Some ideas about musical harmony and chord progressions, especially considered from the point of view of playing six-string guitar in 12-TET, standard tuning.
Chords are sometimes expressed as ordered intervals. Intervals are given enharmonically with the following names in chromatic order: 1, b2, 2, b3, 3, 4, TT, 5, b6, 6, and b7. So for example, a diminished seventh chord is given as {1,b3,TT,b6}, using b6 rather than bb7. Higher intervals are given in first octave terms, so for example a Maj9 chord is {1,3,5,2}, using 2 instead of 9. In chord names, Δ and Maj7 are interchangeable.
Guitar tablature for chords often uses a syntax such as x5467x. I prefer to use dots to indicate muted strings, and to omit any trailing dots. When speaking harmonically, I also use only relative steps, giving whatever is the lowest note as 0, and assuming that the deepest note is the tonic. So for example, x5467x would become just .1023, or .5467 outside of a harmonic context.
Combining the two syntaxes, .1023 is the same as {1,3,7,3}, i.e. a Δno5 chord.
Roman numerals are used as standard, though V-of-V may be given enharmonically as II, for example, and all borrowed chords are marked as flat, so bIII instead of #II. I use bV for the roman numeral, not TT.
iv7 is a good substitution for V7 in IΔ - iii7 - IVΔ - V7.
bii7b5 is a very good chromatic lead in to IΔ.
biii7 can be used as a lead in to V7, or perhaps even a substitution. This is because V7 resolving to I seems to get its tension from {4,7}, a TT step which is {b7,3} using the V as tonic, resolving to {3,1}, which chromatically neighbour {4,7}. The 4 (b7 in V) can be thought of as a kind of sus4 tone since V7 doesn't include the 3-of-I, which is the 6-of-V. To to extend the {4,7} to {3,1} progression, you can add {b3,b2} at the beginning. This most obviously occurs as the {1,b7}-of-biii7, but other chords can be used too. Since {b3,b2} is chromatic from the I, you can also use it after the I, in something like V7 - IΔ - biii7 - ii7 - bii7b5 - IΔ.
Diminished chords sound as they do not just because they are symmetrical, but also because they use dissonant intervals; because they take the {3,5,b7} of a dominant seventh chord's {1,3,5,b7} interval set and move them down a half step. But if you move them down a whole step, you get into some more consonant intervals again: {1,2,4,6}. This sounds extremely suspended without a 3, sort of like a sus2 and sus4 combined, so it resolves to the major triad or major seventh nicely, e.g. Ioo - I. A good guitar inversion for this is 2201, which is {1,4,b6,2}. The b6 is actually a treble-flat, bbb7! If you play I7 - Io - Ioo - IΔ, the character of the Ioo becomes clearer—or rather, is contextualised as a double-diminished more clearly.
Another example of use: IΔ - iii7 - IVΔ - iv7 - Ioo - ii6,9 - IΔ. Also note that the Ioo chord is very similar to the iv6,9 chord, which in turn is like a rootless bVII7,13. This is why the Ioo sounds good after the iv7. You can also come back up the dom/dim kinds by doing IΔ - iv6,9 - viio - V7 - IΔ.
A pentatonic scale is already very rich, which you can hear in the fact that a chord formed from a pentatonic scale sounds a little muddy. Four note chords are common on the guitar, usually a triad and a seventh-colour, or a reduced triad (i.e. {1,3}, {1,5}, or rarely {3,5}) with two colour notes. Yet we often use heptatonic scales. These have considerable scope for rich melodies when compared to four note chord arpeggio-scales. They're almost chromatic.
iii6,9 sounds like it wants to resolve to IIΔ. The reason is that it's a rootless inversion of VI7,13. ii6,9 therefore sounds fine, because it's just a V7,13 which will resolve back to I. For this reason, to use a minor 6,9 chord properly, you may have to be careful to emphasise the minor nature of the chord in order to stop it from sounding too much like a relative IV7 chord. Yet the b3 of the minor 6,9 is also the b7 of the relative IV7 chord. Of course this means that a relative v6,9 chord (V being the inverse of a relative IV) can be used as a substitute for the I7 chord.
A table of harmonic colour tones relative to I would be useful. There is an analysis tool for this sort of thing, but it's not a straightforward reference table.
The II9 chord may have some nice lydian feel to it, here using the standard {1,3,b7,2,5} inversion played as 10101:
Similarly the bVø7 chord, which is the same without the II tonic. The {1,TT,6,2} chord, 1200, can be a beautiful way of establishing a lydian sound in I: it hints at a Maj6,9b5.
A PDF has a nice diagram of harmonic progressions:
According to a recent analysis, the iii - X and X - I arrows at least are right, with the possible exception of the viio, which is kind of unlikely.
iii - IV (59%)
iii - vi (34%)
iii - ii (4%)
(Part 1)
IV - I (35%)
V - I (32%)
ii - I (8%)
vi - I (7%)
(Part 2)
It's surprising that iii - ii only comes out as 4% in the recent survey, and isn't given an arrow in the PDF diagram, because it seems like a very natural way to get back to the I chord.
It would be nice to use U+266D MUSIC FLAT SIGN [So] (♭), but for some reason ♭ displays very wide, as though it's from a monospace font. Perhaps I only have it available in Everson Mono—but on Wikipedia it displays fine. So I use "b", anyway.