In the standard gamut, B/Bb is a flexible degree, appearing in different hexachords as either mi or fa - and likewise E/Eb in the same gamut when transposed by a Bb signature. By around 1200, in addition to using Bb and Eb, composers were sometimes applying the mi-sign - the sign of the "square-B" closely resembling a modern natural sign, and also represented in later medieval and Renaissance notation by the sharp sign - to F (F#) and C (C#).
These signs literally direct that the affected notes should be sung as mi, effectively making a whole-tone (F-G or C-D) into a semitone (F#-G or C#-D). However, there is no standard hexachord in which F or C may be sung as mi - so these symbols call for the "invention" of new or "fictional" hexachords on D (with F# as mi) and A (with C# as mi).
A dramatic case occurs at the opening of Perotin's organum triplum Alleluya posui adjutorium, where the use above the sustained note G in the tenor of C# and F# in the upper parts provides a striking vertical color and tension:
F#4 D4 C#4 D4 G3
Here we have the simultaneous intervals of the tritone and major seventh above the tenor, both counted by 13th-century theorists as among the strongest or "perfect" discords, which alike resolve by oblique motion to a fifth with the tenor.
This example nicely illustrates the element of artistic choice and boldness in 13th-century accidentalism, not always fitting within any convenient "rules": Here, for example, far from avoiding the tritone, the inflections create one.
Additionally, our example illustrates the possible rule of local variations and performer's (or notator's) discretion: While the version of this organum in the Montpellier manuscript indicates both F# and C#, another version in the Florence manuscript omits the C# in the duplum or second voice.
By the early 14th century, theorists were taking note of certain preferences in vertical progressions regularly calling for inflections outside of the standard gamut: Sometime around this epoch, the 12-note keyboard with its five accidentals (typically Eb, Bb, F#, C#, G#) seems to have come into vogue.
By the end of the 14th century, a composer such as Solage can write a piece (Fumeux fumee) calling not only for these accidentals, but also Gb and Db as well as Ab. In the early 15th century, the theorists Prosdocimus of Beldemandis and Ugolino of Orvieto additionally recognize D# and A#, expanding the late medieval system to 17 notes per octave.
Such inflections involve the "invention" not only of new steps, but of new hexachords, and it is mainly on this aspect that our discussion should focus.
One complication of the term musica ficta is that it carries two distinct although not unrelated meanings, one arising from medieval and Renaissance hexachord theory and the other from modern dilemmas of editorial and performance practice.
In the medieval sense, musica ficta involves inflections extra manum or "outside the hand" - the Guidonian hand (Section 1.5) defining the standard hexachords and steps of the gamut. In contrast, musica recta involves hexachords and steps within this gamut - or a transposed version of it (Section 1.6).
Thus singers performing Perotin's Alleluya posui adjutorium, discussed just above, from the Montpellier Codex, would be realizing musica ficta in this sense by singing the F# and C# inflections indicated in the manuscript: There is no Fmi or Cmi in the standard gamut.
In the colloquial modern sense, however, musica ficta can often mean "unwritten accidentals" supplied in performance. Thus the F# and C# inflections shown in the Montpellier Codex would not be musica ficta in this sense, because they are expressly indicated.
However, if we imagine either 13th-century or modern musicians reading the same composition from the Florence manuscript, and singing C# although it is not indicated in this manuscript, then this would be a case of musica ficta in both the medieval sense (an inflection outside of the standard gamut) and modern sense (an unwritten accidental).
In other cases, a given accidental may be musica ficta under the modern definition, but musica recta in medieval terms. Thus if a performer reading a piece without signature chooses to sing or play Bfa (Bb) rather than Bmi (B-natural), in order for example to avoid a vertical or horizontal tritone, this would be an inflection within the regular gamut (musica recta) - but, in the colloquial modern sense, also an exercise in musica ficta (an unwritten inflection).
These two distinct although overlapping senses of musica ficta are both found often in the modern literature, as anyone following discussions on this topic will soon discover. In what follows, the term musica ficta generally carries its medieval meaning of "inflections outside the usual gamut," although as we will see this can often involve the element of unwritten inflections.
