The facts of the relations of tones, the elements, that is,
of melody and harmony, are as follows. We cannot avoid the
observation that certain tones "go together," as the phrase
is, while others do not. This peculiar impression of belonging
together is known as consonance, or harmony. The intervals of
the octave, the fifth, the third, for instance, that is, C-C',
C-G, C-E, in the diatonic scale, are harmonious; while the
interval of the second, C-D, is said to be dissonant.
Consonance, however, is not identical with pleasingness, for
different combinations are sometimes pleasing, sometimes
displeasing. In the history of music we know that the octave
was to the Greeks the most pleasing combination, to medieval
musicians the fifth, while to us, the third, which was once
a forbidden chord, is perhaps most delightful. Yet we should
never doubt that the octave is the most consonant, the fifth
and the third the lesser consonant of combinations. We see,
thus, that consonance, whatever its nature, is independent
of history; and we must seek for its explanation in the nature
of the auditory process.
Various theories have been proposed. That of Helmholtz has
held the field so long that, although weighty objections have
been raised to it, it must still be treated with respect. In
introducing it a short review of the familiar facts of the
physics and physiology of hearing may not be out of place.
The vibration rates per second of the vibrating bodies, strings,
steel rods, etc., which produce those musical tones which are
consonant, are in definite and small mathematical ratios to
each other. Thus the rates of C-C' are as 1:2; of C-G, C-E,
as 2:3, 4:5. In general, the simpler the fraction, the
greater the consonance.
But no sonorous body vibrates in one single rate; a taut
string vibrates as a whole, which gives its fundamental tone,
but also in halves, in fourths, etc., each giving out a
weaker partial tone, in harmony with the fundamental. And
according to the different ways in which a sonorous body
divides, that is, according to the different combination of
partial tones peculiar to it, is its especial quality of tone,
or timbre. The whole complex of fundamental and partial tones
is what we popularly speak of as a tone,--more technically a
clang. These physical agitations or vibrations are transmitted
to the air. Omitting the account of the anatomical path by
which they reach the inner ear, we find them at last setting
up vibrations in a many-fibred membrane, the basilar membrane,
which is in direct connection with the ends of the auditory
nerve. It is supposed that to every possible rate of
vibration, that is, every possible tone, or partial tone, there
corresponds a fibre of the basilar membrane fitted by its
length to vibrate synchronously with the original wave-elements.
The complex wave is thus analyzed into its constituents. Now
when two tones, which we will for clearness suppose to be
simple, unaccompanied by partial tones, sounding together,
have vibration rates in simple ratios to each other, the air-
waves set in motion do not interfere with each other, but
combine into a complex but homogeneous wave. If they have
to each other a complicated ratio, such as 500:504, the air-
waves will not only not coalesce, but four times in the second
the through of one wave will meet the crest of the other, thus
making the algebraic sum zero, and producing the sensation of
a momentary stoppage of the sound. When these stoppages, or
beats, as they are called, are too numerous to be heard
separately, as in the interval, say, 500:547, the effect is
of a disagreeable roughness of tone, and this we call discord.
In other words, any tones which do not produce beats are
harmonious, or harmony is the absence of discord. In the
words of Helmholtz,<1> consonance is a continuous, dissonance
an intermittent, tone-sensation.
<1> _Lehre v.d. Tonempfindungen_, p. 370, in 4th edition.
Aside from the fact that consonance, as a psychological fact,
seems positive, while this determination is negative, two very
important facts can be set up in opposition. As a result of
experimental investigation, we know that the impression of
consonance can accompany the intermittent or rough sound-
sensations we know as beating tones; and, conversely, tones
can be dissonant when the possibility of beats is removed.
Briefly, it is possible to make beats without dissonance,
and dissonance without beats.
The other explanation makes consonance due to the identity
of partial tones. When two tones have one or more partial
tones in common they are said to be related; the amount
of identity gives the degree of relationship. Physiologically,
one or more basilar membrane fibres are excited by both, and
this fact gives the positive feeling of relationship or
consonance. Of course the obvious objection to this view
is that the two tones should be felt as differently consonant
when struck on instruments which give different partial tones,
such as organ and piano, while in fact they are not so felt.
