The interval, described as an APOTOME in the begin ning of the present article, seems to have been so deno minated by Pythagoras, in treating of the Greek scales of music, and by most modern writers, except Dr Boyce, who has denominated it an ABSCISS. It was the HEMI TONUM majus of Boethius. Dr Boyce, (p. 74 of his MSS.) has described this interval, as equivalent to that by which a sharp (*) elevates any given note ; and Dr Callcott, (Gram. art. 213,) describes it as the Chromatic SEMITONE; by which, according to him, a )iic elevates, and a b depresses any given note in the scale ; but other authors have assigned different values to these very common marks in use ;c and See SHARP and FLAT.
The ratio of the apotonte the component primes of which are its common logarithm is 9714811,6927; in the binary logarithms of Euler, or decimals of an octave, it is =094734; in those in which the major comma is the modulus, it is expressed by 5.28612 xc ; and where the schisma is the modulus, we have P=58.188976xE; its value in the new notation is 58E+f-1-5m; it has been found by Mr Farey to be equal to the difference between three-fifths and four minor fourths; and hence, according to the general method invented by him, of tuning any musical inter vals, however small, by means of perfect concords only ; and the apotome may be tuned correctly upon an organ having a sufficient range of pipes. See TUNING.
The following equations exhibit the value of the Apo tome, in terms of the several intervals in the Table, Plate XXX.
The above Tables, by shewing the exact relation which the Apotome has to the many diatonic intervals represented by the several characters, will serve to cor rect the numerous errors into which some ancient wri ters on the theory of music were led, by their having no ready method of comparing ratios toeether, such as either of the four columns of Plate XXX. now furnish, but were forced to have recourse to the actual multipli cation and division of the numbers representing the ra tios, in every case of adding intervals together, or of subtracting them from each other. Even the late Mr had recourse only to this very operose method. except that he sometimes availed himself of the indices of the component primes for shortening his truly labo rious calculations, no part of which can ever need re peating, after his results ; and many others incorporated with them, arc thus exhibited for the amusement and use of the curious in these speculations. It can scarcely perhaps be necessary to mention, that the marks =, and , are the common algebraic signs for equal, more or addition, and less or subtraction, of the quantities that follow them : Thus, P=S+Z signifies that P is equal to I, added to S ; and P=--2fR signifies that P is equal to R, subtracted from 2f. Besides the above, other intervals have by different writers been denominated Apotome, viz.