HARMONY (Gr. [Greek: harmonia], a concord of musical sounds, [Greek:
harmozein] to join; [Greek: harmonike] (sc. [Greek: techne]) meant the
science or art of music, [Greek: mousike] being of wider significance),
a combination of parts so that the effect should be aesthetically
pleasing. In its earliest sense in English it is applied, in music, to a
pleasing combination of musical sounds, but technically it is confined
to the science of the combination of sounds of different pitch.
I. _Concord and Discord._--By means of harmony modern music has attained
the dignity of an independent art. In ancient times, as at the present
day among nations that have not come under the influence of European
music, the harmonic sense was, if not altogether absent, at all events
so obscure and undeveloped as to have no organizing power in the art.
The formation by the Greeks of a scale substantially the same as that
which has received our harmonic system shows a latent harmonic sense,
but shows it in a form which positively excludes harmony as an artistic
principle. The Greek perception of certain _successions_ of sounds as
concordant rests on a principle identifiable with the scientific basis
of concord in simultaneous sounds. But the Greeks did not conceive of
musical simultaneity as consisting of anything but identical sounds; and
when they developed the practice of _magadizing_--i.e. singing in
octaves--they did so because, while the difference between high and low
voices was a source of pleasure, a note and its octave were then, as
now, perceived to be in a certain sense identical. We will now start
from this fundamental identity of the octave, and with it trace the
genesis of other concords and discords; bearing in mind that the history
of harmony is the history of artistic instincts and not a series of
progressive scientific theories.
[Illustration: Ex. 1.--The notes marked * are out of tune.]
The unisonous quality of octaves is easily explained when we examine the
"harmonic series" of upper partials (see SOUND). Every musical sound, if
of a timbre at all rich (and hence pre-eminently the human voice),
contains some of these upper partials. Hence, if one voice produce a
note which is an upper partial of another note sung at the same time by
another voice, the higher voice adds nothing new to the lower but only
reinforces what is already there. Moreover, the upper partials of the
higher voice will also coincide with some of the lower. Thus, if a note
and its octave be sung together, the upper octave is itself No. 2 in the
harmonic series of the lower, No. 2 of its own series is No. 4 of the
lower, and its No. 3 is No. 6, and so on. The impression of identity
thus produced is so strong that we often find among people unacquainted
with music a firm conviction that a man is singing in unison with a boy
or an instrument when he is really singing in the octave below. And even
musical people find a difficulty in realizing more than a certain
brightness and richness of single tone when a violinist plays octaves
perfectly in tune and with a strong emphasis on the lower notes.
Doubling in octaves therefore never was and never will be a process of
harmonization.
Now if we take the case of one sound doubling another in the 12th, it
will be seen that here, too, no real addition is made by the higher
sound to the lower. The 12th is No. 3 of the harmonic series, No. 2 of
the higher note will be No. 6 of the lower, No. 3 will be No. 9, and so
on. But there is an important difference between the 12th and the
octave. However much we alter the octave by transposition into other
octaves, we never get anything but unison or octaves. Two notes two
octaves apart are just as devoid of harmonic difference as a plain
octave or unison. But, when we apply our principle of the identity of
the octave to the 12th, we find that the removal of one of the notes by
an octave may produce a combination in which there is a distinct
harmonic element. If, for example, the lower note is raised by an octave
so that the higher note is a fifth from it, No. 3 of the harmonic series
of the higher note will not belong to the lower note at all. The 5th is
thus a combination of which the two notes are obviously different; and,
moreover, the principle of the identity of octaves can now operate in a
contrary direction and transfer this positive harmonic value of the 5th
to the 12th, so that we regard the 12th as a 5th plus an octave, instead
of regarding the 5th as a compressed 12th.[1] At the same time, the
relation between the two is quite close enough to give the 5th much of
the feeling of harmonic poverty and reduplication that characterizes the
octave; and hence when medieval musicians doubled a melody in 5ths and
octaves they believed themselves to be doing no more than extending and
diversifying the means by which a melody might be sung in unison by
different voices. How they came to prefer for this purpose the 4th to
the 5th seems puzzling when we consider that the 4th does not appear as
a fundamental interval in the harmonic series until that series has
passed beyond that part of it that maintains any relation to our musical
ideas. But it was of course certain that they obtained the 4th as the
inversion of the 5th; and it is at least possible that the singers of
lower voices found a peculiar pleasure in singing below higher voices in
a position which they felt harmonically as that of a top part. That is
to say, a bass, in singing a fourth below a tenor, would take pleasure
in doubling in the octave an alto singing normally a 5th above the
tenor.[2] This should also, perhaps, be taken in connexion with the fact
that the interval of the downward 4th is in melody the earliest that
became settled. And it is worth noticing that, in any singing-class
where polyphonic music is sung, there is a marked tendency among the
more timid members to find their way into their part by a gentle humming
which is generally a 4th below the nearest steady singers.
