Download Er`s Cosmos One of the most influential and fully

This draft excerpt is a from a larger work I wrote entitled, “Harmonic Structures in Kepler’s World.”
Er’s Cosmos
One of the most influential and fully developed musical
cosmologies known to early modern Europeans was Plato’s (ca.
427 - ca. 348 BC) Myth of Er from the Republic.1 It is to this
model that the terms, “heavenly” or “crystalline spheres,” usually
refer.2 This description of cosmic architecture begins with a
rainbow-like shaft or spindle made from adamant,3 an extremely
hard, legendary material with properties similar to diamond and/or
lodestone. This shaft pierces through the centers of eight perfectly nested “whorls,” which are made
from mixtures of adamant and other substances. 4
Aside: Here are some actual “whorls,” (sphandulos (!"#$%&'()) in Greek).
They are used for spinning yarn or thread.
Each one is about 2” in diameter.
I have also seen spherical whorls, but I do not have any pictures.
Each whorl, oftentimes in translation called a “sphere,” is nested one inside another like a Russian
Doll, and each carries a planet5 except for the largest which carries the fixed stars. Each of the eight
1
Plato and Francis Macdonald Cornford, The Republic of Plato, trans. Francis Macdonald Cornford (Oxford: Clarendon
Press, 1941), pp. 340-350. The full chapter title is “The Rewards of Justice After Death, The Myth of Er.”
2
A crystalline sphere studded with stars was previously mentioned by Anaximenes of Miletus (ca.550-ca.475) who was the
teacher of Anaxagoras of Clazomenai (ca.500-ca.428) who flourished in Athens. Anaximenes may also have been a
teacher of Pythagoras.
3
Question: What are the Wolverine’s (X-men) claws made of? Adamantium of course.
4
A sphandulos (!"#$%&'()) is the word usually translated as ‘whorl’ or ‘whirl.’ It also means vertebra or the part of a
top, as in dreidel, that is not the spindle.
5
Planets in this context are the five visible planets (Mercury, Venus, Mars, Jupiter, and Saturn) along with the sun and the
moon. Plato describes a geocentric system.
This draft excerpt is a from a larger work I wrote entitled, “Harmonic Structures in Kepler’s World.”
whorls has an exposed edge or rim, suggesting that they are shaped like bowls or partial spheres (or
like and actual sphandulos as shown above). These circular rims all fit together concentrically and
form a concave shape that I imagine to be something like a stadium.6 See Figure 8.7 The rim of each
whorl is described as having a color and luminous quality (for example the rim for Mars is reddish)
and each also revolves on the shared axis/spindle of adamant in a contrary direction to the daily
rotation of the fixed stars. Each rim is also described as having a particular breadth which is
qualitatively represented in Figures 8 or 9. It is not immediately clear in the Myth of Er why Plato has
chosen this relative arrangement of rim dimensions, but in a notoriously cryptic passage in his book,
Timaeus, there is found a similar circular arrangement for the cosmos derived from “the proportions of
musical harmonia.”8 The description of this harmonia in the Timaeus is lifted directly out of the
mathematical-musical philosophy of his contemporary Archytas of Tarentum, the Pythagorean from
the 4th century B.C. Among other things, Archytas developed theories of tone production as a function
of motion and several varieties of musical scales derived from superparticular ratios9. He arrived at the
musical intervals of fifths (3:2), fourths (4:3), octaves (2:1), tones (9:8), and semitones (256:243) using
not only the stock superparticular ratios, 4:3:2:1, and their interrelations as can be seen in Figure 7,
(Figure 7 has been reproduced at the end of this excerpt) but also came upon them through establishing
6
According to Geminus of Rhodes (fl. ca. 60 BC), the Pythagoreans were the first to assume that the motions of the sun,
moon, stars, and planets were circular and uniform, though I imagine he was biased, being a Pythagorean himself. See Olaf
Pedersen, A Survey of the Almagest (Odense: Odense Universitetsforlag, 1974), p.34. Pederson cites Geminus’ Elementary
Astronomy I, 19. I speculate that Plato’s nested spherical arrangement has this “bite” taken out so that outside forces can
have access to each sphere. Without the “bite” removed the entire system would be sealed up within the sphere of fixed
stars. This vaguely brings to mind the 17th century debates between Newton and Leibniz (among others) over the role of
God in a “clockwork universe.” This also vaguely brings to mind the Death Star.
