The Landini cadence, named after the great Italian composer and organist Francesco Landini (1325-1397) but more generally pervasive in the music of the 14th and earlier 15th centuries, might be described in its most characteristic form as a variation on the harmonic progression in which an unstable sixth (usually major) expands to a stable octave.
In this variation, the upper voice of the sixth momentarily descends to the fifth before the expected resolution to the octave:
e' d' f' d' c' e' g f OR f e M6-5 8 M6-5 8
The Landini cadence invites comparison with another very common medieval idiom where a fifth expands to an octave by way of a sixth, as at the very opening of Francesco Landini's two-voice ballata Chi più le vuol sapere  in 2/4:
1 2 | 1 d' e' f' ... g f ...
In moving from the stable fifth to the octave, the mediating major sixth makes possible a smoother melody in the upper part while also adding directed harmonic tension. The Landini cadence could be seen as a kind of "time-reversed" variation of this, with the cadential sixth followed by the fifth before the octave.
From this perspective, the cadence derives its charm both from the melodic ornament of the descending step followed by the ascending third in the upper voice, and from the momentary harmonic diversion of the fifth coming between the sixth and its expected expansion to the octave. 
In three-voice settings of the Ars Nova or "New Art" (c.1300-1420), the M6-5-8 Landini progression is typically combined with an M3-5 progression to build one of the favorite cadences of the period:
e'-d' f d'-c' e' b' c' a b g f OR f e M6-5 8 M6-5 8 M3 - 5 M3 - 5
As already mentioned, these cadences are by no means unique to Francesco Landini, the greatest Italian composer of the 14th century and a worthy peer of the French composer and poet Guillaume de Machaut. Variations of the Landini cadence may be traced back to at least the 13th century, and it continues to play a vital role in the music of the early Renaissance (1420-1500).
However, the name "Landini cadence" is not inapposite, since no one has used it more beautifully than Landini himself, famed in his own time not only as a composer but as an organist. One 14th-century account tells how his playing on the portative organ -- an instrument of modest technical means requiring the organist to work the bellows with one hand while working the keyboard with the other -- drew down the very birds by its sweetness. The expressive qualities of his many compositions make this story quite believable.
As defined above, the Landini cadence occurs in a number of 13th-century works, as in the final cadence of the conductus-motet Ad veniam/TAMQUAM which may date from the early decades of the century . Here the rhythm is shown as 6/8, with two principal beats to a measure, and the last measure concluding with a rest in all parts:
1 2 3 4 5 6 | 1 2 3 4 5 6 || d'. e' d' f'. r. g . a b c'. r. g . f . r. 5 M6 5 8 1 M2 M3 5
Another example is found in the final cadence of the rondeau A Dieu commant amouretes by Adam de la Hale (c. 1230-c. 1287), one of the first composers to develop the polyphonic French art song . Here the rhythm is expressed as 3/4:
1 2 3 | 1 2 3 || g' f' e' d' f'. d' c' b c'. g f . 8 m7 M6 5 8 5 4 - M3 5
A full appreciation of these and related 13th-century cadence forms would require consideration of the diverse sonorities and progressions which prevail in this period, and also of the constant interplay between vertical and melodic dimensions. These two examples may suggest some of the diverse harmonizations possible.
As we move into the early 14th century, the "modern" theorists of the Ars Nova such as Philippe de Vitry and Johannes de Muris in France and Marchettus of Padua in Italy advocate a simplification of the harmonic language. Under the new system, the most important cadential progressions are those in which an "imperfect concord" (3, 6) resolves to a "perfect concord" (1, 5, 8) by stepwise contrary motion. Further, a preference for progression to the "nearest consonance," with one voice moving by a whole tone and the other by a semitone, causes composers especially to favor m3-1, M3-5, and M6-8.
While just about all the intervals have an essential role in the harmonic system of the 13th century, the Ars Nova "moderns" preferred at least in theory to restrict intervals they considered "discords" to a more ornamental role. Such intervals included all seconds and sevenths (including M2 and m7, considered somewhat "compatible" in the 13th century), and also the fourth in relation to the lowest voice.
