Let the lines AB, ab, represent the times of vibration of two tempered unisons. Whatever be the ratio of AB to ab, whether rational or irrational, it is obvious that the successive vibrations will alternately recede from and approach each other, till they very nearly coincide; and, that during one of these periods, the longer vibration, AB, has gained one of the shorter. Let the points, A, B, &c. represent the middle of the successive times of vibration of the lower; and a, b, &c. those of the higher of the tempered unisons. Let the arc AGN..VA be a part of a circle, representing one period of their pulses, and let the points A, a, be the middle points of the times of those vibrations which approach the nearest to a coincidence. It is obvious that the dislocations bB, cC, &c. of the successive pulses, increase in a ratio which is very nearly that of their distances from A, or a. Now if the pulses exactly coincided, the unisons would be perfect; and the same would be equally true, if the pulses of the one bisected, or divided in any other constant ratio, those of the other; as clearly appears from observation. It is, therefore, not the absolute magnitude, as asserted by Dr. Smith, but the variableness of the successive dislocations, Bb, Cc, &c. which renders the imperfect unisons discordant; and the magnitude of the successive increments of these dislocations is the measure of the degree of discordance heard in the unisons.

If now the time of vibration in each is doubled, AC, ac, &c. will represent the times of vibration of imperfect unisons an octave below, and the successive dislocations will be Cc, Ee, &c. only half as frequent as before. But the unisons AE, ae, will be equally harmonious with AB, ab; because, although the successive dislocations are less frequent than before, yet the coincidences C'c', E'e' of the corresponding perfect unisons are less frequent in the same ratio.

Suppose, in the second place, that the time of vibration is doubled, in only one of the unisons, ab; and that the times become AB and ac, or those of imperfect octaves. These will also be equally harmonious in their kind with the unisons AB, ab. For, although the dislocations Cc, Ee, &c. are but half as numerous as before, the coincidences of the corresponding perfect octaves will be but half as numerous. The dislocations which remain are the same as those of the imperfect unisons; and if some of the dislocations are struck out, and the increments of successive ones thus increased, no greater change is made in the nature of the imperfect than of the perfect consonance.

If, thirdly, we omit two-thirds of the pulses of the lower unison, retaining the octave ac of the last case, we shall have AD, ac, the times of vibration of imperfect Vths, to which, and to all other concords, the same reasoning may be applied as above. It may be briefly exhibited thus; since the intermission of the coincidences C'c', E'e' of the perfect unisons, an octave below A'B', does not render the Vth A'D'G' a'c'e'g' less perfect than the unison A'c' a'c', each being perfect in its kind; so neither does the intermission of the corresponding dislocations Cc, Ee, of the tempered unisons, in the imperfect Vth, ADG, aceg, render it less harmonious in its kind than the tempered unison AB, ab, from which it is derived in exactly the same manner that the perfect Vth is derived from the perfect unison.

The consonances thus derived, as has been shown by Dr. Smith, will have the same periods, and consequently the same beats, with the imperfect unisons. It is obvious, likewise, that they will all be equally tempered. Let m AB, and n ab, be a general expression for the times of vibration of any such consonance. The tempering ratio of an imperfect consonance is always found by dividing the ratio of the vibrations of the imperfect by that of the corresponding perfect consonance. But m AB n ab ÷ m n = AB ab ; which is evidently the tempering ratio of the imperfect unisons.

Hence, so far as any reasoning, founded on the abstract nature of coexisting pulses can be relied on, (for, in a case of this kind, rigid demonstration can scarcely be expected,) we are led to conclude that the harmoniousness of different consonances is proportionally diminished when they are equally tempered.

The remaining part of the proposition, viz. that consonances differently tempered have their harmoniousness diminished, or their harshness increased, in the direct ratio of their temperaments, will be evident, when we consider that the temperament of any consonance is the sole cause of its harshness, and that the effect ought to be proportioned to its adequate cause. We may add, that the rapidity of the beats, in a given consonance, increases very nearly in the ratio of the temperament; and universal experience shows, that increasing the rapidity of the beats of the same consonance, increases its harshness. This is on the supposition that the consonance is not varied so much as to interfere with any other whose ratio is equally simple.

Cor. We may hence infer, that in every system of temperament which preserves the octaves perfect, each consonance is equally harmonious, in its kind, with its complement to the octave, and its compounds with octaves. For the tempering ratio of the complement of any concord to the octave, is the same with that of the concord itself, differing only in its sign, which does not sensibly affect the harmony or the rate of beating; while the tempering ratio of the compounds with octaves is not only the same, but with the same sign.

