Pythagoras: His Life and Teaching, a Compendium of Classical Sources (23 page)

CHAPTER 2

F
IRST
M
USIC IN THE
P
LANETS

T
he names of sounds, in all probability, were derived from the seven stars, which move circularly in the heavens and compass the earth.
⁵⁹¹
(The circular motion of these bodies must of necessity cause a sound, for air being struck from the intervention of the blow sends forth a noise. Nature herself constraining that the violent collision of two bodies should end in sound.
⁵⁹²
)

Now (say the Pythagoreans) all bodies which are carried round with noise—one yielding and gently receding to the other—must necessarily cause sounds different from each other, in the magnitude and swiftness of voice and in place. These, according to the reason of their proper sounds, or their swiftness, or the orbs of repressions, in which the impetuous transportation of each is performed—are either more fluctuating, or on the contrary, more reluctant. But these three differences of magnitude, celerity, and local distance, are manifestly existent in the planets. These planets are constantly with sound circling around through the aetherial diffusion, whence every one is called
[star], as void of
station; and
, always in course; whence God and Aether are called
and
.
⁵⁹³

Moreover the sound which is made by striking the air induces into the ear something sweet and musical, or harsh and discordant. For if a certain observation of numbers moderates the blow, it effects a harmony consonant to itself; but if it be haphazard and not governed by measures, there proceeds a troubled unpleasant noise which offends the ear.
⁵⁹⁴
Now in heaven nothing is produced casually or randomly; but all things there proceed according to divine rules and settled proportions. Whence it may be irrefutably inferred that the sounds which proceed from the conversion of the Celestial Spheres are musical. For sound necessarily proceeds from motion—and the proportion, which is in all divine things causes the harmony of this sound. This Pythagoras, first of all the Greeks, conceived in his mind. He understood that the Spheres sounded something concordant because of the necessity of proportion which never forsakes celestial beings.

From the motion of Saturn, which is the highest and furthest from us, the gravest sound in the
diapason concord
is called
Hypate
, because
signifys highest. From the Lunary, which is the lowest and nearest the earth,
neate
, for
signifys lowest.
⁵⁹⁵
From those which are next these, viz. from the motion of Jupiter who is under Saturn,
parypate;
and of Venus, who is above the Moon,
paraneate.
Again, from the middle, which is the Sun's motion, the fourth from each part,
mese
, which is distant by a
diatessaron
in the
Heptachord
from both extremes according to the ancient way; as the Sun is the fourth from each extreme of the seven planets, being in the middle. Again, from those which are nearest the Sun on each side: from Mars who is placed between Jupiter and the Sun,
hypermese
, which is likewise termed
lichanus;
and from Mercury who is placed between Venus and the Sun,
paramese.

Pythagoras, by Musical proportion, calls that a Tone by how much the Moon is distant from the Earth; from the Moon to Mercury the half of that space; and from Mercury to Venus almost as much. From Venus to the Sun
sesqidulple;
from the Sun to Mars a tone (that is as far as the Moon is from the Earth); from Mars to Jupiter half; and from Jupiter to Saturn half; and thence to the Zodiac sesquiduple. Thus there are made seven tones: which they call a Diapason harmony, that is an universal concord, in which Saturn moves in the
Doric
mood, Jupiter in the
Phrygian
, and in the rest the like.
⁵⁹⁶

The sounds made by the seven planets and the Sphere of Fixed Stars, and that which is above us (termed by them Antichthon), Pythagoras affirmed to be the Nine Muses. But the composition and symphony, and, as it were, connection of them all—whereof as being eternal and unbegotten, each is a part and portion—he named Mnemosyne.

CHAPTER 3

T
HE
O
CTOCHORD

N
ow Pythagoras, first of all, left the middle sound by conjunction, being itself compared to the two extremes, should render only a diatessaron harmony, both to the neate and to the hypate. But that we might have greater variety, the two extremes making the fullest concord each to other, that is to say the concord of diapason, which consists in a double proportion.
⁵⁹⁷
Inasmuch as it could not be done by two Tetrachords, he added an eighth sound, inserting it between the mese and paramese—setting it from the mese a whole tone, and from the paramese a semitone. So that which was formerly the paramese in the Heptachord is still the third from the neate, both in name and place. But that which was now inserted is the fourth from the neate, and has a harmony unto it of diatessaron—which before, the mese had unto the hypate.
⁵⁹⁸

But the tone between them, that is the mese, and the inserted called the paramese, instead of the former, to whichever Tetrachord it be added, whether to that which is at the hypate, being of the lower; or to that of the neate, being of the higher; will render diapente concord. This is either way a system consisting both of the Tetrachord itself, and the additional tone; as the diapente-proportion (viz.
sesquialtera)
is found to be a system of
sesquitertia
, and
sesquioctava;
the Tone therefore is sesquioctava.
⁵⁹⁹
Thus the interval of four chords, and of five, and of both conjoined together, called diapason, and the tone inserted between the two Tetrachords, being after this manner apprehended by Pythagoras, were determined to have this proportion in numbers.

