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?