In the early 14th century, theorists took note of a tendency of which there are signs in certain compositions of the late 13th century (e.g. Adam de la Halle, Petrus de Cruce): a preference for resolutions by stepwise contrary motion from unstable to stable intervals in which one voice moves by a whole-tone and the other by a semitone. The most common cases involve motion from a minor third to a unison (m3-1), a major third to a fifth (M3-5), and a major sixth to an octave (M6-8).
This new situation contrasts with that of the 13th century, which often embraced directed resolutions in which both voices move by whole-tones (e.g. M3-1, m3-5, m6-8, m2-4, M7-5) as well as those in which one of the voices ascends or descends by a semitone. Applying the new preferences to cadences on various steps, we find that regular accidental inflections are often called for; the following examples show some possibilities using the typical inflections Eb, Bb, F#, C#, and G#:
A3 G3 E3 D3 E[b]3 D3 B3 A3 B[b]3 D3 F[#]3 G3 C[#]3 D3 or C3 D3 G[#]3 A3 or G3 D3 (m3-1) (m3-1) (m3-1) (m3-1) (m3-1) F[#]3 G3 C[#]4 D4 G[#]3 A3 G3 A3 D4 E4 D3 C3 A3 G4 E3 D3 or E[b]3 D3 B[b]3 A3 (M3-5) (M3-5) (M3-5) (M3-5) (M3-5) F[#]3 G4 C[#]4 D4 C4 D4 G[#]4 A4 G4 A4 A3 G3 E3 D3 or E[b]3 D3 B3 A3 or B[b]3 A3 (M6-8) (M6-8) (M6-8) (M6-8) (M6-8)
In many cases, obtaining a minor third before a unison, or a major third or sixth before a fifth or octave respectively, might be accomplished either by a mi inflection (F#, C#, or G#), musica ficta; or by a fa inflection (Bb or Eb). The latter solutions involve the musica recta note Bb; or the note Eb, which may become a "naturalized" musica recta tone either by a Bb signature, or at times by a previously indicated Bb inflection in the piece.
If we expand the system to include Ab and D#, as may have been the practice in various localities around 1400 (e.g. England in the epoch of the Old Hall Manuscript), then even more choices would be possible.
In the above examples, the accidentals are shown in brackets: In practice, such inflections were often indicated expressly in the manuscripts, but also often left to the performers.
Interestingly, one report from the early 14th century about the tendency of performers to engage in unwritten inflections comes from a "conservative" of this era: Jacobus of Liege (c. 1325), a champion of the late 13th-century Ars Antiqua or Ars Veterum (the "Old Art," or "Art of the Elders") against the "modern" Ars Nova or "New Art."
In his Speculum musicae or "Mirror of Music," a treatise of seven Books each filling a volume in the modern Latin edition, Jacobus takes an interest not only in Guido's hexachord system, but also in his description of early 11th-century polyphony, citing from it a musical example then about 300 years old. At the same time, he champions the rhythmic subtleties of Petrus de Cruce and other late 13th-century musicians of his youth, and notes that singers sometimes engage in inflections outside the normal gamut.
Thus in Book II, Chapter 80, in a discussion of the ditonus or major third (equal to two whole-tones), Jacobus observes that singers prefer the semiditonus or minor third (a whole-tone plus a diatonic semitone) in approaching the unison :
Et nonne, si duo simul cantent, unus la la la sol, alius la fa fa sol, descendens in fa facit unam falsam? Non ditono, sed potius utitur semiditono, quia voces eius magis placent auditui...
And if two people sing at the same time, one la la la sol and the other la fa fa sol, does not the one descending to fa sing musica falsa? This singer would rather use not the major third but the minor third, because its voices more greatly please the ear...
In the early 14th century, musica falsa is synonymous with musica ficta: The use of steps or mutations outside the standard gamut. If we assume that the two singers of this example are using the natural hexachord of C3-A3, then we might indicate the inflection as:
A3 A3 A3 G3 la la la sol A3 F[#]3 F[#3] G3 la fa fa sol (1 m3 m3 1)
While one of the points made by Jacobus in this passage is that the semiditonus or minor third is somewhat more pleasing in itself than the ditone, he continues by giving other examples with solmization syllables showing how singers prefer this minor third in progressing stepwise from a fifth to a unison. In the natural hexachord of C3-A3, these examples might be realized as follows, in this case using the standard musica recta steps:
A3 G3 F3 G3 F3 E3 la sol fa sol fa mi or D3 E3 F3 C3 D3 E3 re mi fa ut re mi (5 m3 1) (5 m3 1)
While reporting this preference for the minor third before the unison, Jacobus in Book IV, Chapter 11, also notes a preference for the ditone or major third before the fifth, giving the example :
sicut cum quis dicit: re re ut et alius, simul cum illo, dicit: re fa sol; quamvis regulariter inter re fa sit semiditonus, sic tamen intendit fa, ut faciat ibi ditonum et amplius accedit ad diapente.
as when someone sings re re ut and another, singing with that person at the same time, sings re fa sol; although between re fa there is normally a minor third, yet nevertheless there is a stretching of fa so as to make there the major third and more [closely] to approach to the fifth.
