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 B. C.) gives a definition of consonance and dissonance that
could hardly be improved upon, in point of verbal accuracy. The
consonance ([Greek: symphônia]) of two tones, he says, is the mixture,
the blending ([Greek: krasis]) of those two tones; dissonance ([Greek:
diaphônia]), on the other hand, is the incapacity of the tones to blend
([Greek: amixia]), 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 A. D.) 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.
1) In every process of perception, in every observation, the attention
plays a highly important part. We need not look about us long for proofs
of this. You receive, for example, a letter written in a very poor hand.
Do your best, you cannot make it out. You put together now these, now
those lines, yet you cannot construct from them a single intelligible
character. Not until you direct your attention to groups of lines which
really belong together, is the reading of the letter possible.
Manuscripts, the letters of which are formed of minute figures and
scrolls, can only be read at a considerable distance, where the
attention is no longer diverted from the significant outlines to the
details. A beautiful example of this class is furnished by the famous
iconographs of Giuseppe Arcimboldo in the basement of the Belvedere
gallery at Vienna. These are symbolic representations of water, fire,
etc.: human heads composed of aquatic animals and of combustibles. At a
short distance one sees only the details, at a greater distance only the
whole figure. Yet a point can be easily found at which, by a simple
voluntary movement of the attention, there is no difficulty in seeing
now the whole figure and now the smaller forms of which it is composed.
A picture is often seen representing the tomb of Napoleon. The tomb is
surrounded by dark trees between which the bright heavens are visible as
background. One can look a long time at this picture without noticing
anything except the trees, but suddenly, on the attention being
accidentally directed to the bright background, one sees the figure of
Napoleon between the trees. This case shows us very distinctly the
important part which attention plays. The same sensuous object can,
solely by the interposition of attention, give rise to wholly different
perceptions.
If I strike a harmony, or chord, on this piano, by a mere effort of
attention you can fix every tone of that harmony. You then hear most
distinctly the fixed tone, and all the rest appear as a mere addition,
altering only the quality, or acoustic color, of the primary tone. The
effect of the same harmony is essentially modified if we direct our
attention to different tones.
Strike in succession two harmonies, for example, the two represented in
the annexed diagram, and first fix by the attention the upper note _e_,
afterwards the base _e_-_a_; in the two cases you will hear the same
sequence of harmonies differently. In the first case, you have the
impression as if the fixed tone remained unchanged and simply altered
its _timbre_; in the second case, the whole acoustic agglomeration seems
to fall sensibly in depth. There is an art of composition to guide the
attention of the hearer. But there is also an art of hearing, which is
not the gift of every person.
[Music: Fig. 9.]
The piano-player knows the remarkable effects obtained when one of the
keys of a chord that is struck is let loose. Bar 1 played on the piano
sounds almost like bar 2. The note which lies next to the key let loose
resounds after its release as if it were freshly struck. The attention
no longer occupied with the upper note is by that very fact insensibly
led to the upper note.
[Music: Fig. 10.]
Any tolerably cultivated musical ear can perform the resolution of a
harmony into its component parts. By much practice we can go even
further. Then, every musical sound heretofore regarded as simple can be
resolved into a subordinate succession of musical tones. For example, if
I strike on the piano the note 1, (annexed diagram,) we shall hear, if
we make the requisite effort of attention, besides the loud fundamental
note the feebler, higher overtones, or harmonics, 2 ... 7, that is, the
octave, the twelfth, the double octave, and the third, the fifth, and
the seventh of the double octave.
[Music: Fig. 11.]
The same is true of every musically available sound. Each yields, with
varying degrees of intensity, besides its fundamental note, also the
octave, the twelfth, the double octave, etc. The phenomenon is
observable with special facility on the open and closed flue-pipes of
organs. According, now, as certain overtones are more or less distinctly
emphasised in a sound, the _timbre_ of the sound changes--that peculiar
quality of the sound by which we distinguish the music of the piano from
that of the violin, the clarinet, etc.
On the piano these overtones can be very easily rendered audible. If I
strike, for example, sharply note 1 of the foregoing series, whilst I
simply press down upon, one after another, the keys 2, 3, ... 7, the
notes 2, 3, ... 7 will continue to sound after the striking of 1,
because the strings corresponding to these notes, now freed from their
dampers, are thrown into sympathetic vibration.
