Chapter VIII: Front Matter (8)

But there are six concords to be accommodated, instead of two; and it is evident that all the pairs cannot have their temperament inversely as their frequency, since the numbers _a_, _b_, &c. and _m_, _n_, &c. have no constant ratio to each other. This, however, will be the case, at a medium, if _x_ be made such, that the _sum_ of the products of the numbers expressing the frequency of those chords whose temperaments are increased by _x_, into their respective temperaments, shall be equal to the sum of the corresponding products belonging to those chords whose temperaments are diminished by _x_. Applying this principle to the system of temperament in Prop. III, which flattens all the concords, it is plain that raising any given degree by _x_ will increase the temperaments of the concords above that degree, and diminish those of the concords below it. Hence it ought to be raised till

(m - x) a + (n - x)b + (p - x)c = (m′ + x)a + (n′ + x)b′ +
(p + x)c′;

from which _x_ is found

= (am - a′m′ + bn - b′n′ + cp - c′p′) /
(a + a′ + b + b′ + c + c′)

Should either of the temperaments be sharp, the sign of that term of the numerator, in which it occurs, must be changed; and should the total value of the expression be negative, _x_ must be taken below C.

PROPOSITION VI.

To determine that system of temperaments for the concords of
the changeable scale, which will render it, including every
consideration, the most harmonious possible.

We can scarcely expect to find any direct analytical process, which will furnish us with a solution of this complicated problem, at a single operation. We shall therefore content ourselves with a method which gradually approximates towards the desired results. The best position of any given degree, as C, supposing all the rest fixed, is determined by the last proposition. In the same manner it is evident that the constitution of the whole scale will be the best possible, when no degree in it can be elevated or depressed, without rendering the sums of the products there referred to, unequal. We can approximate to this state of the scale, by applying the theorem in Prop. V. to each of the degrees successively. It is not essential in what order the application is made; but for the sake of uniformity, in the successive approximations, we will begin with that degree which has the greatest sum _a_ + _a′_ + _b_ + &c. belonging to it, and proceed regularly to that in which it is least. Making the equal temperament of Prop. III., (in which the Vths, IIIds, and 3ds are flattened, 154, 77 and 77, respectively.) the standard from which to commence the alterations in the scale required by the unequal frequency of different chords, and beginning with D, the theorem gives _x_ = 5. Hence supposing the rest of the degrees in the scale unaltered, it will be in the most harmonious state, when D is raised 5/540 of a comma. For by the last proposition, the temperament of the six concords affected by changing the place of D is best distributed, and that of the other concords is not at all affected. We will now proceed to the second degree in the scale, viz. A; in which the application of the theorem gives _x_ = 13. In this application, however, as D was before raised 5, _m_, the temperament of the Vth below A, must be taken 154 + 5; and in all the succeeding operations, when the exterior termination of any concord has been already altered, we must take its temperament, not what it was at first, but what it has become, by such previous alteration. In this manner, the scale is becoming more harmonious at every step, till we have completed the whole succession of degrees which it contains.

Let us now revert to D, the place where we began. As each of the outer extremities of the chords which are terminated by D has been changed, a new application of the theorem will give a second correction for the place of D; although, as the numbers _a_, _a'_, _b_, &c. continue the same, it will be less than before. Continue the process through the whole scale, and a second approximation to the most harmonious state will be obtained. In this manner let the theorem be applied, till the value of _x_ is exhausted, for every degree; and it will then be in the most harmonious state possible. Three operations gave the following results:

TABLE V.

+------+-----------+-----+-----+
| | 1st | 2d. | 3d. |
|Bases.| Operation.| | |
+------+-----------+-----+-----+
| F♯ | +18 | +5 | +1 |
| F | -20 | -6 | -1 |
| E♯ | +18 | +5 | 0 |
| E | +14 | +5 | 0 |
| E♭ | -69 | -8 | -1 |
| D♯ | +19 | +5 | +1 |
| D | +5 | +2 | +1 |
| D♭ | -45 | -7 | -2 |
| C♯ | +18 | +6 | 0 |
| C | -5 | -5 | -2 |
| B♯ | +18 | +5 | 0 |
| B | +19 | +5 | 0 |
| B♭ | -23 | -10 | -1 |
| A♯ | +18 | +7 | 0 |
| A | +13 | +4 | +1 |
| A♭ | -71 | -7 | -2 |
| G♯ | +17 | +5 | 0 |
| G | -14 | 0 | 0 |
| F♯♯ | +44 | +5 | 0 |
| G♭ | -46 | -5 | 0 |
+------+-----------+-----+-----+

