Commentary from Kevin M. Rooney on the Correct Rendition of the Quilisma (Abridged and Slightly Paraphrased, with Permission):

The mystery of the quilisma is solved.  The quilisma is a rising portamento, Schleifer, or slide, or alternatively a light passing note.  This is more or less the interpretation that Solesmes has always called for.

Performance: The note before the quilisma is always long, as Mocquereau observed.  The note after the quilisma can be short or long, depending on context.  If singing mensurally the early medieval way, the quilisma itself steals a sliver of time from the long note before it.

Fact #1: The note before and below the quilisma is always lengthened, never short.
Mocquereau gives three pages of demonstration, pp. 415-417 of his Le Nombre, English translation.

Fact #2: The quilisma itself never lands on a downbeat.
Mocquereau, p. 415: The rhythmic pulse always falls on the note before the quilisma, never on the quilisma itself.  Cardine, p. 201: The melody always “tends toward” the note that follows the quilisma, in that the paleography consistently places a rhythmic accent upon the upper note that follows the quilisma, but again never on the quilisma itself.

Fact #3: The quilisma itself is never long.
There is not a single instance in the repertory of first-class Gregorian chant manuscripts of a quilisma lengthened by a horizontal episema or by a Romanus letter t for ‘tenete’ = ‘hold’.  This suggests that the quilisma is never a long note.

Fact #4: The quilisma is interchangeable with a light passing note.
We see it sometimes replaced in Laon and Chartres by a breve (short) depicted as a dot.  It happens more often when the ascending interval is larger than a minor third.  In Old Roman notation, which doesn’t depict rhythm but does like to add Italianate decorations, the quilisma is never depicted as any fancy figure, but just an ordinary note on the semitone.

Fact #5: The quilisma is interchangeable with an ascending grace note.
The rising initial auxiliary note of an initio debilis sign found on the second of two syllables is interchangeable with a quilisma bridging the same interval over a single syllable.  Because the first note of an initio debilis sign is functionally a grace note or appoggiatura (per Cardine, Murray, and Van Biezen), it follows that the quilisma is performed similarly, if not the same.

Fact #6: The quilisma is omittable.
It often disappears in parallel places of some Gregorian chant figures (Cardine p. 205), as seen when the quilismatic long pes is replaced by an ordinary long pes and vice versa (Cardine, p. 204; Van Biezen 2016, p. 16).  This can also be seen in parallel passages throughout Old Roman Bodmer C74.  The quilisma is therefore an ornamental note not necessary to the structure of the melody.  Mocquereau, p. 417:

The history of the quilisma during the period of greatest decadence fully confirms this deduction.  One of the most significant and most common characteristics of this epoch was the total omission of the quilisma.  The loss would be inexplicable if a fundamental or lengthened note had been in question.  There are many instances of such omissions.  Where however the quilisma note itself has been retained, it appears as the middle note of the neum of which it formed part, though this again does not imply that it had been originally strong or long.

Thence we can determine from the paleographical evidence alone that the quilisma is never long, never structural, never on a downbeat, and never important; but rather always light, always ornamental, always functioning like a passing note, and perhaps always optional.  Vollaerts 1958/1960, p. 110, gives a list of chants for comparative analysis to confirm the above points.  I’d be glad to give my own as well.

Still, that’s not enough, and so here’s some more evidence, philological and testimonial:

Fact #7: The quilisma symbols of at least three medieval notations are medieval question marks customary in their respective regions.
Cardine’s first page on the quilisma chapter mentions this.  The medieval punctus interrogativus calls implicitly for a rising intonation of the voice.  Well, likewise, a rising portamento or slide is musically equivalent to, even acoustically the same vocal effect as, the rising of the voice at the end of a spoken question.

Fact #8: The German Baroque Schleifer is identical to the St. Gall quilisma.
I have yet to find whether there is a continuity here over five centuries, but, even if there weren’t, it’s a striking coincidence, given the equivalence between the performance of the Schleifer and the quilisma, as supported by all the evidence I just gave.

