Re: Zarlino's harmonic theory (Jon Wild, Paul Erlich)

--------------------------------------------
Zarlino's theory of three-voice sonorities
A dialogue with Jon Wild and Paul Erlich
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"Hearing is much like seeing, and it is just as
peculiar to hear something in place of another
thing as it is to see one thing in place of
another, such as the foundation in place of
the roof, or windows where doors should be.
It is strange to see things arranged contrary
to their natural order and without proportion,
and equally strange to hear a mass of sounds
or consonances combined without proportion and
out of their natural places."
-- Gioseffo Zarlino
_Harmonic Institutions_ (1558)
Book III, Chapter 60

Hello, there, and in response to comments by Jon Wild and Paul Erlich,
I'd like to offer some remarks which might serve as notes for a more
coherent article on Zarlino and his comparison of the different forms
of _harmonia perfetta_ or saturated 5-limit sonorities.

One problem of conventional music history, for the most part, has been
a tendency to approach Zarlino in terms of the presence or absence of
18th-century concepts. As someone oriented to medieval and Renaissance
music, I may have an opposite tendency sometimes to approach him in
terms of 14th-century concepts <grin>, which maybe by a certain law of
averages may give more impartial observers some basis for a balanced
view.

Here I would like to consider Zarlino's approach to three-voice
sonorities, focusing on a concept I might term "conversity" rather
than 18th-century inversion, and to show that his classification of
saturated 5-limit sonorities as "natural" or "artificial"
interestingly leads to a conclusion similar to that of a theory of
inversional affinity, although Zarlino himself to my knowledge does
not himself draw it, at least in Book 3 of the _Harmonic Institutions_
(1558).

Jon Wild writes:

> Speaking of major/minor and triads, it might be important to
> remember that Zarlino considered E-G-C, say, a *minor* sonority
> because it has a minor 3rd and 6th above the bass. There is still no
> recognition in Zarlino of inversional equivalence of triads. Lippius
> has been mentioned by Monz and Margo, and he's the guy to thank for
> that.

As one whose own viewpoint tends to be closer to Zarlino than to
18th-century theory, I would certainly agree that the association of
the minor third with the _minor_ sixth, and the major third with the
_major_ sixth, is an important point on which the two approaches may
differ.

From a dynamic viewpoint, both minor third and minor sixth are
typically contractive intervals, as I call them, the minor third often
resolving by stepwise contrary motion to the unison (m3-1), and the
minor sixth by oblique motion to the fifth (m3-5).

In contrast, the major third and major sixth are expansive, typically
progressing by stepwise contrary motion to fifth (M3-5) or octave
(M6-8).

Affectively, as Zarlino points out, the major third and major sixth
are "bright" and joyful, while the minor third and sixth alike are
somewhat "sad" or "languid."

These tendencies can also apply in various multi-voice sonorities and
progressions. For example, using a MIDI-style notation where C4 is
middle C:

G4 A4 A4 G#3
D4 F4 D4 E4
B3 C4 A3 B3
G3 F3 F3 E3

(M3-5) (M6-8 + M3-5)

Here both the G3-B3-D4-G4 and F3-A3-D4-A4 sonorities are "expansive,"
featuring two-voice progressions to the "nearest consonance" of M3-5, and
also M6-8 in the second example.

At the same time, however, in Zarlino's analysis the first sonority
represents a "natural" arrangement of the consonances, and the second
an "artificial" arrangement -- concepts which may roughly correlate
with the 18th-century distinction between major and minor triads, or
possibly the 20th-century distinction between otonal and utonal
5-limit or higher sonorities.

In distinguishing between natural and artificial arrangements, Zarlino
uses two criteria: the contrast between the harmonic and arithmetic
divisions of an outer interval such as the fifth into major and minor
third; and the ordering of intervals in a saturated 5-limit sonority
as compared to that of the "sonorous numbers."

First let us take a look at Zarlino's scheme of six basic varieties of
three-voice sonorities, with the numbers next to each note showing
string lengths:

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

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

| A3 4 | D4 3 | C4 5
Artificial | (M3) | (4) | (m3)
arrangement 5 | F3 5 M6 | A3 4 m6 | G3 6
| (m3) | (M3) | (4)
| D3 6 | F3 5 | E3 8

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

Gothic and Renaissance theory focuses on what I shall term
_conversity_, the relationship between sonorities sharing the sets of
_intervals_ differently arranged. In our table, this relationship
obtains between the three vertically aligned pairs or columns of
sonorities.

These pairs are the alternative arrangements or permutations of three
basic types of sonorities discussed by Zarlino: a fifth divided into
major and minor third (5,M3,m3); a major sixth divided into major
third and fourth (M6,M3,4); and a minor sixth divided into minor third
and fourth (m6,m3,4).

