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Conversity, Inversion, and Zarlino's Divisions:
A Curious Corollary from Zarlino's Three-Voice Conversities
by Margo Schulter
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1. Introduction to Conversities and Inversions
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Please let me comment that Jacobus, Zarlino, and Lippius seem to me nicely
to address the musical reality here, and specifically as it affects three or
more simultaneous voices. If an 2:1 octave is divided into 3:2 fifth and 4:3
fourth, then they may be arranged in either order. We might say that the
fifth and fourth are "octave complements." Composers such as Perotin use
both arrangements, but with distinctions: thus 8|5_4 (fifth below) is a
favorite closing sonority; while 8|4_5 (fourth below) also occurs
prominently, but very rarely occurs as a final sonority.
In the _quaternario_ or series of the first four natural numbers 1-2-3-4, we
find both the fifth (3:2) and fourth (4:3); and, in practice, if we have an
octave divided to place the fifth below (e.g. D-A-D), we will have a fourth
above; and conversely, the fourth below means the fifth above (e.g. D-G-D).
Jacobus speaks of such pairs of sonorities as being _e converso_, from which
I draw the English word _conversity_. Conversities in this sense, like 8|5_4
(D-A-D) and 8|4_5 (D-G-D), have the same intervals differently ordered.
Inversions as they are often presented, in contrast, have the same pitch
classes differently ordered, e.g. D3-A3-D4 (8|5_4) and A3-D4-A4 (8|4_5). In
this specific example, D3-A3-D4 and A3-D4-A4 are both conversities
(different arrangements of the same intervals 8, 5, 4) and inversions
(different orderings of the pitch classes D, A).
Likewise, if we divide a 3:2 fifth into a 5:4 major third and 6:5 minor
third, then either the major third may be placed below the minor third
(C-E-G) or the converse (C-Eb-G). These two thirds are "fifth complements"
which both occur within the senario, or series of the first six natural
numbers 1-2-3-4-5-6.
In either of these historical Western European divisions, we might thus
speak of a "duality" between the form where a simpler ratio is placed below
a more complex one -- and its conversity where the more complex ratio
precedes or is placed below the simpler one. Thus:
2:3:4 (6:4:3) D-A-D 8|5_4
3:4:6 (4:3:2) D-G-D 8|4_5
-------
4:5:6 (15:12:10) C-E-G 5|Maj3_min3
10:12:15 (6:5:4) C-Eb-G 5|min3_Maj3
-------
If duality means recognizing that a given division of an outer interval like
the octave or fifth into two concordant adjacent intervals permits two
orderings of these adjacent intervals, then, of course, "duality" in this
sense is real.
Jacobus and others around 1300 address it in their discussions of
three-voice sonorities with an outer octave and the fifth either below or
above the fourth, where Jacobus asserts that both forms are legitimately
concordant, but the former more so, and generally preferable.
Zarlino addresses such "duality," or what I would call conversity, in his
analysis not only of the harmonic and arithmetic divisions of the fifth,
whether we choose to apply "harmonic" and "arithmetic" in terms of string or
frequency ratios, but in considering three-voice sonorities where a major
sixth at 5:3 is divided into a 4:3 fourth and 5:4 major third; or likewise a
minor sixth at 8:5 is divided into a 6:5 minor third and 4:3 fourth.
In the context of practical counterpoint in composition, the focus of Part
III of his Harmonic Institutions (1558), he addresses the division of the
3:2 fifth into a major and minor third in Chapters 10 and 31. On the
division of the 5:3 major sixth or 8:5 minor sixth into the two intervals of
a major or minor third plus a fourth, see Chapter 60. For an English
translation, see Guy A. Marco and Claude V. Palisca, tr., _The Art of
Counterpoint: Part Three of Le Istitutioni harmoniche, 1558_ (Norton, 1976),
Chapter 10, pp. 21-23; Chapter 31, pp. 68-81; and Chapter 60, pp. 190-193.
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2. Zarlino's Three Divisions in the _Harmonic Institutions_ (1558)
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In each case, Zarlino distinguishes between a "natural" arrangement that
follows the order of the senario or "sonorous numbers"; and an "artificial"
arrangement that is contrary to, or one might say the converse of, the order
of the senario. We will see that the "natural" arrangement is often, but not
always, synonymous with a harmonic string-ratio division or arithmetic
frequency-ratio division.