The sign _plus_ denotes that the degree to which it belongs is to be raised, and _minus_, that it is to be depressed. The corrections in each succeeding operation are to be added to those in the preceding. The errors, in the 3d approximation, are so trifling, that a 4th would be wholly useless.

NOTE. The foregoing calculations will be rendered much more expeditious and sure, by reducing the theorem, in some sense, to a diagram, as in the first of the following figures; and by applying the successive corrections to the circumference of a circle divided into parts proportioned to the intervals of the enharmonic scale, as in the second.

PROPOSITION VII.

To determine the temperaments and beats of all the concords,
together with the values of the diatonic and chromatic intervals,
and the lengths and vibrations per second of a string producing
all the sounds, of the system resulting from the last proposition.

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.

C +------+------+------+ C
| 2998 | 2998 +------+ B♯
| | | 1772 |
B +------+------+------+ B
| 1831 | | 3033 |
B♭|------+ 4813 | |
| | +------+ A♯
| 2982 | | 1780 |
A +------+------+------+ A
| 1871 | | 3030 |
A♭+------+ 4839 | |
| | +------+ G♯
| 2968 | | 1809 |
G +------+------+------+ G
| 1814 | +------+ F♯♯
| | | 1798 |
G♭+------+ 4820 +------+ F♯
| | | |
| 3006 | | 1824 |
F +------+------+------+ F
| | +------+ E♯
| 2988 | 2988 | |
| | | 1777 |
E +------+------+------+ E
| 1870 | | 3028 |
| | | |
E♭+------+ 4818 | |
| 2948 | +------+ D♯
| | | 1790 |
D +------+------+------+ D
| 1835 | | 3018 |
D♭+------+ 4827 | |
| | +------+ C♯
| 2992 | | 1809 |
C +------+------+------+ C

TABLE VII.