We may describe these paired arrangements compared by Zarlino as
"conversities," because their adjacent intervals are conversely
ordered. Thus the natural arrangement of (5,M3,m3) has the major third
below and the minor third above (e.g. C3-E3-G3), while the artificial
arrangement conversely has the minor third below (e.g. D3-F3-A3).

The two horizontal rows of the table show the three "natural" or more
harmonious arrangements of Zarlino, and the three "artificial" or less
harmonious ones.

In order to illustrate an interesting property of this system which I
am not sure whether Zarlino himself mentioned at any point in his
writings, I have derived all three sonorities of a given row from the
same set of pitch classes -- (C,E,G) for the upper or "natural" row
and (D,F,A) for the lower or "artificial" row.

As this exercise shows, transposing the pitches of a Zarlinan natural
sonorities will produce another natural sonority, e.g. C3-E3-G3,
G3-C4-E4, E3-G3-C4. Similarly, transposing the pitches of an
artificial sonority produces another artificial sonority,
e.g. D3-F3-A3, F3-A3-D4, A3-D4-F4.

Thus we might speak of an "inversional affinity" between Zarlino's
natural and artificial sonorities, while at the same time recognizing
important dynamic and affective contrasts between these different
interval sets. For example, both the natural C3-E3-G3 and the
artificial F3-A3-D4 with their major thirds and sixths are expansive,
while the natural E3-G3-C4 and the artificial A3-D4-F3 with their
minor thirds and sixths are contractive.

How does Zarlino distinguish between natural and artificial
arrangements? He uses two methods, the first of which I shall term the
"harmonic/arithmetic" contrast, the other the "senarial" approach,
from his _senarius_ or _senario_, the series of the first six natural
integers or "sonorous numbers" used to define consonant ratios.

Especially for the interval set (5,M3,m3), Zarlino frequently uses the
harmonic/arithmetic approach, based on string-ratios.

In the harmonic division of the fifth, the ideal form of _harmonia
perfetta_ or complete 5-limit harmony, with the major third below and
the minor third above, we have a string-ratio of 15:12:10, where the
differences between the two two pairs of adjacent terms form a ratio
identical to that between the two outer terms. Note that since we are
dealing with string-ratios, the larger number represents the lower
string and note:

3 : 2
|------------------|
C3 E3 G3
15 12 10
(15-12) (12-10)
3 : 2

In contrast, the "artificial" arrangement with the minor third below
the major third, 6:5:4, has a less harmonious arithmetic division,
with equal differences between adjacent terms:

3 : 2
|------------------|
D3 F3 A3
6 5 4
(6-5) (5-4)
1 : 1

We can also apply this approach to the set (M6,M3,4), where the more
natural harmonic division, 20:15:12, has the fourth below the major
third:

5 : 3
|------------------|
G3 C4 E4
20 15 12
(20-15) (15-12)
5 : 3

In contrast, the artificial arrangement with the major third below the
fourth has the less harmonious arithmetic division 5:4:3 with its
equal differences between adjacent terms:

5 : 3
|------------------|
F3 A3 D4
5 4 3
(5-4) (4-3)
1 : 1

With two arrangements of the set (m6,m3,4), however, this approach
based on the harmonic/arithmetic contrast will not work, because
neither 5-limit division of the 8:5 minor sixth -- 24:20:15 or 8:6:5
-- is an harmonic or arithmetic division. To achieve such a division,
we would need to go to the 13-limit, 104:80:65 (harmonic) or 16:13:10
(arithmetic). Such ratios are beyond the scope of Zarlino's 5-limit
system.

However, Zarlino uses another approach which can apply to all three
interval sets, the criterion of the ordering of the "sonorous
numbers," an approach also used to compare voice-spacings.

In an ideally harmonious sonority, the vertical concords would be
ordered in a series of ratios becoming more complex in ascending
order. Here, again, numbers next to notes show string lengths:

C5 15
8:5
G4 20
6:5
E4 24
5:4
C4 30
4:3
G3 40
3:2
C3 60
2:1
C2 120

The first five ascending intervals of the series follow the order 2:1,
3:2, 4:3, 5:4, and 6:5 in a simple and elegant way; these are the
basic consonances found within the senario itself (1,2,3,4,5,6). To
account for the minor sixth at 8:5 (here E4-C5), however, it is
necessary to add the term 8, not itself within the senario.

Using this ideal sonority based on "the sonorous numbers," Zarlino
finds that in a combination of minor sixth, minor third, and fourth,
the minor third "naturally" occurs _below_ an adjacent fourth, here
E4-G4-C5, i.e. 6:5 below 8:6.

