QuadriviuM Mathematics and
Metaphysics in the Premodern World Assignment 2b Due:
Thursday, February
7^{th} 

Do the following for
Thursday...
Read the following: 
folios 
Plato.
PlatoÕs Cosmology: The Timaeus of Plato. Translation, notes, and
commentary by Cornford. New York:
Liberal Arts Press, 1937. Excerpts. (ca. 30pp of English) 5673:
Bury's translation: Creation, WorldSoul, means, Cosmic Blacksmith 5761,
6675: Cornford's translation of roughly the same section with
commentary. [page number
coincidence] Image:
Chalcidius (Calcidius), and Plato. Commentarius
in Platonis Timaeum. Chalcidius (Calcidius)  MS from Bodlian. 12th c. 
10.5 
Do the following: (just thinking at this point) 
1. As you read these excerpts from Timaeus, think about all the musical math. Think about how the cosmos of Timaeus fits with the cosmos of "The
Myth of Er" from the Republic. Try to do some of the math. Your assignment for next Tuesday will
involve doing a lot of this type of math, so get ready. Play around with the various types of
means (see below). 
Notes on
Bury translation (pp.
5673): Things get mathematically confusing starting on p. 68. Don't get discouraged. [The exercises that I will hand out on
Thursday will help with this section.]
Plato gets back to being only sortofconfusing by p. 71.
Immediately
after p. 73 of Bury's translation I inserted Cornford's translation and
commentary of the same material which have page numbers very close to that of
Bury's. This is a total
coincidence. (...or is it?)
Notes on
Cornford's translation and commentary (start
on p. 57, "The World Soul" and read to 61, skip pp. 6265, then read
6675) This reading can get pretty
mathy and it is rather cryptic.
Cornford's commentary can be useful and it can also be aggravating. If
you rush the reading it will all be aggravating. Savor it like a good singlemalt. Do what you can with it. I'll try to fill in as much as I can in
class.
Notes for inclass material.
If you sing DoSol and then Soldo,
you are going up a fifth (3/2) and then up a fourth (4/3) which combine to be
an octave. It seems like you are adding a fifth to a fourth, but mathematically
you are multiplying.
which is an octave.
So... if going up in intervals is
multiplication. What do you think
going down is? Division, of course.
Go up a fifth, DoSol, then go down a fourth,
SolRe.
The difference between a fifth and a fourth is
This is the definition of a tonus, or tone. It is not technically consonant.
In Timaeus (pp. 6971 in Bury trans. and pp. 6872 in Cornford) Plato
divides all tetrachords, fourths, into two tones (toni) and a semitone, which is defined as follows....
A tetrachord is a fourth,
4/3,... DoFa. [Here comes the
bride.]
It is divided into tonustonussemitonus. Here's how...
Solve for x.
= the semitone.
Check it:
So a
full diatonic scale over an octave
is this:
Do 
Re 
Mi 
Fa 
Sol 
La 
Ti 
do 


9/8 
9/8 
256/243 
9/8 
9/8 
9/8 
256/243 



tone 
tone 
semi 
tone 
tone 
tone 
semi 



tetrachord 
tone 
tetrachord 



fifth (3/2) 
fourth (4/3) 



fourth (4/3) 
fifth (3/2) 


Two tetrachords with a tone
inbetween.
...or a fifth and a fourth.
Plato also uses the harmonic and
arithmetic means to fill up the intervals between his geometric sequence:
1, 2, 3, 4, 8, 9, 27.
The resultant intervals all
superimpose exactly onto a 4+ octave diatonic scale like the one above.
Arithmetic Mean 
c is
linearly equidistant from both a and b. 
The arithmetic mean is
"equidistant between both extremes." AlbertiÕs example: a
: c : b = 4 : 6 : 8 The
common average. 
Geometric Mean 
a x b is a rectangle, c is the
square with the same area. 
"Éthe
dimension of a side [c] that generates a square of equal [area]" to that
of a rectangle with sides measuring a and b. AlbertiÕs example: a : c : b = 4 : 6 : 9 
Harmonic Mean 

"The proportion between
the shortest and the longest dimensions is the same [proportion] as that
[quantity] between the shortest and the middle, and É that [quantity] between
the middle one and the longest." The ratio between the outer numbers
equals the ratio between the differences between the numbers. AlbertiÕs example: a : c : b = 30 : 40 : 60 
The quotations on the right
are from Leon Battista Alberti's description of the means from
De re aedificatoria (1452, On
the Art of Building)