Proposition VII.

The temperaments of all the concords are easily deduced from Table V. The Vth CG, for example, has its lower extremity lowered 12, and its upper extremity 14. Hence it is flatter by 2 than at first, and consequently its temperament=156. The temperaments of all the concords, thus calculated, will be found in the 2d, 3d, and 4th columns of Table VII.

Having ascertained the temperaments, the value of the diatonic and chromatic intervals may be found. The Vth CG being flattened 156, and the Vth FC 139, the major tone FG must be diminished 156 + 139, or be = 4820. By thus fixing the extent of one interval after another, from the temperaments of either of the different kinds of concords, as is most convenient, the intervals in question will be found to have the values exhibited in Table VI.

Let the numbers in this table be added successively, beginning at the bottom, to the log. of 240, the number of vibrations per second of the tenor C, (see Rees's Cyc. Art. Concert Pitch,) and the numbers corresponding to these logarithms will be the vibrations in a second, of a string sounding the several degrees of the scale. They are shown in col. 6, Table VII.

Since the length of a string cÆteris paribus is inversely as its number of vibrations, the lengths in col. 5 may be deduced from the vibrations in col. 6; or more expeditiously, by subtracting the numerical distances from C of the several degrees in Table VI. from O, and taking the corresponding numbers, from the table of logarithms. These numbers, when used as logarithms, must be brought back to the decimal form, agreeably to Scholium 2. Prop. I.

To find the number of beats made in a second by any concord, it is only necessary to take from col. 5 the numbers belonging to the degrees which terminate that concord, and to multiply them crosswise into the terms of its perfect ratio. The difference of the products will be the number of beats made in a second. The 3 last columns contain the beats made by each of the concords, in 10 seconds.

TABLE VI.

TABLE VII.

Proposition VIII.

The aggregate of dissonance, heard in any tempered concord, is as its temperament (Prop. I.) when its frequency of occurrence is given, and as its frequency of occurrence, when its temperament is given: hence, universally, it is as the product of both. The whole amount of dissonance heard in all the concords of the same name must consequently be as the sum of the products of the numbers denoting their temperaments, each into the number in Table IV. denoting its frequency. These products, for the scale of Huygens which divides the octave into 31 equal parts, of which the tone is 5 and the semi-tone 3; for the system of mean tones, and for Dr. Smith's system of equal harmony, compared with the scale of the last proposition, (cutting off the three right-hand figures) stand as follows:

TABLE VIII.

Were we to adhere to Dr. Smith's measure of equal harmony, the rows of products belonging to the Vths, IIIds, and 3ds, must be divided, respectively, by ?, ¹/₁₀, and ¹/₁₃ (the reciprocals of half the products of the terms of their perfect ratios,) before they could be properly added to express the whole amount of dissonance heard in all the concords; but, according to Prop. I. the simple products ought to be added, and the sums at the bottom of the table will express the true ratio of the aggregate dissonance of the systems under which they stand. The last has decidedly the advantage over the first, both in regard to the aggregate dissonance, and the equality of its distribution among the different classes of concords. It has nearly an equal advantage over the second in regard to the first of these considerations; although in regard to the equality of distribution, the latter has slightly the advantage. It has, in a small degree, the advantage over the third, in regard to the aggregate dissonance; while, as it respects the equality of its distribution, it has the decided preference. It is true that the temperaments of the concords of the same name, in the new scale, are not as in the others, absolutely equal; but no one of them is so large as to give any offence to the nicest ear. The largest in the whole scale exceeds the uniform temperament of Dr. Smith's Vths by only ¹/₁₈ of a comma.

Scholium 1.

The above system may be put in practice on the organ, by making the successive Vths CG, GD, DE, &c. beat flat at the rate contained in Table VII., descending an octave, where necessary, and doubling the number of beats belonging to any degree in the table, when the Vth to be tuned has its base in the octave above the treble C. The tenor C must first be made to vibrate 240 in a second, the methods of doing which are detailed at length in various authors. Whenever a IIId results from the Vths tuned, its beats ought to be compared with those required in the table, and the correctness of the Vths thus proved. This system is as easy, in practice, as any other; for no one can be tuned correctly except by counting the beats, and rendering them conformable to what that system requires. The intervals of the first octave tuned ought to be adjusted with the utmost accuracy, by a table of beats. When this is done, the labour of making perfect the other octaves of the same stop, and the unisons, octaves, Vths, &c. of the other stops, is the same in every system. This last, indeed, is so much the most laborious part of the tuning of the organ, that if even much more labour were required than actually is, in adjusting the intervals of the octave first tuned it would occasion little difference in the whole.

Scholium 2.

