ill 





SCALE. 



rj 



In til the uw*t accurate species of drawing, it it better to rely on 

 tables and * really K"xl scale uf equal part* than uu any u{ the > 

 cale*. though the Utter are generally rery good, and will do abundantly 

 well for ordinary purpose*. 



Long Male* of equal parti are mad* with different subdivisions, 

 ranging from the 3uth part of an inch t.> the 50th. If the substance 

 of Uie Male be ivory, an inch will very well bear diviaion into 60 part*, 

 but 50 U more convenient for decimal calculation. A common ivory 

 pale, of a rectangular form, uch a* U usually foun>l in eases of 

 drawing-instrument*, if it have no trigonometrical line* laid down, 

 tuually contains the following aoalea of equal part* : 



1. The quarter of an inch divided into 1 equal parti, each of which ia 

 again subdivided into 10 equal part* by a DIAGONAL SCALE. There are 

 commonly two diagonal Kales, one at each end of the scale of quartern, 

 the on* on the left dividing the 8th of an inch into 100 part*, and thu 



11 the right the quarter. It will easily be aeen that the 400th of 

 an inch i* a lueletaly autall quantity, even when the lines are drawn on 



2. A *et of scale* in which the inch is severally divided into 30, 35, 

 40, 45, 50, and 60 equal part* : 10 of these parts make, in each case, 

 one of the larger aiibdivisions of the scale, and one larger division is 

 also divided into 13 equal part* ; so that, when the larger division is 

 made to represent a foot, feet and inches may be easily laid down. 



8. A set of scale* in which the larger divisions are J, J, |, |, |, {, J, 

 and | of an inch. The larger division ia, as before, divided both into 

 10 and 12 parts. 



n trigonometrical lines are laid down they are usually one or 

 two scslee of cAorrfj, the radius of each of which is found by its chord 

 of 60 degrees ; a scale of mm'u, which U nothing more than a scale of 

 chords, the angular unit being, not a degree, but a point of the 

 compass ; a scale of sines, with one of secants sometimes added ; a 

 scale of tangents, and of semitangents, the latter being really the same 

 scale as the former, but marked with double angles, aemitangent ln-ing 

 a technical term, not for the half of a tangent, but for the tangent of 

 half an angle. We shall have something more to say of these lines 

 under SKCTOH. In Hunter's scale, as it is called, which is a scale of 

 3 feet in length, used in navigation, there are also scales of logarithms, 

 of numbers, sines, tangent*, &c., and also a scale of meridional parts 

 [ Hi MH LINK]; of these logarithmic scales we shall have to speak more 

 purticiunrly under Si. HUM: KULK. 



i i: (Mii-ie). A great deal has been written on this subject, 

 by mathematicians, by musicians, and by those who combine both 

 characters ; but, from various circumstances, hardly anything which is 

 accessible to the young arithmetician wishing for something which 

 may really be a help to him in his musical studies. The Greek scale 

 FHACHOBD], the only fruitless subject of inquiry out of all 

 that U Greek, has exhausted the learning, science, and ingenuity of 

 the best writers, with no result but this, that over-refinements of 

 theory are found either to have hindered practical excellence, or to 

 have arisen out of the want of it ; most likely the latter. The learning 

 however which it was necessary to apply to the explanation uf the 

 Greek writers, hat! made it usual to write on this subject more pro- 

 foundly than on others of the same difficulty : it is our object in the 

 present article to explain the musical scale, if possible, more simply, 

 and in its simplest parts : leaving to the article TtMi'LHAMKNT such 

 considerations as, arising out of the present article, are required by 

 those who would understand the higher practical details of the 

 subject. 



The object of music laeing to please the ear, or the mind through 

 the ear, there is no test of excellence nor criterion of fitness, in any one 

 detail, except the opinion of the best judges. This seems to assume 

 the question, for the best judges can only be described as those who 

 best know what is good music. This circle cannot be avoided, either 

 in speaking of music or any other of the fine arts ; to taste we must 

 appeal, bt not to the taste of every one. All we have here to do with 

 this is to remark, that the mathematical considerations employed in an 

 article like the present are not to be considered as placing the musical 

 Kale u|m a mathematical basis, but simply as showing that there is 

 something like an explanation of those rules which derive their 

 authority, not from the mathematical system which embodies them, 

 but from tho sanction of the majority of cultivated ears. Those things 

 wliii-h nre agreeable in found to be in certain matin ma- 



tical relations to one another which make the theory of the musical 

 scale simple anil interesting : but had it been otherwise, we should 

 have left mathematical simplicity, and preferred a more pleasing 



The sounds which are agreeable to the ear are found to be those 

 which are the consequence of vibrations of equal duration following 

 one another: and the pitch of the note depends on the rapidity of 

 vibration only. [ACOUSTICS.] The note called A, for instance, sounded 

 at the same time on a harp, a flute, and a horn, presents three different 

 character*, three different intensities, but only one species of vi 

 aa to the time of .lasting. If the first instrument communi. 

 vibration* in a second to the air, go does the second instruiin-nt, and 

 also the third. With the difference of intensity or loudness, anil with 

 tin- difference of character, the twang of the harp, or the tone of the 

 I., rn. we have nothing to do in considering the place of the note they 

 sound in the scale; a cultivated ear discovers that they sound tho 



same note, and a mathematician knows that they severally communi- 

 cate to the air the same number of vibrations per second. 