Again using the natural hexachord, this example could be realized:
D3 F[#]3 G3 re fa sol D3 D3 C3 re re ut (1 M3 5)
In his catalogue of cadentiae or progressions in which a more tense or less concordant interval resolves to a less tense or more concordant one (Book IV, Chapter 50), he likewise notes that the minor third seeks the unison but the major third seeks the fifth .
This principle of "closest approach," also typified for 14th-century musicians by the expansion of the major sixth to the octave, becomes an axiom of good technique not only for this era but for Renaissance and Manneristic practice through the early 17th century, when Agazzari (1607) mentions progressing to "the nearest consonance" as a basic precept for players learning how to realize a continuo.
In some of the above examples, Jacobus demonstrates the practice of musica falsa (or ficta) in both the medieval and the colloquial modern sense: Singers produce notes outside the regular gamut such as F#, and do so "intuitively," as one might say, without a need for written indications.
Since Jacobus speaks as a champion of the Ars Antiqua, his remarks as well as some manuscript accidentals from late 13th-century pieces suggest that such inflections (written or otherwise) may have been favored by at least some musicians of this era, although where and how often remains an open question. By the early 14th century, the conservative Jacobus and the Ars Nova moderns agree in recognizing "closest approach" (e.g. m3-1, M3-5, M6-8) and the accidental inflections resulting from this principle.
However, some of the "moderns" carry the new rules and preferences further. Marchettus of Padua (1318), in his Lucidarium, goes so far as to advocate melodic progressions by a chromatic semitone when the vertical context calls for them :
G3 G#3 A4 C3 E3 D4 5 M3 5
Here G3 is needed in the upper voice to form a perfect fifth with C3, while G#3 is needed to form a major third above E3 leading to the fifth D4-A4 (M3-5). The result is a progression in the upper voice of G3-G#3-A4, consisting in a medieval Pythagorean or similar system of intonation of a large chromatic semitone G3-G#3, followed by a usual narrow diatonic semitone G#3-A3 completing the M3-5 resolution.
In the view of Marchettus, such a resolution by stepwise contrary motion has a unifying vertical effect: Both voices contribute to the tension of the unstable interval (here the major third), and both share in the motion required to resolve it. Pythagorean intonation, with its wide major thirds and sixths and its pure fifths and octaves, nicely fits this ethos of "closest approach."
By sanctioning direct chromatic semitones, considered "unsingable" by more orthodox medieval theorists and as late as the mid-16th century by Tomás de Santa María (who would exclude this interval from keyboard music also), Marchettus shows his experimental bent, as he does also by suggesting a variation on the usual Pythagorean tuning with even narrower cadential semitones (and wider major thirds and sixths) for resolutions such as M3-5 and M6-8.
More conventional theorists, however, often agree with his largely vertical approach to musica ficta: Prosdocimus of Beldemandis (1412) and Ugolino of Orvieto (c. 1435?) state that it is motivated by the desire to "color" or "perfect" some consonance, either stable or unstable.
On the one hand, this means obtaining minor thirds before unisons, and major thirds and sixths before fifths and octaves, thus "coloring" or "perfecting" these intervals by making them approach the following stable interval as closely as possible. On the other hand, it means obtaining stable intervals in their perfect rather than augmented or diminished forms (e.g. "perfecting" the diminished fifth B3-F4 by altering it to B3-F#4).
Asserting the viewpoint of the Ars Nova, the composer and theorist Philippe de Vitry declares that musica falsa is indeed "true and necessary." This musical practice involves an expansion of the hexachord system, documented by Ugolino and other theorists.
