triadic inversion another way

Today's discussion is about inverting triads. What is generally meant when inverting triads is the following, done with C major:

That is to say, what we're really dealing with here is a re-ordering of the notes.

The concept of inversion as applied to melodic lines, however, has more to do with the actual meaning of inverting, i.e. turning upside down/placing in an opposite order (like a mirror). For example the following little line

will invert (diatonically, that is not adding any sharps or flats) to:

Now, what if we apply the same idea to chords? Something interesting will happen. We'll invert C major three times, first with C as the axis of symmetry, or mirror line:

The chord we end up with is an F major chord. Now let's use G as the axis:

Now we've produced a G major triad. So the interesting point here is that simply by inverting a triad (let's say melodically) we end up with the IV (subdominant) and V (dominant) chords. Not only that, these inversions have given us all the notes of the key of C major.

Oh, and the last way to invert the triad, with E as axis, produces...

...yeah, we get the same chord right back.

These are, by the way, diatonic inversions. Next time we'll examine what happens when we invert our intervals strictly.

sus4 chords

Superimposing triads over a given harmonic structure is a well-known and -documented phenomenon. I personally love hearing a D major triad over an E minor harmony. And by triads usually meant are the famed major, minor, augmented and diminished. But we shouldn't overlook sus4 chords (or sus2 chords: we'll talk about that, too) as possibilities. As a refresher: a Csus4 chord is comprised of the notes c, f and g, and generalized a sus4 chord is made up of a root, P4 and P5. To a certain extent they can have a "cold" sound as there is no third, major or minor, and are found natively in quartal/quintal harmony.

So as far as use goes there's the obvious: wherever you want! Also here are some conventional usages:

Root of sus4 chord matches root of harmonic chord (e.g. Absus4 over Abmaj7; Esus4 over Emin).

Sus4 chords come from the harmony of a scale implied by the harmonic chord. For example take Dmi7. In a certain context this could be a dorian chord, meaning that we're dealing with a C major scale. In the case of major scales sus4 chords can be built on the 1, 2, 3, 5 and 6 scale degress (yup, you guessed it: a major pentatonic scale!). Concretely: over Dmi7 we could use Csus4, Dsus4, Esus4, Gsus4 and Asus4. Over melodic minor there are less: take sus4 chords built on the 1, 2 and 5 scale degrees. Basically we just have to check the scale tones against those of the sus4 chords and we'll be good.

OK, mention was made of sus2 chords: whassup with them? Let's examine the following 2 chords: Asus4 and Dsus2:
Asus4: a, d, e  
Dsus2: d, e, a
Yeah, the same notes. So we can generalize the situation as: a sus4 chord is the same collection of tones as a sus2 a perfect 4th higher.

As far as that goes, let's look at these notes again, but now starting with e as the root: e, a, d. This can be seen as an E7sus4 without the 5th. So a sus4 chord can be used as a 7sus4 the root of which is a perfect 5th higher.

Hopefully these will add something to your palette...

the enigmatic scale and even some more extended triads

I made a point in the posts about extended triads of keeping only to diminished, perfect and augmented fifths because those were the only ones I could imagine happening in scales. Not so! I recently looked at the enigmatic scale and found that we need to be more inclusive still. Here is the scale along with its triads:


As you can see both the II and V chords go beyond the parameters which I naively set. Since in this scale there are three minor seconds/half steps in a row (A#-B-C-Db) we end up with doubly diminished and doubly augmented fifths. If we consider that each fifth can have four types of 3rd (diminished, minor, major and augmented) then we will end up with 20 (count them twenty) triad types! (As far as symbols go I just simply stacked either plus signs or minus signs to indicate doubly diminished.)

Once again: give the triads a go...the whole point is that hopefully they aid you in expanding your vocabulary.

By the way, check out the wikipedia article on this scale here. You can discover its history (which includes Verdi) and that it actually has a different descending form.

See also extended triads iii, extended triads ii, extended triads i, basic triad types.

extended triad types iii

In this post we're going to look at extended triads in the way in which some might believe we should have done at the beginning: analytically or simply mathematically. The reason we didn't begin this way is because I wanted to demonstrate that there are real (and not just conceptual/theoretical) reasons why one needs the idea of extended triads, viz. the fact that they are produced naturally in certain scales.

