Diatonic scales with CGDEbGC

Another documentation post, really.  In working with this open minor-add-9 tuning (C-G-D-Eb-G-C) I've been exploring the top three strings somewhat methodically, which is a bit amusing since the intervals of Eb-G-C are the same as the standard tuning (G-B-E, two whole steps up), and I've never really studied the standard tuning yet.  Which has led to an observation I'll get to shortly.

First was triads on these strings, done within my usual "box" of three voicings per triad, advancing each string one chord tone to achieve it.  So, in this tuning, at the nut is C/E, C/G is at the fourth and fifth frets, and C in root inversion is at the seventh and ninth frets...et cetera.  Shortly I was running through a standard exercise that has me play, up and down, backward and forward, the C major scale in triad forms, from nut to octave.  So:  C/E, C/G, C, then Dm/F, Dm/A, Dm, then Em/G, Em/B, Em, and so on.  Along with each of the main forms I also tried to note where the sus4 would go, and I'm pretty comfortable that this will shortly become a standard part of the internalized chord library.

Then, I tried adding bass notes to these basic forms, to try and figure out where the fingerings should go.  This is an interesting exercise, since there is often a choice of where a bass note can or should be located, and I'm not by any means done with it yet.  Inadvertently, the little working I've done so far has suggested a specific approach to diatonic seventh forms, wherein the chord is constructed of an appropriate bass note and an appropriate top triad.  This brings back the "tertian arithmetic" idea in a very practical way, and I'm happy to think of a Cmaj7 as an Em triad over the C3 note on my fifth string, or the C2 open sixth...or maybe the Em triad in up-neck inversions over the C4 note on the fourth string.  And that's just with root inversions;  I think I'll dig into that a little more when the library of top-three-string triads becomes more purely autonomic.

It also suggests an interesting idea with respect to tunings:  if I really can master the basic triad and sus forms on the standard tuning's intervals in the top three strings, I might be able to shift the bottom three strings around more at-will, to make available a family of tunings with a common set of top intervals.  So...with a real mastery of "GBE" chord construction, I could take advantage of fifths in the bass with CGDEbGC or CGDGBE, fourths with EADGBE or DGCEbGC (or similar), or a combination like DADFAD.

Which brings me to today.  I was able to work out the seven modes of the major scale on the top three strings, which of course is a great exercise to start to help you to "see" all the notes of the key everywhere on the neck.  I've still got a few fingering decisions to work out, or perhaps to learn them all so that they'll all be available depending on what else I'm doing at the time.  More work on up and down, backward and forward:  my fingers should tell me what is going to be most efficient.  What was encouraging was that I also started to see how I could connect a "lower octave" scale from the bottom three strings, to the "upper octave" scale on the top three strings.  The basic concept here might do well with that "family of tunings with differing bass strings" idea.

More on that as soon as I can.  With the ability to play all the triads and sevenths of a key, and the ability to play all seven modes of the scale, we're really getting somewhere important.  With that in hand, plus the ability to take advantage of the harmonic mayhem of a minor-second interval between fourth and third strings, I should become pretty dangerous in this wacky experiment.  :-)

Working with CGDEbGC

A bit of a documentation post here.  After some nice experiments with the DADFAD tuning over the summer, I somehow ran across the idea of CGDEbGC, and have been quite enjoying it.  This tuning has some interesting features:

• Top three strings are standard tuning intervals.
• Bottom three strings are the fifth intervals I know from Guitar Craft.
• The two three-string groups are offset by a minor second (D-Eb).
• Sixth and first strings are two octaves apart.
• Fifth and second strings are one octave apart.
• Open, the tuning is a Cm add9, which is nice.
• There is an entire G harmonic minor scale in harmonics available on the fifth, fourth, third and first strings;  if you omit the subdominant note (the first harmonic on the first string) this is extremely convenient to the fingers.

Obviously there are going to be some interesting options in chord construction, but I'm not fully there yet.  I think I might get around to it, though--some of the sounds are just gorgeous.  This last week I've turned my attention to the triads on the top three strings (remember, I'm actually new to standard tuning), and thought about how to incorporate a bass note intelligently.  In that regard, having the fourth string offset a whole tone above the first string (D to C), instead of below (as with D to E), it seems more comfortable to finger the three-string triad with middle, ring and pinky fingers and leave the index finger to find the bass note, rather than relegating that role to the pinky.

