end of our article BEATS of our intention to present our readers with an account of all the most celebrated systems or methods of tuning keyed instruments that are known, constituting the most curious and important part of the science of harmonics, we avail ourselves of a communication which Mr. Broadwood lately made to the Monthly Magazine, (Vol. XXXII, p. 106 ; see also pages 238, 321, and 424), to give what that gentleman calls his "practical method" of tuning, which we shall do in his own words ; inserting, in parenthesis, the numbers of beats made in one second of time, by the several tempered fifths that are to be tuned, as they result from our calculation, which will be given at length below, along with some other matters, by way of explanation.

Mr. Broadwood, after mentioniing that most tuners begin their operations with the note C, says, "I prefer tuning from A, the second space in the treble cliff, as being less remote from the two finishing fifths, than any other point of departure : the A being tuned to the forte, (that for this particular temperament should make 403.0445 complete vibrations in one second of time), tune A below an octave ; then E above that octave, a fifth (beating flat .9744 times in one second) ; then B above, a fifth (beating 1.4598) ; then B below, an octave ; then F# a fifth above (beating 1.0929) ; then its octave F# below ; then C# its fifth above (beating .8183) ; then G# its fifth above (beating 1.2258) ; and then G# its octave below.

We then take a fresh departure from A, tuning D its fifth below (beating flat 1.3017) ; then G its fifth below (beating .8692) ; then G its octave above, then C its fifth below (beating 1.1618) ; then C its octave above, then F its fifth below (beating 1.5501) ; then Bb its fifth below (beating 1.0350) ; then Bb its octave above, then F its fifth below (beating 1.03826). The five fifths tuned from notes below, are to be tuned flatter than the perfect fifth, and the six fifths tuned from tones above, must be made sharper than the perfect (i.e. the lower note is to be made sharper than for a perfect fifth, thereby making the interval of the fifth flatter than the perfect as before), in a proportion I will endeavour to explain. If the whole be tuned correctly, the G# with the D# (which is the same tone on the piano-forte as Eb) will be found to make the same concord, that is, possesses the same interval as the other fifths," but we must observe, it is impossible that it should do this, since this bearing or resulting fifth will beat 1.3943 sharp, inslead of .9175 flat, which it would beat if Eb were altered to the same interval as the other fifths (or rather if it were made D#), or .9231 flat if G# were altered to such interval (or rather, made Ab), be in either of these cases , it will be seen, that the former tuning would be undone and spoiled ; but we must return to Mr. Broadwood, who says, though not correctly, p. 107, "the old system of temperament (having a quint wolf, on douzeave instruments) is now deservedly abandoned, and the equal temperament generally adopted," - "suppose two strings B and C in the middle octave of the piano-forte, to be , on a full semitone from the other," (we have here used the major semitone S, or 11/10, which is the interval B C in the natural or diatonic scale of all correct singers and violinists, and on the Rev. Henry Liston's patent organ, without any temperament in its harmony, now exhibiting at Flight and Robson's in London, being VIII-VII. See the Philosophical Magaizine, Vol. XXXCII, p.273), "with your hammer," says Mr. Broadwood, "lower down, or flatten C by the smallest possible gradations, until it becomes unison with B ; with a tolerably steady hand, and a few trials, you will be enabled to enumerate forty gradations of sound, which I call commas." Now, any one unacquainted with the subject, would think that Mr .Broadwood had discovered some hidden property of the full semitone, as he calls it, which disposed it to divide into just 40 smaller intervals, that the ear could appreciate so distinctly as to enable the tuner to make these commas all equal, than which nothing can be farther from the fact. Although he continues, "after having, by a little practice, acquired a distinct and clear idea of the quantity meant to be represented by the term comma, nothing more will be required to make the proper fifth, (for the temperament as above), after having tuned the fifth a perfect, or violin, or singing fifth, than to flatten the said perfect fifth, by lowering the string supposed to be tuning (the upper string), one of the afore-described commas ;" yet we may further add, without fear of being contradicted by the results of impartial trials, that without counting the beats which we have given above for that purpose, it is impossible for any tuner, however practised or expert he may be, to approach this system within tolerable limits : When we say within tolerable limits, we mean such as are essential to the discrimination of one system from another, and of exhibiting the peculiarities of each, which are sufficiently distinguishable, when the tuning is correctly done, by the beats, a monochord will not do it, as we shall shew in the article SONOMETER : Much less can the thing be effected by the ear, directing the "mere mechanical operation" of the tuning-hammer, (or winch used to tune the pegs on which the wires lap,) as Mr. Broadwod maintains, in a subsequent number of the Monthly Magazine, above referred to ; and where, with equal pertinacity, he insists, that an equal temperament is produced by these commas of his : It is true, as Mr. Farey has there observed, that Mr. Broadwood has not expressly defined his "full semitone," to mean the major semitone ; but it is certain, that the ear could not discriminate the semitone or interval (40Σ + 5 1/2 m,) or its parts , of which one-fortieth (1.0006552Σ) is the proper isotonic temperament, nor could it better appreciate another intervaal (48Σ + 4m, or 4d) or its parts, of which one-fortieth or 1.200786Σ (= 1/12 d or []&Sigma+1/1[] m) answers to the system of 12 equally-tempered fifths, but one of them sharp, which just occurs to us, without having been any where described, as far as we know, of which we shall say more under EQUAL-TEMPERED FIFTHS ; and which, it is not very probable that Mr. Broadwood intended, considering the degree of contempt with which he affects to treat the mathematical and only true or satisfactory method of treating this subject, which we are so anxious to see more generally understood by professors of music ingeneral, and which would prevent them from being the dupes of every random or interested proposition respecting temperament, which is brought forwards.