As you know, this sympathetic vibration of the like-pitched strings with
the overtones is really not to be conceived as sympathy, but rather as
lifeless mechanical necessity. We must not think of this sympathetic
vibration as an ingenious journalist pictured it, who tells a gruesome
story of Beethoven's F minor sonata, Op. 2, that I cannot withhold from
you. "At the last London Industrial Exhibition nineteen virtuosos played
the F minor sonata on the same piano. When the twentieth stepped up to
the instrument to play by way of variation the same production, to the
terror of all present the piano began to render the sonata of its own
accord. The Archbishop of Canterbury, who happened to be present, was
set to work and forthwith expelled the F minor devil."
Although, now, the overtones or harmonics which we have discussed are
heard only upon a special effort of the attention, nevertheless they
play a highly important part in the formation of musical _timbre_, as
also in the production of the consonance and dissonance of sounds. This
may strike you as singular. How can a thing which is heard only under
exceptional circumstances be of importance generally for audition?
But consider some familiar incidents of your every-day life. Think of
how many things you see which you do not notice, which never strike your
attention until they are missing. A friend calls upon you; you cannot
understand why he looks so changed. Not until you make a close
examination do you discover that his hair has been cut. It is not
difficult to tell the publisher of a work from its letter-press, and yet
no one can state precisely the points by which this style of type is so
strikingly different from that style. I have often recognised a book
which I was in search of from a simple piece of unprinted white paper
that peeped out from underneath the heap of books covering it, and yet I
had never carefully examined the paper, nor could I have stated its
difference from other papers.
What we must remember, therefore, is that every sound that is musically
available yields, besides its fundamental note, its octave, its twelfth,
its double octave, etc., as overtones or harmonics, and that these are
important for the agreeable combination of several musical sounds.
2) One other fact still remains to be dealt with. Look at this
tuning-fork. It yields, when struck, a perfectly smooth tone. But if you
strike in company with it a second fork which is of slightly different
pitch, and which alone also gives a perfectly smooth tone, you will
hear, if you set both forks on the table, or hold both before your ear,
a uniform tone no longer, but a number of shocks of tones. The rapidity
of the shocks increases with the difference of the pitch of the forks.
These shocks, which become very disagreeable for the ear when they
amount to thirty-three in a second, are called "beats."
Always, when one of two like musical sounds is thrown out of unison with
the other, beats arise. Their number increases with the divergence from
unison, and simultaneously they grow more unpleasant. Their roughness
reaches its maximum at about thirty-three beats in a second. On a still
further departure from unison, and a consequent increase of the number
of beats, the unpleasant effect is diminished, so that tones which are
widely apart in pitch no longer produce offensive beats.
To give yourselves a clear idea of the production of beats, take two
metronomes and set them almost alike. You can, for that matter, set the
two exactly alike. You need not fear that they will strike alike. The
metronomes usually for sale in the shops are poor enough to yield, when
set alike, appreciably unequal strokes. Set, now, these two metronomes,
which strike at unequal intervals, in motion; you will readily see that
their strokes alternately coincide and conflict with each other. The
alternation is quicker the greater the difference of time of the two
metronomes.
If metronomes are not to be had, the experiment may be performed with
two watches.
Beats arise in the same way. The rhythmical shocks of two sounding
bodies, of unequal pitch, sometimes coincide, sometimes interfere,
whereby they alternately augment and enfeeble each other's effects.
Hence the shock-like, unpleasant swelling of the tone.
Now that we have made ourselves acquainted with overtones and beats, we
may proceed to the answer of our main question, Why do certain relations
of pitch produce pleasant sounds, consonances, others unpleasant sounds,
dissonances? It will be readily seen that all the unpleasant effects of
simultaneous sound-combinations are the result of beats produced by
those combinations. Beats are the only sin, the sole evil of music.
Consonance is the coalescence of sounds without appreciable beats.
[Illustration: Fig. 12.]
To make this perfectly clear to you I have constructed the model which
you see in Fig. 12. It represents a claviatur. At its top a movable
strip of wood _aa_ with the marks 1, 2 ... 6 is placed. By setting this
strip in any position, for example, in that where the mark 1 is over the
note _c_ of the claviatur, the marks 2, 3 ... 6, as you see, stand over
the overtones of _c_. The same happens when the strip is placed in any
other position. A second, exactly similar strip, _bb_, possesses the
same properties. Thus, together, the two strips, in any two positions,
point out by their marks all the tones brought into play upon the
simultaneous sounding of the notes indicated by the marks 1.
