For Paul Erlich and others -- Zarlino and Victoria

Hello, there, Paul Erlich and Gene Ward Smith and everyone.

While a recent thread I joined has been titled "7-limit thinking," I want
to address some more basic and general questions raised in that thread
about the roles of melodic and vertical intervals and progressions in
Gothic and Renaissance/Manneristic music of Europe, including for example
the progressions of Tomas Luis de Victoria.

As someone whose background is mainly in European composed music of
around 1200-1600, I may have a different point of view on these
questions than someone oriented mainly to 18th-19th century traditions
of key tonality in Europe, or someone like Haresh Bakshi trained in
the raga system of India, for example.

Since short messages may invite more responses than long ones, I'll
try to keep this brief, as unlikely as it seems <grin> -- that is, not
more than 200 lines.

In dialogues of this kind, I often seem to see a kind of contrast
drawn between isolated melodic lines and 18th-century key progressions
of the kind often notated with Roman numerals.

Here I would like suggest that 13th-16th century European music, and
also to a great extent early 17th-century music, may involve vertical
approaches just as sophisticated as that of the 18th-century key
system, but different.

What I can best describe is how I conceive of some of these concepts
as someone who plays, improvises, and composers in these or similar
styles, and also what some theorists of the Gothic-Manneristic eras
say about them.

Above all, I'd stress that an awareness of vertical progressions in
1200, 1350, 1500, or 1600, means awareness of a complex system quite
different than that of the 18th century -- as does such awareness in
many musical settings of the year 2001.

For an overview of 13th-century verticality, for example, people are
invited to visit my article at Todd McComb's fine Early Music FAQ site
for the Medieval Music and Arts Foundation:

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

As you can read there in more detail, I view 13th-century polyphony as
featuring a set of idiomatic sonorities and cadential progressions for
three or voices, and theorists such as Jacobus of Liege catalogue and
discuss some of those sonorities, which Jacobus terms "partitions."

Since Victoria has been mentioned, I'll say a bit about 16th-century
practice and theory, in my view an independent system of vertical
progressions distinct from that of the 18th century, and having its
own logic.

For example, in the Spanish tradition, Tomas de Santa Maria (1565) in
his treatise on the art of _fantasia_ (one might say "invention" or
improvisation on keyboard or other polyphonic instruments) presents
a technique based on four-voice sonorities and progressions.

He terms these sonorities _differencias_, or "differences," since the
outer two voices of bass and treble define the basic intervals, while
the two inner voices make "differences" or interior intervals
providing additional concords and "filling in the space" between the
outer voices.

This is, of course, only one kind of 16th-century technique, but Santa
Maria's concepts and analysis of the four-voice sonorities or
differences seem to me nicely to fit note-against-note textures in
many vocal as well as instrumental compositions.

When improvising in a Renaissance or "Xeno-Renaissance" style, I often
find myself "instinctively" following some of the patterns described
by Santa Maria: for example, a kind of progression in which the bass
alternates motion by fourths up and steps down, or the like.

He also extensively discusses the modes, and some cadences for three
and four voices, focusing for example on the ascending or descending
semitone motions appropriate for the resolution of different
suspension dissonances.

In his presentation of the "differences" of sonorities for four voices
with different outer intervals between bass and treble (e.g. octave,
tenth, twelfth, fifteenth), he rates them in "grades" by their degree
of euphony, urging composers to use the best grades where the
part-writing permits it.

While a treatise of this kind can hardly capture the subtle artistry
and expertise of composer such as Victoria, it lets us know that
16th-century musicians could insightfully analyze four-voice
sonorities and progressions.

For 18th-century music, of course, one is dealing with a different
kind of four-voice practice, calling for a different analysis,
although the two systems share interesting similarities as well as
differences.

Now we come to 18th-century inversion theory, and analogous but
different 16th-century concepts. Here Zarlino's treatise of 1558 is
illuminating, showing how 16th-century theory makes analytical and
aesthetic distinctions of its own.

Curiously, both Zarlino's 16th-century theory of _harmonia perfetta_
or "complete harmony" (5-limit, in the terms often used here), and the
18th-century theory of inversion with some 17th-century antecedents in
Johannes Lippius (1612) and others, recognize an interesting dichotomy
between two categories of saturated 5-limit sonorities, or "perfect
harmonies," as Zarlino calls them.

Zarlino's "natural" and "artificial" divisions correspond with the
later "major" and "minor" categories, as the following table may
suggest, which shows string ratios for the different arrangements
discussed and compared in his treatise:

------------------------------------------------------------------
Interval set: (5,M3,m3) (M6,M3,4) (m6,m3,4)
------------------------------------------------------------------

| G3 10 | E4 12 | C4 15
Natural | (m3) | (M3) | (4)
arrangement 5 | E3 12 M6 | C4 15 m6 | G3 20
| (M3) | (4) | (m3)
| C3 15 | G3 20 | E3 24

(Frequency (4:5:6) (3:4:5) (5:6:8)
ratios)
------------------------------------------------------------------

| A3 4 | D4 3 | F4 5
Artificial | (M3) | (4) | (m3)
arrangement 5 | F3 5 M6 | A3 4 m6 | D4 6
| (m3) | (M3) | (4)
| D3 6 | F3 5 | A3 8

(Frequency (10:12:15) (12:15:20) (15:20:24)
ratios)

------------------------------------------------------------------

Note that we can use three possible types of "divisions" for saturated
16th-century sonorities, Zarlino's _harmonia perfetta_. We may have
the outer voices at a fifth, divided by a middle voice into a major
third below and minor third above, or the converse; or an outer major
sixth divided into fourth and major third; or an outer minor sixth
divided into minor third and fourth.

Since each division has two arrangements, that makes six arrangements
or types of sonorities in all.

Comparing the two arrangements for each division, Zarlino regards one
sonority of each pair as "natural," and the other as "artificial" -- a
distinction which, as it happens, corresponds with the 18th-century
distinction between "major" and "minor" root-position triads and
inversions.

However, 16th-century theory, following earlier Gothic concepts
developed in a different stylistic setting but still playing an
important role in guiding multi-voice progressions in the newer style,
draws some important distinctions within these "natural" and
"artificial" categories.

For example, Zarlino regards both of the following sonorities as
"artificial":

D4 A4
A3 D4
F3 A3
D3 F3

However, these two sonorities are not equivalent, because the second
features major thirds and sixths above the bass, expansive or
"outgoing" intervals seeking expansion to the fifth (Maj3-5) and
octave (Maj6-8), as in this common progression:

A4 G#4
D4 E4
A3 B3
F3 E3

If we omit the highest voice, in fact, we have a standard 13th-14th
century cadence (Maj6-8 + Maj3-5) resolving to a complete Gothic trine
(2:3:4, here E3-B3-E4), which Pythagorean tuning nicely serves.

In a 16th-century setting, the same two-voice progressions participate
in a different kind of progression from one _harmonia perfetta_ or
5-limit sonority to another, with meantone as a fine realization.

To conclude, I might just observe that Zarlino urges composers ideally
to alternate the natural or "harmonic" division of the fifth --
e.g. C3-E3-G3 -- with the artificial or "arithmetic" division,
e.g. D3-F3-A3. Such alternation is typical in various modes.

This distinct preference in multi-voice harmony produces an aesthetic
rather different from that of the more "polarized" major/minor keys of
the 18th century, where one division tends to prevail.

Most appreciatively,

Margo Schulter
mschulter@value.net