So let's consider triads this way: chords consisting of a root, 3rd and 5th (as opposed to a harmonic structure that has just any 3 tones). As for 3rds we will allow diminished, minor, major and augmented; and for 5ths we'll allow diminished, perfect and augmented. The reason for "will allow" is that these intervals are all found naturally occurring in scales (such as the whole-half diminished and ionian flat-2, etc). We aren't going to examine triads which are comprised of a quadruply diminished 3rd and a quintuply augmented 5th (which triad would produce a third lower than the root and the like -- this could be interesting in a modern physics/mathematical kind of way but it's not extremely useful to us at present.

So based on our options of intervals we can simplify the triad families based on the 5th: there are only three kinds (diminished, perfect and augmented). Each family will have 4 types of third (diminished, minor, major and augmented). You won't have to resort to your calculator to figure, then, that there are, all in all, 12 triads. Here's a 1000 words:


The 3rds in blue denote that that chord is a basic triad type (diminished, minor, major or augmented) -- for example, the min 3rd in the diminished 5th family is blue because that chord is your garden variety diminished triad. And another 1000 words follow here with actual instantiations of all of the chord types:



Finally some issues of nomenclature. A M3P5 chord is simply called a major chord, so all chords containing a major or minor 3rd could be likewise appelled (a major diminished chord for M3dim5). Also since perfect 5ths are somewhat privileged we could drop that part of the name when it occurs: a diminished 3rd chord, an augmented 3rd chord.

Lastly -- and as promised -- let's look at the °3dim chord again. It has another name and is actually an extremely common chord in common practice "classical" music. Here it is in first inversion, showing its outer voice resolution:



Yes, that's right! An It+6 chord! For our purposes, then, the It+6 chord in root position is simply an It°3 chord. More on augmented 6th chords to come.

Play through these chords: some of these triads will undoubtedly be (or become) sonically interesting to you...

see also extended triads i, extended triads ii, basic triads

extended triad types ii

In this post we're going to examine the whole-half diminished scale. It is, like the whole tone scale, a symmetric arrangement of tones, and its name is the formula of the arrangement (i.e. a whole tone followed by a half tone followed by a whole tone...). This scale is different from many, many others in that it has more than 7 tones, in fact it is octatonic (8 toned). Here it is along with its triads:



Since it's composed of more than 7 tones some problems of nomenclature occur. For instance, at some point a scale tone (degree) has to be duplicated. Above we've made it F and F#, though it could've been D and D# (which would've produced an E#, then F#), etc. A massive ramification of this is that there is not a unique, one-to-one correspondence between scale degrees and triads -- some triads will manifest in 2 different forms. Above it is II, IV and VII (because they each contain F, which scale degree also exists as F#).

In this scale there are two triads which are "extended": the I and III. But these extended triads are a bit different from the ones which occurred in the previous post, and that's because they are enharmonically equivalent to basic triads. Put another way, the first two extended triads we encountered have no sonic equivalents, whereas these two that hail from the whole-half diminished scale do. Here they are:



The I chord sounds like a 1st inversion Ab major triad; the III chord sounds like a 2nd inversion G# minor one. But even though this is the case it's still important to be able to analyze these triads as extended types, if only for consistency's sake (most likely from a performer's point of view). But there is precedent for this sort of thing. Consider the Fr+6 chord: it is enharmonically equivalent to a dominant 7th chord, but its function is quite different in common practice harmony (that is its resolution is different from that of a dominant 7th chord).

So, back to the chords at hand: the I chord can be called a minor augmented (min Aug) and the III chord a diminished 3rd augmented (°3 Aug).

extended triad types

Over the next 3 posts we're going to be discussing extended triad types, i.e. triads in addition to the 4 basic types of major, minor, augmented and diminished (which topic was covered in this post). Perhaps unexpectedly this investigation will lead us into taking up the topic of augmented 6th chords.

