130 



Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



Many writers, from Sauveur downwards, have seen the convenience of using the figures of 

 '3010300, the common* logarithm of 2. Thus Sauveur, for one method, divides the octave 

 into 301 parts, so that if the higher of two notes make m vibrations while the lower makes 

 n, the integer in 1000 (log wi — logw) is the number of subdivisions contained in the interval, 

 quam proxime. The tuner of the pianoforte is required to estimate half a subdivision: for the 

 fifth of equal temperament is 175'60 subdivisions, and the perfect fifth is 176"09 subdivisions. 

 Even in practice, then, a smaller subdivision is required: and theory will hardly be content 

 without the representation of the 50th part of the smallest interval in common practical use. 

 I should propose to divide the octave into 30103 equal parts, 2508"6 to a mean semitone. 

 Each part may be called an atom; and we have the following easy rules, which suppose the 

 use of a table of five-figure logarithms. 



To find the number of atoms in the interval from m io n vibrations per second, neglect 

 the decimal point in log m — log n, or in log n — log m, whichever is positive. To find the 

 ratio of the numbers of vibrations in an interval of k atoms, divide by 100,000, and find the 

 primitive to the result as a logarithm. 



To find the number of mean semitones in a number of atoms, divide the number of atoms 



by — logs X 100000, which may be done thus. Multiply by four; deduct the 300th part of this 



product and its 10,000th part, adding one-ninth of this 10,000th part; make four decimal 

 places, and rely on three. Thus a perfect fifth has 100,000 (log 3 — log 2) atoms, or 17609, 

 which multiplied by 4 is 70436. The 300th part of this is 235, and the 10,000th part is 7, of 

 which one-ninth may be called 1. And 70436 — 241 is 70195, whence 70195, say 7"020, is the 

 number of mean semitones in a perfect fifth. 



To find the atoms in a number of mean semitones, multiply by 10,000 ; add to the result 

 its 300th part and its 10,000th part, and divide by 4. Thus 12 mean semitones gives 120,000 

 increased by 400 + 12, or 120412, which divided by 4 gives 30103. This rule is as accurate 

 as the value of log 2; the one which precedes is a near approximation. Both are consequences 

 of the equation 



1 1 / 1 1 \ 



— X -30103 = — 1 -) + . 



12 40 \ 300 10,000/ 



Dr Smith found that the D of his organ, the first space below the lines of the treble, gave 254, 

 262, 268, double vibrationst in the common temperatures of November. September, and August. 



* Euler, and after him Lambert, suggested the use of 

 acoustical logarithms; and proposed systems, of which the 

 bases are 2 and '^2. Prony gave both tables in his Inslruc- 

 tions Elimentaires sur les mot/ens de calculer les intervalles 

 musicaux, Paris, 1832, 4to. The second table shows at once, 

 in log m — log n, the number of mean semitones in the interval 

 whose ratio of vibrations is m : n. Prony has also calculated, 

 but I cannot give the reference, a table of logarithms to the base 



81 



— , which gives the number of commas in m iti, by logm— logn. 



The atom which I have proposed, which is the 540th part of a 

 comma, gives the commas by division by 60 and 9. 1 have my 

 doubts whether any tables will be so convenient as those of 

 common logarithms, used in the way I propose. Special tables. 



for purposes which do not often occur, are of value only when 

 tliey save complicated operations. Such tables are not in 

 the way when wanted ; and when they are found, their struc- 

 ture and rationale have to be remembered. 



It is a sufficient proof of the state of knowledge of the theory 

 of beats that a work which goes so deeply into the formulae 

 connected with musical vibrations as Prony 's makes no allusion 

 to beats. Previously to the use of logarithms, the arithmetical 

 calculations of the scale were very laborious. Mersenne makes 

 58| commas in the octave, the true number being 55J. Nicolas 

 Mercator corrected this in a manuscript seen by Dr Holder, 

 and then proposed an artificial comma of 53 to the octave, 

 which gave all the intervals very nearly integer. 



+ Writers are very obscure in their use of the word vibra- 



