APOTOME, in MUSIC, (P) is a small interval, re maining after a limma is taken from a major tone. The curious relations which the magnitudes of musical in tervals bear to each other, having of late years engaged the attention of several ingenious and learned men, par ticularly the late Mr Marmaduke Overend, the late Dr Boyce, and Dr Callcott; and their manuscripts being now lodged in the library of the Royal Institution in London, we are enabled to present our musical readers with the results of their elaborate researches into the relations which obtain among the several ratios, express ed by the numbers, into whose composition no prime number larger than 5 is allowed to enter. In the ma nuscripts above referred to, certain letters and charac ters are affixed to the most important and useful inter vals in these calculations, in order to be able to exhibit their relations in the form of algebraic equations. Plate XXX. exhibits these characters in its first column, the second of which contains the names of those intervals for which they are substituted, and under which titles the values and relations of these several intervals will be more fully exhibited in the progress of our work. The table in Plate XXX. being calculated to serve as an explanatory index to all the articles which treat of the magnitudes or relations of musical intervals, it may be proper here further to observe concerning it, that the third column contains the several ratios, expressed in the form of vulgar fractions, where the number of figures was not too great for our room ; in which cases, seven at least of the first figures are given ; and it is mentioned how many figures there are in the whole. For the convenience of reading these numbers, we have divided them into periods of three each by commas ; and wherever the decimal point may be wanted, we shall, as Dr Charles Hutton has recommended gene rally to mathematicians to do, distinguish the same by a period reversed ; which thus standing higher in the line than any of the points used in printing, will serve to distinguish the whole and decimal numbers. The fourth column slims, how many times the several primes 2, 3, and 5, are involved in the fractions in the preceding column ; the sign — distinguishing those in dices thereof, which compose the denominator or largest number. For example, against c, we have 4 4 1,
24.5 34 which is equivalent to —; and this, by evolution, gives as in the table.
81 The fifth column contains the reciprocals of the loga rithms; the commas, marking the end of the 7 places of which the common tables consist; and the four figures on the right hand thereof extend the logarithms to 11 places, by which an abundant degree of accuracy may be obtained for the most curious purposes. If the loga rithm itself be wanted instead of its reciprocal, it is readily obtained, by writing clown the complement of every figure to 9, except the last, which is to be the complement of 10. Thus, if I want the logarithms of the minute and the fourth respectively, the operations will stand thus : Recip. . . . 38,5342 2 Recip. . •1249387,3661 Logar. . .9999961,4658 5 Logar. . .8750612,6339 See BINARY Logarithms.
The last column contains the values of the several intervals in the table, expressed in a notation, which Mr Farey first recommended to the notice of musical calculators, through the medium of the Philosophical Magazine, (v. xxviii. p. 140,) viz. by the intervals, schisma (s), lesser fraction (f), and minute (m), being the three smallest intervals known, that are adapted for the purposes of musical computation; and which are found to give the greatest facility in the comparison of almost all intervals with each other, without the incon venience of negative signs. We shall, therefore, re duce all the intervals of which we have to treat, into these terms, as common measures of their magnitudes. One example will fully explain the use of this column, and mode of notation. The medius residual is known to be equal to the sum of the schisma and minor resi dual; and by inspection, it appears in the table, that X-1-22-Ff is equal to 3Z+f. The same truth would be equally apparent, by adding either the logarithms toge the• or the indices of the primes, (regarding the posi tive and negative signs,) or by multiplying the nume rators and denominato17 of the fractions together re spectively.