(This is copy and pasted from the math part of this website)
I have looked into trying to figure what are all the possible types of note set combinations there are and how I would go by listing them if possible. Turns out this is harder than I thought. Here are the 12 notes
1, b2, 2, b3, 3, 4, #4/b5, 5, b6, 6, b7, 7
A type of set would be something like this
The thing is that set has inversion possibilities such as 3,1,5,7 and I do not want to count any of them as part of the list. No repeating notes either such as 1,1,3,5 or 1,b3,5,b3.
I've thought about the inversion case of 3,1,5,7 and convert that same set into 1,b6,b3,5 which is actually the same notes (which I'm trying to avoid as well). But what might be useful gleaming from this is using '1' as a base reference and instead try using each interval in order (from lowest to highest) for every single set on the list for example.. (These are just random examples)
1,b2,2,3 / 1,b2,3,4 / 1,2,3,4 / 1,4,5,b6,b7
I could use 1,b3,5,b6 but then I would have to not list 1,3,5,7 in the list because they are the same 'set type' So in the end its kind of difficult to find which note sets are already there and which are not, which leaves me to believe that I am missing a certain condition. Or maybe rather theres actually an easier way to look at all this that I haven't thought of yet?
So the question is how do I list all the different set types of notes without unintentionally repeating one?
(Edit added the last question to clear things up hopefully)