7 '18 



BROAD WOOD'S TEMPERAMENT. 



wood's 

 Tempera- 

 ment. 



a proportion I will endeavour to explain. If the 

 whole be tuned correctly, the G$C with the D$( which 

 is the same tone on the piano-forte as E[>) will be 

 found to make the si me concord, that is, possesses 

 the same interval as the other Jlfths," but, we must 

 observe, it is impossible that it should do this, since 

 this bearing or resulting fifth will beat 1.3943 sharp, 

 instead of .9175 Jlnt, which it would beat if E|j 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 A]j), 

 but 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 tempera- 

 mtnt generally adopted ;" " suppose two strings 

 B and C in the middle octave of the piano-forte, to 

 be, one a full semitone from the other," (we have 

 here used the major semitone S, or -J-|-> which is the 

 interval B C in the natural or diatonic scale of all 

 correct singers and violinists, and on the Rev. Henry 

 Listen's patent organ, without any temperament in 

 its harmony, now exhibiting at Flight and Robson's 

 in London, being VIII-VII. See the Philosophi- 

 cal Magazine, Vol. XXXVII. 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 toler- 

 ably steady hand, and a few trials, you will be 

 enabled to enumerate forty gradations of sound, 

 which I call commas." Now, any one unac- 

 quainted with the subject, would think from this, 

 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, ac- 

 quired 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 sup- 

 posed 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 es- 

 sential to the discrimination of one system from another, 

 and of exhibiting the peculiarities of each, which are 

 sufficiently distinguishable, when the tuning is cor- 

 rectly done, by the beats, a monochord will not do 

 it, as we shall shew in the article SONOMETEH : 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 Broadwood 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 cer- 

 tain, that the ear could not discriminate the semitone 

 or interval (iOS + Sjm,) or its parts, of which one- 

 fortieth (1.00065522:) is the proper isotonie temper- 

 ament, nor could it better appreciate another interval, 

 ( 48s -l[- 4 m, or 4cl ) or its parts, of which one-fortieth 

 or 1.200786S (= T ^d or f-S + ^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 Broad- 

 wood intended, considering the degree of contempt 

 with which he affects to treat the mathematical and 

 only true or satisfactory method of treating this sub- 

 ject, which we are so anxious to see more generally- 

 understood by professors of music in general, and 

 which would prevent them from being the dupes of 

 every random or interested proposition respecting 

 temperament, which is brought forwards. 



As thi.'i 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 Lon- 

 don, 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 operations 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 

 algebraic signs -(-, , X, -5-, and =r, (foraddition, 

 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 ex- 

 amples of the calculations necessary in considering 

 musical temperaments. 



By a reference to Plate XXX., in Vol. II., and ar- 

 ticle APOTOME, where it is explained, it will be seen that 

 the reciprocal logarithm, or recip. log.of S, or the ma- 

 jor 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, = 1.4297244S : and, from the same Plate 

 we get .1760912,6, (not .17669, &c. as there engra- 

 ved 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 down- 



Broad- 

 wood's 

 Tempera- 

 ment. 



