In 12TET, is there a maximum number of musical scales possible?

Question

For those who don't know, 12 Tone Equal Temperament (12TET) simply means that, for a given frequency $f$, each step between $f$ and $2f$ can be written as $2^{k/12}f$, where $k$ denotes the $k$th step from $f$. In music, these notes are usually denoted as $$\lbrace C,C\#,D,D\#,E,F,F\#,G,G\#,A,A\#,B\rbrace$$Let us call this set $S$. I will define a "scale" to be a nonempty subset of $S$.

My First Thought Process

Taking $S$ as the above set, how many scales can be formed?

My logic is given the set

$\lbrace 1,2,3,4 \rbrace$

Then possible subsets are

$\lbrace 1,2,3,4\rbrace$

$\lbrace 1,2,3\rbrace$

$\lbrace 1,2,4\rbrace$

$\lbrace 1,3,4\rbrace$

$\lbrace 2,3,4\rbrace$

$\lbrace 1,2\rbrace$

$\lbrace 1,3\rbrace$

$\lbrace 1,4\rbrace$

$\lbrace 2,3\rbrace$

$\lbrace 2,4\rbrace$

$\lbrace 3,4\rbrace$

$\lbrace 1\rbrace$

$\lbrace 2\rbrace$

$\lbrace 3\rbrace$

$\lbrace 4\rbrace$

$\lbrace \emptyset \rbrace$

$1 + 4 + 6 + 4 + 1$

So for 12 members, I would get $2^{12}=4096$.

My Question:

What other mathematical ways are there to come to this number $4096$ besides the binomial theorem? Specifically, why might your particular method be interesting from a musical standpoint?

Answer 1

You don't really need the Binomial Theorem to count subsets. When building a subset, you either include or exclude each element. For $n$ elements, that yields $2^n$ possibilities, less one for the empty set.

As for musical validity, that's outside of my brief and a subject for a different forum, I suspect. I would, however disagree with you about silence. As Miles Davis said, "Music is the space between the notes."

Answer 2

The musical scale A minor (A, B, C, D, E, F, G, A) is generally considered different from the scale C major (C, D, E, F, G, A, B, C). In fact the same notes make seven different scales: major (Ionian), minor (Aeolian), Dorian, Phrygian, Lydian, Mixolydian, and Locrian, depending on which note is considered the first note in the scale.

Your method counts these as only one scale.

If you want to consider only scales that start at C, however, in order to avoid the problem of counting different scales on the same notes, then it does not seem right to count sets that do not contain C; for how can a scale start at a note that is not in the scale? In that case you have exactly $2^{11} = 2048$ possible scales, depending on which of the $11$ other notes between C and the next higher C are in the scale.

And that is not considering scales (such as the melodic minor) that are played differently when the frequencies are rising than when they are falling.