Augmented and Diminished chords...

Holy cow - she's looking for a treatise! :) I'm gonna treat this in a couple parts, first the aug/dim and then the A6 chords.

Augmented and diminished chords are unique things; they are both completely symmetrical (at least if you're talking about the º7 chord, which is more common in usage than the º triad). That's important because any note in the chord can be considered the "root"....

C+ = C-E-G#. Each of those notes are two whole steps apart, so those three tones can be C+, E+ (E-G#-Bx) or Ab+ (Ab-C-E)

Cº7 = C-Eb-Gb-Bbb(A). Each of those is three half steps apart, so the same tones can be Cº7, Ebº7 (Eb-Gb-Bbb-Dbb, F#º (F#-A-C-Eb), or Aº (A-C-E-Gb)

The idea that they're symmetrical is important to their usage - because you can go in to one of these chords as if it's one name, and go out of it as if it's another. That's called a "pivot" chord.

As you noted, º is the natural vii chord in a major key. It's also the top three voices of a V7 chord... G-B-D-F. So it's common to use a º triad or a º7 as a substitution for V (if you're using a º7, you use a root a half step higher than the original - G#º7 has the same three upper voices as G7).

So...

Since you can have a progression like ii-V-I - or Dm -> G7 - > C, we can come out of G#/Abº7 into C major. And since that same º7 chord has the upper three voices of three other dominant 7th chords, D7, F7, and B7, we can go in to it as if it was one of them... so you could have a ii-V-I in the key of E - F#m -> B7 -> E.

Now we've got a handy way to switch keys from E to C smoothly:

F#m -> B7 -> Cº7 -> G7 -> C

or we could simply go F#m -> Gbº7 -> C

Augmented chords can be used the same way. A common use of a + chord is as a replacement for the first beat or so of I, as in: G7 -> C+ -> C. You end up with one voice moving from G-G#-G, which is coincidentally the same movement as you get using a º7 substitution for G7. Anyway, that can lead you to combining:

Db7 -> Ab+ -> Ab with G7 -> C+ -> C to get...

Db7 -> C+ -> C

Again we've used the symmetrical chord to effect a key change.

You can also use either chord to smooth out the voices. IV+ is a nice example of that - if you're moving V-IV-I, moving from V-IV+-IV-I gives you a chromatically descending voice.

As far as not diminishing the fifth in a major chord or augmenting the fifth in a minor chord, that's true to a point. But it's just because there are usually clearer labels for that set of tones: "agumenting" a Cm would give you C-Eb-G# = Ab-C-Eb... it's just an iversion of a major triad. And "diminishing" a C major would give you C-E-Gb, which is the same as F#-C-E. Compare that to F#º7 (F#-A-C-E), and it would usually be considered a simplification of a º7.

But whenever minor or augmented triads are part of a larger structure you can have an altered fifth. You could see C9b5 or Cm7+; they just aren't very common.

I'll put this up and start writing about the A6 and N6 chords :)

From my Open University Elements of Music couse books

[A Neapolitan sixth] is formed on the first inversion of the flat supertonic

So, in the key of C Major, the supertonic would be Dm and flatted gives you Db minor. The notes of Dbm are Db, Fb and Ab. First inversion will give you the 3rd in the bass - the Fb - and the chord in built up of Fb, Ab, Db. F anything to D anything is a 6th so it's called a 6th chord because it has a sixth present

Apparently, it was favoured by Italian composers so it's called Neapolitan (from Naples)

Why use it? It takes your melody into the tonic minor key - it says here. Saves going the <common chord> - V- I cadence to change key (or the I-I#dim-ii-V-I method popular in jazz)

A :-)

Ok, augmented sixths are a special family of chords in traditional harmony, and they're closely related to dominant 7th chords. When you look at an augmented sixth chord in a score you'll notice two accidentals... of the four notes* in the chord, only two will be 'native' to the key you're in, and that will be true of all the A6 variations.

*there aren't always four notes - I'll get into this below. But there are always four tones, since A6 chords appear only in four-part harmony.