+-----+-------------------++-------+----------++---------------------+
| |Temperaments of the||Lengths|Vibrations||Beats in 10 S. of the|
|Bases+-------------------+| of | in a |+-------+------+------+
| |Vths♭|IIIds♭|3ds♭||String.| Second. || Vths. |IIIds.| 3ds. |
+-----+------+------+-----++-------+----------++-------+------+------+
| B♯ | | | 77 || 51431 | 466,64 || | | 43,4 |
+-----+------+------+-----++-------+----------++-------+------+------+
| B | 154 | 76 | 93 || 53574 | 447,98 || 47,4 | 39,0 | 57,8 |
+-----+------+------+-----++-------+----------++-------+------+------+
| B♭ | 147 | 35 | 97 || 55880 | 429,49 || 43,5 | 17,7 | 57,4 |
+-----+------+------+-----++-------+----------++-------+------+------+
| A♯ | 156 | | 78 || 57448 | 417,77 || 45,1 | | 46,2 |
+-----+------+------+-----++-------+----------++-------+------+------+
| A | 153 | 71 | 107 || 59852 | 400,99 || 42,5 | 33,5 | 59,4 |
+-----+------+------+-----++-------+----------++-------+------+------+
| A♭ | 154 | 9 | || 62487 | 384,08 || 40,4 | 4,0 | |
+-----+------+------+-----++-------+----------++-------+------+------+
| G♯ | 151 | 76 | 75 || 64177 | 373,97 || 39,1 | 32,9 | 39,2 |
+-----+------+------+-----++-------+----------++-------+------+------+
| G | 132 | 39 | 97 || 66907 | 358,71 || 32,9 | 16,3 | 48,1 |
+-----+------+------+-----++-------+----------++-------+------+------+
| F♯♯ | | | 101 || 68778 | 348,95 || | | 48,5 |
+-----+------+------+-----++-------+----------++-------+------+------+
| G♭ | | 56 | || 69760 | 344,03 || | 21,9 | |
+-----+------+------+-----++-------+----------++-------+------+------+
| F♯ | 154 | 76 | 83 || 71685 | 334,80 || 36,0 | 29,2 | 38,5 |
+-----+------+------+-----++-------+----------++-------+------+------+
| F | 139 | 32 | 130 || 74760 | 321,03 || 30,9 | 11,9 | 57,8 |
+-----+------+------+-----++-------+----------++-------+------+------+
| E♯ | 154 | | 78 || 76874 | 312,20 || 33,2 | | 33,5 |
+-----+------+------+-----++-------+----------++-------+------+------+
| E | 149 | 74 | 110 || 80085 | 299,68 || 30,8 | 25,2 | 45,3 |
+-----+------+------+-----++-------+----------++-------+------+------+
| E♭ | 110 | 13 | 54 || 83608 | 287,05 || 21,7 | 4,1 | 21,5 |
+-----+------+------+-----++-------+----------++-------+------+------+
| D♯ | 154 | 53 | 78 || 85868 | 279,50 || 29,6 | 17,0 | 30,0 |
+-----+------+------+-----++-------+----------++-------+------+------+
| D | 144 | 61 | 112 || 89480 | 268,21 || 26,5 | 18,5 | 41,1 |
+-----+------+------+-----++-------+----------++-------+------+------+
| D♭ | 180 | 50 | || 93342 | 257,12 || 32,0 | 14,8 | |
+-----+------+------+-----++-------+----------++-------+------+------+
| C♯ | 156 | 78 | 82 || 95920 | 250,20 || 26,6 | 22,0 | 28,0 |
+-----+------+------+-----++-------+----------++-------+------+------+
| C | 156 | 46 | 143 ||100000 | 240,00 || 25,8 | 12,8 | 47,5 |
+-----+------+------+-----++-------+----------++-------+------+------+

PROPOSITION VIII.

To compare the harmoniousness of the foregoing system with that
of several others, which have been most known and approved.

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.

+--------------------+---------+------------+-----------+----------+
| Systems. |Huygen's.|Dr. Smith's.|Mean Tones.|New Scale.|
+--------------------+---------+------------+-----------+----------+
| Dissonance { Vths | 825 | 945 | 850 | 786 |
| of the { IIIds | 121 | 382 | 0 | 240 |
| { 3ds | 1049 | 629 | 944 | 683 |
+--------------------+---------+------------+-----------+----------+
| Total | 1995 | 1956 | 1794 | 1709 |
+--------------------+---------+------------+-----------+----------+

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 ⅓, 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.

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.

PROPOSITION IX.

If all the concords of the same name, in a scale of twelve
intervals to the octave, were of equally frequent occurrence, the
best system of temperament would be that of equal semitones.

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.

To determine the best distribution of the temperaments of the
concords in the Douzeave Scale.

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.

+------+------------+-------------+------------+
| | Vths, 4ths,| IIIds, 6ths,| 3ds, VIths,|
|Bases.| and | and | and |
| | Octaves. | Octaves. | Octaves. |
+------+------------+-------------+------------+
| B | 221 | 135 | 1161 |
+------+------------+-------------+------------+
| B♭ | 418 | 654 | 34 |
+------+------------+-------------+------------+
| A | 870 | 568 | 1085 |
+------+------------+-------------+------------+
| G♯ | 57 | 82 | 365⅕ |
+------+------------+-------------+------------+
| G | 1207 | 1197 | 567¼ |
+------+------------+-------------+------------+
| F♯ | 67 | 29½ | 1072 |
+------+------------+-------------+------------+
| F | 639 | 924 | 78 |
+------+------------+-------------+------------+
| E | 548 | 323 | 1151 |
+------+------------+-------------+------------+
| E♭ | 265⅓ | 363½ | 144½ |
+------+------------+-------------+------------+
| D | 1166 | 943 | 569 |
+------+------------+-------------+------------+
| C♯ | 26 | 18 | 581 |
+------+------------+-------------+------------+
| C | 816 | 1131 | 184 |
+------+------------+-------------+------------+

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.