Fact #9: Aurelian of Réôme, circa 850, calls the quilisma ‘tremula,’ or “tremulous inflection.”
He points to the word “canticum” in the verse of Gradual “Exultabunt sancti,” which has a quilisma.  See also Treitler, p. 191.  Or, for Aurelian’s actual text, see the online Thesaurus Musicarum Latinarum.  He uses the Latin word “tremula” (“trembling”) more than twice in his writing, and that suggests the meaning was clear through the word itself.

Robert Fowells, Cardine’s translator, says the following (“Gregorian Semiology: The New Chant.  Part II,” in Sacred Music, vol. 114, no. 3 (1987), p. 9):

Because its appearance looks somewhat like a mordent and early manuscripts say it should be sung with a “tremulous” sound, many musicologists feel that it is a sign for a mordent or a shake, somewhat in the baroque sense.  However, “tremulous” not only means “shaking” but “timid.”  The Solesmes school has always held that the latter interpretation should prevail and that the note should be passed through lightly.

Fact #10: Guido of Arezzo, circa 1022, enumerates ‘tremula' with long and short as a separate category of duration.
Micrologus, chapter 15:

. . . and some notes compared to others have a duration twice as long, or twice as short, or trembling, i.e. a varying duration which, whenever long, is signified by a virgula plana assigned to the letters.

The virgula plana, which means a horizontal line, is the episema.  But the “trembling” or “tremula” note, i.e., the quilisma, is never long, because it is never furnished with an episema.  So it is obvious in Guido’s grammar here that he means the ‘trembling’ note is a third category of note duration that is not long and not exactly short either.

Fact #11: The ‘tremula’ is defined in the medieval theory as two or three small notes.
From the Quid Est Cantus and other medieval treatises, a ‘tremula’ note is:

ex tribus gradibus componitur, id est, ex duabus brevibus et acuto
composed of three pitches, i.e., of two short notes and a high note

The duabus brevibus here are together the single note that we call the quilisma itself.  Elsewhere in the testimony, it is described as “two or three” short notes.  When singing two or three short (enharmonic) notes compressed into the timespan of a single swift note, as the paleographical facts above require, the result is a slide.

Fact #12: Nonantola notation depicts the quilisma not as a question mark but as two rising dots.
So two rising dots in Nonantola are equivalent to the swift note implied by the paleographical facts.  The only resulting interpretation then is exactly what the medieval theorists’ definition calls for.

Conclusion: First note = long.  Second note = slide or swift passing note.  Third note = not restricted by the above evidence; its length depends on the melody.

The interpretation of the quilisma itself as a sliding transition requires the length of the note before it, and since the quilisma cannot be interpreted any other way according to the evidence, it follows that the note before it is required to be long.  If the note before it were short, it would merge into the quilisma and become part of the slide.  This is a more musicologically accurate way to say what Mocquereau always said: that the quilisma retroactively lengthens the note before it.

I don’t have Cardine with me, but I don’t recall him really disagreeing substantially with Mocquereau.  I think his observation that the first note is long on account of its sign is easier to reckon with than Mocquereau’s retroactivity, but that’s splitting hairs.  Mocquereau might have made mistakes in interpreting the data, but in my opinion when it came to data observation he was a fantastic paleographer.  Same with Cardine.

On the other hand, I admit showing the first note is long in and of itself would require good examples.  And showing it is never short is harder, since it would require all examples!

Too many scholars have contended that the quilisma has a vibrato or trill.  This is wrong because:

• There is no practical way to execute a vibrato or trill on a light passing note.  It cannot be done without giving the quilisma a long beat, and there is no evidence of that in the paleography.

• The misunderstanding of tremula as meaning “shaking,” as in shaking the voice up and down, is tied to the St. Gall quilisma wherein the pen undulates upward and downward two or three times.  But other notations do not depict the quilisma in this way.  The Laon/Metz/Messine/Lotharingian notation uses an upside-down question mark, and a question ends with a rising sliding voice, not with a vibration, as Cardine noted.  The standard modern notation for a glissando, also used for a portamento, has any number of serrations, yet no one misinterprets it as a tremolo.