If we only concern ourselves with the major and minor scales (both melodic and harmonic), and even if we throw in the harmonic major as well, we'll never encounter any triads beyond the 4 basic ones already mentioned. But this isn't always the case. Take for example the Double Harmonic scale and the triads which form from the scale degrees:


There are major triads (I and II), minor triads (III and IV) and augmented triads (VI) here, but look at the V and VII chords. They don't fit the pattern of the 4 basic types. The issue is that there isn't any ready-made term for either of these, so let's simply name them according to their intervals (which are some form of 3rd and 5th). The V chord is composed of a major 3rd and a diminished 5th, so we could name it a major diminished triad. Likewise the VII chord reveals a structure of a diminished 3rd and a diminished 5th: let's nominate this one a diminished 3rd diminished. And in abbreviated format: maj dim (V) and °3 dim (VII). Here's how they look built on C:



Why are there these different triad types lurking in this scale? Because scale degress 7, 1 and 2 are 2 consecutive minor 2nd intervals, which adds up to a diminished 3rd (and not a full minor 3rd found in the major and minor scales). This accounts for the double flat in the °3 dim triad built on C.

So now armed with this new knowledge you can analyze all of the following scales and you'll find that they contain the 4 basic triad types plus these 2 new "extended" triads: Neapolitan Major, Neapolitan Minor, Double Harmonic (as above), Ionian flat-2 and the Whole Tone Leading. Next time we're going to delve into the whole-half diminished scale and find even more triad types...

what chord is it?

Let's assume that you've played some sort of harmonic structure and you want to know (for some reason or another) what's the name of this thing I just played? We've discussed before that context has a lot to do with a chord's naming, but for now let's say that we're just dealing with one chord that sounds really cool and we need some kind of name for it. Here's a method which can get you in the game:

1. Write out all the chord tones and remove any duplicated tones.

2. See if there are any triads present (might have to look enharmonically). If no triads
go to step 4.
(a) If there's just one triad then this is likely your chord: go to step 3.
(b) If there is more than one triad pick the one that makes sense to you and go to step 3.

3. If there are any remaining tones they will relate in one of three ways, as:
(a) extensions
(b) additions
(c) suspensions

4. If no triads:
(a) Is one implied (e.g. a major 3rd could imply a major triad)
(b) Is it a power chord?
(c) Is it a stacked interval (stacked 5ths)
(d) Is it a tone cluster?
(e) Does the harmonic structure correspond to/imply any mode?

Needless to say for the above to work we have to have some sort of knowledge of basic triads and extensions, etc. Let's take a few examples and see what happens.

Example 1: f, g, a, c#.
Are there triads present? Yes: f, a, c# is an augmented triad.
Any remaining tones? Yes: g. This relates to f as a 2nd or 9th. Since there's no 7th present (which would be some sort of e) let's call this an F aug add 9 (or F+5 add 9).

Also example 1 can be viewed like this: f might actually be e# enharmonically spelled. That means we would have an A augmented triad: a, c#, e# (f). The remaining g is simply the 7th, so the chord could also be named: A7+5. (This kind of enharmonic spelling is quite common in music, classical or otherwise as music is really a guide for performers and not analyzers.)

Example 2: f#, g, b. Any triads? No. Any triad implied? Yes: g and b can easily give the sense of a G major triad (the perfect 5th is not needed for the ear to hear the "complete" chord). Then f# is simply the 7th which gives us a Gmaj7.

Example 3: bb, a, c, b. Again, no triads, buuut...a and c are enough for an A minor triad. The remaining bs are 9ths: we could call this structure A min add 9 add b9. Clearly, though, this is a form of a chord cluster, which will never yield very willingly to a nomenclature born of tonal music.

Also these 2 rules will help from time to time:
(1) If you ever have both a major 3rd and a minor 3rd treat the major 3rd as the actual 3rd and the minor 3rd as a(n enharmonically epelled) sharp 9th. E.g. the tones e, g#, b, d, g can be viewed as an E7#9 (g = f double sharp).
(2) Similarly in cases where you have a perfect 5th and a diminished 5th the perfect 5th is the real fifth and the diminished 5th can be seen as a sharp 11: c, e, gb, g, b = C maj7 #11 (gb = f#).

