“In short, physical numbers are material forms divided about the subject which receives them. But material powers are the sources of connexion and modi- fication to bodies. For form is one thing, and the power proceeding from it another. For form itself is indeed impartible and essential; but being extended, and becoming bulky, it emits from itself, as if it were a blast, material powers which are certain qualities. Thus, for instance, in fire, the form and essence of it is impartible, and is truly the image of the cause of fire; for in partible natures, the impartible has a subsistence. But from form which is impartible in fire, and which subsists in it as number, an extension of it accompanied with interval takes place about matter, from which the powers of fire are emitted, such as heat, or refrigeration, or moisture, or something else of the like kind. And these qualities are indeed essential, but are by no means the essence of fire. For essences do not proceed from qualities, nor are essence and power the same thing. But the essential every where precedes power. And from this being one the multitude of powers proceeds, and the distributed from that which is undistri-
buted; just as many energies are the progeny of one power.”
P. 79. For Pythagoras always proclaimed, that nothing
admirable pertaining to the Gods, or divine dogmas,
should be disbeheved.
This in the Protreptics forms the fourth symbol, and
is thus explained by Iamblichus:—“ This dogma suffi-
ciently venerates and unfolds the transcendency of the
Gods, affording us a viaticum, and recalling to our
memory that we ought not to estimate divine power
from our judgment. But it is likely that some things
should appear difficult and impossible to us, in con-
sequence of our corporeal subsistence, and from our
being conversant with generation and corruption; from
our having a momentary existence; from being subject
to a variety of diseases; from the smallness of our
habitation; from our gravitating tendency to the
middle; from our somnolency, indigence and repletion;
from our want of counsel and our imbecility; from the
impediments of our soul, and a variety of other circum-
stances, although our nature possesses many illustrious
prerogatives. At the same time however we perfectly
fall short of the Gods, and neither possess the same
power with them, nor equal virtue. This symbol there-
fore in a particular manner introduces the knowledge of
the Gods, as beings who are able to effect all things.
On this account it exhorts us to disbelieve nothing con-
cerning the Gods. It also adds, nor about divine dog-
mas; viz. those belonging to the Pythagoric philosophy.
For these being secured by disciplines and scientific
theory, are alone true and free from falsehood, being
corroborated by all-various demonstration, accompanied
with necessity. ‘The same symbol, also, is capable of
exhorting us to the science concerning the Gods: for
it urges us to acquire a science of that kind, through
which we shall be in no respect deficient in things
asserted about the Gods. It 1s also able to exhort the
same things concerning divine dogmas, and a disciplina-
tive progression. For disciplines alone give eyes to, and produce light about, all things, in him who intends to consider and survey them. For from the participa- tion of disciplines, one thing before all others is effected, viz. a belief in the nature, essence, and power of the Gods, and also in those Pythagoric dogmas, which appear to be prodigious to such as have not been introduced to, and are uninitiated in, disciplines. So that the pre- cept disbelieve not is equivalent to participate and acquire those things through which you will not disbelieve; that is to say, acquire disciplines and scientific demonstra- tions.”
P.65. After this manner therefore it 1s satd that music was discovered by Pythagoras.
The following particulars relative to music are added
for the purpose of elucidating what is said about it in
this chapter.
“Take two brazen chords, such as are used in harps;
for those chords which are made from the intestines of
sheep are for the most part either false or obnoxious
to the change of the air.
A. CBB Ce sD
“Let these chords be perfectly equal, and equally
stretched, so as to be in unison, i.e. so that there may
be only one sound, though there are two strings. But
it is requisite that they should be placed upon some
oblong and polished rule. The ancients called this
rule an harmonic rule, or also a monochord, by which
instrument all consonances and dissonances, and like-
wise musical intervals, were tried. Let now one of
these chords be bisected in E. Afterwards under the
point E place what is vulgarly called the tactus, but which was denominated by the ancients, from its figure, a hemisphere. The tactus, therefore, being placed under E, press there the chord, so that one half of it only, as for instance ED, may be wholly struck and resound. Having therefore struck each of the chords at the same time, viz. the whole of AB, and the half ED, so that they may resound at one and the same time, you will hear the sweetest of all consonances, composed from the sound of the whole chord AB, and the sound of the half ED. This consonance the ancients called diapason, i.e. through all [the chords], because in the musical instruments of the ancients, the two extreme chords, i.e. the most grave, and the most acute of all the chords, contained this consonance; so that, from the gravest chord having made a transition through all the chords to the supreme and most acute of all, they would hear this sweetest consonance. It was, likewise, said to be in a duple ratio of the propor- tion of one sound to the other. For the sound of the chord AB is doubly greater or more grave than the sound of the half ED. For as sounding bodies are to each other, so are their sounds. But the chord AB is the double of ED. This, however, is now commonly called the octave, because from the first sound, and that the gravest, which is called ut, as far as to that sound which corresponds to it in the consonance dia- pason, there are these eight sounds, ut, re, mt, fa, sol, re, mi, fa. And of these the first uz, and the last fa, which is the eighth, produce the consonance diapason, or the double, or the octave.
Again, let the same chord CD be divided into three
equal parts in the points F, G.
A ee a oe eee eee 5
a DD
F
FD, therefore, will be two-thirds as well of the whole
CD as of the whole AB. Let the tactus now be placed
in F, and let AB and FD be struck at the same time,
and a consonance very sweet and perfect will indeed
be heard, yet not so sweet as the diapason. This the
ancients called diapente (i.e. through five chords),
because the first and the fifth chord produce this con-
sonance. But according to proportion it is called ses-
quialter, because the chord AB is sesquialter to FD,
and consequently the sounds of these chords also are
in the same ratio. But sesquialter ratio is when the
greater quantity AB contains the less FD once, and the
half of it besides. It is, indeed, commonly called the
fifth, because it is composed from the first sound ut,
and the fifth, sol.
