BEATS, or BEATINGS, in Music, are an audible phenomenon attending the sounding of two notes at the same time, which approach within certain limits to the producing of a conchord with each other, which the late' Dr Robert Smith, in his Harmonics, has applied, with the happiest effect, to the practical tuning of instruments, according to any given system or arrangement of the intervals. The phenomenon of beats forms also the means, by which practical tuners, unacquainted with theory or the exact comparative magnitudes of intervals, adjust the notes of organs, piano fortes, harps, &c. by the judgment of their ear, in the daily exercise of the tuning profession.
It seems, therefore, of the utmost importance for the advancement of this sublime and beautiful science, ' to exhibit theorems for calculating the number of beats made in a given time, divested as much as pos sible of the difficulties likely to deter the practical tuner and musician from attempting to understand and apply them to use, illustrated by an example in each case. For the satisfaction of such as are unable or unwilling to go into the nice and difficult theory on which these theorems are founded, nothing is so likely to inspire confidence in their truth, as well as in the right application of the rules they furnish to particular cases they may undertake to calculate, as the having several such theorems, involving different data, yet by means of which the same results are to be obtained.
Of the five methods given below, for calculating the beats of any tempered conchord from different data, the two first only have hitherto been published, as far as we are acquainted ; the first is the original method of Dr Smith, Harmonics, prop. and the second is that of Mr Emerson, Algebra, prob. ccii. 1st Method of calculating the Beats of an Imperfect Conchord.
Let the conchord, whose perfect ratio is expressed by —, (n being the least term of the ratio in its lowest terms) be tempered by the fraction 2 of a major com ma, (q being the least term of this fraction ;) also let Al and N be the number of complete vibrations in one second of time, made or excited by the acute and the grave notes of the above tempered conchord re spectively : and let b be the number of beats occasion ed by this temperament in one second.
Then, if the tempera ment be sharp, or the X m 2q xn xM chordgreaterthan per- 161p—q 161p+q feet, Or, if the tempera.
ment be flat, or the --2q x m X N 2qxaxM or chord less than per- — 161p+q 161p—q feet, Example.
If the conchordproposed, bethe minor sixth ( CLA )of Earl Stanhope's monochord system : here g is the ratio of the perfect conchord, and (Phil. Maw. xxvii. 195.) 4-1-=1 is the part of a comma nearly, (not 4-c,', as erroneously printed), by which the same is flatten ed also N=240, the number of complete vibrations of C, the bass note in 1" : and from the first of the lowest of the theorems above, "2X 21 X X 80640 we have = 22.6326, the 161 x 3563 ',eats in 1".
2d Method.
Let the conchord, whose perfect ratio is expressed by (n being the least term of the ratio in its lowest terms) be tempered so that its string, which, for sound ing the treble note of the perfect conchord was S in length, is altered to be s length : also let N be the number of complete vibrations in one second of time made by the bass-note of the conchord ; and let b be the number of beats occasioned by this temperament in one second of time.
Then, if the temperament be shop, — x N x s—S Or, if the temperament be flat, b=—XN Example.
If the conchord proposed, be the minor sixth of Earl Stanhope's monochord system : here i= is the ratio of the perfect conchord, and (Phil. Mag. xxvii. 196, and xxx. p. 1.) S=.625, and s=.6324554, are the lengths of string for sounding this perfect and tempered conchord with the bass note =1, respective ly : also N=240, the vibrations of C the bass-note in 1", and from the second theorem above, we have .