+------+----------+----------+----------+
|Bases.| First | Second | Third |
| |Operation.|Operation.|Operation.|
+------+----------+----------+----------+
| B | -140 | -35 | -2 |
+------+----------+----------+----------+
| B♭ | +308 | +33 | -1 |
+------+----------+----------+----------+
| A | -8 | -23 | +2 |
+------+----------+----------+----------+
| G♯ | -257 | -22 | -2 |
+------+----------+----------+----------+
| G | +107 | +24 | -8 |
+------+----------+----------+----------+
| F♯ | -264 | -7 | 0 |
+------+----------+----------+----------+
| F | +238 | +40 | +6 |
+------+----------+----------+----------+
| E | -80 | -34 | -4 |
+------+----------+----------+----------+
| E♭ | +157 | +2 | -4 |
+------+----------+----------+----------+
| D | +58 | + 8 | 0 |
+------+----------+----------+----------+
| C♯ | -352 | -29 | -1 |
+------+----------+----------+----------+
| C | +176 | +29 | +4 |
+------+----------+----------+----------+

_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._

C +------+------+ C
| 2895 | 2895 |
B +------+------+ B
| | 1991 |
| 4869 +------+ B♭
| | 2878 |
A +------+------+ A
| | 2761 |
| 4865 +------| G♯
| | 2104 |
G +------+------+ G
| | 2903 |
| 4856 +------| F♯
| | 1953 |
F +------+------+ F
| 2911 | 2911 |
E +------+------+ E
| | 2235 |
| 4833 +------+ E♭
| | 2598 |
D +------+------+ D
| | 2957 |
| 4874 +------+ C♯
| | 1917 |
C +------+------+ C

TABLE XII.

+------+--------------------++-------+--------++------------------------+
| |Temperaments of the ||Lengths|Vibrat- ||Beats in 10 Secs. of the|
|Bases.+------+-------+-----+| of |ions per|+------+--------+--------+
| |Vths♭|IIIds ♯|3ds♭||Strings.|Second || Vths | IIIds | 3ds |
+------+------+-------+-----++-------+--------++------+--------+--------+
| B | 143 | 675 | 149 || 53446 | 449,04 || 44,0 | 352,8 | 92,4 |
+------+------+-------+-----++-------+--------++------+--------+--------+
| B♭ | 105 | 69 |1114 || 55954 | 428,92 || 30,8 | 34,0 | 155,2♯ |
+------+------+-------+-----++-------+--------++------+--------+--------+
| A | 138 | 10.♭| 154 || 59787 | 401,42 || 38,6 | 4,6♭ | 85,2 |
+------+------+-------+-----++-------+--------++------+--------+--------+
| G♯ | 387♯ | 833 | 288 || 63712 | 376,79 || 98,7♯| 360,5 | 155,4 |
+------+------+-------+-----++-------+--------++------+--------+--------+
| G | 106 | 43 | 175 || 66874 | 358,88 || 26,4 | 17,6 | 86,8 |
+------+------+-------+-----++-------+--------++------+--------+--------+
| F♯ | 160 | 954 | 150 || 71496 | 335,68 || 37,2 | 372,8 | 69,8 |
+------+------+-------+-----++-------+--------++------+--------+--------+
| F | 124 | 30 | 957 || 74786 | 320,92 || 27,6 | 10,8 | 143,0♯ |
+------+------+-------+-----++-------+--------++------+--------+--------+
| E | 108 | 180 | 151 || 79970 | 300,10 || 22,2 | 66,6 | 62,0 |
+------+------+-------+-----++-------+--------++------+--------+--------+
| E♭ | 136♯ | 311 | 818 || 84194 | 285,06 || 26,6♯| 102,2 | 186,6♯ |
+------+------+-------+-----++-------+--------++------+--------+--------+
| D | 144 | 6 | 174 || 89384 | 268,50 || 26,6 | 2,2 | 64,0 |
+------+------+-------+-----++-------+--------++------+--------+--------+
| C♯ | 52♯ | 1009 | 128 || 95682 | 250,83 || 10,9♯| 295,3 | 44,8 |
+------+------+-------+-----++-------+--------++------+--------+--------+
| C | 135 | 16 | 446 ||100000 | 240,00 || 22,4 | 4,0 | 147,0 |
+------+------+-------+-----++-------+--------++------+--------+--------+

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.

