hard contact with the string, and consequently only imperfectly impeding
the vibrations, Sauveur discovered the harmonic overtones of the string,
at first by the unaided ear, and concluded from this fact that the
string was divided into aliquot parts. The string when plucked, and
when the bridge stood at the third division for example, yielded the
twelfth of its fundamental note. At the suggestion of some
academician[138] probably, variously colored paper riders were placed at
the nodes (_noeuds_) and ventral segments (_ventres_), and the division
of the string due to the excitation of the overtones (_sons
harmoniques_) belonging to its fundamental note (_son fondamental_) thus
rendered visible. For the clumsy bridge the more convenient feather or
brush was soon substituted. . While engaged in these investigations
Sauveur also observed the sympathetic vibration of a string induced by
the excitation of a second one in unison with it. He also discovered
that the overtone of a string can respond to another string tuned to its
note. He even went further and discovered that on exciting one string
the overtone which it has in common with another, differently pitched
string can be produced on that other; for example, on strings having for
their vibrational ratio 3:4, the fourth of the lower and the third of
the higher may be made to respond. It follows indisputably from this
that the excited string yields overtones simultaneously with its
fundamental tone. Previously to this Sauveur's attention had been drawn
by other observers to the fact that the overtones of musical instruments
can be picked out by attentive listening, particularly in the
night.[139] He himself mentions the simultaneous sounding of the
overtones and the fundamental tone.[140] That he did not give the proper
consideration to this circumstance was, as will afterwards be seen,
fatal to his theory.
While studying beats Sauveur makes the remark that they are
_displeasing_ to the ear. He held the beats were distinctly audible only
when less than six occurred in a second. Larger numbers were not
distinctly perceptible and gave rise accordingly to no disturbance. He
then attempts to reduce the difference between consonance and dissonance
to a question of beats. Let us hear his own words.[141]
"Beats are unpleasing to the ear because of the unevenness of the
sound, and it may be held with much plausibility that the reason
why octaves are so pleasing is that we never hear their beats.[142]
"In following out this idea, we find that the chords whose beats we
cannot hear are precisely those which the musicians call
consonances and that those whose beats are heard are the
dissonances, and that when a chord is a dissonance in one octave
and a consonance in another, it beats in the one and does not beat
in the other. Consequently it is called an imperfect consonance. It
is very easy by the principles of M. Sauveur, here established, to
ascertain what chords beat and in what octaves, above or below the
fixed note. If this hypothesis be correct, it will disclose the
true source of the rules of composition, hitherto unknown to
science, and given over almost entirely to judgment by the ear.
These sorts of natural judgment, marvellous though they may
sometimes appear, are not so but have very real causes, the
knowledge of which belongs to science, provided it can gain
possession thereof."[143]
Sauveur thus correctly discerns in beats the cause of the disturbance
of consonance, to which all disharmony is "probably" to be referred. It
will be seen, however, that according to his view all distant intervals
must necessarily be consonances and all near intervals dissonances. He
also overlooks the absolute difference in point of principle between his
old view, mentioned at the outset, and his new view, rather attempting
to obliterate it.
R. Smith[144] takes note of the theory of Sauveur and calls attention to
the first of the above-mentioned defects. Being himself essentially
involved in the old view of Sauveur, which is usually attributed to
Euler, he yet approaches in his criticism a brief step nearer to the
modern theory, as appears from the following passage.[145]
"The truth is, this gentleman confounds the distinction between
perfect and imperfect consonances, by comparing imperfect
consonances which beat because the succession of their short
cycles[146] is periodically confused and interrupted, with perfect
ones which cannot beat, because the succession of their short
cycles is never confused nor interrupted.
"The _fluttering roughness_ above mentioned is perceivable in all
other perfect consonances, in a smaller degree in proportion as
their cycles are shorter and simpler, and their pitch is higher;
and is of a _different kind_ from the _smoother beats_ and
undulations of _tempered consonances_; because we can alter the
rate of the latter by altering the temperament, but not of the
former, the consonance being perfect at a given pitch: And because
a judicious ear can often hear, at the same time, both the
flutterings and the beats of a tempered consonance; sufficiently
distinct from each other.
"For nothing gives greater offence to the hearer, though ignorant
of the cause of it, than those rapid, piercing beats of high and
loud sounds, which make imperfect consonances with one another. And
yet a few slow beats, like the slow undulations of a close shake
now and then introduced, are far from being disagreeable."
Smith is accordingly clear that other "roughnesses" exist besides the
beats which Sauveur considered, and if the investigations had been
continued on the basis of Sauveur's idea, these additional roughnesses
would have turned out to be the beats of the overtones, and the theory
thus have attained the point of view of Helmholtz.
Reviewing the differences between Sauveur's and Helmholtz's theories, we
find the following:
1. The theory according to which consonance depends on the frequent and
regular coincidence of vibrations and their ease of enumeration, appears
from the new point of view inadmissible. The simplicity of the ratios
obtaining between the rates of vibration is indeed a _mathematical_
characteristic of consonance as well as a _physical_ condition thereof,
for the reason that the coincidence of the overtones as also their
further physical and physiological consequences is connected with this
fact. But no _physiological_ or _psychological_ explanation of
consonance is given by this fact, for the simple reason that in the
acoustic nerve-process nothing corresponding to the periodicity of the
sonant stimulus is discoverable.
2. In the recognition of beats as a disturbance of consonance, both
theories agree. Sauveur's theory, however, does not take into account
the fact that clangs, or musical sounds generally, are composite and
that the disturbance in the consonances of distant intervals principally
arise from the beats of the overtones. Furthermore, Sauveur was wrong in
asserting that the number of beats must be less than six in a second in
order to produce disturbances. Even Smith knows that very slow beats are
not a cause of disturbance, and Helmholtz found a much higher number
(33) for the maximum of disturbance. Finally, Sauveur did not consider
that although the number of beats increases with the recession from
unison, yet their _strength_ is diminished. On the basis of the
principle of specific energies and of the laws of sympathetic vibration
the new theory finds that two atmospheric motions of like amplitude but
different periods, _a_ sin(_rt_) and _a_ sin[(_r_ + [rho])(_t_ +
[tau])], cannot be communicated with the same amplitude to the same
nervous end-organ. On the contrary, an end-organ that reacts best to the
period _r_ responds more weakly to the period _r_ + [rho], the two
amplitudes bearing to each other the proportion _a_: [phi]_a_. Here
[phi] decreases when [rho] increases, and when [rho] = 0 it becomes
equal to 1, so that only the portion of the stimulus [phi]_a_ is subject
to beats, and the portion (1-[phi])_a_ continues smoothly onward without
disturbance.
If there is any moral to be drawn from the history of this theory, it is
that considering how near Sauveur's errors were to the truth, it
behooves us to exercise some caution also with regard to the new theory.
And in reality there seems to be reason for doing so.
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.
[Illustration: Fig. 58.]
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.
[Illustration: Fig. 59.]
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.