But it is not after all essential to the aesthetics of music
that the physiological basis of harmony should be fully
understood. The point is that certain tones do indeed seem to
be "preordained to congruity," preordained either in their
physical constitution or their physiological relations, and not
to have achieved congruity by use or custom. Consonance is an
immediate and fundamental impression,--psychologically an
ultimate fact. That it is ultimate is emphasized by Stumpf<1>
in his theory of Fusion. Consonance is fusion, that is, unitary
impression. Fusion is not identical with inability to distinguish
two tones from each other in a chord, although this may be used
as a measure of fusion. Consonance is the feeling of unity, and
fusion is the mutual relation of tones which gives that feeling.
<1> _Beitrage zur Akustik u. Musikwissenschaft_, Heft I,
Konsonanz u. Dissonanz, 1898.
The striking fact of modern music is the principle of tonality.
Tonality is said to be present in a piece of music when every
element in it is referred to, gets its significance from its
relation to, a fundamental tone, the tonic. The tonic is the
beginning and lowest note in the scale in question, and all
notes and chords are understood according to their place in
that scale. But the conception of the scale of course does not
cover the ground, it merely furnishes the point of departure,--
the essential is in the reference of every element to the
fundamental tone. The tonic is the centre of gravity of a
melody.
The feeling of tonality grew up as follows. Every one was
referred to a fundamental, whether or not it made with it an
harmonious interval. The fundamental was imaged TOGETHER WITH
every other note, and when a group of such references often
appeared together, the feelings bound up with the single
reference (interval-feelings) fused into a single feeling,--
the tonality-feeling. When this point is once reached, it is
clear that every tone is heard not as itself alone, but in its
relations; it is not that we judge of tonality, it is a direct
impression, based on a psychological principle that we have
already touched on in the theory of rhythm. The tonality-
feeling is a feeling of form, or motor image, just as the
shape of objects is a motor image. We do not now need to go
through all possible experiences in relation to these objects,
we POSSESS their form in a system of motor images, which are
themselves only motor cues for coordinated movements. So
every tone is felt as something at a certain distance from,
with a certain relation to, another tone which is dimly
imagined. In following a melody, the notes are able to belong
together for us by virtue of the background of the tone to
which they are related, and in terms of which they are heard.
The tonality is indeed literally a "funded content,"--that is,
a funded capital of relation.
These are the general facts of tonality. But what is its
meaning for the nature of music? Why should all notes be
referred to one? Is this, too, an ultimate psychological fact?
In answer there may be pointed out the original basic quality
of certain tones, and the desire we have to return to them.
Of two successive tones, it is always the one which is, in the
ratio of their vibration rates, a power of two, with which we
wish to end.<1> When neither of two successive tones contains
a power of two, we have no preference as to the ending. Thus
denoting any tone by 1, it is always to 1 or 2, or 2n that we
wish to return, from any other possible tone; while 3 and 5, 5
and 7, leave us indifferent as to their succession. In general,
when two tones are related, as 2n:3, 5, 7, 9, 15--in which 2n
denotes every power of two, including 2o=1, with the progression
from the first to the second, there is bound up a tendency to
return to the first. Thus the fundamental fact of melodic
sequence may be said to be the primacy of 2 in vibration rates.
But 2n, in a scale containing 3, 5, etc., is always what we
know as the tonic. The tonic, then, gives a sense of
equilibrium, of rest, of finality, while to end on another tone
gives a feeling of restlessness or striving.
Now tone-relationship alone, it is clear, would not of itself
involve this immediate impulse to end a sequence of notes on
one rather than on another. Nor is tonality, in the all-
pervasive sense in which we understand it, a characteristic of
ancient, or of mediaeval music, while the tendency to end on a
certain tone, which we should to-day call the tonic, was always
felt. Thus, since complete tonality was developed late in the
history of music, while the closing on the tonic was certainly
prior to it, the finality of the tonic would seem to be the
primary fact, out of which the other has been developed.
