must be divided into scales; and it is a rule in the formation of these
scales, that the first must begin with the monad (1) and end with the
duad (2), the second begin with the duad (2) and end with the tetrad (4),
and that the beginning and end of all other scales must be continued in
the same arithmetical progression. These primary elements will then form
the foundation of a series of such scales.
I. (1) (2)
II. (2) (3) (4)
III. (4) (5) (6) (7) (8)
IV. (8) (9) (10) ( ) (12) ( ) (14) (15) (16)
The first of these scales has in (1) and (2) a beginning and an end; but
the second has in (2), (3), and (4) the essential requisites demanded
by Aristotle in every composition, viz., “a beginning, a middle, and
an end;” while the third has not only these essential requisites, but
two intermediate parts (5) and (7), by which the beginning, the middle,
and the end are united. In the fourth scale, however, the arithmetical
progression is interrupted by the omission of numbers 11 and 13, which,
not being multiples of either (2), (3), (5), or (7), are inadmissible.
Such is the nature of the harmonic law which governs the progressive
scales of numbers by the simple multiplication of the monad.
I shall now use these numbers as divisors in the formation of a series
of four such scales of parts, which has for its primary element, instead
of the indivisible monad, a quantity which may be indefinitely divided,
but which cannot be added to or multiplied. Like the monad, however, this
quantity is represented by (1). The following is this series of four
scales of harmonic parts:—
I. (1) (¹⁄₂)
II. (¹⁄₂) (¹⁄₃) (¹⁄₄)
III. (¹⁄₄) (¹⁄₅) (¹⁄₆) (¹⁄₇) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
The scales I., II., and III. may now be rendered as complete as scale
IV., simply by multiplying upwards by 2 from (¹⁄₉), (¹⁄₅), (¹⁄₃), (¹⁄₇),
and (¹⁄₁₅), thus:—
I. (1) (⁸⁄₉) (⁴⁄₅) (²⁄₃) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (¹⁄₃) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) (¹⁄₆) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
We now find between the beginning and the end of scale I. the quantities
(⁸⁄₉), (⁴⁄₅), (²⁄₃), (⁴⁄₇), and (⁸⁄₁₅).
The three first of these quantities we find to be the remainders of the
whole indefinite quantity contained in (1), after subtracting from it
the primary harmonic quantities (¹⁄₉), (¹⁄₅), and (¹⁄₃); we, however,
find also amongst these harmonic quantities that of (¹⁄₄), which being
subtracted from (1) leaves (³⁄₄), a quantity the most suitable whereby
to fill up the hiatus between (⁴⁄₅) and (²⁄₃) in scale I., which arises
from the omission of (¹⁄₁₁) in scale IV. In like manner we find the two
last of these quantities, (⁴⁄₇) and (⁸⁄₁₅), are respectively the largest
of the two parts into which 7 and 15 are susceptible of being divided.
Finding the number 5 to be divisible into parts more unequal than (2)
to (3) and less unequal than (4) to (7), (³⁄₅) naturally fills up the
hiatus between these quantities in scale I., which hiatus arises from the
omission of (¹⁄₁₃) in scale IV. Thus:—
I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) ( ) (¹⁄₃) ( ) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) ( ) (¹⁄₆) ( ) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) ( ) (¹⁄₁₂) ( ) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
Scale I. being now complete, we have only to divide these latter
quantities by (2) downwards in order to complete the other three. Thus:—
I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃) (³⁄₁₀) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) (³⁄₁₆) (¹⁄₆) (³⁄₂₀) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (³⁄₃₂) (¹⁄₁₂) (³⁄₄₀) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
The harmony existing amongst these numbers or quantities consists of the
numerical relations which the parts bear to the whole and to each other;
and the more simple these relations are, the more perfect is the harmony.
