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I’ve noticed that in a melody certain notes stand out and it’s always the root of the current chord or arpeggio. (also, are arpeggios only descending/ascending or can they be in a random order?)
In other words, do the roots actually serve as the basis for what is being done melodically?
Originally I was searching for how to establish tonality but then I realized what I’m trying to understand is actually a tad bit deeper.
Historically, melody existed long before harmony. Before the modern theories of chords were developed, people were writing music that we can analyze harmonically today without any harmonic theory: they analyzed and probably composed this music by concentrating on the intervals between each voice and the melody. That is, you'd start with a melody and write other voices to go with it, sometimes one other voice, frequently two or three, and sometimes even more.
With large numbers of voices especially, it became increasingly common during the renaissance for the lowest voice to move by leaps of a fourth or fifth. By the baroque period, organists and players of other chordal instruments were playing from parts that included only the bass part (originally it included whatever part was lowest at any given time, hence basso continuo -- "continuous bass"). Perhaps because of this, the study of composition began to include writing harmonic accompaniments to a bass line without a melody.
The need to realize a chordal accompaniment naturally led to a theory of chords with which a musician could decide what to play, and that in turn influenced the theory of music composition. In the middle of the baroque period, Rameau published his theory analyzing chords as stacked thirds and identifying for example a chord consisting of E-G-C as an inversion of the chord C-E-G.
When you apply this theory to music that existed at the time you find that the implied bass note -- the root in the modern sense -- moves even more frequently by fourth or fifth. For example, a bass line C-D-E-F-G-C could be harmonized (in modern notation) C G7/D C/E Dm/F G7 C -- one move by step and the others by fourth/fifth.
So, historically, the answer to your question "do the roots actually serve as the basis for what is being done melodically" is no or at least not directly -- a melody that might go with that progression is E-G-E-D-D-C, with only two of six being the same as the bass note and only one being the root in the modern sense.
But in fact it's not clear to me what you mean by "the note that stands out is always the root" -- maybe you are noticing a certain effect when the melody note has the root that is absent when it has the third or fifth. Or maybe you're noticing a tendency for the melody to be harmonized with chords built on the melody note at certain points in the melody's course.
To elaborate: melodies typically do have a certain "pull" toward the tonic note, and there is a sense of rest or arrival when they are on that note, and this is typically harmonized with the tonic chord. So in the sense of trying to understand tonality, you may be heading in the right direction. Chords and melody tones that appear in the middle of the melodic phrase are less likely to be the tonic chord and the melody is less likely to be on the root tone of the chord -- it's more likely to be the third or fifth or seventh or even something more dissonant.
(A broken chord that isn't in ascending or descending order is unlikely to be described as an arpeggio.)
No. A lead melody can be accompanied or harmonized with countless different chords with different roots, and each alternative just feels different. A melody line may "imply" certain chords, but it's subjective, and different options are possible.
So to answer the question logically, since there can be many different root notes depending on which chords an arranger or accompanist happened to choose that time, "the roots" don't have to have anything to do with anything.
Let's say you have a melody or "melody" consisting of only C notes, you can play a lot of different chords as accompaniment. Try it! C, Am, F, G, Dm, Db, Bb, Eb, Ebm, ... just about anything goes. Can you come up with a chord that absolutely, definitely cannot be played together with a C note?
What chords a melody implies, is subjective and can be vague or arbitrary. I can imagine different chords at will, it makes music less boring.
What melody notes "work" over a given chord progression, is subjective and a matter of opinion as well. It's an art.
It's definitely not always the root. For an example, consider this:
This is the verse of Imagine by John Lennon. If you look for chords whose root is featured in the melody, you'll find two: the Dmi (where it pretty much functions as a grace note) and the C near the end. That's it. For all the others, it's their third, fifth, and/or added major seventh.
The fact that it took me only seconds to think of an example where this would be the case for a number of consecutive chords can tell you how ubiquitous it is.
It isn't just the root of the chord that matters, it's the chord as a whole.
In general, you're likely to find a tone (or more) of the underlying chord in the melody. Any tone of the underlying chord. That's how you get all those three-chord songs - a tonic, dominant and subdominant cover the whole scale, so what more can you possibly need? (Answer: Some flavour to spice it with, basically.)
But not even that is strictly always the case. Occasionally you can come across things such as E in the melody (and not just in passing) over a G chord. Effectively this forms a G6 or Emi7 chord together, which is why it can work nicely - but even so, it's far from common.
(I know others have already said it, but I feel like adding an example to illustrate.)
The root of the current chord doesn't necessarily have a special function melodically. It's not obvious what you mean by "stand out," and judging by the typical struggles of learners, you're in the minority if you can pick out the root instantly regardless of voicing. One major exception is the tonic over the I chord, the crucial "restful" state around which western theory revolves.
However, one of the main principles of melody, going back to the common practice era, is that the notes that fall on strong beats are in the current chord. In Baroque music, this was strict, and non-chord tones were always considered transitional, leading back to a chord tone. Modern theory is more flexible, but this rule still forms the basis of the relationship between melody and harmony. It could be that this relationship is what you're trying to describe.
The general tendency is for notes in the melody to reflect in the current chord, or vice-versa. Otherwise, the chord is inappropriate, or the melody is 'out of tune'.
But, as far as just the root is concerned, that's only one of three notes in the underlying triad, or one of four in an extended chord, and so on. Melodies will generally have one, or some, of the chord's notes, but they could just as well be the 3rd, 5th, 7th (major, minor, diminished, augmented). That way, the music fits together sonically.
I doubt there are many pieces which have no chordal notes in them, but there are plenty that don't feature the root or tonic.