Again, let the same chord be cut into four equal parts in the points H, E, I,
A B
C ptt = K LHFMNE GI
so that the chord HD, may be three-fourths of the
whole CD. ‘The tactus, therefore, being placed in H,
let AB and HD be struck at one and the same time,
and a consonance will be heard, indeed, yet more im-
perfect than the preceding two. This was called by
the ancients diatessaron, i.e. through four chords or
sounds, for a similar reason to that by which the former
were denominated. With reference, however, to the
ratio of the chords and sounds, it is called sesquitertian,
because the greater AB contains the less once, and a
third part of it besides. But it is now commonly called
a fourth, because it is found between the first sound
ut, and the fourth fa. If now the point F be added in
the preceding figure, and at one and the same time
two chords HD and FD are compared in arithmetical
ratios, we shall find that the greater HD will have to the less FD a sesquioctave! ratio, and the sound of the greater HD to the less FD will have the same ratio, i.e. in modern terms, that between fa and sol there is a sesquioctave ratio. But if these two sounds are heard together, they will be discordant to the ear. Again, the distance between these sounds fa, sol, or between the chords HD and FD, or between the two harmonic intervals HD and FD, the ratio of which was sesqui- octave, was called by the ancients a tone. Afterwards they divided the whole of CD into nine equal parts, the first of which is divided in K, so that the whole CD may have to the remainder KD, which contains eight of those parts, a sesquioctave ratio. ‘This, in like manner, will be the interval of a tone, the first sound of which, i.e. of the whole CD, is now called uz, but the second sound of the rest of the chord KD is called re. Afterwards they in a similar manner divided the remainder KD into nine parts, the first part of which is marked in the point L. And for the same reason between the chord KD and the chord LD, and their sounds, there will be a sesquioctave ratio. The sound of the chord LD is now called mi; but the interval which remains between the chord LD and the chord HD has not a sesquioctave ratio, but less than it almost by half, and therefore an interval of this kind was called a semitone, and also diesis or a division. But that interval which remains between the points F and E they divided after the same manner as the space between C and H was divided, and they again found the same sounds. Let those divisions be marked by the points M and N; and here, also, between N and E, or between mt and fa, there is in like manner another semitone. These eight sounds, therefore, are ut, re, mi, fa, sol, re, mt, fa, which compose the whole diapason. For as we have before observed, between wt and the last fa is the consonance diapason, or between the chord CD 1 Because } is to § as Q to 8.
or AB, and the chord ED. But from the intervals
which are between the sounds there are two semitones,
viz. one between m1 and fa, denoted by the letters
L, N, and the other between the last mz and fa, denoted
by the letters N, E. The remaining five intervals are
entire tones. It must, also, be observed, that from ut
to the first sol is the consonance diapente, which
contains three tonic intervals, and one semitone;
nevertheless in all there are five sounds, ut, re, mt,
fa, sol.
Again, from sol to the last fa there are four sounds, sol, re, mt, fa, which are perfectly similar to the first four, ut, re, m1, fa. Nevertheless these are more grave, but those are more acute. And as from wt to the first fa is the diatessaron, so likewise from sol to the last fa is another diatessaron; from which, in the last place, it must be observed, it follows that the two conson- ances diatessaron and diapente canstitute the whole diapason; or that the diapason is divided into one diatessaron, and one diapente. For from wt to sol is the diapente, but from sol to the last fa is the dia- tessaron. ‘This will also be the case if we should say that from ut to the first fa is the diatessaron, as is evident from the division of the chord; but from the first fa to the last fa is the diapente, as is evident from the four intervals of the chord, three of which are tones, and the remaining interval is a semitone, which also in the other diapente were contained between ut and sol.
Now again, let the tactus be placed in I; but I is
the fourth part of the whole CD. Let, also, AB and
ID be struck at one and the same time, and the sweetest
consonance, called bisdiapason, will be produced; which
is so denominated, because it is composed from two
diapasons, of which the first is between AB or CD, and
ED, but the second is between ED and ID; for the
ratio of these is double as well as of those. The ratio,
also, of the bisdiapason is quadruple, as is evident from
the division; and is commonly called a fifteenth, be-
cause from the first ut to this sound, which is also
denominated fa, there would be fifteen sounds, if the
interval EI were divided after the same manner as the
first CE is divided.
Farther still, let GD be a third part of the whole CD,
and let the tactus be placed in G. Then at one and the
same time let AB and GD be struck, and a sweet con-
sonance will be heard, which is called diapasondiapente,
because it 1s composed from one diapason contained
by the interval CE, or the two chords CD, ED, and
one diapente, contained by the interval EG, or the
chords ED, GD. For the chord ED is sesquialter to
the chord GD; which ratio constitutes the nature of
the diapente. ‘The proportion, also, of this consonance
is triple. For the chord AB or CD is triple of GD;
and it is commonly called the twelfth, because between
ut and sol, denoted by the letter G, there would be
twelve sounds, if the interval EG received its divisions.
From all which it is manifest by the experience of the
ear, that there are altogether five consonances, three
simple, the diapason, the diapente, and the diatessaron;
but two composite, the bisdiapason, and the diapason-
diapente.
In the last place, it is necessary to observe that those
ancient Greeks differently denominated these sounds,
ut, re, &c. For the first, i.e. the gravest sound or chord,
which is now called ut, they denominated hypate, and
the others in the following order: