Some are a little deaf in their greek ears.

Hear is a an aid to Translating the greek;

Google  "classic greek sculpture discovery", select images, and study them until you can recognize the drape of  clothing in each period.

Then, take your linen tester to the mall (with branch of Needless Markup Department Stores) and do thread counts on wool and linen fabrics that have drape similar to that seen in the fabrics of various periods of Greek Sculpture.

Now, hand spin/hand weave fabrics with that thread count and which have the drape of the fabrics produced in the various periods.   (It is hard to find such nuanced yarns on the commercial market, and ordering spun yarns from a spinner gets expensive.)  However, now you know how fine those hand spun/hand woven Classic Greek fabrics were.  And by now, you will have moved from your single beam warp weighted loom to a double beam loom with (linen) heddles.

With the appropriate use of warp extensions, aprons, lease sticks w/ crosses substituting for warp sticks, and DRS spinning technology,  a fabric sample large to see its drape can be spun, warped, and woven in a few hours.

Now that you know the specifications of Classic Greek Weaving, I expect to shortly see pix of the fine wool Greek and Roman togas that you have hand/ spun and  hand/ woven.

What comes after bragging rights:

http://www.wikihow.com/Put-on-an-Authentic-Roman-Toga

The toga originated from an Etruscan garment called the “tebenna.” The word toga comes from Latin “tegere,” which means “to cover.”

This means that the  Etruscan civilization also had fine weaving, not likely produced on single beam/ warp weighted looms.

Then, there was the Old Kingdom Egyptians weaving their very fine linen on what kind of a loom?For that we look to Roth, noting Figure 37.  It is a 2-beam horizontal loom that uses warp weights. And we note the caption of Figure 9.  We also consider that we do not know the fineness of the fabric produced on the loom illustrated in Figure 36.  Moreover, since Roth had to bring in a textile professional to produce the fabric, we know that Roth does not have a high competence on textile production.

For context on the age of old looms see: https://pdfs.semanticscholar.org/965c/1465787024fd40537ed05f47c65098466fed.pdf

 And look to: The Book of Looms: A History of the Handloom from Ancient Times to the Present by Broudy pg 13 for discussion of  weaving wool at 30 by 38 threads per inch, circa 6,000 years ago.   On page 26, he touches on the Greek loom, and on pg 38, he gets to the horizontal loom.

7,000 year old loom  in Bulgaria; http://archaeologyinbulgaria.com/2016/01/18/archaeologists-find-wooden-wall-four-leaf-clover-amulet-in-prehistoric-settlement-mound-in-bulgarias-petko-karavelovo/

see also ; 

Prehistoric Textiles: The Development of Cloth in the Neolithic and Bronze ...
By E. J. W. Barber pg 271

Do you really think that as metallurgy improved, people did weaving the same way for 3,000 years?

?? http://quod.lib.umich.edu/k/kelsey/x-0000.00.3876/0000_00_3876
?? http://quod.lib.umich.edu/k/kelsey/x-0000.00.3878/0000_00_3878
?? http://quod.lib.umich.edu/k/kelsey/x-0000.02.6544/0000_02_6544

Good Old S. McGee-Russel would have accepted my deductions from the drape of fabric on Greek sculpture to be adequate to demonstrate my point Classic Greek weaving technologies.  He expected his "critters" to go out, and measure stuff, and make deductions that could not be taken from direct observations.  He expected us to know our physics, and chemistry, and calculus.  He expected us to know the world.  He expected us to take risks, and sometimes make mistakes and errors.  If we were not taking the  risks necessary to move the science, we  could not be promoted from "bugs" to "critters".

If you expect to prove everything from step to step without leaps of insight, then you will never move your science or technology forward.  Tomorrow, I intend to make better textiles. I will hand spin Better, Faster, Cheaper.  I will hand weave Better, Faster, Cheaper.  I am not content to stagnate.  I will take risks. I will seek to leap forward. I will advance more than I fall back.