If the above is of only the slightest help then it will have served its purpose: the sound of the chord (and its emotive evocations) is the most important thing; the importance of naming it lies somewhere between a distant second and not completely a worthless endeavor.

See also: triads, seventh chords, slash chords, sixth chords.

creepy chords addendum

I was reading through Ponce's fabulous Guitar Sonata no. 3 last night and realized that in measures 39 and 40 (of the first movement) there is the same chord progression as mentioned in the post on creepy chords. Ponce is in Bb, but the chords qualities are the same: Major 7th +5 and a Minor(Major) 7th:



In this context I wouldn't say that the chords are "creepy", though they are tense. More creepy -- or at least menacing -- are the chords immediately preceding the 2 above (actually starting in measure 35: the structure is repeated):


Here we're really only dealing with one chord per measure: Ponce simply "flips" the outer voices (the 7th and root exchange places) giving him a very nice stepwise-ascending bass line.

And lest we think that these are just contemporary harmonic structures, or something born of jazz, keep in mind that this piece was written in 1927 -- and it's probably a safe bet that Ponce didn't invent this all on his own, so one might be able to trace it back at least to Debussy (maybe even Liszt???)...worthy of investigation.

intervals on the guitar ii

So now we come to the part of tuning the guitar that makes life interesting...the B string. In our last post we conquered the first four strings of the guitar (which are the same as the bass except that the bass being a bass has its strings an octave lower). Check out this diagram: I've made the B string red here to call attention to it:



Under each string is listed the number of frets that that string is distant from its neighbor to the left. If you wanna check this for accuracy consider the following: the low E and high E strings are 2 octaves apart (E2 and E4 respectively). Now look at the numbers: if we simply add up them up we get 5 + 5 + 5 + 4 + 5 = 24 frets. What's the interval corresponding to 24 frets? It's 2 x 12, which is the same as 2 octaves.

So let's take an example: what is the following interval?

We've traveled down the neck 2 frets and laterally 1: but this lateral move has brought us onto the B string. Checking our chart up above we see that that means we've moved the equivalent of 4 frets down the neck. Consequently our interval is (2 + 4 = 6 frets which we know from the last post's first chart is) a tritone. You'll see this show up in A-form dominant chords.

Here's one more:


In this case we've moved laterally 2 strings, one of which is the B string, and backwards 1 fret. The B string is 4 frets, the E is 5 and our "backwards" move is -1, and all of that adds up to 8 frets, a minor 6th.

That really wraps up our discussion. One loose end, though: we've always considered our intervals from lowest to highest, but this may not be the order in which they occur melodically. Just keep in mind that from C up to G is the exact same as from G down to C (a perfect 5th). You can always measure from the bottom up even if you want to know the interval starting from the higher note.

intervals on the guitar/bass

The last post here dealt with seeing the intervals on the piano. This time we're going to look at the intervals on the guitar/electric bass.

If we stick with a single string it's a very straightforward affair: every fret up or down is a semi-tone, aka a minor 2nd. Here's a chart that shows all of the intervals up to (and including) the octave. The left side of the vertical line -- which represents the string, any string -- is how many frets one is measuring from any starting point; to the right side is listed the corresponding interval.

In this chart m = minor, M = major, P = perfect and TT = tritone (augmented 4th/diminished 5th).

So let's take an example: say we're interested in knowing what the interval is when we play a note on the 4th fret and (on the same string) the 11th. Just subtract the lower number from the higher -- which gives us 7 -- and consult that number on the left hand of the chart. Then look at the right hand side for the interval, which in our case is a P5.

Now, an interval like the P5 on the guitar is usually much easier to play using 2 strings. The following will apply completely to the bass and almost completely to the guitar. "Almost" because one of the strings (the B string) is not tuned like the others. For now let's just stick with the first 4 strings on the guitar (E, A, D, G).

The strings are tuned in Perfect 4ths. Consulting our chart that means that if we move from one string to the next (from left to right) we're moving 5 frets. Let's call this movement lateral movement. If we moved laterally 2 strings away we would have moved 10 frets. Actually we all know this because this is how we first started tuning the guitar/bass.

