Re: Landini cadence and intonation (Part I)

Hello, there, and I'd like to address some recent discussions of
14th-century cadences and tunings, specifically the Landini cadence,
named after the great Italian composer Francesco Landini (1325-1397)
and more generally found in many compositions of this era.

Here I shall focus on the question of what a 14th-century Landini
cadence is, a kind of prelude to considering intonational issues. At
the outset, I might express the view that Pythagorean tuning, also
known as 3-limit just intonation, is an attractive solution, but that
other solutions (including the "Xeno-Gothic" option of major thirds
and sixths a Pythagorean comma wider than the usual 81:64 and 27:16,
or ~9:7 and ~12:7) may have appeal in some settings.

In presenting my own approach as a medievalist to this cadence, I
wouldn't want to imply that all other approaches should be excluded:
someone oriented mainly to the dominant-tonic progressions of the late
17th-19th centuries, or Schoenberg's pantonal patterns of the 20th
century, might present a different analysis.

Also, some of the material in this article may tie in with
presentations available on the Web concerning related subjects:

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

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1. A quick definition
---------------------

For those who would like a quick definition, a Landini cadence in its
most typical form is a variant on a favorite 14th-century cadence
combining two favorite resolutions of the period by contrary motion:
major third to fifth (M3-5) and major sixth to octave (M6-8). Let's
look first at this basic cadence, without the "Landini" ornament; here
C4 is middle C, and higher numbers show higher octaves. Incidentally,
knowing how easy it is to type the wrong octave numbers, I should add
that none of the cadences in this article should involve odd octave
leaps; so if you see them, substitute the correct octave number which
avoids such a result:

E4 F4 D4 E4
B3 C4 A3 B3
G3 F3 F3 E3

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

Note that in the first example the upper voices each ascend by a
semitone while the lowest voice descends by a whole-tone; in the
second example, the lowest voice descends by a semitone and the upper
voices ascend by a whole tone. In either version, the outer pair of
voices expand M6-8 while the lower pair expand M3-5. If necessary,
14th-century musicians routinely add accidentals (sometimes written
out, sometimes inferred) to obtain major thirds and sixths for these
resolutions to the fifth and octave respectively.

For example, for a cadence on D, either of these solutions might
occur:

C#4 D4 C4 D4
G#3 A3 G3 A3
E3 D3 Eb3 D3

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

Either choice gives us a major third before the fifth and a major
sixth before the octave.

The distinctive "Landini" figure, in its most typical use, has the
highest voice momentarily descend from the major sixth to the fifth
before the expected resolution to the octave. Let's look at Landini
forms for the above four examples:

E4 D4 F4 D4 C4 E4 C#4 B3 D4 C4 Bb3 D4
B3 C4 A3 B3 G#3 A3 G3 A3
G3 F3 F3 E3 E3 D3 Eb3 D3

Now that we've defined a Landini cadence in its most usual
14th-century form, let's consider the musical context.

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2. Trinic progressions
----------------------

First of all, we might focus on the ideal goal of 13th-14th century
cadences for three or more voices: the complete trine (_trina harmoniae
perfectio_ in Latin, or "threefold perfection of harmony") consisting
of an outer octave, lower fifth, and upper fourth. This sonority
includes all three of the simple nonunisonal stable intervals: the 2:1
octave, the 3:2 fifth, and the 4:3 fourth.

For example, F3-C4-F4 and A4-E4-A4 are complete trines. Such
sonorities represent the medieval ideal of what I might call
"saturated stability": they are the richest and fullest stable
sonorities possible, and thus ideal resolutions for unstable
combinations.

In terms of string-ratios, the usual medieval way of describing
intervals, the complete trine would be 6:4:3, with higher numbers
showing longer strings and thus lower notes. We have a 2:1 octave
between the outer strings, a 6:4 (or 3:2) fifth between the lower
strings, and a 4:3 fourth between the upper strings.

If we apply later frequency-ratios, we get a complete trine of 2:3:4.
A just intonation (JI) advocate might note that this sonority, like
the later complete triad of 4:5:6, involves three adjacent partials,
with the lowest of these partials an octave of the fundamental.

At any rate, taking the complete trine as our ideal cadential goal, we
find that the most popular 13th-14th century cadences typically
involve moving from an unstable sonority to a stable trine (or next
most preferably its most choice interval of the fifth) with stepwise
or sometimes thirdwise motion in all parts.

In these directed cadences, unstable intervals (seconds, thirds,
sixths, and sevenths) ideally resolve to stable ones by conjunct or
near-conjunct contrary motion. That is, either both voices move by
step, or voice moves by step and the other by a third.

In the 13th century, a great variety of progressions fitting this
basic pattern may be heard and relished, with the following a quick
sample. In each case, I show the directed two-voice progressions in
parenthesis below an example:

F4 G4
C4 D4 E4 F4 E4 D4 D4 C4
A3 G3 B3 C4 C4 D4 G3 F3
F3 G3 A3 F4 A3 G3 E3 F3

(M3-1 + m3-5) (M2-5) (m6-8 + m3-5 + M3-1 + m2-4) (m7-5 + m3-1)

In this style one might say that two-voice intervals are the
elementary units of motion, and that these units can be combined to
form various "compounds" -- a metaphor which Howard Hanson uses for
some varieties of 20th-century harmony.

More generally, one might say that these cadences may express a
broader theme in at least Western European traditions of verticality:
the theme of tension-relaxation. An unstable sonority moves
satisfyingly to a richly stable one.