7
I have depicted Er’s cosmos as viewed from outside. It could be argued that this is not possible since there is no outside
in relation to this cosmos.
8
See Cornford’s commentary in Plato and Francis Macdonald Cornford, Plato's Cosmology: The Timaeus of Plato, trans.
Francis Macdonald Cornford, International Library of Psychology, Philosophy, and Scientific Method (London: Routledge
& Kegan Paul Ltd., 1966), p. 57. It is interesting to note that these parts in The Timaeus (Cornford’s translation) are filled
with blacksmithing analogies. Legend had it, Pythagoras noticed the pitch-mass relationship when listening to blacksmiths.
For an example of this legend, see Franchinus Gaffurius and Walter Kurt Kreyszig, The Theory of Music [Theorica
Musice], ed. Claude V. Palisca, trans. Walter Kurt Kreyszig (also introduction and notes), Music Theory Translation Series
(New Haven: Yale University Press, 1993, original 1492), pp. 45-48. Oddly enough this legend makes an error in the masspitch relationship. Vincenzo Galilei, Galileo Galilei’s father, comments on this error. See Claude V. Palisca, Humanism in
Italian Renaissance Musical Thought (New Haven: Yale University Press, 1985), p. 270.
9
Superparticular ratios are those of the form, (n+1)/n, where n is a whole number.
This draft excerpt is a from a larger work I wrote entitled, “Harmonic Structures in Kepler’s World.”
the arithmetic and harmonic means between the intervals of 2n where n is an integer or zero (i.e., 20, 21,
22, 23, …= 1, 2, 4, 8, …) and the intervals between 3n (i.e. 1, 3, 9, 27, … etc.).10 These mean
derivations of harmonic relationships are yet another manifestation of mathematics found lurking in
natural phenomena and they are exactly the derivations used by Plato in his Timaeus, though it is not at
all obvious how these intervals were manifested physically. See Table 1 for a list summarizing these
attributes and Figure 9 for a qualitative depiction of the rim dimensions.
Whorl or Sphere:
1st
Smallest to Largest
(8th)
“Sphere” of… Moon
Relative breadth of
4th
each rims
Color of each rim reflected
from Sun
Rotational speeds fastest
contrary to the
Fixed Stars
2nd
(7th)
Sun
5th
3rd
(6th)
Venus
2nd
4th
(5th)
Mercury
6th
5th
(4th)
Mars
3rd
brightest 2nd whitest yellowish pale red
fast
fast
fast
medium
6th
(3rd)
Jupiter
7th
whitest
7th
(2nd)
Saturn
8th
thinnest
yellowish
8th
(1st)
Fixed Stars
1st
fattest
multicolored
slow
slowest
daily rotation;
lap of Necessity
(or Constraint)
Table 1; Cosmic Information from Plato’s Myth of Er
Figure 9, Rim Relationships with Fates
In Plato’s system, the fixed stars rotate over head in a “twenty-four-hour” period. All other
heavenly bodies are viewed in relation to these fixed stars. So, presumably, the moon will finish its
10
I am using modern notation and concepts to more succinctly explain these ideas. The arithmetic mean between two
2ab
a+b
numbers is what most people now consider an average; ma =
. The harmonic mean, mh =
, is no doubt less
a+b
2
familiar. For example, the arithmetic and harmonic means between 1 and 2 (an octave, 1:2) are 3/2 and 4/3, the fifth and
"3 4
9 8
%
the fourth. The ratio between these two means is 9/8, $ : = : = 9:8' . Archytas’ method of constructing a scale using
#2 3 6 6
! &
!
these methods is clever and also demonstrates the playful nature of Greek mathematics and their fascination with recurring
ratios.
