-------------------------------------------------------
           The Cheshire Cat Cadence in 16th-Century Polyphony: 
                Verdelot, Zarlino, and _Armonia Perfetta_
         -------------------------------------------------------

Here I would like to focus on my view of a standard four-voice cadence
described by Vicentino (1555), for example, and then a modified version
which seems one focus of recent musicological literature and discussions
about 16th-century cadences.

The theme of the Cheshire Cat of _Alice in Wonderland_ fame that can become
invisible except for a distinctive smile is something I associated around
the mid-1980's with the modified four-voice formula I'll describe.


------------------------------------------------------
1. The fully visible cat: The Maj6-8 or min3-1 cadence
------------------------------------------------------

Our cat at issue, made fully visible in the standard cadence with which we
begin, is the classic two-voice resolution by stepwise contrary motion of
major sixth to octave (Maj6-8) -- or, sometimes, minor third to unison
(min3-1). In the final cadence to Philippe Verdelot's _Madonna, qual
certezza_ (from his _First Book of Madrigals, 1533), each voice plays the
melodic role deemed most characteristic for it by Nicola Vicentino (1555).
Here I follow the bracketed and suggested editorial accidentals of Bernard
Thomas, and note that the piece is cantus mollis with a Bb signature in all
voices[1]:

1.......+....+....2......+......|.1
                         #
A4------G4---------------F4-----..G4
E[b]4-------------D4------------..D4
C4-----------Bb3--A3------------..G3
C3------C3--------D3------------..G2

If we look at the last two notes of each voice, the soprano has Vicentino's
"soprano cadence" with an ascending semitone (F#4-G4), while the alto has an
"alto cadence" involving a repeated stationary note (D4-D4). The tenor has a
"tenor cadence" with a descending tone (A3-G3), forming with the soprano the
typical two-voice formula of a 7-6 suspension followed by a Maj6-8
resolution (A3-F#4 to G3-G4). The bass has a "bass cadence," here falling a
fifth (D3-G2), and sometimes rising by a fourth.

Thus our familiar Maj6-8 cat is visible in all her glory between tenor and
soprano, which might be taken as a traditional discant duo around which the
other voices could be shaped. However, Zarlino in 1558 points to another
characteristic feature of multivoice cadences (_Harmonic Institutions_, Part
III, Chapter 61, here assuming a three-voice texture: "Furthermore, in the
principal cadences the parts should be so arranged that the dissonant second
part of a syncopated note is always a fourth or eleventh above the bass and
a second or seventh from the other voice."[2]

Here indeed the suspended G4 in the soprano is an eleventh above the bass
(D3-G4) and a seventh above the other relevant voice, the tenor with A3
(A3-G4). These suspensions resolve, respectively, 11-10 and 7-6. 

Interestingly, in his earlier discussion of two-voice writing, Zarlino
recognizes the progression of the two outer voices, with the 11-10
suspension and then a resolution where the upper voice ascends by a semitone
while the lower voice falls a fifth (as here) or rises a fourth, as a
cadence in its own right, although more typical of pieces for three or more
voices, where the leaping motion of the lowest part is more characteristic.
However, while he would prefer to do otherwise, he recognizes that "it would
be no great error" to begin or end a two-voice composition in this manner.[3]

Thus we can view Verdelot's standard cadence, with each of the four voices
following its characteristic melodic pattern according to Vicentino, as a
harmonious superimposition or union of two basic two voice resolutions: the
classic 7-6 suspension followed by Maj6-8 (tenor-soprano); and the 11-10
suspension followed by Maj10-15 (bass-soprano).

A final consideration is the closing sonority: G2-G3-D4-G4, which in a just
tuning would have a pure ratio of 1:2:3:4, with a 2:1 octave (G2-G3); 3:2
fifth (G3-D4); and 4:3 fourth (D4-G4). This is the "natural series" which is
the basis for stable concord in 13th-14th century European polyphony, as
described in the treatise _Compendium de musica_ sometime in the early to
middle 14th century. The upper three voices would form the most complex
stable sonority and ideal goal for a final cadence in 13th-14th century
polyphony: what Johannes de Grocheio (c. 1300) calls the _trina harmoniae
perfectio_ or "threefold perfection of harmony" with outer octave, lower
fifth, and upper fourth (e.g. G3-D4-G4).
<a href="http://www.bestII.com/~mschulter/Grocheio1.txt></a>
As Jacobus of Liege (c. 1330) notes, this is the best way to end a measured
composition for three or more voices.