Thus Zarlino's most familiar comparison is that of the two divisions of the
fifth:
6 G4 10 4..G4 15
| 6:5 | 5:4
5 E4 12 5 Eb4 12
| 5:4 | 6:5
4 C4 15 6 C4 10
Zarlino famously observes that when these thirds follow the order of the
senario, with the simpler 5:4 major third below and the more complex 6:5
minor third above, then the effect is more "joyful" or "natural"; while
placing the 6:5 minor third below the 5:4 major third is more "sad," "soft,"
or "artificial."
The first division follows the order 4:5:6 in the senario or later harmonic
series, with a harmonic string division of 15:12:10; while the second
division has an arithmetic string division of 6:5:4, producing the frequency
ratios of 10:12:15.
Here it may be helpful to show that 4:5:6 or 15:12:10 involves a harmonic
string ratio, with differences between adjacent terms that are proportionate
to the ratio (that of the 3:2 fifth) between the outer or extreme terms; and
that 10:12:15 or 6:5:4 involves an arithmetic string ratio with equal
differences between adjacent terms:
Natural division Artificial division
15 12 10 6 5 4
C4-------E4-----G4 C4------Eb4------G4
3 2 1 1
---
The question of dividing the 5:3 major sixth is very interesting, because,
as Zarlino notes, it raises questions about certain conventions of
16th-century practice.
5 E4..12 3 C5 20
| 5:4 | 4:3
4 C4 15 4 G4 15
| 4:3 | 5:4
3 G3 20 5 Eb4 12
Zarlino finds that the first division has the 4:3 fourth below the 5:4 major
third, thus following the senario, which 4:3 precedes 5:4. Placing the major
third below the fourth, in contrast, reverses this "natural" order, and thus
may be considered "artificial." We may further note that these two
arrangements represent the harmonic and arithmetic string-ratio divisions of
the 5:3 major sixth:
Natural division Artificial division
20 15 12 5 4 3
G3-------C4-----E4 Eb4------G4------C5
5 3 1 1
With the natural division, we find that the adjacent differences of 5 and 3
are proportioned to the ratio of the outer 5:3 major sixth, a harmonic
division; with the artificial division, the differences are equal, an
arithmetic division.
Zarlino also notes an interesting tension between his analysis, showing that
3:4:5 or 20:15:12 should be more "natural" and pleasing than its conversity,
and usual practice, which treats cautiously the harmonic and natural
division of the major sixth because it has a fourth above the lowest voice,
while it freely admits the arithmetic or "artificial" division where this
fourth occurs between two upper voices. This cautious approach to the fourth
goes back to the earlier 14th century, and I wonder if Zarlino might have
been familiar with the argument of Jacobus (Speculum musicae, c. 1330, Book
VII, Chapters 5-8) that the fourth is a legitimate concord in its own right,
and whether placed above or below the fifth, etc, although placing it above
the fifth is more pleasing.
At any rate, Zarlino notes the prevailing practice, while observing that the
3:4:5 sonority, because of its smooth and "natural" qualities, might well be
treated more freely.
---
Zarlino's third pair of divisions involves the 8:5 minor sixth, the one
concord in his system (adopted by Lippius) which cannot be found within the
senario itself, but requires the additional term 8. Here his verdict accords
with standard period practice, and involves some complications:
8 C5 15 5 Eb4 24
| 4:3 | 6:5
6 G4 20 6 C4 20
| 6:5 | 4:3
5 E4 24 8 G3 15
Zarlino finds that the first arrangement with the 6:5 minor third below and
the 4:3 fourth above is preferable, while the second is "nearly dissonant"
(as it is also in conventional practice, mainly because of the fourth above
the lowest part).
An interesting point is that here, unlike the other two divisions, the more
complex superparticular (n+1:n) ratio of 6:5 is placed below rather than
above the simpler ratio of 4:3 -- in contrast to his preferred 4-5-6 or
3-4-5. Yet, as Zarlino argues, it is true that the terms 5-6-8 occur in this
order in the senario-plus-8 series that he uses (as will Lippius in 1612);
and also that there is no location in this series where the 4:3 fourth is
_directly_ followed by the 6:5 minor third (since there is, of course, an
intervening 5:4 major third). In contrast, 4-5-6 and 3-4-5 involve two
ratios (5:4 and 6:5; or 4:3 and 5:4) that are adjacent in the series.