CHAPTER 4

T
HE
A
RITHMETICAL
P
ROPORTIONS OF
H
ARMONY

P
ythagoras is said to have first found out the proportion and concord of sounds one to another: the Diatessaron in sesquitertia, the Diapente in sesquialtera, the Diapason in duple.
⁶⁰⁰
The occasion and manner is related by Censorinus,
⁶⁰¹
Boethius,
⁶⁰²
Macrobius,
⁶⁰³
and others; but more exactly by Nicomachus
⁶⁰⁴
thus:

Being in an intense thought, whether he might invent any instrumental help for the ear, solid and infallible—such as the sight has by a compass, and a rule, and by a diopter; or the touch by a balance, or by the invention of measures—as he passed by a smith's shop, by a happy chance he heard the iron hammers striking upon the anvil, and rendering sounds most consonant one to another in all combinations except one. He observed in them these three concords: the diapason, the diapente, and the diatessaron. But that which was between the diatessaron and the diapente, he found to be a discord in itself, though otherwise useful for the making up of the greater of them (the diapente).

Apprehending this to come to him from God as a most happy thing, he hastened into the shop. By various trials he found the difference of the sounds to be according to the weight of the hammers—and not according to the force of those who struck, nor according to the fashion of the hammers, nor according to the turning of the iron which was in beating out. Having taken exactly the weight of the hammers,
⁶⁰⁵
he went straightaway home. He tied four strings of the same substance, length, swiftness, and twist
⁶⁰⁶
to a beam on one side of the room, and then extended and fastened the other end of the strings to the wall on the other side of the room (lest any difference might arise from thence, or might be suspected to arise from the properties of several beams). Upon each of them he hung a different weight, fastening it at the lower end, and making the length of the strings altogether equal. Then striking the strings by two at a time interchangeably, he found out the aforesaid concords, each in its own combination.

For that which was stretched by the greatest weight, in respect of that which was stretched by the least weight, he found to sound a diapason. The greatest weight was of twelve pounds, the least of six. Thence he determined that the diapason did consist in double proportion, which the weights themselves did show. Next he found that the greatest to the least but one, which was of eight pounds, sounded a diapente. Whence he inferred this to consist in the proportion called sesquialtera, in which proportion the weights were one to another. But unto that which was less than itself in weight, yet greater than the rest, being of nine pounds, he found it to sound a diatessaron. He discovered that proportionably to the weights, this concord was sesquitertia, which string to nine pounds is naturally sesquialtera to the least. For nine to six is so (viz. sesquialtera) as the least but one, which is eight, was to that which had the weight six, in proportion sesquitertia. And twelve to eight is sesquialtera. And that which is in the middle between diapente and diatessaron, whereby diapente exceeds diatessaron, is confirmed to be in sesquioctava proportion, in which nine is to eight. The system of both was called Diapente, that is, both of the diapente and diatessaron joined together, as duple proportion is compounded of sesquialtera and sesquitertia, such as are two, eight, six. Or on the contrary, of diatessaron and diapente, as duple proportion is compounded of sesquitertia and sesquialtera, as twelve, nine, six being taken in that order.

Applying both his hand and ear to the weights which he had hung—and by them confirming the proportion of the relations—he did ingeniously transfer the common result of the strings upon the cross-beam, to the bridge of an Instrument, which he called
[“Stretched with strings”]. And as for stretching them proportionably to the weights, he did transfer that to an answerable screwing of the pegs. Making use of this foundation as an infallible rule, he extended the experiment to many kinds of instruments: cymbals, pipes, flutes, monochords, triangles, and the like. And he found, that this conclusion made by numbers was consonant without variation in all.

That sound which proceeded from the number six, he named hypate; that which from the number eight, mese, being sesquitertia to the other; that from nine, paramese, being a tone sharper than
the mese, viz. sesquioctave; that from twelve, neate. And supplying the middle spaces according to the diatonic kind, with proportional sounds, he so ordered the Octochord with convenient numbers: duple, sesquialtera, sesquitertia, and (the difference of these two last) sesquioctava.

Arithmetical proportions of harmony
From Thomas Stanley,
The History of Philosophy

Thus he found the progress by a natural necessity from the lowest to the highest according to the diatonical kind; from which again he did declare the Chromatic and Enharmonic kinds.