Part 25

The fact that a musician will never confound a more perfectly consonant chord on a poorly tuned piano with a less perfectly consonant chord on a well tuned piano, although the roughness in the two cases may be the same, is sufficient indication that the degree of roughness is not the only characteristic of a harmony. As the musician knows, even the harmonic beauties of a Beethoven sonata are not easily effaced on a poorly tuned piano; they scarcely suffer more than a Raphael drawing executed in rough unfinished strokes. The _positive physiologico-psychological_ characteristic which distinguishes one harmony from another is not given by the beats. Nor is this characteristic to be found in the fact that, for example, in sounding a major third the fifth partial tone of the lower note coincides with the fourth of the higher note. This characteristic comes into consideration only for the investigating and abstracting reason. If we should regard it also as characteristic of the sensation, we should lapse into a fundamental error which would be quite analogous to that cited in (1).

The _positive physiological_ characteristics of the intervals would doubtless be speedily revealed if it were possible to conduct aperiodic, for example galvanic, stimuli to the single sound-sensing organs, in which case the beats would be totally eliminated. Unfortunately such an experiment can hardly be regarded as practicable. The employment of acoustic stimuli of short duration and consequently also free from beats, involves the additional difficulty of a pitch not precisely determinable.

FOOTNOTES:

[Footnote 134: This article, which appeared in the Proceedings of the German Mathematical Society of Prague for the year 1892, is printed as a supplement to the article on "The Causes of Harmony," at page 32.]

[Footnote 135: The present exposition is taken from the volumes for 1700 (published in 1703) and for 1701 (published in 1704), and partly also from the _Histoire de l'Académie_ and partly from the _Mémoires_. Sauveur's later works enter less into consideration here.]

[Footnote 136: Euler, _Tentamen novae theoriae musicae_, Petropoli, 1739.]

[Footnote 137: In attempting to perform his experiment of beats before the Academy, Sauveur was not quite successful. _Histoire de l'Académie_, Année 1700, p. 136.]

[Footnote 138: _Histoire de l'Académie_, Année 1701, p. 134.]

[Footnote 139: _Ibid._, p. 298.]

[Footnote 140: _Histoire de l'Académie_, Année 1702, p. 91.]

[Footnote 141: From the _Histoire de l'Académie_, Année 1700, p. 139.]

[Footnote 142: Because all octaves in use in music offer too great differences of rates of vibration.]

[Footnote 143: "Les battemens ne plaisent pas à l'Oreille, à cause de l'inégalité du son, et l'on peut croire avec beaucoup d'apparence que ce qui rend les Octaves si agréables, c'est qu'on n'y entend jamais de battemens.

"En suivant cette idée, on trouve que les accords dont on ne peut entendre les battemens, sont justement ceux que les Musiciens traitent de Consonances, et que ceux dont les battemens se font sentir, sont les Dissonances, et que quand un accord est Dissonance dans une certaine octave et Consonance dans une autre, c'est qu'il bat dans l'une, et qu'il ne bat pas dans l'autre. Aussi est il traité de Consonance imparfaite. Il est fort aisé par les principes de Mr. Sauveur qu'on a établis ici, de voir quels accords battent, et dans quelles Octaves au-dessus on au-dessous du son fixe. Si cette hypothèse est vraye, elle découvrira la véritable source des Règles de la composition, inconnue jusqu'à présent à la Philosophie, qui s'en remettait presque entièrement au jugement de l'Oreille. Ces sortes de jugemens naturels, quelque bisarres qu'ils paroissent quelquefois, ne le sont point, ils ont des causes très réelles, dont la connaissance appartient à la Philosophie, pourvue qu'elle s'en puisse mettre en possession."]

[Footnote 144: _Harmonics or the Philosophy of Musical Sounds_, Cambridge, 1749. I saw this book only hastily in 1864 and drew attention to it in a work published in 1866. I did not come into its actual possession until three years ago and then only did I learn its exact contents.]

[Footnote 145: _Harmonics_, pp. 118 and 243.]

[Footnote 146: "Short cycle" is the period in which the same phases of the two co-operant tones are repeated.]

II.

REMARKS ON THE THEORY OF SPATIAL VISION.[147]

According to Herbart, spatial vision rests on reproduction-series. In such an event, of course, and if the supposition is correct, the magnitudes of the residua with which the percepts or representations are coalesced (the helps to coalescence) are of cardinal influence. Furthermore, since the coalescences must first be fully perfected before they make their appearance, and since upon their appearance the inhibitory ratios are brought into play, ultimately, then, if we leave out of account the accidental order of time in which the percepts are given, everything in spatial vision depends on the oppositions and affinities, or, in brief, on the qualities of the percepts, which enter into series.