In contrast, the major third (of M6,M3,4) occurs naturally _above_ an
adjacent fourth, here G3-C4-E4, i.e. 5:4 above 4:3.

Applying this approach to (5,M3,m3), we find likewise that the major
third occurs below the minor third, here C4-E4-G4, i.e. 5:4 below 6:5.

In short, Zarlino's "natural" and "artificial" divisions -- synonymous
for (5,M3,m3) and (M6,M3,4) with the harmonic and arithmetic divisions
-- are apparently analogous to the later concepts of major/minor or
otonal/utonal 5-limit triads.

A full appreciation of Zarlino's system must include a recognition,
for example, that F3-A3-D4 is at once an expansive sonority (M6-8,
M3-5) and an artificial one.

Whether or not Zarlino ever noted it in his writings, the inversional
affinity between the three natural divisions, and likewise between the
three artificial divisions, is an interesting feature of his system.

> It's also interesting that Zarlino can't escape the conclusion that
> the 6/4 chord 3:4:5 "should" be more consonant, according to his
> theory, than the 5/3 4:5:6 - just like Helmholtz, 300 years later. I
> can't quite remember how he squirms out of it - does anyone have it
> handy to check?

Actually I would say that he doesn't go this far, but does conclude in
Chapter 60 of Book III that the "natural" C4-F4-A4, for example, with
the fourth below and the major third above, is more harmonious than
the converse arrangement of these intervals, e.g. C3-E3-A4. However,
while finding the former arrangement "good," he finds the minor third
below the fourth "better," e.g. A3-C4-F4. Both C4-F4-A4 and A3-C4-F4
are natural divisions, but he finds the second more harmonious.

The conclusion that C4-F4-A4 is actually more harmonious than C3-E3-A3
nicely fits his viewpoint that the 4:3 fourth is properly regarded as
a perfect concord in the same category as the 3:2 fifth.

While recognizing that musicians (i.e. of the Renaissance tradition
with which he is familiar) have treated the fourth with considerable
caution, he urges that it reasonably might be treated more freely,
with a sonority such as C4-F4-A4 serving as an example.

His point is that if an "artificial" arrangement like C3-E3-A3 is
freely permitted as concordant, why not the "natural" arrangement with
the fourth below and major third above.

Interestingly, to demonstrate the consonant nature of the fourth he
relies not only on its mathematical simplicity, but also on a bit of
ethnomusicology, pointing out that the Greeks of Venice use it above
the bass in their part-songs, possibly referring to the Greek Orthodox
liturgy.

If he had had available a copy of the Buxheim Organ Book from the
middle to later 15th century, he might also have cited some examples
possibly illustrating the use of Pythagorean schisma thirds and sixths
in the epoch leading up to the introduction of meantone keyboard
temperaments, say 1430-1450.

For example, some pieces in this collection feature phrases ending on
prolonged noncadential sonorities like this one:

E4
C#4
A3
E3

If we play this in a Pythagorean tuning where the written C#4 is
realized as Db4, we get something only a schisma (32805:32768 or ~1.95
cents) from a pure 3:4:5:6. This seems an example of a fourth above
the bass being regarded as part of a euphonious tertian sonority,
albeit an inconclusive one.

While Zarlino's view of the fourth as a perfect concord leads him to
propose its freer or more consonant treatment in practice, he also
proposes that parallel fourths should be avoided on the basis in
Renaissance theory and practice as parallel fifths. He especially
disapproves of what he calls _falsobordone_ in the sense of the
traditional fauxbourdon with its chains of upper parallel fourths.

Vicentino (1555) likewise finds this idiom more tedious than "sweet"
because of the similarity of the sonorities, although he
characteristically is ready to permit it to express the affections of
a text (properly tension rather than sweetness, in his view).

Lippius (1610, 1612) interestingly agrees with Zarlino about the
fourth, classifying it as a perfect concord, and saying that musicians
who regard it as a dissonance suffer from "delirium."

> Clearly, triadic harmonic entropy can't escape that either. Hence I
> invoke an additional concept, "rootedness", which is justified
> partially by the masking phenomena in the ear which make the lowest
> note of each of these triads sound the loudest.

Might this tie in with Zarlino's opinion that C4-F4-A4 is "good," but
C3-E3-A4 is "better"?

Also, of course, Zarlino says that _harmonia perfetta_ normally
consists of "the third plus fifth," i.e. above the lowest note,
although sometimes "the sixth is used in place of the fifth," implying
priority to the former kind of combination.

Most appreciatively,

Margo Schulter
mschulter@value.net