The harmony of the IIIds and 3ds in any of the foregoing systems for the changeable scale is so much finer than it can possibly be in the common Douzeave, that it seems highly desirable that this scale should be introduced into general use. But the increased bulk and expense attendant on the introduction of so many new pipes or strings, together with the trouble occasioned to the performer, in rectifying the scale for music in the different keys, have hitherto prevented its becoming generally adopted. To multiply the number of finger keys would render execution on the instrument extremely difficult; and the apparatus necessary for transferring the action of the same key from one string or set of pipes to another, besides being complicated and expensive, requires such exactness that it must be continually liable to get out of order. This latter expedient, however, has been deemed the only practicable one, and has been carried into effect, under different forms, by Dr. Smith, Mr. Hawkes, M. Loeschman, and others. But Dr. Smith's plan (which is confined to stringed instruments) requires only one of the unisons to be used at once; while those of the two latter nearly double the whole number of strings or pipes. It deserves an experiment, among the makers of imperfect instruments, whether a changeable scale cannot be rendered practicable, at least on the piano forte,[26] without increasing the number of strings, and at the same time allowing both the unisons to be used together—either by an apparatus for slightly increasing the tension of the strings, or by one which shall intercept the vibrations of such a part of the string, at its extremity, as shall elevate its tone, by the diesis of the system of temperament adopted. Were only 4 degrees to the octave, furnishing the instrument with 5 sharps and 4 flats, thus rendered changeable, there is little music which could not be correctly executed upon it.

Scholium 3.

In the same general manner, may be found the best system of intervals, for a scale confined to a less number of degrees than that of the complete Enharmonic scale. In such an investigation, the numbers in Table IV. expressing the frequency of all such adjacent degrees as have but one sound in the given scale, must be united; and the temperaments m, n, &c. of the theorem, when belonging to concords whose terminating degrees are united to those adjacent, must be taken, not what they were in the complete scale, but what they become, considering them as terminated by the substituted adjacent degree.

If, for example, the best temperaments were required for a scale of 15 degrees to the octave, such as is that of some European organs, or in other words, having no Enharmonic intervals except D E, and G A,—the numbers in Table IV. belonging to C and D, E and F, F and G, &c. must be united, and their sums substituted when they occur, for a, a', b, &c. in the theorem; while the temperament, for example, of the IIId on C must not be reckoned 77 as in the complete scale, but 1261 – 77 sharp, since its upper termination has become F, instead of E. With these variations let the same theorem be applied as before, till no value of x can be obtained, and the temperaments for that scale will be the best adjusted possible.

But as the scale which contains but 13 degrees, or 12 intervals, to the octave, is in much more general use than every other, we shall content ourselves with stating how the problem may be solved for scales containing any intermediate number of degrees, and proceed directly to the consideration of that which is so much the most practically important.

Lemma.

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.

Proposition IX.

It is evidently best, so far as the concords of the same name are concerned, that if of equal frequency, they should be equally tempered, unless by rendering them unequal, their medium temperament could be diminished; but this appears, from the Lemma, to be impossible. By tempering them unequally, the aggregate dissonance heard in a given time, by supposition of their equal frequency, would not be diminished, whilst the disadvantage of a transition from a better to a worse harmony would be incurred. Some advocates of irregular systems of temperament have, indeed, maintained this irregularity to be a positive advantage, as giving variety of character to the different keys. But this variety of character is obviously neither more nor less than that of greater and less degrees of dissonance. Now, what performer on a perfect instrument ever struck his intervals false, for the sake of variety? Who was ever gratified by the variety produced in vocal music by a voice slightly out of tune? If this be absurd, when applied to instruments capable of perfect harmony, it is scarcely less so to urge variety of character as being of itself a sufficient ground for introducing large temperaments into the scale. For these large temperaments will have nearly the same effect, compared with the smaller ones, that small temperaments would have, when compared with the perfect harmony of voices and perfect instruments. Possibly a discordant interval, or a concord largely tempered, might, in a few instances, add to the resources of the composer. But when an instrument is once tuned, the situation of these intervals is fixed beyond his control, and by occurring in a passage where his design required the most perfect harmony, it might as often thwart as favour the intended effect.

Since, then, the proposition is true in reference to the Vths, IIIds, and 3ds, when separately considered, it will be equally true when they are considered jointly, that is, as formed into harmonic triads, unless, by rendering the concords of the same name unequal in their temperament, the mean temperament of the Vths could be increased, and that of the IIIds and 3ds proportionally diminished. Could this be done, it might be a question whether the more equal distribution of the temperament among the concords of different names, might not justify the introduction of some inequality among those of the same name. But it is demonstrated in the Lemma, that the sum of the temperaments of each parcel of concords, in the system of equal semitones, is the least possible. Hence no changes in the Vths can diminish the average temperaments of the IIIds and 3ds.