The two strips, placed over the same fundamental note, show that also
all the overtones of those notes coincide. The first note is simply
intensified by the other. The single overtones of a sound lie too far
apart to permit appreciable beats. The second sound supplies nothing
new, consequently, also, no new beats. Unison is the most perfect
consonance.
Moving one of the two strips along the other is equivalent to a
departure from unison. All the overtones of the one sound now fall
alongside those of the other; beats are at once produced; the
combination of the tones becomes unpleasant: we obtain a dissonance. If
we move the strip further and further along, we shall find that as a
general rule the overtones always fall alongside each other, that is,
always produce beats and dissonances. Only in a few quite definite
positions do the overtones partially coincide. Such positions,
therefore, signify higher degrees of euphony--they point out _the
consonant intervals_.
These consonant intervals can be readily found experimentally by cutting
Fig. 12 out of paper and moving _bb_ lengthwise along _aa_. The most
perfect consonances are the octave and the twelfth, since in these two
cases the overtones of the one sound coincide absolutely with those of
the other. In the octave, for example, 1_b_ falls on 2_a_, 2_b_ on 4_a_,
3_b_ on 6_a_. Consonances, therefore, are simultaneous
sound-combinations not accompanied by disagreeable beats. This, by the
way, is, expressed in English, what Euclid said in Greek.
Only such sounds are consonant as possess in common some portion of
their partial tones. Plainly we must recognise between such sounds, also
when struck one after another, a certain affinity. For the second sound,
by virtue of the common overtones, will produce partly the same
sensation as the first. The octave is the most striking exemplification
of this. When we reach the octave in the ascent of the scale we actually
fancy we hear the fundamental tone repeated. The foundations of harmony,
therefore, are the foundations of melody.
Consonance is the coalescence of sounds without appreciable beats! This
principle is competent to introduce wonderful order and logic into the
doctrines of the fundamental bass. The compendiums of the theory of
harmony which (Heaven be witness!) have stood hitherto little behind the
cook-books in subtlety of logic, are rendered extraordinarily clear and
simple. And what is more, all that the great masters, such as
Palestrina, Mozart, Beethoven, unconsciously got right, and of which
heretofore no text-book could render just account, receives from the
preceding principle its perfect verification.
But the beauty of the theory is, that it bears upon its face the stamp
of truth. It is no phantom of the brain. Every musician can hear for
himself the beats which the overtones of his musical sounds produce.
Every musician can satisfy himself that for any given case the number
and the harshness of the beats can be calculated beforehand, and that
they occur in exactly the measure that theory determines.
This is the answer which Helmholtz gave to the question of Pythagoras,
so far as it can be explained with the means now at my command. A long
period of time lies between the raising and the solving of this
question. More than once were eminent inquirers nearer to the answer
than they dreamed of.
The inquirer seeks the truth. I do not know if the truth seeks the
inquirer. But were that so, then the history of science would vividly
remind us of that classical rendezvous, so often immortalised by
painters and poets. A high garden wall. At the right a youth, at the
left a maiden. The youth sighs, the maiden sighs! Both wait. Neither
dreams how near the other is.
I like this simile. Truth suffers herself to be courted, but she has
evidently no desire to be won. She flirts at times disgracefully. Above
all, she is determined to be merited, and has naught but contempt for
the man who will win her too quickly. And if, forsooth, one breaks his
head in his efforts of conquest, what matter is it, another will come,
and truth is always young. At times, indeed, it really seems as if she
were well disposed towards her admirer, but that admitted--never! Only
when Truth is in exceptionally good spirits does she bestow upon her
wooer a glance of encouragement. For, thinks Truth, if I do not do
something, in the end the fellow will not seek me at all.
This one fragment of truth, then, we have, and it shall never escape us.
But when I reflect what it has cost in labor and in the lives of
thinking men, how it painfully groped its way through centuries, a
half-matured thought, before it became complete; when I reflect that it
is the toil of more than two thousand years that speaks out of this
unobtrusive model of mine, then, without dissimulation, I almost repent
me of the jest I have made.