So take the following example:


What is this interval? We've gone up 2 frets and over laterally one string. That lateral move is equivalent to 5 frets, so we've actually gone up in pitch 2 + 5 = 7 frets, which according to our chart way up above is a P5.

Here's another example:


Again, this isn't hard to determine. We've traveled "up" 1 fret and laterally 2 strings, each string being equal to 5 frets. So our total distance is 1 + 5 + 5 = 11. And that corresponds to a M7.

Now let's throw a little wrench in the gears. Consider this interval:


Here we actually have to employ the concept of negative numbers, which really means that we subtract instead of add. Why? Because relative to our starting position we're going in the opposite direction. Here we've moved laterally one string and down the neck (up in the diagram) 2 frets. That is numerically 5 frets + -2 (or just 5 - 2) = 3 frets. Our chart tells us that 3 frets is a minor 3rd.

Let this sink in: next time we'll tackle that B string...

intervals

The keyboard offers a nice way of visualizing intervals (and many other relationships) which is helpful if you're ever having any trouble trying to remember and internalize what they are.

In the following I've taken the basic keyboard, rotated it 90° counter-clockwise (thanks to Photoshop) so that now "up" in pitch is now "up" vertically. What we're going to do is look at all of the intervals of a C major scale as they relate to C. One cool feature of this is that if we look at the intervals ascending from C they are all either perfect or major. If we measure descending from C all of the intervals are either perfect or minor. On the keyboard I've colored blue the reference C. Since we're measuring always from C I've just indicated to the right of the note name the interval. Go ahead, click on the image...it'll enlarge.


So let's see how the keyboard can make the intervals easy to "see" by examining the minor and major 2nds. The minor 2nd is the least amount of distance you can travel on the keyboard without remaining on the same note: there are no keys in between the 2 in question (here in our example C and B). In the major 2nd there is one key in between the notes (C and D). If you know any Latin you'll recall that minor and major are comparative adjectives: they mean "smaller" and "bigger" respectively. And we can easily see why the major 2nd is the "bigger" interval: it takes up more space in terms of keys. On a guitar it will take up more frets. And acoustically the note of the major 2nd will always be a bigger number (in Hertz -- i.e. it will vibrate more times a second) than a minor 2nd related to the same reference note.

One last thing: If you examine and really absorb all of the above intervals you'll notice that almost all of the possible number of keys are covered. E.g. from C up to B is a major 7th, which is 11 keys away from C; from C down to D is a minor 7th, 10 keys. From C up to E is a major 3rd (4 keys distant) and from C down to A is a minor 3rd (3 keys distant). On our chart there is no interval that corresponds to being 6 keys away from C. This is actually a famous interval, and will complete all of the intervals within an octave, and it has several names. If you are thinking about C to F# it is called an Augmented 4th (sometimes +4 or #4) ; C up to Gb is called a Diminished 5th (sometimes b5). It is also known as a tritone because it is made up of 3 (tri) whole steps (tones). Just using the white keys of the piano this interval is found in F - B (because six keys is exactly half of the octave's 12 it is symmetric: from F up or down you'll land on a B if you travel six keys).

put a lyd on it (flat-7, that is...)

The Lydian flat-7 (aka the "Overtone" scale) is the 4th mode of the melodic minor scale. One mood that it is able to evoke (for me, at least) is one of "mechanization". Here are 3 examples, and they're all cartoon related.

The first one is called "Powerhouse" by Raymond Scott. No, it wasn't written for the cartoons, but it got used by Warner Bros. in some Bugs Bunny things, and that's where I heard it as a kid and forever have those 2 things linked in my brain. This is just the first theme (clicking on the images will enlarge them a bit):



The second one is the theme song from the Jetsons.



(The first lick above is instrumental and the theme proper -- "Meet George Jet-son" -- starts on the downbeat of the 2nd measure.)

And the third is from the Simpsons.


Of course there's more going on here than just a certain mode: the rhythm of the melody and the underlying accompaniment have a lot do with projecting that mechanized, futuristic feel. On the other hand, all things being equal (like the rhythm), this does seem to be a good mode to use for those occasions.