From the solmization which Jacobus gives for some of his examples showing inflections to obtain progressions of m3-1 or M3-5, we get the impression that performers of the early 14th century may have been singing musica falsa not only in the sense of using notes outside the standard gamut, but of contradicting some basic patterns of the hexachord system.
For example, let us consider again the example of an inflected m3-1 progression:
A3 A3 A3 G3 la la la sol A3 F[#]3 F[#3] G3 la fa fa sol (1 m3 m3 1)
Here, as Jacobus notes, the interval la-fa in the lower voice is in fact a semiditonus or minor third (A3-F#3), although in the hexachord system it should always be a ditonus or major third. We can also note that in this voice the final interval fa-sol is actually a semitone (F#3-G3) instead of the whole-tone which should always obtain between these syllables, semitones always occurring at mi-fa.
To "regularize the irregular," a remedy for such anomalies is to "invent" not only new notes, but new hexachords in which they can take an ordered place analogous to that of the notes of the standard gamut. Such additional hexachords are known as ficta hexachords.
For the above example, inventing such a new hexachord of D-B permits proper solmization. Our new hexachord is structured like this:
B3 la A3 sol G3 fa F-mi3 [F#] mi E3 re D3 ut
Let us now try the example again, with the singer of the lower part making a mutation from the natural hexachord C-A to this "fictive" hexachord on the first note:
A3 A3 A3 G3 la la la sol A3 F[#]3 F[#3] G3 la-sol mi mi fa (1 m3 m3 1)
Now the minor third A3-F#3 is duly solmized as sol-mi, and the final semitone F#3-G3 as mi-fa.
Our hexachord on D also permits proper solmization of the inflected M3-5 progression given by Jacobus, the upper voice being shown with this revised solmization below the original one:
D3 F[#]3 G3 re fa sol re-ut mi fa D3 D3 C3 re re ut (1 M3 5)
If the singer of the upper part mutates from the natural hexachord of C-A to the new hexachord of D-B on the first note, then the next interval D3-F#3 becomes a proper ut-mi (major third), and the semitone F#3-G3 the expected mi-fa.
In short, the invention of such new hexachords permits the treatment of musica ficta inflections as an extension rather than contradiction of the classic system.
In order to accommodate the usual repertory of mi-inflections in 14th-century music - fmi, cmi, and gmi (f#, c#, g#) - we arrive at an expanded system of seven hexachords. The ficta hexachords of D-B, A-F#, and E-C# accommodate these new mi-degrees; the usual hard (G-E), natural (C-A), and soft (F-D) hexachords comprise the original untransposed musica recta system (Sections 1.3-1.5); and the Bb-G hexachord is part of the musica recta system when transposed by a Bb signature (Section 1.6).
Hexachord ut re mi fa sol la E-C# e f# g# a b c# A-F# a b c# d e f D-B d e f# g a b G-E (hard) g a bmi c d e C-A (natural) c d e f g a F-D (soft) f g a bfa c d Bb-G bb c d eb f g
These are the seven complete hexachords available on a 12-note keyboard tuned to provide the accidentals Eb, Bb, F#, C#, and G#. Such a keyboard in Pythagorean tuning would nicely fit the music of the Robertsbridge Codex (dated by Mark Lindley to around 1335), which calls for all of these accidentals.
Writing sometime in the second quarter of the 15th century, possibly around 1435, Ugolino of Orvieto describes the use of some of the new "imagined" or ficta hexachords in order to, for example, "perfect" or make major a sixth before an octave . Thus he notes that the progression
c-fa d-sol E-mi D-re m6 - 8
should be altered by use of musica ficta to
c[#]-mi d-fa E-mi D-re M6 - 8
and describes the "imagined" hexachord serving this purpose. Such a hexachord, with a mi-fa semitone between cmi (i.e. c#) and dfa, has its "beginning and foundation" on "the second A" (i.e. A3, the note known in the standard gamut as alaremi). "From there by this musica ficta we sing ut on A, re on square-B [B-natural], mi on C, etc."
In examples like this, medieval mi-signs and fa-signs seem equivalent to Renaissance or later sharps and flats: They allow one to make a semitone out of a whole-tone, or vice versa. However, another example from Ugolino to which attention is called by Andrew Hughes  demonstrates that this equivalence is not always exact.
In addition to showing how musica ficta is used to obtain progressions to the nearest stable interval (e.g. m3-1, M3-5, M6-8), Ugolino emphasizes its use to "perfect" stable intervals such as the fifth or octave which would otherwise be "imperfect," i.e. diminished or augmented.