The 8:5 minor sixth differs from the 3:2 fifth or 5:3 major sixth in being
an interval which not only is outside the senario proper, but is incapable
within Zarlino's realm of acceptable concords (those involving odd factors
no greater than 5, as with 2:1, 4:3, 3:2, 5:4, 6:5, 5:3, and 8:5) of either
a harmonic or arithmetic division. Looking at the differences of terms for
Zarlino's two divisions of 8:5 will confirm that neither is a harmonic or
arithmetic division:
Natural division Artificial division
24 20 15 8 6 5
E4------G4-------C5 G3--------C4----Eb4
4 5 2 1
A harmonic division of the 8:5 minor sixth would have adjacent differences
with a ratio likewise of 8:5, while an arithmetic division would have equal
differences. Here the 24:20:15 string division has differences of 4 and 5,
while the 8:6:5 division has differences of 2 and 1.
Another approach might be to argue that if we focus on frequency ratios,
then the "natural" 5:6:8 seems simpler than the "artificial" 15:20:24, just
as 4:5:6 is simpler than 10:12:15, and 3:4:5 is simpler than 12:15:20.
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3. Zarlino's Comparisons of Divisions and Conversities
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While Zarlino's contrast between the more "natural" or "cheerful" 4:5:6 (or
15:12:10), i.e. 5|Maj_min3 in the senario, and the more "artificial" or
"sad" or "soft" 6:5:4 (or 10:12:15), i.e. 5|min3_Maj3, are well known, his
comparison of the concordant sixth sonorities are also noteworthy.
Zarlino sums up his views from one perspective by saying:
"[T]he fourth accompanied by a minor third beneath or a major third
above may always be used with good effect, but the opposite
arrangements will be somewhat poor." (Marco and Palisca, Chapter 60,
pp. 191-192.)
Thus Zarlino prefers Maj6|4_Maj3 (3:4:5 or 20:15:12) to Maj6|Maj3_4
(10:12:15 or 5:4:3); and min6|min3_4 (5:6:8 or 24:20:15 to min6|4_min3
(8:6:5 or 15:20:24). To explain these contrasts, Zarlino compares music to
the visual arts, or hearing to seeing:
"This should not seem strange to anyone. Hearing is 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." (Ibid., p. 192.)
In dealing with divisions of the fifth and the major and minor sixth,
Zarlino more generally has observed: "[W]henever consonances are placed in
an order other than the natural one of the sonorous numbers, a certain
dissonance will result." However, such an element of "dissonance," or we
might say of less ideal concord, may be justified in Zarlino's view for at
least two reasons.
The first is the desire for artistic diversity of concords and their
arrangements, as with the frequent alternation of the harmonic and
arithmetic divisions of the fifth with their different ordering of 5:4 major
and 6:5 minor thirds (e.g. Chapter 31, p. 70). The second is the desire to
fit the music to the meaning or emotional qualities of a text, as opposed to
a situation where "sad music would be set to happy words." In Chapter 57, at
p. 177, Zarlino notes how melodic steps, simple intervals, the placement of
vertical intervals in multivoice sonorities, can help to express a happy,
soft or sad, or "harsh" affect.
Returning to our immediate focus on sixth sonorities for three voices, we
find that Zarlino, in addition to preferring Maj6|4_Maj3 to Maj6|Maj3_4 and
min6|min3_4 to min6|4_min3, makes some comparisons between three pairs of
sonorities which are not conversities. Thus we have:
(p. 192, Ex. 135)
F4 A4
C4 E4
A3 C3
"Good" "Less good"
Here the comparison is between sonorities with an outer sixth and a lower
third, generically 6|3_4, where A3-C4-F4 or min6|min3_4 (5:6:8 in the
senario) is preferable to C3-E4-A4 or Maj6|Maj3_4 (12:15:20, the arithmetic
division of the 5:3 major sixth). In conventional 16th-century practice,
both arrangements would be considered concords, with a lower third and a
fourth between the two upper voices.