All A6 chords have the same two chromatic alterations - the fourth note is raised, and the sixth note is lowered. That's the first source of confusion, because the name "augmented" generally means bigger... and the sixth is actually smaller in an "augmented" sixth chord. We call them augmented because of the way they're usually voiced.

A6 built on a C root is going to include both F# and Ab. But the chord was originally (and still most commonly) voiced in third inversion - which puts that Ab on the bottom, and the F# on the top. The distance between those two voices happens to be an augmented sixth interval, hence the name.

But they function as dominant chords, so understanding their structure is easiest in the context of a dominant 7th in root position. So let's start with C7: C-E-G-Bb. The difference between that an a basic augmented sixth chord is in spelling: if we write the b7 enharmonically, we get C-E-G-A#.

This set of tones is called the "German" augmented sixth, 1-3-5-#6.

The other A6 chords differ only in the fifth; if you leave the fifth out completely (1-3-#6) you have an "Italian" A6*; if you flat the fifth (1-3-b5-#6), you have a "French" A6. There's also an "English" A6 in some texts, but it's really just a re-spelling of the German as 1-3-x4-#6.

*told ya I'd get into it below.... in the Italian A6 you've got three tones, but four notes - one gets doubled to keep it in four parts.

Now you're probably thinking that's all well and good, but none of those chords have a sharp 4. And that's where the inversions come in.

A6 chords usually appear in exactly the order I've shown them. But they're really in third inversion... so that C note isn't the root - the E note is! And now we have to think about what key we're really in...

A6 chords are usually used as secondary dominants. That means in practice they'll be the V/V, or some chromatic variation like the bV/V or the #IV/V. Here we have an E root A6 chord, and E is the V of A... which is the V of D. So we're in the key of D. And that makes our C note the b6 in the key, and our G note the #4 (Fx). We could also be in the key of Db or Eb, but we'd get the same end result - two accidentals, falling on the 4 and 6 of the chord as written. Which doesn't mean the fourth and sixth of the chord, since it's inverted...

Sheesh. No wonder people get confused by this :)

On to the N6. That one's not really a sixth chord at all - it's also called that because of the inversion. It's really just a major triad built on the bII of the scale; if you're in C, N6 is a Db triad, Db-F-Ab. But it's voiced with the F in the bass, giving us a 6th up to the Db.

As far as guitar goes, both A6 and N6 chords are really coming from four-part harmonic analysis... and for the average guitarist they fall into the "ya really don't need this stuff" part of theory. But if you're writing string quartets, I'd bone up on 'em - Piston devotes chapter 26 to N6 chords, and chapter 27 to the A6 variations and their uses.

Thanks guys :) :) :)

I'm printing this off so I can go over it in more detail tomorrow, but a lot of sunk in on the first read-through. Noteboat, serious props for the lengthy and thorough response.

Could I ask what the Piston text is?

One of my goals is to start writing more for different instruments, so I've been going over the 4-part harmony stuff I did for A level (hence this question cropping up, lol) and this definitely seems like something I could benefit from knowing inside out.

It's "Harmony" by Walter Piston. When I was in college it was the standard text, and had been since the 1940s or so - although a lot of schools are now moving to other, more current, harmony books. Mine's a hardcover fourth edition that set me back about $50 at the time (school books are never cheap!)

It's now in a 5th edition, but I'm sure the older ones are available for a lot less in used bookstores. I think they also publish a workbook for the 5th edition... don't know what that's like, since they didn't have on for the fourth.

Wow, i wanna hang out with noteboat. I can think of maybe one other person I know who has that sort of theory knowlege, and that's my theory teacher at uni. This has been easily the most interesting, informative and exciting post I've read through since joining this forum. You are my new guitarnoise idol, well i never really had one, but very impressed by the crazy theory styles. We're doing basic Four part harmony at school now, got an exam in a couple of weeks, just doing up to diatonic progressions with inversions. Was really interesting to read ahead a little bit, and see some more advanced stuff. Theory is my favourite thing in the world, which some people would say is a little strange, like I'm sick or something, but whatever. Awesome stuff.