Melodically, I'm finding that I'm gravitating toward the "ignore the third string" philosophy, and thinking of the basic minor scale in terms of fifths intervals for the first octave and fourths intervals for the second.

The next step with chords, once I've really got the triads down, is to tackle sevenths.  I suspect that is where I'll find out whether this can be a general purpose tuning, or should be relegated to tunes specifically composed to take advantage of it.  Depending on how the fingerings work with that fourth string, I may look at solving the problem with a polychord approach, and I admit that forcing myself to use that concept as a core portion of my playing might not be a bad thing at all.

In addition to the ongoing tuning thoughts, I've been working a little bit more deliberately on a controlled three-string burst with the right hand, ascending (i-m-a).  For whatever reason, my fingers have always seemed to be pretty comfortable with such a cascade descending (a-m-i), but I've been impatient with the reverse and haven't given it the attention it deserves.  In trying to incorporate it into my triad work on the top three strings, my ear has something much more specific to focus on and it's suddenly started to improve. 

Yay!

Take Five: the reunion

I am really happy to be working with a mandolin student (I'll call him G.) again.  Not only does it give me a way to help with all the noise in my head, but it gives me an excuse to start looking for repertoire that will be useful in multiple contexts:  performing, technique, theory, and improvisation.

This is where Dave Brubeck's (actually Paul Desmond's) "Take Five" is an absolute masterpiece.  For starters, it's already a masterpiece;  probably the catchiest tune counted in five that most people will ever hear, it's a beautiful vamp with a brilliant chorus, and capped off by an astonishingly beautiful and compact alto sax solo.  But it's also a total treasure trove of examples and illustrations;  if you need to explain a concept in basic music theory,  this tune may well have it. 

Like many string players, I prefer it transposed up a half-step (to Em) for convenience, but a great way to test yourself once you're comfortable with the piece, is to go try it in the original E-flat minor (which is what's convenient for the alto sax).  Most of us find that using all closed shapes takes a real conscious effort, and it's worth testing yourself on.  (I'll consider it G.'s 'final exam' for the piece.  Poor guy.  :-)

So tonight, after a long time away from Take Five (which I first approached with Guitar Craft Standard Tuning guitar, CGDAEG), I sat down with the mandolin and hashed a bunch of things out.  First and foremost:  man, what a fantastic piece of music that is.  Improvising on it is simply intoxicating. 

I worked through the melody in the first two available octaves, and will eventually work into the third.  Here, I let the horn phrasing determine the right fingerings, which can be a little more convenient on the smaller instrument, and it will be good to be able to show G. an example of how you get to choose which fingerings you want to use.

G. is currently working through diatonic triads, and I will probably just use the "trust me" logic in trying to go through the vamp chords (Em and Bm7);  I can't decide whether it would be better to explain the Bm7 itself, or just tell him to play a D chord instead.  (The root-in-bass voicing of the Bm7 is easy;  it's the other inversions that may get confusing at speed.)  At any rate, there are plenty of Em and Bm7 chords available on the neck.

I'll have G. improvise over this simple two-chord vamp;  that alone could take a goodly amount of time and lots of questions.  Not only does he have to figure out where his Em notes are, but he can't fall back on a four-count as a crutch!  (On the other hand, if he can hold in 4, I just might jump in with a 5 and we'll get us a polyrhythm going.  :-)

The chorus, of course, will take a little time;  first we'll work the chords:

Triads first:
C - Am/C - Bm - Em/B - Am - D/A - G - G
C - Am/C - Bm - Em/B - Am - D/A - F#m - B

Then, the chords as written:
Cmaj7 - Am6 - Bm7 - Em7 - Am7 - D7 - Gmaj7 - Gmaj7
Cmaj7 - Am6 - Bm7 - Em7 - Am7 - D7 - F#m7 - B7
(on the mandolin, we do tweak voicings a bit to deal with "only" four strings)

and once he's got that, we can go over exactly what that beautiful descending movement of chords is actually doing.  On the melody side, there's a chromatic example in every measure which is great for listening, and again we'll try it in at least two octaves, ideally with different fingerings.

