THE MUSICAL OCTAVE.

BY C. STANILAND WAKE.

In his very interesting work, "Plato and the Other Companions of Socrates," Grote, the English historian of Greece, gives an analysis of Plato's Timaeus, in which is described the creation of the Kosmos by the world-architect, Demiurgos. We are there told that after forming the Kosmic sphere and endowing its parts with certain motions, Demiurgos made it a soul by the mixture of three ingredients-the Same or Identical, which is the indivisible and unchangeable essence of ideasthe Different, that is, the Plural or Divisible essence of bodies and of the elements—and a third principle compounded out of the other two. These three ingredients were thoroughly blended, and the whole was then divided into parts which were afterwards united once more in certain harmonic numerical proportions-complicate and difficult to follow, as remarked by Grote, who reproduces a "musical diagram" given by Plutarch to aid in the understanding of Plato's ideas. this diagram throws little light on the subject in the absence of Plutarch's own account of Plato's theory, which may be found in his essay, entitled "On the Procreation of the Soul." It is quite possible that Plato, who purposely refrained from giving information on certain subjects, intended that only the initiated should understand the significance of his statements, which, according to Taylor the Platonist, assume an acquaintance with the difference between arithmetical, geometrical and harmonic progression.

But

Let us see, however, what explanation is given by Plutarch of the Greek philosopher's statements. He tells us that the quaternary of numbers set down by Plato, "contains the unit, the common original of all even and odd numbers. Subsequent to which are 2 and 3, the first plane numbers; then 4 and 9, the first squares, and next 8 and 27, the first cubical numbers." Hence he supposes it was Plato's intention that (1) See Plutarch's "Moralia;" Goodwin's English Translation, Vol I.

the numbers should be placed, not in a single line, but the even numbers together in one line and the odd numbers together in another line opposite to each other. To illustrate this explanation he arranges the numbers according to a triangular figure, of which the following is a copy:

Plutarch then refers to the fact that by the addition of those pairs of numbers we obtain as products, 5, 13 and 35, three numbers with which the Pythagoreans associated certain ideas. Thus 5, which formed the basis of the Greek numeral system, was called by them the Nourisher, as they believed the fifth to be the first of all the intervals of tones which could be sounded. As to 13, the Pythagoreans called it the Remainder, "despairing of being able to divide a tone into equal parts." And as to the number 35, they named it Harmony, "as consisting of the two cubes 8 and 27, the first that rise from an odd and an even number, as also of four numbers, 6, 8, 9, 12, comprehending both harmonical and arithmetical proportions." That his meaning may be the better understood, Plutarch gives a "scheme for the eye," which is as follows:

In explanation of this diagram Plutarch says: "Admit a right-angled parrallelogram, A B C D, the lesser side of which, A B, consists of 5, the longer side, A C, contains 7 squares. Let the lesser division be unequally divided into two

and three squares marked by E, the larger division is two unequal divisions made of 3 and 4 squares marked F. Thus, A E F G comprehends 6, E B GI9, FGCH 8, and G I HD 12." The Greek writer adds: "By this means the whole parallelogram, containing 35 little square areas, comprehends all the proportions of the first concords of music in the numbers of these little squares."

In another of his essays, "Concerning Music," Plutarch explains the construction of the musical octave, showing how the numbers 6, 8, 9, and 12 enter into it. He states that the octave has a double proportion between its two extremes, which he takes to be 6 and 12. The intermediate numbers must, he adds, hold a proportion to the extremes of one and a third and one and a half. Such are the numbers 8 and 9; for 8 contains one and a third of 6, and 9 contains one and a half of 6, giving one extreme. The other extreme is 12, which contains 9 and a third part of 9 and 8 and a half part of 8. The Greek writer puts these figures into the form of a double proportion, thus: as 6:8::9:12, and as 6:9::8:12. This means only that as 6 contains three twos, and 8 four twos, so 9 contains three threes, and 12 four threes; and as 6 contains two threes, and 9 three threes, so 8 contains two fours and 12 three fours. In the former case we have the proportion of 4:3, characteristic of the diatessaron, and in the latter case, the proportion 3:2, which is distinctive of the diapente, the two intervals of which the octave was said by the Greeks to consist. Plutarch does not explain how the above numbers were arrived at, but the intermediate numbers 8 and 9 appear among the squares and cubes referred to in "The Procreation of the Soul," and as the two extremes must equal the two means, which give 72 as their product, the former must be 6 and 12, which also give 72.

We have seen that 35 was called Harmony by the Pythagoreans, as being the product of the addition of the four numbers, 6, 8, 9 and 12, which enter into the construction of the musical octave, and it might be thought that the octave ought to be divided into thirty-five parts to give the relations represented by those numbers. That is not so, however, and the smallest intervals used by the ancient Greeks would seem to have been the quarter tones obtained by dividing the tones into

four equal parts, a scale of 24 dieses being thus formed. *Although the number 24 can be derived from the above figures only in an indirect way, it would seem to furnish the true division of the octave, as I have endeavored to prove in a previous article. **The number 24 is obtainable by multiplying the numbers 6, 8 and 12, by 4, 3 and 2 respectively, and the three latter numbers added together make 9, the fourth of the Platonic numbers. But 9 is the cube of 3, which multiplied by 24 gives 72, the product of the two extremes, and also of the two means, of Plutarch's proportion. Probably, however, twenty-four is to be regarded merely as the duplication of twelve, one of the extremes of the proportion, a number which was anciently of great importance, as is evident from its association with the Zodiac, which is divided into 12 signs, and with the foot measure of 12 inches.

There does not appear to be anything to show that the Pythagoreans had any special regard for the number 12, and yet there are reasons for believing that it formed the real numerical basis of the musical octave. But before considering this point, let us see whether the division into twenty-four parts will furnish the numerical principles referred to by Plu

tarch. This division is noted in the musical staff of Fig. 3, on which are marked the recognized intervals of the modern diatonic scale.

If we analyze this scale in accordance with the principles of the triad, we find that it first divides into three equal parts,

fig. 4.

giving the intervals C F, F A, and A C, each of which contains eight subdivisions of the scale, as shown in Fig. 4.

But the elements of the staff may be so re-arranged as to *" The Modes of Ancient Greek Music. By D. R. Munro, page 53. ** MUSIC, December, 1896.

make the two poles of the octave C and G change places, as represented in Fig. 5.

Here we have the octave of G, which includes the trial GB D, as shown by the next figure.

fig. 6

In this figure we find number 9 twice, that is in G B and D G, while in the interval B D, which completes the scale, we have 6 subdivisions. Thus we have found within the limits of the octave the two mean numbers 8 and 9, and also the first extreme, number 6, which appears twice, moreover, in the triad CE G, as exhibited in the following figure:

E

fiq.7

The intervals C E and E G of this staff both contain six subdivisions, giving one of Plutarch's extremes, and the other extreme, 12, is furnished by the simple polar division of the octave into two parts, that is, by the intervals C G, G C. It is in this polar arrangement that we must really seek for the origin of the octave, which may be described as the product of the combination of two triads, which stand towards each other in a polar relation. The discovery that the ancient orientals used the pentatonic, or five-toned scale, corresponding to our F, G, A, C, D, which is still retained by the Chinese and most of the uncultured peoples shows that the musical diatonic scale has gone through a process of development. It is not necessary at present to consider this point, however, as we have to deal with the existing scale, which has been reached through the operation of certain fundamental principles of Nature.

To understand the polar construction of the octave based on