In order to compare the foregoing system with those which have been most generally approved, the temperaments of all the concords have been calculated, in the system of equal semitones; in that of Earl Stanhope, which has had considerable celebrity; in that of Dr. T. Young; in that of Mr. Hawkes; in that of Kirnberger, which has been extensively adopted in Germany; and in that which is described by Rousseau and D'Alembert as generally practised in France. If these temperaments be multiplied into the corresponding numbers of Table IX., agreeably to what was shown under Prop. VIII., and those products which belong to the several concords of the same name be added, the sums, after the three right-hand figures are cut off, will be as follows:

TABLE XIII.

+-------------+------+--------+--------+-------+-------+--------+------+
| Systems. | Mean |Young's.|Kirnber-|French.| Stan- |Hawkes'.| New |
| | Temp.| | ger's. | |hope's.| |Scale.|
+-------------+------+--------+--------+-------+-------+--------+------+
|Dissonance of| | | | | | | |
| the { Vths | 309 | 494 | 681 | 561 | 595 | 665 | 810 |
| { IIIds| 2184 | 1541 | 1397 | 1346 | 1175 | 925 | 530 |
| { 3ds | 2740 | 2448 | 2019 | 2121 | 1992 | 1676 | 1363 |
+-------------+------+--------+--------+-------+-------+--------+------+
| Total | 5233 | 4483 | 4097 | 4028 | 3762 | 3266 | 2703 |
+-------------+------+--------+--------+-------+-------+--------+------+

From an inspection of the sums at the foot of the table, it will be seen that the amount of dissonance heard in a given time is decidedly less in the new scale than in either of the others; and that it is scarcely more than half as great as in the scale of equal semitones. On the other hand, the temperament is very unequally distributed, which must be admitted, cæteris paribus, to be a disadvantage. It is even somewhat greater than in the scheme of Mr. Hawkes, although by no means in the same ratio, as the aggregate dissonance is less. It contains one Vth, which will be somewhat harsh, and four IIIds and three 3ds, which will be quite harsh. But these, as will appear from an inspection of Table IX., are, of all others, of by far the most unfrequent occurrence; so that the unpleasant effect of a transition from a better to a much worse harmony will be very seldom felt. In the six simplest keys of the major, and in the three of most frequent occurrence in the minor mode, they are _never_ heard, except in occasional modulations; and even then, generally no one, and rarely more than one is heard. Now these nine keys, as will appear from Table III., comprise more than five times as much of the music examined as all the rest. The same remarks might be extended to three other minor keys, were it not that the sharp seventh is so generally used, that it deserves to be considered as an essential note of the key.

But there are two important considerations, more than counterbalancing the objection to this system, derived from the greater inequality in the distribution of its temperaments, which have not been hitherto noticed, as not being susceptible of mathematical computation.

1st. We have gone on the supposition that tunes on the more difficult keys are as often performed, according to their number, as those on the simpler keys; and have taken for the measure of dissonance, in different systems, what would be actually heard, if the 1600 scores, whose signatures were examined, were all played in succession, and on the keys to which they are set. But the fact is, that those pieces which are set to the simpler keys are oftener played, and with fuller harmony, on account of the greater ease of execution, than those in which many of the short finger keys must be used.

2d. Pieces on the more difficult keys are often played on the adjacent easier keys, but the contrary is seldom or never done.

Giving to these two considerations no more than a reasonable weight, they will counterbalance the objection, and will render it evident that the sums under the several systems in the table may be taken as a true exhibition of their respective merits, without any injustice to the more equal systems at the left-hand of the table.

_Cor._ We may hence draw a comparison between the systems in common use. Their merits, when every consideration is taken into view, are nearly in the inverse ratio of the sums denoting their aggregate dissonance. That of Mr. Hawkes is the best, and, in many respects, has a remarkable analogy to the one derived from the preceding investigations.

