COUNTERPOISE. 



COUNTERPOISE. 



200 



In Florid Counterpoint two or more notes are written against each 

 note of the subject, or canto fa-mo, and discords are admissible. 

 Examples from Fux. 



Double Counterpoint, says the German theorist whose examples we 

 have quoted, " is a species of composition in which any of the parts 

 may be transposed into the tenth above or below, omitting some notes, 

 Counterpoint. 



the subject remaining stationary." Or, in other words, it is an inver- 

 sion of the parts, so that the base may become the subject, the subject 

 the base, &c., thus producing new melodies and new harmonies, " in 

 au artificial and wonderful manner," adds Pedro Cerone, who seems to 

 have been quite delighted with the contrivance ; at a period, however 

 (1613), when subtleties of all kinds were more valued than in the 





m 



Oar.lo fermo. 

 Canto fermo. 



,-^-Tf-f 



'jtBt- 





C"0Mf7K>Ifl. 



present day. Fux gives the subjoined examples of double counter- 

 point, from which it will be seen that this is little more than an ex- 

 tension of florid counterpoint. In the first example the canto fermo is 

 Ex. 1. 



the base, the counterpoint the treble. In the second, the canto fermo \s 

 the upper part, while the counterpoint is transposed a tenth lower, 

 and becomes the base. In the third, the canto fermo and counterpoint 





r -T~=- TI I Q. 



Ex. 2. 



3E 



Ex. 3. 



-ESE 



!SJh 



ig^g^gz 



remain as in the first example, while the latter, transposed a tenth 

 lower, is adopted as a base, as in Ex. 2 ; and thus is formed a com- 

 l>s:tinn in three parts. 



The study of Counterpoint is necessary as part of the education of 

 a good musician, though now much neglected ; it has, nevertheless, 

 been sometimes carried to such a length as to become pure pedantry. 

 For the principal rules of Counterpoint, and also for some brief 

 remarks on the question, ".whether the art of writing in parta," was 

 understood by the ancients, see HARMONY. 



COUNTERPOISE is, generally, a mass of brass or iron so disposed 

 as to keep a part of some instrument or machine m equilibria. 



The instruments employed in practical astronomy are generally 

 mounted so that their centres of gravity are supported, in which case 

 they require no counterpoise; but a reflecting circle, for example, when 

 placed in any position except one which is horizontal, has its centre of 

 gravity on one side of the pillar which supports it ; and a mass of 

 brass, so connected with the axis of the circle as to be always on the 

 i the pillar which is opposite the latter, serves the purpose of 

 keeping the instrument steady. 



A transit instrument, or a mural circle of considerable magnitude 



and weight, whose pivots would press heavily on their supports, is 



sometimes provided with counterpoises, one for each point which is to 



be relieved ; this is applied at one end of a lever which is supported 



by the pillar or stand of the instrument, the other end being con- 



1 , near the pivot, with the axle ; and thus the pressure is dimin- 



1 or wholly removed. 



A bridge which is capable of being turned on horizontal joints at 



one of it* extremities [DHAWUIIIDGE] usually has its weight relieved, 



"r aluiort wholly removed, by a counterpoise, so that the machinery 



yed to raise it has little except the resistance arising from friction 



. The counterpoise is at one extremity of a chain, which 



passes over a pulley at the top of a pillar, and is attached at the other 



extremity to some part of the bridge. 



Now, since a drawbridge, in being raised or lowered, exerts on the 

 suspending chains a strain which increases as the bridge declines more 



ARTS AND SCI. DIV. VOL. IH. 



from a vertical position, it is necessary, in order that this strain may 

 be always in equilibria with the constant weight serving as a counter- 

 poise, that the latter should move on a surface whose figure is deter- 

 mined consistently with that condition. The rules for the resolution 

 of forces give expressions for the strains to which the two parts of the 

 chain are subject in the directions of their lengths ; and, making these 

 equal to one another, there is obtained an equation which may be 

 shown to be that of an cpitrochoid ; such, consequently, is the figure 

 which should be given to the surface on which the counterpoise is to 

 slide. It should be observed that, in the investigation, the effective 

 weight of the drawbridge is not its absolute weight (acting at the centre 

 of gravity) ; but is equal to this latter weight diminished in the inverse 

 ratio of the distances of the fulcrum, or joint, from the centre of 

 gravity and from the point at which the chain is attached to the bridge, 

 generally the opposite extremity of the latter. 



If the pulley over which the chain passes is close to the elevated 

 extremity of the bridge when the latter is in a vertical position, the 

 length of the chain will, of course, be equal to the diagonal of a square 

 of which the length of the bridge is one side ; then, a circle whose 

 centre corresponds to the joint of the drawbridge, and whose radius is 

 the whole length of the chain, being supposed to revolve on the cir- 

 cumference of a fixed circle of equal radius ; a point supposed to be 

 on the produced radius of the revolving circle, or epicycle, at a distance 

 from its centre equal to the difference between twice the length of the 

 radius, or chain, and the length of the bridge will, by revolving with 

 the epicycle, describe the required curve. 



The great telescope of Lord Rosse, at Parsonstown, Ireland, turns 

 upon a joint at its lower extremity ; and, being intended to decline 

 from a vertical position both towards the north and the south, it is 

 provided with two counterpoises, in order to facilitate its elevation or 

 depression. One of these weights is on a chain which is attached, at 

 one extremity, to a point near the object end of the telescope, and, at 

 the other, to a fixed point at the top of the wall ; this point being so 

 situated that, in elevating the telescope, the weight descends, and 

 describes, by its gravity, a circular arc coinciding nearly with the 



u 



