The best scheme of temperament for the changeable scale, on supposition that all the concords were of equally frequent occurrence, is investigated in Prop. III. But it is shown, in the last Proposition, that some chords occur in practice far more frequently than others. Hence it becomes necessary to ascertain what changes in the scale above referred to, this different frequency requires. Any given degree, as C, terminates six different concords; a Vth, IIId, and 3d above, and the same intervals below it. Let the numbers denoting the frequency of these chords below C be denoted by _a_, _b_, and _c_, and their temperaments, before the position of C is changed, by _m_, _n_, and _p_: and let the frequency of the chords above C be denoted by _a′_, _b′_, and _c′_, and their temperaments by _m′_, _n′_, and _p′_, respectively. If, now, we regard any two of these 6 chords, whose temperaments would be diminished by moving C opposite ways, and of which the sum of the temperaments is consequently fixed, it is manifest that the more frequent the occurrence, the less ought to be the temperament. Were we guided _only_ by the consideration of making the aggregate of dissonance heard in them in a given time, the least possible, we should make the one of most frequent occurrence perfect, and throw the whole of the temperament upon the other. Let, for example, _a_ be greater than _a′_, and let _x_ be any variable distance to which C is moved, so as to diminish the temperament _m_, of the chord whose frequency is expressed by _a_. Then the temperament of _a_ will become = _m_ ~ _x_, and that of _a′_ = _m′_ + _x_. Hence, as the dissonance head in each, in a given time, is in the compound ratio of its frequency of occurrence and its temperament, their aggregate dissonance will be as a · (m ~ x) + a′ · (m′ + x); a quantity which, as _a_ is supposed greater than _a′_, evidently becomes a minimum when _x_ = _m_, or the chord, whose frequency is _a_, is made perfect. But in this way we render the harmony of the chords very unequal, which is, cæteris paribus, a disadvantage. As these considerations are heterogeneous, it must be a matter of judgment, rather than of mathematical certainty, what precise weight is to be given to each. We will give so much weight to the latter consideration, as to make the temperament of each concord _inversely as its frequency_. We have then a : a′ :: 1/(m - x) : 1/(m′ + x); which gives x = (am - a′m′)/(a + a′).