Answer: When we see a melody, how do we know which harmonies make an appropriate accompaniment? Being able to harmonize a tune this way is a useful skill. Very good at parties, if you ever chance to attend a party at which people stand by the piano and sing. Which is to say, a real party.

This is discussed in more detail in Chapter 10 of Exploring Theory with Practica Musica, but here's the basic idea:Harmony normally changes only on a beat, and most frequently on strong beats. And notes that come on rhythmically strong positions (for example, 1 and 3 in 4/4 time, or 1 in 3/4) are the most important for harmony and are most likely to be members of the prevailing harmony.

There are exceptions that come under the heading of "non-harmonic tones" in voice-leading. A note on a strong beat might move by second to a chord tone, for example. But if a note on a strong beat moves by leap it is almost certain to be moving between two notes of the implied chord.

So, if you see melody notes that move by leap from a strong beat, they are probably outlining part of the prevailing implied harmony, and if they move by step then usually the strong note is the harmony tone and the second one is an "in-between" or "passing" tone, but occasionally it will be the reverse, with the second tone being the chord tone. A passage consisting of a number of successive scale steps is almost always one in which the strong notes are in the chord and the weak notes are passing tones moving to another chord tone.

If you add to those very basic principles an understanding of which chords typically follow each other, e.g. I and V, I and IV, iii and vi, ii and V, etc., then it's not so difficult to come up with a believable chord accompaniment for almost any tonal melody.