Let us then suppose a string to be mounted, and stretched at both 

 r, better still perha|w, suspended v,-rti. -ally by our end, and 

 bearing a weight at the other. If this string be then set in M 

 by the finger or by the bow of a violin, a musical (that is, a pleasant) 

 sound is produced, if the string be not too long, nor stretched by too 

 small a weight. With the phenomena of vibration, as connected 'with 

 the length, material, and stretching weight of the string, we ha\ 

 nothing to do [Conn] except to remark, 1, That the ear observes that, 

 material and tension remaining the same, the loug< r tin -trim; the 

 l.iwir tin- tone, and vict mid. 2, That tin- knows 



that, ctrleru jiarllitu, the longer the string the fewer the numb, i ..i" 

 vibrations in a given time, in inverse proportion to the length. Tims 

 if a certain string, stretched by a certain weight, give loo vibrations 

 per second, a string of half the length, stretched by the same weight, 

 will give 200 vibrations per second. If a vibration mean a double 

 motion of the string, once backwards and once forwards, the effects 

 begin to be musical soon after the string is short enough, or stretched 

 enough, to give 30 vibrations per second. 



The number of musical tones is, theoretically, infinite : that is, 

 between any two tones as many different tones as we please 

 interposed, no one of which is so high as the higher, nor so low- 

 lower. Highness and lowness of tone are terms which are purely 

 , and refer to an effect upon the ear which does not admit of 

 definition ; common terms usually distinguish onl; 

 thus, a tone disagreeably high is a squeak, and one disagreeably low is 

 a growl. There is no absolute reason why we should call the ' 

 high and the latter low, rather than the contrary ; and in fact tin- 

 earlier Greeks (naming them after the jarts of the throat in which 

 iln-y thought they were produced) called the squeaking soun 

 and the growling ones high. But while we endeavour to separate 

 names from things, we must not forget that there is much which all 

 men acknowledge of real connexion between the associations whirh 

 accompany sounds and those derived from other sensible phen 

 For instance, it would be impossible to persuade any one, that if light 

 and darkness were to be imitated by musical tones, the light ... 

 be represented by low notes, and the darkness by high notes : 

 composer who should accompany words expressive of transition from 

 darkness to light by a marked descent from the higher part of the 

 ' tho lower, would be thought to mean irony or burlesque. Xo 

 satisfactory explanation has ever come to our knowledge as to what 

 associations are awakened by the lower notes of the scale vvh id i con- 

 nect them with darkness; but that this connexion doe- exist is 

 certain. 



Taking such a string or inouochord (sinyle siring) as abovi 

 it is immediately found that any alteration of its length produce 

 alteration of the tone. If the change be very slight, a dull or 

 unpractised ear may not readily perceive it ; but let the alteration bo 

 carried a little further, and there con be no difficulty. Sueh ton.-;, 

 near to one another, when sounded together, have a 

 jarring effect, accompanied by beats [Acoustics] ; but when the 

 second string has been considerably shortened (say that this is .1 me 

 gradually), the disagreeable effect ceases almost at once, and at tin- 

 moment when the shortened string is to the longer one as five ; 

 Two sounds are then heard which harmonise together, and on their 

 joint eU'ect the ear dwells with pleasure until it becomes monotonous 

 \this very common word is itself derived, as to its common sit 

 tion [here used, from the wearying effect of the same tone, or 

 tones, long continued). In the mean while, and during the shon 

 of the string, the joint effect, though always disagreeable, is not 

 equally so throughout; and there is one place in particular where the 

 effect, though not agreeable to the beginner, is bearable fora little 

 while, and highly agreeable to the practised ear, which knov. 

 full compensation is at hand in what is called the n-.-nlution 

 discord, or transition to a more harmonious combination in a manner 

 which seems peculiarly natural. This intermediate and more to.. 

 phase of sound takes place when the shortened string is to the other 

 as eight to nine. Moreover.it maybe ol lat this la.-t 



bination, hardly bearable, is rendered perfectly so if the two 

 instead of being sounded together, arc mode to follow each other in 

 succession, no matter how rapidly. In both these cases i 

 will observe that the proportion of the lengths of the 

 some small numbers, five to six, and eight to nine. And [\ 

 of experiment, that the more simple the proportions of tl; 

 two strings (stretched by tin- same weight), tho moi 16 com- 



bination in music it is usual to say, the more agiv. ied b\- 



itself; but to this we cannot subscribe, as we 



ears the more complicated combination of a third (presently to be 

 described) is more agreeable than the less conip 1 oi n'lifili. 



Instead of speaking of the lengths of the 



itive numbers of vibrations in a second, which are inversely as 

 the lengths. Thus, two strings of ten and seven feet, stretched by tho 



Woolhousc ' On Musieul Intervals,' p. 04. The author repeated the exprri- 

 iin-nt- of Fischer [Am. -in-, ami founil n tmmochord thus constructed better 

 than the common one for tho purpose. Hiu result wan that A (the second space 

 of tho treble clef) made 424 vibrations in one second. 