Thus he gives the example of a two-voice progression where the lowest voice begins on the "first A" of the regular gamut (Are) and moves up a whole-tone to Bmi, while the upper voice begins at the octave on "the second A" (alaremi) with the syllable la (natural hexachord C-A), and descends to "the first F" (F3, Ffaut), in this hexachord mi. The result is an "imperfect" (diminished) fifth in which the solmization syllables mi and fa are pronounced simultaneously by the two voices, and from which "arises the greatest discord":
a-la f-fa A-re B-mi 8 d5
In order to perfect this fifth, Ugolino explains that musica ficta was invented, and here involves the use of a hexachord based on "the first D" (D3, Dsolre) as ut and G (G3, gsolreut) as fa. The F-mi of this hexachord - equivalent to F# - forms the required perfect fifth:
a-la f[#]-mi A-re B-mi 8 5
However, rather than indicating this alteration with a mi-sign for the affected F, Ugolino actually places a fa-sign on G before the affected note in the upper part in order to define a hexachord with F[#] as mi. If there were a literal equivalence between fa-signs and flats, then this sign should show Gb and only Gb - a note which Ugolino indeed recognizes as part of his extended tuning system with 17 notes per octave (Gb-A#), but does not intend here.
In fact, as Hughes points out, medieval accidentals are more flexible hexachord markers - in contrast to Renaissance and later systems where sharps and flats are held consistently to "raise" or "lower" by a semitone the notes to which they are applied.
Thus Ugolino's fa-sign on G might have two interpretations, both consistent with the basic hexachord meaning that this G represents fa in some hexachord, a note a semitone above the next lower step mi. In Ugolino's example, the indicated G-fa itself is not altered, but rather the F below it, which becomes Fmi (indicated by modern F#). As Ugolino himself explains, this interpretation of G-fa involves a hexachord with ut on D:
B la A sol G fa F[#] mi E re D ut
However, in at least one composition of Solage, and on Ugolino's proposed organ keyboard of 17 notes, the same sign G-fa can also indicate the same meaning as the later Gb: a note a chromatic semitone lower than G. This note is found as fa in a hexachord with Db as ut:
Bb la Ab sol Gb fa F mi Eb re Db ut
In Ugolino's example, where the fa-sign on G indicates the more usual ficta hexachord of D-B and calls for an F-mi equivalent to F#, we see another aspect of the "accidental as hexachord marker" concept. The fa-sign is placed at G on the staff, but there is no actual note G in the example: only A followed by the affected note F[#] in the relevant upper part, and A an octave below followed by B in the lower part.
Therefore the G-fa marker does not necessarily serve as the equivalent of a Renaissance or later Gb "lowering G by a chromatic semitone" - here there is no G to be "lowered" - but more generally and flexibly directs: "Sing in a hexachord where G is fa."
This example also may illustrate a subtle point concerning the distinction between musica recta and musica ficta. Rather than "perfecting" the impure fifth by altering the F in the upper voice, we might alter the B in the lower voice to B-fa (Bb):
a-la f-fa A-re B[b]-fa 8 5
At first blush, the alternation in the lower voice might seem to be musica recta, since generally B/Bb is a fluid degree in the standard hexachord system, with either Bfa or Bmi available. However, we are dealing here with "the first" B, B2, located where there is only one recta hexachord: the hard hexachord G2-E3. In this first hexachord the normal gamut includes only Bmi (B-natural), not Bfa (Bb). Thus the altered note Bfa (Bb2) would be a musica ficta inflection, like Ugolino's alternate solution of Fmi (F#3) in the upper voice.
While the unavailability of this low Bb as a recta note may seem a fine point, Richard Hoppin for example observes that this factor may have led to the transposition of certain chants up a fifth in order to relocate this note to a normal musica recta F (F3, Ffaut). 
The two categories of musica ficta which Ugolino describes, serving either to obtain stable concords in their perfect rather than diminished or augmented forms, and to make unstable intervals approach more closely to their stable goals (e.g. m3-1, M3-5, M6-8), are sometimes referred to respectively as causa necessitatis ("cause of necessity") and causa pulchritudinis ("cause of beauty").
