I'm an artist working on an installation of photos.  For the
installation I need to stretch two cables horizontally for a great
distance and then clip photos snugly between the two.  For aesthetic
reasons it is very important that these cables can maintain a true (or
very close to true) horizontal level throughout their length.  The
total length is approximately 1,000 feet, and I would like to know how
often I will need to add new vertical brace units to keep the cable
vertical and taut, and what the formula would be to calculate (or
approximate) that number.  Thanks!

Hello rleitzell, thanks to our favorite search engine's new book
scanning project I was able to find your answer.

http://books.google.com/books?hl=en&id=qF3AxLlv1o4C&pg=PA196&lpg=PA196&dq=tension+in+horizontal+cable&prev=http://books.google.com/books%3Fq%3Dtension%2Bin%2Bhorizontal%2Bcable%26lr%3D&sig=2tJuXZa1c9TW__H34ISCfCs0X_0

The formula you need is:

f = (w x L^2) / (8 x H)

Where:
f is the vertical sag
w is the weight of the cable per unit horizontal length
L is the horizontal span between supports
H is the horizontal component of the cable tension

You don't say what you plan to use for cable. I will choose 3/8 IWRC
from this table:

http://www.wrca.com/wr_619_636.html

Looking in the table 3/8 IWRC weighs .26 #/ft. So that we will end up
with sag in inches we need to use weight in #/in. or .26/12 = .022
#/in.

Also, from the table the minimum breaking strength is 6.56 tons x
2,000 #/ton = 13,120#

So, if we tension the 3/8 cable to 13,120# we can calculate the
expected sag in a 100 foot = 1,200 inch span.

f = .022 x 1,200^2 / 8 x 13,120 = 0.3 inches

Notice that the sag increases as the square of the span. So,
increasing the span by a factor of two you will get 4 times the sag.

If you want to choose a cable size and type, ask for a clarification
and I will calculate the sag for you. Or, you may want to do the
calculation yourself. Either way I will be glad to answer any question
you might have.

Good luck with your art project, Redhoss

Do you mean 1,000 non-stop feet of horizontal cable?

I expect that you will have to use trial and error to solve this.  You
will definitely need many vertical supports if you want the photos to
hang without noticable gaps at the bottom.  Sounds like the project
would be very sensitive to wind.

3 Things:

1: Don't forget to include the weight of the pictures into the
equation above for the top cable (if they are hung) or half the weight
of the pictures into each cable if they are attached to both cables
(ie the weight of the pictures is evenly distributed into each cable).

2: Allow for adjustment, the cables WILL stretch.

3: You may also want to consider a substantial safty factor (sometimes
called design factor).  It is not wise to load the cable to its yield
strength (breaking strength) with people around.  A safty factor of 2
means that you would pretend the yield strength is half of the book
value (6.56 tons for 3/8 cable according to redhoss).  So your maximum
tension allowable would become 3.28 tons with a SF of 2 and 2.18 tons
with a SF of 3.  If this is displayed outside, exposed to the
elements, you must also worry about corrosion, increase your safty
factor accordingly.

For aesthetic reasons the total sag at center should be a FRACTION of
vertical dimensions of the pictures. Therefore in my opinion it is an
impossible task with a set of two wires over 1000 ft.
A set of three wire scenario will give a perfectly horizontal layout of pictures. 
To visualise this just imagine 
1. any suspension bridge most notably Golden Gate Bridge and its twin
decks http://www.goldengatebridge.org/
2. Overhead wires on Railway lines - catenary 
http://en.wikipedia.org/wiki/Image:Tile_Hill_train_550.jpg

The catenary wire size reduces drastically if catenary profile is
allowed to be deep. To add aesthetics to the scene, the drop wires can
be transparent and very thin as in fishing lines.

Xplusak-ga,
That certainly sounds like a better solution that vertical supports,
especially when one considers that the underlying ground will almost
surely not be perfectly level, so that adjusting their heights to
provide support for a "perfectly" straight cable would be extremely
difficult.  Of course, adjusting the droplines would also be a tedious
project, and the "pylons" for the catenary cable and their support
will also be a problem.  1000 meters - we are talking about a major
suspension bridge span.  And you have 1000 meters of photos ...?

(I am gaining respect for Christo!)

If the project does materialize, I hope the construction reaps
appropriate recognition.

First, It is 1,000 ft ... not 1,000 meters. This simplifies the
problem by a factor 3 raised to power ... something ... > 2; I mean
substantially. That difference keeps the probelm into light
engineering domain as against heavy engineering project.

Second, Drop lines height adjustment is to be done on lower end of
drop lines. Their initial length (estimate) can be calculated once the
profile of catenary is fixed (designed). So adjustment happens at the
same level where you hang the pictures, I mean no issues about
accessibility. Fine tuning of drop lines heights to get a horizontal
layout of pictures will be necessary otherwise it gets complex.

Third, Even after considering the catenary type design, the cable for
hanging the picture need be under good tension, if number of drop
lines are to be limited, else this cable can be same as fishing line
used for drop lines.

Fourth all aspects discussed above can be put under calculations.

Finally, one aspect missing from original question - it is outdoors?
You have to consider winds and it get murky there.