The general theorem of Prop. V. is equally applicable to the determination of the approximate place for any degree in this scale, considering the numbers in the above table as those to be substituted for _a_, _a′_, _b_, &c.; and _m_, _n_, and _p_, in the first instance, as 49, -343 and 392, the uniform temperaments of the Vths, IIIds, and 3ds, in the scale of equal semitones. Since, however, the temperaments of the IIIds in this scale are sharp, which would require the signs of the 3d and 4th terms in the numerator of the general formula to be continually changed, it will be rendered more convenient for practice, if they are changed at first, so that it will stand thus: x = (am - a′m′ - bn + b′n′ + cp - c′p′) / (a + a′ + b + b′ + c + c′) Three successive applications of this theorem to each degree in the scale, in the manner described Prop. VI., will bring them very near to the required position, as appears by the smallness of the corrections in the 3d column below, where the results of the several operations are exhibited at one view. TABLE X. +------+----------+----------+----------+ |Bases.| First | Second | Third | | |Operation.|Operation.|Operation.| +------+----------+----------+----------+ | B | -140 | -35 | -2 | +------+----------+----------+----------+ | B♭ | +308 | +33 | -1 | +------+----------+----------+----------+ | A | -8 | -23 | +2 | +------+----------+----------+----------+ | G♯ | -257 | -22 | -2 | +------+----------+----------+----------+ | G | +107 | +24 | -8 | +------+----------+----------+----------+ | F♯ | -264 | -7 | 0 | +------+----------+----------+----------+ | F | +238 | +40 | +6 | +------+----------+----------+----------+ | E | -80 | -34 | -4 | +------+----------+----------+----------+ | E♭ | +157 | +2 | -4 | +------+----------+----------+----------+ | D | +58 | + 8 | 0 | +------+----------+----------+----------+ | C♯ | -352 | -29 | -1 | +------+----------+----------+----------+ | C | +176 | +29 | +4 | +------+----------+----------+----------+ _Cor._ Hence we may deduce, in the same manner as in Prop. VII., the diatonic and chromatic intervals, the lengths of a string and their vibrations in a second, and the temperaments and beats of all the concords for the scale which results from the foregoing computations. They may be seen in the two following tables: TABLE XI. _DIATONIC AND CHROMATIC INTERVALS._