Re: diminished fourth == schismic third

>
> 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??

Yes, but (slap forhead here), you are calling A-Db a third when it is
a diminished fourth. The diminished fourth in a pure pythagorean tuning
is very near to 5:4.

My question, to Margo particularly, do we have notation in any
pieces during the transitional period that specifically use B-Eb (or
similar diminished fourth) or mention that a major third (for instance
B-D#) should be sung "as if" it were B-Db?

Back to Joseph, one of the 'features' of 72 being 6 bicycle chains of
12 is that, even though you have an improved major third available, it
is not on the normal spiral of fifths as it would be in a schismic tuning.
In 29, 41, or 53, after drawing your circle of fifths, you'll find that
C-E is your wide major third and C-Fb is your 5:4.

Bob Valentine

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:

/tuning/topicId_28515.html#28515

> >
> > 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??
>
> Yes, but (slap forhead here), you are calling A-Db a third when it
is a diminished fourth. The diminished fourth in a pure pythagorean
tuning is very near to 5:4.
>

Hello Bob!

Thanks so much for the help... I guess what confused me in the
Schulter article... which I found *really* interesting as is obvious
from my posts was the following paragraph:

Margo Schulter:
/tuning/topicId_28294.html#28367

"Lindley has observed, more specifically, that the treatment of
intermediate cadences and phrasing may suggest that Dufay often
composed with expectations based on some kind of keyboard instrument
(an organ or clavichord) for 12 notes tuned in the Pythagorean Gb-B
arrangement. He notes that sonorities such as A3-C#4-E4 often have
what he calls a "stable" and I might call a "prolonged noncadential"
role, the sonorities where schisma thirds (e.g. A3-Db4-E4 as realized
on a Gb-B keyboard) would have a smooth, quasi-5-limit, effect."

Here it seems that Db and C# are used almost interchangeably...

Obvious they are different pitches in the Pythagorean system and are
*non* enharmonic equivalents...

So, I guess it appears that possibly the composers were tuning a
Pythagorean C# to a schmatic Db and using a diminished fourth while
writing it as a third...

At least that's the impression I'm getting from this... but I could
just be misunderstanding it.... ??

Thanks!

________ ________ _______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> So, I guess it appears that possibly the composers were tuning a
> Pythagorean C# to a schmatic Db and using a diminished fourth while
> writing it as a third...
>
> At least that's the impression I'm getting from this... but I could
> just be misunderstanding it.... ??

You're understanding Lindley's thesis just fine, Joseph!