!
This draft excerpt is a from a larger work I wrote entitled, “Harmonic Structures in Kepler’s World.”
cycle through the fixed stars more quickly than the sun, and much more quickly than Saturn, which
makes its circuit among the fixed stars very slowly.11 This system adequately explains in broad strokes
the apparent “wanderings” of the planets through the stars, but it does not explain the more subtle
behaviors such as seasons, retrograde motions or non-uniform orbital velocities.12 It did, however,
incorporate a different set of features that continued to inspire philosophers for more than two
millennia. Plato writes in the Myth of Er, “Upon each of its circles [rims] stood a Siren, who was
carried round with its movement, uttering a single sound on one note, and that all the eight made up the
concords of a single scale.”13 In addition to these singing sirens there were the three Fates, “the
daughters of Necessity,” sitting on thrones equally spaced around the rims. See Figure 9.14 Each Fate
chants to the tones made by the sirens: “Lachesis of things past, Clotho of the present, and Atropos of
things to come.”15 The Fates are also responsible for regulating the rotations of the revolving whorls.
Plato writes, “…from time to time Clotho lays her right hand on the outer rim of the Spindle [the rim
of the sphere of fixed stars] and helps to turn it.” Clotho is apparently responsible for maintaining the
daily rotation of the heavens. “Atropos turns the inner circles likewise with her left [in a direction
contrary to the fixed star revolution], and Lachesis with either hand takes hold of inner and outer
alternately.”16
Before the advent of mechanical clocks the cyclic movements of the heavens were the principle
rhythms used in keeping time. Such simple periodic occurrences as the day-night cycle or the
11
The fact that our earth has a daytime and nighttime (caused by the sun’s orbit, assuming a geocentric perspective) is, of
course, a constant annoyance making the stars invisible roughly half of the time. This makes observations of Mercury, who
is always wandering in the neighborhood of the sun, particularly difficult.
12
When a planet exhibits retrograde motion it appears to stop and move backwards and then stop and continue in its
original motion. This phenomena is most noticeable with the inner planets. In his Timaeus, Plato states his awareness of
the irregularities of the motions of the outer planets but claims that few men have measured directly their wanderings,
“bewildering as they are in number and of surprisingly intricate pattern.” See Plato and Cornford, Plato's Cosmology: The
Timaeus of Plato, pp. 105-119.
13
Plato and Cornford, The Republic of Plato, p. 346.
14
For the Fates to be equally spaced we can assume that they are separated from one another by 120˚.
15
From Joscelyn Godwin, ed., Music, Mysticism, and Magic: A Sourcebook (London; New York: Routledge & Kegan Paul,
1986), p. 6.
16
Plato and Cornford, The Republic of Plato, p. 346.
This draft excerpt is a from a larger work I wrote entitled, “Harmonic Structures in Kepler’s World.”
changing seasons or lunar phases are manifestations of astronomical rhythms.17 More accurate
measurements of these periodic cycles can be made by the use of a gnomon (a sun dial) or an astrolabe
combined with the keeping of detailed records. For Plato, Clotho is the Fate who chants of the present
and regulates the daily rhythm of the outermost sphere of fixed stars, the basis of all motion in the
heavens. This motion is the most noticeable of all the cycles and is responsible for the “24 hour” daynight cycle.18 The sun, moon, and planets may all have their own subtle motions, but their gross
motions are determined by the sphere of the fixed stars inside of which they are all carried round.
With her right hand Clotho regulates the speed of rotation of this sphere as if pushing a merry-goround or a roulette wheel.
