Hcfyk

This is a question that readers may find confusing. It is a bit confusing to me too, but I'll try to explain and illustrate it as best as I can.

What are the harmonic functions (in terms of Roman-numeral scale degrees) normally associated with a stepwise intonation of a diatonic scale (upwards or downwards)?

There are of course tons of examples of such scale-based phrases in classical music, but I would use here the examples of

A. Kalinnikov's Serenade for Strings, the 7-step descent (in minor) at about 1:01; and

B. the full (8-step) downward major scale upon which the incredibly-powerful Pas de Deux (from Tchaikovsky's Nutcracker) is built.

Interestingly, while the scale descent in A. implies/needs a harmonic change with every note, B. only seems to require three: one at the beginning (the ii, I think), one at the middle (the V), and one at the end note of the scale (the tonic).

In both cases, intonation of the scale itself describes (or closes) the melody completely. But what exactly is the music-theoretical mechanism that requests (or not) a change of harmony at each further step along this melody? Can this harmonic progression be regarded as a cadence, and if so, why are the harmonic functions that the two examples (cadences) go through so different?

There are two common ways to harmonize a step-wise, diatonic bass.

• The Rule of the Octave
  

Example from Fenaroli...

• And harmonizing the scale with chord of the sixth (first inversion triads), some call this Fauxbourdon...

...what exactly is the music-theoretical mechanism...

If you read Gjerdingen's article on the rule of the octave, you will see the basic theoretically (really a stylistic preference) is to use stable root position chords on the tonic and dominant and chords of the sixth on the other scales degrees. A sort of pure theory explanation of why there would be such a preference is the tonic and dominant chords define the key and so should be heard clearly and as cadence points they need to be stable. All the other chords should be unstable inversions to create the dynamic instability that leads to the tonic and dominant.

By comparison, Fauxbourdon will harmonize the dominant degree in the bass with an unstable and tonally weak iii6 chord. That lends a lot to the 'drifting' sound of Fauxbourdon. In classical style these parallel sixth chord progression will normally break of at some point and form some kind of cadential harmony. You can think of it as a very elaborate move to a stable conclusion.

The difference is due to rhythm. In every song there's a perceived rhythmic pulse: beats, stronger beats and weaker beats, on top of which melody notes land. Notes that are perceived as being on strong beats feel like making a harmonic statement, and notes that are not on strong beats feel more like embellishments. Notes that are on strong beats can be considered "chord tones" and notes on weaker beats "passing tones". How you perceive the rhythmic pulse depends on many things like tempo, dynamics, articulations and what the backing instruments are playing.

In your first example, all of the notes are emphasized quite strongly, so that they all create strong beats. In the second example, only a few of the scale notes land on strong beats, implying fewer changes in harmony.

Here are two examples of descending lines which should make you feel differently about harmonic changes. I have deliberately omitted barlines and other things. The relevant thing to notice is the dynamic emphasis.

I don't think it has anything to do with the direction of the line either. I think the same phenomenon works with ascending lines just the same.

Does it even have to be a linear sequence? How about this:

The harmonic implications of a melody line greatly depend on how you emphasize the notes.

Compare these two:

In my ear the lines imply different chords or at least different inversions. The same notes in the same order, but with different emphasis, leads to a different harmony.

There's lots of ways to do it. Trivially, the whole C major scale could be harmonised by Cmaj13. I guess what I'm trying to say is that you don't need to actually change chords every time the melody moves up or down, and you actually could harmonise the major scale with some non-diatonic chords! For example, the note G could be part of an E♭ chord, which would work well if the G isn't supposed to be part of a dominant chord.