One of my motivations for working on chant is my fascination with the elusive nature of this music. Lacking defined rhythm, form, tonality or harmonic function, perception of structure in this music seems vague and subjective, yet endlessly compelling. I have been encouraged by several people to set down my observations on the musical structure of chant, and I will attempt a beginning here. Comments and observations are always welcome: please drop me a note at firstname.lastname@example.org.
Antiphon: Orietur in diebus p. 
Due to the amorphous nature of the music it is hard to know where to begin, so I will simply begin with the observation that in typical antiphons for the office it usually seems possible to reduce the neumatic style to a syllabic style by omitting ornamental notes. I find that this can help in discerning the underlying structure and at the same time helping to secure a satisfying performance. So, I shall address an antiphon that I happen to be looking at right now, Orietur in diebus, Antiphon 6 at Matins on Sundays in Advent. page . Here I would reduce ‘Orietur’ to F.D.C.F. To be specific, I would consider E a lower neighbour, and D of the third syllable an appoggiatura. This would be related to Schenker’s ‘Boundary-play’ idea. One could go into a more sophisticated analysis of the music of this first word, but my point here is really only to set up the idea of looking at the possibility of an underlying syllabic structure for antiphons. Moving along, I would see ‘diebus’ as F.G.G, ‘ejus’ as G.G, ‘pacis’ as A.G. ‘eum’ is particularly interesting; it could be understood as F.F. or G.F; or as two levels beyond the surface, an initial level of G.F. where G leans on F as an appoggiatura, and a deeper level of F.F, where the appoggiatura is removed. ‘omnes’ would be F.D; ‘reges’ F.C; ‘gentes’ simply F.F, and ‘servient’ would be G.G.G. (It is worthwhile to sing the antiphons reduced to one syllable per note. This can confirm one’s perceptions of the structure.)
Moving along we could then see the underlying pentatonic basis of the melody, C-D-F-G-A. and the basic contour would be movement through the third F-G-A-G-F. (April 17, 2020)
The primary function of a psalm-tone ending is to effect a convincing return to the beginning of an antiphon. Besides this, a psalm-tone ending can effect a motivic ‘linkage’ to the beginning of an antiphon. So for example when the antiphon Dominus defensor vite mee (Noted Breviary A-04, p. ) is used at matins on Mondays, the gesture GAG of the opening is foreshadowed in the ending of the psalm-tone. We should be aware, however, that this sort of thing is not found consistently. The antiphon Alleluya iii. for sext on ferias in Eastertide (Noted Breviary B-33, p. 1347) contains the same (F.)GAG gesture. Here the connection is to be again with the end of the psalm-tone, F.GA.G.(F), but the gesture seems to be more of an echo than a direct repetition.
Heinrich Schenker, who greatly admired this technique in the works of the nineteenth century masters, might have been surprised to find it also here and there in the simple chant that he generally dismissed.
Linear progressions and tonal structure
I am not prepared to make bold generalizations concerning linear progressions and tonal structure in Gregorian chant. But I will point out that these concepts are certainly relevant. The invitatory antiphon Alleluya iii. for matins on Mondays after the octave of Easter (Noted Breviary B-33, p. 1338) displays very clearly an initial ascent F.G.A. over the course of the first two alleluyas; the final alleluya begins with a rapid retracing of this rising third (a ‘slide’ or double appoggiatura), and concludes with A-G-F, a ‘3-2-1’ descent. G is decorated with a turn, and F with an appoggiatura.
In larger terms the Venite itself (Noted Breviary D-1, page 35*.) is based on the upper neighbour, Bb and its accompanying G, which resolve back to A and F respectively at the end of the verse. The final gesture of the verse repeats the initial ascent found in the antiphon. That is, the verses of the Venite are based on the upper neighbour, while the antiphon is based on the principel note, A.
Antiphon: Veniet fortior me p. 116
This mode VIII antiphon is unusual in that it is full of B-flats, except for the first two very Bs. The B-naturals are lower neighbours to C; the B-flats are upper neighbours to A. To our thinking the profusion of B-flats seems to deny the mode. On the other hand, the ‘structural tones of the antiphon are C, A, and G, which confidently assert mode VIII. Compositionally, the piece seems to be based on CDCA and its sequenced repetition, AB-flatAF, after which the A falls to the final G. The sequencing of C-A to A-F forces the music into an F hexachordal structure. It would seem that the conflict, such as it is, is between the G mode and the F hexachord. There may be conflict in the theoretical ideas of mode and hexachord, but there is no conflict in the music of this chant. Further, the antiphon is entirely in accord with the psalm tone: GAC-DC : C-ACDC. (In contrast, tone VIII.i. includes B natural.)
Antiphon: Accipite Spiritum Sanctum p. 1531
In most Sarum sources this mode VII antiphon is transposed to a C finalis, giving a ‘major’ scale instead of the putative mixolydian scale. Nevertheless, the psalm-tone retains its usual shape, with B-flat in its transposed form. (This does not appear in the cue for the psalm-tone, but it may be observed in the Sarum Psalm-Tones. p. 60*.) Most of the continental sources available give this antiphon in the normal transposition, and thus in a ‘pure’ mixolydian mode.