Again, the 5th, whose perfect ratio has been generally laid down as 45 : 64, but which is in reality 25 : 36,[6] cannot be sharpened more than ⅓ of a comma, before it becomes more harmonious by having its temperament increased, as approaching nearer the simpler ratio 7 : 10. At the same time, the effect of this interval in melody would not be sensibly varied. The limits, within which the harmoniousness of the IVth is inversely as its temperament, are still narrower.
Hence it appears that no inference can be drawn from the temperaments of such consonances as the 7th, 5th, IVth, &c. respecting their real harmoniousness. The other perfect ratios which have nearly the same value with those of these chords, and which are in equally simple terms, are so numerous that by increasing their temperament they alternately become more and less harmonious; and in a manner so irregular, that to attempt to subject them to calculation, with the concords, would be in vain. Even when unaltered, they may be considered either as greater temperaments of more simple, or less temperaments of more complex ratios. Suppose the 5th, for example, to be flattened ⅕ of a comma: shall it be considered as deriving its character from the perfect ratio 25 : 36, and be regarded as flattened 108; or shall it be referred to the perfect ratio 7 : 10, and considered as sharpened 239? No one can tell.—On the whole, it is manifest that no consonances more complex than those included in the proposition, can be regarded in adjusting the temperaments of the scale.
Proposition III.
The best scale of sounds, which renders the harmony of all the concords as nearly equal as possible, is that in which the Vths are flattened 2/7, and the IIIds and 3ds, each 1/7 of a comma.
The octave must be kept perfect, for reasons which have satisfied all theoretical and practical harmonists, how widely soever their opinions have differed in other respects. Admitting equal temperament to be the measure of equal harmony, the complements of the Vth, IIId, and 3d, to the octave, and their compounds with octaves will be equally harmonious in their kinds with these concords respectively; according to the corollary of Prop I.
Hence we have only to find those temperaments of the Vths, IIIds, and 3ds, in the compass of one octave, which will render them all, as nearly as possible, equally harmonious. The temperaments of the different concords of the same name ought evidently to be rendered equal; since, otherwise, their harmony cannot be equal. This can be effected only by rendering the major and minor tones equal, and preserving the equality of the two semitones. If this is done, the temperament of all the IIIds will be equal, since they will each be the sum of two equal tones. For a similar reason the 3ds, and consequently the Vths, formed by the addition of IIIds, and 3ds, will be equally tempered.
In order to reduce the octave to five equal and variable tones, and two equal and variable semitones, we will suppose the intervals of the untempered octave to be represented by the parts CD, DE, &c. of the line Cc. Denoting the comma by c, we will suppose the tone DE, which is naturally minor, to be increased by any variable quantity, x; then, by the foregoing observations, the other minor tone, GA, must be increased by the same quantity. As the major tones must be rendered equal to the minor, their increment will be x – c. As the octave is to be perfect, the variation of the two semitones must be the same with that of the five tones, with the contrary sign; and as they are to be equally varied, the decrement of each will be 5x – 3c 2 ; or what amounts to the same thing, the increment of each will be 3c – 5x 2 .
The several concords of the same name in this octave are now affected with equal and variable temperaments. The common increment of the IIIds will be 2x – c; that of the 3ds ½ · c – 3x; and consequently that of the Vths ½ · x – c.