And think of how much we still lack! When, several thousand years hence,
boots, top-hats, hoops, pianos, and bass-viols are dug out of the earth,
out of the newest alluvium as fossils of the nineteenth century; when
the scientists of that time shall pursue their studies both upon these
wonderful structures and upon our modern Broadways, as we to-day make
studies of the implements of the stone age and of the prehistoric
lake-dwellings--then, too, perhaps, people will be unable to comprehend
how we could come so near to many great truths without grasping them.
And thus it is for all time the unsolved dissonance, for all time the
troublesome seventh, that everywhere resounds in our ears; we feel,
perhaps, that it will find its solution, but we shall never live to see
the day of the pure triple accord, nor shall our remotest descendants.
Ladies, if it is the sweet purpose of your life to sow confusion, it is
the purpose of mine to be clear; and so I must confess to you a slight
transgression that I have been guilty of. On one point I have told you
an untruth. But you will pardon me this falsehood, if in full repentance
I make it good. The model represented in Fig. 12 does not tell the whole
truth, for it is based upon the so-called "even temperament" system of
tuning. The overtones, however, of musical sounds are not tempered, but
purely tuned. By means of this slight inexactness the model is made
considerably simpler. In this form it is fully adequate for ordinary
purposes, and no one who makes use of it in his studies need be in fear
of appreciable error.
If you should demand of me, however, the full truth, I could give you
that only by the help of a mathematical formula. I should have to take
the chalk into my hands and--think of it!--reckon in your presence. This
you might take amiss. Nor shall it happen. I have resolved to do no more
reckoning for to-day. I shall reckon now only upon your forbearance, and
this you will surely not gainsay me when you reflect that I have made
only a limited use of my privilege to weary you. I could have taken up
much more of your time, and may, therefore, justly close with Lessing's
epigram:
"If thou hast found in all these pages naught that's worth the thanks,
At least have gratitude for what I've spared thee."
FOOTNOTES:
[Footnote 10: Sauveur also set out from Leibnitz's idea, but arrived
by independent researches at a different theory, which was very near
to that of Helmholtz. Compare on this point Sauveur, _Mémoires de
l'Académie des Sciences_, Paris, 1700-1705, and R. Smith,
_Harmonics_, Cambridge, 1749. (See _Appendix_, p. 346.)]
THE VELOCITY OF LIGHT.
When a criminal judge has a right crafty knave before him, one well
versed in the arts of prevarication, his main object is to wring a
confession from the culprit by a few skilful questions. In almost a
similar position the natural philosopher seems to be placed with respect
to nature. True, his functions here are more those of the spy than the
judge; but his object remains pretty much the same. Her hidden motives
and laws of action is what nature must be made to confess. Whether a
confession will be extracted depends upon the shrewdness of the
inquirer. Not without reason, therefore, did Lord Bacon call the
experimental method a questioning of nature. The art consists in so
putting our questions that they may not remain unanswered without a
breach of etiquette.
Look, too, at the countless tools, engines, and instruments of torture
with which man conducts his inquisitions of nature, and which mock the
poet's words:
"Mysterious even in open day,
Nature retains her veil, despite our clamors;
That which she doth not willingly display
Cannot be wrenched from her with levers, screws, and hammers."
Look at these instruments and you will see that the comparison with
torture also is admissible.[11]
This view of nature, as of something designedly concealed from man, that
can be unveiled only by force or dishonesty, chimed in better with the
conceptions of the ancients than with modern notions. A Grecian
philosopher once said, in offering his opinion of the natural science of
his time, that it could only be displeasing to the gods to see men
endeavoring to spy out what the gods were not minded to reveal to
them.[12] Of course all the contemporaries of the speaker were not of
his opinion.
Traces of this view may still be found to-day, but upon the whole we are
now not so narrow-minded. We believe no longer that nature designedly
hides herself. We know now from the history of science that our
questions are sometimes meaningless, and that, therefore, no answer can
be forthcoming. Soon we shall see how man, with all his thoughts and
quests, is only a fragment of nature's life.
Picture, then, as your fancy dictates, the tools of the physicist as
instruments of torture or as engines of endearment, at all events a
chapter from the history of those implements will be of interest to you,
and it will not be unpleasant to learn what were the peculiar
difficulties that led to the invention of such strange apparatus.