We speak to-day, for instance, of dissonant chords, which call
for a resolution--and are inclined to interpret them as
dissonant just because they do so call. But the desire for
resolution is historically much later than the distinction
between consonance and dissonance.... "What we call resolution
is not change from dissonant to consonant IN GENERAL, but the
transition of definite tones of a dissonant interval into
DEFINITE TONES of a consonant."<1> The dissonance comes from
the device of getting variety, in polyphonic music, by letting
some parts lag behind, and the discords which arose while they
were catching up were resolved in the final coming together;
but the STEPS were all PREDETERMINED.<2> Resolution was
inevitably implied by the very principle on which the device
is founded. That is, the understanding of a chord as something
TO BE RESOLVED, is indeed part of the feeling of tonality; but
the ending on the tonic was that out of which this resolution-
feeling grew.
<1> Stumpf, op. Cit., p. 33.
<2> Grove, _Dict. Of Music and Musicians_. Art. "Resolution."
Must we, then, say that the finality of the tonic is a unique, inexplicable phenomenon? giving up the nature of melody as a
problem if not insoluble, at least unsolved?
The feeling of finality in the return to 2n is explained by
Lipps and his followers, from the fact that the two-division
is most natural, and so tones of 2n vibrations would have the
character of rest and equilibrium. This explanation might hold
if we were ever conscious of the two-division as such, in tones
--which we are not; so that it would seem to depend on the
restful character of a perception which by hypothesis is never
present to the mind at all.
The experience is, on the contrary, immediate,--an impression,
not a perception; and this immediacy points to the one ultimate
fact in musical feeling we have so far discovered. The whole
development of the scale, and the complex feeling of tonality,
is an expression of the desire for consonance. Every change
and correction in the scale has gone to make every note more
consonant with its neighbors. And naturally the tonic is the
tone with which all other tones have the most unity. Now this
"return" phenomenon is a simpler case of the desire for the
feeling of unity. The tonic is the epitome of all the most
perfect feelings of consonance or unity which are possible in
any particular sequence of tones, and is therefore the goal
or resting-place after an excursion. The undoubted feeling
of equilibrium or repose which we have in ending on the tonic
is thus explained. Not that consonance itself, the feeling of
unity, is explained. But at any rate consonance is the root
of the "return," and of its development into complete tonality.
The history of music is then the explicit development of
acoustic laws implicit in every stage of musical feeling. That
feeling covers an ever wider field. When Mr. Hadow says that
the terms concord and discord are wholly relative to the ear
of the listener,<1> and that the distinction between them is
not to be explained on any mathematical basis, or by any a
priori law of acoustics,--that it is not because a minor
second is ugly that we dislike it, for it will be a concord
some day,--he is only partly right. The minor second may be
a "concord," that is, we may like it, some day; but that will
be because w have extended our feeling of tonality to include
the minor second. When that day comes the minor second will
be so closely linked with other fully consonant combinations
that we shall hear it in terms of them, just as to-day we
hear the chord of the dominant seventh in terms of its
resolution. But the basis will not be convention or custom,
except in so far as custom is the unfolding of natural law.
The course of music, like that of every other art, is away
from arbitrary--though simple--convention, to a complexity
which satisfies the natural demands of the organism. The
"natural persuasion" of the ear is omnipotent.
<1> W.H. Hadow, _Studies in Modern Music_, 1893.
V
It has been said that the feeling of tonality is a motor image
or "form-quality" and that the image of the tonic persists
throughout every sequence of tones in a melody. Now these are
not only felt as having a certain relation to the tonic; that
relation is an active one. It was said that we had a positive
desire to end on a certain tone, and that a tendency to pass
to that tone was bound up with the hearing of another tone.
The degree of this tendency is determined by their relation.
The key, the tonality, is determined by the consensus of
intervals which have been felt as more or less consonant.
Then steps in this scale which come near to the great salient
points--that is, the points of greatest consonance, which is
unity, which is rest--are felt as suggesting them. This is
the reason why a semitone progression is felt as so compelling.
In taking the scale upward, C to C', that element in the tone-
Space already clearly foreshadowed by the previous tones is C';
B is so near that it is almost C'--it seems to cry aloud to be
completed by C'. Then the tendency to move from B to C' is
especially strong. In the same way a chromatic note suggests
most strongly the salient point in the scheme to which it is
nearest--and "tends" to it as to a point of comparative rest.
The difference between the major and minor scales may be found
in the lesser definiteness<1> with which the tendency to
progression, in the latter, is felt--"a condition of hovering,
a kind of ambiguity, of doubt, to which side the movement
shall proceed." We may then understand a melody as ever tending
with various degrees of urgency, of strain, to its centre of
gravity, the tonic.