The following are the numerical harmonic ratios which the parts bear to
the whole:—
I. (1:1) (8:9) (4: 5) (3: 4) (2: 3) (3: 5) (4: 7) (8:15) (1: 2)
II. (1:2) (4:9) (2: 5) (3: 8) (1: 3) (3:10) (2: 7) (4:15) (1: 4)
III. (1:4) (2:9) (1: 5) (3:16) (1: 6) (3:20) (1: 7) (2:15) (1: 8)
IV. (1:8) (1:9) (1:10) (3:32) (1:12) (3:40) (1:14) (1:15) (1:16)
The following are the principal numerical relations which the parts in
each scale bear to one another:—
(¹⁄₂):(⁴⁄₇) = (7:8)
(⁴⁄₅):(⁸⁄₉) = (9:10)
(²⁄₃):(⁴⁄₅) = (5:6)
(⁴⁄₇):(²⁄₃) = (6:7)
(⁸⁄₁₅):(⁴⁄₇) = (14:15)
(¹⁄₂):(⁸⁄₁₅) = (15:16)
Although these relations are exemplified by parts of scale I., the same
ratios exist between the relative parts of scales II., III., and IV.,
and would exist between the parts of any other scales that might be added
to that series.
These are the simple elements of the science of that harmony which
pervades the universe, and by which the various kinds of beauty
æsthetically impressed upon the senses of hearing and seeing are
governed.
THE SCIENCE OF BEAUTY AS APPLIED TO SOUNDS.
It is well-known that all sounds arise from a peculiar action of the
air, and that this action may be excited by the concussion resulting
from the sudden displacement of a portion of the atmosphere itself, or
by the rapid motions of bodies, or of confined columns of air; in all
which cases, when the motions are irregular, and the force great, the
sound conveyed to the sensorium is called a noise. But that musical
sounds are the result of equal and regular vibratory motions, either
of an elastic body, or of a column of air in a tube, exciting in the
surrounding atmosphere a regular and equal pulsation. The ear is the
medium of communication between those varieties of atmospheric action and
the seat of consciousness. To describe fully the beautiful arrangement
of the various parts of this organ, and their adaptation to the purpose
of collecting and conveying these undulatory motions of the atmosphere,
is as much beyond the scope of my present attempt as it is beyond my
anatomical knowledge; but I may simply remark, that within the ear, and
most carefully protected in the construction of that organ, there is a
small cavity containing a pellucid fluid, in which the minute extremities
of the auditory nerve float; and that this fluid is the last of the media
through which the action producing the sensation of sound is conveyed
to the nerve, and thence to the sensorium, where its nature becomes
perceptible to the mind.
The impulses which produce musical notes must arrive at a certain
frequency before the ear loses the intervals of silence between them,
and is impressed by only one continued sound; and as they increase
in frequency the sound becomes more acute upon the ear. The pitch of
a musical note is, therefore, determined by the frequency of these
impulses; but, on the other hand, its intensity or loudness will depend
upon the violence and the quality of its tone on the material employed in
producing them. All such sounds, therefore, whatever be their loudness
or the quality of their tone in which the impulses occur with the same
frequency are in perfect unison, having the same pitch. Upon this the
whole doctrine of harmonies is founded, and by this the laws of numerical
ratio are found to operate in the production of harmony, and the theory
of music rendered susceptible of exact reasoning.
The mechanical means by which such sounds can be produced are extremely
various; but, as it is my purpose simply to shew the nature of harmony
of sound as related to, or as evolving numerical harmonic ratio, I shall
confine myself to the most simple mode of illustration—namely, that of
the monochord. This is an instrument consisting of a string of a given
length stretched between two bridges standing upon a graduated scale.
Suppose this string to be stretched until its tension is such that,
when drawn a little to a side and suddenly let go, it would vibrate at
the rate of 64 vibrations in a second of time, producing to a certain
distance in the surrounding atmosphere a series of pulsations of the same
frequency.
