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.

In adjusting these variable temperaments, so as to render the harmony of the concords of different kinds, as nearly equal as possible, we immediately discover that, as the Vth is composed of the IIId and 3d, the temperaments of the three cannot all be equal. When the temperaments of the IIId and 3d have the same sign, that of the Vths must be equal to their sum; and, when they have contrary signs, to their difference. Hence the temperament of one of these three concords is necessarily equal to the sum of that of the other two. This being fixed, the temperaments, and consequently, (by Prop. I.) the discordance of the different consonances is the most equably divided possible, when the two smaller temperaments, whose sum is equal to the greater, are made equal to each other. The problem contains three cases.

1. When the temperaments of the IIId and 3d have the same sign, they ought to be equal to each other. Making 2x – c = ½ · c – 3x, we obtain x = ³/₇ c, which, substituted in the general expressions for the temperaments of the Vth, IIId, and 3d, makes their increments equal to –²/₇ c, –¹/₇ c, –¹/₇ c, respectively.

2. Let the temperaments of the IIId and 3d have contrary signs: and first, let that of the IIIds be the greater. Then the former ought to be double of the latter, in order that the temperament of the Vths and and 3ds may be equal. Hence we have 2x – c = – 2 · ½ · c – 3x; whence x is found = 0; and by substitution as before, the required temperament of the IIId = – c; of the Vth – ½ c, and of the 3d ½ c.

3. Let the temperaments of the IIId and 3d have contrary signs, as before; and let that of the 3d be the greater. Making ½ · c – 3x = –2 · 2x – c, we obtain x = ³/₅ c; which gives, by substitution, the temperaments of the 3d, Vth, and IIId – ²/₅ c, – ¹/₅ c, and ¹/₅ c, respectively.

Each of these results makes the harmony of all the consonances as nearly equal as possible; but as the sum of the temperaments in the first case is much the least, it follows that the temperaments stated in the proposition constitute the best scheme of intervals for the natural scale, in which the harmony of all the different consonances is rendered as nearly equal as possible.

Cor. 1. In the same manner it may be shown that these temperaments are the best, among those which approach as nearly as possible to equal harmony, for the artificial scale; provided that it is furnished with distinct sounds for all the sharps and flats in common use. By inserting a sound between F and G, making the interval F

G equal to either of the semitones found above, the intervals, reckoned from G as a key note, will be exactly the same in respect to their temperaments, as the corresponding ones reckoned from C. The same thing holds, whatever be the number of flats and sharps. It is supposed, however, that the flat of a note is never used for the sharp of that next below, or the contrary; and hence this scheme of temperament would only be adapted to an instrument, furnished with all the degrees of the enharmonic scale; or, at least, with as many as are in common use.

Cor. 2. This scale will differ but little in practice from the one deduced, with so much labour, by Dr. Smith, from his criterion of equal harmony; which flattens the Vths ⁵/₁₈, the IIIds ¹/₉, and the 3ds ¹/₆ of a comma. The several differences are only ¹/₁₂₆, ²/₆₃, and ¹/₄₂ of a comma. Hence, as his measure of equal harmony differs so widely from that of Proposition I. we may infer that the consideration of equalizing the harmony of the concords of different names can have very little practical influence on the temperaments of the scale. Should it, therefore, be maintained that the criterion laid down in Prop. I. is not mathematically accurate; yet, as it must be allowed, in the most unfavourable view, to correspond far better with the decisions of experience than that of Doctor Smith, the chance is, that, at the lowest estimate, the temperaments deduced from it approach much more nearly to correctness. Hence it is manifest that equal temperament may be made, without any sensible error in practice, the criterion of equal harmony.

Scholium 3.

Although the foregoing would be the best division of the musical scale, if our sole object were to render the harmony of its concords as nearly equal as possible, yet the two other considerations, stated at the beginning of the essay, must by no means be neglected, as has been done by Dr. Smith. It seems to be universally admitted, that the sum of the temperaments may be increased to a certain extent, in order to equalize the harmony of the concords; otherwise the natural scale of major and minor tones, which makes the sum of the temperaments of the Vths, IIIds, and 3ds but 2 commas, ought to be left unaltered. Yet how far this principle ought to be carried, may be a matter of doubt. If we make the IIIds perfect, and flatten the Vths and 3ds each ¼ c, according to the old system of mean tones, we shall have the smallest aggregate of temperaments which admits of the different concords of the same name being rendered equally imperfect; but this amounts to 2½ commas. Thus far, however, it seems evidently proper to proceed. If we go still farther, and endeavour to equalize the harmony of the concords of different names, it may be questioned whether nearly as much is not lost as gained; for the aggregate temperaments are increased, in Dr. Smith's scale, to 2⅔ c, and in that of the above proposition to 2⁵/₇ c. The system of mean tones, although more unequal in its harmony when but two notes are struck at once, yet when the chords are played full, as they generally are on the organ, never offends the ear by a transition from a better to a worse harmony. For every triad is equally harmonious; being composed of a perfect IIId, and a Vth and 3d, tempered each ¼ c, or of their complements to, or compounds with octaves, which, in their kinds, are equally harmonious.

Again, if different chords, in practice, vary in the frequency of their occurrence, this will be a sufficient reason for deviating from the system of equal temperament. Suppose, for example, that a given sum of temperament is to be divided between two Vths, one of which occurs in playing ten times as often as the other: there can be no doubt that the greater part of the temperament ought to be thrown upon the latter. Hence it becomes an important problem to ascertain, with some degree of precision, the relative frequency with which different consonances occur in practice. Before proceeding to a direct investigation of this problem, it may be observed, in general, that such a difference manifestly exists. In a given key, it cannot have escaped the most superficial observer, that the most frequent combination of sounds is the common chord on the tonic; that the next after this is that on the dominant, and the third, that on the subdominant. Perhaps scarcely a piece of music can be found, in which this order of frequency does not hold true. It is equally true that some signatures occur oftener than others. That of one sharp will be found to be more used, in the major mode, than any other; and, in general, the more simple keys will be found of more frequent occurrence than those which have more flats or sharps. These differences are not the result of accident. The tonic, dominant, and subdominant, are obviously the most prominent notes in the scale, and must always be the fundamental bases of more chords than either of the others; while the greater ease of playing on the simpler keys will always be a reason with composers for setting a larger part of their music on these, than on the more difficult keys. It is observable, that the greater part of musical compositions, whether of the major or minor mode, is reducible to two kinds: that in which the base chiefly moves between the tonic and its octave, and that in which the base moves between the dominant and subdominant of the key. The former class, in the major mode, are almost universally set on the key of one sharp; the latter, generally on the natural key, or that of two sharps. In the minor mode, the former class have usually the signature of two flats, or the natural key; the latter, that of one flat. Hence the three former keys will comprise the greater part of the music in the major mode, and the three latter, of that in the minor mode, in every promiscuous collection. But if we were even to suppose each of the chords in the same key, and each of the signatures, of equally frequent occurrence, some chords would occur much oftener, as forming an essential part of the harmony of more keys than others. The Vth DA, for example, forms one of the essential chords of six different keys; while the Vth G

forms a part only of the single key of four sharps.