The limited compass of voices soon caused modifications in the medieval
parallelisms of 4ths and 5ths, and the introduction of independent
ornaments into one or more of the voices increased to an extent which
drew attention to other intervals. It was long, however, before the true
criterion of concord and discord was attained; and at first the notion
of concord was purely acoustic, that is to say, the ear was sensitive
only to the difference in roughness and smoothness between combinations
in themselves. And even the modern researches of Helmholtz fail to
represent classical and modern harmony, in so far as the phenomena of
beats are quite independent of the contrapuntal nature of concord and
discord which depends upon the melodic intelligibility of the motion of
the parts. Beats give rise to a strong physical sense of discord akin to
the painfulness of a flickering light (see SOUND). Accordingly, in the
earliest experiments in harmony, the ear, in the absence of other
criteria, attached much more importance to the purely acoustic roughness
of beats than our ears under the experience of modern music. This, and
the circumstance that the _imperfect_ concords[3] (the 3rds and 6ths)
long remained out of tune owing to the incompleteness of the Pythagorean
system of harmonic ratios, sufficiently explain the medieval treatment
of these combinations as discords differing only in degree from the
harshness of 2nds and 7ths. In the earliest attempts at really
contrapuntal writing (the astonishing 13th and 14th-century motets, in
which voices are made to sing different melodies at once, with what
seems to modern ears a total disregard of sound and sense) we find that
the method consists in a kind of rough-hewing by which the concords of
the octave, 5th and 4th are provided at most of the strong accents,
while the rest of the harmony is left to take care of itself. As the art
advanced the imperfect concords began to be felt as different from the
discords; but as their true nature appeared it brought with it such an
increased sense of the harmonic poverty of octaves, 5ths and 4ths, as
ended in a complete inversion of the earliest rules of harmony.
The harmonic system of the later 15th century, which culminated in the
"golden age" of the 16th-century polyphony, may be described as follows:
Imagine a flux of simultaneous independent melodies, so ordered as to
form an artistic texture based not only on the variety of the melodies
themselves, but also upon gradations between points of repose and points
in which the roughness of sound is rendered interesting and beautiful by
means of the clearness with which the melodic sense in each part
indicates the convergence of all towards the next point of repose. The
typical point of repose owes its effect not only to the acoustic
smoothness of the combination, but to the fact that it actually
consists of the essential elements present in the first five notes of
the harmonic series. The major 3rd has thus in this scheme asserted
itself as a concord, and the fundamental principle of the identity of
octaves produces the result that any combination of a bass note with a
major 3rd and a perfect 5th above it, at any distance, and with any
amount of doubling, may constitute a concord available even as the final
point of repose in the whole composition. And by degrees the _major
triad_, with its major 3rd, became so familiar that a chord consisting
of a bare 5th, with or without an octave, was regarded rather as a
skeleton triad without the 3rd than as a concord free from elements of
imperfection. Again, the identity of the octave secured for the
combination of a note with its minor 3rd and minor 6th a place among
concords; because, whether so recognized by early theorists or not, it
was certainly felt as an inversion of the major triad. The fact that its
bass note is not the fundamental note (and therefore has a series of
upper partials not compatible with the higher notes) deprives it of the
finality and perfection of the major triad, to which, however, its
relationship is too near for it to be felt otherwise than as a concord.
This sufficiently explains why the minor 6th ranks as a concord in
music, though it is acoustically nearly as rough as the discord of the
minor 7th, and considerably rougher than that of the 7th note of the
harmonic series, which has not become accepted in our musical system at
all.
[Illustration: Ex. 2.]
[Illustration: Ex. 3.]
