Proposition IV.

To find a set of numbers, expressing the ratio of the probable number of times that each of the different consonances in the scale will occur, in any set of musical compositions.

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

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.


Bases.	Common Chords.		Flat Fifths.		7ths.		9-sevenths.
Bases.	Major	Minor	Major.	Minor.	Major.	Minor.	Major.	Minor.
	mode.	mode.	Major.	Minor.	Major.	Minor.	Major.	Minor.
B III	5	8	—	—	7	—	—	—
B	3	—	163	55	11	17	2	—
B	4	4	—	—	—	—	—	—
A VII	—	—	—	—	—	—	3	—
A III	19	8	—	—	7	2	—	—
A	166	588	2	1	26	5	2	—
G	—	—	3	38	—	—	—	—
G 3	18	15	—	—	—	—	—	—
G	965	93	—	—	178	15	3	—
F	—	—	46	4	11	2	—	—
F	352	60	—	—	11	12	7	3
E III	26	271	—	—	1	25	—	—
E	32	25	5	1	8	—	1	4
D III	—	—	2	1	—	—	—	—
D	—	—	—	4	—	—	—	—
D III	29	4	—	—	49	7	—	—
D	120	129	—	—	55	18	6	1
C	—	—	2	4	1	—	—	—
C 3	2	—	—	—	—	—	—	—
C	1769	275	—	—	5	1	4	1

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

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.


Bases.	Vths, 4ths, and Octaves.		IIIds, 6ths, and Octaves.		3ds, VIths, and Octaves.
Bases.	Major.	Minor.	Major.	Minor.	Major.	Minor.
B	8	8	10	8	1141	214
B	3	6	22	19	——	——
A	195	607	22	10	626	663
G	——	——	——	——	32	310
G	1088	116	1090	125	22	23
F	——	——	——	——	78	10
F	395	78	486	301	——	——
E	59	308	40	284	1828	308
E	——	——	2	——	——	——
D	——	——	——	——	7	9
D	197	156	60	7	403	213
C	——	——	——	——	26	12
C	1807	278	1959	870	4	1

Bases.	5ths, IVths, and Octaves.		7ths, IIds, and Octaves.		VIIths, 2ds, and Octaves.
Bases.	Major.	Minor.	Major.	Minor.	Major.	Minor.
B	256	265	25	17	——	——
B	——	——	——	——	——	——
A	2	1	34	7	3	——
G	10	53	——	——	——	——
G	——	——	188	20	——	——
F	74	7	1	2	——	——
F	——	——	——	——	17	16
E	10	1	20	27	——	——
E	——	——	——	——	——	——
D	7	5	——	——	——	——
D	——	——	123	27	——	——
C	9	10	1	——	——	——
C	——	——	5	1	10	1

Besides the following chromatic intervals:

	{	8 extreme sharp 5ths on C
Major mode.	{	1 ————————— D
	{	1 extreme flat 7th —— G

	{	4 extreme sharp 6ths on F
Minor mode.	{	4 extreme flat 7ths on C
	{	3 ————————— G

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.

Signatures.	Major Mode.	Minor Mode.
4s	42	2
3s	95	6
2s	200	13
1	322	72
	176	121
1	180	97
2s	70	77
3s	116	8
4s	0	3
Ratio of their sums 1201 :		399

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 C

, 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.


Bases.	Vths and 4ths.	IIIds and 6ths.	3ds and VIths.	Bases.	Vths and 4ths.	IIIds and 6ths.	3ds and VIths.
F	67	29	1072	B	——	——	4
F	639	924	66	B	221	135	1161
E	——	——	12	B	418	654	5
E	548	323	1151	A	——	——	29
E	265	363	½	A	870	568	1085
D	⅓	½	144	A	52	78	⅕
D	1166	943	569	G	5	4	365
D	1	6	——	G	1207	1197	567
C	25	12	581	F	——	——	¼
C	816	1131	180	G	——	½	——

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].)