Re: Trines, rootedness, etc. (Paul Erlich)

> [....] However, since Partch was largely concerned with intervals
> between pitch classes regardless of octave transpositions, and he
> coined the terms otonal and utonal, it seems better to use other
> terms to describe the distinction between the two trines, which as
> you note, can be regarded either as analogous to major and minor, or
> instead, as analogous to the different inversions of the major
> chord.

Hello, and here I'm much in agreement. Thank you very much for
educating me on this point.

In fact Joseph Yasser uses both analogies we're discussing in the
third part of his article on "Medieval Quartal Harmony" in the
_Musical Quarterly_ (v. 24, 1938, pp. 351-385). While I have lots of
differences with many of Yasser's specific conclusions about medieval
polyphonic style, nevertheless his presentation of medieval
verticality as an independent system in its own right has had immense
influence on me.

Anyway, in proposing a system for Gregorian chant harmonization based
mainly on the fourth as a primary consonance, Yasser notes that one
may sometimes conclude on a sonority with a fifth above the lowest
tone for a more "resonant" ending -- as opposed to an "irresonant"
ending on a fourth.

He refers to this liberty in his system as a _Quint de Picardie_ --
explicity drawing a parallel with the tertian _Tierce de Picardie_, a
term adopted at some point for the very widespread 16th-century and
later practice of preferring that thirds in cadential sonorities of
arrival be major. Thus he contrasts the "resonant" 3-limit fifth and
5-limit major third with the "irresonant" 3-limit fourth and 5-limit
minor third.

Elsewhere, Yasser also expressly treats the fourth and fifth as
inversions comparable to thirds and sixths in tertian harmony.
Curiously, he regards the fourth as the primary form and the fifth as
the "inversion" -- possibly a tenable concept in styles like that of
Guido d'Arezzo in the early 11th century where the fourth is preferred
to the fifth as a basic consonance. He thus describes parallel fifths
in some medieval contexts as "inverted" sonorities comparable to the
parallel sixths of early 15th-century _fauxbourdon_ (the latter being
favored by Dufay and Binchois, for example, in what I would call the
transition era from medieval to Renaissance style and also from
3-limit to 5-limit).

Anyway, thanks both for alerting me to an important point about
Partch's outlook, and for giving me an opportunity to mention
Yasser. Reading him again, I see that he argues that a fourth above
the bass may be in tension with both the third partial (at the twelfh)
and the fifth partial (at the major 17th).

Your points about comparisons between 5-limit sonorities also call for
lots of discussion, but I wonder if maybe first I should write a piece
on Zarlino's analysis of these sonorities which I'm planning. Trying
to frame an immediate response on this point, I found myself raising
issues which maybe would be clearer against the background of such an
article.

Most appreciatively,

Margo Schulter
mschulter@value.net