There are many musicians, amateur and professional, who are not music theory conversant.
Most popular and folk musics are played by musicians who get in a groove, get used to working in keys and things, and do things in a way over and over that their cpacities are expressed habitually.
Theory often becomes relevant when you want to step out of the box so to speak as your OP does because it is asking about chord progressions outside a normal 7 note diatonic scale system which is, I presume, what you have normally operated in (other than a few accidentals or passing notes here and there). You asked about things in the context of a chromatic scale which by strict definition is a sequence of notes with every semi tone in it - nothing missing. Therefore it follows that every note in such a scale will have every possible chord type at its disposal (because every possible interval is available in a chromatic scale).
The mathematics isn’t that heavy either.
For the fun of it pplease see my ruler analogy …
We are both baby boomer generation from Australasia so I am going to use a 12 inch ruler. remember the lazy hazy days of feet and inches?
So we have a 12 inch ruler in front of us and we are going to mark the major scale on it - 7 notes and the first one repeated for the octave.
I am going to call the notes as Doh Ray Mi etc so as not get into key names. Every major scale regardless of what key it is in has the same relationship beiween the notes and so I will do this as a universal with doh ray mi etc.
MAJOR SCALE
Doh is at 0
Ray is at 2 inches
Mi is at 4 inches
Fah is at 5 inches
So is at 7 inches
La is at 9 inches
Ti (or Si) is at 11 inches
Doh (octave) is at 12 inches
Chords using a major scale:-
the most harmonic chords are ones where the root has a corresponding perfect 5th (perfectly harmonious) in it.
If Doh is root, the perfect 5th, is of course So. So is 7 inches from Doh.
Now we need the note in the middle for the chord triad. There is nothing at 3 and a half inches, the nearest being Mi at 4 inches. Therefore Doh Mi and So make up a major chord Had the Mi been at 3 inches we could have a minor chord but that is not happening here.
Now lets look at the Ray chord. To get a perfect harmonic triad we need to find the note 7 inches from Ray. That is the note La. There is no note exactly at 3 and half inches from Ray and the nearest one is Fah (3 inches) That will be the middle note and it describes a shorter interval from Ray Root than Doh Root’s middle inetrval. Therefore Ray is Root for a minor chord.
If you keep doing this with all the other notes you will get Maj, min min Maj Maj min sequence with the notes Doh to La.
Now when you get to Ti you will see that there is no note 7 inches from it (pretend you are extending your ruler by another 12 inches and repeating the previous pattern on it).
The nearest thing to 7 inches from Ti is Fah at 5 inches (1 inch from Ti to Octave and 5 inches to Fa= only 6 inches)
So when you pick this note as the other end of your chord triad you get a freaky sound, hardly complete in harmony.
The middle note for this triad is only 3 inches away and so it is a minor middle note. It is, in fact, a diminished chord.
It is harmonically diminished! It cannot be described as either minor or major, because the fundamental DNA of Root and Perfect 5th isn’t there as its main frame. The the thing has a Root (ti) and diminished 5th (not a perfect 5th) interval as its main frame.
Hopefully the ruler analogy will show you that the maths isn’t heavy at all. It is pretty basic arithmetic.
Its just that you haven’t probably turned your mind on it very much because you have taken
the keys and stuff you work in for granted and were more focussed and busy with the physical and aesthetic practicalites of playing the instrument.
In my case I do melodic composition in a tradition not restricted to the major scale and its modes and one in which pieces will often be transposed according to the needs of the singers.
I am also often involved in cross cultural arrangements and therefore I have been somewhat “forced” to go out of the box so to speak and be “versatile” in my approaches.
(I have had to spread my response over several posts because my PC dosn’t like long posts …)
Now, to complete the analogy lets us mark our ruler “chromatically”.
0 doh
1 inch Ray 1
2 ins Ray 2
3 ins Mi 1
4 ins Mi 2
5 ins Fah
6 ins Diablo
7 ins So
8 ins La1
9 ins La 2
10 ins Ti 1
11 ins Li 2
12 ins Octave
you can see from this that evry semitone is there and every note in this scale can form any type of chord possible.
The scale is “keyless”, not “locked in”. Even the idea of calling something by a tonal value name as Doh itself is oxymoronic.
Dunno if this will be useful for you, but I happened upon it, and remembered this thread.
http://www.musicafter50.com/2010/01/reading-roman-numerals-on-lead-sheets/