However, both Ugolino and his apparently older contemporary Prosdocimus writing earlier in the 15th century extend the second category of inflections well beyond the usual "closest approach" resolutions to include inflections such as the following :
C#4 D4 E4 G3 F#4 G4 A3 F3 E3 Bb3 A3 G3 M3 M6 8 M6 M6 8 (Prosdocimus) (Ugolino)
In the first example, the expected M6-8 resolution occurs without any need for inflections; but Prosdocimus specifies the note C4-mi (C#4), although there is no immediate "closest approach" progression here.
In the second example, while the F4-mi (F#4) inflection in the upper voice is required for an M6-8 resolution, the preceding B3-fa (Bb3) inflection in the lower voice has no such conventional explanation, as Hughes observes . However, since this Bb3 is musica recta (part of the regular gamut), one could argue that its choice might reflect the traditional fluidity of BfaBmi as well as the liberal use of discretionary ficta inflections illustrated in the example of Prosdocimus.
Another feature of 14th-century polyphony promoting artistic choice is the way that cadences to some steps of the gamut may attain "closest approach" (e.g. M3-5, M6-8) by the use of either mi-inflections or fa-inflections - generally equivalent to later sharps or flats. This aspect of fluidity continues to play a role in Renaissance and Manneristic polyphony through the early 17th century.
For example, let us consider a cadence on A in a composition without signature, with a regular gamut of hexachords on G, C, and F. Performers might achieve the desired M6-8 and M3-5 progressions by using either Bfa (Bb, a recta tone) or D-mi and G-mi (D#4 and G#4, ficta tones):
G4 A4 G#4 A4 D4 E4 D#4 E4 Bb3 A3 or B3 A3 (M6-8 + M3-5) (M6-8 + M3-5)
If we join modern scholars such as Margaret Bent and Andrew Hughes in adhering rather closely to the medieval axiom that ficta should only be resorted to "where necessary" - or where explicitly indicated - then the first interpretation would generally be preferred in the case of a manuscript with no signature or indicated accidentals for such a cadence. 
Yet more possibilities for cadential discretion are opened by the use "partial signatures" - sometimes also known as "contrasting signatures" - common in medieval and early Renaissance polyphony of the 13th-15th centuries, where different parts have different gamut signatures. For example, a signature of Bb in the lowest part and no signature in the upper parts might serve to make available a cadence with a descending semitone to D, since the Bb signature (transposed regular gamut with hexachords on Bb, F, and C) makes E/Eb a flexible degree in the lowest part, and thus Eb musica recta, like Bb in the untransposed gamut (see Section 1.6):
(Bb signature, lowest part) (no signatures) C4 D4 C#4 D4 G3 A4 G#3 A3 Eb3 D4 E3 D3 (M6-8 + M3-5) (M6-8 + M3-5)
If all parts have no signature, then a solution with sharps would seem more likely, at least in the repertory studied and transcribed by Bent and Hughes: The pattern here seems to be that flats are very rarely introduced as ficta, e.g. here an Eb without any previous Bb to "naturalize" E/Eb as a flexible degree.
These last examples involve alternative solutions within the usual rule of "closest approach"; but we should not necessarily assume that this rule was followed by composers or performers in all cases. In the rhythmically intricate ballade S'aincy estoit, Solage concludes with a final cadence having a manuscript accidental of Bb:
Bb3 C4 F3 G3 D3 C3 (m6-8 + m3-5)
Such a progression with m6-8 and m3-5, routine in the 13th century, may have had a deliberately striking effect in the composition from around the end of the 14th century, the epoch of the Ars subtilior or "subtler art" when "closest approach" was the general norm. Many such progressions may have occurred at less prominent places in 14th-century pieces, likely depending in good part on the taste of the performers.
Another norm evidently subject to exceptions is that of avoiding "mi contra fa in perfect consonances" - that is, the simultaneous singing of "mi against fa" in unisons, fifths, or octaves, producing augmented or diminished versions of these normally stable intervals. Here are a few examples:
B3mi F3fa C#4mi Bb3fa B4mi Bb3fa B2mi F3fa B2mi Bb3fa (A1) (d5) (A5) (d8) (A8)
Whether this rule also applies to fourths is an interesting question: While typically viewed in the 13th century as relatively stable concords ranking with but after fifths, simple fourths or fourths above the lowest part are often treated as dissonances in 14th-century theory. Indeed, this is a point on which Jacobus dissents vigorously and at some length from the early 14th-century "moderns," taking a traditional 13th century view that the fourth is a true concord in itself although it becomes more pleasant when supported by a fifth below (Speculum musicae, Book VII, Chapters 5-8). 
