Since, however, we are narrating the wisdom employed by Pythagoras in
instructing his disciples, it will not be unappropriate to relate that
which is proximate in a following order to this, viz. how he invented
the harmonic science, and harmonic ratios. But for this purpose we must
begin a little higher. Intently considering once, and reasoning with
himself, whether it would be possible to devise a certain instrumental
assistance to the hearing, which should be firm and unerring, such as
the sight obtains through the compass and the rule, or, by Jupiter,
through a dioptric instrument; or such as the touch obtains through the
balance, or the contrivance of measures;—thus considering, as he was
walking near a brazier’s shop, he heard from a certain divine casualty
the hammers beating out a piece of iron on an anvil, and producing
sounds that accorded with each other, one combination only excepted. But
he recognized in those sounds, the diapason, the diapente, and the
diatessaron, harmony. He saw, however, that the sound which was between
the diatessaron and the diapente was itself by itself dissonant, yet,
nevertheless, gave completion to that which was the greater sound among
them. Being delighted, therefore, to find that the thing which he was
anxious to discover had succeeded to his wishes by divine assistance, he
went into the brazier’s shop, and found by various experiments, that the
difference of sound arose from the magnitude of the hammers, but not
from the force of the strokes, nor from the figure of the hammers, nor
from the transposition of the iron which was beaten. When, therefore, he
had accurately examined the weights and the equal counterpoise of the
hammers, he returned home, and fixed one stake diagonally to the walls,
lest if there were many, a certain difference should arise from this
circumstance, or in short, lest the peculiar nature of each of the
stakes should cause a suspicion of mutation. Afterwards, from this stake
he suspended four chords consisting of the same materials, and of the
same magnitude and thickness, and likewise equally twisted. To the
extremity of each chord also he tied a weight. And when he had so
contrived, that the chords were perfectly equal to each other in length,
he afterwards alternately struck two chords at once, and found the
before-mentioned symphonies, viz. a different symphony in a different
combination. For he discovered that the chord which was stretched by the
greatest weight, produced, when compared with that which was stretched
by the smallest, the symphony diapason. But the former of these weights
was twelve pounds, and the latter six. And, therefore, being in a duple
ratio, it exhibited the consonance diapason; which the weights
themselves rendered apparent. But again, he found that the chord from
which the greatest weight was suspended compared with that from which
the weight next to the smallest depended, and which weight was eight
pounds, produced the symphony diapente. Hence he discovered that this
symphony is in a sesquialter ratio, in which ratio also the weights were
to each other. And he found that the chord which was stretched by the
greatest weight, produced, when compared with that which was next to it
in weight, and was nine pounds, the symphony diatessaron, analogously to
the weights. This ratio, therefore, he discovered to be sesquitertian;
but that of the chord from which a weight of nine pounds was suspended,
to the chord which had the smallest weight [or six pounds,] to be
sesquialter. For 9 is to 6 in a sesquialter ratio. In like manner, the
chord next to that from which the smallest weight depended, was to that
which had the smallest weight, in a sesquitertian ratio, [for it was the
ratio of 8 to 6,] but to the chord which had the greatest weight, in a
sesquialter ratio [for such is the ratio of 12 to 8.] Hence, that which
is between the diapente and the diatessaron, and by which the diapente
exceeds the diatessaron, is proved to be in an epogdoan ratio, or that
of 9 to 8. But either way it may be proved that the diapason is a system
consisting of the diapente in conjunction with the diatessaron, just as
the duple ratio consists of the sesquialter and sesquitertian, as for
instance, 12, 8, and 6; or conversely, of the diatessaron and the
diapente, as in the duple ratio of the sesquitertian and sesquialter
ratios, as for instance 12, 9, and 6. After this manner, therefore, and
in this order, having conformed both his hand and his hearing to the
suspended weights, and having established according to them the ratio of
the habitudes, he transferred by an easy artifice the common suspension
of the chords from the diagonal stake to the limen of the instrument,
which he called _chordotonon_. But he produced by the aid of pegs a
tension of the chords analogous to that effected by the weights.
Employing this method, therefore, as a basis, and as it were an
infallible rule, he afterwards extended the experiment to various
instruments; viz. to the pulsation of patellæ or pans, to pipes and
reeds, to monochords, triangles, and the like. And in all these he found
an immutable concord with the ratio of numbers. But he denominated the
sound which participates of the number 6 _hypate_: that which
participates of the number 8 and is sesquitertian, _mese_; that which
participates of the number 9, but is more acute by a tone than mese, he
called _paramese_, and _epogdous_; but that which participates of the
dodecad, _nete_. Having also filled up the middle spaces with analogous
sounds according to the diatonic genus, he formed an octochord from
symphonious numbers, viz. from the double, the sesquialter, the
sesquitertian, and from the difference of these, the epogdous. And thus
he discovered the [harmonic] progression, which tends by a certain
physical necessity from the most grave [i. e. flat] to the most acute
sound, according to this diatonic genus. For from the diatonic, he
rendered the chromatic and enharmonic genus perspicuous, as we shall
some time or other show when we treat of music. This diatonic genus,
however, appears to have such physical gradations and progressions as
the following; viz. a semitone, a tone, and then a tone; and this is the
diatessaron, being a system consisting of two tones, and of what is
called a semitone. Afterwards, another tone being assumed, viz. the one
which is intermediate, the diapente is produced, which is a system
consisting of three tones and a semitone. In the next place to this is
the system of a semitone, a tone, and a tone, forming another
diatessaron, i. e. another sesquitertian ratio. So that in the more
ancient heptachord indeed, all the sounds, from the most grave, which
are with respect to each other fourths, produce every where with each
other the symphony diatessaron; the semitone receiving by transition,
the first, middle, and third place, according to the tetrachord. In the
Pythagoric octachord, however, which by conjunction is a system of the
tetrachord and pentachord, but if disjoined is a system of two
tetrachords separated from each other, the progression is from the most
grave sound. Hence all the sounds that are by their distance from each
other fifths, produce with each other the symphony diapente; the
semitone successively proceeding into four places, viz. the first,
second, third, and fourth. After this manner, therefore, it is said that
music was discovered by Pythagoras. And having reduced it to a system,
he delivered it to his disciples as subservient to every thing that is
most beautiful.[29]