-------------------------------------------------
2. The Cheshire Cat: An ascending semitonal smile
-------------------------------------------------

Having met our Maj6-8 cat in a familiar guise, let us now turn to a cadence
on G in Verdelot from another madrigal of his First Book, _Donna leggiadr'
e bella_. Here the largely declamatory, note-against-note texture, the
arrival at the cadence of all voices at the end of a poetic line, _Il di`
che la mia stella_, and a subsequent rest in each voice make the cadential
nature of the passage clear[4]:

...|1......+......2......+......3.....+......|.1....+.....2......|..1
   |E4-------------------E4-----F4----E4------.D4---------D4-----...r
   |C4-------------------C4-----B4----C4------.A3---------B3-----...r 
G3-|-------G3-----A3-----G3-----F3----G3-----------F[#]3--G3-----...r
...|C3-------------------C3-----D3----C3------.D3---------G2-----...r

Here three of the four voices proceed with Vicentino's melodic formulae,
albeit with some roles interchanged, as he notes may happen: thus the
soprano has an "alto cadence" with a repeated note (D4-D4); the tenor has a
"soprano cadence" ascending by a semitone (F#3-G3); while the bass has its
usual "bass cadence," here again falling by a fifth (D3-G2).

We also find Zarlino's characteristic element of multivoice cadences, the
suspension of a fourth in the tenor (D3-G3) resolving to a major third
(D-F#3), which then in turn cadences on an octave (G2-G3).

How about our familiar cat, the Maj6-8 or min3-1 cadence, introduced by a
7-6 or 2-3 suspension, that could be viewed as the traditional nucleus of
this four-voice close? We indeed find that, just as Zarlino described, the
suspended note G3 in the tenor forms not only a fourth with the bass, but a
second with the alto (G3-A3), and that this dissonance has its usual 2-3
resolution to D3-F#3. So far, so good.

However, the alto part, rather than having Vicentino's "tenor cadence"
moving down a tone to complete a min3-1 resolution with the tenor (F#3-A3 to
G3-G3), moves up a step to B3 -- instead of min3-1 by stepwise contrary
motion, we have parallel thirds (F#-A, G3-B3). If the middle voices alone
were the lines of a two-voice composition, we would call this some kind of
averted or evaded cadence.

However, the bass-tenor pair with its formula of a 4-3 suspension, and then
Maj3-8, remains as a distinct two-voice cadence -- and the modified motion
of a step up rather than a step down in the alto makes possible arrival at a
sonority of G2-G3-B3-D4. Assuming a modern 16th-century intonation of the
kind described by Zarlino and approximated on keyboards through some shading
of meantone temperament, this sonority under Zarlino's analysis would follow
the series of natural numbers 2-4-5-6. That is, we have a 4:2 (or 2:1)
octave G2-G3; a 5:4 major third G3-B3; and a 6:5 minor third B3-D4. In terms
of string ratios, the lengths would be 30:15:12:10.

As Zarlino explains, a composer writing for three or more voices should seek
to maintain _armonia perfetta_, in which the bass or lowest voice has above
it as consistently as possible both a third and a fifth, or sometimes a
third and a sixth -- or else octave extensions or one or both notes.[5] In
G2-G3-B3-D4, both the major tenth G2-B3 (an extension of the major third),
and the twelfth G2-D4 (an extension of the fifth G2-G3) are present, so we
have indeed arrived at a full _armonia perfetta_.

In the process, however, our fully visible Maj6-8 cat of the first Verdelot
cadence, which here would have been min3-1 between the middle pair of voices
(with its typical 2-3 suspension), here becomes the "Cheshire cat" smile of
the expected 2-3 suspension followed by the usual "soprano cadence" or
ascending semitone in the tenor part -- with the middle voices proceeding in
parallel thirds rather than arriving at the customary closing perfect
consonance, the unison. It is the reduction of the cat to the Cheshire Cat
smile that provides a harmonious major third between these voices at the
close -- and a euphonious major tenth between bass and alto (G2-B3).


----------------------------------------------------------------
3. Armonia perfetta in four parts: Only the Cheshire Cat will do
----------------------------------------------------------------

As it turns out, in four-voice writing, only the Cheshire Cat modification
makes it possible to arrive, with the otherwise customary cadential motions
and gestures, at a closing sonority with both the third and the fifth, or
their octave extensions. As Verdelot shows in the final cadence of another
madrigal from his First Book, _Piove dagli occhi della donna mia_, it is
possible to keep the traditional Maj6-8 resolution and have a third in
the closing sonority -- but then the fifth must be omitted[6]! 