Scholium 1.

There is no point in harmonics, concerning which theorists have been more divided in opinion than in regard to the true measure of equal harmony, in consonances of different kinds. Euler maintains, that the more simple a consonance is, the less temperament it will bear; and this seems to have ever been the general opinion of practical musicians.[4] Dr. Smith, on the contrary, asserts, and has attempted to demonstrate, that the simpler will bear a much greater temperament than the more complex consonances. The foregoing proposition has, at least, the merit of taking the middle ground between these discordant opinions. If admitted, it will greatly simplify the whole subject, and will reduce the labour of rendering all the concords in three octaves as equally harmonious as possible, which occupies so large a portion of Dr. Smith's volume, to a single short proposition. Dr. Smith's measure of equal harmony, viz. equal numbers of short cycles in the intervals between the successive beats, seems designed, not to render the different consonances proportionally harmonious, but to reduce the simpler to an absolute level, in point of agreeableness, with the more complex; which, as has been shown, is not the object to be aimed at in adjusting their comparative temperaments. But, in truth, his measure is far more favourable to the complex consonances than equal harmony, even in this sense, would require; and, in a great number of instances, leads to the grossest absurdities. Two consonances, according to him, are equally harmonious, when their temperaments are inversely as the products of the least numbers expressing their perfect ratio. If so, the VIII + 3d, whose ratio is ⁵/₁₂, when tempered ¹/₂₀ of a comma, and the unison, whose ratio is ¹/₁, when tempered 3 commas, are equally harmonious. But all who have the least experience in tempered consonances will pronounce, at once, that the former could scarcely be distinguished by the nicest ear from the corresponding perfect concord, while the latter would be a most offensive discord. One instance more shall suffice. The temperaments to render the VIII + Vth, and the VIII + 6th equally harmonious, are laid down in his tables to be as 80 : 3. We will now suppose an instrument perfectly tuned in Dr. Smith's manner, and furnished with all the additional sounds which constitute his changeable scale. In this system, the IIIds, and consequently the VIII + 6ths, are tempered ¹/₉ of a comma; which, so far from being offensive, will be positively agreeable to the ear. This cannot be doubted by those who admit that the VIII + 6ths in the common imperfect scales, when tempered at a medium nearly seven times as much, make tolerable harmony. Yet, according to the theory which we are opposing, the VIII + Vth will be equally harmonious when tempered nearly a minor semitone. Now let any one, even with the common instruments, whenever an VIII + Vth occurs, strike the semitone next above or below: for example, instead of playing C, g, let him play C, g; instead of A, e, let him play A, e, &c. and compare the harmony of these with that of the VIII + 6ths, if he wants any farther evidence that Dr. Smith's measure of equal harmony is without foundation.

It may be thought, that even the measure of equal harmony laid down in the proposition, is more favourable to the complex consonances than the conclusions of experience will warrant. But when it is asserted by practical musicians, that the octave will bear less tempering than the Vth, the Vth less than the IIId, &c., they doubtless intend to estimate the temperament by the rate of beating, and to imply, that when different consonances to the same base are made to beat equally fast, the simpler are more offensive than the more complex consonances. This is entirely consistent with the proposition; for when equally tempered, the more complex consonances will beat more rapidly than the more simple; if on the same base, very nearly in the ratio of their major terms. (Smith's Har. Prop. XI. Cor. 4.) If, for example, an octave, a Vth, and a IIId on the same base were made to beat with a rapidity which is as the numbers 2, 3, and 5, no unprejudiced ear would probably pronounce the octave less harmonious in its kind than the IIId.

To those, on the other hand, who may incline to a measure of equal harmony between that laid down in the proposition and that of Dr. Smith, on account of the rapidity of the beats of the more complex consonances, it maybe sufficient to reply, that if the beats of a more complex consonance are more rapid than those of a simpler one, when both are equally tempered, those of the latter, cÆteris paribus, are more distinct. It is the distinctness of the undulations, in tempered consonances, which is one of the principal causes of offence to the ear.

Scholium 2.

