(from Nicomachus)
While describing Pythagoras's wisdom in instructing his disciples, we must not fail to note that he invented the harmonic science and ratios. But to explain this we must go a little backwards in time.
Once as he was intently considering music, and reasoning with himself whether it would be possible to devise some instrumental assistance to the sense of hearing, so as to systematize it, as sight is made precise by the compass rule, and [telescope], or touch is made reckonable by balance and measures, - so thinking of these things Pythagoras happened to pass by a brazier's shop, where providentially he heard the hammers beating out a piece of iron on the anvil, producing sounds that harmonized, except one. But he recognized in these sounds, the concord of the octave, the fifth, and the fourth. He saw that the sound between the fourth and the fifth, taken by itself, was a dissonance, and yet completed the greater sound among them. Delighted, therefore, to find that the thing he was anxious to discover had by divine assistance succeeded, he went into the smithy, and by various experiments discovered that the difference of sound arose from the magnitude of the hammers, but not from the force of the strokes, nor from the shape of the hammers, nor from the change of position of the beaten iron. Having then accurately examined the weights and the swing of the hammers, he returned home, and fixed one stake diagonally to the walls, lest some difference should arise from there being several of them, or from some difference in the material of the stakes. From this stake he then suspended four gut-strings, of similar materials, size, thickness and twist. A weight was suspended from the bottom of each. When the strings were equal in length, he struck two of them simultaneously, he reproduced the former intervals, forming different pairs.
He discovered that the string stretched by the greatest weight, when compared with that stretched by the smallest weight, the interval of an octave. The weight of the first was twelve pounds, and that of the latter six. Being therefore in a double ratio, it formed the octave, which was made plain by the weights themselves. Then he found that the string from which the greater weight was suspended compared with that from which was suspended the weight next to the smallest, and which weight was eight pounds, produced the interval known as the fifth. Hence he discovered that this interval is in a ratio of one and a half to one, or three to two, in which ratio the weights also were to each other. The he found that the string stretched by the greatest weight produced, when compared with that which was next to it, in weight, namely, nine pounds, the interval called the fourth, analogous to the weights. This ratio, therefore, he discovered to be in the ratio of one and a third to one, or four to three; while that of the from string from which a weight of nine pounds was suspended to the string which had the smallest weight, again in a ratio of three to two, which is 9 to 6. In like manner, the string next to that from which the small weight was suspended, was to that which had the smallest weight, in the ratio of 4 to 3 (being 8 to 6) but to the string which had the greatest weight, in a ratio of 3 to 2, being 12 to 8. Hence that which is between the fifth and fourth, and by which the fifth exceeds the fourth is proved to be as nine is to eight. But either way it may be proved that the octave is a system consisting of the fifth in conjunction with fourth, just as the double ratio consists of three to two, and four to three; as for instance 12, 8 and 6; or, conversely of the fourth and the fifth, as in the double ratio of four to three and three to two, as for instance, 12, 9 and 6 therefore, and in this order, having confirmed both his hand and hearing to the suspended weights, and having established according to them the ratio of the proportions, by an easy artifice he transferred the common suspension of the strings from the diagonal stake to the head of the instrument which he called "chordotenon," or string-stretcher. Then by the aid of pegs he produced a tension of the strings analogous to that effected by the weights. Employing this method, therefore, as a basis, and as it were an infallible rule, he afterward extended the experiment to other instruments, namely, the striking of pans, to pipes and [r----], to monochords, triangles, and the like in all of which he found the same ratio of numbers to obtain. Then he named the sound which participates in the number 6, tonic; that which participates of the number 8, and is four to three, sub-dominant; that which participates of the number 9, and is one tone higher then the sub-dominant, he called, dominant, and 9 to 8; but that which participates of the number 12, octave.
Then he filled up the middle spaces with analogous sounds in diatonic order, and formed an octochord from symmetric numbers; from the double, the three to two, the four to three, and from the difference of these, the 8 to 9. Thus he discovered the harmonic progression, which tends by a certain physical necessity from the lowest to the most acute sound, diatonically. Later, from the diatonic he progressed to the chromatic and enharmonic orders, as we shall later show when we treat of music. This diatonic scale however, seems to have the following progression, a semi-tone, a tone, and a tone; and this is the fourth, being a system consisting of two tones, and of what is called a semi-tone. Afterwards, adding another tone, we produce the fifth, which is a system consisting of three tones and a semi-tone. Next to this is the system of a semi-tone, a tone, and a tone, forming another fourth, that is, another four to three ratio. Thus in the more ancient octave indeed, all the sounds from the lowest pitch which are with respect to each other fourths, produce everywhere with each other fourths; the semi-tone, by transition, receiving the first, middle and third place, according to that tetrachord. Now in the Pythagoric octave, however, which by conjunction is a system of the tetrachord and pentachord, but if disjoined is a system of two tetrachords separated from each other, the progression is from the gravest to the most acute sound. Hence all sounds that by their distance from each other are fifths, with each other produce the interval of the fifth; the semi-tone successively proceeding into four places, the first, second, third, and fourth.
This is the way in which music is said to have been discovered by Pythagoras. Having reduced it to a system, he delivered it to his disciples to utilize it to produce things as beautiful as possible.
(Iamblichus - This story of the smithy is an ancient error, as pieces of iron give the same note whether struck by heavy or light hammers. Pythagoras may therefore have brought the discovery with him from Egypt, though he may also have developed the further details mentioned in this chapter.)
posted by Mikela @ 8:18 AM
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