I hope I put this thread in the correct spot!
This is my first post and let me start by saying I am definitely not a mathematician. I am in fact a musician and I started thinking last night about just how many possible combinations of 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 notes there are above a given note and within one octave. I'm not sure if not knowing about music will affect anyone's ability to understand what I'm talking about so I'll try to fill in the gaps now.
There are twelve notes in the octave in western (European) music - that is, if you start on C and play every note up to the next C you will have played 12 notes (the next C being the 13th note). Up until the 20th century most chords consisted of 3-5 notes, and scales 7 notes (the notes of the scales being what chords are built up from). In a chord or a scale there is never two of the same note, that is C+E+G is not considered any different to C+E+G+C. The same goes for scales, you can't have two D flats in one scale. At the turn of the 20th century, composers began experimenting with bigger chords (chords with more notes) and new scales. So we now have music using scales and chords that have anywhere from 1 to 12 notes. Essentially, anything is possible, but what I want to know is exactly what is possible.
(If you do know a lot about music please ignore the fact that in tonal music chords are constructed directly in relation the the scales upon which the music is based and the effect a chord has in music is entirely dependent on context- I know that already. For this exercise I want to consider chords a scale as independent sonorities.)
I figured out (I think) how many combinations there where of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 notes within an octave above a given note. Here is what I got. There is a definite pattern here and I would love to know if there was a simpler way of expressing it;
For a single note there is possibility - the root note,
2 note combinations there are 11 - the root note plus each of the other 11 notes,
3 notes = 10+9+8+7+6+5+4+3+2+1 = 55 - the root note plus all combinations of 2 notes within the remaining 11 notes,
4 notes = 9+(8x2)+(7x3)+(6x4)+(5x5)+(4x6)+(3x7)+(2x8)+(1x9) = 165 - the root note plus all combinations of 3 notes within the remaining 11 notes,
5 notes = 8+(7x(1+2))+(6x(1+2+3))+(5x(1+2+3+4))+(4x(1+2+3+4+ 5))+(3x(1+2+3+4+5+6))+(2x(1+2+3+4+5+6+7))+(1x(1+2+ 3+4+5+6+7+8)) = 330
6 notes = 7+(6x(1+(1+2)))+(5x(1+(1+2)+(1+2+3)))+(4x(1+(1+2)+ (1+2+3)+(1+2+3+4)))+(3x(....))+(2x(....))+(1x(.... )) = 462
7 notes = 6+(5x(....))+(4x(....)).... = 462
8 notes = 5+(4x(....))+(3x(....)).... = 330
9 notes = 4+(3x(....))+(2x(....)).... = 165
10 notes = 3+(2x(....))+(1x(....)) = 55
11 notes = 2+(1x9) = 11
12 notes = 1
Grand total = 2048
I really hope I have written this in a way that makes sense. Like I said, I am no mathematician.
I would like to know two things. First; Is there a more elegant way of expressing this? Second. If I wanted the extend it into two octaves, keeping in mind that if the notes in the second octave are identical the those in the first it wouldn't count as different (as long as one note is different then it is okay), how would I do that?