Let us see how the theory stands with respect to the special facts involved.

1. If intersecting series only, running anteriorly and posteriorly, are requisite for the production of spatial sensation, why are not analogues of them found in all the senses?

2. Why do we measure differently colored objects and variegated objects with one and the same spatial measure? How do we recognise differently colored objects as the same in size? Where do we get our measure of space from and what is it?

3. Why is it that differently colored figures of the same form reproduce one another and are recognised as the same?

Here are difficulties enough. Herbart is unable to solve them by his theory. The unprejudiced student sees at once that his "inhibition by reason of form" and "preference by reason of form" are absolutely impossible. Think of Herbart's example of the red and black letters.

The "help to coalescence" is a passport, so to speak, made out to the name and person of the percept. A percept which is coalesced with another cannot reproduce all others qualitatively different from it for the simple reason that the latter are in like manner coalesced with one another. Two qualitatively different series certainly do not reproduce themselves because they present the same order of degree of coalescence.

If it is certain that only things simultaneous and things which are alike are reproduced, a basic principle of Herbart's psychology which even the most absolute empiricists will not deny, nothing remains but to modify the theory of spatial perception or to invent in its place a new principle in the manner indicated, a step which hardly any one would seriously undertake. The new principle could not fail to throw all psychology into the most dreadful confusion.

As to the modification which is needed there can be hardly any doubt as to how in the face of the facts and conformably to Herbart's own principles it is to be carried out. If two differently colored figures of equal size reproduce each other and are recognised as equal, the result can be due to nothing but to the existence in both series of presentations of a presentation or percept which is qualitatively _the same_. The colors are different. Consequently, like or equal percepts must be connected with the colors which are yet independent of the colors. We have not to look long for them, for they are the like effects of the muscular feelings of the eye when confronted by the two figures. We might say we reach the vision of space by the registering of light-sensations in a schedule of graduated muscle-sensations.[148]

A few considerations will show the likelihood of the rôle of the muscle-sensations. The muscular apparatus of _one_ eye is unsymmetrical. The two eyes together form a system which is vertical in symmetry. This already explains much.

1. The _position_ of a figure influences its view. According to the position in which objects are viewed different muscle-sensations come into play and the impression is altered. To recognise inverted letters as such long experience is required. The best proof of this are the letters d, b, p, q, which are represented by the same figure in different positions and yet are always distinguished as different.[149]

2. It will not escape the attentive observer that for the same reasons and even with the same figure and in the same position the fixation point is also decisive. The figure seems to change _during_ the act of vision. For example, an eight-pointed star constructed by successively joining in a regular octagon the first corner with the fourth, the fourth with the seventh, etc., skipping in every case two corners, assumes alternately, according to where we suffer the centre of vision to rest, a predominantly architectonic or a freer and more open character. Vertical and horizontal lines are always differently apprehended from what oblique lines are.

3. The reason why we prefer vertical symmetry and regard it as something special in its kind, whereas we do not recognise horizontal symmetry at all immediately, is due to the vertical symmetry of the muscular apparatus of the eye. The left-hand side _a_ of the accompanying vertically-symmetrical figure induces in the left eye the same muscular feelings as the right-hand side _b_ does in the right eye. The pleasing effect of symmetry has its cause primarily in the repetition of muscular feelings. That a repetition actually occurs here, sometimes sufficiently marked in character as to lead to the confounding of objects, is proved apart from the theory by the fact which is familiar to every one _quem dii oderunt_ that children frequently reverse figures from the right to the left, but never from above downwards; for example, write [epsilon] instead of 3 until they finally come to notice the slight difference. Figure 50 shows how pleasing the repetition of muscular feelings may be. As will be readily understood, vertical and horizontal lines exhibit relations similar to symmetrical figures which are immediately disturbed when oblique positions are chosen for the lines. Compare what Helmholtz says regarding the repetition and coincidence of partial tones.

I may be permitted to add a general remark. It is a quite universal phenomenon in psychology that certain qualitatively quite different series of percepts mutually awaken and reproduce one another and in a certain aspect produce the appearance of sameness or similarity. We say of such series that they are of like or of similar form, naming their abstracted likeness _form_.

