Re: Pythagorean practice

Hello, there, and the discussion about Pythagorean tuning in practice
prompts me to offer a few comments.

First of all, in response to Joe Monzo: to argue that Pythagorean
tuning was a vitally important or even "standard" tuning in medieval
European music, as some of us do, is not to assert that it was
necessarily a _universal_ standard. I wouldn't be at all surprised to
learn, for example, that medieval bagpipes were often tuned with what
would now be called neutral thirds (~11:9), and it would take a lot of
Pythagorean notes per octave to approximate such an interval.
Similarly, theorists such as Theinred of Dover (13th c.?), Anonymous
IV (c. 1275), and Walter Odington (c. 1300) suggest that certain
dialects of English music featured a more pervasive use of thirds than
in most Western European styles of the time, and that performers may
have tended to tune such thirds as approximating 5:4 and 6:5.

However, the question remains: does it follow that because Pythagorean
was evidently not _universal_ in Gothic practice for all regions of
Western Europe and styles of music, that therefore it has little
practical importance? Please let me emphasize that I do not ascribe to
you this position, but sense that it could be widespread.

By the same token, we could argue that meantone tuning is really of
little practical importance to Renaissance practice, because as early
as the 14th century there is an account of tuning some intervals on a
pipe organ in septimal ratios, and a normal minor seventh in 1/4-comma
meantone is nowhere near 7:4.

A rebuttal might be that while 7:4 is a fine tuning for many purposes,
it is not necessarily the best for a minor seventh in a 16th-century
setting where this interval is generally regarded as an unequivocal
dissonance, and the typical meantone tuning not too far from 9:5 may
better fit its role.

Similarly, as Mark Lindley has eloquently argued, a major third around
81:64 and a major sixth around 27:16 very nicely fit the needs of
Gothic music, where these intervals are typically points of
instability inviting resolution to stable concords.

One interesting statement on the practicality of Pythagorean tuning
for medieval polyphony was made by Easley Blackwood in his study on
recognizable diatonic tunings, in which he shows the contradictions
that would result from attempting to sing a passage from Machaut's
Mass in 5-limit just intonation, and concludes that Pythagorean
(3-limit JI) is the best choice.

These observations tie in with a more general point: tuning _all_
intervals as concordantly as possible is not necessarily the objective
of just intonations, Pythagorean or otherwise.

Most respectfully,

Margo Schulter
mschulter@value.net