...|1...+...+.....2..+...+.....|.1....+.....2....3.....||
                                      #
   |D4------E4-------D4--D4----|.-----C4---.D4----------
A3-|----B3--C3----A3-----G3----|.A3---------F[#]3-------
---|F3------G3----F3-----D3----|.E3---------D3----------
D3-|--------C3----A3-----B[b]2-|.A2---------D3----------

This variation has our Maj6-8 cat fully visible between the tenor-soprano
pair, with their respective usual motions as described by Vicentino; the
bass likewise makes its usual "bass cadence," here rising by a fourth. It is
the alto that is altered: rather than remaining stationary on A3, it
descends by a minor third to F#3, a major third above the bass and tenor at
a unison on D3.

We also have the other typical two-voice cadence, between the outer voices,
with an 11-10 suspension and then Maj10-8 (A2-C#4 to D3-D4). On a keyboard
tempered in 1/4-comma meantone, it might be argued that this formula rather
than Zarlino's _armonia perfetta_ is in a sense the most ideally consonant,
since D3-F#3-D4 will have pure rations of 4:5:8, with the octave divided
into a pure 5:4 major third D3-F#3 and a pure 8:5 minor sixth F#3-D4. In
contrast, the fifth in this temperament is narrow by 5.377 cents, as
compared to 1.955 cents in the 12-tone equal temperament used on many
20th-century keyboards. I have found the somewhat "bouncy" meantone fifth in
a full _armonia perfetta_ close a pleasant trademark of a "Renaissance"
sound, but I wonder if some keyboard pieces around 1600 favoring a close
with a major third but no fifth may be going for a purely intoned sound.

However, with four voices, if both the third and fifth are desired at a
close, then the classic and fully visible Maj6-8 or min3-1 cat must become a
Cheshire Cat, aurally smiling upon the musicians and listeners with the
expected 7-6 or 2-3 suspension and resolution, but then parallel sixths or
thirds rather than Maj6-8 and min3-1, and more specifically the ascending
semitone customarily a part of that two-voice closest approach progression.


----------------------------------------------------------------
4. Zarlino goes a Cheshire Catting: Armonia perfetta triumphant
----------------------------------------------------------------

In lieu of Maj6-8 or min3-1, the governing directed two-voice progression
from an imperfect to a perfect concord becomes the combination of
Vicentino's "bass cadence" and "soprano cadence," that is Zarlino's
two-voice cadence recognized as an option for two voices, but held to be
more characteristic for three or more voices: a 4-3 or 11-10 suspension,
followed by Maj3-8, Maj3-1, Maj10-8, Maj10-15, etc. The semitonal ascent of
an upper voice, together with the bass falling a fifth or rising a fourth,
become the defining nucleus of the four-voice cadence.

At least in his First Book of Madrigals, Verdelot uses the Cheshire Cat
cadence internally and prominently, but not as a final close. However, the
ideal of what Zarlino will term _armonia perfetta_ is fulfilled in a number
of final cadences where the bass falls a fourth or rises a fifth, or where
all four voices move by step.

In 1558, while Zarlino prefers the smooth and melodically conjunct qualities
of the traditional min3-1 or Maj6-8 cadence with the customary 2-3 or 7-6
suspension dissonance for two-voice writing, he seems to have no problem
embracing the Cheshire Cat cadence for four voices, where he sees the 4-3 or
11-10 dissonance and the Maj3-8 or Maj3-1 resolution (or octave extensions,
of course) as standard procedures in multivoice writing. Thus an
illustration of "Cadences in four voices," Part III, Chapter 61, is filled
with examples like the following, in a context showing how one cadence can
lead swiftly to another[7]:

|1......+.....2..+..+.....|.1....+.....2....+.....|...
|C5-----G4----A4----E4----|.G4--------------A4----|...
|G4-----------C4----------|.D4---------E4---C4----|...
|E4-----E4-------D4-C4-----------B3----C4---A3----|...
|C3-----C4----A3----A3----|-G3---------C3---F3----|...

