It’s safe to assume that most users of this site already know the music theory behind the exercises, but in case you’re aware of a few gaps in your understanding, or you’re new to this genre, or just want to know what’s behind the thinking here, then this page is for you.
I’ll explain some of the exercises below, but first I’ll take a moment to talk about “rock theory” (an oxymoron if there ever was one), and mandolin blues as I see it.

Rock Theory
Jazz theory talks a lot about the Circle of Fifths, because so many “standards” move around the circle, for example a common III–VI–II–V–I progression, which in C would be E–A–D–G–C, or some variation like Em7–A7–Dm7–G7–Cmaj7. Classical theory talks about the “V of the V” – in the key of C major, in a D–G–C progression, D is the “V of the V” because D is the fifth step of the scale of G major. Rock theory doesn’t really exist, but if it did, it would talk about the “IV of the IV” – rock likes to move around the Circle of Fifths as much as jazz does, but it likes to go the other direction: G–D–A–E is a very common progression. D–A–E is used all the time (know why that sounds so satisfying, so full of truth? It’s because of the little-acknowledged fact that the notes of the basic-position guitar chords (not counting the D’s low A, if played) are all harmonics of the bass’s notes). Hey Joe’s and Deep Purple’s Hush are extreme examples of the Circle of Fifths, using a IV of the IV of the IV of the IV of the I.
I think that pretty much covers Rock Theory – got any more? Please contact me.

Blues Theory (for mandolinists)
There’s a lot of How To Play The Blues On Mandolin on the Internet, but none of it (as far as I know) goes further than how to play Blues Mandolin as a solo instrument – if you play any of what they’re demonstrating at a real blues jam, with guitarists, you’ll sound pretty cheezy, or cute at best. There are two reasons for this: (1) guitarists are usually playing low triads, seventh chords or ninth chords, and the mandolin, playing its own version of these chords, is playing everything an octave higher, and not blending in with the guitars’ voicings; (2) they often teach blues as four bars of I, two bars of IV, two bars of I, two bars of V (or one of V and one of IV), and two bars of I. Well, yeah, kinda, but really, that’s only what the bassist and the low notes of the guitar chords are playing – the melody, the vocal line, the heart of the song usually plays against that, sticking (in the simplest form) to the minor pentatonic scale of the root – that’s why guitarists play ninth chords (especially on the IV and V), to make their upper notes blend with the vocal/melody line.
What does this mean for a mandolinist trying to fit in? Pretty often, it means that you’ll simply play the V minor of whatever chord the guitar is playing, eg your Em triad will sound great with the guitarist’s A9. Trouble is, that’s kinda weird to practice by yourself, trying to imagine that your F#m–Em–Bm progression is really everyone else’s B9–A9–E7, but trust me, if you want to play with the big boys, this is what you’ve gotta get down. And if you want to get fancy, you’ve got to learn your minor b5 chords – a Bm7b5 sounds great over a G. I don’t remember who said it, but it went something like “The biggest misconception about the Blues is that it’s easy” – that’s especially true for mandolinists.

The Theory Behind The Exercises

Major & Minor Triads
These, like most of the triads on this site, are in what’s called “open harmony”. A C major arpeggio would go C-E-G-C-E-G-C (however high you wish to go). “Closed harmony” is every note of that arpeggio in order – a C triad in closed harmony would be C-E-G, E-G-C or G-C-E, with intervals of thirds and a fourth. Open harmony is every other note of the arpeggio – C – G – E, E – C – G or G – E – C, with intervals of sixths and a fifth. Basic mandoliny-sounding stuff – the common G chord is in open harmony.
The third and fifth time around the keys, the bass has dropped down a third. What’s going on here? Your triad has become the upper three notes of a seventh chord – your G – E – C over an A bass notes gives the listener an Am7, made up of A-C-E-G. The major triads become the upper notes of a minor 7, and the minor triads become the upper notes of a major 7. Again, basic mandoliny-sounding stuff – it’s pretty common to not play the root note of a seventh or ninth chord, leaving that to someone with lower notes.

Harmonic Minor Pattern II
Natural minor, harmonic minor, melodic minor – what’s the difference, and why the names? In the key of C (the “people’s key”, as Garrison Keillor calls it), the common V-I progression, a G chord to a C chord, has the half-step B-to-C going on. We like this, we like the strength of the resolution. If we move down a third, to the relative key of A minor, and play a V–I (Em chord to an Am chord), if we want that same resolution we have to turn the Em into an E major, to get that G#-to-A half-step movement. So, for harmonic reasons, we change the natural A minor scale (A-B-C-D-E-F-G-A) into the A Harmonic Minor scale (A-B-C-D-E-F-G#-A). Not the prettiest-sounding scale, with that F-G# jump, so for melodic reasons we also sharp (sharpen?) the F, giving us the A Melodic Minor scale (A-B-C-D-E-F#-G#-A) which, if it wasn’t for that C, the makes-everything-sound-minor A-C minor third, would be an A major scale.

Sexatonic Pattern III
Okay, I invented the term “sexatonic” (I’ve since heard the term “hexatonic” used) to describe, depending on how you want to look at it, either a minor pentatonic scale with an extra note, or a minor-ish scale that is neither dorian nor minor, and it’s just too useful to be ignored. One nice feature is that every other note produces a chord, so if you have any kind of synced delay set up, it sounds pretty cool. It’s the skipping of the sixth step that makes it work in so many settings – a minor scale or a dorian scale will too often pull a song in a direction that it doesn’t want to commit to.

