Proposition V.
To determine that position of any degree in the scale, which will render all the concords terminated by it, at a medium, the most harmonious; supposing their relative frequency given, and all the other degrees fixed.
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′ .
But there are six concords to be accommodated, instead of two; and it is evident that all the pairs cannot have their temperament inversely as their frequency, since the numbers a, b, &c. and m, n, &c. have no constant ratio to each other. This, however, will be the case, at a medium, if x be made such, that the sum of the products of the numbers expressing the frequency of those chords whose temperaments are increased by x, into their respective temperaments, shall be equal to the sum of the corresponding products belonging to those chords whose temperaments are diminished by x. Applying this principle to the system of temperament in Prop. III, which flattens all the concords, it is plain that raising any given degree by x will increase the temperaments of the concords above that degree, and diminish those of the concords below it. Hence it ought to be raised till (m – x) a +(n – x)b + (p – x)c = (m′ + x)a + (n′ + x)b′ + (p + x)c′ ; from which x is found = am – a′m′ + bn – b′n′ + cp – c′p′ a + a′ + b + b′ + c + c′ . Should either of the temperaments be sharp, the sign of that term of the numerator, in which it occurs, must be changed; and should the total value of the expression be negative, x must be taken below C.