It will be proper to explain, in this place, the notation of musical intervals, which will be adopted in the following pages. It is well known that musical intervals are as the logarithms of their corresponding ratios. If, therefore, the octave be represented by .30103, the log. of 2, the value of the Vth will be expressed by .17509; that of the major tone by .05115; that of the comma by .00540, &c. But in order to avoid the prefixed ciphers, in calculations where so small intervals as the temperaments of different concords are concerned, we will multiply each of these values by 100,000, which will give a set of integral values having the same ratio. The octave will now become 30103, the comma 540, &c.; and, in general, when temperaments are hereafter expressed by numbers, they are to be considered as so many 540ths of a comma. Had more logarithmic places been taken, the intervals would have been expressed with greater accuracy; but it was supposed that the additional accuracy would not compensate for the increased labour of computation which it would occasion. This notation has been adopted by Dr. Robinson, in the article Temperament, (Encyc. Brit. Supplement;) and for every practical purpose, is as much superior to that proposed by Mr. Farey, in parts of the Schisma, lesser fraction and minute,[5] as all decimal measures necessarily are, to those which consist of different denominations.

Proposition II.

It has been generally assigned as the reason for neglecting the consonances, usually termed discords, in ascertaining the best scheme of temperament, that they are of less frequent occurrence than the concords. This, however, if it were the only reason, would lead us, not to neglect them entirely, but merely to give them a less degree of influence than the concords, in proportion as they are less used.

A consideration which seems not to have been often noticed, renders it impossible to pay them any regard in harmonical computations. All such computations must proceed on the supposition that within the limits to which the temperaments of the different consonances extend, they become harsher as their temperaments are increased. It is evident that any consonance may be tempered so much as to become better by having its temperament increased, in consequence of its approaching as near to some other perfect ratio, the terms of which are equally small; or perhaps much nearer some perfect ratio whose terms are not proportionally larger. For example, after we have sharpened the Vth more than 3 commas, it becomes more harmonious, as approaching much nearer to the perfect ratio ⁵/₆. In this, however, and the other concords, the value of the nearest perfect ratios in small numbers, varies so much from the ratios of these concords, and the consequent limits within which the last part of Prop. I. holds true, are so wide that there is no hazard in making it a basis of calculation. And if there be a few exceptions to this, in some systems, in which the temperaments of a few of the concords become so large as to approach nearer to some other perfect ratio, whose terms are nearly as small as those of the perfect concord, although they might become more harmonious, by having their temperament increased, yet their effect in melody would be still more impaired; so that the concords may all be considered as subjected to the same rule of calculation.

But the limits within which the second part of Prop. I. holds true, with regard to the more complex consonances, are much more limited. We cannot, for instance, sharpen the 7th, whose ratio is 9 : 16 more than ½ a comma, without rendering it more harmonious, as approaching nearer another perfect ratio which is simpler; that of 5 : 9. Yet the difference between these two 7ths is so trifling that they have never received distinct names; and, indeed, their effect on the ear in melody would not be sensibly different.

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 ? 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 ? 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 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.

This can be done only by investigating their actual frequency of occurrence in a collection of pieces for the instrument to be tuned, sufficiently extensive and diversified to serve as a specimen of music for the same instrument in general. This may appear, at first view, an endless task; and it would be really such, were we to take music promiscuously, and count all the consonances which the base makes with the higher parts, and the higher parts with each other. But it appears, from Prop. I. Cor. that all the positions and inversions of a chord, when the octaves are kept perfect, are equally harmonious with the chord itself. The Vth, for example, which makes one of the consonances in a common harmonic triad, is equally harmonious in its kind, with the V + VIII, which takes its place in the 3d position of this triad, and with the 4th in its second inversion. Hence, instead of counting single consonances, we have only to count chords; and this is done with the greatest ease, by means of the figures of the thorough base. The labour will be still farther abridged by reducing the derivative chords, such as the 6, the ⁶/₄, &c. to their proper roots, as they are taken down. But even after these reductions, the labour of numbering the different chords in a sufficiently extensive set of compositions, to establish, with any degree of certainty, the relative frequency of the different signatures, would be very irksome. A method, however, presents itself, which renders it sufficient to examine the chords in such a set of pieces only as will give their chance of occurrence in two keys—a major, and its relative minor.