But the major triad and its inversion are not the only concords that
will be produced by our flux of melodies. From time to time this flux
will arrest attention by producing a combination which, while it does
not appeal to the ear as being a part of the harmonic chord of nature,
yet contains in itself no elements not already present in the major
triad. Theorists have in vain tried to find in "nature" a combination of
a note with its minor 3rd and perfect 5th; and so long as harmony was
treated unhistorically and unscientifically as an a priori theory in
which every chord must needs have a "root," the minor triad, together
with nearly every other harmonic principle of any complexity, remained a
mystery. But the minor triad, as an artistic and not purely acoustic
phenomenon, is an inevitable thing. It has the character of a concord
because of our intellectual perception that it contains the same
elements as the major triad; but its absence of connexion with the
natural harmonic series deprives it of complete finality in the simple
system of 16th-century harmony, and at the same time gives it a
permanent contrast with the major triad; a contrast which is
acoustically intensified by the fact that, though its intervals are in
themselves as concordant as those of the major triad, their relative
position produces decidedly rough combinations of "resultant tones."
By the time our flux of melodies had come to include the major and minor
triads as concords, the notion of the independence of parts had become
of such paramount importance as totally to revolutionize the medieval
conception of the perfect concords. Fifths and octaves no longer formed
an oasis in a desert of cacophony, but they assumed the character of
concord so nearly approaching to unison that a pair of consecutive 5ths
or octaves began to be increasingly felt as violating the independence
of the parts. And thus it came about that in pure 16th-century
counterpoint (as indeed at the present day whenever harmony and
counterpoint are employed in their purest significance) consecutive 5ths
and octaves are strictly forbidden. When we compare our laws of
counterpoint with those of medieval discant (in which consecutive 5ths
and octaves are the rule, while consecutive 3rds and 6ths are strictly
forbidden) we are sometimes tempted to think that the very nature of the
human ear has changed. But it is now generally recognized that the
process was throughout natural and inevitable, and the above account
aims at showing that consecutive 5ths are forbidden by our harmonic
system for the very reason which inculcated them in the system of the
12th century.
II. _Tonality._--As soon as the major and minor triad and their first
inversions were well-defined entities, it became evident that the
successions of these concords and their alternations with discord
involved principles at once larger and more subtle than those of mere
difference in smoothness and artificiality. Not only was a major chord
(or at least its skeleton) necessary for the final point of repose in a
composition, but it could not itself sound final unless the concords as
well as the discords before it showed a well-defined tendency towards
it. This tendency was best realized when the penultimate concord had its
fundamental note at the distance of a 5th or a 4th above or below that
of the final chord. When the fundamental note of the penultimate chord
is a 5th above or (what is the same thing) a 4th below that of the final
chord, we have an "authentic" or "perfect" cadence, and the relation
between the two chords is very clear. While the contrast between them is
well marked, they have one note in common--for the root of the
penultimate chord is the 5th of the final chord; and the statement of
this common note, first as an octave or unison and then as a 5th,
expresses the first facts of harmony with a force which the major 3rds
of the chords can only strengthen, while it also involves in the bass
that melodic interval of the 4th or the 5th which is now known to be the
germ of all melodic scales. The relation of the final note of a scale
with its upper 5th or lower 4th thus becomes a fundamental fact of
complex harmonic significance--that is to say, of harmony modified by
melody in so far as it concerns the succession of sounds as well as
their simultaneous combination. In our modern key-system the final note
of the scale is called the _tonic_, and the 5th above or 4th below it is
the _dominant_. (In the 16th century the term "dominant" has this
meaning only in the "authentic" modes other than the Phrygian, but as an
aesthetic fact it is present in all music, though the theory here given
would not have been intelligible to any composers before the 18th
century). Another penultimate chord asserts itself as the converse of
the dominant--namely, the chord of which the root is a 5th below or a
4th above the final. This chord has not that relationship to the final
which the dominant chord shows, for its fundamental note is not in the
harmonic series of the final. But the fundamental note of the final
chord is in its harmonic series, and in fact stands to it as the
dominant stands to the final. Thus the progression from _subdominant_,
as it is called, to tonic, or final, forms a full close known as the
"plagal cadence," second only in importance to the "perfect" or
"authentic cadence." In our modern key-system these three chords, the
tonic, the dominant and the subdominant, form a firm harmonic centre in
reference to which all other chords are grouped. The tonic is the final
in which everything ultimately resolves: the dominant stands on one side
of it as a chord based on the note harmonically most closely related to
the tonic, and the subdominant stands on the other side as the converse
and opposite of the dominant, weaker than the dominant because not
directly derived from the tonic. The other triads obtainable from the
notes of the scale are all minor, and of less importance; and their
relationship to each other and to the tonic is most definite when they
are so grouped that their basses rise and fall in 4th and 5ths, because
they then tend to imitate the relationship between tonic, dominant and
subdominant.
[Illustration: Ex. 4.]
[Illustration: Ex. 5.]
[Illustration: Ex. 6.