At some point we'll deconstruct the solo too, especially Desmond's delicious phrasing, and the various techniques that one can use to play that on the mandolin (yes, vibrato is possible on the little beastie, and worth it!). 

It will be interesting to see how he handles it--and I have to remember, myself, not to try and do too much too fast.  First, a score he can read and chords he can see.  Then, setting the vamp.  Then the melody.  Then improv.  Then the chorus, either melody or chords first (don't know which yet) and finally the solo and improv over the chorus.

All followed by E-flat minor, which all but removes his open strings as options.  (I'm a stinker, that way.)

Tertian arithmetic - the basics

This post lays out the example of looking at extended chords (ninths, elevenths, thirteenths) as the product of two simpler chords. The philosophical idea behind this is discussed here.

Okay, so think in the box of a seven-note scale and its diatonic, tertian chords. I'll stick with the Western major scale for simplicity--the example will be good old C major.

Our givens:

Diatonic triads are built by stacking two thirds on top of one another. The sequence and nature of the component thirds determines the type of triad: a diminished triad is two minor thirds (BDF), a major triad is a minor third on top of a major third (CEG), a minor triad is a major third on top of a minor third (DFA), and an augmented triad is two major thirds (unavailable in the Western major scale, this would be CEG#). In C major, then, the diatonic triads are C major (CEG), D minor (DFA), E minor (EGB), F major (FAC), G major (GBD), A minor (ACE), B diminished (BDF).

Diatonic sevenths take these triads and add another diatonic third onto each one. The combinations become much more complex, and C major only yields four of the possible variants. The diatonic seventh chords for C major, then, are Cmaj7 (CEGB), Dm7 (DFAC), Em7 (EGBD), Fmaj7 (FACE), G7 (GBDF), Am7 (ACEG), and Bm7b5 (BDFA), also known as the "half-diminished seventh".

Ninths add another diatonic third onto the seventh, making them a stacking of four thirds. Elevenths add a fifth third, and thirteenths are six thirds stacked together. (At this point, you are playing every note in the scale simultaneously, so there is no such thing as a fifteenth or beyond.) The possible permutations quickly get ridiculous here, and so does the harmonic density. After a few too many of these chords, even the most diehard jazz lover can appreciate the clarity and resolution of a triad or power chord.

Great, so where is all this going?

Consider C major: C-E-G. Now, consider E minor: E-G-B. What happens if you get your buddy to play a C major at the same time you play an E minor? Together, you're playing Cmaj7, C-E-G-B, and are overlapping on the E and the G. Huh.

You may also have heard that Em is a good "substitute chord" for C major. The logic goes that the bass player is probably covering the root note, so if you play an Em somewhere in the upper register, it should sound "right", and a little fuller. What you're doing is extending the basic chord by another diatonic third, making a more complex chord in the process - Cmaj7.

Now, what makes Em a suitable substitute chord for C? Here is the simple, master-key answer: it's a diatonic third up from C.

From here, everything starts to fall into place. If you understand diatonic triads in your scale, you can easily create sevenths by playing two triads a third apart. If you understand diatonic sevenths in your scale, you can easily create diatonic ninths by playing a "root" seventh chord and the diatonic seventh one-third above it. (Three tones will overlap.) That is, you can easily create Cmaj9 by playing Cmaj7 and having your buddy play Em7 over it. C-E-G-B plus E-G-B-D equals C-E-G-B-D, or Cmaj9. Work up the scale: Dm7 + Fmaj7 = Dm9, Em7 + G7 = Em7b9, etc.

Now, here's the real kicker: consider those overlapping tones. In the case of Cmaj9, the notes are C-E-G-B-D. If you start to break apart the ninth into constituent chords, do you notice that you could also play the G triad over the C triad, and come up with Cmaj9? C-E-G plus G-B-D equals C-E-G-B-D. The G triad is two thirds up from C, and now you're playing a ninth from two triads with one note still overlapping. Why not go a step further? Three thirds up from C is the B diminished triad, B-D-F. C major (C-E-G) plus B diminished (B-D-F) yields C-E-G-B-D-F, or Cmaj11. Imagine playing eleventh chords simply from knowing your triads, and knowing how to go up three diatonic thirds up from a root chord. Not bad!

