I came across a chord progression called the Mixolydian Vamp denoted as : I bVII IV I, but I don’t recognize bVII. What is its relation to the diminished 7th diatonic triad (vii degree)?
I think there are three ways to think of the symbol
♭VII depending on the analysis or musical style.
In a very common style of analysis key is given with Roman numerals and upper case means chord with major third and lower case means chord with minor third,
o means diminished triad. Diatonic triads in, for example,
C major key are
C: I ii iii IV V vi viio. For
C minor the symbols are
Cm: i iio III iv V VI VII. Notice that no flat signs are given on the Roman numerals. That is because sharps or flats are understood from the key indication -
Cm - and all chords are understood to be diatonic.
Cm: VIIis a
B♭major triad, which is the diatonic triad of the subtonic scale degree in minor.
In that system you can do analysis to show mode mixture or borrowed chords. When you do that sharps and flats are given with the Roman numerals to indicate a change of chord root from whatever is the diatonic root of the given key. So, for example in
C: I ♭VII IV I, means the seventh scale degree, which in
C major is a
B natural, is chromatically altered down a half step to
B♭. The Roman numerals are upper case so the chord third is major, the final chord is a
B♭ major triad.
C: ♭VIIis a
B♭major triad, which is the subtonic triad in major borrowed from the parallel minor key.
In pop/rock/jazz a lot of mode mixture is used and it often gets analyzed differently. A basic, diatonic major key tonality is common, but various chromatic tones are added. Tones like the lowered seventh scale degree. In, for example,
C major, the lowered seventh scale degree is
B♭. In terms of scales that gives us
C D E F G A B♭ C, which is the same set of tones for a
C mixolydian scale. You could call this "borrowing" from mixolydian, or mixolydian "coloring", etc.
- The final point is then pop/rock/jazz will often consider
♭VIIas some kind of borrowing from mixolydian but the first two methods I describe consider the borrowing to be from the parallel minor key.
You could call the mixolydian view "pop" or "jazz" analysis and the parallel minor "classical" analysis. Personally, I don't like those labels, but they are commonly used. The difference is technical, but if you aren't aware, it can lead to confusion.
...bVII. What is its relation to the diminished 7th diatonic triad (vii degree)?
Comparing the two I would say this. Comparing either major or minor key or common modes like mixolydian and dorian, they all have the same
^4 scale degree a major second and perfect fourth above their tonics. Relative to a tonic of
^4 degrees will be
D natural and
F natural in major, minor, mixolydian, and dorian modes (basically all the common modes except phrygian and locrian, and locrian has never been used as a common tonality.)
In both chords -
♭VII and the diminished diatonic triad
viio - their chordal thirds and fifths are the same. The only thing differing is their roots. How you get one chord or the other is largely a matter of how the seventh scale degree is handled. In tonal music the "default" seventh scale degree is one half step below the tonic and is called the leading tone. When that scale degree is a whole step below the tonic it is called the "lowered seventh" or the subtonic.
So the relationship between the two chords is one of an alteration of the chord root, an alteration of the seventh scale degree.
But I think it's better to think of the relative relationship between the two chords and the tonic. The root of
♭VII is whole step below the tonic, and the root of
viio is a half step below the tonic.
The relationship is two notes identical. But the sound is rather different.
♭VII in key C is B♭ major, whereas vii° is B°.
The first is non-diatonic, the second diatonic.
The diminished triad tends not to stand up on its own, as it's neither major nor minor, thus is often used as a transitionary chord. ♭VII is 'out of key', and often followed by IV, as it could be heard as IV/IV.
♭VII tips its hat towards Rock and Blues, due to its flat 7, but it's rare to find a diminished chord in those genres.
As already stated, in key C, vii° is B (m3>) D F, whereas ♭VII is B♭ (M3>) D F.