As this temperament of Mr. Broadwood's of which we are treating, or some other, which perhaps by chance, and without any fixed principle, his tuners practise, has obtained considerable celebrity in London, and being also the first that has occured to be described in our work, we trust that we shall be excused by oour more learned readers, for setting down the whole of the operations necessary for obtaining 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 algebraic signs +, -, X, ÷, and =, (for addition, subtraction, 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 their 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. S, or the major semitone, is -.280287,2. 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, = 1.4297244Σ ; and, from the same Plate, we get .1760912,6, (not .17669, &c. 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, according to the preceding directions, the tuning of it is upwards, 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 working backwards, such that the note C may have a log. of 2.3802112,1, answering to the number 240 of vibrations , 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, according 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 G# be deducted from that of Eb, which, in the present case, will give .1779140,6 for this bearing or resulting fifth, from which, taking the perfect fifth .1780912,6, we get .0018228,0, the recip. log. of the quint wolf or sharp and fifth in Mr. Broadwood's system, = 3.719106Σ ; and by reference to Mr. Farey's 15th corollary 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 x .0007007,2-.00058851,4 = .0018227,8 ; which differing only 2 in the eighth place of logarithms, shews that all the several operations in this column have been correctly performed ; otherwise they must have been gone over again and corrected. We next proceed carefully to take out the numbers in the logarithmic Tables, answering to the several notes marked by the letters in front of the first column, and place them opposite in column two, after the sign = ; the next operation is, to halve all these numbers where an octabe has been tuned downwards, as from A, B, F# and G#, which are to be placed below (and opposite to their respective logs.) in the first half of 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 table, as the letters placed after the second column indicate.<;/p>

We now turn to the new and correct theorem for calculating BEATS by our 4th method, and multiply the least number of vibrations by 3, and the greatest number by 2, *the terms of the ratio, 3.2) in each corresponding pair of vibrations of the fifths, 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 operation may be preserved for future use or revision. The products for G# and Eb, at the two extremities of the parts of the Table, may easily be deducted to obtain the beats, where they stand, and without transcribing. Methods so very simle 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 hop, 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 practise, the index and decimal point of the logarithm, in column 1, may very well be dispensed with. (t)

Richard Moody, "James Broadwood and ET, 1811." 2006(?)

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