Now, combine a seventh with a triad. Hold down your Cmaj7 and have your buddy play a D minor (D-F-A). Poof! Cmaj13 (C-E-G-B-D-F-A), no overlapping notes. You could do the same playing a Bm7b5 over a C major triad.

There will be practical limits to this, of course, but this is a powerful concept, and once you have mastered triads and are comfortable with sevenths, these extended chords are almost instantly available to you. Commit these ideas to immediate recall:

Look at complex chords as a series of thirds stacked together in the scale. That is, for Cmaj13, look at the chord as C-E-G-B-D-F-A. Do you suddenly see all the triads in there? Just break them up, assign them and play them!

To extend a chord by one "factor" (triad to seventh, seventh to ninth, ninth to eleventh, eleventh to thirteenth), you need to add a third. You can do that by creating a polychord from a root chord of appropriate complexity and a chord a third above of appropriate complexity. (Here, we start to get limited by terminology. I'm working on this!)

You can extend a chord by more than one "factor" at a time (thereby keeping constituent chords simpler and fewer notes overlapping) by going up two or three thirds for your "upper" component of the polychord, and/or by extending the upper chord by one or more "factors".

In general, you can cover any diatonic ninth, eleventh or thirteenth chord with one seventh and one triad.

Hopefully I will get clearer with this each time I explain it. For now, here are some practical examples that illustrate the explanation. Try these things!

Cmaj9 (CEGBD) = C (CEG) + Em7 (EGBD) (also w/ Cmaj7)
Cmaj9 (CEGBD) = C (CEG) + G (GBD) (also w/ Cmaj7)

Cmaj11 (CEGBDF) = C (CEG) + G7 (GBDF) (also w/ Cmaj7)
Cmaj11 (CEGBDF) = C (CEG) + Bdim (BDF) (also w/ Cmaj7)

Cmaj13 (CEGBDFA) = Cmaj7 (CEGB) + Bm7b5 (BDFA)
Cmaj13 (CEGBDFA) = Cmaj7 (CEGB) + Dm (DFA)

Note that in each of those pairs of examples, the first suggestion goes up the minimum number of thirds and uses a seventh form; the second suggestion goes up an additional third from the root, but simplifies the upper chord form.

There is a little arithmetic involved here, sure, but it is very logical, and I imagine that mastering it would make one a very powerful improviser. I can envision a situation in which I'm playing mandolin and reading a chart which vamps on a dominant chord for a few bars. The band has the G7 covered, and I want to add tension on top of the mix. For the first bar I play G7, then I play Dm (making a G9), then F (making a G11), then cap it off with Am (G13)--just by knowing to go up in thirds.

I know I'm a geek, but that's cool.

Tertian arithmetic - the idea

Okay, this has been stewing around in my head for a little while now, and I think there's something significant in it, but I have not yet found the pow! way to explain it. So, please forgive me if this discussion still seems a bit formative--it is.

The topic is looking at extended complex chords (ninths, elevenths, thirteenths) as polychords. The intent here is to come up with a deliberate device to understand these complex chords better, by viewing them as the products of multiple simpler chords (triads, sevenths) that one may already have in the hands. What I like is that there is a very simple logic at work that makes this daunting subject rather easy to grasp--at least in a practical way.

And that is my goal. Many of us find ourselves in a group context in which we are one voice among several, and so this concept of reducing a complex chord into two simpler chords, each to be played by a separate person, has a great deal of appeal. And for the improvising musician, the ability to add harmonic complexity spontaneously (without having to know where, say, a G13 or an Am11 is) is a real bonus.

You've probably heard the phrase "chords are built in thirds", and the term "tertian chords". It turns out that there's a lot more to this than you might think. Most of our chords are indeed built from stacked thirds, but the stacking can be applied further to whole chords a diatonic third apart. Somewhere in there is a sort of "tertian arithmetic" that can be described, that just unlocks the whole mess like a master key. It is the description of this arithmetic that I continue to struggle with; I can see it by permutation, but the teacher in me wants to be able to distill out the best way to 'splain it. (I'll get there.)

So, here we go. The next post will lay out the examples.