1. Of spatial figures we have already spoken.

2. We call two melodies like melodies when they present the same succession of pitch-ratios; the absolute pitch (or key) may be as different as can be. We can so select the melodies that not even two partial tones of the notes in each are common. Yet we recognise the melodies as alike. And, what is more, we notice the form of the melody more readily and recognise it again more easily than the key (the absolute pitch) in which it was played.

3. We recognise in two different melodies the same rhythm no matter how different the melodies may be otherwise. We know and recognise the rhythm more easily even than the absolute duration (the tempo).

These examples will suffice. In all these and in all similar cases the recognition and likeness cannot depend upon the qualities of the percepts, for these are different. On the other hand recognition, conformably to the principles of psychology, is possible only with percepts which are the same in quality. Consequently there is no other escape than to imagine the qualitatively unlike percepts of the two series as necessarily connected with other percepts which are qualitatively alike.

Since in differently colored figures of like form, like muscular feelings are necessarily induced if the figures are recognised as alike, so there must necessarily lie at the basis of all forms also, and we might even say at the basis of all abstractions, percepts of a peculiar quality. And this holds true for space and form as well as for time, rhythm, pitch, the form of melodies, intensity, etc. But whence is psychology to derive all these qualities? Have no fear, they will all be found, as were the sensations of muscles for the theory of space. The organism is at present still rich enough to meet all the requirements of psychology in this direction, and it is even time to give serious ear to the question of "corporeal resonance" which psychology so loves to dwell on.

Different psychical qualities appear to bear a very intimate mutual relation to one another. Special research on the subject, as well also as the demonstration that this remark may be generally employed in physics, will follow later.[150]

FOOTNOTES:

[Footnote 147: This article, designed to illustrate historically that on Symmetry, at page 89, first appeared in Fichte's _Zeitschrift für Philosophie_, for 1865.]

[Footnote 148: Comp. Cornelius, _Ueber das Sehen_; Wundt, _Theorie der Sinneswahrnehmung_.]

[Footnote 149: Comp. Mach, _Ueber das Sehen von Lagen and Winkeln_. _Sitzungsb. der Wiener Akademie_, 1861.]

[Footnote 150: Comp. Mach, _Zur Theorie des Gehörorgans_. _Sitsungsber, der Wiener Akad._, 1863.--_Ueber einige Erscheinungen der physiolog. Akustik._ _Ibid._, 1864.]

INDEX.

Absolute, temperature, 162; time, 204; forecasts, have no signification in science, 206.

Abstract, meaning of the term, 240.

Abstraction, 180, 200, 208, 231.

Acceleration, organ for forward, 299 et seq.

Accelerations, 204, 216, footnote, 225-226, 253.

Accident, logical and historical, in science, 160, 168, 170, 213; in inventions and discoveries, 262 et seq.

Accord, the pure triple, 46.

Accumulators, electrical, 125 et seq.; 132, footnote.

Acoustic color, 36.

Acoustics, Sauveur on, 375 et seq.

Action and reaction, importance of the principle of, 191.

Adaptation, in organic and inorganic matter, 216, 229; in scientific thought, 214-235.

Æsthetics, computation as a principle of, 34; researches in, 89, footnote; repetition, a principle of, 91.

Africa, 186, 234, 237.

Agreeable effects, due to repetition of sensations, 92, 97 et seq.

Agriculture, transition to, 265.

Air-gun, 135.

Alcohol and water, mixture of oil and, in Plateau's experiments, 4.

Algebra, economy of, 196.

Alien thoughts in science, 196.

All, the, 88.

Amontons, 174, 346.

Ampère, the word, 314.

Ampère's swimmer, 207.

Analogies, mechanical, 157, 160; generally, 236-258.

Analogy, defined, 250.

Analysis, 188.

Analytical geometry, not necessary to physicians, 370, footnote.

Anatomic structures, transparent stereoscopic views of, 74.

Anatomy, character of research in, 255.

Andrieu, Jules, 49, footnote.

Animals, the psychical activity of, 190, 231; the language of, 238; their capacity for experience, 266 et seq.

Animism, 186, 187, 243, 254.

Anisotropic optical fields, 227.

Apparatus for producing movements of rotation, 287 et seq.

Arabesque, an inverted, 95.

Arabian Nights, 219.

Arago, 270.

Aral, the Sea of, 239.

Archæopteryx, 257.

Archimedes, 4, 237.

Arcimboldo, Giuseppe, 36.

Area, principle of least superficial, 10 et seq.

Ares, the bellowing of the wounded, 272.

Aristotelians, 283.

Aristotle, 348, 296.

Art, development of, 28 et seq.

