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of all the other keys, there would be little difficulty; as a mere bisection of the intervals would, probably, answer every practical purpose. Thus 1 or 2 might be represented by √(1 ×); 2# = 32 by ( X ) and so on.
b. But, in practice, no preference is given to this particular key, which is denoted in music by the letter C; on the contrary, variety is purposely studied. If the ear absolutely required perfect concords, there could be no music, or but a very limited and monotonous one. But this is not the case. Perfect harmony is never heard, and if heard would probably be little valued, except by the most refined ears; and it is this fortunate circumstance which renders musical composition, in the exquisite and complicated state in which it at present exists, possible.
112. The limits, to which the ear will bear a deviation from exact consonance of musical vibrations, may, however, be judged of by the alternate reinforcements and subsidences of the sound called beats, which take place when two notes nearly, but not quite, in unison or concord are sounded together.
The nearer the sounds of the strings approach to exact unison, the longer is the interval between the beats. When the unison is complete, no beats are heard.
On the other hand, when it is very defective, they have the effect of a rattle of a very unpleasant kind. The complete destruction of the beats affords the best means of attaining by trial a perfect harmony.
a. Conceive two strings, exactly equal and similar, and equally drawn out from the straight line, to be let go at the same instant; and suppose one to make 100 vibrations per second, the other 101; let them be placed side by side, and at the same distance from the ear.
Beats. Resultant Sounds.
Their first vibrations will conspire in producing a sound-wave of double force, and the impression on the ear will be double.
But, at the 50th vibration, one has gained half a vibration on the other, so that the motions of the aerial particles, in virtue of the two coexistent waves emanating from either string, are not now in the same, but in opposite, directions; and the two waves, being by supposition of equal intensity, will, instead of conspiring, exactly destroy each other; and this will be very nearly the case for several vibrations on either side of the 50th. Consequently, in approaching the 50th vibration, the joint sound will be enfeebled; there will be a moment of perfect silence, and then the sound will again increase till the 100th; when the one string having gained a whole vibration on the other, the motions of the particles of air in the two waves will again completely conspire, and the sound will attain its maximum.
b. If we call n the number of vibrations, in which one string gains or loses exactly one vibration on the other, and m the number of vibrations per second made by the quicker, will be the interval between two consecutive beats.
c. Beats will likewise be heard when other concords, as fifths, are imperfectly adjusted.
Suppose one string to make 201 vibrations, while the other makes 300; then, at and about the 100th of one, and the 150th of the other, the former will have gained half a vibration, and those vibrations of the one, which fall exactly on those of the other, (see fig. 31,) being performed with contrary motions, will destroy each other; those which fall intermediate only partially. The beats will then be heard, but with less distinctness than in the case of unisons.
113. We may here notice an effect which takes place in perfect concords, and only in those which are very perfect, viz., the production of a grave sound by the mere concurrence of two or more acute ones.
a. If we examine the figure 31, which represents the succession of vibrations in a perfect, fifth, we shall see that every third of the one coincides exactly with every second vibration of the other. These coincidences, (so delicate is the ear,) are remarked by it, and a sound is heard, besides the two actually sounded, of a pitch determined only by the frequency of the precise coincidences; that is, in this case, a precise octave below the lowest tone of the concord.
b. In general, if one note makes m vibrations and the other n, while another, of which they may be both regarded as harmonics, makes one, that one will be the resultant tone, provided m and n be prime to each other; so that the only difficulty, in determining the resultant of two notes, is to determine of what they are both harmonics.
This will be done by reducing m and n, if fractions, to a common
denominator and ; then, if m' and n' have no common fac
will represent the fundamental tone. If, then, m and n be
integers, and without any common factor, the resultant will be represented by 1.
c. Hence follows a very curious fact, viz., that if several strings, or pipes, be tuned exactly to be harmonics of one of them, or to have their vibrations in the ratios 1, 2, 3, 4, 5, &c., then, if they be all, or any number of them, from the first onward, sounded together, there will be heard but one note, viz., the fundamental
For they are harmonics of the first note 1; and, moreover, if we combine them all two and two, we shall find comparatively but few which will give other resultants, so that these will be lost, as well as the individual sounds of the strings, all but the first, in the united effect of all the resultant unit sounds.
But to produce this effect, the strings, or pipes, must be very perfectly tuned to be strict harmonics. The effect can never take place by touching the keys of a piano-forte corresponding to the harmonic notes, because they are always of necessity tempered.
Iso-Harmonic Scale. System of Equal Temperament.
114. To return to the subject of temperament; the most simple method of distributing the imperfections of the chromatic scale is, to make the successive infervals between the notes all equal; and the scale thus obtained is called the scale of equal intervals.
This scale must, obviously, be the result of a system of equal temperament, which consists in making all the octaves perfect, and all the fifths and thirds equally imperfect.
a. If we count the semitones in the chromatic scale between (1) and (8), we shall find the number of such intervals 12. If, then, we would have a scale exactly similar to itself in all its parts, and which would admit of our playing equally well in every key; we have only to divide the whole interval from (1) to (8) into 12 equal parts. But, as the interval between two notes is nothing but the ratio, or quotient of the number of vibrations of the second divided by those of the first; these ratios are here all equal, or the numbers of vibrations form a geometrical series, whose first term is 1, last term 2, and number of terms 12 + 1 = 13. The following must, therefore, be the series,
1= 2o, 212, 214, 21
and the values of the fractions may be computed by logarithms, and we shall thus obtain the scale of equal intervals.
b. The impossibility of forming a scale, exactly similar to itself in all its parts, of 12 or any other limited number of notes, in which the fifths and thirds, as well as the octaves, should all be perfect, appears from the consideration; that, if we start from any note and obtain its fifth and then the fifth of this fifth, and again the fifth of this new note, and so on indefinitely, we shall never arrive at a note contained among the octaves of either of the notes before obtained; but shall form an endless series of new notes which must all be inserted in the scale. The series of fifths will, indeed, be
}, } × } = (})2, (§)2 × 3 = (4)3, (2)1... &c.,
Iso-Harmonic Scale. System of Equal Temperament.
and will consist of the different powers of, while the series of octaves consists of the different powers of 2; but, since no power of 2 is exactly the same with any power of, it is evident that no note can be found in one series which is contained in the other.
c. Theoretically speaking, the iso-harmonic scale is the simplest that could be devised; and, practically, though fastidious ears may profess to be offended by it, it must produce no contemptible harmony. It has, however, one radical fault; it gives all the keys one character. In any other system of temperament some intervals, though of the same denomination, must differ by a minute quantity from each other; and this difference, falling in one part of the scale in one key, in another in another key, gives a peculiarity of quality to each key, which the ear seizes and enjoys extremely.
This fact, in which, we believe, all practical musicians will agree, is alone sufficient to prove, that perfect harmony is not necessary for the fuli enjoyment of music. Most practical musicians seem to have no fixed or certain system of temperament; at least very few of them, when questioned, appear to have any distinct ideas on the subject.
115. It is a mistake to suppose, as some have done, that temperament applies only to instruments with keys and fixed notes. Singers, violin-players, and all others, who can pass through every gradation of tone, must all temper, or they could never keep in tune with each other or with themselves.
Any one, who should keep on ascending by perfect fifths, and descending by octaves or thirds, would soon find his fundamental pitch grow sharper and sharper, till he could at last neither sing nor play; and two violin players accompanying each other, and arriving at the same note by different intervals, would find a continual want of agreement.
116. Many different systems of temperament have been proposed; and some fruitless attempts have been