_Cor._ 2. As the aggregate dissonance of the changeable scale is calculated on the same principles, in Prop. VIII., as that of the Douzeave in this, a comparison of the results in Table VIII. with those in Table XIII., will furnish us with the relative dissonance of different systems for these different scales. The relative dissonance of the two systems which form the object of this essay, is nearly as 17 : 27. Hence it appears, that by inserting eight new sounds between those of the common octave, the harshness of the music executed, at a medium of all the keys, may be diminished by more than one third of the whole, while the transition from a better to a worse harmony will never be perceived.

ART. XXII. _Notice of Colonel Trumbull's Picture of the Declaration of Independence._

It is proper that some mention of this great national work should be made, in publications less transient than newspapers; and as the fine arts are included within the design of this Journal, it may with propriety be noticed here. This is the greatest work which the art of painting has ever produced in the United States. The picture is magnificent both in size and in execution. The dimensions of the canvass are eighteen feet by twelve.

"This picture forms one of a series long since meditated by Mr. Trumbull, in which it was intended to represent the most important events, civil and military, of the American revolution, with portraits of the most distinguished actors in the various scenes. The materials for this purpose were collected many years ago, and two plates have been engraved from paintings of the deaths of Gen. Warren and Gen. Montgomery;[27] but the work was suspended, in consequence of the political convulsions, which, during twenty-five years, were so fatal to the arts of peace.

"The government of the United States have ordered four of the subjects originally proposed by Mr. Trumbull, to be painted by him, and to be deposited in the capitol.

"No event in human history ever shed a more salutary influence over the destinies of so great a mass of mankind: the wisdom of no political act was ever so soon and so powerfully demonstrated, by such magnificent consequences. And justly may the nation be proud of the act itself; and of those eminent men, its authors, whose patriotism (rising above enthusiasm, and the passions which have so often bewildered mankind) was calm, dignified, persevering, and always under the guidance of reason and virtue.

"The painting represents the congress at the moment when the committee advance to the table of the president to make their report.

"It contains faithful portraits of all those members who were living when the picture was begun, and of all others of whom any authentic representation could be obtained. Of a small number, no trace could be discovered; and nothing was admitted which was not authentic."

This picture is now, by permission of government, exhibited in the Academy of Arts in New-York, and will probably be shown in some of our other principal cities, before it receives its final location at Washington.

It exhibits the interior of the then Congress Hall at Philadelphia. Most of the members are represented as sitting in their respective chairs, or, in various instances, as standing in different parts of the room. Almost all the portraits were taken by Colonel Trumbull _from the living men_, and their accuracy may therefore be relied on.

The president, John Hancock, sitting at a table, and elevated somewhat by a low platform, is receiving the report of the committee declaring the independence of the colonies; that committee, individually illustrious, and in this august transaction collectively memorable, was composed of Franklin, Adams, Sherman, Jefferson, and Livingston. Mr. Jefferson, in the prime of life, is in the act of laying upon the table the great charter of a nation's liberties; while his companions support him by their silent but dignified presence, and the venerable Franklin, in particular, imposes new obligations on his country's gratitude.

The figures are as large as the life; and it may safely be said, that the world never beheld, on a similar occasion, a more noble assemblage. It was the native and unchartered nobility of great talent, cultivated intelligence, superior manners, high moral aim, and devoted patriotism. The crisis demanded the utmost firmness of which the human mind is capable--a firmness not produced, for the moment, by passion and enthusiasm, but resting on the most able comprehension of both duties and dangers, and on a _principled_ determination to combat the one and to fulfil the other.

This moral effect has been produced in the fullest and finest manner by this great painter; and no true American can contemplate this picture without gratitude to the men who, under God, asserted his liberties, and to the artist who has commemorated the event, and transmitted the very features and persons of the actors to posterity. Such efforts of the pencil tend powerfully also to invigorate patriotism, and to prompt the rising generation to emulate such glorious examples.

The composition and execution in this picture are in a masterly style. The grouping of so many full length portraits, in a scene in which there could scarcely be any action, and in such a manner as to dispose of them without monotony, was an attainment of no small difficulty. The painter could not even avail himself of the adventitious relief of splendid costume and furniture, and of magnificence or rich decorations in architecture; for on this occasion both were characterized by an elegant simplicity only, such however as became the actors and the crisis.

