Ptolemy and tonality diamonds

The 11-limit tonality diamond is listed in the Scala archives as
partch_29.scl, and the header says "Partch/Ptolemy 11-limit Diamond".
What is the connection with Ptolemy?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> The 11-limit tonality diamond is listed in the Scala
> archives as partch_29.scl, and the header says
> "Partch/Ptolemy 11-limit Diamond".
> What is the connection with Ptolemy?

Ptolemy was the first music-theorist whose work implied
the ratios of 11 "as a body", to use Partch's words.

i have some data on Ptolemy's tunings here:

http://tonalsoft.com/enc/index2.htm?ptolemy.htm

but 18/11 and 11/7 are the only 11-limit ratios
you'll see there, because i'm only discussing the
"reference" tetrachord (_meson_). (and the ratios
for that tetrachord are actually 9/11 and 11/14.)

there would also be:

- replicates of those, an "8ve" higher in the
_hyperbolaion_ tetrachord,

- another "8ve" pair a "5th" higher in the _diezeugmenon_
and a "4th" lower in the _hypaton_, and

- another, a "4th" higher in the _synemmenon_ .

Ptolemy's _diatonon homalon_ ("even diatonic") and
_chromatikon syntonon_ ("tense chromatic") were
the two genera of his which used 11-limit ratios.

the "even diatonic" and "tense chromatic" have in
each tetrachord, respectively:

- hyperbolaion: 18/11, 11/7
- diezeugmenon: 27/22, 33/28
- synemmenon: 12/11, 22/21
- meson: 9/11, 11/14
- hypaton: 27/11, 33/14

Ptolemy's "tense chromatic" is particular important
because embedded within it is the tetrachord of
Qutb al-Din al-Shiriz (1236-1311), which is
1/1 - 12/11 - 14/11 - 4/3.

according to Daniel Wolf (posted here several years ago),
this tetrachord:

>> 'has metamorphosed thru many approximations by both
>> Pythagorean cycles and equal-division of both the
>> "4th" and the "8ve" ... but remains the basic tetrachord
>> of both theory and practice in the Islamicate world.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

duh ... after all that, it was stupid of me not to
give the links to pages which show the entire tetrachords:

http://tonalsoft.com/enc/index2.htm?diatonic.htm&equable

http://tonalsoft.com/enc/index2.htm?chromatic.htm&ptolemy-tense

-monz

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > The 11-limit tonality diamond is listed in the Scala
> > archives as partch_29.scl, and the header says
> > "Partch/Ptolemy 11-limit Diamond".
> > What is the connection with Ptolemy?
>
>
>
> Ptolemy was the first music-theorist whose work implied
> the ratios of 11 "as a body", to use Partch's words.
>
>
> i have some data on Ptolemy's tunings here:
>
> http://tonalsoft.com/enc/index2.htm?ptolemy.htm
>
>
> <big snip>