How does an 81:64 feel? -- reply to Gary Morrison

Hello, there.

Recently Gary Morrison raised a very interesting question about how
different people perceive a major third at the Pythagorean or 3-limit
tuning of 81:64 (around 407.82 cents).

> Using a related 3-limit interval as another example, I personally
> have never managed to attribute any intuitive meaning to 81:64. To
> me it sounds like an off-5:4 much more than anything meaningful in
> itself. It's just too complicated a pitch relationship very close
> to a vastly more obvious one.

Here I'd like to have a try at describing how an 81:64 feels to me in
some typical musical settings where I encounter it.

First, I might note that an 81:64 can take on different qualities
depending upon the timbre, also true to an extent of a 5:4 major
third, for example. Using a somewhat regal-like or crumhorn-like
registration (Yahama TX-802 preset voice A22, for the curious), I find
that an 81:64 major third can be quite "strident," but with a more
subdued organ-like registration (e.g. a combination of TX-802 voices
A17 "cello" and A23 "flute"), it can sound quite "sweet."

Generally, the medieval term _ditonus_ or "ditone" is intuitively
descriptive: this major third consists of two pure 9:8 whole-tones.

In polyphonic music of the 11th and 12th centuries, I feel this
"ditonal" quality in the very common cadence where two voices contract
from a major third to a unison, each moving by a whole-tone in
conjunct contrary motion. It's a kind of convergence or flowing
together, and an 81:64 nicely suits this process, making the major
third before a unison a bit more "complicated" and interesting.
Incidentally, someone has already observed on the Web that the beats
of an 81:64 add cadential interest before a unison, so this isn't
necessarily the most original observation .

Around 1200, as polyphony expands to three or four independent voices,
this major third also makes itself heard in cadences like this:

d' -- +204 -- e'
(702,294) (702,702)
b -- -204 -- a
(408) (0)
g -- +204 -- a

Here, again, the contraction of the ditone to a unison by whole-tone
motion in both lower voices, complemented in this cadence by the
similar m3-5 resolution between the upper voices, has what I might
describe as a certain stately quality.

By the late 13th century to some extent, and in the 14th century, the
81:64 typically takes on a more "incisive" cadential character,
expanding to a fifth with one voice moving by a semitone and the other
by a whole tone. It's especially characteristic for the 81:64 to team
together with a 27:16 major sixth, expanding to fifth and octave
respectively:

c#' -- +90 -- d'
(906,498) (1200,498)
g# -- +90 -- a
(408) (702)
e -- -204 -- d

How does this feel -- I would say "hot, passionate, intense,
expressive," with accidental inflections often required to make the
third and sixth major further underscoring the power of the cadence.

It's beautiful the way that a scale yielding pure fifths and fourths
also seems to lead just the right amount of tension to those major
thirds and sixths to really make this progression ideal. I once
compared its expansiveness in a poem to the Big Bang.

Of course, this isn't to say that 81:64 and 27:16 are the _only_
tunings that can "feel right" for this cadence. In a Xeno-Gothic
tuning, where the major third and major sixth can be made a
Pythagorean comma wider than usual (close to 9:7 and 12:7), this same
progression in certain timbres can have a very convincing "ring" for
me:

c#' -- +67 -- d'
(930,498) (1200,498)
g# -- +67 -- a
(430) (702)
e -- -204 -- d

Here I'd describe the usual Pythagorean version with 81:64 and 27:16
as "classic," and this version as more "jazzy," although my sense of
"neo-Gothic jazzy" might not be the same as other people's .

Additionally, a sonority with an 81:64 often occurs in 14th-century
music as a kind of half-cadence, and the "classic" edge to this major
third emphasizes the musical message that this is a mild but
_unstable_ sonority, and that there's more to come. Of course, this is
a matter of _musical_ expectation that would hold in 12-tet or even
meantone: thirds are inconclusive in this style! However, that bit of
extra tension nicely adds emphasis to this point, and heightens my
sense of "a _partial_ repose" still seeking harmonic "perfection" and
moving toward what follows next.

> There is another possible explanation though: Exactly opposite of
> Margo, I have almost no time whatsoever with 81:64, or 3-limit
> tunings in general. Historically, I've been much more interested in
> the other end of the spectrum: 7s, 11s, 13s, and such.

Please let me admit that even when considering intervals such as 9:7
or 12:7, I tend to approach them from a kind of 3-limit viewpoint: "A
9:7 is sort of like a superactive 81:64 inviting expansion to a
fifth." Also, I may like the idea of combining 9:7 and 12:7 (or their
Xeno-Gothic approximations of 81:64-plus-comma and 27:16-plus-comma)
because the two ratios together give a pure 4:3 fourth between the
upper voices.

> In fact, I may even be able to count the total number of times I've
> intentionally confronted myself with 81:64 on my fingers and toes.
> If I were to listen to it more an intuitive meaning for it might
> become apparent.

Please let me, in turn, caution that I'm oriented to 81:64 in a Gothic
or neo-Gothic setting, both in performing and in composing or
improvising. This leaves open all kinds of new styles and applications
for this interval which might be quite different from either the
medieval tradition or from my musical experience largely based upon
it, however imperfectly transmitted across the centuries. I certainly
wouldn't want to leave the impression that Gothic music defines _the_
way to use an interval such as 81:64; it's just one possibility.

Of course, if the fact that this interval _has_ been used to make some
very impressive music encourages people to try using it now in ways
both old and new, then history and creativity may prove happy allies.

Most respectfully,

Margo Schulter
mschulter@value.net