A treatise of around 1300 mentions that one purpose of accidentals is to obtain pure octaves, fifths, or fourths where they would otherwise be impure; whether 14th-century musicians would include the fourth in the scope of such inflections causa necessitatis remains an open question. In certain cadences, the application of the principle of closest approach also results in the avoidance of diminished fifths and/or augmented fourths:
F[#]4 G4 B3 C4 F[#]4 G4 B3 C4 F[#]3 G3 B3 C4 F[#]3 G3 D3 C3 D3 C3 D3 C3 (M6-8 + M3-5) (M10-12 + M6-8) (M10-12 + M6-8 + M3-5)
In the first progression, the F-mi (F#) alteration achieves an M3-5 rather than m3-5 resolution between the lower voices while also resulting in a perfect fourth F#3-B4 between the upper voices rather than the tritone F3-B4 (Ffa-Bmi). In the second progression, F# in the highest voice likewise achieves M10-12 while resulting in the perfect fifth B3-F#4 between the upper voices rather than the diminished fifth B3-F4 (Bmi-Ffa). In the third progression, the F# inflections realize M10-12 and M3-5, and also result in the perfect fifth B3-F#4 and perfect fourth F#3-B4 instead of diminished and augmented versions of these intervals (B3mi-F4fa, F3mi-B4fa).
Since these inflections are routinely justified by the norm of closest approach, it may be a moot question whether and to what extent they might also be motivated by a desire to avoid impure fifths or fourths.
At any rate, one 14th-century theorist, Johannes Boen (1357), includes not only perfect fourths but also augmented or diminished fourths in a special category of consonantia per accidens or "consonance by circumstance." These intervals have a concordant effect in sonorities for three (or more) voices where they are supported by an appropriate lower interval.
While Johannes Boen's example of the perfect fourth placed above a fifth (e.g. A3-E4-A4) fits with earlier Ars Nova theory and practice, his examples of the augmented fourth above a minor third and the diminished fourth above a major third involve mi contra fa:
B3(mi) C4 (fa) F3(fa) G#3(mi) D3 E3
In practice, the diminished fourth might also occur in two-voice pieces, for example in this type of cadence with an ornamented m3-1 resolution favored by some Italian composers:
E3-mi F3-fa D3-mi C3[#]-mi m3 d4 1
Here the upper voice of the minor third C[#]3-E3 momentarily touches on F3 before resolving as expected to a unison on D3. 
While Boen's approach places perfect as well as augmented or diminished fourths in something of a special category, there are passages in 14th-century pieces where intervals such as fifths might be performed in augmented form (mi contra fa) also.
We may recall (Section 2.2) that Marchettus (1318) endorses direct melodic chromaticism in order to obtain a perfect fifth followed by an M3-5 resolution in this progression:
G3 G#3 A3 C3 E3 D3
In practice, however, performers may have often accepted augmented fifths in cadences such as this:
C#4-mi D4 F3-fa E3 D3 A5 M6 8
Here the first note in the upper voice remains stationary while the lower voice descends from the fifth to the sixth; a 6-8 resolution follows. The likely performer's inflection of this note as C#4-mi achieves an M6-8 progression - and also results in the augmented fifth F3fa-C#4mi. Leaving the note uninflected would avoid the augmented fifth, but result in an m6-8 resolution not fitting the ideal of "closest approach."
Some performers, faced with this dilemma, might devise a Marchettan solution of breaking this single written note into two, the first uninflected and the second inflected:
C4 [C#4] D4 F3 E3 D3 5 M6 8
While Hughes does recognize such a solution of "half-sharpening" a note as tenable in rare instances from the Old Hall Manuscript , one is hesitant to propose it as general practice. From a contrapuntal view, the version with an augmented fifth attractively lends emphasis to the melodic and rhythmic independence of the two lines: This interval results from the figure of two notes in the lower voice against one in the upper voice preceding their united cadential motion of M6-8:
C#4 D4 F3 E3 D4 5 M6 8
Such intervals add an engaging element of melodic and vertical color to Ars Nova music, an element which it is tempting to conclude was often accepted and relished by 14th-century performers.