Atropos with her left hand is responsible for regulating the rotations of the inner spheres. In a
sense she is maintaining the speed of a roulette ball, or in this case 7 balls with differing radial
trajectories, thrown against the spin of the outer roulette wheel. Atropos is the Fate who chants of the
future and perhaps her responsibilities reflect this. By knowing how she motivates each inner sphere,
predictions of the future could be made. For example, she regulates the movements of the sun’s sphere
within the larger starry sphere. Though the sun’s daily appearance was controlled by Clotho, the more
subtle manipulation of the suns longer term movements was under the control of Atropos. She thus
controlled the seasonal cycle, a very important thing to be able to predict for an agrarian society.
Interpreting the role of Lachesis, who chants of the past and alternately controls the inner and outer
spheres both with her left and right hands (allowing for acceleration and deceleration) is somewhat
more problematic, but nonetheless possible. I think it reasonable to assume that Plato is referring to
17
Where there is a steady rhythm, there is a clock. The rhythmic quality of astronomy is another musical element that has
yet to be investigated so far as I am aware. As luck would have it, the vibrational theory of sound, developed in the 17th
century, was and still is based on rhythmic principles. The equivalence of the macrocosm to the microcosm is again
reinforced.
18
Plato writes in the Timaeus, “In virtue, then, of this plan and intent of the god for the birth of Time, in order that Time
might be brought into being, Sun and Moon and five other stars-- ‘wanders’, as they are called [wanderers meaning
planets]-- were made to define and preserve the numbers of Time.” Plato and Cornford, Plato's Cosmology: The Timaeus
of Plato, p. 105.
This draft excerpt is a from a larger work I wrote entitled, “Harmonic Structures in Kepler’s World.”
what were the less predictable motions in the heavens: retrograde motions, and apparent speed
variations of the planets and the sun. As Plato mentions in his Timaeus, in section 38C-39E as titled
by Cornford, “The Planets as instruments of Time,” The sun and the moon have definite cycles
demarcating the year and the month, but the other planets he distinguishes from the sun and the moon
by calling them “wanderers.”19 About these wanderers he writes, “The periods of [the 5 planets] have
not been observed by men, save for a few; and men have no names for them,20 nor do they measure
one against another by numerical reckoning. They barely know that the wanderings of these others are
time at all, bewildering as they are in number and of surprisingly intricate pattern.”21 Because the
wanderers have such intricate movements, probably referring to retrograde motions and the like, they
are not useful for predictions into the future. Their movements are not understood by “numerical
reckoning.” Thus I suspect they are more likely useful for locating events in the past, hence their
association with Lachesis, who chants of the past. For example, Mars may have been in Libra when
an earthquake hit Athens x number of years ago. Since ‘they’ do not “measure one against another”
(wanderer/planet against sun or moon) it precludes predicting where and when a wanderer will be in
relation to the predictable times and places of the sun and the moon. Concerning the specific actions
of Lachesis, Plato is admittedly vague.
Combined with the cosmos as described in the Timaeus it is possible that the tones chanted by each
Siren were determined by the harmonic radii discussed above and several modern scholars have
explored this connection.22 See Table 2 for a reconstruction of Plato’s cosmos if read as a pitch-radius
19
*'+$,-ı) = planetos
… no names for their periods such as month or year.
21
Plato and Cornford, Plato's Cosmology: The Timaeus of Plato.
22
See Godwin, ed., Music, Mysticism, and Magic: A Sourcebook, pp. 295-299, Joscelyn Godwin, ed., The Harmony of the
Spheres: A Sourcebook of the Pythagorean Tradition in Music (Rochester, Vt.: Inner Traditions International, 1993), pp.
405-406, Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Stony Brook, New York: Nicolas Hays,
Ltd., 1978), pp. 47-70, Bruce Stephenson, The Music of the Heavens: Kepler's Harmonic Astronomy (Princeton, N.J.:
Princeton University Press, 1994), pp. 19-20. Directly merging the cosmos from the Myth of Er with that from the Timaeus
does not appear to be an easy one-to-one task.