At the opening of the second measure, we have a 4-3 suspension in the tenor
followed by a Maj3-8 resolution; in Vicentino's terms, we have a bass
cadence (down a fifth) in the bass, and a soprano cadence (up a semitone) in
the tenor. The soprano, with its stationary G4, has an alto cadence. The
middle voices have a 2-3 suspension -- but with the minor third, instead of
resolving to a unison (B3-D4 to C4-C4), moving by parallel thirds to C4-E4.
The alto, which in Vicentino's tenor cadence descending by a tone would
proceed D4-C4, instead makes what I might call a "Cheshire Cat tenor
cadence" ascending by a tone instead, D4-E4.

This altered part-writing, here in the alto, admirably serves Zarlino's
dedication to _armonia perfetta_, with the voices arriving at a sonority of
C3-C4-E4-G4, or from his point of view the consonances represented by the
"sonorous numbers" 2:4:5:6, with the senario of natural numbers 1-6, plus
their octave extensions, representing the basis of musical concord. Here we
have a 1:2 (or 2:4) octave, then a 4:5 major third, and then a 6:5 major
third, arranged in their "natural" order[8].

Zarlino's immediately following example features two Cheshire Cat cadences
which he finds less than ideal, and indeed which he instructs must be
treated "with caution." Their imperfection and potential risk of outright
error, however, results not from the omission of the classic Maj6-8 or
min3-1, but from the absence of a suspension dissonance: "But I must not
neglect to say that a cadence without dissonance lacks the grace and charm
found in those employing it."[9]

.....+.....|.1..+..+.....2...+...|1..||.....+....|.1.....2...+....|.1..||
.r----D5---|----C5-B4--------A4--.B4.||.G4-------|.G4--------F#4---.G4.||
.F#4-------|.G4----G4--------F#4-.G4.||.E4-------|.D4--------D4----.D4.||
.A3--------|.B3----D4--------D4--.D4.||.r---.C4--|.B3--------A3----.B3.||
.D3--------|.G3----G3--------D3--.G3.||.C3-------|.G3--------D3----.G2.||

Both examples involve cadences on G. In the first, the two upper voices have
a minor third F#4-A4 that proceeds in parallel motion to G4-B4 rather than
making a min3-1 cadence to G4-G4; the bass-alto pair have a Maj10-8 cadence
that is, in a sense, the nucleus of the progression. In the second example,
the tenor and soprano have major sixth A3-F#4 that, instead of making a
Maj6-8 expansion to G3-G4, proceeds in parallel motion to B3-G4. Here,
otherwise, the parts have their characteristic motions as in Vicentino: a
bass cadence down a fifth, D3-G2; an alto cadence D4-D4; and a soprano
cadence F#4-G4.

Both cadences, although less than ideal from the point of view of both
rhythmic and contrapuntal variety (compromised alike by the absence of the
usual suspension dissonance), arrive at the aptly spaced _armonia perfetta_
sonorities that the Cheshire Cat makes possible in four voices. In the
first, we have G3-D4-G4-B4, a 2:3:4:5 sonority with a 2:3 fifth, 3:4 fourth,
and 4:5 major third. In the second, we have G2-B3-D4-G4, a 2:5:6:8 sonority
with a 2:5 major tenth, 5:6 minor third, and 6:8 fourth. Both sonorities
show Zarlino's preference for the more euphonious major tenth above the bass
rather than a simple major third, following the series of "sonorous numbers"
where the 2:1 octave occurs before the 5:4 major third.[10]

Apart from the imperfection that these cadences lack a suspension
dissonance, Zarlino warns that having a voice fail to move at the right time
may give rise to an unintended augmented or diminished interval, for
example.[11] However, we can easily make modifications to include a
suspension dissonance while keeping the Cheshire Cat feature that Zarlino
has repeatedly embraced in his immediately preceding illustration of
cadences well handled:

.....+.....|.1..+..+.....2...+...|1..||.....+....|.1......2....+....|.1..||
.r----D5--------C5-B4----A4------.B4.||.G4-------|.G4----------F#4---.G4.||
.F#4-------|.G4----G4--------F#4-.G4.||.E4-------|.D4-----D4---------.D4.||
.A3--------|.B3----D4----D4------.D4.||.r---.C4--|.B3-----A3---------.B3.||
.D3--------|.G3----G3----D3------.G3.||.C3-------|.G3-----D3---------.G2.||

The first example has a suspended G4 in the alto forming a 2-3 resolution
with the soprano and an 11-10 resolution with the bass. In the second, the
suspended G4 in the soprano takes part in a 7-6 resolution with the tenor
and another 11-10 resolution with the bass. Thus Zarlino's imperfection is
mended, with the Cheshire Cat grinning as broadly as ever.