(p. 192, Ex. 136)
A4 F4
F4 D4
C4 A3
"Good" "Not good"
Here the comparison is between sonorities with an outer sixth and a lower
fourth, generically 6|4_3, where the fourth above the lowest voice calls for
caution in standard 16th-century practice. Zarlino prefers C4-F4-A4 or
3:4:5, the harmonic division of the 5:3 major sixth, to A3-D4-F4 or
15:20:24, which does not follow the order of the senario.
In these comparisons, as we might expect, Zarlino prefers a "natural"
arrangement to one he considers "artificial," as judged by the order of
intervals in the senario or series of "sonorous numbers" (including 8 when
the 8:5 minor sixth is involved). However, a third comparison is between two
"natural" divisions:
(p. 193, Ex. 137)
A4 F4
F4 C4
C4 A3
"Good" "Better"
The sonorities here are the "good" Maj6|4_Maj3, the 3:4:5 harmonic division
of the major sixth; and the "better" min6|min3_4, 5:6:8, which likewise
follows the order of the "sonorous numbers" although it is neither a
harmonic nor an arithmetic division. It is interesting that Zarlino prefers
the second form, which has the fourth above rather than below the third, and
so would also be considred more concordant in standard practice.
Zarlino sums up his conclusions from these comparisons by indicating that he
generally agrees with standard usage, but finds that the Maj6|4_Maj3
division or 3:4:5 should logically be accepted as agreeably concordant also:
"In sum, fourths are best used when a fifth or third is heard below
them, as I have shown. A third above may also be used if it is major,
though this use has received little acceptance by musicians. If the
fourth accompanied by a major third below, which is not very
consonant, is tolerated, there is no reason for denying a place to the
fourth with major third above, a coupling that is better, as
experience always shows." (Chapter 60, p. 193.)
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4. A Corollary of Zarlino's Approach: Inversional Affinity
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Zarlino's approach, like that of the earlier Jacobus in a medieval context,
generally focuses on conversity rather than inversion: that is, different arrangements
of the same intervals rather than the same pitch classes. However, Zarlino's
discussion of natural and artifical divisions of the outer fifth, major
sixth, and minor sixth, often but not always synonymous with harmonic and
arithmetic string-ratio divisions respectively, has an interesting
corollary.
Let us begin with a harmonic division of the fifth, e.g. C4-E4-G4, and an
arithmetic division, e.g. C4-Eb4-G4 (the examples I used in Section 2
above). Then, applying the mostly later concept of inversion, we can derive
these forms:
Natural Artificial
6 G4 10 4 G4 15
| 6:5 | 5:4
5 E4 12 5 Eb4 12
| 5:4 | 6:5
4 C4 15 6 C4 10
5|Maj3_min3 5|min3_Maj3
8 C5 15 3 C5 20
| 4:3 | 4:3
6 G4 20 4 G4 15
| 6:5 | 5:4
5 E4 24 5 Eb4 12
min6|min3_4 Maj6|Maj3_4
5 E4 12 5 Eb4 24
| 4:3 | 4:3
4 C4 15 6 C4 20
| 6:5 | 5:4
3 G3 20 8 G3 15
Maj6|4_Maj3 min6|4_min3
Note that if a given arrangement is natural (here C4-E4-G4) or artificial
(here C4-Eb4-G4), then other arrangements or inversions of the same pitch
classes will also yield the same type of arrangement. We might thus speak of
"inversional affinity."
Conversity is a central concept for Jacobus, and later for Zarlino.
Inversion would seem to be mainly a later focus, but the fact that in
Zarlino's system what we might call inversions of a given natural or
artificial sonority always produce another natural or artificial
sonority, respectively, is worth noting.
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5. An Aside: Arithmetic and harmonic divisions of the 8:5 minor sixth
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Using the recognized consonances of Zarlino and Lippius, the 8:5 minor
sixth, unlike the 3:2 fifth or 5:3 major sixth, cannot be divided
harmonically or arithmetically into two concordant intervals. Of course, the
divisions of 5:6:8 (preferred) or 8:6:5 are possible. The latter, which
Zarlino regards as "almost dissonant" because the intervals are the converse
of their "natural" arrangement of 5:6:8 and because the fourth is below
rather than above the third, may from another perspective be ideal at a
point where consideration tension may be welcome, as in this cadential
idiom:
C4 ---- B3 -------- C#4
A3 --------- G#3 -- A3
E3 ---------------- A2
Here the sonority E3-A4-C4 at 8:6:5 or 15:20:24 has a tension that nicely
leads into the following suspended sonority of E3-A4-B3 (6:8:9 in JI), with
a 4-3 resolution arriving at a harmonic division of the fifth (E3-G#3-B3)
followed by a resolution of the kind described by Zarlino, where the bass
and one of the upper voices progress from a major third or tenth to a
unison, octave, or fifteenth, with the bass falling a fifth or rising a
fourth (here E3-A2) while this upper voice ascends by a semitone (G#3-A3).