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3. "Closest approach" and the 14th century
------------------------------------------

In the early 14th century, another artistic element comes into play:
the ideal of "closest approach." This ideal specifically favors
elementary two-voice resolutions in which one voice moves by a
whole-tone and the other ascends or descends by a semitone. While
progressions of this kind are also very common in the 13th century,
so are progressions where both voices move by a whole-tone.

Let's consider the progressions 3-1, 3-5, 6-8, and 7-5 to see how
the "closest approach" ideal leads specifically to a preference for
m3-1, M3-5, M6-8, and m7-5:

------------------------- ------------------------
3-1 progressions 3-5 progressions

E3 D3 G3 F3 F3 E3 G3 A3 B3 C4 A3 B3
C3 D3 E3 F3 D3 E3 E3 D3 G3 F3 F3 E3

M3-1 m3-1 m3-1 m3-5 M3-5 M3-5

------------------------- ------------------------
6-8 progressions 7-5 progressions

C4 D4 E4 F4 D4 E4 E4 D4 D4 C4 C4 B3
E3 D3 G3 F3 F3 E3 F3 G3 E3 F3 D3 E3

m6-8 M6-8 M6-8 M7-5 m7-5 m7-5

In each group of progressions, the first cadence is common in
13th-century practice but does _not_ follow the 14th-century ideal of
"closest approach" because both voices move by whole-step. The
remaining cadences in each group do follow this ideal, with one the
voices moving by a semitone, which may be either ascending (second
example in each group) or descending (third example).

As mentioned above, 14th-century musicians typically use accidentals
in order to achieve "closest approach" progressions where they would
not occur naturally. The most commonly used accidentals for this
purpose are the five likely available on many 12-note Pythagorean
keyboards of the era: Eb, Bb, F#, C#, and G#. However, some cadences
naturally featuring (m6-8 + m3-5) might be left uninflection, and
composers sometimes deliberately specify such progressions, e.g.

Bb3 C4
F3 G3
D3 C3

(m6-8 + m3-5)

This consideration of trinic progressions and the 14th-century norm --
although not invariable rule -- of closest approach may put the
Landini cadence in some perspective.

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4. The Landini cadence and its variants
---------------------------------------

From one viewpoint, a Landini cadence might be seen as combining some
of the features of two popular cadences in the 13th century. The first
of these cadences, which we have already encounters, involves a union
of the 6-8 and 3-5 resolutions by contrary motion; in certain late
13th-century compositions, manuscript accidentals suggest some
tendency to prefer the M6-8 and M3-5 forms regularly favored in the
14th century. Four examples:

E3 F4 F#4 G4 D4 E3 G4 A4
B3 C4 C#4 D4 A3 B3 D4 E4
G3 F3 A3 G3 F3 E3 Bb3 A3

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

In another common 13th-century cadence, the lower voices also have a
3-5 resolution, but the outer voices move from fifth to octave. In the
cadence, the first unstable sonority is what Jacobus of Liege calls a
_quinta fissa_ or "split fifth": an outer fifth "split" by the middle
voice into the mildest unstable intervals of a major third below and a
minor third above, or vice versa. Here are some examples with the M3-5
progression increasingly preferred around 1300:

D4 F4 E3 G4 C4 E4 F4 A4
B3 C4 C#3 D4 A3 B3 D4 E4
G3 F3 A3 G3 F3 E3 Bb3 A3

(M3-5) (M3-5) (M3-5) (M3-5)

From the viewpoint of "efficiency," one might argue that this is not
quite as ideal a progression as (M6-8 + M3-5). Both progressions
arrive at a complete trine, but here the upper voice leaps by a third,
rather than all voices moving by stepwise motion. Also, while the
unstable third between the lower two voices resolves by ideally
efficient stepwise contrary motion (M3-5), the third between the upper
pair of voices resolves by less ideal similar motion (m3-4).

However, the thirdwise motion in the highest voice gives this cadence
its own flavor which can be quite "lyrical" (at least to me).

The Landini cadence might be said to combine both these forms by
having a sixth in the upper voice descend momentarily to the fifth
before this interval expands to the octave:

E4 D4 F4 F#4 E4 G4 D4 C4 E4 G4 F4 A4
B3 C4 C#4 D4 A3 B4 D4 E4
G3 F3 A3 G3 F3 E3 Bb3 A3

Many variants can and do occur: while such cadences may sometimes be
found in the 13th century, they become very popular in the 14th
century, with Landini certainly being one the most famous composers to
enjoy this figure.

While I have given three-voice examples, three parts being the norm in
13th-14th century writing, the Landini idiom is also very common in
two-voice compositions of the 14th century. To see typical two-voice
Landini cadences, we need only look at the outer pair of voices in the
last example, disregarding the middle part.

While the Landini figure is most often applied to the 6-8 resolution,
it may also be applied to the 3-5 resolution, as in an idiom popular
during the early 15th century in which both progressions are
ornamented:

E4 D4 F4
B3 A3 C4
G3 F3

From an aesthetic viewpoint, given that the favorite 14th-century
cadence uniting M6-8 and M3-5 represents the ideal of directed
vertical action, the Landini figure might be described as a pleasant
and lyrical prolongation of the inevitable. One early 14th-century
treatise asserts that the major sixth "seeks the octave, and without
fail"; by a diverting dalliance with the fifth before the anticipated
expansion to the octave, this idiom adds melodic interest and
decoration to one of the most beautiful cadences of all time.

In this presentation I have focused on basic musical patterns and
ideals such as "closest approach" which may be relevant in considering
issues of intonation -- the topic for the concluding portion of this
article.

Most respectfully,

Margo Schulter
mschulter@value.net