Here are some thoughts after working through every exercise in Gradus ad Parnassum. These are not intended as summaries of the treatise, or even as accurate representations of the syntax being described there. Rather, they are my own reactions to, and interpretations of, that syntax.
1. Some modes are unstable as to pitch content, others stable. This is similar to a difference between major and minor, where the latter has options at la and ti, while the major is fixed (alternative pitches being heard as such, as outside the scale).
The unstable modes are:
Dorian (variable la – B natural/B flat)
Lydian (variable fa – B natural/B flat)
Mixolydian (variable ti – F natural/F sharp)
In all three modes, the instability results from the desire to avoid the B-F tritone, and stems from voice-leading situations common to the mode. The tritone is avoided horizontally, but not vertically: The augmented fourth is treated the same as a perfect fourth (dissonant only as against the bass), and diminished fifths are treated as consonances. (Note that only in Locrian mode, which Fux ignores, would a diminished fifth appear in the final cadential triad.)
The stable modes are:
Phrygian, Aeolian, Ionian.
Note that in these three modes, the “normal” resolution of the tritone B-F gives C-E, which includes the tonic in Phrygian (E) and Ionian (C), and the two non-tonic tones of the tonic triad in Aeolian (C-E). By contrast, C-E aren’t in the Dorian or Mixolydian tonic triads, and give only the fifth (C) of the Lydian tonic.
The modes fail to exhaust the possible combinations of M3/m3, M6/m6, and up-moving leading tone on seventh versus down-moving leading tone on second (these being the most characteristic variables for distinguishing modes).
Vertical and horizontal intervals behave very differently:
Horizontal intervals: motion by second is strongly preferred. Several intervals are prohibited: tritone, major sixth, descending minor sixth, and minor/major sevenths. Repeated notes are not used, but octave jumps are.
Vertical: All intervals are allows, but some are characterized as “dissonant” and mandate specific following movements. These required motions can generate chain reactions, especially with ligatures, which can devolve into extended descending sequences as dissonant notes resolve down a second, which then in turn becomes dissonant and requires another downward resolution, etc.
Local syntax is based on pitch-class sets. The emphasis is on sounding complete triads. There is a distinct lack of emphasis on harmonic rhythm or harmonic motion. Any triad can succeed any other triad. The pitch-class sets are based on set types (0,3,7), (0,4,7), and (0,3,6) [aka tone + m/M3 + m/M3]. (0,4,8) sets (augmented triads) don’t exist in this system.
Dissonances act independently – they can occur in more than one voice at a time. The resulting vertical sets are not considered as functional harmonies (e.g. 7th or 9th chords), and any vertical set is allowed.