Cor. Hence we derive an important practical conclusion: that whatever irregularities are introduced into the scale, must be such as are demanded by the different frequency of occurrence of the several concords. If we make any alterations in the scale of equal semitones, this must be our sole criterion. A given system of temperament is eligible, in proportion to the accuracy with which it is deduced from the different frequency of the different concords. And those who maintain that the frequency of different intervals does not sensibly vary, or that it is of such a nature as not to be susceptible of calculation, must, to be consistent, adhere to the scale of equal semitones.

Proposition X.

As the scale of equal semitones has been demonstrated to be the best, on supposition that all the concords of the same name occurred equally often, it ought to be made the standard from which all the variations, required by their unequal frequency, are to be reckoned. To find a set of numbers expressing the relative frequency of the several concords in the common scale, we have only to unite the numbers in Table IV. standing against those adjacent degrees which have but one sound in this scale. They will then stand as in the following table:

TABLE IX.

The general theorem of Prop. V. is equally applicable to the determination of the approximate place for any degree in this scale, considering the numbers in the above table as those to be substituted for a, a', b, &c.; and m, n, and p, in the first instance, as 49, –343 and 392, the uniform temperaments of the Vths, IIIds, and 3ds, in the scale of equal semitones. Since, however, the temperaments of the IIIds in this scale are sharp, which would require the signs of the 3d and 4th terms in the numerator of the general formula to be continually changed, it will be rendered more convenient for practice, if they are changed at first, so that it will stand thus:
x = am – a'm' – bn + b'n' + cp – c'p' a + a' + b + b' + c + c' .

Three successive applications of this theorem to each degree in the scale, in the manner described Prop. VI., will bring them very near to the required position, as appears by the smallness of the corrections in the 3d column below, where the results of the several operations are exhibited at one view.

TABLE X.

Cor. Hence we may deduce, in the same manner as in Prop. VII., the diatonic and chromatic intervals, the lengths of a string and their vibrations in a second, and the temperaments and beats of all the concords for the scale which results from the foregoing computations. They may be seen in the two following tables:

TABLE XI.

DIATONIC AND CHROMATIC INTERVALS.

TABLE XII.

Nothing in the above tables will need explanation, except the anomalous sharp beats of the 3ds, in the last column. These are derived from the perfect ratio 6 : 7, because these 3ds are, in reality, much nearer to the ratio of 6 : 7 than to that of 5 : 6; and hence could their beats be counted, they would be those of the table, and not those which would be derived from considering these 3ds as having flat temperaments of the ratio 5 : 6. But although the beats are slower, the nearer they approach the ratio 6 : 7, this ought not to be regarded as any sufficient reason for admitting so large temperaments into the scale, were it not absolutely necessary, in order to accommodate those 3ds which are of far more frequent occurrence. Although the beats of these 3ds grow slower as their temperaments are increased, yet they are losing their character in melody; and become, in this respect, more and more offensive, the more they are tempered. Hence the harmony and melody of the several intervals, jointly considered, are to be judged of rather from their temperaments, in the three first columns, than from their beats, in the three last.

Scholium 1.

It will be perceived, from a comparison of the temperaments in Table XII. with the corresponding numbers in Table IX., that the harshness of the several concords, especially of the IIIds and 3ds, is, in general, nearly in the inverse ratio of their frequency. The contending claims of the different concords render it impossible that this ratio should hold exactly. Including the Vths, the harmony of the concords is much more nearly equal, than the principle of rendering the temperament of each inversely as its frequency, could it be carried into complete effect, would require.

Scholium 2.

The foregoing system may be put in practice, on the organ, by making the Vths beat flat, with the exception of those on C, E, and G, which must beat sharp, at the rate required in the table; proving the correctness of the temperaments of the Vths, by comparing the beats of the IIIds, as they rise, with those required by column two. Should less accuracy be required, the IIIds on C, D, and A, might be made perfect, without producing any essential change in the system. This would reduce the labour of counting the beats to eight degrees only.

Scholium 3.

To show that the computations of the different frequency of occurrence of the different concords, on which this system of temperament is founded, may be relied on as practically correct, for music in general, it may be proper to state, that a similar series of calculations had been before made, from an enumeration of the concords in fifty scores of music entirely different from that made use of in Prop. IV. They were not, indeed, made with the same accuracy, for the music of which the chords were counted, was too generally of the simpler kind, and the numbers corresponding to those in the two columns under each concord in Table II., and those belonging to the major and to the minor signatures, corresponding to the numbers in Table III., were added, before the products were taken, instead of keeping the modes distinct, which is necessary to perfect accuracy. Yet the resulting scheme of temperament was essentially the same throughout, with the one which has been just described. It had the same anomalous temperaments, viz. the Vths on C, E, and G; and the IIId on A; and these anomalies were similar in degree. The greatest difference between any two corresponding temperaments, was between those of the 3d on E; the first computation making it only 702, while the last has it 818.

Proposition XI.

The aggregate of dissonance, heard in a given time, in the system of temperament unfolded in the last Proposition, will be less than in either of the systems generally practised.