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!