These pulsations will communicate through the ear a musical note which
would, therefore, be the fundamental note of such a string. Now, the
phenomenon said to be discovered by Pythagoras is well known to those
acquainted with the science of acoustics, namely, that immediately after
the string is thus put into vibratory motion, it spontaneously divides
itself, by a node, into two equal parts, the vibrations of each of which
occur with a double frequency—namely, 128 in a second of time, and,
consequently, produce a note doubly acute in pitch, although much weaker
as to intensity or loudness; that it then, while performing these two
series of vibrations, divides itself, by two nodes, into three parts,
each of which vibrates with a frequency triple that of the whole string;
that is, performs 192 vibrations in a second of time, and produces a
note corresponding in increase of acuteness, but still less intense than
the former, and that this continues to take place in the arithmetical
progression of 2, 3, 4, &c. Simultaneous vibrations, agreeably to the
same law of progression, which, however, seem to admit of no other primes
than the numbers 2, 3, 5, and 7, are easily excited upon any stringed
instrument, even by the lightest possible touch of any of its strings
while in a state of vibratory motion, and the notes thus produced are
distinguished by the name of harmonics. It follows, then, that one-half
of a musical string, when divided from the whole by the pressure of the
finger, or any other means, and put into vibratory motion, produces a
note doubly acute to that produced by the vibratory motion of the whole
string; the third part, similarly separated, a note trebly acute; and
the same with every part into which any musical string may be divided.
This is the fundamental principle by which all stringed instruments are
made to produce harmony. It is the same with wind instruments, the sounds
of which are produced by the frequency of the pulsations occasioned in
the surrounding atmosphere by agitating a column of air confined within
a tube as in an organ, in which the frequency of pulsation becomes
greater in an inverse ratio to the length of the pipes. But the following
series of four successive scales of musical notes will give the reader
a more comprehensive view of the manner in which they follow the law of
numerical ratio just explained than any more lengthened exposition.
It is here requisite to mention, that in the construction of these
scales, I have not only adopted the old German or literal mode of
indicating the notes, but have included, as the Germans do, the note
termed by us B flat as B natural, and the note we term B natural as
H. Now, although this arrangement differs from that followed in the
construction of our modern Diatonic scale, yet as the ratio of 4:7
is more closely related to that of 1:2 than that of 8:15, and as it
is offered by nature in the spontaneous division of the monochord,
I considered it quite admissible. The figures give the parts of the
monochord which would produce the notes.
I. { (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)*
{ C D E F G A B H _c_
II. { (¹⁄₂)* (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃)* (³⁄₁₀) (²⁄₇) (²⁄₁₅) (¹⁄₄)*
{ _c_ _d_ _e_ _f_ _g_ _a_ _b_ _h_ _c′_
III. { (¹⁄₄)* (²⁄₉) (¹⁄₅)* (³⁄₁₆) (¹⁄₆)* (³⁄₂₀) (¹⁄₇)* (²⁄₁₅) (¹⁄₈)*
{ _c′_ _d′_ _e′_ _f′_ _g′_ _a′_ _b′_ _h′_ _c′′_
IV. { (¹⁄₈)* (¹⁄₉)* (¹⁄₁₀)* (³⁄₃₂) (¹⁄₁₂)* (³⁄₄₀) (¹⁄₁₄)* (¹⁄₁₅)* (¹⁄₁₆)*
{ _c′′_ _d′′_ _e′′_ _f′′_ _g′′_ _a′′_ _b′′_ _h′′_ _c′′′_
The notes marked (*) are the harmonics which naturally arise from the
division of the string by 2, 3, 5, and 7, and the multiples of these
primes.
Thus every musical sound is composed of a certain number of parts called
pulsations, and these parts must in every scale relate harmonically
to some fundamental number. When these parts are multiples of the
fundamental number by 2, 4, 8, &c., like the pulsations of the sounds
indicated by _c_, _c′_, _c′′_, _c′′′_, they are called tonic notes, being
the most consonant; when the pulsations are similar multiples by 3, 6,
12, &c., like those of the sounds indicated by _g_, _g′_, _g′′_, they are
called dominant notes, being the next most consonant; and multiples by
5, 10, &c., like those of the sounds indicated by _e_, _e′_, _e′′_, they
are called mediant notes, from a similar cause. In harmonic combinations
of musical sounds, the æsthetic feeling produced by their agreement
depends upon the relations they bear to each other with reference to the
number of pulsations produced in a given time by the fundamental note of
the scale to which they belong; and it will be observed, that the more
simple the numerical ratios are amongst the pulsations of any number of
notes simultaneously produced, the more perfect their agreement. Hence
the origin of the common chord or fundamental concord in the united
sounds of the tonic, the dominant, and the mediant notes, the ratios and
coincidences of whose pulsations 2:1, 3:2, 5:4, may thus be exemplified:—
[Illustration]
In musical composition, the law of number also governs its division
into parts, in order to produce upon the ear, along with the beauty of
harmony, that of rhythm. Thus a piece of music is divided into parts
each of which contains a certain number of other parts called bars,
which may be divided and subdivided into any number of notes, and the
performance of each bar is understood to occupy the same portion of time,
however numerous the notes it contains may be; so that the music of art
is regularly symmetrical in its structure; while that of nature is in
general as irregular and indefinite in its rhythm as it is in its harmony.