By the way, in improvisation one can see a rationale for the Lydian b7 (even if it weren't a mode that existed all on its own, anyway) by looking at the regular old Lydian. That is, the Lydian is used somewhat in place of the Ionian to get a raised 4th; similarly in the Mixolydian one might also desire a raised 4th...et voila! the Lydian flat-7.

Oh, yes, you can easily find all of the above tunes on youtube, etc, if they aren't a permanent part of your memory yet.

the neapolitan sixth chord

The Neapolitan Sixth chord (usually N⁶) is a major triad built on the lowered 2nd degree of a major or minor scale/key. The sixth, as you'll recall from this post, meaning that the triad is in first inversion. It still functions as a II chord: that is it is usually followed by the dominant (or I⁶₄), and like any II chord it can occur in either major or minor settings.

Here's an example from Coste's Quadrille No. 2, op. 3 -- this one's in E minor, meaning that the Neapolitan chord is an F major, and since it's in first inversion it has an A as its lowest note. (click on the image to make it bigger):


And here's a nice extended usage from the end of Tárrega's Maria (Gavotte).


The key here is A minor, so the Neapolitan sixth is a B-flat major triad (with D in the bass).

Here's another example which uses the Neapolitan as a pivot chord in modulating. Because the Neapolitan is a major triad it is found in other keys diatonically. The following is from Carcassi's Etude no. 9 op. 60. It's in A minor and goes to the N⁶ chord, but instead of going on to cadence in A minor it pivots here to D minor (B-flat is the sixth degree of D minor).


(N.B. In the above Roman numeral analysis both of the VII chords are diminished and probably should have little circles next to them, too, to help call attention to that fact.)

So these are some uses of the Neapolitan: they are, in fact, widespread, and even though the above examples don't illustrate it they are used also in major keys. Also they aren't always used in the first inversion: Segovia in his Remembranza (Etude II) uses the Neapolitan in 2nd inversion (i.e. N⁶₄). Molino in his first Guitar Sonata (2nd movement) uses the Neapolitan in root position (the N/V...an F major chord going to E major, overall in the key of A major).

That appellation of Neapolitan has no significance, btw...just like the augmented sixth chords which are called Italian, German and French. More on those in a later post.

some creepy chords

I've always associated the minor-major 7th chord with jazz tunes, South American (mainly Brazilian) music and cliff-hanger moments in James Bond movies and 70s TV shows like Charlie's Angels. But I just watched The Machinist (2004) a Spanish made film with Christian Bale and Jennifer Jason Leigh and it turns out that they can be used to convey creepiness, too (which I guess, after all, is an extension of tension). The moment I'm talking about here occurs at ca. 40min 37sec.


Note also that the 2nd "creepy" chord (1st chord in the 2nd measure) is an augmented triad with a major 7th. (You could also look at this as a C/Ab, which could easily come from the Harmonic Major mode.) From the first chord to this one the only changing part is the middle two voices: a major third that rises a half step. The last chord in that measure is also an augmented triad: really just the "upper" part of the Ab min/Maj7th.

Other elements which add to the creepiness (which seem to be universal): a slow tempo -- somewhere around a quarter note equaling something in the 50s -- and a lot of reverb.

Music by Roque Baños.

the many flavors of 7th chords

Seventh chords are triads that have had a third added to their uppermost member, the 5th (a triad contains some form of 1, 3 and 5). The third will 'skip' a note if you're thinking alphabetically, meaning that it skips a scale degree -- there the 6th -- and so we end up with a tone that is a 7th above the root of the original triad.

OK, my apologies for such a convoluted definition. Since a picture is worth a thousand words here's a C major seventh (the gray notes are not part of the chord):


Looking at this it's clear that a seventh chord can also be seen as a triad with the interval of a 7th added to the root of that triad. Regarding this approach the first thing to keep in mind is that there are two main intervals of a 7th: major and minor. (There's actually another one that comes into play and we'll talk about it in a minute.) So if we recall that there are 4 basic triads we should be able to produce (4 triads x 2 types of sevenths =) 8 types of 7th chords.

So the major triad gives us a major 7th and a dominant 7th.

