a little bit of math: 7 note scales

I've been considering scales and modes lately, and have been wondering how many possibilities there are out there. I started writing out some lists (based on the major/Ionian scale, such as 1234b567, 1#2345b6b7, etc). At a certain point, however, I started to consider using any combination of 7 notes from the total chromatic of 12. Here writing out by hand started to become futile, so I wondered how to go about determining the actual number of possibilities. So here's a little math about that.

If we are concerned with 7 notes from a total of 12, and are not concerned about order -- we're looking for a set of tones, not a melodic sequence -- then what we want to find is known mathematically as combinations. There's a simple formula for determining them which is shown in the following image (which image was swiped from wikipedia -- thanks, guys!!!):

(In case you're not familiar with it, that ! doesn't indicate a loud, demanding or angry number: it's a factorial. 4! = 4 x 3 x 2 x 1 = 24. It's better if your calculator has a factorial button, because 12! =

479,001,600...best to do that in one keystroke!)

In our case n = 12 and k = 7. If you work through the equation you'll see that 12 tones taken 7 at a time can be arranged 792 different ways! (That exclamation is not a factorial). Some of these modes will be quite strange beasts from a typical scale point of view: c, c#, d, d#, e, f, g# is not the most common mode around. But if we want to know the exact, finite number then here we have it.

And here's something interesting, too, very, very interesting: if we want to know how many pentatonic scales there are we will find that there are 792, the exact number of septatonic scales (start to work it out and you'll see why). Hexatonic scales, by the way, produce the highest number of combinations: 924.

So if you're wondering if there are any more modes/scales out there to investigate the answer is most probably YES!

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...