313 



SCALE. 



SCALE. 



same weight, vibrate so that the one of ten feet makes seven vibrations 

 while that of seven feet makes ten. 



We now proceed to consider the most simple combinations ; and 

 first, that of two to one. Let the second string be half the first, or 

 make two vibrations while the first makes one : there is then not only 

 a joint effect which is agreeable, but a peculiar sameness of the two 

 notes, insomuch that two instruments made to play together in such 

 manner that the notes of the second shall always be of twice as many 

 vibrations as the simultaneous notes of the first, would be universally 

 admitted to be playing the same air, with no more difference than of 

 that sort which is heard when a man and a boy attempt to sing the 

 same air together. This perfect sameness, for so it will be called, 

 though the two instruments never sound the same tone together, 

 admits of no explanation ; for though the ratio of the simultaneous 

 vibrations is the simplest possible (two to one), there is no perceptible 

 reason why, because simple ratios generally give harmonious com- 

 binations, the most simple of all should produce an absolute feeling of 

 identity of character in the two tones. To this circumstance however 

 we owe the most material simplification of the musical scale ; for let 

 it be settled, for instance, what strings give agreeable notes between 

 those of 20 and 10 feet long, and division by two will give all the 

 strings which can be admitted between those of 10 and 5 feet ; thus, 

 if it be proper to admit a string of eight feet in the former set, one of 

 four feet must also take its place in the latter. 



Again, it is observed that the relative effect of two tones is always 

 the same as those of other two, when the numbers of vibrations made 

 in a given time in the first pair are in the tame proportion as the corre- 

 ng numbers in the second pair. Thus, suppose that in a given 

 time the numbers of vibrations made by four strings are 12, 18, 40, 

 and 60. Then, since we see that 



13 : 18 : : 40 : 60 or || = { 



we may say that, according as the first and second sounded 

 together are pleasant or unpleasant, so are the third and fourth ; also 

 if an air beginning on the first string require an immediate transition 

 to the second, then the same air begun on the third string will require 

 an immediate transition to the fourth. 



A inHtiral intrrcal, then, is given when the fraction which expresses 

 the proportion of the vibrations of its two notes in a given time is 

 given. By the interval | we mean that of two notes, the higher of 

 which makes three vibrations while the lower makes two. Thus, if 

 . and 30 be the numbers of vibrations made by three strings in 

 the same time, and we wish to find a fourth note which is as much 

 above the third as the second is above the first, we must not make a 

 string of 35 vibrations in the same time (as the beginner might do), 

 that is, not one of 30 + 23 18, but one of 30 x JJ, or 38 J vibrations in 

 the same time. 



Let us now take a string, and call the note sounded by it C, and let 

 the string of twice as many vibrations (or half its length) have the 

 feme name, with a difference (for the reason above given) ; call it C'. 

 Let us now seek for the simplest fractions which lie between 1 and 2. 

 Take the numbers up to 8 (the ear does not so well agree with 7 and 

 all higher prime numbers, why of course cannot be told, but sim- 

 plicity must end somewhere, and, by the constitution of the ear, ratios 

 in which 7 and higher primes occur are not agreeable), and form 

 every fraction out of them which lies between 1 and 2 ; we have 



l.i.l. f. (t = l). ! 



Put these down, with 1 and 2, in order of magnitude, and we have 

 ' { $ I J { * 



Take such a set of strings that while the first makes one vibration the 

 second make* J of a vibration, the third J of a vibration, and so on 

 up to the last, which makes 2 vibrations ; or take a set of strings 

 in illy stretched, of which the length of the first being 1, that of the 



I is L tc., and of the last J. Every one of the notes thus pro- 

 duced will I* agreeable when sounded with the first, and if the first 



r, tlm musician will have the following part of the scale before 

 liim in its most natural form : 



C TSf E F O A C 



1 5 } i i , 2 



Them intervals liare the following names; why, will presently >>< 



' T; 



f minor third. } Ifth. 



f major third. | major sixth. 



} fonrth. 2 octarc or eighth. 