Mixolydian Major Triads
“Mixolydian” because there’s a flatted 7th step of an otherwise normal major scale – blues harmonica players use a key of D harmonica to play an A blues. It’s interesting that many Irish Trad players know their modes – the “ABC” system of notation used states the key + mode at the beginning, and then just writes the letter names, ignoring the sharps and flats.
Modes are always taught with a major scale as the “grid”, with the whatever mode starting on a different note (eg D dorian is a C major scale, starting on a D). I think that it’s important to be able to think in the other direction as well, for the four commonly-used modes, eg C major has no flats, C mixolydian has one, C dorian has two, and C minor has three – you’re moving around the Circle of Fifths, getting more minor-ish as you go.

Scales Zigzag 5
Not really theory, but: This scale exercise is all over the neck, but not in all positions – only in first position, and if you’re up the neck at all, you either start on a first finger or a fourth finger. Why not, as Dennis Sandole said, “Every fret, every finger”? Well, if you’re somewhere up the neck and you start a major scale on your second finger, the first string change you encounter involves a half-step – your fourth finger on the third step of the scale, your first finger on the next string up for the fourth step of the scale. This is a stretch of seven frets, and ya, maybe you could and should practise this position, but I find that if I start a scale-ish lick in this position, I’ll slide up (or down) a fret pretty soon because the next position up (or down) is so much more comfortable. So, for more efficient practising, the awkward positions have been ignored.
In this exercise you sometimes go above the twelfth fret – what’s the pattern dictating this? It’s to avoid “redundant” movement above the twelfth fret – no going past that just to repeat something that you’ll also play in a lower position. If it’s in a new, higher register, then the fifteenth fret is the limit.

more to follow…

Extras

Father Charles Goes Down And Ends Battle; Battle Ends And Down Goes Charles’s Father – it’s the order of sharps and flats in the key signatures. I’d always thought that this was as common as cows eating grass until I played violin under a conductor who’d never heard of it, so there it is.

Equal Temperament and the Circle of Fifths
Musicians use the Circle of Fifths, but I don’t imagine that anyone’s 100% comfortable with it – there’s something fishy about it. How to understand it? Well, firstly, picture the key signatures as a number line, with C being 0, G (with one sharp) being 1, D being 2 etc, and, going the other way, F (with one flat) being -1 etc. Theoretically, it goes infinitely in both directions, and it’s almost like keys with flats exist in a different dimension than keys with sharps. Secondly, there’s Equal Temperament to grasp – here’s the deal, with some math involved: a note vibrates at x times per second, the note an octave above it (the first harmonic) vibrates at 2x times per second, the note a twelfth (an octave and a fifth) above it (the second harmonic) vibrates at 3x times per second. The note an octave below the 3x times per second note vibrates at 1.5x times per second – this note is, of course, a fifth above the original note, so you end up with these two facts: (1) go up an octave, the frequency doubles, and (2) go up a fifth, the frequency goes up by 1.5. Okay, so, let’s say we start at the lowest C on the piano – it vibrates at 32.703 times per second, but for the purpose of this explanation (so you won’t have to get out your calculator to play along), we’re going to pretend that it vibrates at 10 times per second. The top C on the piano (a normal 88-key piano, that is) is seven octaves higher, so it vibrates at 1280 (10 x 2 x 2 x 2 x 2 x 2 x 2 x 2) times per second. Start back on the low C, and now go up by fifths: C G D A etc, multiplying by 1.5. You’ll eventually end up, conveniently, on the same top C, but something’s wrong – your math now shows that it should be vibrating at 1297.5 times per second (ya, you’ll need your calculator anyways). Why didn’t this work out? It’s because you went up by fifths, you travelled along the number line to the right, and you didn’t really land on a C – you landed on a B#. You actually went C G D A E B F# C# G# D# A# E# B#. So, aha, a B# is not the same as a C! Well, we want a B# to be the same as a C, we want an A# to be the same as a Bb, and we want an F# to be the same as a Gb. How can we make this happen? Easy – we tune the B# down to 1280, and we tune everything down by the same percentage. All the fifths are now a little flat, everything’s a little out of tune, but not so much that we can’t live with it. This is called Equal Temperament. And now, we can do something a little dodgy – we can clip our number line in two places, at the F# and the Gb, form a circle that has the F# and the Gb in the same spot, and voila, we have the Circle of Fifths. What’s fishy about it is that, at some point, like going from the F# to the Db, we aren’t going up a fifth at all, we’re going up a diminished sixth. That’s where we feel a little uncomfortable about the whole thing – it’s not actually a circle of fifths.
(The equal temperament explanation also reveals why guitarists who tune using harmonics can’t seem to be able to get their instruments in tune!)

If you start on C, and go up a major third, a minor third, a major third, a minor third, and keep going, interesting things happen, not the least of which is that you start hitting sharps a la Father Charles: C E G B D F# etc. This is useful for mandolinists to be aware of because these “out there” notes occur so naturally as you get higher that they’re sort of in a different class than the “altered” notes of a scale. If a Cmaj7 is what a guitarist is playing, you can put a first-position Gmaj7 (G D B F#, or any voicing that has that high-enough F#) on top of that without offending anyone – you can even play a Dmaj7 if it’s high enough up the neck (eg same first-position, seventh fret). Impress your friends!