Thus, at least for composers who strictly followed the new theory, good harmony became an artful alternation of stable 8/5 combinations and mildly unstable 5/3 and 6/3 sonorities. Given these constraints, it is not surprising that the two most popular cadences should be the direct progression from M6/M3 to 8/5, with its two aptly united resolutions (M6-8 + M3-5), and the Landini variant interposing a momentary 5/3 sonority:
Basic M6/M3-8/5 forms Landini forms --------------------- ------------- f#'-g' g' -a' f#'-e'-g' g'-f'-a' c#'-d' d' -e' c#' d' d -e' a -g OR bb -a a g OR bb -a M6 8 M6 8 M6- 5 8 M6-5 8 M3 5 M3 5 M3 - 5 M3 - 5
As these examples show, notes were often inflected to obtain a major third before a fifth, or a major sixth before an octave. These inflections might be expressly indicated by manuscript accidentals, but were often left to the performers. Forms where the lowest voice descends by a whole tone while the upper voices ascend semitonally (possibly with the Landini variation) serve as the most popular final cadences (French clos, Italian chiuso), while forms where the lowest voice descends semitonally occur almost exclusively as internal cadences (French ouvert, Italian overto), where they are much favored.
Landini's ballata L'alma mia piange uses two variations on the same melodic motif for the sectional cadences ending the first and second repeats of the B section (the form is ABBAA). This example illustrates Landini's fluid melody and harmony while showing how the Landini cadence on E (descending semitone in lowest part) in the first version is reworked into a direct M6/M3-8/5 on D for the second version (ascending semitones in upper parts). Here the rhythm is represented as 2/4, with a dotted line indicating tied notes:
1 & 2 & | 1 & 2 & | 1 2 ------ d' e' f' e' e' d' c' e' d' c' b a b g a g f e 5 - M6 m6 -5 M6-5 M6-5 8 5 - m3 - M3 M3 5 1 & 2 & | 1 & 2 & | 1 2 ------ # d' e' f' e' e' d' c' d' ---------- # d' a a g a g a f e d 5- M6 m6 -5 M7-M6 m7-M6 8 5 1 M3 M3 5
The second version, as it happens, features a new 14th-century idiom that was to become a central aspect of Renaissance style: the suspension, here illustrated by two 7-6 resolutions leading up to the final M6/M3-8/5.
In practice, these M6/M3-8/5 and related Landini cadences predominate even in the works of composers such as Guillaume de Machaut and some of his English contemporaries who mixed the "modern" Ars Nova approach with a continuing taste for the bolder liberties of the 13th century in using M2, m7, and M9 as essential ingredients of harmony.
However, there are stimulating variants such as what might be called a "double Landini" form in which both the upper and middle voice descend a step before ascending a third to the resolving sonority, creating a momentary 5/2 combination. This stirring cadence concludes the famous Agincourt song, Deo gracias, Anglia (c. 1420), celebrating the English victory of 1415 under King Henry V . The rhythm is represented as 3/4, with a bit of syncopation, and unwritten accidentals seem likely to facilitate M3-5 and M6-8:
1 & 2 & 3 & | 1 2 3 || # # d'. c' c' b d'. # # a . b g f a . f d e d . M6 8 M7 M6 5 8 M3 5 M6 M3 M2 5
The epoch beginning around 1420 is often taken to mark the transition from the Ars Nova to the early Renaissance. During this epoch, the influence of John Dunstable (?-1453) upon such Continental composers as Guillaume Dufay (c. 1397-1474) and Gilles Binchois (c.1400-1460) lead to a new "English countenance" style with thirds and sixths becoming more and more preponderant in the texture.