Thus I have endeavoured briefly to explain the manner in which the law of
numerical ratio operates in that species of beauty perceived through the
ear.
The definite principles of the art of music founded upon this law have
been for ages so systematised that those who are instructed in them
advance steadily in proportion to their natural endowments, while those
who refuse this instruction rarely attain to any excellence. In the
sister arts of form and colour, however, a system of tuition, founded
upon this law, is still a desideratum, and a knowledge of the scientific
principles by which these arts are governed is confined to a very few,
and scarcely acknowledged amongst those whose professions most require
their practical application.
THE SCIENCE OF BEAUTY AS APPLIED TO FORMS.
It is justly remarked, in the “Illustrated Record of the New York
Exhibition of 1853,” that “it is a question worthy of consideration how
far the mediocrity of the present day is attributable to an overweening
reliance on natural powers and a neglect of the lights of science;”
and there is expressed a thorough conviction of the fact that, besides
the evils of the copying system, “much genius is now wasted in the
acquirement of rudimentary knowledge in the slow school of practical
experiment, and that the excellence of the ancient Greek school of
design arose from a thoroughly digested canon of form, and the use
of geometrical formulas, which make the works even of the second and
third-rate genius of that period the wonder and admiration of the present
day.”
That such a canon of form, and that the use of such geometrical formula,
entered into the education, and thereby facilitated the practice of
ancient Greek art, I have in a former work expressed my firm belief,
which is founded on the remarkable fact, that for a period of nearly
three centuries, and throughout a whole country politically divided into
states often at war with each other, works of sculpture, architecture,
and ornamental design were executed, which surpass in symmetrical
beauty any works of the kind produced during the two thousand years that
have since elapsed. So decided is this superiority, that the artistic
remains of the extraordinary period I alluded to are, in all civilised
nations, still held to be the most perfect specimens of formative art in
the world; and even when so fragmentary as to be denuded of everything
that can convey an idea of expression, they still excite admiration and
wonder by the purity of their geometric beauty. And so universal was
this excellence, that it seems to have characterised every production of
formative art, however humble the use to which it was applied.
The common supposition, that this excellence was the result of an
extraordinary amount of genius existing among the Greek people during
that particular period, is not consistent with what we know of the
progress of mankind in any other direction, and is, in the present state
of art, calculated to retard its progress, inasmuch as such an idea
would suggest that, instead of making any exertion to arrive at a like
general excellence, the world must wait for it until a similar supposed
psychological phenomenon shall occur.
But history tends to prove that the long period of universal artistic
excellence throughout Greece could only be the result of an early
inculcation of some well-digested system of correct elementary
principles, by which the ordinary amount of genius allotted to mankind
in every age was properly nurtured and cultivated; and by which, also, a
correct knowledge and appreciation of art were disseminated amongst the
people generally. Indeed, Müller, in his “Ancient Art and its Remains,”
shews clearly that some certain fixed principles, constituting a science
of proportions, were known in Greece, and that they formed the basis of
all artists’ education and practice during the period referred to; also,
that art began to decline, and its brightest period to close, as this
science fell into disuse, and the Greek artists, instead of working for
an enlightened community, who understood the nature of the principles
which guided them, were called upon to gratify the impatient whims of
pampered and tyrannical rulers.