Artillery, practical, 334-335.

Artistic value of scientific descriptions, 254.

Arts, practical, 108.

Ascent, heights of, 143-151.

Asia, 234.

Assyrians, the art of, 79.

Astronomer, measures celestial by terrestrial distances, 136.

Astronomy, antecedent to psychology, 90; rigidity of its truths, 221.

Atomic theories, 104.

Atoms, 207.

Attention, the rôle of, in sensuous perception, 35 et seq.

Attraction, generally, 226; of liquid particles, 13-14; in electricity, 109 et seq.

Aubert, 298.

Audition. See _Ear_.

Austrian gymnasiums, 370, footnote.

Axioms, instinctive knowledge, 190.

Babbage, on the economy of machinery, 196.

Bach, 20.

Backwards, prophesying, 253.

Bacon, Lord, 48, 280.

Baer, C. E. von, 235.

Balance, electrical, 127, footnote; torsion, 109, 168.

Balloon, a hydrogen, 199.

Barbarism and civilisation, 335 et seq.

Bass-clef, 101.

Bass, fundamental, 44.

Beats, 40-45, 377 et seq.

Beautiful, our notions of, variable, 99.

Beauty, objects of, in nature, 91.

Becker, J. K., 364, 369.

Beethoven, 39, 44.

Beginnings of science, 189, 191.

Belvedere Gallery at Vienna, 36.

Bernoulli, Daniel, on the conservation of living force, 149; on the vibrations of strings, 249.

Bernoulli, James, on the centre of oscillation, 149.

Bernoulli, John, on the conservation of living force, 149; on the principle of virtual velocities, 151.

Bible, parallel passages from, for language study, 356.

Binocular vision, 66 et seq.

Black, his theory of caloric, 138, 162; on quantity of heat, 166, 174; on latent heat, 167, 178; researches in heat generally, 244.

Blind cat, 303.

Bodies, heavy, seek their places, 224 et seq.; rotating, 285.

Body, a mental symbol for groups of sensations, 200-203; the human, our knowledge of, 90.

Boltzmann, 236.

Booth, Mr., 77.

Borelli, 217.

Boulder, a granite, 233.

Bow-wave of ships and moving projectiles, 323 et seq.

Boys, 317.

Bradley, 273.

Brahman, the, 63.

Brain, localisation of functions in, 210.

Breuer, 272, 282 et seq., 293, 298, 300, 301, 303, 306.

Brewster, his stereoscope, 73.

Bridge, invention of the, 264, 268.

British Association, 108.

Brooklyn Bridge, 75, footnote.

Brown, Crum, 293, 301.

Building, our concepts directions for, 253; facts the result of, 253; science compared to, 257.

Building-stones, metrical units are, 253.

Busch, 328.

Business of a merchant, science compared to the, 16.

Butterfly, a, 22.

Calculating machines, their economical character, 196.

Caloric, theory of, stood in the way of scientific advancement, 138, 167.

Calypso, the island of, 351.

Canterbury, Archbishop of, 39.

Cantor, M., 361, footnote.

Capacity, electrical, 116 et seq., 123; thermal, 123; specific inductive, 117.

Capulets and Montagues, 87.

Cards, difficult games of, 357.

Carnot, S., excludes perpetual motion in heat, 156, 162; his mechanical view of physics, 156; on thermodynamics, 160 et seq.; his principle, 162; also, 191.

Carus, Dr. Paul, 265, footnote.

Casselli's telegraph, 26.

Cassini, 51.

Cauchy, character of the intellectual activity of a, 195.

Causal insight, awakened by science, 357.

Causality, 157-159, 190, 198 et seq., 221 et seq., 237, 253, 254.

Cause and effect, 198 et seq. See also _Causality_.

Centimetre-gramme-second system, 111.

Centre of gravity, must lie as low as possible for equilibrium to subsist, 15; Torricelli's principle of, 150 et seq.

Centre of oscillation, 149.

Change, method of, in science, 230.

Changeable character of bodies, 202.

Changes, physical, how they occur, 205.

Character, a Universal Real, 192.

Character, like the forms of liquids, 3; persons of, 24.

Charles the Fifth, 369.

Chemical, elements, 202; symbols, 192; current, 118.

Chemistry, character of research in, 255; the method of thermodynamics in, 257.

Child, a, modes of thought of, 223; looking into a moat, 208.

Child of the forest, his interpretation of new events, 218-219.

Childish questions, 199-200.

Children, the drawings of, 201-202.