We hare not yet, however, got a sufficiently agreeable scale, and the 

 reason* why the ear will not be contented with the preceding most 

 simple concords, must be derived from observation, from which it 

 ap[x<nn 



1. That a frequent repetition of sounds very near to one another 



pleading to the uncultivated ear. Now the interval from tlir 



minor to the major third U u follows : the first makes f of a vibration 



while the second makes f, or the first makes 1 vibration while the 

 second makes f x |, or g. This is much too near to a unison for con- 

 tinual repetition. 



2. That a frequent repetition of sounds too far from each other is 

 not pleasing to the ear, after a little cultivation. If we look at the 

 intervale from the fourth to the fifth, and from the fifth to the sixth, 

 we find | and ' for their representatives, while from the fundamental 

 note to the minor third, and also from the sixth to the octave, the 

 interval is !j, much larger than the preceding intervals. 



Both these defects, as must easily be seen arithmetically, and as the 

 ear finds out for itself, may be remedied by inserting a note between 

 C and E in place of E |>, which shall make a better division of the 

 interval C E, and by placing an additional note between A and C 1 . 

 But how are we to choose these additional notes 1 If we cannot have 

 any more very simple consonances with the fundamental note, we 

 must take those tones which make the simplest consonances with 

 other notes, and the more they make the better. We have already a 

 repetition of some consonances ; for instance, 



Interval FC 1 is 2 

 Interval GC 1 U 9 

 Interval FA Is J 



J = 4 , or a fifth. 



i; = ^ , or a fourth. 



$ = f , or a major third. 



Now since |xj=|, we see that a notejj, or one which makes 

 vibrations while the fundamental note C makes 8, will be a fourth 

 below G, and J divides C and E well, the three notes 1, S, f, giving the 

 intervals jj , ^>, already found in another part of the scale. This note 

 is D. Again, observe the interval from E to F, or i, and take a 

 fifth above E, or f x | or y : this fraction falls between J and 2, and 

 looking at the intervals of |, y, and 2, we find J and }, both of them 

 intervals already found. This note \f, or which makes 1 5 vibration* 

 while the fundamental note makes 8, is B, and the usual scale of 

 civilised nations, called the diatonic scale, is now complete in the 

 following 



CDEFGAliC 1 



t 1 I I V i 



This diatonic scale seems then to be the scale of the simplest con- 

 cords of the fundamental note, with one alteration on account of the 

 too great proximity of two concordant notes, and one interpolation on 

 account of the too great distance of two others. If we examine all 

 its intervals, we shall find both repetition and variety as follows 

 landing for the interval from C to D, &c.), some new appel- 

 lations being added : 



We observe here the consonances mentioned before, two inhar- 

 monious intervals, a new species of consonance (the flat seventh) 

 standing as it were between the more perfect consonances and the 

 others, and new varieties of a tone, of a minor third, and of a fifth, 

 ilitliTing froui those already described, and flatter by the interval jjj. 

 This interval is called a comma, and though the ear can distinguish 

 a difference between the tones of two strings, one of which vibrates 

 81 times while the other vibrates 80, yet the difference is so slight as 

 to produce no prejudicial effect. With regard to the comparatively 

 harmonious character of the flat seventh, observe that 'j, is very 

 nearly equal to , differing only by the interval Jj. 



We have also the diatonic semitone, {J, which is incorrectly ii.-i-.in -.1, 

 since, if beginning with 1, we repeat the interval of a semitone twice, 

 we have JJ x Q, or gj, which is very near tu ?, sharper (that is, higher, 

 as flatter means lower) than a major tone by the interval gj and than a 

 minor tone by 55, very nearly. 



We shall presently resume the diatonic scale, but we now proceed 

 to mention two varieties of it. It seems to have been offensive to the 

 ears of rude nations to hear any semitones at all. If we deprive the 

 diatonic scale of K and B, the notes which make semitones with their 

 nearest neighbours, we have C, D, E, G, A, C, for all the sounds which 

 remain in the octave. This unfinished scale, as we should call it, is the 

 original scale of the Chinese, Avans, Hindus, and Eastern Islands, the 



* An inharmonious interval, when the notes are sounded together, 

 f Decidedly more harmonious than the seventh. 