By being instructed in this science of proportion, the Greek artists
were enabled to impart to their representations of the human figure
a mathematically correct species of symmetrical beauty; whether
accompanying the slender and delicately undulated form of the Venus,—its
opposite, the massive and powerful mould of the Hercules,—or the
characteristic representation of any other deity in the heathen
mythology. And this seems to have been done with equal ease in the minute
figure cut on a precious gem, and in the most colossal statue. The same
instruction likewise enabled the architects of Greece to institute those
varieties of proportions in structure called the Classical Orders of
Architecture; which are so perfect that, since the science which gave
them birth has been buried in oblivion, classical architecture has been
little more than an imitative art; for all who have since written upon
the subject, from Vitruvius downwards, have arrived at nothing, in
so far as the great elementary principles in question are concerned,
beyond the most vague and unsatisfactory conjectures. For a more clear
understanding of the nature of this application of the Pythagorean law of
number to the harmony of form, it will be requisite to repeat the fact,
that modern science has shewn that the cause of the impression, produced
by external nature upon the sensorium, called light, may be traced to a
molecular or ethereal action. This action is excited naturally by the
sun, artificially by the combustion of various substances, and sometimes
physically within the eye. Like the atmospheric pulsations which produce
sound, the action which produces light is capable, within a limited
sphere, of being reflected from some bodies and transmitted through
others; and by this reflection and transmission the visible nature of
forms and figures is communicated to the sensorium. The eye is the
medium of this communication; and its structural beauty, and perfect
adaptation to the purpose of conveying this action, must, like those of
the ear, be left to the anatomist fully to describe. It is here only
necessary to remark, that the optic nerve, like the auditory nerve,
ends in a carefully protected fluid, which is the last of the media
interposed between this peculiarly subtle action and the nerve upon
which it impresses the presence of the object from which it is reflected
or through which it is transmitted, and the nature of such object made
perceptible to the mind. The eye and the ear are thus, in one essential
point, similar in their physiology, relatively to the means provided
for receiving impressions from external nature; it is, therefore, but
reasonable to believe that the eye is capable of appreciating the exact
subdivision of spaces, just as the ear is capable of appreciating the
exact subdivision of intervals of time; so that the division of space
into exact numbers of equal parts will æsthetically affect the mind
through the medium of the eye.
We assume, therefore, that the standard of symmetry, so estimated, is
deduced from the simplest law that could have been conceived—the law
that the angles of direction must all bear to some fixed angle the same
simple relations which the different notes in a chord of music bear to
the fundamental note; that is, relations expressed arithmetically by the
smallest natural numbers. Thus the eye, being guided in its estimate by
direction rather than by distance, just as the ear is guided by number
of vibrations rather than by magnitude, both it and the ear convey
simplicity and harmony to the mind without effort, and the mind with
equal facility receives and appreciates them.
_On the Rectilinear Forms and Proportions of Architecture._
As we are accustomed in all cases to refer direction to the horizontal
and vertical lines, and as the meeting of these lines makes the right
angle, it naturally constitutes the fundamental angle, by the harmonic
division of which a system of proportion may be established, and the
theory of symmetrical beauty, like that of music, rendered susceptible of
exact reasoning.
Let therefore the right angle be the fundamental angle, and let it be
divided upon the quadrant of a circle into the harmonic parts already
explained, thus:—
Super- Sub- Sub- Sub- Semi-sub-
Right tonic Mediant dominant Dominant mediant tonic tonic Tonic
Angle. Angles. Angles. Angles. Angles. Angles. Angles. Angles. Angles.
I. (1) (⁸⁄₉) (⁴⁄₅) (³⁄₄) (²⁄₃) (³⁄₅) (⁴⁄₇) (⁸⁄₁₅) (¹⁄₂)
II. (¹⁄₂) (⁴⁄₉) (²⁄₅) (³⁄₈) (¹⁄₃) (³⁄₁₀) (²⁄₇) (⁴⁄₁₅) (¹⁄₄)
III. (¹⁄₄) (²⁄₉) (¹⁄₅) (³⁄₁₆) (¹⁄₆) (³⁄₂₀) (¹⁄₇) (²⁄₁₅) (¹⁄₈)
IV. (¹⁄₈) (¹⁄₉) (¹⁄₁₀) (³⁄₃₂) (¹⁄₁₂) (³⁄₄₀) (¹⁄₁₄) (¹⁄₁₅) (¹⁄₁₆)
In order that the analogy may be kept in view, I have given to the parts
of each of these four scales the appropriate nomenclature of the notes
which form the diatonic scale in music.