20
This draft excerpt is a from a larger work I wrote entitled, “Harmonic Structures in Kepler’s World.”
system.23 Another plausible argument could also be made that the pitches are related to the orbital
speeds similar to a model by Cicero. (Cicero’s description is not discussed in this excerpt.)
“Sphere” of…
Radii measured from
earth’s center
Reconstructed intervals
due to preceding planet
Musical intervals
lowest to highest
earth
n.a.
Moon
1
sun
2
Venus
3
Mercury
4
Mars
8
Jupiter Saturn Fixed Stars
9
27
?
silent
1
1:2
2:3
3:4
4:8
8:9
9:27
(1:0)
(1:2)
(1:3)
n.a.
unison? octave
fifth
fourth
octave
tone
octave
tonus?
+ fifth
Table 2, Stephenson’s Reconstruction of Plato’s Musical Cosmos
27:?
?
The Myth of Er is not simply a pagan description of the celestial mechanism. The Myth of Er is
also a Last Judgment-like story in which the deeds of men are evaluated and suitable punishments or
rewards meted out. Though I shall not go into any further detail, the astrological and Christian
interpretations that this model inspired should not go unacknowledged.
Here is an illustration referred to in the text which was not part of this excerpt. What is shown is a side
view of a monochord (like a guitar with one string) and the placement of frets that would sound out a
“Pythagorean scale.”
Figure 7, Eratosthenes’ Diatonic Scale (Ptolemy’s Diatonic Ditoniaion or ‘Pythagorean Intonation’) 24
23
This table is taken directly from Stephenson, p. 20. I am very dubious of this interpretation. Not only are the radii
presented in a nonstandard order (for reasons having to do with its derivation, this series is usually written 1,2,3,4,9,8,27)
but Plato makes no explicit indication that this was his concept. Stephenson, as well, makes it clear that he is not
committed to this chart.
24
James Murray Barbour, Tuning and Temperament: A Historical Survey, 2d ed. (East Lansing: Michigan State College
Press, 1953), pp. 16-21. Also note that (9/8)5.(256/243)2 = 2 (Barbour)
This draft excerpt is a from a larger work I wrote entitled, “Harmonic Structures in Kepler’s World.”
Bibliography for this excerpt
All illustrations and photos were made by the author.
Barbour, James Murray. Tuning and Temperament: A Historical Survey. 2d ed. East Lansing: Michigan State
College Press, 1953.
Gaffurius, Franchinus, and Walter Kurt Kreyszig. The Theory of Music [Theorica Musice]. Translated by
Walter Kurt Kreyszig (also introduction and notes) Music Theory Translation Series, ed. Claude V.
Palisca. New Haven: Yale University Press, 1993, original 1492.
Godwin, Joscelyn, ed. Music, Mysticism, and Magic: A Sourcebook. London; New York: Routledge & Kegan
Paul, 1986.
________, ed. The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music. Rochester,
Vt.: Inner Traditions International, 1993.
McClain, Ernest G. The Pythagorean Plato: Prelude to the Song Itself. Stony Brook, New York: Nicolas Hays,
Ltd., 1978.
Palisca, Claude V. Humanism in Italian Renaissance Musical Thought. New Haven: Yale University Press,
1985.
Pedersen, Olaf. A Survey of the Almagest. Odense: Odense Universitetsforlag, 1974.
Plato, and Francis Macdonald Cornford. The Republic of Plato. Translated by Francis Macdonald Cornford.
Oxford: Clarendon Press, 1941.
________. Plato's Cosmology: The Timaeus of Plato. Translated by Francis Macdonald Cornford International
Library of Psychology, Philosophy, and Scientific Method. London: Routledge & Kegan Paul Ltd.,
1966.
Stephenson, Bruce. The Music of the Heavens: Kepler's Harmonic Astronomy. Princeton, N.J.: Princeton
University Press, 1994.