--------------
5. Conclusion
--------------

During the entire era of what may be termed multivoice discantic polyphony,
from the epoch of Perotin around 1200 when writing for three or four parts
became common, through the developments of the earlier 17th century, there
is a creative interplay between favored two-voice directed resolutions and
the ideal of saturated concordant sonorities being maintained as
consistently as practical -- except, of course, where instability is
desired.

In most Continental European styles of the 13th-14th centuries, the standard
of vertical saturation, or Johannes de Grocheio's _consonantia
perfectissima_ ("most perfect concord") is clear: it is the "threefold
perfection of harmony" (_trina harmoniae perfectio_), or in compact English,
"trine," with outer octave, lower fifth, and upper fourth.

Directed two-voice progressions by stepwise contrary motion, or sometimes
"near-conjunct" contrary motion (with one voice moving by step, and the
other by a third), are superimposed or united so as to arrive at the common
goal of a complete trine, or sometimes a simple fifth as the most harmonious
single interval of this three-voice combination. These directed progressions
include 2-4, 2-5, 3-1, 3-5, 6-8, 6-4, and 7-5.

By around 1300, the principle of "closest approach" especially prefers the
use of such directed progressions where one voice moves by a tone, and the
other by a semitone: thus min3-1, Maj3-5, Maj6-8, min7-5, and Maj2-4. During
the 14th century, we find in Machaut and also for example in some English
composers through the Old Hall repertory a prominent use of min7-5 and/or
Maj2-4, although the modern theory of discant and the new art of
counterpoint relegates all seconds and sevenths to a more incidental use,
and by the early to middle 15th century, practice more consistently runs in
this direction also.

Another restriction introduced for simple two-voice writing as an element of
14th-century discant and counterpoint, but applied more and more
consistently between all pairs of voices by the mid-15th century, is the
prohibition against parallel fifths and octaves. As Don Randel has observed,
consistent observation of this prohibition in a multivoice texture itself
has a shaping effect on the acceptable progressions which remain.

All of these factors play a role in shaping the style described by Aron in
his writings of the 1520's, and developed by Verdelot in his early
madrigals. Here the standard 14th-century closest approach resolution of
Maj6-8 or min3-1, and the restrained but expressive suspension dissonance of
7-6 or 2-3 accompanying these directed resolutions as an element of
Renaissance style, serve as elements of a musical equilibrium quite
different than that of the Ars Antiqua and Ars Nova, but equally compelling
in its own way.

During the 1520's and 1530's, we also encounter evidence for a new
development which affects multivoice cadences: a desire to arrive at a goal
sonority which satisfies the desire for what Zarlino will term _armonia
perfetta_, "complete or perfect harmony" including above the bass both a
third and a fifth or sixth, or their octave extensions.

While a number of four-voice cadential formulae where all voices proceed by
step and which feature traditional 14th-century directed resolutions, or
where the bass rises by a fifth or falls a fourth, can easily achieve this
new sonorous ideal, the type of cadence involving a classic Maj6-8 or min3-1
where the bass falls a fifth or rises a fourth requires a bit of adjustment
to fulfill this purpose.

Thus the Cheshire Cat modification: changing the expected Maj6-8 or min3-1
resolution following a 7-6 and 2-3 suspension (which itself is retained as
before) to a motion of these voices by parallel sixths or thirds,
respectively, so that the voice customarily ascending by a semitone
continues to do so (the smile of the Cheshire Cat), while the other ascends
rather than descends by a tone.

The reduction of the traditional Maj6-8 or min3-1 to a Cheshire Cat
variation leaves a newer two-voice directed resolution as in a sense the
nucleus of the four-voice progression: a 4-3 or 11-10 suspension in one of
the upper voices, followed by Maj3-8, Maj3-1, Maj10-15, or Maj10-8, etc.
Even in two-voice textures, as Zarlino notes, this formula with its more
disjunct voice-leading may sometimes be used, although he prefers the
traditional progressions by stepwise contrary motion as smoother. For
multivoice textures, however, both the 4-3 or 11-10 suspension, and the
disjunct bass motion, are commonplaces. Zarlino thus feels free to write and
recommend Cheshire Cat cadences, taking no special note of the fact that a
traditional Maj6-8 or min3-1 resolution never quite occurs, and only
exhorting that such cadences are better with the inclusion of a suspension
dissonance.

