Re: Trines, triads, octave-affinity and "rootedness" (Erlich)

Hello, there, and in Tuning Digest 656, Paul Erlich writes:

> Ultimately, I have to agree with Margo, and the psychoacoustical
> explanation cannot be found in the concepts of roughness or
> tonalness. There appears to be an additional phenomenon,
> "rootedness", which gives the lowest sounding note an extra
> probabilistic boost in its likelihood of being interpreted as
> octave-equivalent to the fundamental. This would explain why, for
> example, 4:5:6 is more "consonant" than 3:4:5.

In turn, Paul, please let me express my agreement with your earlier
point that Partch's "otonal/utonal" distinction may be intended to
address a different issue that the one I'm focusing on: the parallel
between 3-limit trinic and 5-limit triadic conversities, or sonorities
with the same intervals in "converse" or "mirror-reflection"
arrangements.

Historically, Western European theorists have tended to focus on two
kinds of affinities between sonorities. What I term the "conversities"
of 13th-16th century theory share the same set of intervals in
different arrangements, while 17th-19th century "inversions" share the
same set of pitch classes in different octave transpositions. Neither
approach necessarily exhausts all the relevant parameters and
variations.

An interesting conclusion which might be drawn from your comment above
is that not all octave transpositions are equally "equivalent" -- not
only are the 3-limit trines 2:3:4 and 3:4:6 distinct (here I use
frequency-ratios in the usual modern manner), which are conversities
as well as inversions, but also the 5-limit forms 4:5:6 and 3:4:5,
which are inversions but not conversities.

As a medievalist often tending to think in terms of "octave affinity"
rather than exact equivalence, I can come up with other examples. For
example, I find 4:6:9 in a Gothic context as relatively blending but
still unstable, with considerable "tension" -- yet 1:3:9 or 1:6:9 (the
latter being easier for one person to play on a conventional keyboard)
sounds like a curiously stable sonority, with enough acoustical
"space" for all the tones to coexist without such tension. Jacobus of
Liege may be making the same point around 1325 when he terms 9:4 an
"intermediate concord," but 9:1 a "perfect concord."

Getting back to 2:3:4 vs. 3:4:6, and 4:5:6 vs. 3:4:5, some
20th-century theory speaks of a "perfect fifth anchor" which tends to
reinforce what you term a sense of "rootedness." At the same time,
with 2:3:4 and 4:5:6, there's the factor of the lowest tone as an even
octave of the fundamental that we've both noted.

From another angle, using "smooth" in a way which may not necessarily
be synonymous with your "smoothness," I'm tempted to guess that 2:3:4
may be more "smooth" than 3:4:6 because the fourth above the lowest
note of the latter sonority may be in tension with the third partial
of the lowest note. Likewise, might this tension come into play with
the lower fourth of a 3:4:5 sonority vis-a-vis a 4:5:6 sonority?

Analogously, it has been said that the lower minor third of 10:12:15 is in
tension with the fifth partial of the lower note -- like the lower
fourth of 3:4:6 in relation to the third partial of the lower
note. Joseph Yasser, as I recall, in his "Medieval Quartal Harmony"
therefore refers to 2:3:4 and 4:5:6 as "resonant" forms, and to 3:4:6
and 10:12:15 as "irresonant" forms.

A parallel here is that 3:4:6 is in tension with the third partial
(3-limit), while 10:12:15 is in tension with the fifth partial
(5-limit).

Anyway, this is a very interesting dialogue. You're evidently coming
from a perspective in part of 18th-century inversion theory, while I'm
coming from a perspective of what I might term 13th-16th century
conversity theory -- but in either case, more recent theorists about
concord/discord and the like can add new elements.

Most appreciatively,

Margo Schulter
mschulter@value.net