one final question

Really, this is going to be my last post... well, for right now,
anyway.

I just started thinking a bit more about Margo's post...

Maybe, Paul, you can just fire off a "quickie" answer on this one:

OK... here is the Pythagorean chain Margo mentions:

Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B

Now... am I to assume that just in *the nature* of this chain we have
two different kinds of thirds, if we bring all the pitches down into
one octave?? So, the G-B third is large, or Pythagorean and the A-Db
third is schismatic??

So it just "naturally" works out that way without a continuing chain
of enharmonics or anything of the kind??

Is that possibly because, lets say in the A-Db third, one is taking
part of the chain that is *LOWER* from the A (the Db) to make the
schismatic third, whereas the G-B pitches are in strictly *ascending*
order in the chain?? Does that have something to do with it...?? It
seems like it might...

Thanks!!!

_________ ________ ________
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> Really, this is going to be my last post... well, for right now,
> anyway.
>
> I just started thinking a bit more about Margo's post...
>
> Maybe, Paul, you can just fire off a "quickie" answer on this one:
>
> OK... here is the Pythagorean chain Margo mentions:
>
> Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B
>
> Now... am I to assume that just in *the nature* of this chain we
have
> two different kinds of thirds, if we bring all the pitches down
into
> one octave??

Yes . . . any 12-tone chain of identical fifths will give two
different kind of major thirds.

> So, the G-B third is large, or Pythagorean and the A-Db
> third is schismatic??

Yes.
>
> Is that possibly because, lets say in the A-Db third, one is taking
> part of the chain that is *LOWER* from the A (the Db) to make the
> schismatic third, whereas the G-B pitches are in strictly
*ascending*
> order in the chain?? Does that have something to do with it...??

Yup! When one "wraps around" the chain from B to Gb, one is losing a
Pythagorean comma . . . so that when one constructs a major third
that goes over the "wrap", one ends up with a major third narrowed by
a Pythagorean comma . . . very nearly a just major third.

It appears that medieval Arabic music made use of this phenomenon
(via 7-tone scales from a 17-tone Pythagorean chain), though in what
ways, we may never know.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_28510.html#28545

> --- In tuning@y..., jpehrson@r... wrote:
> > Really, this is going to be my last post... well, for right now,
> > anyway.
> >
> > I just started thinking a bit more about Margo's post...
> >
> > Maybe, Paul, you can just fire off a "quickie" answer on this one:
> >
> > OK... here is the Pythagorean chain Margo mentions:
> >
> > Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B
> >
> > Now... am I to assume that just in *the nature* of this chain we
> have two different kinds of thirds, if we bring all the pitches
down into one octave??
>
> Yes . . . any 12-tone chain of identical fifths will give two
> different kind of major thirds.
>
> > So, the G-B third is large, or Pythagorean and the A-Db
> > third is schismatic??
>
> Yes.
> >
> > Is that possibly because, lets say in the A-Db third, one is
taking part of the chain that is *LOWER* from the A (the Db) to make
the schismatic third, whereas the G-B pitches are in strictly
> *ascending* order in the chain?? Does that have something to do
with it...??
>
> Yup! When one "wraps around" the chain from B to Gb, one is losing
a Pythagorean comma . . . so that when one constructs a major third
> that goes over the "wrap", one ends up with a major third narrowed
by a Pythagorean comma . . . very nearly a just major third.
>
> It appears that medieval Arabic music made use of this phenomenon
> (via 7-tone scales from a 17-tone Pythagorean chain), though in
what ways, we may never know.

Thanks, Paul! Glad I guessed that one right for a change...

_________ _______ _______
Joseph Pehrson