Having presented this hypothesis as to a likely motivation for the Cheshire
Cat cadence starting by the second quarter of the 16th century, I would
emphasize that a conceptual framework where the bass motion down a fifth or
up a fourth becomes part of a basic two-voice progression (M3-8 or M3-1,
etc.) does not mean that the style is equivalent in practice or theory to
that of major/minor tonality in the late 17th-19th centuries. Rather, these
are two different styles sharing certain common assumptions and constraints
(the standard of saturated vertical concord, the rule against parallel
perfect consonances), but diverging greatly in other aspects.

Just as correlation need not imply causation, so artistic evolution that
happens through a number of steps to develop in a certain way need not imply
teleology. Music of the era 1530-1640 or so, Cheshire Cat cadence and all,
presents a range of discant progressions (e.g. major third to fifth) often
yielding beautiful results quite different from those of tonal harmony; and
likewise the fluidity of accidental inflections, for example.

In short, the Cheshire Cat cadence heralds the recognition of a new standard
of vertical saturation. In place of the medieval trine which so well served
composers from Perotin to Machaut, Landini, Ciconia, and the early Dufay,
and as a result of compositional and intonational changes through the 15th
and early 16th centuries, what Zarlino will term _armonia perfetta_ now
becomes a shaping factor in up-to-date polyphonic writing, including the
details of cadences.

This has been an overview from a theorist's or composer's perspective. The
equally important perspective of the ensemble performer reading a single
line deserves its own consideration, which I do not attempt here, but hope
that my hypothesis may encourage.


---------
  Notes
---------

1. Bernard Thomas, ed., _The Italian Madrigal, Volume 3: Philippe Verdelot,
twenty-two madrigals for four voices of instruments_ (London Pro Musica
Edition, LPM MA3, c. 1980), Madrigal No. 4, at p. 15, mm. 47-48.

2. Gioseffo Zarlino, tr. Guy A. Marco and Claude V. Palisca. _The Art
of Counterpoint: Part Three of Le Istitutione harmoniche 1558_
(Norton, 1976), Chapter 61, p. 200.

3. Ibid., Chapter 53, pp. 147-148. "We should not use this type of cadence
often in two-voice writing, since such ascending and descending leaps are more
suitable to the lowest voice of a composition for more voices, where such
cadences are common."

4. Bernard Thomas, ed., cited  in n. 1 above, Madrigal No. 3, p. 12, mm. 7-8.

5. Gioseffo Zarlino, cited in n. 2 above, Chapter 59, pp. 186-190. Zarlino
realistically recognizes that one cannot always maintain this "truly perfect"
harmony at all times, especially in compositions for only three voices.
"However, in four-voice work, the error of omitting one of the two consonances
would be greater, because the extra part facilitates obedience to the rule. As
the number of parts increases, the obligation to follow the rule grows
proportionately." He adds that when parts have rests, causing a pleasing
variety in the texture, then the rule is of necessity relaxed.

6. Bernard Thomas, ed., cited in n. 1 above, Madrigal No. 18, p. 47,
mm. 28-29.

7. Gioseffo Zarlino, cited in n. 2 above, Chapter 61, pp. 201-202, ex. 146,
mm. 8-9.

8. Ibid., p. 8, n.4; and 195-197.

9. Ibid., p. 203, ex. 147.

10. Ibid., Chapter 61, pp. 195-197. After noting that in the ideal arrangement
major third 5:4 would come after the double octave (4:1), as a major 17th
(5:1); and commenting on the indifference of some musicians to questions of
the euphonious arrangement or spacing of voices, Zarlino offers a practical
suggestion: "I shall say only this: when a third is desired in a composition,
let it be placed always above the octave rather than within it. The tenth, in
other words, gives a better effect than the third." There follows an example
(p. 197, ex. 140) to show the more pleasing qualities in a three-voice texture
of some parallel tenths with the bass as compared to parallel thirds.

11. Ibid., p. 203-204 and ex. 148, Zarlino demonstrates the perils "of one
voice not moving to cadence with the others" where the passage proceeds in
such a way as to cause a singer to supply as a matter of course an accidental
inflection that the composer does not in this special context desire,
resulting in the example in the perfect fifth F-C being performed as an
augmented fifth F-C#.

Margo Schulter
28 October 2015