Pythagorean tunings

Hi Brett,

<<
Marion, I think your mistake is thinking that all intervals are measured from
the 1/1.
>>

Gee, maybe I was wrong about the harmony police:)

<<
So the 81/64 would not be heard against the 1/1, it would be heard against
the 27/16, for example.
>>

Are you saying that it was against the rules to actually use the interval
27:16. They just kept it in the scale as some kind of convenient
placeholder? With such a small number of tones, it would seem that a
musician would be reluctant to give up harmonic opportunities, especially
since it is so close to the 5/3 which some people would say is the next most
harmonious interval after 3/2. Actually, this and other points you have made
sound like a reiteration of the point I was trying to make which was that the
scale they were using did not describe what they were actually doing, and
that we still suffer from the same problem today with 12et. Perhaps our
viewpoints are not so dissimilar as would first appear.

<<
And the 27/16 would not be tuned 5/3 because it would be heard against the
9/8, which would not be tuned 10/9 because it would be heard against the 3/2.
A chain of fifths, i.e., Pythagorean tuning, is clearly the best way to tune
such music. Sure, vocalists could use both 5/4 and 81/64 in the course of a
piece of music, depending on immediate harmonic context, but if we had to
choose one for these musics, it would certainly be 81/64. Melodically, also,
the Pythagorean versions of the pentatonic and heptatonic scale are very
smooth and easily learned.

You can't take the major third (5/4 or 81/64) and analyze it out of the
context of the rest
of the scale and the type of harmony it is to be used for!
>>

With all due respect, you are in no position to tell me what I can or can't
do.

I assure you that I have, and routinely use, a variety of tools for analyzing
every possible harmonic relationship in a scale.

To really describe what's going on we need a scale with a lot more than five
or seven tones. It must have both 5/4 and 81/64, as well as a lot of others.
I still find it extremely difficult to believe that the interval 81/64 is
really used, but I since there is currently no way to measure such things,
with sufficient accuracy, we will have to each hold our separate opinions
until technology improves?

The limitations of the chain of fifths idea have been discussed in this forum
many times. It seems to me that lattices are just an extension of this
concept and suffer from the same limitations. These concepts, a kind of
musical parallelism, are useful, but I find other methods better, especially
in the context of discretely-tuned electronic musical instruments.

<<
Also, the scales of Pythagorean times had 7, not 5, tones!
>>

You are correct, and thank you for reminding me of that.

While I am grateful for your comments, I find myself still unable to
completely understand your viewpoint, and suspect you may have similar
difficulties with mine. Let's not allow these problems to interfere with our
good humor.

Marion

Marion wrote,

> <<
> So the 81/64 would not be heard against the 1/1, it would be heard against
> the 27/16, for example.
> >>
>
> Are you saying that it was against the rules to actually use the interval
> 27:16. They just kept it in the scale as some kind of convenient
> placeholder? With such a small number of tones, it would seem that a
> musician would be reluctant to give up harmonic opportunities, especially
> since it is so close to the 5/3 which some people would say is the next most
> harmonious interval after 3/2.

But the major sixth was considered a dissonance, so tuning it to a consonance would
be contrary to its musical function. Rather than being a missed harmonic
opportunity, the dissonant sixth gave the music impetus to move forward and
resolve.

> To really describe what's going on we need a scale with a lot more than five
> or seven tones. It must have both 5/4 and 81/64, as well as a lot of others.
> I still find it extremely difficult to believe that the interval 81/64 is
> really used, but I since there is currently no way to measure such things,
> with sufficient accuracy, we will have to each hold our separate opinions
> until technology improves?

I don't know what you mean. There's plenty of technology that can measure such
things to sufficient accuracy. Unless you mean technology that can analyze music
that was played 1000 years ago, which is of course far more relevant to this
discussion; in that case we need a significant improvement in technology :)

> The limitations of the chain of fifths idea have been discussed in this forum
> many times. It seems to me that lattices are just an extension of this
> concept and suffer from the same limitations. These concepts, a kind of
> musical parallelism, are useful, but I find other methods better, especially
> in the context of discretely-tuned electronic musical instruments.

Oh?

> While I am grateful for your comments, I find myself still unable to
> completely understand your viewpoint, and suspect you may have similar
> difficulties with mine. Let's not allow these problems to interfere with our
> good humor.

I wholeheartedly agree.

P.S. I think Margo Schulter addressed your comments in a much more polite but also
much more historically informed way. How would you respond to her?