The composition has all the variety of which it is susceptible; and there is also enough of it in the style of dress and of features to relieve the eye from any danger of satiety.

It is believed, that in this picture, the United States possess a treasure to which there is no parallel in the world. In no instance, within our knowledge, is there an exhibition to an equal extent, of the actual portraits of an illustrious assembly, concerned in so momentous a transaction.

It was a great thing to assert, _in principle_, the liberties of this country; but it was also a great thing to vindicate them by arms; and we rejoice that Colonel Trumbull is still to proceed, under the sanction of government, to delineate other scenes, in which Washington and his illustrious American coadjutors, and the flower of French chivalry, were the actors. In the maturity of his experience, skill, and fame--possessed, as he is, of the portraits of most of the great men of that period, taken principally from the life, and having been himself largely and personally conversant with them in their great deeds, we trust that the government will promptly second what we doubt not the united voice of the nation will demand--that the illustrious artist should dedicate the evening of his life to his country's honour and glory.

INTELLIGENCE.

ART. XXIII. _An Address to the People of the Western Country._

A number of the citizens of Cincinnati have recently instituted a society for the collection, preservation, exhibition, and illustration of natural and artificial curiosities, particularly those of the _western country_. The first efforts of the managers will be directed to the establishment of a permanent museum, on a scale so comprehensive as to receive specimens of every thing curious which they may be able to procure. In attempting to form this repository, they must of course solicit the aid of their fellow-citizens in all quarters of the extensive region, whose ancient works and natural history they propose to illustrate. The following are the classes of objects that will especially attract their attention, and to which they are desirous, at an early period, of directing the views of the community:

1. Our metals and minerals generally, including petrifactions.

2. Our indigenous animals, embracing the remains of those which are now extinct.

3. The relics of the unknown people who constructed the ancient works of the western country.

4. The various articles manufactured, for ornament or use, by the present savage tribes.

The subjects of the first class are considered by the Society as extremely interesting. Every citizen of the western country must _feel_ the necessity of a speedy developement of its mineral resources. To find beneath our own soil an adequate supply of the various minerals which are now imported at an enormous expense, must be regarded by all as a matter of the first and greatest importance. The managers are anxious to be instrumental in the advancement of this useful work, and earnestly solicit the co-operation of the public. They will be thankful for specimens of all the rare or curious minerals that may be discovered in this country. To every specimen that may be transmitted, a label should be attached, stating either the kind of rock or stratum to which it belonged, or its precise locality. Whenever it is required, the managers will have a part of any specimen which is sent to them, analyzed, and a correct report made of its nature, thus affording to the discoverer a full opportunity of availing himself of all the pecuniary advantages that may attend the discovery.

As objects of scientific interest, the managers intend, as early as possible, to commence the formation of a cabinet of petrifactions. The rocks of few other countries contain a greater number and variety of these animal remains of the ancient ocean, than the limestone districts of the Ohio and Mississippi. They both astonish and confound most of the travellers through this region; and although objects of familiar examination to ourselves, they have not been collected or described by our citizens. An extensive and well arranged cabinet of these extraneous fossils would afford, both to the zoologist and geologist, an exquisite feast. It is hoped that every specimen sent to the Society will be accompanied by a label, stating the place where it was found.

It is the wish of the Society to obtain and preserve specimens of all the native animals of this country. Most of the larger quadrupeds having receded before the unceasing extension of our settlements, are now so rare as to be unknown to all but our oldest emigrants. Measures will be taken by the managers to procure from the general retreat in the northwest, and exhibit to the people in the Ohio countries, a specimen of every quadruped which lately inhabited them; and while engaged in this enterprise, they hope to import from the same distant wilderness, a variety of the animals which are peculiar to it.

Our native birds have not retreated, like our quadrupeds, and are, therefore, within our reach. The managers hope to see the Society, in due time, in possession of a large collection of these beautiful animals. In the accomplishment of this undertaking, it is easy to perceive that the Society may be powerfully aided by the community: and a sanguine hope is entertained, that no backwardness or indifference will be manifested by those who may fortunately have it in their power to forward specimens.