Popular scientific lectures. Ernst Mach
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But it is a stupendously ingenious lock. Not only one tone, but every tone makes it open; but each one differently. To each tone it replies with a different sensation.
More than once it has happened in the history of science that a phenomenon predicted by theory, has not been brought within the range of actual observation until long afterwards. Leverrier predicted the existence and the place of the planet Neptune, but it was not until sometime later that Galle actually found the planet at the predicted spot. Hamilton unfolded theoretically the phenomenon of the so-called conical refraction of light, but it was reserved for Lloyd some time subsequently to observe the fact. The fortunes of Helmholtz's theory of Corti's fibres have been somewhat similar. This theory, too, received its substantial confirmation from the subsequent observations of V. Hensen. On the free surface of the bodies of Crustacea, connected with the auditory nerves, rows of little hairy filaments of varying lengths and thicknesses are found, which to some extent are the analogues of Corti's fibres. Hensen saw these hairs vibrate when sounds were excited, and when different notes were struck different hairs were set in vibration.
I have compared the work of the physical inquirer to the journey of the tourist. When the tourist ascends a new hill he obtains of the whole district a different view. When the inquirer has found the solution of one enigma, the solution of a host of others falls into his hands.
Surely you have often felt the strange impression experienced when in singing through the scale the octave is reached, and nearly the same sensation is produced as by the fundamental tone. The phenomenon finds its explanation in the view here laid down of the ear. And not only this phenomenon but all the laws of the theory of harmony may be grasped and verified from this point of view with a clearness before undreamt of. Unfortunately, I must content myself to-day with the simple indication of these beautiful prospects. Their consideration would lead us too far aside into the fields of other sciences.
The searcher of nature, too, must restrain himself in his path. He also is drawn along from one beauty to another as the tourist from dale to dale, and as circumstances generally draw men from one condition of life into others. It is not he so much that makes the quests, as that the quests are made of him. Yet let him profit by his time, and let not his glance rove aimlessly hither and thither. For soon the evening sun will shine, and ere he has caught a full glimpse of the wonders close by, a mighty hand will seize him and lead him away into a different world of puzzles.
Respected hearers, science once stood in an entirely different relation to poetry. The old Hindu mathematicians wrote their theorems in verses, and lotus-flowers, roses, and lilies, beautiful sceneries, lakes, and mountains figured in their problems.
"Thou goest forth on this lake in a boat. A lily juts forth, one palm above the water. A breeze bends it downwards, and it vanishes two palms from its previous spot beneath the surface. Quick, mathematician, tell me how deep is the lake!"
Thus spoke an ancient Hindu scholar. This poetry, and rightly, has disappeared from science, but from its dry leaves another poetry is wafted aloft which cannot be described to him who has never felt it. Whoever will fully enjoy this poetry must put his hand to the plough, must himself investigate. Therefore, enough of this! I shall reckon myself fortunate if you do not repent of this brief excursion into the flowered dale of physiology, and if you take with yourselves the belief that we can say of science what we say of poetry,
"Who the song would understand,
Needs must seek the song's own land;
Who the minstrel understand
Needs must seek the minstrel's land."
ON THE CAUSES OF HARMONY.
We are to speak to-day of a theme which is perhaps of somewhat more general interest—the causes of the harmony of musical sounds. The first and simplest experiences relative to harmony are very ancient. Not so the explanation of its laws. These were first supplied by the investigators of a recent epoch. Allow me an historical retrospect.
Pythagoras (586 BC) knew that the note yielded by a string of steady tension was converted into its octave when the length of the string was reduced one-half, and into its fifth when reduced two-thirds; and that then the first fundamental tone was consonant with the two others. He knew generally that the same string under fixed tension gives consonant tones when successively divided into lengths that are in the proportions of the simplest natural numbers; that is, in the proportions of 1:2, 2:3, 3:4, 4:5.
Pythagoras failed to reveal the causes of these laws. What have consonant tones to do with the simple natural numbers? That is the question we should ask to-day. But this circumstance must have appeared less strange than inexplicable to Pythagoras. This philosopher sought for the causes of harmony in the occult, miraculous powers of numbers. His procedure was largely the cause of the upgrowth of a numerical mysticism, of which the traces may still be detected in our oneirocritical books and among some scientists, to whom marvels are more attractive than lucidity.
Euclid (300 BC) gives a definition of consonance and dissonance that could hardly be improved upon, in point of verbal accuracy. The consonance (συμφωνία) of two tones, he says, is the mixture, the blending (κρᾶσις) of those two tones; dissonance (διαφωνία), on the other hand, is the incapacity of the tones to blend (ἀμιξία), whereby they are made harsh for the ear. The person who knows the correct explanation of the phenomenon hears it, so to speak, reverberated in these words of Euclid. Still, Euclid did not know the true cause of harmony. He had unwittingly come very near to the truth, but without really grasping it.
Leibnitz (1646–1716 AD) resumed the question which his predecessors had left unsolved. He, of course, knew that musical notes were produced by vibrations, that twice as many vibrations corresponded to the octave as to the fundamental tone, etc. A passionate lover of mathematics, he sought for the cause of harmony in the secret computation and comparison of the simple numbers of vibrations and in the secret satisfaction of the soul at this occupation. But how, we ask, if one does not know that musical notes are vibrations? The computation and the satisfaction at the computation must indeed be pretty secret if it is unknown. What queer ideas philosophers have! Could anything more wearisome be imagined than computation as a principle of æsthetics? Yes, you are not utterly wrong in your conjecture, yet you may be sure that Leibnitz's theory is not wholly nonsense, although it is difficult to make out precisely what he meant by his secret computation.
The great Euler (1707–1783) sought the cause of harmony, almost as Leibnitz did, in the pleasure which the soul derives from the contemplation of order in the numbers of the vibrations.[10]
Rameau and D'Alembert (1717–1783) approached nearer to the truth. They knew that in every sound available in music besides the fundamental note also the twelfth and the next higher third could be heard; and further that the resemblance between a fundamental tone and its octave was always strongly marked. Accordingly, the combination of the octave, fifth, third, etc., with the fundamental tone appeared to them "natural." They possessed, we must admit, the correct point of view; but with the simple naturalness of a phenomenon no inquirer can rest content; for it is precisely this naturalness for which he seeks his explanations.
Rameau's remark dragged along through the whole modern period, but without leading to the full discovery of the truth. Marx places it at the head of his theory of composition, but makes no further application of it. Also Goethe and Zelter in their correspondence were, so to speak, on the brink of the truth. Zelter knew of Rameau's view. Finally, you will be appalled at the difficulty of the problem, when I tell you that till very recent times even professors of physics were dumb when asked what were the causes of harmony.
Not till quite recently did Helmholtz find the solution of the question. But to make this solution clear to you I must first speak of some experimental principles of physics and psychology.
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