Chinese language, economy of, 192; study of, 354.

Chinese philosopher, an old, 186.

Chinese, speak with unwillingness of politics, 374; the art of, 79-80.

Chosen, many are called but few are, 65.

Christ, saying of, 65.

Christianity, Latin introduced with, 311.

Christians and Jews, monotheism of the, 187.

Church and State, 88.

Cicero, 318.

Circe, 372.

Circle, the figure of least area with given periphery, 12.

Circular polarisation, 242.

Civilisation and barbarism, 335 et seq.

Civilisation, some phenomena of, explained by binocular vision, 74.

Civilised man, his modes of conception and interpretation, 219.

Clapeyron, 162.

Class-characters of animals, 255.

Classical, culture, the good and bad effects of, 347; scholars, not the only educated people, 345.

Classics, on instruction in, 338-374; the scientific, 368.

Classification in science, 255.

Clausius, on thermodynamics, 165; on reversible cycles, 176.

Claviatur, Mach's, 42-43.

Club-law, 335.

Cochlea, the, a species of piano-forte, 19.

Cockchafer, 86.

Coefficient of self-induction, 250, 252.

Colophonium, solution of, 7.

Color, acoustic, 36.

Color-sensation, 210.

Color-signs, their economy, 192.

Colors, origin of the names of, 239.

Column, body moving behind a, 202.

Communication, its functions, import and fruits, 197, 238 et seq.; by language, 237; high importance of, 191 et seq.

Comparative physics, 239.

Comparison in science, 231, 238 et seq.

Computation, a principle of æsthetics, 34.

Concepts, abstract, defined, 250-252; metrical, in electricity, 107 et seq.

Conceptual, meaning of the term, 240.

Conceptual thought, 192.

Concha, 18.

Condensers, electrical, 125 et seq. 132, footnote.

Conductors and non-conductors. See _Electrical_, etc.

Conformity in the deportment of the energies, 171-175.

Confusion of objects, cause of, 95.

Conic sections, 257.

Conical refraction, 29, 242.

Conservation of energy, 137 et seq. See _Energy_.

Conservation of weight or mass, 203.

Consonance, connexion of the simple natural numbers with, 33; Euclid's definition of, 33; explanation of, 42; scientific definition of, 44; and dissonance reduced to beats, 376, 370, 383.

Consonant intervals, 43.

Constancy of matter, 203.

Constant, the dielectric, 117.

Constants, the natural, 193.

Continuum of facts, 256 et seq.

Cornelius, 388, footnote.

Corti, the Marchese, his discovery of minute rods in the labyrinth of the ear, 19.

Coulomb, his electrical researches, 108, 109, 113; his notion of quantity of electricity, 173; his torsion-balance, 168.

Crew, Prof. Henry, 317, footnote.

Criticism, Socrates the father of scientific, 1, 16.

_Critique of Pure Reason_, Kant's, 188.

Crucible, derivation of the word, 49, footnote.

Crustacea, auditory filaments of, 29, 272, 302.

Cube of oil, 5.

Culture, ancient and modern, 344.

Currents, chemical, 118; electrical, 118; galvanic, 132; measurement of electrical, 135-136; of heat, 244, 249-250; strength of, 250.

Curtius, 356.

Curved lines, their asymmetry, 98.

Curves, how their laws are investigated, 206.

Cycles, reversible, Clausius on, 176.

Cyclical processes, closed, 175.

Cyclops, 67.

Cyclostat, 298.

Cylinder, of oil, 6; mass of gas enclosed in a, 179.

D'Alembert, on the causes of harmony, 34; his principle, 142, 149, 154; also 234, 279.

Danish schools, 338, footnote.

Darwin, his study of organic nature, 215 et seq.; his methods of research, 216.

Deaf and dumb, not subject to giddiness, 299.

Deaf person, with a piano, analyses sounds, 27.

Death and life, 186.

Definition, compendious, 197.

Deiters, 19.

Delage, 298, 301, 302.

Democritus, his mechanical conception of the world, 155, 187.

Demonstration, character of, 362.

Deportment of the energies, conformity in the, 171-175.

Derivation, laws only methods of, 256.

Descent, Galileo's laws of, 193; generally, 143 et seq., 204, 215.

Description, 108, 191, 236, 237; a condition of scientific knowledge, 193; direct and indirect, 240; in physics, 197, 199.

Descriptive sciences, their resemblance to the abstract, 248.

Determinants, 195.

Diderot, 234.

Dielectric constant, the, 117.

Difference-engine, the, 196.