It will be evident to all who are much conversant with musical compositions, that the internal structure of all pieces in the same mode, whatever be their signature, is much the same. There is scarcely more difference, for example, in the relative frequency of different chords in the natural key, and in that of two sharps, or two flats, than there is in different pieces on the same key. If the Vth CG on the tonic has to the Vth EB on the mediant in the natural key, any given ratio of frequency m : n, the relative frequency of the Vth DA on the tonic, and the Vth F C on the mediant in the key of two sharps, will not sensibly differ from that of m : n. Hence, if we examine a sufficient number of pieces to establish the relative frequency of the different consonances in one major and its relative minor key, and, by a much more extensive investigation, ascertain the relative frequency of occurrence of the different signatures, it is evident, that by multiplying this last series of numbers into the first, and adding those products which belong to chords terminated by the same letters, we shall have a series of numbers expressing the chance of occurrence in favour of each of the consonances of the scale, when all the keys are taken into view.

It was judged that 200 scores, taken promiscuously from all the varieties of music for the organ,[7] would afford a set of numbers expressing, with sufficient accuracy, the chance that a given consonance will occur in a single major, and its relative minor key. Accordingly 200 scores were examined, 150 in the major, and 50 in the minor mode, (as it will appear hereafter that this is nearly the ratio of their frequency) of the various species of music for the organ, comprising a proper share both of the simpler and of the more rapid and chromatic movements. As the selecting and reducing to their proper keys all the occasional modulations which occur in the same piece would render the labour of ascertaining the relative frequency of different signatures very tedious, it was thought best to consider all those modulations which are too transient to be indicated by a new signature, as belonging to the same key. This will account for the occurrence of the chords in the following table, which are affected by flats and sharps.

The minim, or the crotchet, was taken for unity, according to the rapidity of the movement. Bases of greater or less length had their proper values assigned them; although mere notes of passage, which bore no proper harmony, were generally disregarded. The scores were taken promiscuously from all the different keys; and were reduced, when taken down, to the same tonic; the propriety of which will evidently appear from the foregoing remarks. The following table contains the result of the investigation.

TABLE I.

The following anomalous chords were found in the major mode, and are subjoined, to make the list complete:

8 5ths on C, and 1 on D.
5 ⁵/₄ths on D, 2 on E, and 1 on G.

The left hand column of the foregoing table contains the fundamental bases of the several chords. When any number is annexed to the letter denoting the fundamental, it denotes the quality of some other note belonging to the chord. E III, for example, denotes that the various chords on E, which stand against it, have their third sharped; G 3, that the third, which is naturally major, is to be taken minor, &c. Of the two columns in each of the four remaining pairs, the left contains the number of chords belonging to each root, of the kind specified at the top, which were found in 150 scores in the major mode; and the right, the corresponding results of the examination of 50 scores in the minor mode. The diminished triad, which is used in harmonical progression like the other triads, has its lowest note considered as its fundamental. The diminished 7th, in the few instances in which it occurred, was considered as the first inversion of the ⁹/₇th, agreeably to the French classification, and was accordingly reduced to that head.

From this table, the number of times that each consonance of two notes would actually occur, were the 200 scores played, is easily computed. We will suppose three notes, besides octaves, to be played to each chord. The octaves played it is unnecessary to take into the computation, as it would only multiply the number of consonances whose temperament is the same, in the same ratio, and would have no effect on the ratio of the numbers expressing the frequency of the different consonances. In the chord of the 7th, which naturally consists of four notes, we will suppose, for the sake of uniformity, that one is omitted; and as the 7th ought always to be struck, we will suppose the Vth and IIId of the base to be omitted, each half the number of times in which this chord occurs. Considered as composed of three distinct notes, neither of which is an octave of either of the others, each chord will contain three distinct consonances. The common chord on C, for example, will contain the Vth CG, the IIId CE, and the 3d EG. The ⁹/₇ on C will contain the VII CB, the IX, or (which must have the same temperament) the IId CD, and the 3d BD. Reducing all these consonances to their proper places, and adding those of the same name which have the same degree for their base, we obtain the following results:

TABLE II.

Besides the following chromatic intervals:

It was thought best to exhibit a complete table of all the consonances which occurred in the 200 scores examined; although (Prop. II.) only the concords in the upper half of the table can be regarded in forming a system of temperament. For the more frequent consonances, this table may be regarded as founded on a sufficiently extensive induction to be tolerably accurate. For the more unfrequent chords, and especially for those which arise from unusual modulations, it expresses the chance of occurrence with very little accuracy; and it is doubtless the fact that a more extensive investigation would include some chords not found at all in this list. But it must be recollected, on the other hand, that the influence of these unusual chords on the resulting system of temperament would be insensible, could their chance of occurrence be determined with the greatest accuracy.