As you noted, º is the natural vii chord in a major key. It's also the top three voices of a V7 chord... G-B-D-F. So it's common to use a º triad or a º7 as a substitution for V (if you're using a º7, you use a root a half step higher than the original - G#º7 has the same three upper voices as G7).

So...

Since you can have a progression like ii-V-I - or Dm -> G7 - > C, we can come out of G#/Abº7 into C major. And since that same º7 chord has the upper three voices of three other dominant 7th chords, D7, F7, and B7, we can go in to it as if it was one of them... so you could have a ii-V-I in the key of E - F#m -> B7 -> E.

Now we've got a handy way to switch keys from E to C smoothly:

F#m -> B7 -> Cº7 -> G7 -> C

or we could simply go F#m -> Gbº7 -> C

this is great! does this mean that you can substitute a dominant chord for any the ii, IV, and vii chords of the major scale? so for the key C, you could use a D7, F7, and B7? or is this usually used for moving to and fro with different keys?

thanks for the time you put into this.

I' m afraid it's not quite that simple...

The diminished chord works in place of a dominant chord, because they both have tension. In any major scale, there are two notes - the 4th and 7th notes of the scale - that are a tritone (b5) apart. Those two notes want to resolve by moving, genarally by moving apart (to the 3rd and 8th/root) - that's what drives a V-I progression: in G7 -> C, the F and B notes in the G7 chord "want" to move to the C and E notes in C major.

Dominant chords have the tritone between their 3rd and 7th, and diminished triads have it between the root and 5th. So you can freely substitute them; you're trading tension for tension.

In a ii or IV chord you don't have a tension. So you'd be changing the character of the chord progression.

But that doesn't mean you won't find II7 or IV7 chords. The II7 is often used as a "secondary dominant"... which basically means it's the dominant chord of your dominant chord - in theory terms, it's the V/V (pronounced "five of five"). D7 -> G7 -> C sounds just fine, because the tritone in D7 (F#-C) wants to resolve to G-B, and that's the root/3rd of G7. Then the tritone of G7 (B-F) wants to move to the C-E of the I chord. You're basically stepping into another key for part of the chord change - in this case, you're in the key of C, but you're briefly going into the key of G.

IV7 chords "want" to resolve to the bVII. Since that's outside the key, they're not all that common generally... but they're often used in the blues (because blues is full of dissonance, it works)

Does that help?

ahh notebook that makes perfect sense, thanks again. i have read about the secondary dominant principle and i do very much like the sound it creates. and i think i might just buy your book notebook, so you don't have to keep answering all these strange questions i have.

Sorry for the late reply.

There's no mistake - but in re-reading it, I could have worded it better. Aug 6 chords have an altered 4 and 6. As I pointed out, there's plenty of confusion over what the root is, because it's usually presented in inversion; if it wasn't, there would be no point in calling them augmented sixth chords.

In a nutshell, here's the point I was trying to make: all augmented 6 chords are called augmented sixths because there's an A6 interval between the bass and another note - in the case of the German sixth I used to illustrate (C-E-G-A#), that's C-A#.

Augmented sixth chords are usually presented in inversion, so the root of that chord is E. Seen from the root, all augmented sixths, in any "nationality" have an altered 4 and 6, with 4 raised and 6 lowered:

French: C-E-A#; C=b6, A=#4
German: C-E-G-A#; C=b6, A=#4
Italian: C-E-F#-A#; C=b6, A=#4

Where I think I lost you guys was by not being clear that there are two different ways to look at it, from the root and from the bass. Earlier in the post I said the German sixth C-E-G-A# is enharmonic to the "English" sixth C-E-Fx-A#. So you can see this as an altered 4 and 6 from either direction:

C-E-G-A# = b6-R-5-#4 (from the root)
C-E-Fx-A# = B-3-x4-#6 (from the bass)

In any chord analysis, the only method that gives you consistent results is to work from the root. Unfortunately, when you're looking at A6 chords, there's no augmented sixth interval from the root... so because of confusion in naming them (which dates back centuries), they have to be looked at differently - from the bass (to explain the A6 interval/name), and from the root (to discover the function).