<1> F. Weinmann, _Zeitschr. f. Psychol._, Bd. 35, p. 360.
It is from this point of view that we can see the cogency of
Gurney's remark, that when music seems to be yearning for
unutterable things, it is really yearning only for the next
note. "In this step from the state of rest into movement and
return, the coming again to rest; on what circuitous ways,
with what reluctances and hesitations; whether quick and
decisively or gradually and unnoticed--therein consists the
nature of melody."<1>
<1> Weinmann, op. cit.
Or in Gurney's more eloquent description, "The melody may begin
by pressing its way through a sweetly yielding resistance to a
gradually foreseen climax; whence again fresh expectation is
bred, perhaps for another excursion, as it were, round the same
centre but with a bolder and freer sweep,...to a point where
again the motive is suspended on another temporary goal; till
after a certain number of such involutions and evolutions, and
of delicately poised leanings and reluctances and yieldings,
the forces so accurately measured just suffice to bring it home,
and the sense of potential and coming integration which has
underlain all our provisional adjustments of expectation is
triumphantly justified."<1>
<1> Op. cit., p. 165.
This should not be taken as a more or less poetical account
under the metaphor of motion. These "leanings" are literal
in the sense that one note does imply another as its natural
complement and satisfaction and we seek to reach or make it.
The striving is an intrinsic element, not a by-product for our
understanding.
There is another point to note. The "sense of potential and
coming integration" is a strong factor of melody. If it cannot
be said that the first note implies the last, it is at least
true that from point to point the next step is dimly foreseen,
and this effect is cumulative. If melody is an ever-hindered
striving for the goal, at least the hindrances themselves are
stations on the way, each one as overcome adding to the final
momentum with which the goal is reached. It is like an
accumulation of evidence, a constellation of associations. AB
foretells C; but ABCDEF rushes yet more strongly upon G. So
it is that the irresistibleness, the "unalterable rightness"
of a piece of music increases from beginning to end.
The significance of this essential internal necessity of
progression cannot be overestimated. The unalterable rightness
of music is founded on natural acoustic laws, and this
"rightness" is fundamental. A melody is not right because it
is beautiful, it is beautiful because it is right. The natural
tendencies point out different paths to the goal; and thus
different ways of being beautiful; but the nature of the
relation between point and point, the nature of the progression,
that is, the nature of melody, is the same.
Up to this point we have consistently abstracted from the
element of rhythm in melody. Strictly speaking, however, it
is impossible to do so. The individuality of a melody is
absolutely dependent on its rhythm, that is, on the relative
time-value of its tones. Gurney has devoted some amusing pages
to showing the trivial, dragging, lustreless tunes that result
from ever so slight a change in the rhythm of noble themes, or
even in the distribution of rhythmical elements within the bar.
The reason for this is evident. The nature of melody in the
sense of sequence consists in the varied answers to the demands
of the ear as felt at each successive point. Now it is clear
that such "answer" can be emphasized, given indifferently, held
in suspense, in short, subjected to all kinds of variation as
well by the rhythmical form into which it is cast, as by the
different choice of possibilities for the tone itself. The
rhythm helps out the melody not only by adding to it an
independently pleasing element, but, and this is indeed the
essential, by reinforcing the intrinsic relations of the notes
themselves. Thus it is in the highest degree true that in
melody and rhythm we do not have content and form, but that,
strictly speaking, the melody is tone-sequence in rhythm.
The intimate bondage of tone-sequence and rhythm is grounded
in the identity of their inner nature; both are varieties of
the objective conditions of embodied expectation. It is not
of the essence of music to satisfy explicit and conscious
expectation--to satisfy the understanding. It meets on the
contrary a subconscious, automatic need which becomes conscious
only in the moment of its contenting. Every moment of progress
in a beautiful melody is hailed like an instinctive action
performed for the first time. Rhythm is the ideal satisfaction
of attention in general with all its bodily concomitants and
expressions. Tone-sequence is the satisfaction of attention
directed to auditory demands. But the form-quality of rhythm,
the form-quality of tonality, is an all but subconscious
possession. Together, reinforcing each other in melody, they
furnish the ideal arrangement of the most poignant of sense-
stimulations.