Lemma.
No arrangement of the intervals in the common scale of 12 degrees, which renders none of the Vths or 3ds sharp, and none of the IIIds flat, can make any change in the aggregate temperaments of all the concords of the same name.
We will conceive the 12 Vths of the Douzeave scale to be arranged in succession, as CG, GD, DA, &c. embracing 7 octaves. Let them at first be all equal: they will each be flattened 49. I say that no change in these Vths which preserves the two extreme octaves perfect, and renders none of them sharp, can alter the sum of their temperaments. Let a, b, c, &c. be any quantities, positive or negative, by which the points C, G, D, &c. may be conceived to be raised above the corresponding points, belonging to the scheme of equal Vths. Then as the mean temperament Vth = V – 49, the first Vth in the supposed arrangement will be V – 49 + a. The distance from C to D will be, in like manner, 2 · (V – 49) + b; and consequently the Vth GD will be V – 49 + b – a. In the same manner the third Vth DE will be V – 49 + c – b, &c. Hence the temperament of CG = -49 + a, of GD = -49 + b – a, of DA = -49 + c – b, &c. Adding the 12 temperaments together, we find their sum = -12 × 49 + a + b + &c. – a – b – &c. in which all the terms except the first destroy each other, and leave their sum = –12 × 49 which is the aggregate temperament of the twelve equal Vths in the scheme of equal semitones.
The same reasoning holds good if we bring these Vths within the compass of an octave; since, if the octave be kept perfect, all the Vths on the same letter, in whatever octave they are situated, must have the same temperament.
The reasoning is precisely the same for the IIIds and 3ds, considering the former as forming 4 distinct series of an octave each, beginning with C, C
, D and E
; and the latter as forming 3 distinct series of an octave each, beginning with C, C
and D. If the former be made all equal, each will be sharpened 343; if the latter be made equal, each will be flattened 392. In every system which renders none of the former flat, and none of the latter sharp, the sum of their temperaments will be 12 × 343, and 12 × 392, respectively.
Cor. The demonstration holds equally true, whatever be the magnitude of a, b, c, &c.: only if they be such that the difference –a + b, –b + c, &c. of any two successive ones be greater than the temperament of the corresponding concord in the system of equal semitones, the temperament of that chord must be reckoned negative, and the sum, in the enunciation of the proposition, must be considered as the excess of those temperaments which have the same sign with those of the same concords in the system of equal semitones, above those which have the contrary sign. Hence it is universally true that the excess of the flat above the sharp temperaments of the Vths is equal to 12 × 49; that the excess of the sharp above the flat temperaments of the IIIds is equal to 12 × 343; and that the excess of the flat above the sharp temperaments of the 3ds is 12 × 392. Hence likewise we have a very easy method of proving whether the temperaments of any given system have been correctly calculated. It is only to add those which have the same sign; and if the differences of the sums be equal to the products just stated, the work is right.