This three-voice formula also illustrates a nuance which becomes more common
around the second third of the 16th century in pieces for three or more
voices. In a more traditional version of this cadence, the Maj3-8 or similar
resolution of the bass and middle voice would be supplemented by a customary
min3-1 or Maj6-8 resolution between the upper voices:
C4 ---- B3 -------- A3
A3 --------- G#3 -- A3
E3 ---------------- A2
Here we have the superimposed directed resolutions (Maj3-8 + min3-1), with
min3-1 and Maj6-8 as standard medieval resolutions codified in the 14th
century as "closest approach" progressions, and endorsed by Zarlino as the
ideal closes for two-voice compositions (see Marco and Palisca, Chapter 53,
pp. 141-151). However, the disadvantage of this typical three-voice formula
from the late 15th or early 16th century is that it concludes with a simple
octave, rather than realizing or approaching the _armonia perfetta_ or
complete sonority sought be Zarlino and his contemporaries such as Willaert,
with a third and fifth or their octave extensions above the bass.
Having the highest voice conclude B3-C#4 rather than B3-A3 permits arrival
at the sonority A2-A3-C#4, or 2:4:5, with an outer 5:2 major tenth divided
into a lower 2:1 octave and upper 5:4 major third. As a result, the two
upper voices, which might be expected to resolve min3-1, instead move in
parallel thirds, G#3-B3 to A3-C#4.
However, let us move from 16th-century style to the question of arithmetic
and harmonic divisions for 8:5. If we are thinking in terms of string
ratios, then the arithmetic division is easy to find:
359.5 454.2
16:13 13:10
|------|------|
16 13 10
|-------------|
8:5
814.7
In modern JI or tempered styles of music, the ratio of 16:13 at 359.5 cents
may represent a medium large middle or neutral third, a common ratio going
back to Ibn Sina (c. 980-1037) in Near Eastern theory. A ratio of 13:10 or
454.2 cents is in the region somewhere between a very large major third
(e.g. 9:7, 435.1 cents) and a very narrow fourth (e.g. 21:16, 470.7 cents),
and might be perceived as either depending on the context.
In a harmonic string division (or arithmetic frequency division), the ratios
follow the order of the harmonic series, thus 10:13:16. If we think in terms
of frequency ratios, this is very easy to find, being the conversity of our
16:13:10 sonority:
454.2 359.5
10:13 13:16
|------|------|
10 13 16
|-------------|
8:5
814.7
However, if we wish to calculate the string lengths for this harmonic
division, there is a handy shortcut which may go back to classic
times. We start with the terms for the arithmetic division of
16:13:10, and find the three terms by multiplying the middle term by
the greatest term (208); the middle term by the least term (130); and
then the largest and smallest terms to find the harmonic mean (160).
Thus we have 208:160:130, or this division:
454.2 359.5
10:13 13:16
|----------------|----------|
208 160 130
48 30
Here we may confirm that the difference of the larger pair of terms, or 48,
has a ratio to the difference of the smaller pair of terms, 30, of 48:30 or
8:5, the same as that of the outer minor sixth being divided.
While this division is correct, it is not expressed on the smallest possible
terms, something which can be achieved by halving the terms:
454.2 359.5
10:13 13:16
|----------------|----------|
104 80 65
24 15
Another method for finding the harmonic string division for 8:5 is to take
the sum of these terms, 8+5 or 13, and multiply each term by this number to
obtain the correspondencing string lengths for the outer notes of the string
division, here 8x13 or 104 and 5x13 or 65. Note that the difference of these
lengths is 39. Now, from the longer length of 104, we subtract 8/13 of this
difference, or 24, arriving at 80 for the harmonic mean.
Margo Schulter
12 July 2018