It has also been shewn by that gentleman, that each of the columns of any notation of small terms, or inter vals, so that negative signs are avoided, except when they affect a whole column, furnishes a set of artificial commas, as must, indeed, be evident, since there can be no carrying forwards of whole numbers from one co lumn of notation to another, in adding, or borrowing in performing subtraction, as in the columns of pounds shillings and pence in money calculations, &c. ; hut each column, when there arc no decimals, concurs in sheaving, independently, the same results, as far as they can be expressed therein ; yet carrying and bor rowing are sometimes used, at the rate of .14966096, &c. between/ and 2, and of .007862405, &c. between in and E.
The in and f, in Mr Farcy's notation, are the small est intervals that arc yet known, we believe ; but two others, d and F, occur in our Table, Plate XXX. Vol. II. between the latter and 2, the largest term of this no tation ; which term was fixed upon, on account of the many important relations which the schisms (2) bears to other intervals; but Mr Farcy has tried other nota tions of these small intervals, as follows, viz. 612d+ 1848f--559m=VIII, 612F+624f -559m=VIII, 306r -294f+ 53m = VIII, 559E+53d+ 17 If=V1II, 2791r +53d--1081f=V111, 65r + 4292+53F = VIII, 65c -12R-312= VIII, - =-. VI I I,53c+ 292+12f = + 12r+52=--V1II, 412=VIII, 12Jc-1-53c-312=VIII, 125.c+53c-19Z =VIII, 12ir+ 41c-- 19E, = VIII, 12ll= VIII, 12f +29e+222=VIII, &c. Whence several different sets of artificial commas, of 1848, 624, 612, 559, 429, 306, 294, 2791, 171, 1081, 65, 53, 41, 31, 29, 22, 19, 12, respectively,'in the octave, might now readily be calculated by that gentleman's manuscript tables of intervals. The first of which, 1848, would give a set of artificial commas, considerably more exact in the very smaller intervals, owing to the largeness of the numbers, than that by 2, adopted by Mr Farey ; but the notation, whence it is derived, for other reasons be sides the negative sign in all its terms, is less adapted to general use than E,f, and in, and they would not prove in the least degree more exact, for all intervals larger than E. The notation that is adopted, besides furnish ing Mercator's and Farey's artificial commas, contains another set of these (or of artificial half notes rather) in its middle column, which slims the number of de grees, or half-notes, or the finger-key, (12 in the oc tave,) to which any interval belongs, and which is of very considerable use in the practice of musical corn putations.
The musical student must, however, be on his guard in using the above, and all other artificial commas, in his calculations ; always remembering, that when .z..12r/c cumber; of one interval only are used, that the same have no Fixed or precise values, but within small limits, (if the number in the octave is considerable,) in expressing every different interval to which they are applied. Thus, if 1 of l'arey's artificial commas be supposed exactly to represent the schisma, or 1', (and these commas cannot represent any smaller interval) : then in 5, the alto comma or 1 is 1.000000 5; but in 10 for the minor comma, or C, the same is 1.0007862•; in I I for c, each is 1.0007020; in 12 for (j. each is 1.0006552 ; in 21 for 1, each is 1.007488 ; in 57 for S, each is 1.0033154 ; in 104 for T, each is 1.003558j ; in 197 for III, each is 1.0037173 ; in 358 for V, each is 1.0036072 ; in 612 for VIII, each is 1.0036154, Sze. differing in every interval, and so they would do, on tile supposition that 1-'6 E, IV, ci, &c. any one of them exactly represented this artificial comma ; and vet the curious and admirable properties of these numbers arc such, that, by adding and subtracting, they truly re present, and with the utmost facility perform, all the operations with intervals, as certainly as the logarithms, or the multiplication or division of the ratios can do, except with intervals smaller than, or very nearly equal to, the unit, or which do not differ from one another by more than this unit, or by some fraction of it ; and hence the great advantage of a series of large numbers, like 612 in the VIII. But multiplication can only cor
rectly be performed on these artificial commas in some cases, and division in still fewer cases, and only where there are no remainders after such divisions ; and, ge nerally, where fractions of intervals arise, as in all tem pered systems must be the case, they cannot be relied on, or used, except as a rough check on more correct and difficult calculations ; but in all such cases, either the three columns, as E f, and m, or logarithms of some sort, (as the decimals of any one interval always are,) to a sufficient number of places, must be used in all calculations relating to temperaments, or other frac tional intervals. See ?ristoxenian COMMON MEASURES. MEASURES, .4ristoxenian,in music. These were, as Dr Holder expresses it, (Treatise, 1st edit. p. 149,) a Irrational contrivances for expressing the se veral intervals." Aristoxenus, and many others among the ancients, having vainly imagined, that by dividing all the principal degrees of the scale, or consonances, into all their aliquot parts, that some one, in each of all these sets of parts, would be found exactly equal to each other ; and this small interval, to be found in the aliquot parts or every other one, was the least common measure of all the intervals, which they laboured in cessantly to find. Had the ancients been able as quick ly and accurately to extract the 12th, 30th, 72d, &c. roots of numbers, as we now can, and even of still higher numbers, almost without limit, by the help of a Table of Logarithms, as we have shewn, as to the 104th, 197th, 358th, and 612th parts of the T, ill, V, and VIII, respectively, in our article CONIMON MEA SURES above, this delusion must quickly have vanished; and it would have appeared, as we have shewn in the article above referred to, that an artificial comma has a different value in every different interval to which it is applied. But owing to the tedious and difficult labour of extracting the roots of the terms of musical ratios by arithmetical operations, and the impossibility almost of avoiding errors therein, it was not difficult for those engaged tit these calculations, to persuade themselves and their followers, that they had made out certain of their roots, thus obtained, to agree exactly. And thus it should seem, that the 12th part of the major tone, the 30th part of diatessaron, or fourth, and the 72d part of the octave, were known by the common name of the Aristoxenian common measure, and were supposed by many to be equal, and adapted accurately to measure these and all other intervals ; and though others showed the absurdity and impossibility of this abstractedly, or in some particular cases, yet the doctrine could not be exploded entirely, until after the invention.of logarithms, when it quickly vanished, like a mist before the sun. We shall, however, subjoin the most useful particulars of these measures, viz.
Which Table may prove of considerable use to those who wish to examine the ancient musical writings, and see the extent of the errors into which such authors may have been led, by assuming the three intervals above to be one and the same. The last of the above intervals has been called the duodecimal of a tone, and supposed to equal the first of them ; from which it differs nearly one in the fourth place of corn. logs. or not much more than : the middle one appears to be less than an arithmetical mean between the other two. (?)