• The misunderstanding of tremula as meaning “shaking” may also be due to a confusion in the High Middle Ages between the quilisma (a rising slide) and the salicus (interpreted by Van Biezen as a composite containing a rising mordent).  The first two notes of the three-note salicus are sometimes on the same pitch or one pitch apart, and some singers seem to have a natural tendency to put a mordent between them to keep them distinct.  Although many notations write the plain punctum of a scandicus instead, the symbol for the second note is generally considered to be a form of the oriscus, which in the notation of Laon is practically the same sign as a turn (gruppetto) in modern notation.  This may indeed indicate a “trembling,” i.e., a shaking, of the voice, but it cannot be said for certain.  The oriscus may be merely a melodic indication initiating upward or downward motion; it is never followed by a unison note unless a new phrase begins.  The confusion between trembling = timid and trembling = shaking is a result of the rising quilisma figure and rising oriscus/salicus figure being interchangeable in many circumstances because of their rhythmic time equivalence, two long beats when sung mensurally, the lack of notational differentiation between quilisma and oriscus in Chartres 47, and their status as ornaments—the quilisma certainly and the oriscus possibly.

Conclusion: The quilisma is either a rising passing note or a slide, between two beats.

The notion that chant was actually measured is demonstrable, and has been demonstrated before.  The reason so many honest God-fearing people doubt it is their trust in Solesmes’ repeated denial thereof ever since the Vatican edition.  The reasons for their denial are sundry but nonetheless unjustified.  If Solesmes admitted their interpretations were novel, we’d have no problem, since their music can still be lovely.  But, since they have claimed to have a (mostly) historically accurate interpretation, per Cardine, it behooves us, in my opinion, to correct that mistake in our understanding.

According to Guido of Arezzo’s Micrologus (c. 1022), chapter 15, paraphrased, says the duration of a note may be long or short or tremula (which refers to the quilisma), and whenever it’s long it is marked by a horizontal line that doubles the note.  The term measure is inherent in the medieval word for chant modulatio.  The theory for proportional concord between rhythmic units in cantus melodies reaches all the way back to Greco-Roman antiquity and is consistently held down to the Carolingian era, not vanishing till the second millennium.  The a priori supposition that chant should be free of rhythm is due to (1) the chant’s loss of rhythm in the 1000s, (2) Solesmes’ historical bias against fixed rhythm, and (3) the a priori assumption that all chant melody evolved from oratorical prosaic rhythm-free psalmody.  It’s a big topic, and I recommend another thread with a better title.  Still, to start somewhere, John Rayburn's Gregorian Chant: A History of the Controversy Concerning Its Rhythm, 1964, is required reading.

In Early Medieval performance practice, which Solesmes claims is its gold standard:

• quilisma’s first note is always lengthened, if it is present.  Paleography proves this.

• quilisma’s second note is never lengthened.  Paleography and theory both prove this.

• quilisma’s third note is sometimes lengthened.  Depends on the neum.

In my opinion, the reason people are against one interpretation or another is nowadays more due to unawareness than anything.

Tempo, according to the medieval theorists, is constant in Early Medieval performance practice, except at the last few notes.

Episema isn’t always doubled; that’s just the default meaning.  Most of the time it is.  But, if you follow the theory of Van Biezen, Gregorian chant also has a bit of notes inegales.  For example, the figure short-short-long-short-long-long is then  𝅘𝅥𝅮 𝅘𝅥𝅮 𝅘𝅥𝅮.𝅘𝅥𝅯 𝅘𝅥  𝅘𝅥  [8th, 8th, dotted 8th, 16th, quarter, quarter], which makes sense only if you keep the plausus (pulse) encoded by St. Augustine and enjoined by the Enchiriadis writings.  It’s a relatively new discovery that removes the chief obstacle that Cardine rejected mensuralism over in 1964, namely that mensural fixity is incompatible with the “looseness” (what does that even mean?) of the neums.  Fluidity and aesthetics are not compromised; quite the opposite rather.