While cadences with more disjunct motion in the lowest part came into prominence, the traditional 6/3-8/5 and its Landini variants retained an important role -- but with an apparent predilection for forms combining the traditional M6-8 progression with an m3-5 progression, giving a tritonic flavor. An internal cadence from Dufay's Quel fronte signorille  illustrates this likely preference in a "double Landini" context. The rhythm is represented as 3/4, and the crossing of the lower parts slightly complicates the task of following the intervals:
1 & 2 & 3 & | 1 2 3 | # d' e' g' f' e' g' r | g c' b d' r b a g r 5 M6 m7 M6 5 8 M3 - m3 - M2 5
By the late 15th-century epoch of Johannes Ockeghem (c. 1410-1497), newer cadential idioms were overshadowing the older forms of the Ars Nova as a four-voice texture with consistent avoidance of parallel fifths became the norm. Yet the Landini idiom could be adapted to this transformed harmonic environment as one of the optional ornaments to the more and more prevalent cadence formula featuring a 4-3 suspension. The final cadence from Ockeghem's Petite camusette shows this adaptation , and also the central role of the suspension in the emergent Renaissance practice (tied notes indicated by dashes):
1 & 2 & | 1 & 2 & | 1 & 2 & | 1 2 | 1 2 || ------------ -------- # # c' e'. e' d' f' d' d' c' c' b d' g. a b a b c' a g a a e g a d e d b e e c B A d 8 M10 M9 12 M10 11- M10 - M9 m6 - 5 - 4- 5 8 M6 M7- M6 8 - 8 m3 - 4 m3 M6 - M3 - 5 - 5
In the third measure, the top part presents a typical 4-M3 (or 11-M10) suspension and resolution, followed by the Landini tone a ninth above the bass before the final sonority. If we look at this part and the tenor (next to lowest part) as a two-voice pair, we interestingly find the traditional Ars Nova progression of M6-5-8, preceded by a 7-6 suspension. 
Although rare in the 16th century as a final close, the Landini cadence appears "fairly frequently" at other points according to Alec Harman, the editor of Thomas Morley's famous A Plain & Easy Introduction to Practical Music (1597), where it appears in examples of two-voice discant. 
This brief survey may give some sense of the rich history of the Landini cadence as part of the changing tapestry of medieval and Renaissance harmony. With its intermingled harmonic and melodic aspects, this cadence marks a creative intersection of the vertical and horizontal dimensions, an arena of stylistic perfection and stylistic change.
In Leo Schrade, ed., Polyphonic Music of the Fourteenth Century, v. 4: The Works of Francesco Landini (Monaco, 1958), p. 55, mm. 1-2. The upper part has a Bb signature.
The Landini cadence is sometimes known as the "Landini sixth," a reference not to the vertical interval of the sixth, but to a melodic description. The octave of the resolution is taken as defining the outer notes of a scale in which the distinctive "Landini tone" is the sixth degree. Taking this view, we might write the Landini formula as follows, showing that the "Landini sixth" is the note forming the interposed fifth of the M6-5-8 progression:
7-6-8 2 1
The Landini cadence is often described in a similar vein as the "under-third" cadence.
In Hans Tischler, A Medieval Motet Book: A Collection of 13th Century Motets in Various Vocal and Instrumental Combinations (New York, 1973), pp. 10-14, mm. 42-43.
In Nigel Wilkins, ed., The Lyric Works of Adam de la Hale, Corpus Mensurabilis Musicae 44 (American Institute of Musicology, 1977), pp. 52-53, mm. 10-11.
In W. Thomas Marrocco and Nicholas Sandon, eds., The Oxford Anthology of Music: Medieval Music (London, 1977), pp. 174-175, mm. 51-56.
In Don Anselm Hughes, ed. (gen. ed. Gerald Abraham), The History of Music in Sound, v. 3: Ars Nova and the Renaissance (New York, 1966), p. 27, ex. 6(b), mm. 38-39.
Compare Thurston Dart, ed., Invitation to Medieval Music, v. 1: Music of the Earlier Fifteenth Century (London, 1967), pp. 16-17, mm. 11-12.
In Martin Picker, ed., The Chanson Albums of Marguerite of Austria (Berkeley, 1965), pp. 437-439, mm. 44-47.
This closing passage illustrates an observation by Todd McComb that in approaching Renaissance music one can look either ahead to the practice and theory of the late Baroque and Classical periods, or back to the medieval tradition of harmony. An analyst oriented to an 18th-century framework might describe Ockeghem's close as a "V-I" cadence, preceded by "dominant preparation" in the last half of the second measure. A medievalist might eagerly note that the lower three voices moving from the end of the second to the beginning of the third measure form a typical M6/M3-8/5 progression of the 14th century (Bb-d-g to A-e-a), while the concluding progression can be regarded as a new harmonization of another very traditional M6-8 progression, or M6-5-8 with the Landini decoration. One person's "iv-V-I cadence" can be another person's artful linking of two M6-8 progressions, the first with descending and the second with ascending semitonal motion.
Thomas Morley, A Plain & Easy Introduction to Practical Music Alec Harman, ed., foreword by Thurston Dart (New York, 1973), pp. 160 and n. 1; pp. 170 and n. 1.
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