I thought that the answer to both these questions amounts to steps one needs to take on the circle of fifths...
You are conflating the two circle of fifths.
Root progression by descending fifth is a common harmonic progression, it is commonly called a circle of fifths, but that isn't the circle of fifth of key signatures.
Circle of fifths progression is visually represented like this:
Circle of fifth key signatures is visually represented like this:
Is there an easy way to visualise... And to also use the circle to 'measure' how distant a given out-of-key chord is to our home key? For instance, B-flat is only 2 steps away from C, whereas the in-key chord E is 4 steps away.
Roman numerals are used to label chords in relation to the key signature not the circle of fifths.
C major the
E minor is the diatonic triad build on the third scale degree. It's labelled like this:
In that case the tonic is
C, but you can make a sort of temporary or secondary reference using one of
C major's diatonic chords using a slash notation. If we are in
C major and then use a
D major chord, we can say
D major is the dominant of
G major which is the dominant of
C major, and then label it like this:
If you compare that to the circle of fifths of keys, it is two steps away.
You can have other secondary chords. An
A major chord is the dominant of
D minor is the
ii chord. The secondary label is then:
C: V/ii ii
You would say that out loud as "five of two" resolving to the "two chord."
A major chord is three steps away on the circle of fifths of keys, but that is more or less coincidental. The important thing is the secondary dominant relationship, and that is an immediate relationship to the
You can look at two elements in the case of
C: V/ii ii or
A major to
D minor in the key of
A major chord can be associated with the key
A major which as 3 sharps and is three steps away from
C major on the circle of fifths of keys.
A major is the dominant of
D minor, it is a chord from the key of
D minor. It is the
V chord of
D minor. The key of
D minor has one flat and on the circle of fifths of keys it is only one step away from
In harmonic analysis the second point is the important one!
Relate chord to the keys to which they belong. Don't relate chord to their coincidental place on the circle of fifths of keys.
B flat major in the key
C major presents an interesting and different kind of relationship which can be looked at two ways: a secondary chord or a borrowed chord.
You can view that as a secondary subdominant chord. In
C major the subdominant is
F major. In
F major the subdominant is
B flat major. So,
B flat is the subdominant of the subdominant in
C major. You can label it as:
...the "four of four."
Borrowed chord are chord that can be found in the parallel key.
C minor is the parallel minor of
C major. In
B flat major is the
VII chord build on the lowered seventh scale degree. In Roman numerals we use upper case to show the chord is major and add a flat sign to show the chord root has been lowered from the major mode:
In terms of measuring distance the secondary subdominant comes from a key which is one step away on the circle of fifth:
B flat comes from the key of
F which is one flat different from
When thinking of the measure of the borrowed chord, I suppose you could say there is no distance because the tonic doesn't change from
C major to
C minor. Or you could say it is fairly distant, because
C minor is three flats different from
C major and that is three steps along the circle of fifths of keys. Probably many think of the borrowed chords as colorful because of their chromatic nature, but they aren't distant because the tonic isn't changed.