The minor triad: a minor/major 7th and a minor 7th.

The augmented: a major 7th +5 and a dominant 7th +5

The diminished...well, let's talk about this one for a second. Here we really need to look at the first way (of stacking thirds) of constructing sevenths to see how these come about. We could slap a major 7th interval onto the root of a diminished triad but in practice this doesn't happen: it's enharmonically equivalent to a B major triad with a b9. But the real reason that it doesn't "happen" is that seventh chords developed historically from voice leading and from functional harmonic concerns. The diminished triad is a chord that came about as a VII chord, and in neither the major nor the minor modes will we get a VII chord that is diminished and has a major seventh.

So what chords do we get? If we add a minor seventh to the triad we get a (classically appelled) half-diminished chord. On modern charts you're most likely to see this called a minor 7 b5.

Then there is a fully diminished seventh chord: it results from stacking minor thirds up from the root. In the key of C we could start on B and get:
b, d, f, a-flat.
And if we added another minor 3rd we'd be back to b. (If we examine the sort of 7th here we don't have a major or a minor variety: in this case it's a diminished 7th.)

This chord is unique because any of it's tones could be the root (and because of this in 19th century music it was used a pivot chord in modulations). Mo' on dat in a later post...

There is one more seventh that is not a basic triad with an added third. It kind of results from whole tone harmony, or it can be seen as an analogue of the dominant 7th +5:
the dominant 7th b5:

You won't really see a C-E-Gb chord floating around out there all on it's own, but this seventh chord is not uncommon at all (and not just in jazz: it comes up in Granados' Valses Poeticos if memory serves).

Play the sevenths, use them, they're beautiful...

slash chords

We kind of broached the subject of slash chords on the post concerning 6th chords. In that case we talked about one specific use of slash chords, viz. representing inversion, or that the chord in question's root is not the lowest, 'bass' note. That doesn't cover completely what slash chords do, so let's now generalize what a slash chord is:

A slash chord is a way of representing a harmonic structure that has a bass note which is other than the root of the chord indicated. A slash chord is written in the form X/Y (pronounced "X over Y") where X is some chord (usually a triad) and Y is to be understood as a single note: the bass note.

There are 2 broad categories of slash chords:

1. The bass note is a member of the chord in question (though not the root).
2. The bass note is not a note found in the chord in question.

Examples of (1) above: C/E is a C major triad with E as its lowest member (in classical terminology this is C major in first inversion). G/D represents G with its 5th as the lowest member ("second inversion"), and so on.

Examples of (2): A/Bb (Bb is not found in the A major triad), D/C, F#/F, E/C and so on...

Furthermore, in this category of slash chords there is at least one big subdivision. Take D/C: it can be seen as an inversion of a seventh chord (in this case D7 but with the 7th in the bass). But D/C may just indicate a C lydian situation, so let's note that context has everything to do with slash chords.

For instance, slash chords are a great way of indicating bass lines. Here's a common one:

C | G/B | Amin |

The bass here is a stepwise line descending (C - B - A).

Also slash chords are an easy way to indicate pedals. Take "Someday My Prince Will Come" four measures from the end (in the Real Book):

Bb/F | Cmin7/F F7 |

Obviously these 2 measures are an F pedal.

a modal question

So here's a question about modes not often asked as far as I can tell. As a bit of prologue let's assume that we all know a bit about modes: their structures and the sorts of chords (triads/sevenths) that they produce. Armed with such knowledge we could easily address the following question: what is the mode of the following progression?

G7 | Fmaj7/G | G7 | Fmaj7/G | (repeated)

This is fairly unambiguous: it's G mixolydian. An analytic play-by-play might go like this (here the paths are many, but this one). The notes of the chords in question are:

G7: g, b, d, f
Fmaj7/G: f, a, c, e (with a g in the bass)

Now we put these notes in order to see if they form a scale. We could start anywhere, but since we notice that G is prominent (it's the bass note in both chords) let's put it first. We then get:

g, a, b, c, d, e, f

We now have a 7-note scale. Since there are no sharps or flats we can easily conclude that it must be some kind of mode of C major. G is the 5th tone of C major, and the mode on that tone is the mixolydian. (We could also have deduced mixolydian from the tones themselves simply by analyzing their relationship: 1 2 3 4 5 6 b7).

