In order to avoid negative signs, which are indispensible in the use of such large intervals as S, t, and T, the small est that any writer bad previously used as the terms of their notations, it occurred to Mr Farey to select the three smallest intervals that mr Orerend had di:,covcred, viz. f and m ; but this notation, as well as that by V, f and m, proving to have negative signs to nt, in every instance, as remarked in our arliele ('om MON Mi.InseuF,s Of Musi cal Intervals, he next tried Z, and in, which has been found in its most extended use, the best adapted by far, than any others of the numerous ones subsequently tried, for a general notation of musical intervals; and, as such, we have in our work adopted and used it.
The methods by which Mr Farcy at first deduced the expression for each interval' in his notation, from the mul tifarious ones of Mr Overcnd, were far from the most di rect or easy ; but, assuming the notes of the common chord, Ill, V, and V11I, to be known to be 197Z+ 4 f + 17 in, 3581+71+31 in, and 612X+ 12 f+ 53 m, respec tively, as may easily be proved to be true, by either the indices of the primes, or the logarithms of these intervals, in our Table, Plate XXX. Vol. II. by adding together 197 times x, 4 times f, and 17 times in, and so of the others, the following resolutions and compositions would give us his expressions for all the intervals in the 'Fable : thus, tar intervals, and their logs, which we must rcluctant:y omit here for want of room.
All such intervals in Mr Farcy's general Table, as do not conform to the above, with regard to the number of f and of in's that they contain with their s's, have been deno minated irregular intervals, such as d, F, r, 5c, c, R aml II, none of which ought to have either I or in, but con sist of Z's constitute them regular intervals ; so 0, 1), and ought to have no 1; and I' to have an f, and only two in's, &c. and these changes' may be effected, and any intervals be brought into a regular form, by means of de cimals in the schisma column, equivalent to the f's or m's, that may be added or taken away ; reckoning, each f as .14966096 z, and each m_as .00786241 Z ; thus, for exam ple, din a regular Table, will be .5583795 Z, = 10.149661 + m,f= 31.8582014 Z + 1+2 m, &c.
If we consider, that whenever the prime number 2 is found involved or multiplied in the numerator, or least term of any musical ratio, it is equivalent to deducting an octave .? from it ; and if, in the denominator, it answers to the addition of VIll; so 3 denotes the subtraction or addi tion of a major twelfth (-1) ; 4 of a double octave (a); 5 of a major seventeenth ( '), &c. it will thence appear evident that a Table of the intervals &c. expressed in the new notation, will enable us to find the expresssion for any interval whose ratio is given. We can, on the present oc casion, only find room for the first 25 numbers of Mr Fa rey's Table of this kind, viz.
After several hundred intervals, expressed in this new notation, had been collected from Mr Overend's MS. and many other sources, and arranged in a Table, according to their magnitudes, as before mentioned, it was observed, that all but a very few of them formed a regular series, in each of the three columns formed a separately in creasing series of numbers; such, that f, 1, 2, 3, 4, &e. first appeared with E, 23, 80, 127,174, 231, 278, 325, 382, 429, 486, 533, 590, 637, &c. respectively ; and m, 1, 2, 3, 4. &c.
first appeared associated with :2, 8, 19, 34, 44, 55, 66, 76, 91, 102, 112, 123, 138, 149, 159, 170, 185, 195, 206, 217, 227, 242, 253, 263, 274, 289, 300. 310, 321, 336, 346, 357, 368, 378, 393, 404, 414, 425, 440, 450, 461, 472, 482, 497, 508, 518, 529, 544, 554, 565, 576. 586. 601, 612, 622, 633, 648, 658, 669, &c. respectively. The intervals of the com mencement of the f's, being sometimes S and sometimes +, commencing with 23 E+f+2 to ;. and the intervals of the beginnings of the m's, either C, or commencing with 8 X + in's, by help of which the musical student may readily construct for himself a Table of the above 71 regu From this Table the primes 7, 11, 13, 17, 19, &c. an their multiples, are not excluded, because intervals involv ing these do sometimes require to be calculated : the rea son why two values are affixed to 7, and to some others of the primes differing by in or f, is, in order that a regular interval may be made between every adjacent number, as 1, 4, 4, &c. A simple subtraction will give the value, whenever the terms of the ratio are found in the first co lumn ; and when this is not the case, the multipliers of each term must be sought, and the corresponding notations of each added together, and then the sums are to be subtract ed ; thus if the value of the minor tone, were wanted, once of which sums, is 93 E + 2 f + 8 m, the value sought.
Since, in the use of this notation, a carrying or borrow ing to or from one column to another, never takes place, in whole numbers at least, as with columns of pounds, shil lings, and pence, or yards, feet, and inches, &c. but each column separately agrees in the result, it is plain, that ei ther of the columns may be separately used, but with dif ferent degrees of exactness in some cases.
The middle column having 12 f's to the octave, it is evi dent that the number of f's in the regular expression for any interval, will spew, to which of the 12 finger-keys, or notes of the vulgar half-tone system, the same belongs : thus all intervals less than 23 E-1-f+2 m may be classed with the unison, or first degree, and considered as temperaments ; all between this and 80 E+2 f+7 in, may be considered of the second degree, or as minor seconds, &c.
In like manner, the third or in column of 53 parts in the octave, are the Artificial Commas of Mercator, as is particu larly explained in our article CoarnoN Measures of inter vals, and by help of which commas, the calculations of most intervals, except those near to or less than a comma, may be correctly performed. And, in like manner, the first column, separately considered, of a table of regular intervals, constitute Mr Farey's ArtVcial Commas, 612 to the octave : by means of which, the utmost facility, and every requisite degree of accuracy, is given to the calculations of all real or diatonic intervals larger than E, (except some times confounding R and c, and also fj and ,p), mostly in whole numbers; and in the calculations of temperaments, or where decimal or vulgar fractional parts of this artifi cial comma are used, even the smallest intervals, as well as the largest, arc represented by them and decimals, with greater accuracy than it is practical to make experiments, or to apply musical calculations in practice. •