ancient Greek intervals (was: [metatuning] Music of the distant future)

(i realize that this thread belongs on metatuning, but
the recent responses are pushing it back into tuning-list
territory ... i've altered the subject line, and if
any responses to this post should be posted to tuning.
i'm sending a copy of this to tuning.)

--- In metatuning@yahoogroups.com, "stephenszpak" <stephen_szpak@...>
wrote:
>
> --- In metatuning@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> >
> >
> > My point was that these ratios have been understandable for
> > a long time. The ancient Greeks would have had no trouble
> > defining what they are, or what it means for a number to
> > be between 1 and 2, because they looked at the thing
> > geometrically. They didn't have the "number line",
> > but they had its equivalent. Hence, in one sense to say
> > that any new intervals have been concocted is wrong.
>
>
> Gene
>
>
> +++++ They didn't have the whole Kyle Gann division of the
> ocatave, did they? (or did they)

There is no provable way for us to know what tuning was
used in the few extant examples of actual ancient Greek music.
All we can go on is the writings of the theorists which
have survived.

In brief, i can say that the ratios of their JI systems
included prime-factors 3, 5, 7, 11, 13, 19, 23, and 31:

* all the rational theorists used 3;

* Eratosthenes, Didymus, and Ptolemy used 5;

* Archytas and Ptolemy used 7;

* Erastosthenes used 13 and 19;

* Ptolemy used 11 and 23;

* Didymus used 31.

The intervals of Aristoxenos and his followers were
measured as acoustically linear "parts" of a tone, and
most likely indicated something resembling 144-edo.

see these Encyclopedia entries:

http://tonalsoft.com/enc/p/ptolemy.aspx
http://tonalsoft.com/enc/d/diatonic-genus.aspx
http://tonalsoft.com/enc/c/chromatic-genus.aspx
http://tonalsoft.com/enc/e/enharmonic-genus.aspx
http://tonalsoft.com/monzo/aristoxenus/318tet.htm

-monz
http://tonalsoft.com
Tonescape microtonal music software