OK, so far so good. But here's the question part: What if the chord progression is changed just slightly to this:

G7 | Fmaj7 | G7 | Fmaj7 | (repeated)

That is we no longer have a grounding G bass note -- instead we have 2 different chords. If they're both of equal length (and consequently equally prominent) what is the mode now? Can we really safely say that it's G mixolydian? It seems that it might just as likely be F lydian. And really the first or last chords may not give any aid in determining: the first chord might likely lead to the real tonal area later, and the last chord might produce some kind of unresolved, "hanging" effect.

Again, we could have a chord progression as above that does emphasize one modal area, by rhythm or perhaps even the melody. But my main point is that in no way would we be on sure footing in certain circumstances when attempting to answer the question of mode.

And, by the way, why would we ask such a question? Because if we're improvising we have to have some way of dealing with the music at hand. And this goes to my over-arching view of practicality. If someone were to say, well, given the 2nd chord progression above I'd just play a C major scale, I don't see how that's a real problem. Also if one were to say that there's no single mode and that s/he would switch between mixolydian and lydian, also not a problem (though if the tune were up-tempo it might be difficult to manage 'switching').

The question is also interesting from an historical point of view: in music around Mozart's time an ambiguous key center would have to be deliberate (like in a developmental section) or it would just be bad music because projecting a key center was the name of the game. In our time that's just not the case: ambiguity abounds and we can bask in it, but also have to deal with its implications...

some points about triads

Triads are 3 note chords built out of 3rds. Here are the 4 basic types:

These are in root position and close voicing, but triads can be spaced out more, inverted (i.e. the root needn't be the lowest note) and can have more than 3 notes so long as there aren't more than 3 different notes. The following are all A major chords:

I tend to regard the Augmented and Major triads as related (both contain major 3rds) and the Minor and Diminished as related (both having minor 3rds). But actually the Major and Minor are related as they both contain a Perfect 5th. The Augmented and Diminished triads have no commonality at all -- in fact they're kind of "opposites":



Traditionally triads ended up having harmonic functions as they conveyed a sense of key. They acquired names based upon their root as it related to its parent scale:

I tonic
II supertonic
III mediant
IV subdominant
V dominant
VI submediant
VII leading tone

(The 'sub' label does mean under: a subdominant chord is a fifth below just as the dominant is a fifth above the tonic. The submediant is a third below the tonic, just as the mediant is a third above.)

Keys were established by a strong V - I relationship, usually by a
II - V - I
(root movement of a fifth being felt strongest). In the 20th century triads were used by composers much, much more freely as keys were less important than modality or color. You'll find Ponce using progressions like D min to Eb min, F min to B maj, C maj to Gb maj to C maj (all in his fabulous piece Variations sur "Folia de Espana" and Fugue -- vid. variations vii and viii). Of course the voicing is important, but more on that later.

See also Sixth Chord.

beyond the 13th chord

Yeah, I know what you're thinking: a 13th chord contains all the notes of its scale, so how can you go beyond it? That's a good point, but if you consider that a 13th chord consists of 7 different tones and that there are an available 12 from the total chromatic then there are obviously 5 tones remaining which could be used harmonically (if so desired). Vincent Persichetti goes over this in his fabulous book Twentieth-Century Harmony: Creative Aspects and Practice. He does state that the 'unwieldy' terms of 19th or so aren't normally employed (that is the descriptions, not the chords) but let's just see if we can wield one -- even a bigger one -- anyway.

Consider the following chord:



From the G2 up to the E4 we have a G13 (dominant) chord. When we get back to the G it is sharpened, the sharp 15th. Next in the series is B, but here flattened, so a flat 17th. And so on. It is quite unwieldy to label such a beast:
G13#15b17#19#21#25
(There is no possibility in this type of chord of altering the 23rd to produce a new tone: can you see why?) In this case on a chart it would seem better to indicate something like "TC" (total chromatic) or some such, but seeing a symbol like G13#15 shouldn't pose a real problem...