Positive tunings and adjectives

Hello, there, everyone, and thank you all for making such a valuable
forum possible. Already, some of you have offered me invaluable
assistance and dialogue, and in fact this hospitality is one factor
drawing me here.

As a curious topic for a first post, why don't I focus on a question
of language and stereotyping -- happily not of people, but of a
somewhat undervalued group of intervals: Pythagorean thirds and
sixths. Also, as a kind of coda, I suggest that Ludmila Ulehla's
concept of "dual-purpose" sonorities as a third category somewhere
between full concord and urgent discord might enrich discussions of
various tunings and their characteristic intervals.


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1. Gothic and neo-Gothic interval aesthetics
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While it is generally agreed that a "just" major third has a ratio of
5:4, and that a "pure" major third likewise implies this simplest
ratio, how about some friendly adjectives for the Pythagorean major
third of 81:64, or for that matter the major sixth of 27:16?

Here my special interest is Pythagorean tuning in the context of the
Gothic polyphony of the 13th and 14th centuries in Europe, where these
Pythagorean intervals play an integral role in the harmonic
style. Carl Dahlhaus and Mark Lindley, for example, have written
eloquently on this point.

In a Gothic setting an M3 of 81:64 (about 407.82 cents) or an M6
of 27:16 (about 905.87 cents) have the virtue of expanding very
efficiently to the stable goal of a fifth and octave respectively.
How can we express the positive qualities of these intervals in a
medieval setting while making clear their differences from the 5:4 and
5:3 forms favored in tertian styles of harmony?

These same issues may apply with even more force to tuning systems
such as 17-tet, which offer major thirds and sixths _wider_ than
Pythagorean, which expand even more efficiently to the fifth and
octave with diatonic semitones even keener than the usual 90-cent
limma of 256:243.

Marchettus of Padua (c. 1318) advocates cadential leading tones of
only around 1/5-tone or 2/9-tone (depending on one's interpretation of
his unorthodox division of the whole-tone) for the M6-8 and M3-5
resolution. While a literal reading would produce a cadential M3 at
around 450 cents, and M6 at around 950 cents, we might take him more
freely to recommend a cadential tuning of these intervals somewhat
wider than there usual Pythagorean forms, possibly comparable to those
of 17-tet (about 423.53 cents and 917.64 cents).

The following standard Gothic cadence, tuned in regular Pythagorean
and in 17-tet, may illustrate these points, with apologies for slight
rounding discrepancies. Numbers in parentheses represent vertical
intervals above the lowest voice, and other numbers represent the
melodic motion of each part up or down, all measured in cents.

Classic Pythagorean 17-tet

e'-- +90.22 -- f' e'-- +70.59 -- f'
(905.87) (1200.00) (917.64) (1200.00)
b -- +90.22 -- c' b -- +70.59 -- c'
(407.82) (701.96) (423.53) (705.88)
g -- -203.91 f g -- -211.76 f

M6 8 M6 8
M3 5 M3 5

In our classic Pythagorean example, and even more dramatically in
17-tet, the active M3 and M6 expansively seek the fifth and octave,
The generous whole-tone motions and keen semitonal motions of the
parts contribute to the total harmonic effect of active tension very
efficiently resolved. We arrive at the ideally concordant three-voice
combination of the Gothic with outer octave, lower fifth, and upper
fourth (string-ratio 6:4:3, frequency ratio 4:3:2).


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2. Positive adjectives for positive tunings
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Here I would like to suggest that Pythagorean thirds and sixths in a
Gothic context may felicitously be described as "active," "vibrant,"
"bright," and "dynamic." They are also, of course, "unstable," but at the
same time regarded in theory and practice as _relatively_ blending. They
are points of motion, or of diversion, not points of arrival.

In contrast, pure thirds and sixths in a Renaissance context might be
described as "restful," "optimally blending," "smooth," "tranquil."

This kind of contrast applies not only to these intervals, but to
others whose _musical_ uses change along with stylistic developments,
and thus also, quite possibly, their optimal tuning for a given
purpose.

One interesting reflection of a Gothic viewpoint, possibly relevant to
lines of experimentation with 17-tet and other positive tunings (my
special interest), is that a major third or sixth of full Pythagorean
proportions is said to be "perfected," because it approaches as
closely as possible its cadential goal of the fifth or octave. This
"striving" to expand to stability is an element of the Gothic musical
language which maybe might be emphasized more often in discussions
about the Pythagorean intervals and their aesthetic qualities.


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3. Dual-purpose sonorities: a xenharmonic perspective
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Finally, an aside which might apply to many musical styles, but I
suspect especially to contemporary just intonation (Pythagorean or
otherwise) and xenharmonic settings.

Often there is a tendency to divide intervals and combinations into
two categories: consonance/dissonance. However, whether we are
analysing Gothic or impending 21st-century music, a middle category
might go far to increase the subtlety of the discussion.

Ludmila Ulehla has proposed the term "dual-purpose" sonority to
describe an interval or combination which may be somewhere between
stable concord and urgent discord; the medieval concept of "imperfect
concord" or "imperfect discord" ("imperfect" meaning "partial, semi-")
seems to convey a similar concept. Although Ulehla focuses on the era
from around 1750 to the 20th century, her approach may nicely fit
earlier practice and theory as well as new xenharmonic developments.

In a 13th-century setting, for example, a interval such as 81:64 or
three-voice combination such as 9:6:4 seems to be regarded as
relatively blending and euphonious, but unstable; and the same
category might fit many of the sonorities of extended just intonation
or xenharmonics in certain settings. To describe such sonorities
simply as "concordant" may not convey their instability, while the
adjective "discordant" may not convey their perceived level of
independent euphony or "coloristic" appeal apart from any immediate
resolution. Of course, both dual-purpose sonorities and stronger
"discords"

In these terms, the apparent transition during the 15th century from
Pythagorean to meantone keyboard tunings reflects a shift in the role
of M3 and M6 from active dual-purpose sonorities (81:64, 27:16) to
more restful concords (5:4, 5:3). Similarly, the possible tunings of
the minor seventh at 16:9, 9:5, and 7:4 may represent different levels
of tension desired in divergent styles, as well the exigencies of
fitting m7 into a larger scheme of things.

In conclusion, I should emphasize that 17-tet, for example, is by no
means limited to "neo-Gothic" applications, but might be very
congenial to such applications. Also, 81:64 might have xenharmonic
applications quite different from those obtaining in medieval
polyphony, which suggest only one very effective use of this
interval.

Most respectively,

Margo Schulter
mschulter@value.net