But none of the numbers in the foregoing table by any means expresses the chance that a given interval will occur, considering all the keys in which it is found. For example, the Vth CG on the tonic of the natural key, in music written on this key, is the one of most frequent occurrence, its chance being expressed by 1807; but in the key of two flats, it becomes the Vth on the supertonic, and its chance of occurrence is only as 197. Hence the problem can be completed only by finding a set of numbers which shall express, with some degree of accuracy, the relative frequency of different signatures.

An examination of 1600 scores, comprising four entire collections of music for the organ and voice, by the best European composers, besides many miscellaneous pieces, afforded the results in the following table:

TABLE III.

The chance of occurrence for any chord varies as the frequency of the key to which it belongs, and as the number belonging to the place which it holds, as referred to the tonic, in Table II., jointly. Hence the chance of its occurrence in all the keys in which it is found, is as the sum of the products of the numbers in Table III., each into such a number of Table II. as corresponds to its place in that key. To give a specimen of the manner in which this calculation is to be conducted, the numbers belonging to the major mode in the three first divisions of Table II. are first to be multiplied throughout by 176, which expresses the relative frequency of the major mode of the natural key. They are then to be multiplied throughout by 322, which expresses the frequency of the key of one sharp. But the first product, which expresses the frequency of the Vth on the tonic, now becomes GD, and must be added, not to the first, but to the fifth, in the last row of products. The product into 59, expressing the frequency of the Vth on the mediant, becomes BF, an interval not found among the essential chords of the natural key. In general, the products of the numbers in Table III. into those in Table II. are to be considered as belonging, not to the letters against which these multipliers stand, but to those which have the same position with regard to their successive tonics, as these have with regard to C. Whenever an interval occurs, affected with a new flat or sharp, it is to be considered as the commencement of a new succession of products. The IIId CE, for example, does not occur at all till we come to the key of two sharps, and even then only in occasional modulations, corresponding to the IIId on B in the natural key, whose multiplier is 10. In the key of 3 sharps it becomes another accidental chord, answering to the IIId on E in the key of C, and consequently has 40 for its multiplier. It is only in the key of 6 sharps, that it becomes a constituent chord of the key; when if that key were ever used, it would correspond to the IIId GB on the dominant of the natural key.

After all the products have been taken and reduced to their proper places, in the manner exemplified above, a similar operation must be repeated with the numbers in the second column of Table III. and those in the second columns in the three first divisions of Table II.

The necessity of keeping the major, and its relative minor key, distinct, will be evident, when we consider that the several keys in the minor mode do not follow the same law of frequency as in the major; as is manifest from the observations in Schol. Prop. III. and as clearly appears from an inspection of Table III.

But in order to discover the relative frequency of the different chords on every account, the results of the two foregoing operations must be united. Now, as the numbers in the two columns of Table II. at a medium, are as 3 : 1, and those in Table III. are in the same ratio, although the factors are to each other in only the simple ratio of the relative frequency of the two modes, yet their products will, at a medium, be in the duplicate ratio of that frequency. Hence, to render the two sets of results homologous, so that those which correspond to the same interval may be properly added, to express the general chance of occurrence for that interval in all the major and minor keys in which it is found, this duplicate ratio must be reduced to a simple one, either by dividing the first, or by multiplying the last series of results, by 3. We will do the latter, as it will give the ratios in the largest, and, of course, the most accurate terms. Then adding those results in each which belong to the same interval, and cutting off the three right hand figures, (expressing in the nearest small fractions those results which are under 1000) which will leave a set of ratios abundantly accurate for every purpose; the numbers constituting the final solution of the problem will stand as follows:

TABLE IV.

Note. In this table, as well as the last, the Vths, IIIds, and 3ds are to be taken above, and the 4ths, 6ths, and VIths, their complements to the octave, below the corresponding degrees in the first column. And, in general, whenever the Vths, IIIds, and 3ds are hereafter treated as different classes of concords, each will be understood to include its complement to the octave and its compounds with octaves.

Scholium.

The foregoing table exhibits, with sufficient accuracy, the ratio of the whole number of times which the different chords would occur, were the 1600 scores, whose signatures were examined, actually played in succession, on the keys to which they are set, and with an instrument having distinct sounds for all the flats and sharps. Had the examination been more extensive, the results might be relied on with greater assurance as accurate; but the general similarity, not only in the structure of different musical compositions, but in the comparative frequency of the different keys in different authors; is so great, that a more extensive examination was thought to be of little practical importance.

(To be continued.)