Re: Pythagorean tuning -- another voice

Hello, there, and in response to some previous discussion, I'd like
briefly to comment on why the Pythagorean major third or ditone at
81:64 (~407.82 cents) can fit so nicely into the kinds of complex
3-limit Gothic and neo-Gothic styles which I practice.

Here there may be two sides to the question: the nature of the
historical tradition, and my more personal experiences and impressions
as I practice it. Maybe the second aspect is a better place to begin.

In a Gothic or neo-Gothic setting where fifths and fourths and
complete trines (2:3:4) set the standard of rich and saturated
concord, I find that prominent sonorities with Pythagorean thirds are
at once relatively concordant or blending and yet unstable, nicely
whetting my anticipation of motion to a new stable sonority.

Please let me make it very clear that prominent cadential and other
sonorities including Pythagorean major and minor thirds -- the latter
at 32:27 (~294.13 cents) -- are a routine and essential feature of
multi-voice Gothic polyphony from Perotin to Ciconia and his
colleagues, or about 1200-1420.

These sonorities are unstable, dynamic, and absolutely essential to
the vertical color and cadential action of the music -- as integral an
element as the various sonorities with sevenths from Corelli to
Beethoven.

In a relatively recent post, Kraig Grady discussed how the language
used by musicians of a given tradition (e.g. Javanese or Balinese
gamelan) may ideally express that artistic framework. In trying to
express somewhat ineffable musical experiences, which may be radically
alien to others in concept and perception, I can empathize with the
dilemmas of Pierre Lamothe in confronting a different kind of natural
language barrier.

To me, the sonority 64:81:96 or 54:64:81 is what Jacobus terms it: a
_quinta fissa_ or "split fifth" (e.g. F3-A3-C4, A3-C4-E4) divided into
two relatively blending and pleasantly unstable thirds.

Many writers of the 13th and 14th centuries describe these thirds as
"imperfect concords," or sometimes (Jacobus) "medial concords" -- they
"are heard to differ greatly, and yet concord" (Franco of Cologne,
c. 1260?).

The modern term "semi-concord" seems very felicitous to me also. As
Johannes Boen so aptly writes in 1357, thirds and also sixths are
"messengers and handmaidens" heralding the arrival of their purer
sister concords, the fifth and octave.

Since this discussion has largely focused on the _quinta fissa_, I
might just cite a cadence from a setting of _Haec dies_ maybe a bit
after 1200 in a more or less Perotinian style which beautifully
illustrates this point:

D4 E4
B3 A3
G3 A3

(M3-1 + m3-5)

Here the musical grammar, which has been "second nature" to me for
some 30 years of listening, improvising, and composing, makes the
first sonority the pleasant and stirring _avant-coureur_ of a stable
sonority to follow, here a fifth (the choicest trinic interval). This
passage virtually sent shivers up my spine when I heard it performed
as part of a series on music featuring Yehudi Menuhin maybe 20 years
ago.

While the grammar and logic does not depend on the precise tuning, the
Pythagorean third gives this passage "bounce" or "natural reverb," as
I might put it.

In forms such as the 13th-century conductus and motet we may sometimes
have a chain of two or three _quinta fissa_ sonorities leading up to a
standard cadence:

1 & 2 3 | 1
E4 D4 C4 B3 A3
B3 A3 G3 A3
E3 F3 E3 D3

(M3-1 + m3-5)

Interestingly, my ears may focus on the relatively blending quality of
the thirds at least as much as on their element of tension -- which
seems to me quite natural, given that they are semi-concordant but
unstable.

Major and minor thirds also occur in a variety of sonorities, some
very important for cadences, where they are mixed with more tense
intervals including seconds, sixths, and sevenths. For much more on
this, please see:

http://www.medieval.org/emfaq/harmony/13c.html
http://www.medieval.org/emfaq/harmony/pyth.html

To me, both the instability of thirds in this music and their pleasant
quality as Pythagorean "semi-concords" are routine features of a style
deeply ingrained within me. Might these two perceptions, or
conceptions, be intimately related -- and likewise the perplexity of
listeners who may take the stability of thirds as "second nature," and
find Gothic musical style _and_ intonation "out of tune" with their
deeply ingrained expectations?

Might musicians accustomed to Paul Erlich's decatonic music with its
stable 4:5:6:7 tetrads ask with some trepidation or at least
bewilderment how European musicians could remain satisfied with "such
egregiously out-of-tune minor sevenths at 9:5 or 16:9"?

Would they analyze the factors of pitch range, beat rates, and melodic
ornamentation in Mozart which might make his "5-limit tetrads" somehow
more bearable?

Might some passionate lover of Mozart gently point out that referring
to these same sonorities as "seventh chords" is not just a matter of
18th-century "musicological correctness," but a difference in musical
perception and experience -- however ineffable and possibly
untransmissible?

The term egregious, for me, can also have its original Latin meaning:
_egregi-us/a/um_ means "standing out from the herd or flock," much
like the English "outstanding."

In this sense, I find Pythagorean thirds in the _quinta fissa_ and
other Gothic sonorities to be outstandingly suited to the music.
This is a happy feature of Pythagorean JI: two pure 9:8 tones just
happen to make a very nice semi-concordant major third.

For some other people, however, these same intervals can be egregious
in the usual Modern English sense -- which I might here translate as
"outstandingly out-of-tune with musical and intonational expectations
premised on stable thirds."

Of course, there are interesting collateral matters: the very relative
nature of pitch in the medieval era; the various ranges which
ensembles of men, women, boys, portative organs, and other instruments
may have used in performance; the total range of about G2-D5 (G-d'')
for pieces in the Montpellier Codex, with the largest usual vertical
interval or overall range for a given piece rarely exceeding about a
twelfth.

However, I suspect that the basic issue is one of deep acculturation
and habit -- something very different from crude bias or prejudice, I
would strongly emphasize.

Similarly, I must confess my comic difficulties with basic French
phonology, quite natural to native French speakers at least since the
era around 814 when clerics recognized the _lingua rustica Romana_ as
a distinct tongue whose use would promote intelligible communication.
Native speakers of many languages may have yet greater difficulties
with English phonology.

Please let me add that my own perceptions may be rather "bilingual."
Pythagorean thirds (and sixths) seem "as natural as the air" to me in
most Gothic polyphony -- but curiously "ruffled" or "peculiar" for
16th-century music, telling me that I neglected to switch the
synthesizer to meantone.

For me, either intonational phonology is "natural" for its musical
style or language.

Most respectfully,

Margo Schulter
mschulter@value.net