For medieval theorists such as Prosdocimus and Ugolino, as Hughes observes, musica ficta is motivated by vertical considerations - or, as they themselves put it, for the "perfection" or "coloration" of some consonance. As we have seen, this includes the "perfection" of what would otherwise be "imperfect" stable intervals: unisons, octaves, fifths, and more problematically fourths; and also the alteration of unstable intervals in order to obtain progressions to the nearest stable consonance (e.g. m3-1, M3-5, M6-8). Ugolino draws a distinction between "perfection," in which a third expanding to a fifth or a sixth to an octave is made major, and "coloration" in which a third contracting to a unison is made minor, etc.
There are hints in certain 14th-century treatises, however, that singers sometimes may have made musica ficta inflections motivated by melodic considerations, and at times without following the usual rules of solmization (where a semitone is always sung mi-fa or fa-mi).
Viewing such practices with disapproval, the Quatuor principalia reports :
"[M]any [singers] in modern times are faulty ... since when they pronounce sol fa sol or re ut re they place a semitone there instead of a tone, thus confusing the diatonic genus and falsifying the plainsong."
Such alterations might also be made by performers of secular monophonic songs, such Machaut's virelai Douce dame jolie, which has this closing phrase, with solmization shown in the natural hexachord of C-A:
... 2 | 1 2 & | 1 2 | 1 G3 F3. E3 D3 C3 D3 sol fa mi re ut re
Since there is only a single voice, the conventional vertical motivations for musica ficta do not apply; but this passage nicely illustrates an occasion where some performers might sing the concluding figure D3-C3-D3 (re-ut-re) with a semitone rather than whole-tone - that is, with an inflection would would be written with a mi-sign on C3, equivalent to C#3.
In practice, some 14th-century performers may have sung and played a whole-tone and others a semitone, and modern performers may choose either alternative.
Another 14th-century observer, the author of the fifth treatise in the Berkeley Manuscript or Paris Anonymous (a portion of which bears the date 1375), also reports the use of such semitones, but with evident approval. 
Having proposed a division of the whole-tone into three equal parts - a scheme which, if carried out systematically, would result in a form of temperament quite different from the usual medieval Pythagorean system and identical to a Renaissance division of the octave into 19 equal parts proposed by Guillaume Costeley in 1570  - the author discusses accidental signs and their effect on tones and semitones.
A usual semitone of mi-fa or fa-mi (e.g. b-c, e-f) is called a semitonus, equal to 2/3-tone. A semitone indicated by the sign of the "square-B" resembling a modern natural sign changes a whole-tone such as fa-sol to a semitonium of only 1/3-tone. While stating that such an inflection is most readily placed at fa-sol, the author adds that the same sign and inflection are applicable "from sol to la, from ut to re, and from re to mi."
Having introduced this mi-sign and explained its intonation, the author adds :
"Although it may not be used in plainsong, nevertheless a [square-B sign] is always comprehended at the end of any ascent between the next-to-the-last and the [ultimate] note, as is shown here:"
In quoting the example which follows, which shows natural or accidental semitones involving steps of the gamut, I use an "h"-sign (following the German notation for B-natural) to show the "square-B" sign indicating the narrow semitonium of 1/3-tone, and the usual sharp sign (#) to show the sharp-like sign which this author apparently uses to indicate the usual semitonus of 2/3-tone between E3 and F3 (mi-fa) :
Fh3-G3 Gh3-A3 E#3-F3 Fh3-G
From the perspective of solmization and accidentals, this example is interesting in part because it shows the use of a "#" sign simply to confirm that the marked note E3 should be sung as mi (2/3-tone below the next higher step fa, here F3).
More generally, however, the author suggests that cadential semitones are to be "comprehended" or understood in monophonic music, as well as in polyphonic music where the vertical rule of "closest approach" would generally call for ascending or descending semitonal motion in at least some of the voices. 
The Quatuor principalia and this treatise of the Berkeley Manuscript thus report that performers might sometimes sing or play "understood" musica ficta semitones in plainsong and other monophonic music as well as polyphony. The different attitudes of the two authors regarding this practice indicate divergent views on musical propriety and taste which may serve as a basis for diverse interpretations by modern performers also.
To Section 3 - Renaissance and Manneristic approaches.
To Table of Contents.Margo Schulter